https:///astrobaki/api.php?action=feedcontributions&user=Zaki&feedformat=atom AstroBaki - User contributions [en] 2022-09-25T08:56:38Z User contributions MediaWiki 1.35.1 Programming Resources 2015-05-11T19:36:58Z <p>Zaki: </p> <hr /> <div>These are some useful programming resources. They vary from introductions to python to more advanced topics and exercises. <br /> <br /> * MIT 600: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-00-introduction-to-computer-science-and-programming-fall-2008/<br /> * UCB CS9H course: http://www-inst.eecs.berkeley.edu/~selfpace/class/cs9h/<br /> * Enthought: A self contained python package distribution. You can install it by going here: https://store.enthought.com<br /> * python.org for a python tutorial. Take a look at the getting started section for a programmers/non-programmers tutorial: https://www.python.org/about/<br /> * Python BootCamp : https://astrocompute.wordpress.com/2013/09/02/the-berkeley-python-boot-camp-fall-2013/<br /> * Project Euler : Useful programming exercises to get your feet wet: https://projecteuler.net</div> Zaki 10/23/14 Quadratic Estimators I 2015-04-04T01:02:53Z <p>Zaki: </p> <hr /> <div>&lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> <br /> \def\hf{\frac12}<br /> \def\dcube#1{{d^{3}\mathbf{#1}}}<br /> \def\ddtau#1{{\frac{d#1} {d\tau}}}<br /> \def\bra#1{\langle #1|}<br /> \def\ket#1{|#1\rangle}<br /> \def\expval#1{\langle #1 \rangle}<br /> \def\rhor{\rho(\mathbf{r})}<br /> <br /> \def\deltar{\delta(\mathbf{r})}<br /> \def\deltarp{\delta(\mathbf{r}^{\prime})}<br /> \def\deltaru#1{\delta_{#1}(\mathbf{r})}<br /> <br /> \def\deltak{\delta(\mathbf{k})}<br /> \def\deltaku#1{\delta_{#1}(\mathbf{k})}<br /> <br /> \def\fdeltak{\tilde{\delta}(\mathbf{k})}<br /> \def\fdeltakp{\tilde{\delta}(\mathbf{k}^{\prime})}<br /> \def\fdeltaku#1{\tilde{\delta}_{#1}(\mathbf{k})}<br /> \def\vec#1{\mathbf{#1}}<br /> <br /> <br /> \newcount\colveccount<br /> \newcommand*\colvec{<br /> \global\colveccount#1<br /> \begin{pmatrix}<br /> \colvecnext<br /> }<br /> \def\colvecnext#1{<br /> #1<br /> \global\advance\colveccount-1<br /> \ifnum\colveccount&gt;0<br /> \\<br /> \expandafter\colvecnext<br /> \else<br /> \end{pmatrix}<br /> \fi<br /> }<br /> <br /> \usepackage{fullpage}<br /> \usepackage{amsmath}<br /> \usepackage{commath}<br /> \usepackage{amsfonts}<br /> \usepackage[margin=1in]{geometry}<br /> \usepackage{graphicx,rotating}<br /> <br /> \begin{document}<br /> <br /> \title{Quadratic Estimators: Part I}<br /> <br /> \section{preamble} <br /> We have learned, in the previous lectures, about the power spectrum. By<br /> definitiion we have that, in general, <br /> <br /> \begin{equation}<br /> \expval{\tilde{T}^{*}_{b}(\mathbf{k}) \tilde{T}^{*}_{b}(\mathbf{k}^{\prime})} =<br /> (2\pi)^{3}\delta^{D}(\mathbf{k} - \mathbf{k}^{\prime})P(\mathbf{k}).<br /> \end{equation}<br /> <br /> And for the CMB, that is for a 2 dimensional shell, we can describe the power<br /> spectrum by $C_{l}$ from from a spherical harmonic decompostion <br /> <br /> \begin{equation}<br /> \expval{a_{lm}a_{l^{\prime}m^{\prime}}} = C_{l}\delta_{ll^{\prime}}\delta_{mm^{\prime}}<br /> \end{equation}<br /> <br /> We will now develop some formalism for quadratic estimators. This is especially<br /> useful for the power spectrum, a quadratic quantity.<br /> <br /> \section{Quadratic Estimator Formalism}<br /> Let us take our data vector $\mathbf{x}$, given by, <br /> <br /> \begin{equation}<br /> \mathbf{x} = \begin{pmatrix} {\mathbf{x_{1}}} \\ {\mathbf{x_{2}}} \\ {.} \\ {.} \\{.}\\<br /> \end{pmatrix}.<br /> \end{equation}<br /> <br /> where each ${\mathbf{x_{i}}}$ can be temperature data (${\mathbf{T_{i}}}$),<br /> visibility data (${\mathbf{V_{i}}}$), or some other quantity.<br /> The quadratic estimator is defined as <br /> <br /> \begin{equation}\label{eqn:quadest}<br /> \hat{p}_{\alpha} \propto \vec{x}^{\dagger}E^{\alpha}\vec{x} - b^{\alpha}.<br /> \end{equation}<br /> <br /> In equation \ref{eqn:quadest}, $\hat{}$ means the quantity hatted is the<br /> estimator, $p_{\alpha} \equiv P({k_{\alpha}})$, $\vec{x}$ is the data input vector,<br /> $E^{\alpha}$ is some matrix which is the data analysts choise, and $b^{\alpha}$ is<br /> the bias correction. <br /> <br /> \section{What's with the ${E^{\alpha}?}$}<br /> Now that we have defined the quadratic estimator in equation \ref{eqn:quadest},<br /> we can discuss what a sensible choise for $E^{\alpha}$ may be. The choice for<br /> $E^{\alpha}$ depends on the choice of basis $\vec{x}$ is expressed in. Note that<br /> $E^{\alpha}$ is a family of matrices, one for each $\alpha$. Therefore, to<br /> measure the the power at some $k$-mode $k_{i}$, we must use the $E^{\alpha}$<br /> matrix that corresponds to that $k$-mode.<br /> <br /> \subsection{example 1: $T(\vec{r_{i}})$}<br /> Suppose that the data vector $\vec{x} = ( T(\vec{r}_{1}), T(\vec{r}_{2}),<br /> ...)$. Let us guess that <br /> <br /> \begin{equation}<br /> E^{\alpha}_{ij} \sim e^{i\vec{k}_{\alpha}(\vec{r}_{i} - \vec{r}_{j})}.<br /> \end{equation}<br /> <br /> Why is this a good guess? Well we want to make an estimate of the power spectrum<br /> from temperature data. Back from the power spectrum lectures, we know that we<br /> need to fourier transform temperature data (a function of $\vec{r}$) in order<br /> to go into $k$-space. Hence $E^{\alpha}$ should look something like a fourier<br /> transform.<br /> <br /> In order to check this, let us compute the estimator,<br /> \begin{align}<br /> \vec{x}^{\dagger} E^{\alpha} \vec{x} &amp;= \sum_{ij}{x_{i}E_{ij}^{\alpha}x_{j}}\\<br /> &amp;= \sum_{ij}{T(\vec{r}_{i})e^{i\vec{k}_{\alpha}(\vec{r}_{i} - \vec{r}_{j})}T(\vec{r}_{j})}\\<br /> &amp;= ( \sum_{i}{e^{i\vec{k}_{\alpha}\vec{r}_{i}}T(\vec{r}_{i})} ) ( \sum_{j}{e^{i\vec{k}_{\alpha}\vec{r}_{j}}T(\vec{r}_{j})} )\\<br /> &amp;= \mid\sum_{j}{e^{-i\vec{k} _{\alpha}\vec{r}_{j}}T(\vec{r}_{j})\mid^{2}.<br /> \end{align}<br /> <br /> Hence, the choice of $E^{\alpha}$ above gives us a fourier tranform-ish result.<br /> This result would be a true fourier transform if the data were on a regular<br /> grid. <br /> <br /> \subsection{example 2: $\tilde{T}(\vec{k})$}<br /> In this case $\vec{x} = ( \tilde{T}(\vec{k}_{1}), \tilde{T}(\vec{k}_{2}),<br /> ...)$. We can then choose an $E^{\alpha}$ that extracts particular ${k}$-modes.<br /> Remember that $\alpha$ is a particular $k$-mode. Specifically, we have that <br /> $E^{alpha=k} = \delta{k}\delta_{i}\delta{j}$, so that the matrix is equal to 1<br /> only when $i=j=k$.