https:///astrobaki/api.php?action=feedcontributions&user=WikiSysop&feedformat=atomAstroBaki - User contributions [en]2023-06-03T11:28:08ZUser contributionsMediaWiki 1.35.1Undergraduate Radio Lab2020-01-21T21:15:42Z<p>WikiSysop: /* Class Programmatics */</p>
<hr />
<div>This course consists of four laboratory experiments that concentrate on radio instrumentation and laboratory techniques. We will build receiving, observing, and data analysis systems for two telescopes: a single-dish 21-cm line system, and a 10.7-GHz interferometer. We will use these telescopes for astronomical observing projects including structure of the Milky Way galaxy, precise position measurement of several radio sources, and measurement of the radio brightness distributions of the sun and moon with high angular resolution. There is a heavy emphasis on digital data acquisition, software development in the Python language, and high-quality written reports.<br />
<br />
=== Class Programmatics ===<br />
* Class Code Repository: [http://github.com/AaronParsons/ugradio http://github.com/AaronParsons/ugradio]<br />
* [https://github.com/AaronParsons/ugradio/raw/master/schedule/syllabus_2020_spring.docx Syllabus]<br />
* Class Hours:<br />
** Tuesday/Thursday 1:00-2:30pm<br />
* Office Hours:<br />
** Aaron (aparsons at berkeley): TBD<br />
** Emily (emily_ramey at berkeley): TBD<br />
** Jackson (jsipple at berkeley): TBD<br />
** Frank Latora (fjlatora at berkeley)<br />
* Lab Groups:<br />
**<br />
<br />
=== Experiments ===<br />
* Lab 1: [https://github.com/AaronParsons/ugradio/blob/master/lab_mixers/allmixers.pdf Exploring Digital Sampling, Fourier Transforms, and both DSB and SSB Mixers]<br />
* Lab 2: [https://github.com/AaronParsons/ugradio/blob/master/lab_bighorn/bighorn.pdf Astronomy with the 21cm Line; Some Microwave Electronics]<br />
* Lab 3: [https://github.com/AaronParsons/ugradio/blob/master/lab_interf/interf.pdf Radio Interferometry at X Band]<br />
* Lab 4: [https://github.com/AaronParsons/ugradio/blob/master/lab_dish/HI1.pdf Mapping the HI Line: the Galaxy and Supershells]<br />
<br />
=== General Skills Used Through-Out Course ===<br />
<br />
* [[Python Installation and Basic Programming]]<br />
* [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/python_intro.ipynb Introduction to Python and Plotting]<br />
* [[Unix Primer]]<br />
* [[Unix Text Editors]]<br />
* [[LaTeX]]<br />
* [[Revision Control]]<br />
* [[Unit Testing]]<br />
<br />
== Topics by Date ==<br />
<br />
=== Lab 1 ([https://github.com/AaronParsons/ugradio/blob/master/lab_mixers/allmixers.pdf Exploring Digital Sampling, Fourier Transforms, and both DSB and SSB Mixers]), Due Feb 11, 1:00p ===<br />
==== Lab 1, Week 1 (Jan 21): Sampling and Power Spectra ====<br />
* Resources and Handouts<br />
** [[Nyquist Sampling]]<br />
** [[Fourier Transform]]<br />
** [[Unix Primer]]<br />
** [[Python Installation and Basic Programming]]<br />
** [[Revision Control]]<br />
** [[Unix Text Editors]]<br />
* Demos and Tutorials<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/python_intro.ipynb Introduction to Python and Plotting]<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/aliasing_demo.ipynb Aliasing Demo]<br />
* In class:<br />
** Astrobaki, [https://github.com/AaronParsons/ugradio/blob/master/schedule/syllabus_2019_spring.docx Syllabus], Office Hours<br />
** Getting accounts (premade by Bill Boyd, change password)<br />
** Lab access (email Mark Hayden)<br />
** EM waves from sky to wire<br />
** [[Nyquist Sampling]] and aliasing<br />
** [[Fourier Transform]] <br />
** Lab Hardware<br />
*** PicoScope 2206a<br />
<br />
==== Lab 1, Week 2 (Jan 28): DSB and SSB Mixers ====<br />
* Theory and Background:<br />
** [[Heterodyne Mixers]]<br />
** [[Convolution Theorem]]<br />
** [[Fast Fourier Transform]]<br />
* Demos and Tutorials<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/Sec%204.%20In%20the%20mind.ipynb In the Mind: Introduction to the Fourier Transform ] with handout on [https://github.com/AaronParsons/ugradio/blob/master/dft_intro/fourierc.pdf DFTs with DFTs]<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/datatype_demo.ipynb Mommy Fortuna's Midnight Carnival of Python Oddities]<br />
** [[Data Representations]], with handout on [https://github.com/AaronParsons/ugradio/blob/master/pythonprimer/datatypes.pdf Data Types and Organizational Structures] <br />
* In class:<br />
** Lecture: Introduction to DSB and SSB Mixers<br />
** Show and Tell (10m per group)<br />
** Lecture: Discrete Fourier Transforms and the Convolution Theorem<br />
<br />
==== Lab 1, Week 3 (Feb 4): More Mixers, and Lab Reports ====<br />
* Theory and Background<br />
** [[LaTeX]]<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab1/python_intro.ipynb Introduction to Python and Plotting], second pass<br />
* In Class:<br />
** Lecture: One more pass on [[Convolution Theorem]], [[Heterodyne Mixers]], and [https://github.com/AaronParsons/ugradio/blob/master/dft_intro/fourierc.pdf DFTs]<br />
** Show and Tell<br />
** Writing Lab Reports<br />
* Lab 1 Due Feb 11, 1:30p<br />
<br />
---------<br />
<br />
=== Lab 2 ([https://github.com/AaronParsons/ugradio/blob/master/lab_bighorn/bighorn.pdf Astronomy with the 21cm Line; Some Microwave Electronics]), due Mar 3, 1:00p ===<br />
==== Lab 2, Week 1 (Feb 11): 21cm Line and Waveguides ====<br />
* Theory and Background:<br />
** [https://github.com/AaronParsons/ugradio/blob/master/lab_bighorn/cal_intensity.pdf Calibrating the Intensity and Shape of Spectral Lines]<br />
** [[Specific Intensity]]<br />
** [[21cm Transition]]<br />
** [[Central Limit Theorem]] and [[Random Walks]]<br />
* Demos and Tutorials<br />
** Python Tutorial Part 3: Functions, Modules, and Objects<br />
** [https://github.com/AaronParsons/ugradio/blob/master/jupyter_tutorials/lab2/Central%20Limit%20Theorem%20and%20Averaging.ipynb Central Limit Theorem and Averaging]<br />
* In class:<br />
** Introduction to Horn and Receiver<br />
** Time<br />
** Doppler Corrections, ugradio.doppler<br />
<br />
==== Lab 2, Week 2 (Feb 18): Collect and Analyze Data ====<br />
* Theory and Background<br />
** [[Coordinates]]<br />
** [https://github.com/AaronParsons/ugradio/blob/master/least_squares/lsfit_lite.pdf Least Squares Lite for the Budding Aficionado: Art and Practice]<br />
** [http://ugastro.berkeley.edu/radio/2017/handout_links/RWvD.pdf Fields and Waves in Communication Electronics (Ramo, Whinney, and Van Duzer)]<br />
* Demos and Tutorials<br />
** Matrix Math with Numpy<br />
* In Class:<br />
** Show and Tell<br />
** Waveguides, Transmission Lines, and Rope<br />
** Fitting Gaussians and Polynomials, ugradio.gauss<br />
<br />
==== Lab 2, Week 3 (Feb 25): Write Lab Report ====<br />
* Theory and Background<br />
** [https://github.com/AaronParsons/ugradio/blob/master/least_squares/lsfit_2008.pdf Least-Squares and Chi-Square for the Budding Aficionado: Art and Practice]<br />
* In Class:<br />
** Show and Tell<br />
** Least Squares Part 2<br />
* Lab 2 Due Mar 3, 1:00p<br />
<br />
---------<br />
<br />
=== Lab 3 ([https://github.com/AaronParsons/ugradio/blob/master/lab_interf/interf.pdf Radio Interferometry at X Band]), due Apr 7, 1:00p ===<br />
<br />
==== Lab 3 Week 1 (Mar 3): Interferometer ====<br />
* Theory and Background<br />
** [[Coordinates]]<br />
** [[Basic Interferometry]]<br />
* In Class:<br />
** Tour of Rooftop Interferometer<br />
** Exercise Ball Coordinates<br />
** Interferometry with Strings<br />
<br />
==== Lab 3 Week 2 (Mar 10): Collect and Analyze Data ====<br />
* Theory and Background<br />
** [[Measurement Equation]]<br />
** [[Telescope Field of View/Resolution]]<br />
* In Class:<br />
** Show and Tell<br />
** Controlling the Telescope<br />
** Tracking the Sun<br />
** Scheduling Observations<br />
<br />
==== Lab 3 Week 3 (Mar 17): Collect and Analyze Data ====<br />
* Theory and Background<br />
** [[Basic Interferometry II]]<br />
** [[Least Squares]]<br />
* In Class:<br />
** Show and Tell<br />
** Linear Least-Squares in Python<br />
** Minimizing Chi-Square<br />
** Noise in Observations<br />
<br />
==== No class (Mar 24, 26) ====<br />
<br />
==== Lab 3 Week 4 (Mar 31): Write Lab Report ====<br />
* Theory and Background<br />
*** [[Correlators]]<br />
*** [[Units of radiation]]<br />
* In Class:<br />
** Show and Tell<br />
** Photon bucket demo<br />
* Lab 3 Due Apr 7, 1:00p<br />
<br />
<br />
---------<br />
=== Lab 4 ([https://github.com/AaronParsons/ugradio/blob/master/lab_dish/HI1.pdf Mapping the HI Line: the Galaxy and Supershells]), due May 5, 1:00p ===<br />
<br />
==== Useful Links ====<br />
[https://teamup.com/ksd715ee7fb766a108/#/ Link to Observing Calendar]<br />
<br />
[http://leuschner.berkeley.edu:8080 Leuschner WebCam]<br />
<br />
==== Lab 4 Week 1 (Apr 7): Leuschner Dish ====<br />
* Theory and Background<br />
** [[Specific Intensity]]<br />
** [[Radio Sky]]<br />
** [[Screen Command]]<br />
* In Class:<br />
** Trip to Leuschner: Class will go later than usual<br />
** drive mechanism (how the dish moves)<br />
** feed (notice probes) and cables<br />
** IF setup (one channel for OH, one for HI)<br />
** show spectrum<br />
** demo action<br />
** spatial sampling with dish<br />
<br />
==== Lab 4 Week 2 (Apr 14): Collect and Analyze Data ====<br />
* Theory and Background<br />
** [[21cm Transition]]<br />
** [[Black-Body Radiation]]<br />
** [[Line Profile Functions]]<br />
* In Class:<br />
** Show and Tell<br />
** pointing control<br />
** how to take data<br />
** computing doppler width for a line of sight (assuming circular motion)<br />
<br />
==== Lab 4 Week 3 (Apr 21): Collect and Analyze Data ====<br />
* Theory and Background<br />
** [[Emission Line Observing]]<br />
** [[Radiative Transfer Equation]]<br />
** [[Optical Depth]]<br />
** [[Estimating Atomic Transition Strengths]]<br />
* In Class:<br />
** Show and Tell<br />
** close-out plan<br />
** calibrating spectra<br />
** converting spectra to hydrogen<br />
<br />
==== Lab 4 Week 4 (Apr 28): Write Lab Report ====<br />
* Theory and Background<br />
** [https://github.com/AaronParsons/ugradio/blob/master/lab_bighorn/cal_intensity.pdf Calibrating the Intensity and Shape of Spectral Lines (Carl Heiles)] <br />
* In Class:<br />
** Show and Tell<br />
** displaying information in image form<br />
** Lab 4 Due May 5, 1:00p<br />
<br />
<br />
<!-- === Lab 2 (Digital Lab) Due Mar 3 6:00p ===<br />
* Lab 2, Part 1 (Feb 11)<br />
** Quiz 4 on:<br />
** Breakout Sessions:<br />
** Discrete Fourier Transform<br />
*** Convolving and Smoothing<br />
*** Aliasing<br />
*** Intro to Lab 2, pick groups<br />
<br />
* Lab 2, Part 2 (Feb 18)<br />
** Quiz 5 on:<br />
*** [[Data Representations]]<br />
*** [[Digital Down Conversion]]<br />
** Breakout Sessions:<br />
*** Data Representations demo<br />
*** Mixing, Filtering, Decimation, and the FFT<br />
*** [[Intro to ROACH]]<br />
<br />
* Lab 2, Part 3 (Feb 25)<br />
** Quiz 6 on:<br />
*** [[Synchronous and Asynchronous Logic]]<br />
*** [[Processor Architectures]]<br />
*** [[FIR Filters]]<br />
** Breakout Sessions:<br />
*** Acting out various processors<br />
*** Simulating an FIR filter<br />
*** Lab feedback<br />
** Lab 2 (Digital Lab) Due Mar 3 6:00p<br />
--><br />
<br />
== Unused but Useful? ==<br />
<br />
* [[Radiometer Equation]]<br />
* [[Quantization and Rounding]]<br />
* [[Reciprocity Theorem]]<br />
* [[Dipole Antennas]]<br />
* [[Impedance of Free Space]]<br />
* [[Radiometer Equation Applied to Telescopes]]<br />
* [[Radiometer Equation Applied to Interferometers]]<br />
* [[Fringe Stopping]]<br />
* [[Direction Dependent Beams]]<br />
* [[Self Calibration]]<br />
* [[Flux Calibration]]<br />
* [[Gridding]]<br />
* [[Earth Rotation Synthesis]]<br />
* [[Delay Imaging]]</div>WikiSysopAipyPhilosophy2015-05-11T22:08:18Z<p>WikiSysop: Created page with '== AIPY Philosophy (e.g. Pedantry) == === AIPY is: === * A module adding tools for interferometry to Python (not visa versa) * An amalgam of pure-Python and wrappers around C…'</p>
<hr />
<div>== AIPY Philosophy (e.g. Pedantry) ==<br />
<br />
=== AIPY is: ===<br />
* A module adding tools for interferometry to Python (not visa versa)<br />
* An amalgam of pure-Python and wrappers around C++ and Fortran<br />
* Object-Oriented<br />
* A collection of low- to mid-level operations (you write the program)<br />
* A toolkit (i.e. a grocery store, not a restaurant)<br />
<br />
=== AIPY is not: ===<br />
* A solution (it is just a tool to help you find it)<br />
* A one-stop shop (it makes use of other open-source projects)<br />
* A replacement for other interferometry packages<br />
* Wedded to a file format (but only MIRIAD-UV, FITS currently supported)<br />
<br />
== Why Another Interferometry Package? ==<br />
<br />
=== New Problems to Solve ===<br />
* Wide fields of view<br />
* Large relative bandwidths<br />
* Huge numbers of antennas<br />
* Non-tracking primary beams<br />
* Real-time processing<br />
* Source separation<br />
* Ionospheric distortion<br />
<br />
=== Why Python? ===<br />
* Interpreted (up to 5x more productive than compiled languages, according to Burton Group study)<br />
* Object-Oriented<br />
* Readable<br />
* Fast, high-level data types<br />
* Large community of programmers<br />
* Fast-growing community of numerical/scientific/astronomy programmers<br />
<br />
=== Performance ===<br />
* Generally, only a small fraction of code needs to run fast<br />
* Python's profiler can tell you where the bottlenecks are<br />
* Bottlenecks can be recoded in C/C++/ Fortran and wrapped into Python <br />
* NumPy, the foundation of numerical/vectorized processing in Python, is coded in C and runs on average only 1.5 times slower than pure C<br />
* You should only be allowed to worry about speed while your code is actually running<br />
<br />
== Zen of AIPY (with apologies to [https://www.python.org/dev/peps/pep-0020/ The Zen of Python]) ==<br />
* Explicit is better than implicit<br />
* Simple is better than complex<br />
* Complex is better than complicated<br />
* Special cases aren't special enough to break the rules<br />
* Practicality beats purity<br />
* There should be one (and preferably only one) obvious way to do it, although that way may not be obvious at first unless you're Dutch<br />
* Now is better than never<br />
* In the face of ambiguity, refuse the temptation to guess<br />
* Don't handicap the programmer; allow them all the rope they want<br />
* Don't hide data; return it (or at least grant access) at every turn</div>WikiSysopAIPY2015-05-11T21:56:36Z<p>WikiSysop: /* Documentation */</p>
<hr />
<div>= Astronomical Interferometry in PYthon =<br />
<br />
This package collects together tools for radio astronomical interferometry. In addition to pure-python phasing, calibration, imaging, and deconvolution code, this package includes interfaces to MIRIAD (a Fortran interferometry package) and HEALPix (a package for representing spherical data sets), and some math/fitting routines from SciPy.<br />
<br />
The primary driver for this software is the [http://arxiv.org/abs/0904.2334 Precision Array for Probing the Epoch of Reionization (PAPER)], an experiment for detecting the first stars and galaxies that formed in the universe via the effect of their radiation on intergalactic hydrogen. This experiment presents many new challenges, including widefield imaging, non-tracking antennas, high data-rates, a large fractional bandwidth, and power-spectrum detection.<br />