<br /> <br /> \section{The general form of $E$}<br /> <br /> Suppose that we have a power spectrum that was binned up. That is we model<br /> $P(k)$ as piecewise constant. The value of $P(k_{\alpha}) \equiv \alpha^{th}$ band<br /> power $\equiv P_{\alpha}$.<br /> Now to relate this to the data, consider the covariance matrix of the data: <br /> $C \equiv \expval{\vec{x}\vec{x}^{\dagger}}$; $C_{ij} =<br /> \expval{x_{i}x^{*}_{j}}$. The covariance matrix $C$ consists of two parts:<br /> <br /> \begin{equation}<br /> C = N + \sum_{\alpha}{p_{\alpha}\frac{\partial{C}}{\partial{p_{\alpha}}}}<br /> \equiv N + \sum_{\alpha}{p_{\alpha}C_{,\alpha}},<br /> \end{equation}<br /> <br /> where the first term $N$ is the noise term and consists of instrumental noise,<br /> systematics, etc.... The second term, the partial derivative of the covariance<br /> matrix with respect to $\alpha$ is the crucial link between theory and data.<br /> This term is related to $E^{\alpha}$. <br /> <br /> We note that the unvierse is not random and hence the power spectrum<br /> ($p_{\alpha}$) is not random. However, the quadratic estimator<br /> $\hat{p}_{\alpha}$ is random due to our instrument. Because of this we can<br /> measure the variance in our estimate of the power spectrum. Lets do that... <br /> The variance of the power spectrum in the $\alpha$-th bin is given by <br /> <br /> \begin{align}<br /> \expval{\hat{p}_{\alpha}} &amp;= \expval{\vec{x}^{\dagger}E^{\alpha}\vec{x}} -<br /> b_{\alpha}\\<br /> &amp;= \sum_{ij}\expval{x^{*}_{i}E_{ij}^{\alpha}x_{j}} - b_{\alpha}\\<br /> &amp;= \sum_{ij}E_{ij}^{\alpha}\expval{x_{j}x^{*}_{i}} - b_{\alpha}\\<br /> &amp;= \sum_{ij}E^{\alpha}_{ij}C_{ij} - b_{\alpha}\\<br /> &amp;= tr(E^{\alpha}C) - b_{\alpha}.<br /> \end{align}<br /> <br /> Plugging in for the covariance matrix, we get that <br /> \begin{align}<br /> \expval{\hat{p}_{\alpha}} &amp;= tr[E^{\alpha}\sum_{\beta}p_{\beta}C_{,\beta}] +<br /> tr[E^{\alpha}N] - b_{\alpha}\\<br /> &amp;= \sum_{\beta}tr[E^{\alpha}C_{,\beta}]p_{\beta} + tr[E^{\alpha}N] - b_{\alpha}.<br /> \end{align}<br /> <br /> Note that the second term, $tr(E^{\alpha}N)$ is the additive (noise) bias term.<br /> If we let $b_{\alpha} = tr[E^{\alpha}N]$ then we can correct fo this additive<br /> term. Hence, we are left with $\expval{\hat{p_{\alpha}}} = \sum{w_{\alpha\beta}p_{\beta}}$ where $w_{\alpha\beta} =<br /> tr[E^{\alpha}C_{,\beta}]$. This $w_{\alpha\beta}$ matrix is the window function<br /> matrix (or taper function matrix). The $\alpha$-th row of $w_{\alpha\beta}$<br /> gives linear cominations of $P(k_{\beta})$ probed by $\hat{p}_{\alpha}$. This<br /> gives us horizontal errorbrs.<br /> <br /> \section{Vertical Error Bars}<br /> The window (or taper) functions above gave us the horizontal error bars, which<br /> is to say for each power spectrum measurement in the $\alpha$-th bin, the amount<br /> other bins contributed to that measurement. <br /> <br /> Now, we discuss how to get the vertical error bars in the power spectrum<br /> estimate, which is basically the covariance of $\hat{p}_{\alpha}$. Let us<br /> calculate this, <br /> <br /> \begin{align}<br /> V_{\alpha\beta} &amp;= \expval{\hat{p}_{\alpha}\hat{p}_{\beta}} -<br /> \expval{\hat{p}_{\alpha}}\expval{\hat{p}_{\beta}}\\<br /> &amp;=<br /> \sum_{ijkl}{\expval{x_{i}E_{ij}^{\alpha}x_{j}x_{k}E^{\beta}_{kl}x_{l}}} -<br /> \sum_{ijkl}{\expval{x_{i}x_{j}}E_{ij}^{\alpha}}<br /> \sum_{ijkl}{\expval{x_{k}x_{l}}E_{kl}^{\beta}}, <br /> \end{align}<br /> <br /> where terms $\expval{x_{i}x_{j}}$ are $C_{ij}$. If we assume that x is Gaussian<br /> distributed, we can use wicks theorem (or Issirelli's Theorem; see wikipedia) to calculate the<br /> 4 point function above in terms of two point functions. This gives us that the<br /> vertical error bars are <br /> <br /> \begin{equation}<br /> V_{\alpha\beta} = 2tr[CE^{\alpha}CE^{\beta}]<br /> \end{equation}<br /> <br /> <br /> <br /> \section{A Concrete Example : The 1-D Universe and a noiseless instrument}<br /> Consider a 1-D universe from $\frac{-L}{2}$ to $\frac{L}{2}$ with some<br /> pixelation (meaning discrete bins of size $\Delta{r}$). Then a measurement in this universe,<br /> $x_{i}$, is given by <br /> \begin{equation}<br /> x_{i} = \int{dr T(r) \Phi_{i}(r)}<br /> \end{equation}<br /> where $\Phi_{i}$ is some pixelization function. We take it to be 1 inside the<br /> $i$-th cell and 0 outside. Let us now estimate $p_{\alpha}$ in this universe.<br /> <br /> $$<br /> \hat{p}_{\alpha} = \vec{x}^{\dagger}E^{\alpha}\vec{x}<br />$$<br /> Note that we are not including the noise bias $b_{\alpha}$ here and therefore<br /> are not considering noise.<br /> <br /> A good choice for $E_{\alpha}$ in this situation would be one that is<br /> proportional to $C_{,\alpha}$. Previously we had that the expectaion value of<br /> the estimator is <br /> <br /> \begin{align}<br /> \expval{\hat{p}} &amp;= \sum_{ij}{E^{\alpha}_{ij}C_{ji}} \\<br /> &amp;\propto \sum_{ij}{(C_{,\alpha})_{ij}C_{ij}}<br /> \end{align}<br /> <br /> Now, lets compute the error bars. Lets start by first going into fourier space.<br /> Then, <br /> \begin{align}<br /> \vec{x}_{i} &amp;= \int{dr\Phi_{i}(r)}\int{\frac{dk}{2\pi}\tilde{T}(k)e^{ikr}}\\<br /> &amp;= \int{\frac{dk}{2\pi}\tilde{T}(k)}\int_{r_{i} -<br /> \frac{\Delta{r}}{2}}^{r_{i}+\frac{\Delta{r}}{2}}{e^{ikr}}\\<br /> &amp;=<br /> \int{\frac{dk}{2\pi}\tilde{T}(k)\Delta{r}e^{ikr_{i}}sinc(k\frac{\Delta{r}}{2})}, <br /> \end{align}<br /> by noting that $sinc(nx) = \frac{1}{2n}\int{e^{ixt}dt}$.<br /> <br /> Then the covariance ($C_{ij} = \expval{x_{i}x_{j}^{*}}$) is given by,<br /> <br /> \begin{align}<br /> C_{ij} &amp;=<br /> \int{\frac{dk}{2\pi}\frac{dk^{'}}{2\pi}\expval{\tilde{T}(k)\tilde{T}^{*}(k^{'})}(\Delta{r})^{2}e^{ikr_{i}}e^{-ik^{'}r_{j}}sinc(k\frac{\Delta{r}}{2})sinc(k^{'}\frac{\Delta{r}}{2})}\\<br /> &amp;=\int{\frac{dkdk^{'}}{2\pi}\delta(k-k^{'})P(k)(\Delta{r})^{2}e^{ikr_{i}}e^{-ik^{'}r_{j}}sinc(k\frac{\Delta{r}}{2})sinc(k^{'}\frac{\Delta{r}}{2})}\\<br /> &amp;=\int{\frac{dk}{2\pi}P(k)(\Delta{r})^{2}e^{ik(r_{i}-r_{j})}[sinc(k\frac{\Delta{r}}{2})]^{2}}\\<br /> &amp; \text{Then assuming}\, \Delta{k} \text{ is very small, and } P(k) \text{is piecewise and pulling } p_{\alpha} \text{ out of he integral,}\\<br /> &amp;\approx \sum_{\alpha}{p_{\alpha}\frac{\Delta{k}}{2\pi}(\Delta{r})^{2}e^{ik_{\alpha}(r_{i}-r_{j})}[sinc(k_{\alpha}\frac{\Delta{r}}{2})]^{2}}<br /> \end{align}<br /> <br /> Therefore, $C_{,\alpha} = \frac{\partial{C}}{\partial{p_{\alpha}}} \propto<br /> e^{ik_{\alpha}(r_{i} - r_{j}}[sinc(k_{\alpha}\frac{\Delta{r}}{2})]^{2}$.<br /> Plugging into the equation for the taper functions we get, <br /> <br /> \begin{align}<br /> w_{\alpha\beta} &amp;\propto tr[C_{,\alpha}C_{,\beta}]\\<br /> &amp;\propto \mid\sum_{\alpha\beta}{e^{i(k_{\alpha} -k_{\beta})r_{i}}[sinc(k_{\alpha}\frac{\Delta{r}}{2})]^{2}][sinc(k_{\beta}\frac{\Delta{r}}{2})]^{2}}\mid<br /> \end{align}<br /> <br /> Remember, that the $\alpha$-th row tells us what linear combination of $k$ calue<br /> are probed by $\hat{p_{\alpha}}$.<br /> So, we want the $w_{\alpha\beta}$ large if $k_{\alpha} \approx k_{\beta}$ and<br /> that $w_{\alpha\beta}$ small if $k_{\alpha}$ is very different from $k_{\beta}$.<br /> This works out with the above window (taper) functions above. <br /> We then normalize $w_{\alpha\beta}$ so that each row sums to 1.<br /> <br /> <br /> \end{document}</div> Zaki 10/23/14 Quadratic Estimators I 2015-04-04T00:14:11Z <p>Zaki: Created page with '&lt;latex&gt; \documentclass[11pt]{article} \def\hf{\frac12} \def\dcube#1{{d^{3}\mathbf{#1}}} \def\ddtau#1{{\frac{d#1} {d\tau}}} \def\bra#1{\langle #1|} \def\ket#1{|#1\rangle} \def\ex…'</p> <hr /> <div>&lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> <br /> \def\hf{\frac12}<br /> \def\dcube#1{{d^{3}\mathbf{#1}}}<br /> \def\ddtau#1{{\frac{d#1} {d\tau}}}<br /> \def\bra#1{\langle #1|}<br /> \def\ket#1{|#1\rangle}<br /> \def\expval#1{\langle #1 \rangle}<br /> \def\rhor{\rho(\mathbf{r})}<br /> <br /> \def\deltar{\delta(\mathbf{r})}<br /> \def\deltarp{\delta(\mathbf{r}^{\prime})}<br /> \def\deltaru#1{\delta_{#1}(\mathbf{r})}<br /> <br /> \def\deltak{\delta(\mathbf{k})}<br /> \def\deltaku#1{\delta_{#1}(\mathbf{k})}<br /> <br /> \def\fdeltak{\tilde{\delta}(\mathbf{k})}<br /> \def\fdeltakp{\tilde{\delta}(\mathbf{k}^{\prime})}<br /> \def\fdeltaku#1{\tilde{\delta}_{#1}(\mathbf{k})}<br /> \def\vec#1{\mathbf{#1}}<br /> <br /> <br /> \newcount\colveccount<br /> \newcommand*\colvec{<br /> \global\colveccount#1<br /> \begin{pmatrix}<br /> \colvecnext<br /> }<br /> \def\colvecnext#1{<br /> #1<br /> \global\advance\colveccount-1<br /> \ifnum\colveccount&gt;0<br /> \\<br /> \expandafter\colvecnext<br /> \else<br /> \end{pmatrix}<br /> \fi<br /> }<br /> <br /> \usepackage{fullpage}<br /> \usepackage{amsmath}<br /> \usepackage{commath}<br /> \usepackage{amsfonts}<br /> \usepackage[margin=1in]{geometry}<br /> \usepackage{graphicx,rotating}<br /> <br /> \begin{document}<br /> <br /> \title{Quadratic Estimators: Part I}<br /> <br /> \section{preamble} <br /> We have learned, in the previous lectures, about the power spectrum. By<br /> definitiion we have that, in general, <br /> <br /> \begin{equation}<br /> \expval{\tilde{T}^{\*}_{b}(\mathbf{k}) \tilde{T}^{\*}_{b}(\mathbf{k}^{\prime})} =<br /> (2\pi)^{3}\delta^{D}(\mathbf{k} - \mathbf{k}^{\prime})P(\mathbf{k}).<br /> \end{equation}<br /> <br /> And for the CMB, that is for a 2 dimensional shell, we can describe the power<br /> spectrum by $C_{l}$ from from a spherical harmonic decompostion <br /> <br /> \begin{equation}<br /> \expval{a_{lm}a_{l^{\prime}m^{\prime}}} = C_{l}\delta_{ll^{\prime}}\delta_{mm^{\prime}}<br /> \end{equation}<br /> <br /> We will now develop some formalism for quadratic estimators. This is especially<br /> useful for the power spectrum, a quadratic quantity.<br /> <br /> \section{Quadratic Estimator Formalism}<br /> Let us take our data vector $\mathbf{x}$, given by, <br /> <br /> \begin{equation}<br /> \mathbf{x} = \colvec{5}{\mathbf{x_{1}}}{\mathbf{x_{2}}}{.}{.}{.}, <br /> \end{equation}<br /> <br /> where each ${\mathbf{x_{i}}}$ can be temperature data (${\mathbf{T_{i}}}$),<br /> visibility data (${\mathbf{V_{i}}}$), or some other quantity.<br /> The quadratic estimator is defined as <br /> <br /> \begin{equation}\label{eqn:quadest}<br /> \hat{p}_{\alpha} \propto \vec{x}^{\dagger}E^{\alpha}\vec{x} - b^{\alpha}.<br /> \end{equation}<br /> <br /> In equation \ref{eqn:quadest}, $\hat{}$ means the quantity hatted is the<br /> estimator, $p_{\alpha} \equiv P({k_{\alpha}})$, $\vec{x}$ is the data input vector,<br /> $E^{\alpha}$ is some matrix which is the data analysts choise, and $b^{\alpha}$ is<br /> the bias correction. <br /> <br /> \section{What's with the ${E^{\alpha}?}$}<br /> Now that we have defined the quadratic estimator in equation \ref{eqn:quadest},<br /> we can discuss what a sensible choise for $E^{\alpha}$ may be. The choice for<br /> $E^{\alpha}$ depends on the choice of basis $\vec{x}$ is expressed in. Note that<br /> $E^{\alpha}$ is a family of matrices, one for each $\alpha$. Therefore, to<br /> measure the the power at some $k$-mode $k_{i}$, we must use the $E^{\alpha}$<br /> matrix that corresponds to that $k$-mode.<br /> <br /> \subsection{example 1: $T(\vec{r_{i}})$}<br /> Suppose that the data vector $\vec{x} = ( T(\vec{r}_{1}), T(\vec{r}_{2}),<br /> ...)$. Let us guess that <br /> <br /> \begin{equation}<br /> E^{\alpha}_{ij} \sim e^{i\vec{k}_{\alpha}(\vec{r}_{i} - \vec{r}_{j})}.<br /> \end{equation}<br /> <br /> Why is this a good guess? Well we want to make an estimate of the power spectrum<br /> from temperature data. Back from the power spectrum lectures, we know that we<br /> need to fourier transform temperature data (a function of $\vec{r}$) in order<br /> to go into $k$-space. Hence $E^{\alpha}$ should look something like a fourier<br /> transform.<br /> <br /> In order to check this, let us compute the estimator,<br /> \begin{align}<br /> \vec{x}^{\dagger} E^{\alpha} \vec{x} &amp;= \sum_{ij}{x_{i}E_{ij}^{\alpha}x_{j}}\\<br /> &amp;= \sum_{ij}{T(\vec{r}_{i})e^{i\vec{k}_{\alpha}(\vec{r}_{i} - \vec{r}_{j})}T(\vec{r}_{j})}\\<br /> &amp;= ( \sum_{i}{e^{i\vec{k}_{\alpha}\vec{r}_{i}}T(\vec{r}_{i})} ) ( \sum_{j}{e^{i\vec{k}_{\alpha}\vec{r}_{j}}T(\vec{r}_{j})} )\\<br /> &amp;= \abs{\sum_{j}{e^{-i\vec{k} _{\alpha}\vec{r}_{j}}T(\vec{r}_{j})}}^{2}.<br /> \end{align}<br /> <br /> Hence, the choice of $E^{\alpha}$ above gives us a fourier tranform-ish result.<br /> This result would be a true fourier transform if the data were on a regular<br /> grid. <br /> <br /> \subsection{example 2: $\tilde{T}(\vec{k})$}<br /> In this case $\vec{x} = ( \tilde{T}(\vec{k}_{1}), \tilde{T}(\vec{k}_{2}),<br /> ...)$. We can then choose an $E^{\alpha}$ that extracts particular ${k}$-modes.