<br />
For an overview of the latest changes to this package, see the [http://setiathome.berkeley.edu/~aparsons/aipy/CHANGELOG CHANGELOG].<br />
<br />
You are invited to edit this wiki.<br />
<br />
== Download ==<br />
<br />
If you have [http://peak.telecommunity.com/DevCenter/setuptools setuptools] and [http://numpy.scipy.org numpy] is already installed, then with root permissions, you can just type:<br />
<br />
<source lang="bash"><br />
easy_install aipy<br />
</source><br />
<br />
Otherwise, you may download from http://pypi.python.org/pypi/aipy.<br />
<br />
If you want the bleeding-edge source tree, or are interested in source development, you can checkout a copy from http://github.com/AaronParsons/aipy using [http://git.or.cz GIT]. See [[GitAipy]] for more information about using GIT with AIPY.<br />
<br />
<br />
== Installation ==<br />
<br />
=== Resolving Dependencies ===<br />
This is primarily a *nix package. With some trouble it can install on intel-based Macs. It probably doesn't install on Windows. You need to have python >= 2.4. AIPY depends of the following Python packages:<br />
<br />
<source lang="bash"><br />
numpy >= 1.2<br />
pyephem >= 3.7.2.3<br />
pyfits >= 1.1<br />
*matplotlib >= 0.98<br />
*matplotlib-basemap >= 0.99<br />
</source><br />
<br />
(* installation can proceed without these, but some scripts will choke)<br />
<br />
To resolve these dependencies, your options are:<br />
* (safest) -- Manually install the dependencies.<br />
* (experimental) -- Open up the AIPY download, and with network connectivity and root access, type:<br />
<br />
<source lang="bash"><br />
install_required.sh<br />
</source><br />
<br />
and then (if you want matplotlib/basemap):<br />
<br />
<source lang="bash"><br />
install_recommended.sh<br />
</source><br />
<br />
=== Installing with Root Permission ===<br />
If you used easy_install, and everything worked, then you're set. Otherwise, download the AIPY package, open it up, and then run (with root permission):<br />
<br />
<source lang="bash"><br />
python setup.py install<br />
</source><br />
<br />
If any of the dependencies fails to install, you should download the package source and build it manually. <br />
<br />
=== Installing without Root Permission ===<br />
To resolve dependencies using setuptools, but without root access, you'll need to follow the instructions at http://peak.telecommunity.com/DevCenter/setuptools, and apply them to each line in install_dependencies.sh. Once dependencies have been solved, run the following command:<br />
<br />
<source lang="bash"><br />
python setup.py install --install-lib "module_dir" --install-scripts "scripts_dir"<br />
</source><br />
<br />
where "module_dir" and "scripts_dir" are 2 directories you define. "module_dir" will hold the python module for the package you install, and "scripts_dir" will hold any scripts associated with the package. If you take this second route, you should also run (for bash)<br />
<br />
<source lang="bash"><br />
export PYTHONPATH="module_dir"<br />
</source><br />
<br />
so that python can find the modules. You should also add "scripts_dir" to your path:<br />
<br />
<source lang="bash"><br />
export PATH=$PATH:"scripts_dir"<br />
</source><br />
<br />
To avoid having to type these lines every time you open a console, add them to your .bashrc file.<br />
<br />
=== Additional Packages ===<br />
Here's a short list of additional packages that I find generally useful when working in Python:<br />
* [http://ipython.scipy.org IPython]: A slick interactive command-line interface.<br />
* [http://www.scipy.org SciPy]: General-purpose math, signal processing, fitting, and science stuff.<br />
<br />
== Documentation ==<br />
<br />
There are several options for obtaining information about the structure and usage of AIPY:<br />
* a [http://astro.berkeley.edu/~aparsons/aipy.pdf tutorial] that includes primers on various packages, code documentation, and some examples<br />
* [https://github.com/AaronParsons/aipy/blob/master/doc/source/tutorial.rst online code documentation] generated automatically from docstrings in the source code<br />
* this wiki, which you are welcome to join and edit, especially the list of topics below:<br />
** [[AipyPhilosophy]] - pedantry you may or may not find useful<br />
** [[AipyFaq]] - a place to post (and answer) questions<br />
** [[AipyCookBook]] - a repository for example code snippets and explanations<br />
* you may also send emails to aparsons at astron berkeley edu (with dots between astron, berkeley, and edu).<br />
<br />
== Contributing Code ==<br />
<br />
AIPY is open-source software released under the GNU Public License. It may be freely used and modified. That said, the interferometry community as a whole stands to benefit from cooperative, organized development. A general model for contributing to the AIPY code-base goes as follows:<br />
* If you find a bug, write an [[AipyBugReport]] to help get it fixed.<br />
* Write an [[AipyEnhancementProposal]] that details what you're interested in changing, how it used to work, how it will work after your enhancment, and why the enhancement is better<br />
* Get a copy of the latest source code git (see [[GitAipy]] for instructions)<br />
* Make your edits (writing unit tests for any new features), and then check that you haven't broken any other unit tests.<br />
* Contact a primary developer (currently, Aaron Parsons: aparsons at astron berkeley edu) with an emailed patch. These patches, which may be produced automatically with git, will be merged into the primary source code branch.<br />
<br />
----------------------------</div>WikiSysopIntroduction Papers2015-05-11T21:04:48Z<p>WikiSysop: /* 21cm Cosmology Reading List */</p>
<hr />
<div>=21cm Cosmology Reading List=<br />
<br />
Conceptual Overviews:<br />
* Avi Loeb’s [https://www.cfa.harvard.edu/~loeb/sciam.pdf Scientific American article] <br />
* Loeb and Pritchard’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..772P/ explanatory supplement in Nature].<br />
<br />
Experimental papers:<br />
* The [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..796B/ EDGES experimental results] <br />
* The [http://labs.adsabs.harvard.edu/adsabs/abs/2014ApJ...782L...9V/ SCI-HI experiment]<br />
<br />
Theory work:<br />
* Pritchard & Loeb’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010PhRvD..82b3006P/ technical article] <br />
<br />
Review Articles:<br />
* Pritchard & Loeb, [http://iopscience.iop.org/0034-4885/75/8/086901 21cm Cosmology in the 21st Century]: (on [http://arxiv.org/pdf/1109.6012.pdf arXiv]), 2011.<br />
* Morales & Wyithe, [http://arxiv.org/abs/0910.3010 Reionization and Cosmology with 21cm Fluctuations], 2010.<br />
* Furlanetto, Oh, & Briggs, [http://arxiv.org/abs/astro-ph/0608032 Cosmology at Low Frequencies: The 21 cm Transition and the High-Redshift Universe], 2006.<br />
<br />
Background:<br />
* Loeb & Furlanetto, [http://press.princeton.edu/titles/9914.html The First Galaxies in the Universe], 2013.<br />
<br />
= Interferometry Reading List =<br />
<br />
* Thompson, Moran, and Swenson, "[http://books.google.com/books/about/Interferometry_and_Synthesis_in_Radio_As.html?id=AwBN5bpuEU0C Interferometry and Synthesis in Radio Astronomy]"<br />
* [http://www.phys.unm.edu/~gbtaylor/astr423/s98book.pdf Synthesis Imaging in Radio Astronomy II], ed. Taylor, Carilli, Perley<br />
* Rick Perley's slides on [http://www.aoc.nrao.edu/events/synthesis/2010/lectures/2010Interferometry.pdf Fundamentals of Radio Interferometry] from the VLA summer school</div>WikiSysopIntroduction Papers2015-05-11T20:57:22Z<p>WikiSysop: </p>
<hr />
<div>=21cm Cosmology Reading List=<br />
<br />
Light conceptual overviews:<br />
* Avi Loeb’s [https://www.cfa.harvard.edu/~loeb/sciam.pdf Scientific American article] <br />
* Loeb and Pritchard’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..772P/ explanatory supplement in Nature].<br />
<br />
Experimental papers:<br />
* The [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..796B/ EDGES experimental results] <br />
* Another [http://labs.adsabs.harvard.edu/adsabs/abs/2014ApJ...782L...9V/ experiment]<br />
<br />
Theory work:<br />
* Pritchard & Loeb’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010PhRvD..82b3006P/ technical article] <br />
* Pritchard & Loeb, [http://iopscience.iop.org/0034-4885/75/8/086901 21cm Cosmology in the 21st Century]: (on [http://arxiv.org/pdf/1109.6012.pdf arXiv])<br />
<br />
= Interferometry Reading List =<br />
<br />
* Thompson, Moran, and Swenson, "[http://books.google.com/books/about/Interferometry_and_Synthesis_in_Radio_As.html?id=AwBN5bpuEU0C Interferometry and Synthesis in Radio Astronomy]"<br />
* [http://www.phys.unm.edu/~gbtaylor/astr423/s98book.pdf Synthesis Imaging in Radio Astronomy II], ed. Taylor, Carilli, Perley<br />
* Rick Perley's slides on [http://www.aoc.nrao.edu/events/synthesis/2010/lectures/2010Interferometry.pdf Fundamentals of Radio Interferometry] from the VLA summer school</div>WikiSysopIntro to Research Resources2015-05-11T19:59:13Z<p>WikiSysop: /* Global Signal Interferometer */</p>
<hr />
<div>* [[Programming Resources]]<br />
* [[Introduction Papers]]<br />
<br />
== Projects ==<br />
<br />
===Global Signal Interferometer===<br />
<br />
Measuring 21-cm hyperfine emission from neutral hydrogen at cosmological distances is one of the most promising techniques for probing our early universe. A positive detection would provide direct observations of key unexplored epochs of our cosmic history, including the cosmic dark ages before the universe’s first stars formed, and the reionization of the bulk of the hydrogen in the universe by starlight once star formation became widespread. Measuring the spherically averaged 21 cm brightness temperature as a function of redshift (the "global signal") provides unique information about our universe that cannot be accessed any other way.<br />
<br />
Several experiments are attempting to measure the global signal with a single antenna, reasoning that for a globally present signal, directionality is not a critical aspect of the experiment. I believe that this is incorrect, and have begun investigating how the directionality of an interferometer could be leveraged to improve such experiments. The conventional wisdom is that interferometers are not sensitive to a global signal, owing to the differential nature of their measurements. However, in [http://arxiv.org/abs/1501.01633 Presley, Liu, and Parsons (2015)], we demonstrate that this is incorrect. We are now working to design and build a radio interferometer capable of carrying out this experiment and measuring the cosmic dawn of our universe.<br />
<br />
There are many ways for students to get involved in this project, ranging from physically assembling and testing analog electronics. to designing the digital correlator using supercomputing hardware, to programming in Python to calibrate and analyze data, to applying sophisticated linear algebra techniques to extract the cosmological signal. A good way to get started on the science is to read [http://arxiv.org/pdf/1109.6012.pdf “21-cm cosmology in the 21st Century” by Pritchard & Loeb (2011)]. For understanding the principles of the analog system, [http://arxiv.org/abs/1211.3800 “SARAS measurement of the Radio Background at long wavelengths” by Patra et al. (2015)] is an good reference. For the digital correlator, I recommend [http://arxiv.org/abs/0809.2266 “A Scalable Correlator Architecture Based on Modular FPGA Hardware, Reuseable Gateware, and Data Packetization” by Parsons et al. (2008)].<br />
<br />
This work is supported by an [http://www.nsf.gov/awardsearch/showAward?AWD_ID=1352519 NSF CAREER award (#1352519)] and is designed to be a student-led project in order to help train our next generation of instrument builders in radio astronomy.</div>WikiSysopIntroduction Papers2015-05-11T19:44:04Z<p>WikiSysop: </p>
<hr />
<div>=21cm Cosmology Reading List=<br />
<br />
Light conceptual overviews:<br />
* Avi Loeb’s [https://www.cfa.harvard.edu/~loeb/sciam.pdf Scientific American article] <br />
* Loeb and Pritchard’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..772P/ explanatory supplement in Nature].<br />
<br />
Experimental papers<br />
* The [http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..796B/ EDGES experimental results] <br />
* Another [http://labs.adsabs.harvard.edu/adsabs/abs/2014ApJ...782L...9V/ experiment]<br />
<br />
Theory work:<br />
* Pritchard & Loeb’s [http://labs.adsabs.harvard.edu/adsabs/abs/2010PhRvD..82b3006P/ technical article] <br />
* Pritchard & Loeb, [http://iopscience.iop.org/0034-4885/75/8/086901 21cm Cosmology in the 21st Century]: (on [http://arxiv.org/pdf/1109.6012.pdf arXiv])</div>WikiSysopIntroduction Papers2015-05-11T19:34:16Z<p>WikiSysop: </p>
<hr />
<div>=21cm Cosmology Reading List=<br />
<br />
Light conceptual overviews:<br />
* Avi Loeb’s Scientific American article: https://www.cfa.harvard.edu/~loeb/sciam.pdf<br />
* Loeb and Pritchard’s explanatory supplement in Nature: http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..772P/. This is actually a companion piece to…<br />
<br />
Experimental papers<br />
* The EDGES experimental results: http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..796B/<br />
* Another experiment: http://labs.adsabs.harvard.edu/adsabs/abs/2014ApJ...782L...9V/<br />
<br />
Theory work:<br />
* Pritchard & Loeb’s technical article: http://labs.adsabs.harvard.edu/adsabs/abs/2010PhRvD..82b3006P/<br />
* Pritchard & Loeb "21cm Cosmology in the 21st Century": http://iopscience.iop.org/0034-4885/75/8/086901</div>WikiSysopIntroduction Papers2015-05-11T19:26:04Z<p>WikiSysop: Created page with '=21cm Cosmology Reading List= Light conceptual overviews: * Avi Loeb’s Scientific American article: https://www.cfa.harvard.edu/~loeb/sciam.pdf * Loeb and Pritchard’s explan…'</p>
<hr />
<div>=21cm Cosmology Reading List=<br />
<br />
Light conceptual overviews:<br />
* Avi Loeb’s Scientific American article: https://www.cfa.harvard.edu/~loeb/sciam.pdf<br />
* Loeb and Pritchard’s explanatory supplement in Nature: http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..772P/. This is actually a companion piece to…<br />
<br />
Experimental papers<br />
* The EDGES experimental results: http://labs.adsabs.harvard.edu/adsabs/abs/2010Natur.468..796B/<br />
* Another experiment: http://labs.adsabs.harvard.edu/adsabs/abs/2014ApJ...782L...9V/<br />
<br />
Theory work:<br />
* Pritchard & Loeb’s technical article: http://labs.adsabs.harvard.edu/adsabs/abs/2010PhRvD..82b3006P/</div>WikiSysopIntro to Research Resources2015-05-11T18:38:37Z<p>WikiSysop: </p>
<hr />
<div>* [[Programming Resources]]<br />
* [[Introduction Papers]]<br />
<br />
== Projects ==<br />
<br />
===Global Signal Interferometer===<br />
<br />
Measuring 21-cm hyperfine emission from neutral hydrogen at cosmological distances is one of the most promising techniques for probing our early universe. A positive detection would provide direct observations of key unexplored epochs of our cosmic history, including the cosmic dark ages before the universe’s first stars formed, and the reionization of the bulk of the hydrogen in the universe by starlight once star formation became widespread. Measuring the spherically averaged 21 cm brightness temperature as a function of redshift (the "global signal") provides unique information about our universe that cannot be accessed any other way.<br />