<br /> Remember that $\alpha$ is a particular $k$-mode. Specifically, we have that <br /> $E^{alpha=k} = \delta{k}\delta_{i}\delta{j}$, so that the matrix is equal to 1<br /> only when $i=j=k$.<br /> <br /> \section{The general form of $E$}<br /> <br /> Suppose that we have a power spectrum that was binned up. That is we model<br /> $P(k)$ as piecewise constant. The value of $P(k_{\alpha}) \equiv \alpha^{th}$ band<br /> power $\equiv P_{\alpha}$.<br /> Now to relate this to the data, consider the covariance matrix of the data: <br /> $C \equiv \expval{\vec{x}\vec{x}^{\dagger}}$; $C_{ij} =<br /> \expval{x_{i}x^{*}_{j}}$. The covariance matrix $C$ consists of two parts:<br /> <br /> \begin{equation}<br /> C = N + \sum_{\alpha}{p_{\alpha}\frac{\partial{C}}{\partial{p_{\alpha}}}}<br /> \equiv N + \sum_{\alpha}{p_{\alpha}C_{,\alpha}},<br /> \end{equation}<br /> <br /> where the first term $N$ is the noise term and consists of instrumental noise,<br /> systematics, etc.... The second term, the partial derivative of the covariance<br /> matrix with respect to $\alpha$ is the crucial link between theory and data.<br /> This term is related to $E^{\alpha}$. <br /> <br /> We note that the unvierse is not random and hence the power spectrum<br /> ($p_{\alpha}$) is not random. However, the quadratic estimator<br /> $\hat{p}_{\alpha}$ is random due to our instrument. Because of this we can<br /> measure the variance in our estimate of the power spectrum. Lets do that... <br /> The variance of the power spectrum in the $\alpha$-th bin is given by <br /> <br /> \begin{align}<br /> \expval{\hat{p}_{\alpha}} &amp;= \expval{\vec{x}^{\dagger}E^{\alpha}\vec{x}} -<br /> b_{\alpha}\\<br /> &amp;= \sum_{ij}\expval{x^{*}_{i}E_{ij}^{\alpha}x_{j}} - b_{\alpha}\\<br /> &amp;= \sum_{ij}E_{ij}^{\alpha}\expval{x_{j}x^{*}_{i}} - b_{\alpha}\\<br /> &amp;= \sum_{ij}E^{\alpha}_{ij}C_{ij} - b_{\alpha}\\<br /> &amp;= tr(E^{\alpha}C) - b_{\alpha}.<br /> \end{align}<br /> <br /> Plugging in for the covariance matrix, we get that <br /> \begin{align}<br /> \expval{\hat{p}_{\alpha}} &amp;= tr[E^{\alpha}\sum_{\beta}p_{\beta}C_{,\beta}] +<br /> tr[E^{\alpha}N] - b_{\alpha}\\<br /> &amp;= \sum_{\beta}tr[E^{\alpha}C_{,\beta}]p_{\beta} + tr[E^{\alpha}N] - b_{\alpha}.<br /> \end{align}<br /> <br /> Note that the second term, $tr(E^{\alpha}N)$ is the additive (noise) bias term.<br /> If we let $b_{\alpha} = tr[E^{\alpha}N]$ then we can correct fo this additive<br /> term. Hence, we are left with $\expval{\hat{p_{\alpha}}} = \sum{w_{\alpha\beta}p_{\beta}}$ where $w_{\alpha\beta} =<br /> tr[E^{\alpha}C_{,\beta}]$. This $w_{\alpha\beta}$ matrix is the window function<br /> matrix (or taper function matrix). The $\alpha$-th row of $w_{\alpha\beta}$<br /> gives linear cominations of $P(k_{\beta})$ probed by $\hat{p}_{\alpha}$. This<br /> gives us horizontal errorbrs.<br /> <br /> \section{Vertical Error Bars}<br /> The window (or taper) functions above gave us the horizontal error bars, which<br /> is to say for each power spectrum measurement in the $\alpha$-th bin, the amount<br /> other bins contributed to that measurement. <br /> <br /> Now, we discuss how to get the vertical error bars in the power spectrum<br /> estimate, which is basically the covariance of $\hat{p}_{\alpha}$. Let us<br /> calculate this, <br /> <br /> \begin{align}<br /> V_{\alpha\beta} &amp;= \expval{\hat{p}_{\alpha}\hat{p}_{\beta}} -<br /> \expval{\hat{p}_{\alpha}}\expval{\hat{p}_{\beta}}\\<br /> &amp;=<br /> \sum_{ijkl}{\expval{x_{i}E_{ij}^{\alpha}x_{j}x_{k}E^{\beta}_{kl}x_{l}}} -<br /> \sum_{ijkl}{\expval{x_{i}x_{j}}E_{ij}^{\alpha}}<br /> \sum_{ijkl}{\expval{x_{k}x_{l}}E_{kl}^{\beta}}, <br /> \end{align}<br /> <br /> where terms $\expval{x_{i}x_{j}}$ are $C_{ij}$. If we assume that x is Gaussian<br /> distributed, we can use wicks theorem (or Issirelli's Theorem; see wikipedia) to calculate the<br /> 4 point function above in terms of two point functions. This gives us that the<br /> vertical error bars are <br /> <br /> \begin{equation}<br /> V_{\alpha\beta} = 2tr[CE^{\alpha}CE^{\beta}]<br /> \end{equation}<br /> <br /> <br /> <br /> \section{A Concrete Example : The 1-D Universe and a noiseless instrument}<br /> Consider a 1-D universe from $\frac{-L}{2}$ to $\frac{L}{2}$ with some<br /> pixelation (meaning discrete bins of size $\Delta{r}$). Then a measurement in this universe,<br /> $x_{i}$, is given by <br /> \begin{equation}<br /> x_{i} = \int{dr T(r) \Phi_{i}(r)}<br /> \end{equation}<br /> where $\Phi_{i}$ is some pixelization function. We take it to be 1 inside the<br /> $i$-th cell and 0 outside. Let us now estimate $p_{\alpha}$ in this universe.<br /> <br /> $$<br /> \hat{p}_{\alpha} = \vec{x}^{\dagger}E^{\alpha}\vec{x}<br />$$<br /> Note that we are not including the noise bias $b_{\alpha}$ here and therefore<br /> are not considering noise.<br /> <br /> A good choice for $E_{\alpha}$ in this situation would be one that is<br /> proportional to $C_{,\alpha}$. Previously we had that the expectaion value of<br /> the estimator is <br /> <br /> \begin{align}<br /> \expval{\hat{p}} &amp;= \sum_{ij}{E^{\alpha}_{ij}C_{ji}} \\<br /> &amp;\propto \sum_{ij}{(C_{,\alpha})_{ij}C_{ij}}<br /> \end{align}<br /> <br /> Now, lets compute the error bars. Lets start by first going into fourier space.<br /> Then, <br /> \begin{align}<br /> \vec{x}_{i} &amp;= \int{dr\Phi_{i}(r)}\int{\frac{dk}{2\pi}\tilde{T}(k)e^{ikr}}\\<br /> &amp;= \int{\frac{dk}{2\pi}\tilde{T}(k)}\int_{r_{i} -<br /> \frac{\Delta{r}}{2}}^{r_{i}+\frac{\Delta{r}}{2}}{e^{ikr}}\\<br /> &amp;=<br /> \int{\frac{dk}{2\pi}\tilde{T}(k)\Delta{r}e^{ikr_{i}}sinc(k\frac{\Delta{r}}{2})}, <br /> \end{align}<br /> by noting that $sinc(nx) = \frac{1}{2n}\int{e^{ixt}dt}$.<br /> <br /> Then the covariance ($C_{ij} = \expval{x_{i}x_{j}^{*}}$) is given by,<br /> <br /> \begin{align*}<br /> C_{ij} &amp;=<br /> \int{\frac{dk}{2\pi}\frac{dk^{'}}{2\pi}\expval{\tilde{T}(k)\tilde{T}^{*}(k^{'})}(\Delta{r})^{2}e^{ikr_{i}}e^{-ik^{'}r_{j}}sinc(k\frac{\Delta{r}}{2})sinc(k^{'}\frac{\Delta{r}}{2})}\\<br /> &amp;=<br /> \int{\frac{dkdk^{'}}{2\pi}\delta(k-k^{'})P(k)(\Delta{r})^{2}e^{ikr_{i}}e^{-ik^{'}r_{j}}sinc(k\frac{\Delta{r}}{2})sinc(k^{'}\frac{\Delta{r}}{2})}\\<br /> &amp;=<br /> \int{\frac{dk}{2\pi}P(k)(\Delta{r})^{2}e^{ik(r_{i}-r_{j})}[sinc(k\frac{\Delta{r}}{2})]^{2}}\\<br /> &amp; \text{Then assuming $\Delta{k}$ is very small, and $P(k)$ is<br /> piecewise and pulling $p_{\alpha}$ out of the integral,}\\<br /> &amp;\approx<br /> \sum_{\alpha}{p_{\alpha}\frac{\Delta{k}}{2\pi}(\Delta{r})^{2}e^{ik_{\alpha}(r_{i}-r_{j})}[sinc(k_{\alpha}\frac{\Delta{r}}{2})]^{2}}<br /> \end{align*}<br /> <br /> Therefore, $C_{,\alpha} = \frac{\partial{C}}{\partial{p_{\alpha}}} \propto<br /> e^{ik_{\alpha}(r_{i} - r_{j}}[sinc(k_{\alpha}\frac{\Delta{r}}{2}]^{2}$.