<br />
Several experiments are attempting to measure the global signal with a single antenna, reasoning that for a globally present signal, directionality is not a critical aspect of the experiment. I believe that this is incorrect, and have begun investigating how the directionality of an interferometer could be leveraged to improve such experiments. The conventional wisdom is that interferometers are not sensitive to a global signal, owing to the differential nature of their measurements. However, in Presley, Liu, and Parsons (2015), we demonstrate that this is incorrect. We are now working to design and build a radio interferometer capable of carrying out this experiment and measuring the cosmic dawn of our universe.<br />
<br />
There are many ways for students to get involved in this project, ranging from physically assembling and testing analog electronics. to designing the digital correlator using supercomputing hardware, to programming in Python to calibrate and analyze data, to applying sophisticated linear algebra techniques to extract the cosmological signal. A good way to get started on the science is to read “21-cm cosmology in the 21st Century” by Pritchard & Loeb (2011). For understanding the principles of the analog system, “SARAS measurement of the Radio Background at long wavelengths” by Patra et al. (2015) is an good reference. For the digital signal processing, I recommend “A Scalable Correlator Architecture Based on Modular FPGA Hardware, Reuseable Gateware, and Data Packetization” by Parsons et al. (2008).<br />
<br />
This work is supported by an NSF CAREER award (#1352519) and is designed to be a student-led project in order to help train our next generation of instrument builders in radio astronomy.</div>WikiSysopAIPY2014-06-24T17:36:35Z<p>WikiSysop: /* Documentation */</p>
<hr />
<div>= Astronomical Interferometry in PYthon =<br />
<br />
This package collects together tools for radio astronomical interferometry. In addition to pure-python phasing, calibration, imaging, and deconvolution code, this package includes interfaces to MIRIAD (a Fortran interferometry package) and HEALPix (a package for representing spherical data sets), and some math/fitting routines from SciPy.<br />
<br />
The primary driver for this software is the [http://arxiv.org/abs/0904.2334 Precision Array for Probing the Epoch of Reionization (PAPER)], an experiment for detecting the first stars and galaxies that formed in the universe via the effect of their radiation on intergalactic hydrogen. This experiment presents many new challenges, including widefield imaging, non-tracking antennas, high data-rates, a large fractional bandwidth, and power-spectrum detection.<br />
<br />
For an overview of the latest changes to this package, see the [http://setiathome.berkeley.edu/~aparsons/aipy/CHANGELOG CHANGELOG].<br />
<br />
You are invited to edit this wiki.<br />
<br />
== Download ==<br />
<br />
If you have [http://peak.telecommunity.com/DevCenter/setuptools setuptools] and [http://numpy.scipy.org numpy] is already installed, then with root permissions, you can just type:<br />
<br />
<source lang="bash"><br />
easy_install aipy<br />
</source><br />
<br />
Otherwise, you may download from http://pypi.python.org/pypi/aipy.<br />
<br />
If you want the bleeding-edge source tree, or are interested in source development, you can checkout a copy from http://github.com/AaronParsons/aipy using [http://git.or.cz GIT]. See [[GitAipy]] for more information about using GIT with AIPY.<br />
<br />
<br />
== Installation ==<br />
<br />
=== Resolving Dependencies ===<br />
This is primarily a *nix package. With some trouble it can install on intel-based Macs. It probably doesn't install on Windows. You need to have python >= 2.4. AIPY depends of the following Python packages:<br />
<br />
<source lang="bash"><br />
numpy >= 1.2<br />
pyephem >= 3.7.2.3<br />
pyfits >= 1.1<br />
*matplotlib >= 0.98<br />
*matplotlib-basemap >= 0.99<br />
</source><br />
<br />
(* installation can proceed without these, but some scripts will choke)<br />
<br />
To resolve these dependencies, your options are:<br />
* (safest) -- Manually install the dependencies.<br />
* (experimental) -- Open up the AIPY download, and with network connectivity and root access, type:<br />
<br />
<source lang="bash"><br />
install_required.sh<br />
</source><br />
<br />
and then (if you want matplotlib/basemap):<br />
<br />
<source lang="bash"><br />
install_recommended.sh<br />
</source><br />
<br />
=== Installing with Root Permission ===<br />
If you used easy_install, and everything worked, then you're set. Otherwise, download the AIPY package, open it up, and then run (with root permission):<br />
<br />
<source lang="bash"><br />
python setup.py install<br />
</source><br />
<br />
If any of the dependencies fails to install, you should download the package source and build it manually. <br />
<br />
=== Installing without Root Permission ===<br />
To resolve dependencies using setuptools, but without root access, you'll need to follow the instructions at http://peak.telecommunity.com/DevCenter/setuptools, and apply them to each line in install_dependencies.sh. Once dependencies have been solved, run the following command:<br />
<br />
<source lang="bash"><br />
python setup.py install --install-lib "module_dir" --install-scripts "scripts_dir"<br />
</source><br />
<br />
where "module_dir" and "scripts_dir" are 2 directories you define. "module_dir" will hold the python module for the package you install, and "scripts_dir" will hold any scripts associated with the package. If you take this second route, you should also run (for bash)<br />
<br />
<source lang="bash"><br />
export PYTHONPATH="module_dir"<br />
</source><br />
<br />
so that python can find the modules. You should also add "scripts_dir" to your path:<br />
<br />
<source lang="bash"><br />
export PATH=$PATH:"scripts_dir"<br />
</source><br />
<br />
To avoid having to type these lines every time you open a console, add them to your .bashrc file.<br />
<br />
=== Additional Packages ===<br />
Here's a short list of additional packages that I find generally useful when working in Python:<br />
* [http://ipython.scipy.org IPython]: A slick interactive command-line interface.<br />
* [http://www.scipy.org SciPy]: General-purpose math, signal processing, fitting, and science stuff.<br />
<br />
== Documentation ==<br />
<br />
There are several options for obtaining information about the structure and usage of AIPY:<br />
* a [http://astro.berkeley.edu/~aparsons/aipy.pdf tutorial] that includes primers on various packages, code documentation, and some examples<br />
* [https://github.com/AaronParsons/aipy/blob/master/doc/source/tutorial.rst online code documentation] generated automatically from docstrings in the source code<br />
* this wiki, which you are welcome to join and edit, especially the list of topics below:<br />
** [[AipyFaq]] - a place to post (and answer) questions<br />
** [[AipyCookBook]] - a repository for example code snippets and explanations<br />
* you may also send emails to aparsons at astron berkeley edu (with dots between astron, berkeley, and edu).<br />
<br />
== Contributing Code ==<br />
<br />
AIPY is open-source software released under the GNU Public License. It may be freely used and modified. That said, the interferometry community as a whole stands to benefit from cooperative, organized development. A general model for contributing to the AIPY code-base goes as follows:<br />
* If you find a bug, write an [[AipyBugReport]] to help get it fixed.<br />
* Write an [[AipyEnhancementProposal]] that details what you're interested in changing, how it used to work, how it will work after your enhancment, and why the enhancement is better<br />
* Get a copy of the latest source code git (see [[GitAipy]] for instructions)<br />
* Make your edits (writing unit tests for any new features), and then check that you haven't broken any other unit tests.<br />
* Contact a primary developer (currently, Aaron Parsons: aparsons at astron berkeley edu) with an emailed patch. These patches, which may be produced automatically with git, will be merged into the primary source code branch.<br />
<br />
----------------------------</div>WikiSysopRC Filters2011-12-14T20:30:23Z<p>WikiSysop: /* Short Topical Videos */</p>
<hr />
<div>=== Short Topical Videos ===<br />
* [http://youtu.be/TBhdzzdT3cU RC Filters (by Karol Sanchez)]<br />
<br />
===Reference Material===<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{Resistors}<br />
\section*{Capacitors}<br />
\section{RC Filters}<br />
\subsection*{Low-Pass Filter}<br />
\subsection*{High-Pass Filter}<br />
<br />
</latex></div>WikiSysopFast Fourier Transform2011-11-30T15:58:21Z<p>WikiSysop: Created page with '=== Short Topical Videos === * [http://youtu.be/L4gpMr_OHnA Introduction to a Fast Fourier Transform Algorithm (by Aaron Parsons)] ===Reference Material=== * [http://en.wikipedi…'</p>
<hr />
<div>=== Short Topical Videos ===<br />
* [http://youtu.be/L4gpMr_OHnA Introduction to a Fast Fourier Transform Algorithm (by Aaron Parsons)]<br />
===Reference Material===<br />
* [http://en.wikipedia.org/wiki/Fast_Fourier_transform Wikipedia: Fast Fourier Transform]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-11-22T00:17:04Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
* [[Fisher Matrices]]<br />
<br />
Signal Path<br />
* [[RC Filters]]<br />
* [[Transistors]]<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Radiometer Equation]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** Working on class project.<br />
** [[Units of radiation]]<br />
* Oct 12:<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
** [[Data Representations]]<br />
* Oct 26:<br />
** [[21cm Transition]]<br />
* Nov 02:<br />
** [[Principle Component Analysis]]<br />
* Nov 09:<br />
* Nov 16:<br />
** [[Fisher Matrices]]<br />
* Nov 23:<br />
** [[RC Filters]]<br />
* Nov 30:<br />
** [[Fast Fourier Transform]]<br />
** [[Transistors]]</div>WikiSysopRC Filters2011-11-22T00:14:12Z<p>WikiSysop: </p>
<hr />
<div>=== Short Topical Videos ===<br />
* [http://youtu.be/nbjqptZUVec RC Filters (by Karol Sanchez)]</div>WikiSysopRC Filters2011-11-22T00:13:29Z<p>WikiSysop: Created page with '=== Short Topical Videos === * [http://youtu.be/nbjqptZUVec| RC Filters (by Karol Sanchez)]'</p>
<hr />
<div>=== Short Topical Videos ===<br />
* [http://youtu.be/nbjqptZUVec| RC Filters (by Karol Sanchez)]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-11-22T00:10:43Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
* [[Fisher Matrices]]<br />
<br />
Signal Path<br />
* [[RC Filters]]<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Radiometer Equation]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** Working on class project.<br />
** [[Units of radiation]]<br />
* Oct 12:<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
** [[Data Representations]]<br />
* Oct 26:<br />
** [[21cm Transition]]<br />
* Nov 02:<br />
** [[Principle Component Analysis]]<br />
* Nov 09:<br />
* Nov 16:<br />
** [[Fisher Matrices]]<br />
* Nov 23:<br />
** [[RC Filters]]<br />
* Nov 30:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-11-16T22:05:03Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
* [[Fisher Matrices]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Radiometer Equation]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** Working on class project.<br />
** [[Units of radiation]]<br />
* Oct 12:<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
** [[Data Representations]]<br />
* Oct 26:<br />
** [[21cm Transition]]<br />
* Nov 02:<br />
** [[Principle Component Analysis]]<br />
* Nov 09:<br />
* Nov 16:<br />
** [[Fisher Matrices]]<br />
* Nov 23:<br />
* Nov 30:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-11-16T22:02:10Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
* [[Fisher Matrices]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]] (again)<br />
** [[Units of radiation]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** [[Units of radiation]]<br />
** Working on class project.<br />
* Oct 12:<br />
** [[21cm Transition]]<br />
** [[Data Representations]]<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
** [[Principle Component Analysis]]<br />
* Nov 16:<br />
** [[Fisher Matrices]]<br />
* Nov 23:<br />
* Nov 30:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-11-02T21:18:16Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]] (again)<br />
** [[Units of radiation]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** [[Units of radiation]]<br />
** Working on class project.<br />
* Oct 12:<br />
** [[21cm Transition]]<br />
** [[Data Representations]]<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Nov 30:</div>WikiSysopBasic Interferometry2011-10-19T22:32:58Z<p>WikiSysop: FIxed typos</p>
<hr />
<div>===Prerequisites===<br />
* [[Convolution Theorem | Convolution_Theorem]]<br />
* [[Correlators | Correlators]]<br />
<br />
<br />
===Short Topical Videos===<br />
* [http://www.youtube.com/watch?v=xavu2z2o33U Basic Radio Interferometry I (by Karol Sanchez)]<br />
<br />
===Reference Material===<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{Basic Interferometry}<br />
<br />
Interferometry is the practice of using a two-or-more-element radio telescope array to observe astronomical sources. The array itself, along with the electronics used to synthesise the signals detected by the telescopes, are what we call the interferometer. <br />
<br />
\subsection*{The Two-Element Interferometer}<br />
%%INSERT PICTURE OF INTERFEROMETER<br />
<br />
<br />
A two-element interferometer consists of two telescopes separated by a vector $\mathbf{b}$, called the baseline. Both antennas receive electromagnetic radiation from an astronomical source in the sky, which is in the direction of the unit vector $\mathbf{\hat{s}}$. Because astronomical sources are far away, the radiation received by the antennas is in the form of plane waves. As the plane waves reach the antennas, one will actually receive the signal first and the second will not receive the signal until a certain amount of time has passed. The time that needs to pass is called the \emph{geometric delay}, and is denoted by the variable $\tau$.<br />
<br />
In order to find the value of $\tau$, we need to know the extra distance that the plane waves had to travel to reach the second telescope. This distance equal to the baseline vector $\mathbf{b}$ dotted with the unit vector $\mathbf{\hat{s}}$:<br />
\begin{align}<br />
\mathbf{b}\cdot\mathbf{\hat{s}} \ .<br />
\end{align}<br />
<br />
Since $\mathbf{\hat{s}}$ is a unit vector, this distance is actually the projection of the baseline vector, $\mathbf{b}$ onto the vector $\mathbf{\hat{s}}$ and is given by:<br />
<br />
\begin{equation}<br />
\mathbf{b}\cdot\mathbf{\hat{s}} = \mid \mathbf{b} \mid \cos\theta , <br />
\end{equation}<br />
where $\theta$ is the angle between the baseline and direction vectors.<br />
<br />
Knowing the distance, we just divide by the velocity of the plane waves, which is the speed of light, \emph{c}:<br />
<br />
\begin{equation}<br />
\tau = \frac{\mathbf{b}\cdot\mathbf{\hat{s}}}{\emph{c}} \ .<br />
\end{equation}<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=12in]{Interferometry_1_src.jpg} <br />
\caption{The basic two-element interferometer.}<br />
\end{figure}<br />
<br />
<br />
Suppose there were another source in the sky, in a different direction from the first. This second source would also cause a geometric delay between the two antennas; however, it would be different from the geometric delay of the first source because they would have different directional unit vectors:<br />
<br />
\begin{equation}<br />
\tau_1 = \frac{\mathbf{b}\cdot\mathbf{\hat{s_1}}}{\emph{c}} \ , \tau_2 = \frac{\mathbf{b}\cdot\mathbf{\hat{s_2}}}{\emph{c}} \ .<br />
\end{equation}<br />
<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=12in]{Interferometry__src.jpg} <br />
\caption{Two antennas detecting two sources.}<br />
\end{figure}<br />
%\subsection*{Why Interfeormetry?}<br />
<br />
Now each antenna is receiving a signal from each source, and what they detect is actually the total signal of the two sources. In order to separate the signals from each source at each antenna, we need to correlate these total signals with each other.<br />
<br />
\subsection*{Correlation}<br />
First, we define the total signal detected by each antenna as :<br />
<br />
\begin{equation}<br />
E_{i}(t)=e_{1}(t)+e_{2}(t) \ ,<br />
\end{equation}<br />
<br />
\begin{equation}<br />
E_{j}(t)=e_{1}(t-\tau_{1})+e_{2}(t-\tau_{2}) \ , <br />
\end{equation}<br />
<br />
where $e_{1}$ and $e_{2}$ are the signals given by each source at times $t$ and $t$ minus either $\tau_{1$} or $\tau_{2}$.<br />
<br />
<br />
We will use the correlation equation <br />
<br />
\begin{equation}<br />
(f \star g)(\tau) = \int{f(t)g^*(t-\tau)} dt<br />
\end{equation}<br />
<br />
and substitute <br />
<br />
\begin{equation}<br />
f=E_{i} \ , g=E_{j}. <br />
\end{equation}<br />
<br />
The substitution and the expansion are<br />
<br />
<br />
<br />
\begin{equation}<br />
(f \star g)(\tau) = \int{[e_{1}(t)+e_{2}(t)][e_{1}(t-\tau_{1}-\tau)+e_{2}(t-\tau_{2}-\tau)]} \ dt<br />
\end{equation}<br />
\begin{equation}<br />
(f \star g)(\tau) = \int{e_{1}(t)e_{1}(t-\tau_{1}-\tau)+e_{1}(t)e_{2}(t-\tau_{2}-\tau)+e_{2}(t)e_{1}(t-\tau_{1}-\tau)+e_{2}(t)e_{2}(t-\tau_{2}-\tau)} \ dt \ .<br />
\end{equation}<br />
<br />
Notice that we are working in the real-valued time domain, which is why $g^*=g$.<br />
Also, because $e_{1}$ and $e_{2}$ are different and independent of each other, the 2nd and 3rd term integrate away by averaging to zero as random noise.<br />
We are then left with<br />
<br />
<br />
\begin{equation}<br />
(f \star g)(\tau) = \int{e_{1}(t)e_{1}(t-\tau_{1}-\tau)+e_{2}(t)e_{2}(t-\tau_{2}-\tau)} \ dt \ .<br />
\end{equation}<br />
<br />
<br />
In order for us to extract any meaningful signal information out of the above integral, they must not both average to zero as well. This is possible when $\tau$ is equal either to $-\tau_{1}$ or $-\tau_{2}$.<br />
<br />
Taking the case when $\tau=-\tau_{1}$, the above integral becomes<br />
<br />
\begin{equation}<br />
(f \star g)(\tau) = \int{e_{1}(t)e_{1}(t-\tau_{1}-(-\tau_{1})) \ dt \ = \int{e_{1}(t)e_{1}(t) \ dt \ =\ <e_{1}^2> \ .<br />
\end{equation}<br />
<br />
Here the $e_{2}$ term averages to zero because $\tau_{2}-(-\tau_{1})$ do not cancel out.<br />
<br />
<br />
The end result is the average power received by the antennas for either source 1 or 2, depending on the geometric delay, $\tau$. So for a set of geometric delays, we have a set of power values, which rise above the signals that could not be correlated and are essentially noise.<br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=12in]{Interferometry4.jpg} <br />
\caption{Different geometric delays give different power values.}<br />
\end{figure}<br />
<br />
<br />
<br />
We just showed in a very basic way how an interferometer works: the telescopes collect the astronomical signals, and the correlator matches the signal functions from each antenna with each other so that they are maximized to give us a power value.<br />
<br />
The correlation gives us a one dimensional image of the sky in a direction parallel to the baseline of the telescopes. To form a more complete, 2-Dimensional picture, we would need another pair of telescopes, for example forming a baseline in a perpendicular direction to the first pair, to give us information about the sources in another direction. <br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=12in]{Test.jpg} <br />
\caption{Obtaining a 2D image.}<br />
\end{figure}<br />
<br />
<br />
This is more advanced interferometry, which will be covered in another section.<br />
<br />
<br />
</latex></div>WikiSysopData Representations2011-10-13T22:39:32Z<p>WikiSysop: /* Short Topical Videos */</p>
<hr />
<div>===Prerequisites===<br />
<br />
None<br />
<br />
===Short Topical Videos===<br />
<br />
* [http://www.youtube.com/watch?v=-WNkBkIE560| (Computer) Data Types and Representations, Part I (by Aaron Parsons)]<br />
* [http://www.youtube.com/watch?v=KsKWwtBJjGw| (Computer) Data Types and Representations, Part II (by Aaron Parsons)]<br />
<br />
===Reference Material===<br />
<br />
* [http://en.wikipedia.org/wiki/ASCII| ASCII reference]<br />
* [http://en.wikipedia.org/wiki/Floating_point| Floating-point reference]<br />
* [http://en.wikipedia.org/wiki/Fixed-point_arithmetic| Fixed-point reference]</div>WikiSysop21cm Transition2011-10-13T22:38:38Z<p>WikiSysop: /* Short Topical Videos */</p>
<hr />
<div>===Prerequisites===<br />
* [[Quantum Mechanics]]<br />
* [[Radiation Transfer]]<br />
* [[Black-Body Radiation]]<br />
<br />
===Short Topical Videos===<br />
<br />
* [http://www.youtube.com/watch?v=yZYpEtF2H-k| Basics of the 21cm Hyperfine transition (by Dyas Utomo)]<br />
<br />
===Reference Materials===<br />
<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{Hyperfine transition of hydrogen atoms}<br />
<br />
The ground state of atomic hydrogen split into two hyperfine levels by the interactions between the spins of electron and proton. Parallel spin has higher energy than antiparallel spin. The energy difference between these two levels is ${6 \times 10^{-6} eV}$ which corresponds with photons with frequency ${\nu} =$ 1.4204 GHz or wavelength 21.105 cm.<br />
<br />
\section*{Einstein coefficient}<br />
<br />
The Einstein coefficient ${A_{21}}$ is the probability for a system in the excited level ${E_{2}}$ to return spontaneously to the lower level ${E_{1}}$. Therefore, if ${N_{2}}$ is the electron density in level ${E_{2}}$ then ${N_{2}}{A_{21}}$ is the number of such spontaneous transition per second per unit volume.<br />
<br />
The probability that incoming photon is absorbed is ${B_{12}{U}$ where ${U} = 4\pi{I}/{c}$ is the average energy density of the radiation field. So, the number of photons absorbed by electron in level ${E_{1}}$ to jump to ${E_{2}}$ is ${N_{1}}{B_{12}}{U}$. There is another emission process proportional to ${U}$ that need to include: ${N_{2}{B_{21}{U}$ which equal to the number of photons emitted by ''stimulated emission''.<br />
<br />
For system in stationary state, the number of absorbed and emitted photons must be equal, so<br />
<br />
$$ {N_{2}}{A_{21}} + {N_{2}}{B_{21}}{U} = {N_{1}}{B_{12}}{U} $$<br />
<br />
For hyperfine transition,<br />
<br />
$${A_{10}} = 2.86888(7) \times 10^{-15} s^{-1}$$.<br />
<br />
This transition probability is about $10^{23}$ smaller than that of an allowed optical transition.<br />
<br />
Characteristic time for hyperfine transition is<br />
<br />
$${t_{1/2}} \approx 1/{A_{10}} = 3.49 \times 10^{14} s \approx 1.11 \times 10^7 yr$$.<br />
<br />
\section*{Spin temperature}<br />
<br />
Spin temperature ${T_{s}}$ describes the ratio of atoms in the excited states (${N_1}$) to the ground state (${N_0}$). According to Boltzmann distribution:<br />
<br />
$$ \frac{N_1}{N_0} = \frac{g_1}{g_0} \exp (-\frac{h\nu}{kT_s})$$<br />
<br />
There are three processes which determine the population of the hyperfine levels in ground state of hydrogen: collisions, 21 cm radiation, and Lyman-$\alpha$ radiation. Their relationship with spin temperature is<br />
<br />
$${T_s} = \frac{T_R + y_c T_K + y_L T_L}{1 + y_c + y_L}$$<br />
<br />
where ${T_R}$, ${T_K}$, and ${T_L}$ is the brightness temperature of 21 cm radiation, kinetic temperature, and the temperature of Lyman-$\alpha$, respectively, while ${y_c}$ and ${y_L}$ are coefficients which determine the relative efficiencies of the processes.<br />
<br />
Spin temperature becomes equal to thermal temperature after many interactions between atoms via collisions or radiative transfer.<br />
<br />
\section*{HI Column Density}<br />
<br />
HI column density ${N_{H}}$ is the number of neutral hydrogen atoms per unit area of line of sight. If the spin temperature ${T_{s}$ is constant along the line of sight then<br />
<br />
$${N_{H}} = 1.8224(3) \times 10^{18} (\frac{T_{s}}K) \int{\tau(\upsilon)d(\frac{\upsilon}{km s^{-1}}) cm^{-2}$$<br />
<br />
If the line is gaussian then this approximation is useful:<br />
<br />
$${N_{H}} \approx 1.94 \times 10^{18} (\frac{T_{s}}K) {\tau_0} (\frac{\Delta V}{km s^{-1}}) cm^{-2}$$<br />
<br />
where ${\tau_0}$ is the Gaussian peak and ${\Delta V}$ is the Full Width Half Maximum (FWHM) of the Gaussian.<br />
<br />
\section*{Wouthuysen Field Effect}<br />
<br />
Wouthuysen Field Effect is coupling of 21 cm to Lyman-$\alpha$ radiation. This effect becomes important for the Epoch of Reionization. See, for example, a paper from [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Furlanetto, et al. 2006]</div>WikiSysopStatistics in Python2011-10-13T22:37:37Z<p>WikiSysop: Created page with '===Prerequisites=== ===Short Topical Presentations=== ===Reference Material=== * [http://oneau.wordpress.com/2011/02/28/simple-statistics-with-scipy/| Simple statistics with Sc…'</p>
<hr />
<div>===Prerequisites===<br />
<br />
===Short Topical Presentations===<br />
<br />
===Reference Material===<br />
* [http://oneau.wordpress.com/2011/02/28/simple-statistics-with-scipy/| Simple statistics with SciPy]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-10-13T22:36:39Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
* [[Statistics in Python]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]] (again)<br />
** [[Units of radiation]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** [[Units of radiation]]<br />
** Working on class project.<br />
* Oct 12:<br />
** [[21cm Transition]]<br />
** [[Data Representations]]<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopData Representations2011-10-13T22:23:57Z<p>WikiSysop: Created page with '===Prerequisites=== None ===Short Topical Videos=== * [http://www.youtube.com/watch?v=-WNkBkIE560| (Computer) Data Types and Representations, Part I] * [http://www.youtube.c…'</p>
<hr />
<div>===Prerequisites===<br />
<br />
None<br />
<br />
===Short Topical Videos===<br />
<br />
* [http://www.youtube.com/watch?v=-WNkBkIE560| (Computer) Data Types and Representations, Part I]<br />
* [http://www.youtube.com/watch?v=KsKWwtBJjGw | (Computer) Data Types and Representations, Part II]<br />
<br />
===Reference Material===<br />
<br />
* [http://en.wikipedia.org/wiki/ASCII| ASCII reference]<br />
* [http://en.wikipedia.org/wiki/Floating_point| Floating-point reference]<br />
* [http://en.wikipedia.org/wiki/Fixed-point_arithmetic| Fixed-point reference]</div>WikiSysop21cm Transition2011-10-13T22:14:28Z<p>WikiSysop: /* Prerequisites */</p>
<hr />
<div>===Prerequisites===<br />
* [[Quantum Mechanics]]<br />
* [[Radiation Transfer]]<br />
* [[Black-Body Radiation]]<br />
<br />
===Short Topical Videos===<br />
<br />
[http://www.youtube.com/watch?v=yZYpEtF2H-k| Basics of the 21cm Hyperfine transition]<br />
<br />
===Reference Materials===<br />
<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{Hyperfine transition of hydrogen atoms}<br />
<br />
The ground state of atomic hydrogen split into two hyperfine levels by the interactions between the spins of electron and proton. Parallel spin has higher energy than antiparallel spin. The energy difference between these two levels is ${6 \times 10^{-6} eV}$ which corresponds with photons with frequency ${\nu} =$ 1.4204 GHz or wavelength 21.105 cm.<br />
<br />
\section*{Einstein coefficient}<br />
<br />
The Einstein coefficient ${A_{21}}$ is the probability for a system in the excited level ${E_{2}}$ to return spontaneously to the lower level ${E_{1}}$. Therefore, if ${N_{2}}$ is the electron density in level ${E_{2}}$ then ${N_{2}}{A_{21}}$ is the number of such spontaneous transition per second per unit volume.<br />
<br />
The probability that incoming photon is absorbed is ${B_{12}{U}$ where ${U} = 4\pi{I}/{c}$ is the average energy density of the radiation field. So, the number of photons absorbed by electron in level ${E_{1}}$ to jump to ${E_{2}}$ is ${N_{1}}{B_{12}}{U}$. There is another emission process proportional to ${U}$ that need to include: ${N_{2}{B_{21}{U}$ which equal to the number of photons emitted by ''stimulated emission''.<br />
<br />
For system in stationary state, the number of absorbed and emitted photons must be equal, so<br />
<br />
$$ {N_{2}}{A_{21}} + {N_{2}}{B_{21}}{U} = {N_{1}}{B_{12}}{U} $$<br />
<br />
For hyperfine transition,<br />
<br />
$${A_{10}} = 2.86888(7) \times 10^{-15} s^{-1}$$.<br />
<br />
This transition probability is about $10^{23}$ smaller than that of an allowed optical transition.<br />
<br />
Characteristic time for hyperfine transition is<br />
<br />
$${t_{1/2}} \approx 1/{A_{10}} = 3.49 \times 10^{14} s \approx 1.11 \times 10^7 yr$$.<br />
<br />
\section*{Spin temperature}<br />
<br />
Spin temperature ${T_{s}}$ describes the ratio of atoms in the excited states (${N_1}$) to the ground state (${N_0}$). According to Boltzmann distribution:<br />
<br />
$$ \frac{N_1}{N_0} = \frac{g_1}{g_0} \exp (-\frac{h\nu}{kT_s})$$<br />
<br />
There are three processes which determine the population of the hyperfine levels in ground state of hydrogen: collisions, 21 cm radiation, and Lyman-$\alpha$ radiation. Their relationship with spin temperature is<br />
<br />
$${T_s} = \frac{T_R + y_c T_K + y_L T_L}{1 + y_c + y_L}$$<br />
<br />
where ${T_R}$, ${T_K}$, and ${T_L}$ is the brightness temperature of 21 cm radiation, kinetic temperature, and the temperature of Lyman-$\alpha$, respectively, while ${y_c}$ and ${y_L}$ are coefficients which determine the relative efficiencies of the processes.<br />
<br />
Spin temperature becomes equal to thermal temperature after many interactions between atoms via collisions or radiative transfer.<br />
<br />
\section*{HI Column Density}<br />
<br />
HI column density ${N_{H}}$ is the number of neutral hydrogen atoms per unit area of line of sight. If the spin temperature ${T_{s}$ is constant along the line of sight then<br />
<br />
$${N_{H}} = 1.8224(3) \times 10^{18} (\frac{T_{s}}K) \int{\tau(\upsilon)d(\frac{\upsilon}{km s^{-1}}) cm^{-2}$$<br />
<br />
If the line is gaussian then this approximation is useful:<br />
<br />
$${N_{H}} \approx 1.94 \times 10^{18} (\frac{T_{s}}K) {\tau_0} (\frac{\Delta V}{km s^{-1}}) cm^{-2}$$<br />
<br />
where ${\tau_0}$ is the Gaussian peak and ${\Delta V}$ is the Full Width Half Maximum (FWHM) of the Gaussian.<br />
<br />
\section*{Wouthuysen Field Effect}<br />
<br />