<br /> Plugging into the equation for the taper functions we get, <br /> <br /> \begin{align*}<br /> w_{\alpha\beta} &amp;\propto tr[C_{,\alpha}C_{,\beta}] \\<br /> &amp;\propto \abs{\sum_{\alpha\beta}{e^{i(k_{\alpha} -<br /> k_{\beta})r_{i}}[sinc(k_{\alpha}\frac{\Delta{r}}{2}]^{2}][sinc(k_{\beta}\frac{\Delta{r}}{2}]^{2}}}<br /> \end{align*}<br /> <br /> Remember, that the $\alpha$-th row tells us what linear combination of $k$ calue<br /> are probed by $\hat{p_{\alpha}}$.<br /> So, we want the $w_{\alpha\beta}$ large if $k_{\alpha} \approx k_{\beta}$ and<br /> that $w_{\alpha\beta}$ small if $k_{\alpha}$ is very different from $k_{\beta}$.<br /> This works out with the above window (taper) functions above. <br /> We then normalize $w_{\alpha\beta}$ so that each row sums to 1.<br /> <br /> <br /> \end{document}</div> Zaki 21cm Cosmology 2015-04-04T00:07:50Z <p>Zaki: /* Adrian's Class Notes */</p> <hr /> <div>== Adrian's Class Notes ==<br /> <br /> * [[6/13/14 Linear Algebra Representation]]<br /> * [[6/16/14 Linear Algebra Representation Cont. + Fisher Matrix]]<br /> * [[6/25/14 Power Spectra]]<br /> * [[7/2/14 Power Spectra Cont. + Interferometry]]<br /> * [[10/23/14 Quadratic Estimators I ]]<br /> <br /> == Accompanying Videos ==<br /> * [https://www.youtube.com/watch?v=QCqsJVS8p5A Generating correlated random variables]<br /> * [https://www.youtube.com/watch?v=QtF7Sz6KhSg Drawing Likelihood contours]</div> Zaki 21cm Cosmology 2015-04-04T00:04:26Z <p>Zaki: /* Adrian's Class Notes */</p> <hr /> <div>== Adrian's Class Notes ==<br /> <br /> * [[6/13/14 Linear Algebra Representation]]<br /> * [[6/16/14 Linear Algebra Representation Cont. + Fisher Matrix]]<br /> * [[6/25/14 Power Spectra]]<br /> * [[7/2/14 Power Spectra Cont. + Interferometry]]<br /> * [[10/23/14 Power Spectra Cont. + Interferometry]]<br /> <br /> == Accompanying Videos ==<br /> * [https://www.youtube.com/watch?v=QCqsJVS8p5A Generating correlated random variables]<br /> * [https://www.youtube.com/watch?v=QtF7Sz6KhSg Drawing Likelihood contours]</div> Zaki Intro to Research Resources 2015-01-22T22:06:51Z <p>Zaki: </p> <hr /> <div>* [[Programming Resources]]<br /> * [[Introduction Papers]]</div> Zaki Programming Resources 2015-01-22T22:04:55Z <p>Zaki: Created page with 'These are some useful programming resources. They vary from introductions to python to more advanced topics and exercises. * MIT 600 * UCB CS9H course * Enthought: A self conta…'</p> <hr /> <div>These are some useful programming resources. They vary from introductions to python to more advanced topics and exercises. <br /> <br /> * MIT 600<br /> * UCB CS9H course<br /> * Enthought: A self contained python package distribution. You can install it by going here. <br /> * python.org for a python tutorial. <br /> * Python BootCamp : XXXneed link<br /> * Project Euler : Useful programming exercises to get your feet wet.</div> Zaki Intro to Research Resources 2015-01-22T22:00:52Z <p>Zaki: Created page with '* Programming Resources'</p> <hr /> <div>* [[Programming Resources]]</div> Zaki Main Page 2015-01-22T21:59:57Z <p>Zaki: /* Lectures and Materials */</p> <hr /> <div>__TOC__<br /> <br /> == What is AstroBaki ==<br /> <br /> AstroBaki is a wiki for current and aspiring scientists to collaboratively build pedagogical materials such as videos, lecture notes, and textbooks. Textbooks in particular take a lot of work to write and get right. The more people that participate in the learning, teaching, and writing process, the better the result. And the fruits of this labor should be open and free, because science works by being open and free. Except for writing grants.<br /> <br /> AstroBaki was inspired by the story of [http://en.wikipedia.org/wiki/Nicolas_Bourbaki Nicolas Bourbaki]. From the 1930s to the 70s, a bunch of French mathematicians got together and rewrote math collaboratively. No individual got to claim credit for the work (hence the psuedonym Bourbaki). The result were a set of useful pedagogical texts and a rejuvenation of math throughout France. Nicolas Bourbaki was a wiki ahead of its time.<br /> <br /> The AstroBaki wiki augments the standard MediaWiki engine to accept LaTeX for rendering. The goal is for texts to be written in LaTeX for anyone to download, compile, and view, and for the very same LaTeX code to be rendered into HTML by the wiki engine for a reasonable viewing experience online. All of the standard wiki tools of collaborative writing, revision control, and attribution (if desired) apply to the LaTeX code. <br /> <br /> This work is supported in part by [http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0901961 funding from the National Science Foundation].<br /> <br /> == Who is AstroBaki ==<br /> <br /> In this section, we aim to acknowledge special contributors to this project. The edit history of documents on this wiki will record contributions for posterity, but in some cases, this is not enough. To more adequately acknowledge the hard work and generosity of members who have labored to put their work in the public domain here, we have the following list:<br /> * [http://astro.berkeley.edu/~aparsons '''Prof. Aaron Parsons'''], of the University of California, Berkeley<br /> ** transcribed into latex (in the classroom, on the fly) the lectures for [[Radiative Processes in Astrophysics]], [[Cosmology]], [[Galactic Dynamics and Stellar Formation]], [[Fluid Dynamics]], and [[Order-of-Magnitude Physics]]<br /> ** Summarized and distilled from lectures the summary for [[Interstellar Medium]]<br /> ** authored [[Radio Astronomy: Tools and Techniques]]<br /> ** Created this wiki, along with the MediaWiki extension that performs wholesale conversion of latex into MediaWiki syntax<br /> ** is teaching the Spring 2014 [[Undergraduate Radio Lab]]<br /> * '''Prof. Eugene Chiang''', of the University of California, Berkeley<br /> ** authored the original 26 lectures of [[Radiative Processes in Astrophysics]]<br /> ** authored the original 22 lectures of [[Order-of-Magnitude Physics]]<br /> * '''Prof. Chung-Pei Ma''', of the University of California, Berkeley<br /> ** authored the original 