Wouthuysen Field Effect is coupling of 21 cm to Lyman-$\alpha$ radiation. This effect becomes important for the Epoch of Reionization. See, for example, a paper from [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Furlanetto, et al. 2006]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-10-12T19:47:41Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]] (again)<br />
** [[Units of radiation]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
** [[Coordinates]]<br />
** [[Units of radiation]]<br />
** Working on class project.<br />
* Oct 12:<br />
** [[21cm Transition]]<br />
** [[Data Representations]]<br />
** Working on class project. Report on [[Homemade Interferometer]] deployment.<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-28T02:20:26Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[General software tools]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers <br />
* [[Basic Interferometry]]<br />
* [[Basic Interferometry II]]<br />
* Units<br />
** [[Coordinates]]<br />
** [[Units of radiation]] <br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[21cm Transition]]<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[General software tools|Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
** [[Basic Interferometry II]] (again)<br />
** [[Units of radiation]]<br />
** Discuss class project: [[Parallel Computing]]<br />
** Create plan of work for each class project. Catch up on class software.<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopBasic Interferometry II2011-09-28T02:10:50Z<p>WikiSysop: </p>
<hr />
<div>===Prerequisites===<br />
* [[Basic Interferometry | Basic Interferometry]]<br />
<br />
<br />
===Short Topical Videos===<br />
* [http://www.youtube.com/watch?v=JaaPH_6Dj4A Basic Radio Interferometry II, part1 (by Aaron Parsons)]<br />
* [http://www.youtube.com/watch?v=39g0hTTn5n8 Basic Radio Interferometry II, part2 (by Aaron Parsons)]<br />
* [http://www.youtube.com/watch?v=quZfMV0Ai3A Basic Radio Interferometry II, part3 (by Aaron Parsons)]<br />
<br />
===Reference Material===<br />
<latex><br />
\documentclass[]{article}<br />
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}<br />
\usepackage{amsmath}<br />
\usepackage{graphicx}<br />
\usepackage{natbib}<br />
<br />
\begin{document}<br />
\title{Interferometry II}<br />
<br />
\section{Recap of Interferometry I}<br />
<br />
You'll recall from the Interferometry I lecture that for two antennas $i,j$, we call the vector separating<br />
them $\vec b_{ij}$ the {\bf baseline}. If a source in the sky is in direction $\hat s$, then we derived<br />
the time delay between the two antennas from the source's point of view was<br />
$\tau_{ij}=\frac{\vec b_{ij}\cdot\hat s}{c}$. We then showed how the fact that these delays as a function<br />
of baseline could in principle be used to reverse-engineer where sources were on the sky. In this lecture,<br />
we will begin there and develop a bit more formalism about how this works, and then walk through how<br />
imaging works, to first order.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=4in]{Interferometry2_1src.jpg}<br />
\caption{The basic two-element interferometer.}<br />
\end{figure}<br />
<br />
\section{The Visibility Equation}<br />
<br />
Let's begin by restricting our discussion to a single frequency $\nu$, with corresponding wavelength $\lambda$.<br />
Then just as $\frac{\vec b_{ij}\cdot\hat s}{c}$ was the time delay between antennas $i,j$, we can also describe<br />
the number of wavelengths this is:<br />
\begin{equation}<br />
\tau_{ij}\nu = \frac{\vec b\cdot\hat s}{\lambda}<br />
\end{equation}<br />
Knowing the number of wavelengths between the two antennas, we can now say that for a signal of a particular<br />
frequency eminating from a source in direction $\hat s$, the complex phase difference between that signal measured<br />
at antenna $i$ and the signal at antenna $j$ will be $e^{-i\theta}$, where $\theta$ is the angle swept out by<br />
the wave as it propagates from $i$ to $j$. Using that $\theta$ is just $2\pi$ times the number of wavelengths,<br />
we have the phase difference $\Delta\phi$ is:<br />
\begin{equation}<br />
\Delta\phi = e^{-2\pi i\tau_{ij}\nu} = e^{-2\pi i\frac{\vec b\cdot\hat s}{\lambda}}<br />
\end{equation}<br />
The phase difference $\Delta\phi$ is, of course, frequency dependent. And at a given frequency, it also varies<br />
with position on the sky. The pattern that this complex phase traces on the sky (see below for a graph of just<br />
the real component) is called the ``fringe pattern'' of an interferometer.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{fringe_pattern.png} <br />
\caption{A graph of the real component of $e^{-2\pi i\frac{\vec b\cdot\hat s}{\lambda}}$ at a fixed<br />
frequency, as a function of direction on the sky. The complex response of a baseline along the<br />
sky is called the ``fringe pattern'', and it is suspiciously close to a sine wave.}<br />
\end{figure}<br />
<br />
Now we will define a few variables that will help us extrapolate from a single baseline in a single direction<br />
to a picture of how a whole array might respond to the whole sky that falls within the primary beam of the<br />
correlated antennas. First, we will define coordinates representing the length of a baseline in units of<br />
wavelength:<br />
\begin{equation}<br />
\frac{\vec b}{\lambda}\equiv (u,v,w),<br />
\end{equation}<br />
where $u$ is the east-west component of the baseline, $v$ is the north-south component, and $w$ is the vertical<br />
(up-down) component. We will also split the source direction vector $\hat s$ into its components:<br />
\begin{equation}<br />
\hat s\equiv (l,m,\sqrt{1-l^2-m^2}),<br />
\end{equation}<br />
where $l$ is the east-west direction on the sky, $m$ is the north-south direction, and the third component comes<br />
from the fact that we restrict $\hat s$ to have unit length (it's a direction vector).<br />
<br />
Using these components, we can now write down the response of a baseline (called the ``visibility'' $V$) as<br />
a function of the $u,v,w$ separation of the antennas, integrating over all the source intensity $I$ on the<br />
sky as a function of $l,m$:<br />
\begin{equation}<br />
V(u,v)=\int\!\!\int{A(l,m)\cdot I(l,m)\cdot e^{-2\pi i(ul+vm+w\sqrt{1-l^2-m^2})}dl\ dm}.<br />
\end{equation}<br />
The equation above is the full form of the ``visibility equation'', otherwise known as the ``measurement equation''<br />
of an interferometer. The only variable that we haven't yet defined is $A$, which is the response of the<br />
primary beams of the antennas as a function of direction on the sky. In general, $A$ and $I$ are always grouped<br />
together, because the sky is always seen through the filter of the primary beam. The product $A\cdot I$ is<br />
sometimes called the ``perceived intensity''.<br />
<br />
\section{Understanding the Visibility Equation as a Fourier Transform}<br />
<br />
The equation we derived above can be much easier to understand if we make a simplifying assumption, known as<br />
the ``flat-sky'' approximation. This approximation is either that $w=0$, or alternately, that the primary beam<br />
$A(l,m)$ is sufficiently small that $l,m\ll1$, making $\sqrt{1-l^2-m^2}\approx1$. In either case, we are asking<br />
that the response of a baseline not need to account for the fact that the sky is a curved surface of a sphere.<br />
Under this assumption, the term $e^{-2\pi i w\sqrt{1}}$ is no longer a function of $l,m$, and can be removed<br />
from the integral to give us:<br />
\begin{equation}<br />
V(u,v)=e^{-2\pi iw}\int\!\!\int{A(l,m)\cdot I(l,m)\cdot e^{-2\pi i(ul+vm)}dl\ dm}.<br />
\end{equation}<br />
This formulation of the Visibility Equation is much more illuminating. It says that when phased<br />
to a ``phase center'' via a choice of a corresponding $e^{-2\pi iw}$, with $w$ being the baseline<br />
component along the direction toward the phase center in wavelengths, the visibility $V(u,v)$<br />
is just the Fourier Transform of the perceived sky.<br />
<br />
So in addition to thinking about the fringe-pattern of a baseline on the sky, we can equivalently think<br />
of the following process.<br />
We take an image of the sky in $l,m$ coordinates:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{true_sky.jpg} <br />
\caption{The true image of the sky, in $l,m$ coordinates.}<br />
\end{figure}<br />
<br />
and Fourier Transform it:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{true_uvplane.jpg} <br />
\caption{The true uv-plane.}<br />
\end{figure}<br />
<br />
The result is called the ``uv-plane'', and its coordinates are inverse angles.<br />
An inverse angle is the same thing as a wavelength, so the uv-plane has coordinates (not surprisingly)<br />
of $u,v$.<br />
<br />
Next, this uv-plane is sampled at particular $u,v$-coordinates by various baselines in an antenna array.<br />
The sampling pattern can be computed from the antenna configuration by choosing all of the antenna-to-antenna<br />
spacings.<br />
(Interestingly, this sampling pattern is the convolution of the antenna placement pattern with itself):<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{uvplane_sampling.jpg} <br />
\caption{The array sampling pattern in the uv-plane.}<br />
\end{figure}<br />
<br />
Note that for each pair of antennas you get two samples: one at $u,v$, and one at $-u,-v$. Because the<br />
sky is real-valued (no complex fluxes), these two Fourier components are related by a complex conjugate.<br />
That is, if you measure $V(u,v)$ at $u,v$, you will measure $V^*(u,v)$ at $-u,-v$.<br />
<br />
Now, the sampling of the uv-plane is simply multiplying the true uv-plane by the sampling pattern you<br />
just computed. This is what you would get if you took visibilities recorded from an interferometer,<br />
and then placed each measured visibility $V(u,v)$ at the corresponding $u,v$ (and $-u,-v$) coordinates of<br />
a matrix. Finally, if you take the inverse Fourier Transform of this sampled uv-plane, you get an image:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{dirty_image.jpg} <br />
\caption{The ``dirty image''.}<br />
\end{figure}<br />
<br />
As you may notice, this image is somewhat degraded from our original. In fact, it is usually called a<br />
``dirty image''. Why is it dirty? Because we lost information when we sampled the uv-plane. We<br />
multiplied the true uv-plane by our sampling function. This is equivalent to convolving the true sky<br />
by the Fourier Transform of our sampling function:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{dirty_beam.jpg} <br />
\caption{The ``dirty beam''.}<br />
\end{figure}<br />
<br />
The Fourier transform of our sampling function is often called the ``dirty beam''. The dirty beam is<br />
what convolved the true sky to yield the dirty image.<br />
It is possible to recover something resembling the true sky by attempting to deconvolve the dirty image<br />
by the dirty beam. This is a complex process that will be described in detail in another lecture. Broadly,<br />
deconvolving attempts to compensate for the information that was lost by only sampling part of the uv-plane<br />
by injecting prior information about the sky. This information might be along the lines of ``I know<br />
the sky is just point-sources'' or ``I want the smoothest sky that fits the data to the level of noise''.<br />
Either way, deconvolution is not a well-posed problem until you decide exactly what prior information you have<br />
about the sky.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{clean_image.jpg} <br />
\caption{A ``cleaned'' dirty image.}<br />
\end{figure}<br />
<br />
</latex></div>WikiSysopFile:Interferometry2 1src.jpg2011-09-21T21:10:16Z<p>WikiSysop: </p>
<hr />
<div></div>WikiSysopBasic Interferometry II2011-09-21T21:09:23Z<p>WikiSysop: Created page with '<latex> \documentclass[]{article} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} \usepackage{amsmath} \usepackage{graphicx} \usepackage{natbib} \begin{document} \t…'</p>
<hr />
<div><latex><br />
\documentclass[]{article}<br />
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}<br />
\usepackage{amsmath}<br />
\usepackage{graphicx}<br />
\usepackage{natbib}<br />
<br />
\begin{document}<br />
\title{Interferometry II}<br />
<br />
\section{Recap of Interferometry I}<br />
<br />
You'll recall from the Interferometry I lecture that for two antennas $i,j$, we call the vector separating<br />
them $\vec b_{ij}$ the {\bf baseline}. If a source in the sky is in direction $\hat s$, then we derived<br />
the time delay between the two antennas from the source's point of view was<br />
$\tau_{ij}=\frac{\vec b_{ij}\cdot\hat s}{c}$. We then showed how the fact that these delays as a function<br />
of baseline could in principle be used to reverse-engineer where sources were on the sky. In this lecture,<br />
we will begin there and develop a bit more formalism about how this works, and then walk through how<br />
imaging works, to first order.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
\includegraphics[width=4in]{Interferometry2_1src.jpg}<br />
\caption{The basic two-element interferometer.}<br />
\end{figure}<br />
<br />
\section{The Visibility Equation}<br />
<br />
Let's begin by restricting our discussion to a single frequency $\nu$, with corresponding wavelength $\lambda$.<br />
Then just as $\frac{\vec b_{ij}\cdot\hat s}{c}$ was the time delay between antennas $i,j$, we can also describe<br />
the number of wavelengths this is:<br />
\begin{equation}<br />
\tau_{ij}\nu = \frac{\vec b\cdot\hat s}{\lambda}<br />
\end{equation}<br />
Knowing the number of wavelengths between the two antennas, we can now say that for a signal of a particular<br />
frequency eminating from a source in direction $\hat s$, the complex phase difference between that signal measured<br />
at antenna $i$ and the signal at antenna $j$ will be $e^{-i\theta}$, where $\theta$ is the angle swept out by<br />
the wave as it propagates from $i$ to $j$. Using that $\theta$ is just $2\pi$ times the number of wavelengths,<br />
we have the phase difference $\Delta\phi$ is:<br />
\begin{equation}<br />
\Delta\phi = e^{-2\pi i\tau_{ij}\nu} = e^{-2\pi i\frac{\vec b\cdot\hat s}{\lambda}}<br />
\end{equation}<br />
The phase difference $\Delta\phi$ is, of course, frequency dependent. And at a given frequency, it also varies<br />
with position on the sky. The pattern that this complex phase traces on the sky (see below for a graph of just<br />
the real component) is called the ``fringe pattern'' of an interferometer.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{fringe_pattern.png} <br />
\caption{A graph of the real component of $e^{-2\pi i\frac{\vec b\cdot\hat s}{\lambda}}$ at a fixed<br />
frequency, as a function of direction on the sky. The complex response of a baseline along the<br />
sky is called the ``fringe pattern'', and it is suspiciously close to a sine wave.}<br />
\end{figure}<br />
<br />
Now we will define a few variables that will help us extrapolate from a single baseline in a single direction<br />
to a picture of how a whole array might respond to the whole sky that falls within the primary beam of the<br />
correlated antennas. First, we will define coordinates representing the length of a baseline in units of<br />
wavelength:<br />
\begin{equation}<br />
\frac{\vec b}{\lambda}\equiv (u,v,w),<br />
\end{equation}<br />
where $u$ is the east-west component of the baseline, $v$ is the north-south component, and $w$ is the vertical<br />
(up-down) component. We will also split the source direction vector $\hat s$ into its components:<br />