24 lectures of [[Cosmology]]<br /> * '''Prof. Leo Blitz''', of the University of California, Berkeley<br /> ** authored the original 26 lectures of [[Galactic Dynamics and Stellar Formation]]<br /> * '''Prof. Al Glassgold''', of the University of California, Berkeley<br /> ** co-taught with James Graham the 30 lectures from which the summary of the [[Interstellar Medium]] was drawn<br /> * '''Prof. Eliot Quataert''', of the University of California, Berkeley<br /> ** authored [[Stellar Structure]] (transcribed by James McBride)<br /> * '''Prof. James Graham''', of the University of California, Berkeley<br /> ** co-taught with Al Glassgold the 30 lectures from which the summary of the [[Interstellar Medium]] was drawn<br /> ** authored the original 3 lectures of [[Fluid Dynamics]]<br /> * '''Prof. Carl Heiles''', of the University of California, Berkeley<br /> ** authored the original labs and related hand-outs for the [[Undergraduate Radio Lab]]<br /> <br /> == Lectures and Materials ==<br /> <br /> To start off AstroBaki, here are lecture notes from introductory graduate astrophysics classes, typed on-the-fly in class. They are coarse, poorly edited, and possibly incorrect in places (usually because of transcription error), but make up for that in sheer content. Though the elegance of the latex rendering (and writing) varies dramatically, there should be plenty of examples off of which contributors can base new contributions. Please add and edit!<br /> <br /> * [[Radiative Processes in Astrophysics]]<br /> * [[Cosmology]]<br /> * [[Galactic Dynamics and Stellar Formation]]<br /> * [[Fluid Dynamics]]<br /> * [[Order-of-Magnitude Physics]]<br /> * [[Interstellar Medium]]<br /> * [[Introduction to Digital Signal Processing]]<br /> * [[Radio 101]]<br /> * [[Undergraduate Radio Lab]]<br /> * [[Radio Astronomy: Tools and Techniques]]<br /> * [[Stellar Structure]]<br /> * [[21cm Cosmology|Cosmological Statistics with an Emphasis on 21cm Cosmology]]<br /> * [[Intro to Research Resources]]</div> Zaki Main Page 2015-01-22T21:59:35Z <p>Zaki: /* Lectures and Materials */</p> <hr /> <div>__TOC__<br /> <br /> == What is AstroBaki ==<br /> <br /> AstroBaki is a wiki for current and aspiring scientists to collaboratively build pedagogical materials such as videos, lecture notes, and textbooks. Textbooks in particular take a lot of work to write and get right. The more people that participate in the learning, teaching, and writing process, the better the result. And the fruits of this labor should be open and free, because science works by being open and free. Except for writing grants.<br /> <br /> AstroBaki was inspired by the story of [http://en.wikipedia.org/wiki/Nicolas_Bourbaki Nicolas Bourbaki]. From the 1930s to the 70s, a bunch of French mathematicians got together and rewrote math collaboratively. No individual got to claim credit for the work (hence the psuedonym Bourbaki). The result were a set of useful pedagogical texts and a rejuvenation of math throughout France. Nicolas Bourbaki was a wiki ahead of its time.<br /> <br /> The AstroBaki wiki augments the standard MediaWiki engine to accept LaTeX for rendering. The goal is for texts to be written in LaTeX for anyone to download, compile, and view, and for the very same LaTeX code to be rendered into HTML by the wiki engine for a reasonable viewing experience online. All of the standard wiki tools of collaborative writing, revision control, and attribution (if desired) apply to the LaTeX code. <br /> <br /> This work is supported in part by [http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0901961 funding from the National Science Foundation].<br /> <br /> == Who is AstroBaki ==<br /> <br /> In this section, we aim to acknowledge special contributors to this project. The edit history of documents on this wiki will record contributions for posterity, but in some cases, this is not enough. To more adequately acknowledge the hard work and generosity of members who have labored to put their work in the public domain here, we have the following list:<br /> * [http://astro.berkeley.edu/~aparsons '''Prof. Aaron Parsons'''], of the University of California, Berkeley<br /> ** transcribed into latex (in the classroom, on the fly) the lectures for [[Radiative Processes in Astrophysics]], [[Cosmology]], [[Galactic Dynamics and Stellar Formation]], [[Fluid Dynamics]], and [[Order-of-Magnitude Physics]]<br /> ** Summarized and distilled from lectures the summary for [[Interstellar Medium]]<br /> ** authored [[Radio Astronomy: Tools and Techniques]]<br /> ** Created this wiki, along with the MediaWiki extension that performs wholesale conversion of latex into MediaWiki syntax<br /> ** is teaching the Spring 2014 [[Undergraduate Radio Lab]]<br /> * '''Prof. Eugene Chiang''', of the University of California, Berkeley<br /> ** authored the original 26 lectures of [[Radiative Processes in Astrophysics]]<br /> ** authored the original 22 lectures of [[Order-of-Magnitude Physics]]<br /> * '''Prof. Chung-Pei Ma''', of the University of California, Berkeley<br /> ** authored the original 24 lectures of [[Cosmology]]<br /> * '''Prof. Leo Blitz''', of the University of California, Berkeley<br /> ** authored the original 26 lectures of [[Galactic Dynamics and Stellar Formation]]<br /> * '''Prof. Al Glassgold''', of the University of California, Berkeley<br /> ** co-taught with James Graham the 30 lectures from which the summary of the [[Interstellar Medium]] was drawn<br /> * '''Prof. Eliot Quataert''', of the University of California, Berkeley<br /> ** authored [[Stellar Structure]] (transcribed by James McBride)<br /> * '''Prof. James Graham''', of the University of California, Berkeley<br /> ** co-taught with Al Glassgold the 30 lectures from which the summary of the [[Interstellar Medium]] was drawn<br /> ** authored the original 3 lectures of [[Fluid Dynamics]]<br /> * '''Prof. Carl Heiles''', of the University of California, Berkeley<br /> ** authored the original labs and related hand-outs for the [[Undergraduate Radio Lab]]<br /> <br /> == Lectures and Materials ==<br /> <br /> To start off AstroBaki, here are lecture notes from introductory graduate astrophysics classes, typed on-the-fly in class. They are coarse, poorly edited, and possibly incorrect in places (usually because of transcription error), but make up for that in sheer content. Though the elegance of the latex rendering (and writing) varies dramatically, there should be plenty of examples off of which contributors can base new contributions. Please add and edit!