\begin{equation}<br />
\hat s\equiv (l,m,\sqrt{1-l^2-m^2}),<br />
\end{equation}<br />
where $l$ is the east-west direction on the sky, $m$ is the north-south direction, and the third component comes<br />
from the fact that we restrict $\hat s$ to have unit length (it's a direction vector).<br />
<br />
Using these components, we can now write down the response of a baseline (called the ``visibility'' $V$) as<br />
a function of the $u,v,w$ separation of the antennas, integrating over all the source intensity $I$ on the<br />
sky as a function of $l,m$:<br />
\begin{equation}<br />
V(u,v)=\int\!\!\int{A(l,m)\cdot I(l,m)\cdot e^{-2\pi i(ul+vm+w\sqrt{1-l^2-m^2})}dl\ dm}.<br />
\end{equation}<br />
The equation above is the full form of the ``visibility equation'', otherwise known as the ``measurement equation''<br />
of an interferometer. The only variable that we haven't yet defined is $A$, which is the response of the<br />
primary beams of the antennas as a function of direction on the sky. In general, $A$ and $I$ are always grouped<br />
together, because the sky is always seen through the filter of the primary beam. The product $A\cdot I$ is<br />
sometimes called the ``perceived intensity''.<br />
<br />
\section{Understanding the Visibility Equation as a Fourier Transform}<br />
<br />
The equation we derived above can be much easier to understand if we make a simplifying assumption, known as<br />
the ``flat-sky'' approximation. This approximation is either that $w=0$, or alternately, that the primary beam<br />
$A(l,m)$ is sufficiently small that $l,m\ll1$, making $\sqrt{1-l^2-m^2}\approx1$. In either case, we are asking<br />
that the response of a baseline not need to account for the fact that the sky is a curved surface of a sphere.<br />
Under this assumption, the term $e^{-2\pi i w\sqrt{1}}$ is no longer a function of $l,m$, and can be removed<br />
from the integral to give us:<br />
\begin{equation}<br />
V(u,v)=e^{-2\pi iw}\int\!\!\int{A(l,m)\cdot I(l,m)\cdot e^{-2\pi i(ul+vm)}dl\ dm}.<br />
\end{equation}<br />
This formulation of the Visibility Equation is much more illuminating. It says that when phased<br />
to a ``phase center'' via a choice of a corresponding $e^{-2\pi iw}$, with $w$ being the baseline<br />
component along the direction toward the phase center in wavelengths, the visibility $V(u,v)$<br />
is just the Fourier Transform of the perceived sky.<br />
<br />
So in addition to thinking about the fringe-pattern of a baseline on the sky, we can equivalently think<br />
of the following process.<br />
We take an image of the sky in $l,m$ coordinates:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{true_sky.jpg} <br />
\caption{The true image of the sky, in $l,m$ coordinates.}<br />
\end{figure}<br />
<br />
and Fourier Transform it:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{true_uvplane.jpg} <br />
\caption{The true uv-plane.}<br />
\end{figure}<br />
<br />
The result is called the ``uv-plane'', and its coordinates are inverse angles.<br />
An inverse angle is the same thing as a wavelength, so the uv-plane has coordinates (not surprisingly)<br />
of $u,v$.<br />
<br />
Next, this uv-plane is sampled at particular $u,v$-coordinates by various baselines in an antenna array.<br />
The sampling pattern can be computed from the antenna configuration by choosing all of the antenna-to-antenna<br />
spacings.<br />
(Interestingly, this sampling pattern is the convolution of the antenna placement pattern with itself):<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{uvplane_sampling.jpg} <br />
\caption{The array sampling pattern in the uv-plane.}<br />
\end{figure}<br />
<br />
Note that for each pair of antennas you get two samples: one at $u,v$, and one at $-u,-v$. Because the<br />
sky is real-valued (no complex fluxes), these two Fourier components are related by a complex conjugate.<br />
That is, if you measure $V(u,v)$ at $u,v$, you will measure $V^*(u,v)$ at $-u,-v$.<br />
<br />
Now, the sampling of the uv-plane is simply multiplying the true uv-plane by the sampling pattern you<br />
just computed. This is what you would get if you took visibilities recorded from an interferometer,<br />
and then placed each measured visibility $V(u,v)$ at the corresponding $u,v$ (and $-u,-v$) coordinates of<br />
a matrix. Finally, if you take the inverse Fourier Transform of this sampled uv-plane, you get an image:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{dirty_image.jpg} <br />
\caption{The ``dirty image''.}<br />
\end{figure}<br />
<br />
As you may notice, this image is somewhat degraded from our original. In fact, it is usually called a<br />
``dirty image''. Why is it dirty? Because we lost information when we sampled the uv-plane. We<br />
multiplied the true uv-plane by our sampling function. This is equivalent to convolving the true sky<br />
by the Fourier Transform of our sampling function:<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{dirty_beam.jpg} <br />
\caption{The ``dirty beam''.}<br />
\end{figure}<br />
<br />
The Fourier transform of our sampling function is often called the ``dirty beam''. The dirty beam is<br />
what convolved the true sky to yield the dirty image.<br />
It is possible to recover something resembling the true sky by attempting to deconvolve the dirty image<br />
by the dirty beam. This is a complex process that will be described in detail in another lecture. Broadly,<br />
deconvolving attempts to compensate for the information that was lost by only sampling part of the uv-plane<br />
by injecting prior information about the sky. This information might be along the lines of ``I know<br />
the sky is just point-sources'' or ``I want the smoothest sky that fits the data to the level of noise''.<br />
Either way, deconvolution is not a well-posed problem until you decide exactly what prior information you have<br />
about the sky.<br />
<br />
\begin{figure}[!h]<br />
\centering<br />
%\includegraphics[width=4in]{clean_image.jpg} <br />
\caption{A ``cleaned'' dirty image.}<br />
\end{figure}<br />
<br />
</latex></div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-21T18:38:07Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
** [[Basic Interferometry]]<br />
** Discus class project: the [[Homemade Interferometer]]<br />
** Create some software for simulating a visibility. That is, given two antennas (with x,y,z positions in equatorial coordinates) and a source (with x,y,z also in equatorial coordinates), compute the phase that you would measure as a function of frequency.<br />
* Sep 21:<br />
** [[21cm Transition]]<br />
** [[Basic Interferometry II]]<br />
** Discuss class project: [[Aggregating Software Tools]]<br />
** Extend visibility simulation software to handle many sources (with spectra), and many antennas (with passbands).<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopConvolution Theorem2011-09-15T21:19:01Z<p>WikiSysop: /* Short Topical Videos */</p>
<hr />
<div>===Prerequisites===<br />
* Fourier Transforms (link?)<br />
* Integral Calculus (link?)<br />
<br />
===Short Topical Videos===<br />
* [http://www.youtube.com/watch?v=a0IdGLczoAA The Convolution Theorem] by Aaron Parsons<br />
* [http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations Introduction to the Convolution] by Khan Academy<br />
<br />
===Reference Material===<br />
<br />
<latex><br />
\documentclass[]{article}<br />
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}<br />
\usepackage{amsmath}<br />
\usepackage{graphicx}<br />
\usepackage{natbib}<br />
<br />
\begin{document}<br />
\title{Convolution Theorem}<br />
<br />
\section*{Fourier Transform}<br />
<br />
Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$)<br />
Fourier transforms:<br />
\begin{align}<br />
f(t)&= \mathcal{F}(\hat f(\omega)) = \frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}d\omega}\\<br />
\hat f(\omega)&= \mathcal{F}^{-1}(f(t)) = \int{f(t)e^{-i\omega t}dt}<br />
\label{eq:fourier_transform}<br />
\end{align}<br />
where $\omega\equiv2\pi\nu$ is the angular frequency coordinate that is the Fourier complement of time $t$,<br />
and a top-hat is generally used to denote Fourier-domain quantities.<br />
<br />
\section*{Convolution Theorem}<br />
<br />
The {\it convolution} is a useful operation with applications ranging from photo editing to<br />
crystallography to astronomy. In words, the convolution of two functions $f,g$ is what you get when you smooth one function ($f$) by another ($g$). Note that the order of $f$ and $g$ does not matter, though people often call the latter the ``kernel''. Smoothing $f$ by $g$ means that you slide $g$ along $f$, and at each step along the way, you sum up all of the parts of $f$ with weights drawn from the value of $g$ at the point you slid it to. In essence, you are blurring $f$ by $g$.<br />
<br />
Mathematically, this is described as: <br />
<br />
\begin{align}<br />
\left[f*g\right](\tau)&\equiv \int{f(t)g(\tau-t)dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)e^{i\omega_1 t}d\omega_1\,<br />
\hat g(\omega_2)e^{i\omega_2(\tau-t)}d\omega_2\,dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)\hat g(\omega_2)e^{i(\omega_1-\omega_2) t}<br />
e^{i\omega_2\tau}d\omega_1\,d\omega_2\,dt}\\<br />
& = \frac1{2\pi}\int{\hat f(\omega)\hat g(\omega) e^{i\omega\tau}d\omega}.<br />
\end{align}<br />
Renaming $\tau$ to be $t$ (which we are totally free to do), we get a statement of the<br />
{\it convolution theorem}:<br />
\begin{equation}<br />
f(t)*g(t) = \int{\hat f(\omega) \hat g(\omega) e^{i\omega t}d\omega} <br />
= \mathcal{F}^{-1}\!\!\left(\mathcal{F}(f)\cdot\mathcal{F}(g)\right).<br />
\label{eq:conv_thm}<br />
\end{equation}<br />
<br />
\subsection*{Convolution vs. Correlation}<br />
<br />
{\it Correlation} is very similar to convolution, and it is best defined through its<br />
equivalent ``correlation theorem'':<br />
\begin{equation}<br />
f(t)\star g(t) = \int{\hat f(\omega) \hat g^*(\omega) e^{i\omega t}d\omega}<br />
\label{eq:corr_thm}.<br />
\end{equation}<br />
The difference between correlation and convolution is that<br />
that when correlating two signals, the Fourier transform of the second<br />
function ($\hat g(\omega)$ in equation \ref{eq:corr_thm}) is conjugated before<br />
multiplying and integrating.<br />
Using that<br />
\begin{align}<br />
g^*(-t)&=\frac1{2\pi}\int{\hat g^*(\omega)e^{-i\omega(-t)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat g^*(\omega)e^{i\omega t}d\omega},<br />
\end{align}<br />
we can show that correlating $f(t)$ and $g(t)$ is equivalent to convolving $f(t)$ with a<br />
conjugated, time-reversed version of $g(t)$:<br />
\begin{equation}<br />
f(t)*g^*(-t) = f(t)\star g(t).<br />
\label{eq:conv_corr_relation}<br />
\end{equation}<br />
Although this relation between convolution and correlation is often mentioned<br />
in the literature, I don't personally find it very intuitively illuminating.<br />
I much prefer the ``correlation theorem'' in equation (\ref{eq:corr_thm}), because<br />
when it is combined with the expression of a time-shifted signal in Fourier domain:<br />
\begin{align}<br />
f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega(t-\tau)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}e^{-i\omega\tau}d\omega}<br />
\label{eq:delay_freq},<br />
\end{align}<br />
it shows that correlating a flat-spectrum signal with a time-shifted version of itself yields<br />
a measure of the power of the signal at the delay corresponding to the time shift:<br />
\begin{align}<br />
f(t)\star f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)\hat f^*(\omega)e^{-i\omega\tau}e^{i\omega t}d\omega}\\<br />
&=\frac1{2\pi}\int{|f|^2e^{i\omega (t-\tau)}d\omega}\\<br />
&=|f|^2\cdot\delta(t-\tau).<br />
\end{align}<br />
<br />
\end{document}<br />
</latex><br />
<br />
===Related Subjects===<br />
* [[Interferometric Imaging]]</div>WikiSysopBasic Interferometry2011-09-13T19:31:12Z<p>WikiSysop: </p>
<hr />
<div>===Prerequisites===<br />
* <br />
<br />
===Short Topical Videos===<br />
* [http://www.youtube.com/watch?v=xavu2z2o33U Basic Radio Interferometry I (by Karol Sanchez)]<br />
<br />
===Reference Material===<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{Basic Interferometry}<br />
<br />
Interferometry is the practice of using a two-or-more-element radio telescope array to observe astronomical sources. The array itself, along with the electronics used to synthesise the signals detected by the telescopes, are what we call the interferometer. <br />
<br />
\subsection*{The Two-Element Interferometer}<br />
%%INSERT PICTURE OF INTERFEROMETER<br />
<br />
A two-element interferometer consists of two telescopes seperated by a vector $\mathbf{b}$, called the baseline. Both antennas receive electromagnetic radiation from an astronomical source in the sky, which is in the direction of the unit vector $\mathbf{\hat{s}}$. Because astronomical sources are far away, the radiation received by the antennas is in the form of plane waves. As the plane waves reach the antennas, one will actually receive the signal first and the second will not receive the signal until a certain amount of time has passed. The time that needs to pass is called the \emph{geometric delay}, and is denoted by the variable $\tau$.<br />
<br />
In order to find the value of $\tau$, we need to know the extra distance that the plane waves had to travel to reach the second telescope. This distance equal to the baseline vector $\mathbf{b}$ dotted with the unit vector $\mathbf{\hat{s}}$:<br />
\begin{align}<br />
\mathbf{b}\cdot\mathbf{\hat{s}} \ .<br />
\end{align}<br />
<br />
Since $\mathbf{\hat{s}}$ is a unit vector, this distance is actually the projection of the baseline vector, $\mathbf{b}$ onto the vector $\mathbf{\hat{s}}$ and is given by:<br />
<br />
\begin{equation}<br />
\mathbf{b}\cdot\mathbf{\hat{s}} = \mid \mathbf{b} \mid \cos\theta , <br />
\end{equation}<br />
where $\theta$ is the angle between the baseline and direction vectors.<br />
<br />
Knowing the distance, we just divide by the velocity of the plane waves, which is the speed of light, \emph{c}:<br />
<br />
\begin{equation}<br />
\tau = \frac{\mathbf{b}\cdot\mathbf{\hat{s}}}{\emph{c}} \ .<br />
\end{equation}<br />
<br />
<br />
%\subsection*{Why Interfeormetry?}<br />
<br />
<br />
\end{document}<br />
<br />
</latex></div>WikiSysopConvolution Theorem2011-09-10T23:49:28Z<p>WikiSysop: /* Reference Material */</p>
<hr />
<div>===Prerequisites===<br />
* Fourier Transforms (link?)<br />
* Integral Calculus (link?)<br />
<br />
===Short Topical Videos===<br />
* [http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations Introduction to the Convolution] by Khan Academy<br />
<br />
===Reference Material===<br />
<br />
<latex><br />
\documentclass[]{article}<br />
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}<br />
\usepackage{amsmath}<br />
\usepackage{graphicx}<br />
\usepackage{natbib}<br />
<br />
\begin{document}<br />
\title{Convolution Theorem}<br />
<br />
\section*{Fourier Transform}<br />
<br />
Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$)<br />
Fourier transforms:<br />
\begin{align}<br />
f(t)&= \mathcal{F}(\hat f(\omega)) = \frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}d\omega}\\<br />
\hat f(\omega)&= \mathcal{F}^{-1}(f(t)) = \int{f(t)e^{-i\omega t}dt}<br />
\label{eq:fourier_transform}<br />