<br /> <br /> * [[Radiative Processes in Astrophysics]]<br /> * [[Cosmology]]<br /> * [[Galactic Dynamics and Stellar Formation]]<br /> * [[Fluid Dynamics]]<br /> * [[Order-of-Magnitude Physics]]<br /> * [[Interstellar Medium]]<br /> * [[Introduction to Digital Signal Processing]]<br /> * [[Radio 101]]<br /> * [[Undergraduate Radio Lab]]<br /> * [[Radio Astronomy: Tools and Techniques]]<br /> * [[Stellar Structure]]<br /> * [[21cm Cosmology|Cosmological Statistics with an Emphasis on 21cm Cosmology]]<br /> *[[Intro to Research Resources]]</div> Zaki File:Hamming.png 2012-12-13T08:31:25Z <p>Zaki: </p> <hr /> <div></div> Zaki File:Hanning.png 2012-12-13T08:31:04Z <p>Zaki: </p> <hr /> <div></div> Zaki File:Timesignals.png 2012-12-13T08:30:29Z <p>Zaki: uploaded a new version of &quot;File:Timesignals.png&quot;</p> <hr /> <div></div> Zaki File:Timesignals.png 2012-12-13T08:28:25Z <p>Zaki: uploaded a new version of &quot;File:Timesignals.png&quot;</p> <hr /> <div></div> Zaki File:Freqleakage.png 2012-12-13T08:13:21Z <p>Zaki: </p> <hr /> <div></div> Zaki File:Timesignals.png 2012-12-13T08:12:33Z <p>Zaki: </p> <hr /> <div></div> Zaki Windowing 2012-12-13T08:08:54Z <p>Zaki: </p> <hr /> <div>===Short Topical Videos===<br /> * [to come] (by Zaki Ali)]<br /> <br /> ===Reference Material===<br /> * [http://en.wikipedia.org/wiki/Window_function http://en.wikipedia.org/wiki/Window_function ]<br /> * [http://www.bores.com/courses/advanced/windows/files/windows.pdf bores.com]<br /> <br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \setlength\parindent{0 in}<br /> \setlength\parskip{0.1 in}<br /> \usepackage{fullpage}<br /> \usepackage{amsmath}<br /> \usepackage{graphicx}<br /> <br /> \begin{document}<br /> \title{Windowing Functions}<br /> \maketitle<br /> <br /> \section{Spectral Leakage}<br /> <br /> Suppose we are taking a discrete Fourier transform(DFT) of a signal.<br /> When taking the DFT for the signal, we can only sample it for a finite time<br /> (the Fourier transform of a function is defined for infinite time). This leads<br /> to some unwanted features. Because we are only sampling the signals for a finite<br /> time, we need to assume something about the signal outside of this interval. The<br /> DFT implicitly assumes that the signal repeats. This causes discontinuities at<br /> the edges of the sampled signal. Fourier transforms of discontinuities lead to<br /> broad frequency structure. For an example, see figure \ref{fig:samplewave} and<br /> \ref{fig:specleak}. Notice the leakage of power in the neighboring spectral<br /> bins. This is &quot;spectral leakage&quot;. <br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{timesignals.png}<br /> \caption{The time series signals. The top window shows the real time signal<br /> in the 'infinite' limit. The middle window shows that actual sampled wave that<br /> we feed into the FFT. The bottom window shows what the FFT thinks the signal is<br /> : infinitely repeated sampled signal.}<br /> \label{fig:samplewave}<br /> \end{figure}<br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{freqleakage.png}<br /> \caption{The Fourier transform of the sine wave that does not have sharp<br /> discontinuities (top). The FFT of the sampled sine wave, which has sharp<br /> discontinuities.}<br /> \label{fig:specleak}<br /> \end{figure}<br /> <br /> <br /> \section{Windowing}<br /> Windowing is a way to minimize spectral leakage by tapering off the ends of the<br /> sampled waveform to 0. This is nothing new, and in fact when we take a regular<br /> ol' FFT (the fast Fourier transform algorithm that implements a DFT<br /> efficiently), we are in fact multiplying by a windowing function. This windowing<br /> function is a square windowing function which is described as being 1 if in the<br /> sampled window, and 0 otherwise. Hence, we can write the Fourier transform as <br /> <br /> $$<br /> \hat{f} = \int{W(t)f(t)e^{-2\pi{j}\omega{t}}} = \hat{W} \star \hat{f},<br />$$<br /> <br /> where $\star$ is the convolution function and $\hat{}$ denotes the<br /> Fourier transform. $W$ is the windowing function. Note that the windowing<br /> function can either be defined in time space or frequency space. The same term<br /> refers to both spaces. The result above follows from the convolution theorem.\\<br /> <br /> As we noted above, the sharp discontinuities at the ends of the sampled waveform<br /> are what cause spectral leakage. To minimize this we can use a different<br /> windowing function that tapers off to zero at the ends. This will minimize<br /> discontinuities. Some popular windowing functions include the Hanning and<br /> Hamming windowing functions, see figure \ref{fig:hann} and \ref{fig:hamm}. \\<br /> <br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{hanning.png}<br /> \caption{The hanning window and its frequency response. The frequency<br /> response is the Fourier transform of the window function on a test tone.}<br /> \label{fig:hann}<br /> \end{figure}<br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{hamming.png}<br /> \caption{The hamming window and its frequency response. The frequency<br /> response is the Fourier transform of the window function on a test tone.}<br /> \label{fig:hamm}<br /> \end{figure}<br /> <br /> One way to think about spectral leakage is that it is a spreading out of<br /> energy/power over spectral bins by the windowing function. The contribution of<br /> the spectral bin to its neighbors is the weight of the window function (in<br /> frequency space) centered at the given frequency component evaluated at the FFT<br /> bin in question. Another way to think about it is to think about the FFT as a<br /> series of filters centered on a specific spectral channel. Then the FFT filters<br /> response at that spectral channel is given by the windowing function (in<br /> frequency space). Therefore, each spectral bin includes signal from all other<br /> frequencies in the filters bandwidth.\\<br /> <br /> \section{Noise Equivalent Bandwidth}<br /> <br /> Note that for non square window functions are tapering off at the ends and<br /> hence we are in effect throwing away information about the signal.