\end{align}<br />
where $\omega\equiv2\pi\nu$ is the angular frequency coordinate that is the Fourier complement of time $t$,<br />
and a top-hat is generally used to denote Fourier-domain quantities.<br />
<br />
\section*{Convolution Theorem}<br />
<br />
The {\it convolution} is a useful operation with applications ranging from photo editing to<br />
crystallography to astronomy. In words, the convolution of two functions $f,g$ is what you get when you smooth one function ($f$) by another ($g$). Note that the order of $f$ and $g$ does not matter, though people often call the latter the ``kernel''. Smoothing $f$ by $g$ means that you slide $g$ along $f$, and at each step along the way, you sum up all of the parts of $f$ with weights drawn from the value of $g$ at the point you slid it to. In essence, you are blurring $f$ by $g$.<br />
<br />
Mathematically, this is described as: <br />
<br />
\begin{align}<br />
\left[f*g\right](\tau)&\equiv \int{f(t)g(\tau-t)dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)e^{i\omega_1 t}d\omega_1\,<br />
\hat g(\omega_2)e^{i\omega_2(\tau-t)}d\omega_2\,dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)\hat g(\omega_2)e^{i(\omega_1-\omega_2) t}<br />
e^{i\omega_2\tau}d\omega_1\,d\omega_2\,dt}\\<br />
& = \frac1{2\pi}\int{\hat f(\omega)\hat g(\omega) e^{i\omega\tau}d\omega}.<br />
\end{align}<br />
Renaming $\tau$ to be $t$ (which we are totally free to do), we get a statement of the<br />
{\it convolution theorem}:<br />
\begin{equation}<br />
f(t)*g(t) = \int{\hat f(\omega) \hat g(\omega) e^{i\omega t}d\omega} <br />
= \mathcal{F}^{-1}\!\!\left(\mathcal{F}(f)\cdot\mathcal{F}(g)\right).<br />
\label{eq:conv_thm}<br />
\end{equation}<br />
<br />
\subsection*{Convolution vs. Correlation}<br />
<br />
{\it Correlation} is very similar to convolution, and it is best defined through its<br />
equivalent ``correlation theorem'':<br />
\begin{equation}<br />
f(t)\star g(t) = \int{\hat f(\omega) \hat g^*(\omega) e^{i\omega t}d\omega}<br />
\label{eq:corr_thm}.<br />
\end{equation}<br />
The difference between correlation and convolution is that<br />
that when correlating two signals, the Fourier transform of the second<br />
function ($\hat g(\omega)$ in equation \ref{eq:corr_thm}) is conjugated before<br />
multiplying and integrating.<br />
Using that<br />
\begin{align}<br />
g^*(-t)&=\frac1{2\pi}\int{\hat g^*(\omega)e^{-i\omega(-t)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat g^*(\omega)e^{i\omega t}d\omega},<br />
\end{align}<br />
we can show that correlating $f(t)$ and $g(t)$ is equivalent to convolving $f(t)$ with a<br />
conjugated, time-reversed version of $g(t)$:<br />
\begin{equation}<br />
f(t)*g^*(-t) = f(t)\star g(t).<br />
\label{eq:conv_corr_relation}<br />
\end{equation}<br />
Although this relation between convolution and correlation is often mentioned<br />
in the literature, I don't personally find it very intuitively illuminating.<br />
I much prefer the ``correlation theorem'' in equation (\ref{eq:corr_thm}), because<br />
when it is combined with the expression of a time-shifted signal in Fourier domain:<br />
\begin{align}<br />
f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega(t-\tau)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}e^{-i\omega\tau}d\omega}<br />
\label{eq:delay_freq},<br />
\end{align}<br />
it shows that correlating a flat-spectrum signal with a time-shifted version of itself yields<br />
a measure of the power of the signal at the delay corresponding to the time shift:<br />
\begin{align}<br />
f(t)\star f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)\hat f^*(\omega)e^{-i\omega\tau}e^{i\omega t}d\omega}\\<br />
&=\frac1{2\pi}\int{|f|^2e^{i\omega (t-\tau)}d\omega}\\<br />
&=|f|^2\cdot\delta(t-\tau).<br />
\end{align}<br />
<br />
\end{document}<br />
</latex><br />
<br />
===Related Subjects===<br />
* [[Interferometric Imaging]]</div>WikiSysopRadiometer Equation Applied to Telescopes2011-09-10T23:37:23Z<p>WikiSysop: </p>
<hr />
<div>===Prerequisites===<br />
* [[Central Limit Theorem]]<br />
* [[Basic_Interferometry | Basic Interferometry]] (for Radiometer equation applied to interferometers)<br />
* [[Radio Astronomy Units]]<br />
* [[Black-Body Radiation]]<br />
<br />
===Short Topical Videos===<br />
* [http://www.youtube.com/watch?v=dwT-tMoscsY The Radiometer Equation (by Chat Hull)]<br />
<br />
===Reference Material===<br />
<br />
<latex><br />
\documentclass[11pt]{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{The Radiometer Equation}<br />
<br />
The radiometer equation is really all about signal to noise. Here's the answer:<br />
<br />
$$\frac{S}{N} = \frac{T_{src}}{T_{rms}} = \frac{T_{src}}{T_{sys}} \sqrt{\tau \Delta\nu}$$<br />
<br />
where:<br />
\begin{itemize}<br />
\item $T_{src}$ is the signal of the source you're observing<br />
\item $T_{sys}$ is your system temperature<br />
\item $T_{rms}$ is the noise in your system, or the RMS fluctuations in your system temperature<br />
\item $\Delta\nu$ is the bandwidth of your correlator (in kHz, MHz, GHz...)<br />
\item $\tau$ is integration time (seconds)<br />
\end{itemize}<br />
<br />
<br />
\subsection*{What's the deal with all the temperatures?}<br />
<br />
All of the temperatures in the above equation have their origin in the Rayleigh-Jeans limit of the blackbody equation:<br />
<br />
$$I_\nu \approx \frac{2kT}{\lambda^2}$$<br />
<br />
Radio astronomers like to refer to the above temperature $T$ as<br />
the "brightness temperature," $T_B$, of a source. $T_B$ is the temperature of the blackbody needed<br />
to produce the observed specific intensity at that frequency:<br />
$$T_B(\nu) = \frac{I_\nu \lambda^2}{2k}$$<br />
<br />
Now, back to specific intensity. $I_\nu$ is defined as:<br />
<br />
$$I_\nu = \frac{\Delta E}{\Delta \Omega \Delta A \Delta t \Delta \nu}$$<br />
where:<br />
\begin{itemize}<br />
\item $\Delta t$ is the exposure time<br />
\item $\Delta \nu$ is a small interval in frequency<br />
\item $\Delta E$ is the energy emitted in bandwidth $\Delta \nu$ over time $\Delta t$<br />
\item $\Delta A$ is the area of the telescope<br />
\item $\Delta \Omega$ is the observed solid angle on the source, or your beam size (steradians)<br />
\end{itemize}<br />
<br />
So, <br />
$$\boxed{I_\nu = \frac{dE}{dt d\nu dA d\Omega} = [erg\ s^{-1}\ cm^{-2}\ Hz^{-1}\ Sr^{-1}]}$$<br />
<br />
It is common to report $I_\nu$ in Jy/beam, where the beam has units of $Sr^{-1}$, and Janskys (Jy) are the units of flux density $S_\nu$, defined below.<br />
<br />
To arrive at other quantities, we must integrate over $I_\nu$:<br />
\begin{itemize}<br />
\item {\bf Flux Density:} $S_\nu = \int I_\nu d\Omega = [erg\ s^{-1}\ cm^{-2}\ Hz^{-1}]$. The fundamental unit of flux density is the Jansky. $1 \textrm{Jy} = 10^{-23} erg\ s^{-1}\ cm^{-2}\ Hz^{-1}$<br />
\item {\bf Power received:} $P_\nu = \int S_\nu dA = [erg\ s^{-1}\ Hz^{-1}]$. <br />
\end{itemize}<br />
<br />
\subsection*{Source Temperature, $T_{src}$}<br />
The source temperature is defined as the brightness temperature associated with the power<br />
received by the telescope from the source you're observing:<br />
$$\begin{aligned}<br />
P_\nu & = I_\nu dA d\Omega \\<br />
& = \frac{2kT_{src}}{\lambda^2} dA d\Omega\\<br />
& = 2kT_{src}\\<br />
\end{aligned}$$<br />
The last step comes from the Antenna Theorem, which states that $dA d\Omega = \lambda^2$.<br />
<br />
\subsection*{K / Jy, or "forward gain"}<br />
Now that we've related the power $P_\nu$ received by the antenna at a given frequency, <br />
to the source temperature, $T_{src}$, and the observed intensity, $I_\nu$, we can arrive at a conversion between <br />
flux density and brightness temperature:<br />
$$\begin{aligned}<br />
P_\nu & = I_\nu A_e \Omega_a \\<br />
2kT_A & = S_\nu A_e\\<br />
T_A & = \left(\frac{A_e}{2k}\right) S_\nu\\<br />
\end{aligned}$$<br />
Therefore, the conversion between K and Jy, also known as the "forward gain" of an antenna, is just <br />
$$K / Jy = \frac{A_e}{2k}$$<br />
<br />
Thus, brightness temperature is just another measure of the brightness of a source. The forward gain is a physical property of a telescope, which dictates the telescope receivers' response to a given increase in Janskys.<br />
<br />
\subsection*{The System Temperature}<br />
While the source temperature $T_{src}$ describes the energy received from the source you are interested in, <br />
the system temperature, $T_{sys}$ describes the actual power received due to both the sky ($T_{sky}$) and the receivers ($T_{Rx}$):<br />
$$T_{sys} = T_{sky} + T_{Rx} $$<br />
The main component is the receiver temperature, $T_{Rx}$, which comes from the [http://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise<br />
| thermal (or Johnson) noise] from the receiver electronics. Depending on the observation waveband, receivers are sometimes cooled to reduce $T_{Rx}$.<br />
<br />
$T_{sky}$ is the power generated by everything you \textit{don't} want to be looking at on the sky: background sources, water vapor in the atmosphere (especially in mm-wave astronomy), galactic backgrounds (especially in long-wavelength astronomy), etc.<br />
<br />
\subsubsection*{Effective Area $A_e$}<br />
<br />
Above we have written the area of an antenna as $A_e$, or the "effective area." <br />
<br />
$$A_e = \eta_a A_p$$ <br />
<br />
where ($\eta_a$ is the aperture efficiency and $A_p$ is the projected area of the telescope). $\eta_a$ is a number less than one, and signifies that not all radiation incident upon the telescope actually makes it to the receiver because of the dish's finite reflectivity. This loss of signal requires that the signal later be amplified. There are two scenarios that can be encountered:<br />
<br />
\begin{itemize}<br />
\item $T_{sys}$ is dominated by $T_{Rx}$: in this case aperture efficiency matters, because in amplifying the signal, noise from the receiver is also amplified.<br />
\item $T_{sys}$ is dominated by $T_{sky}$: in this case, which might happen on a cloudy day at a millimeter telescope where the noise from the atmosphere is dominant, aperture efficiency doesn't matter. This is because the dominant sky noise is first cut down by a low aperture efficiency, and then multiplied back up to its original level by the amplifiers.<br />
\end{itemize}<br />
<br />
\subsection*{Detecting the signal...}<br />
<br />
Typically $T_{sys} >> T_{src}$, so how do we detect <br />
the source we are interested in? Beat down the noise!<br />
<br />
In order to detect the source, we need $T_{src} > T_{rms}$, where the observation-dependent quantity $T_{rms}$ is the noise in our measurement of the observation-independent quantity $T_{sys}$:<br />
<br />
$$T_{rms} = \frac{T_{sys}}{\sqrt{N}}$$<br />
<br />
where $N$ is the number of independent data points.<br />
<br />
For a telescope, the number of independent samples is $\Delta \nu \cdot \tau$, where $\Delta \nu$ is the <br />
bandwidth (Hz) and $\tau$ is the integration time (seconds). With a bandwidth $\Delta \nu$, the signal is <br />
statistically independent over a time interval $1/\Delta\nu$, so that the number of independent samples is <br />
just $\tau$ divided by $1/\Delta\nu$, so $N = \tau \Delta\nu$. Therefore,<br />
$$T_{rms} = \frac{T_{sys}}{\sqrt{\tau \Delta\nu}}$$<br />
<br />
\subsection*{The Radiometer Equation}<br />
Now we can write down an expression for the signal to noise ratio (the radiometer equation):<br />
$$\frac{S}{N} = \frac{T_{src}}{T_{rms}} = \frac{T_{src}}{T_{sys}} \sqrt{\tau \Delta\nu}$$<br />
Typical values might be $\Delta \nu = 10$ MHz, $\tau = 1$ sec so that $ \sqrt{\tau \Delta\nu} \sim 3\times10^3$.<br />
Typical values for $T_{sys}$ are 40 - 200 K.<br />
<br />
\subsubsection*{SEFD}<br />
The SEFD is the `system equivalent flux density', which is the flux density equivalent of $T_{sys}$:<br />
<br />
$$\textrm{SEFD} = \frac{T_{sys}}{(K/Jy)} = \frac{T_{sys}}{A_e / 2k} = \frac{2kT_{sys}}{A_e}$$<br />
<br />
The SEFD is a useful way to compare the sensitivity of two different systems since it folds in both<br />
$T_{sys}$ and $A_e$. This also greatly simplifies the sensitivity calculation: if you know the <br />
flux in Jy of the source you want to detect, and you know the SEFD, then you can easily calculate<br />
the integration time you need to make a given S/N detection (for an unresolved source):<br />
$$\frac{S}{N} = \frac{S_\nu (Jy)}{\textrm{SEFD}} \sqrt{\tau \Delta\nu}$$<br />
<br />
After substituting in the temperature-based Radiometer Equation for $S \ N$, we arrive at an intuitive expression for the RMS variations in flux density $S_{\nu, rms}$:<br />
<br />
$$S_{\nu, rms} = \frac{\textrm{SEFD}}{\sqrt{\tau \Delta\nu}}$$<br />
<br />
In the temperature-based Radiometer Equation, signal-to-noise \textit{increases} with increased bandwidth and integration time. In the flux-density-based Radiometer equation, RMS flux density varations \textit{decrease} with increased bandwidth and integration time.<br />
<br />
\subsubsection*{Extending to interferometric arrays}<br />
<br />
You should refer to the lecture on [[Basic_Interferometry]] for explanations of the multi-antenna concepts presented here.<br />
<br />
Making more independent measurements is one way to increase signal-to-noise. With an array of $N$ antennas, every baseline (involving two antennas) adds two more independent measurements: a measurement of the signal's amplitude, and its phase. Thus, for an array of antennas, the Radiometer Equation becomes:<br />
<br />
$$\begin{aligned}<br />
S_{\nu, rms} & = \frac{\textrm{SEFD}}{\sqrt{ \frac{N(N-1)}{2} \tau (2\Delta\nu)}} \\<br />
& = \frac{\textrm{SEFD}}{\sqrt{ N(N-1) \tau \Delta\nu}}<br />
\end{aligned}$$<br />
<br />
The factor of 2 in the denominator comes from the amplitude and phase measurements, and $N(N-1) / 2$ is the number of baselines between the $N$ antennas.<br />
<br />
\end{document}<br />
</latex><br />
<br />
====Other Links====<br />
* Much of the preceding is based on the Radio 101 single-dish lecture, [[Single_Dish_Basics | Single Dish Basics]].<br />
* [[Media:Radiometer_equation_notes.pdf | Handwritten notes]] used to make the [http://www.youtube.com/watch?v=dwT-tMoscsY video by Chat Hull]<br />
* [http://www.cv.nrao.edu/course/astr534/Radiometers.html The NRAO Course on Radiometers]<br />
<br />
===Related Subjects===<br />
* [[Basic Interferometry]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-07T21:03:41Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
** Choosing a Lab Project<br />
** Begin a Python project (radioastro), revision-controlled under GIT, that<br />
*** has a convolution module (conv.py) with functions for<br />
**** performing 1D and 2D convolutions of two provided functions<br />
*** has a module (noise.py) with brightness-temperature/jansky conversions<br />
**** should take beam size and wavelength as arguments<br />
**** should predict noise levels for observations of given bandwidth, time, number of antennas, etc. <br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-07T20:50:07Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[Revision Control]]<br />
** [[Radiometer Equation]]<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-06T21:49:59Z<p>WikiSysop: /* Topics */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing / Fourier Analysis<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[GIT Revision Control]]<br />
** [[Radiometer Equation]]<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-06T21:48:54Z<p>WikiSysop: </p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
== Topics by Date ==<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[GIT Revision Control]]<br />
** [[Radiometer Equation]]<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-06T21:48:02Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
====Topics by Date====<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[GIT Revision Control]]<br />
** [[Radiometer Equation]]<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-06T21:47:36Z<p>WikiSysop: /* Topics by Date */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
===Topics by Date===<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
** Getting an account<br />
* Sep 07:<br />
** [[GIT Revision Control]]<br />
** [[Radiometer Equation]]<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopRadio Astronomy: Tools and Techniques2011-09-06T21:45:45Z<p>WikiSysop: /* Topics */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Algorithms<br />
* [[Fast Fourier Transform]]<br />
* [[Markov-Chain Monte Carlo]]<br />
<br />
Software Development<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
<br />
Computing<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Signal Path<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
* [[Receivers and Amplifiers]]<br />
<br />
Pedagogy of Radio Astronomy / Meta-Information<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
Science of Radio Astronomy<br />
* [[Black-Body Radiation]]<br />
* [[21cm Transition]]<br />
<br />
===Topics by Date===<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
* Sep 07:<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopConvolution Theorem2011-08-31T21:00:43Z<p>WikiSysop: </p>
<hr />
<div>===Prerequisites===<br />
* Fourier Transforms (link?)<br />
* Integral Calculus (link?)<br />
<br />
===Short Topical Videos===<br />
* [http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations Introduction to the Convolution] by Khan Academy<br />
<br />
===Reference Material===<br />
<br />
<latex><br />
\documentclass[]{article}<br />
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}<br />
\usepackage{amsmath}<br />
\usepackage{graphicx}<br />
\usepackage{natbib}<br />
<br />
\begin{document}<br />
\title{Convolution Theorem}<br />
<br />
\section*{Fourier Transform}<br />
<br />
Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$)<br />
Fourier transforms:<br />
\begin{align}<br />
f(t)&= \mathcal{F}(\hat f(\omega)) = \frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}d\omega}\\<br />
\hat f(\omega)&= \mathcal{F}^{-1}(f(t)) = \int{f(t)e^{-i\omega t}dt}<br />
\label{eq:fourier_transform}<br />
\end{align}<br />
where $\omega\equiv2\pi\nu$ is the angular frequency coordinate that is the Fourier complement of time $t$,<br />
and a top-hat is generally used to denote Fourier-domain quantities.<br />
<br />
\section*{Convolution Theorem}<br />
<br />
The {\it convolution} is a useful operation with applications ranging from photo editing (blurring) to<br />
crystallography to astronomy.<br />
<br />
<br />
\begin{align}<br />
[f*g](\tau)&\equiv \int{f(t)g(\tau-t)dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)e^{i\omega_1 t}d\omega_1\,<br />
\hat g(\omega_2)e^{i\omega_2(\tau-t)}d\omega_2\,dt}\\<br />
& = \frac1{(2\pi)^2}\int\!\!\!\int{\hat f(\omega_1)\hat g(\omega_2)e^{i(\omega_1-\omega_2) t}<br />
e^{i\omega_2\tau}d\omega_1\,d\omega_2\,dt}\\<br />
& = \frac1{2\pi}\int{\hat f(\omega)\hat g(\omega) e^{i\omega\tau}d\omega}.<br />
\end{align}<br />
Renaming $\tau$ to be $t$ (which we are totally free to do), we get a statement of the<br />
{\it convolution theorem}:<br />
\begin{equation}<br />
f(t)*g(t) = \int{\hat f(\omega) \hat g(\omega) e^{i\omega t}d\omega} <br />
= \mathcal{F}^{-1}\!\!\left(\mathcal{F}(f)\cdot\mathcal{F}(g)\right).<br />
\label{eq:conv_thm}<br />
\end{equation}<br />
<br />
\subsection*{Convolution vs. Correlation}<br />
<br />
{\it Correlation} is very similar to convolution, and it is best defined through its<br />
equivalent ``correlation theorem'':<br />
\begin{equation}<br />
f(t)\star g(t) = \int{\hat f(\omega) \hat g^*(\omega) e^{i\omega t}d\omega}<br />
\label{eq:corr_thm}.<br />
\end{equation}<br />
The difference between correlation and convolution is that<br />
that when correlating two signals, the Fourier transform of the second<br />
function ($\hat g(\omega)$ in equation \ref{eq:corr_thm}) is conjugated before<br />
multiplying and integrating.<br />
Using that<br />
\begin{align}<br />
g^*(-t)&=\frac1{2\pi}\int{\hat g^*(\omega)e^{-i\omega(-t)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat g^*(\omega)e^{i\omega t}d\omega},<br />
\end{align}<br />
we can show that correlating $f(t)$ and $g(t)$ is equivalent to convolving $f(t)$ with a<br />
conjugated, time-reversed version of $g(t)$:<br />
\begin{equation}<br />
f(t)*g^*(-t) = f(t)\star g(t).<br />
\label{eq:conv_corr_relation}<br />
\end{equation}<br />
Although this relation between convolution and correlation is often mentioned<br />
in the literature, I don't personally find it very intuitively illuminating.<br />
I much prefer the ``correlation theorem'' in equation (\ref{eq:corr_thm}), because<br />
when it is combined with the expression of a time-shifted signal in Fourier domain:<br />
\begin{align}<br />
f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega(t-\tau)}d\omega}\\<br />
&=\frac1{2\pi}\int{\hat f(\omega)e^{i\omega t}e^{-i\omega\tau}d\omega}<br />
\label{eq:delay_freq},<br />
\end{align}<br />
it shows that correlating a flat-spectrum signal with a time-shifted version of itself yields<br />
a measure of the power of the signal at the delay corresponding to the time shift:<br />
\begin{align}<br />
f(t)\star f(t-\tau)&=\frac1{2\pi}\int{\hat f(\omega)\hat f^*(\omega)e^{-i\omega\tau}e^{i\omega t}d\omega}\\<br />
&=\frac1{2\pi}\int{|f|^2e^{i\omega (t-\tau)}d\omega}\\<br />
&=|f|^2\cdot\delta(t-\tau).<br />
\end{align}<br />
<br />
\end{document}<br />
</latex><br />
<br />
===Related Subjects===<br />
* [[Interferometric Imaging]]</div>WikiSysopCreating Short Topical Presentations2011-08-31T19:56:56Z<p>WikiSysop: /* Topical Videos */</p>
<hr />
<div><br />
In practice, people have attention spans of 10-15min at a stretch for consuming media at the level of thought and analysis required for technical and scientific learning. Breaking up teaching sessions, adding variety to presentation formats, and changing how a subject is presented are all techniques that are scientifically demonstrated to improve learning.<br />
<br />
For this reason, we aim to keep topical presentations to 10 to 15 minutes. If a subject absolutely requires more time than that, it should be broken up into two related presentations. Usually, though, there is a lot of time that can be trimmed out of a video presentation relative to a live classroom performance. Here are some time-saving tips:<br />
* Avoid repetition. The viewer can always rewind.<br />
* Pick up the pace and avoid stops. The viewer can always pause when they need to.<br />
* Keep it punchy. Topical presentations are meant to introduce a subject and provide intuition. Printed media are more appropriate for detailed technical content that would be useful as a reference for later. Avoid incorporating too much reference material into a video.<br />
<br />
===Materials===<br />
There are many ways to do this, but I've chosen to copy what [https://sites.google.com/a/khanacademy.org/forge/for-translators/reference-page Salman Khan of Khan Academy uses]:<br />
* A [http://www.staples.com/Wacom-CTL460-Bamboo-Pen-Tablet/product_812911?cid=PS:SBD:GS:E:N:PLA:71000000000227348:58000000007550141:812911 Wacom Bamboo Tablet]<br />
* Quicktime Player (on my Mac)<br />
<br />
Feel free to experiment, though.<br />
<br />
===Topical Videos===<br />
* [http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html Salman Khan on Khan Academy]: On using instructional videos instead of lecture<br />
* [http://www.youtube.com/watch?v=QZJAhfaZnUA How to make] a Khan Academy-style video</div>WikiSysopRadio Astronomy: Tools and Techniques2011-08-31T19:37:18Z<p>WikiSysop: /* Topics */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Computing<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Statistics<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Electronics<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
<br />
Pedagogy / Meta<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
===Topics by Date===<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
* Sep 07:<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysopCreating Short Topical Presentations2011-08-31T01:14:02Z<p>WikiSysop: Created page with ' ===Topical Videos=== * [http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html Salman Khan on Khan Academy]: Videos instead of Lecture * [http://www.you…'</p>
<hr />
<div><br />
===Topical Videos===<br />
* [http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html Salman Khan on Khan Academy]: Videos instead of Lecture<br />
* [http://www.youtube.com/watch?v=QZJAhfaZnUA How to make] a Khan Academy-style video</div>WikiSysopPython Installation and Basic Programming2011-08-31T01:10:21Z<p>WikiSysop: Created page with 'Here we will assemble resources for learning Python, and for getting it and other programming-related software installed on your computer. For a scientific programmer in Python,…'</p>
<hr />
<div>Here we will assemble resources for learning Python, and for getting it and other programming-related software installed on your computer.<br />
<br />
For a scientific programmer in Python, the absolute basics you need to have installed are:<br />
* [http://www.python.org Python] 2.X (note that 3.X exists and is maturing, but a lot of scientific code and packages are not yet ported)<br />
* [http://numpy.scipy.org/ NumPy]: a package for fast numerical array processing<br />
* [http://matplotlib.sourceforge.net/ Matplotlib/Pylab]: a package for generating publication-quality plots<br />
* [http://git-scm.com/ GIT]: a revision-control program for keeping tabs on the changes you make to your code. Not just for python.<br />
<br />
==Python==<br />
===Topical Videos===<br />
<br />
* [http://www.khanacademy.org/video/introduction-to-programs-data-types-and-variables?playlist=Computer%20Science Introduction to Programs Data Types and Variables] by Khan Academy<br />
* [http://www.khanacademy.org/video/python-lists?playlist=Computer%20Science Python Lists] by Khan Academy<br />
* [http://www.khanacademy.org/video/for-loops-in-python?playlist=Computer%20Science For Loops in Python] by Khan Academy<br />
* [http://www.khanacademy.org/video/while-loops-in-python?playlist=Computer%20Science While Loops in Python] by Khan Academy<br />
* [http://www.khanacademy.org/video/fun-with-strings?playlist=Computer%20Science Fun with Strings] by Khan Academy<br />
* [http://www.khanacademy.org/video/writing-a-simple-factorial-program---python-2?playlist=Computer%20Science Writing a Simple Factorial Program] by Khan Academy<br />
* ... there are lots more by Khan Academy ...<br />
* [http://www.khanacademy.org/video/simpler-insertion-sort-function?playlist=Computer%20Science Simpler Insertion Sort Function]<br />
<br />
===Links===<br />
* Josh Bloom's [http://sites.google.com/site/pythonbootcamp Python Boot Camp] offers a wealth of resources for getting started with Python. In particular, see:<br />
** [http://sites.google.com/site/pythonbootcamp/preparation/software Instructions] for installing all of the packages mentioned above.<br />
** [http://sites.google.com/site/pythonbootcamp/links Useful links] to Python resources<br />
* A [http://code.google.com/edu/languages/google-python-class/index.html Google Class on Python]<br />
* The [http://docs.python.org/tutorial/ Python Tutorial]<br />
<br />
==GIT==<br />
===Topical Videos===<br />
* [http://www.youtube.com/watch?v=OFkgSjRnay4 Longish (1hr) GIT Introduction]<br />
<br />
===Links===<br />
* For an overview of common GIT commands, see [http://ktown.kde.org/~zrusin/git/git-cheat-sheet-medium.png this cheat sheet] and [http://www.sourcemage.org/Git_Guide this guide]. You can also get help on any GIT command by typing:<br />
<source lang="bash"><br />
git help {command}<br />
</source><br />
* [http://www.kernel.org/pub/software/scm/git/docs/gittutorial.html GIT Tutorial]</div>WikiSysopRadio Astronomy: Tools and Techniques2011-08-31T00:26:07Z<p>WikiSysop: /* Radio Astronomy: Tools and Techniques */</p>
<hr />
<div>This is course is aimed at<br />
graduate students, advanced undergraduates, and interested third<br />
parties who:<br />
* would like to understand radio astronomy better<br />
* would like to develop technical skills (programming, signal processing, instrumentation, algorithms, pedagogy, etc) to help them in their research<br />
* would like to be involved, and involve their peers, in developing concrete tools to help their research<br />
<br />
This class will follow a flexible, non-traditional format whereby each<br />
week, a group of students and I will work together to prepare<br />
public-domain pedagogical materials on a subject that will be<br />
distributed to the rest of the class in advance of each meeting.<br />
Class time will be split between discussing the subject informally,<br />
and working in groups to develop tools and address on-going research<br />
questions that each student brings to the class.<br />
<br />
My hope is that this class will be moderately time-consuming, but that<br />
the tools, collaborations, and research developed inside the class<br />
will have a broad enough scope that it can double-count as<br />
research/work time. All of our activities are aimed at creating tools<br />
(both pedagogical and research-oriented) that will have value beyond<br />
the classroom.<br />
<br />
=== Topics ===<br />
<br />
Here is a (non-exhaustive) list of topics that we will consider covering in this class. Eventually, it would be nice to link in as many topics as possible and begin to organize subjects by their prerequisites and relatedness.<br />
<br />
Computing<br />
* [[Python Installation and Basic Programming]]<br />
* [[Revision Control]]<br />
* [[Programming Models]]<br />
* [[Processor Architectures]]<br />
* [[Data Representations]]<br />
* [[Network Programming]]<br />
<br />
Signal Processing<br />
* [[Convolution Theorem]]<br />
* [[Windowing]]<br />
* [[Correlators]]<br />
* [[Deconvolution]]<br />
<br />
Interferometers<br />
* [[Basic Interferometry]]<br />
* [[Units]]<br />
* [[Advanced Interferometry]]<br />
* [[Interferometric Imaging]]<br />
<br />
Noise<br />
* [[Central Limit Theorem]]<br />
* [[Radiometer Equation]]<br />
* [[Bayesian Statistics]]<br />
<br />
Electronics<br />
* [[Transmission Lines]]<br />
* [[Antennas and Feeds]]<br />
<br />
Pedagogy / Meta<br />
* [[Creating Short Topical Presentations]]<br />
* [[Using AstroBaki]]<br />
<br />
===Topics by Date===<br />
* Aug 31: <br />
** [[Convolution Theorem]]<br />
** [[Creating Short Topical Presentations]]<br />
** [[Python Installation and Basic Programming]]<br />
** Brainstorming Lab Projects<br />
** Choosing a Topic to Present<br />
* Sep 07:<br />
* Sep 14:<br />
* Sep 21:<br />
* Sep 28:<br />
* Oct 05:<br />
* Oct 12:<br />
* Oct 19:<br />
* Oct 26:<br />
* Nov 02:<br />
* Nov 09:<br />
* Nov 16:<br />
* Nov 23:<br />
* Dec 03:</div>WikiSysop