<br /> Specifically, we are decreasing signal to noise. One way to quantify this, is<br /> the talk about the Noise Equivalent Bandwidth : the bandwidth (of the non<br /> rectangular windowed function) that gives us the same amount of noise power as<br /> the rectangular window did. Numerically, <br /> <br /> $$<br /> NEB = \frac{B}{B_{eq}}<br />$$<br /> <br /> There is also a concept of the Signal Equivalent bandwidth. By the radiometer<br /> equation, this is just given by the square root of the Noise Equivalent<br /> Bandwidth. Hence the NEB is always greater than the SEB. Therefore, there is<br /> a crucial trade off for windowing functions. The narrower your window is, the<br /> greater your NEB is and hences the lower your signal to noise is going to be.<br /> Depending on your needs, you need to pick your windowing function wisely.<br /> <br /> \end{document}<br /> <br /> &lt;/latex&gt;</div> Zaki Windowing 2012-12-13T08:06:57Z <p>Zaki: Created page with '===Short Topical Videos=== * [to come] (by Zaki Ali)] ===Reference Material=== * [http://en.wikipedia.org/wiki/Window_function] * [http://www.bores.com/courses/advanced/windo…'</p> <hr /> <div>===Short Topical Videos===<br /> * [to come] (by Zaki Ali)]<br /> <br /> ===Reference Material===<br /> * [http://en.wikipedia.org/wiki/Window_function]<br /> * [http://www.bores.com/courses/advanced/windows/files/windows.pdf]<br /> <br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \setlength\parindent{0 in}<br /> \setlength\parskip{0.1 in}<br /> \usepackage{fullpage}<br /> \usepackage{amsmath}<br /> \usepackage{graphicx}<br /> <br /> \begin{document}<br /> \title{Windowing Functions}<br /> \maketitle<br /> <br /> \section{Spectral Leakage}<br /> <br /> Suppose we are taking a discrete Fourier transform(DFT) of a signal.<br /> When taking the DFT for the signal, we can only sample it for a finite time<br /> (the Fourier transform of a function is defined for infinite time). This leads<br /> to some unwanted features. Because we are only sampling the signals for a finite<br /> time, we need to assume something about the signal outside of this interval. The<br /> DFT implicitly assumes that the signal repeats. This causes discontinuities at<br /> the edges of the sampled signal. Fourier transforms of discontinuities lead to<br /> broad frequency structure. For an example, see figure \ref{fig:samplewave} and<br /> \ref{fig:specleak}. Notice the leakage of power in the neighboring spectral<br /> bins. This is &quot;spectral leakage&quot;. <br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{timesignals.png}<br /> \caption{The time series signals. The top window shows the real time signal<br /> in the 'infinite' limit. The middle window shows that actual sampled wave that<br /> we feed into the FFT. The bottom window shows what the FFT thinks the signal is<br /> : infinitely repeated sampled signal.}<br /> \label{fig:samplewave}<br /> \end{figure}<br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{freqleakage.png}<br /> \caption{The Fourier transform of the sine wave that does not have sharp<br /> discontinuities (top). The FFT of the sampled sine wave, which has sharp<br /> discontinuities.}<br /> \label{fig:specleak}<br /> \end{figure}<br /> <br /> <br /> \section{Windowing}<br /> Windowing is a way to minimize spectral leakage by tapering off the ends of the<br /> sampled waveform to 0. This is nothing new, and in fact when we take a regular<br /> ol' FFT (the fast Fourier transform algorithm that implements a DFT<br /> efficiently), we are in fact multiplying by a windowing function. This windowing<br /> function is a square windowing function which is described as being 1 if in the<br /> sampled window, and 0 otherwise. Hence, we can write the Fourier transform as <br /> <br /> $$<br /> \hat{f} = \int{W(t)f(t)e^{-2\pi{j}\omega{t}}} = \hat{W} \star \hat{f},<br />$$<br /> <br /> where $\star$ is the convolution function and $\hat{}$ denotes the<br /> Fourier transform. $W$ is the windowing function. Note that the windowing<br /> function can either be defined in time space or frequency space. The same term<br /> refers to both spaces. The result above follows from the convolution theorem.\\<br /> <br /> As we noted above, the sharp discontinuities at the ends of the sampled waveform<br /> are what cause spectral leakage. To minimize this we can use a different<br /> windowing function that tapers off to zero at the ends. This will minimize<br /> discontinuities. Some popular windowing functions include the Hanning and<br /> Hamming windowing functions, see figure \ref{fig:hann} and \ref{fig:hamm}. \\<br /> <br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{hanning.png}<br /> \caption{The hanning window and its frequency response. The frequency<br /> response is the Fourier transform of the window function on a test tone.}<br /> \label{fig:hann}<br /> \end{figure}<br /> <br /> \begin{figure}<br /> \centering<br /> \includegraphics[width=6in]{hamming.png}<br /> \caption{The hamming window and its frequency response. The frequency<br /> response is the Fourier transform of the window function on a test tone.}<br /> \label{fig:hamm}<br /> \end{figure}<br /> <br /> One way to think about spectral leakage is that it is a spreading out of<br /> energy/power over spectral bins by the windowing function. The contribution of<br /> the spectral bin to its neighbors is the weight of the window function (in<br /> frequency space) centered at the given frequency component evaluated at the FFT<br /> bin in question. Another way to think about it is to think about the FFT as a<br /> series of filters centered on a specific spectral channel. Then the FFT filters<br /> response at that spectral channel is given by the windowing function (in<br /> frequency space). Therefore, each spectral bin includes signal from all other<br /> frequencies in the filters bandwidth.\\<br /> <br /> \section{Noise Equivalent Bandwidth}<br /> <br /> Note that for non square window functions are tapering off at the ends and<br /> hence we are in effect throwing away information about the signal.<br /> Specifically, we are decreasing signal to noise. One way to quantify this, is<br /> the talk about the Noise Equivalent Bandwidth : the bandwidth (of the non<br /> rectangular windowed function) that gives us the same amount of noise power as<br /> the rectangular window did. Numerically, <br /> <br /> $$<br /> NEB = \frac{B}{B_{eq}}<br />$$<br /> <br /> There is also a concept of the Signal Equivalent bandwidth. By the radiometer<br /> equation, this is just given by the square root of the Noise Equivalent<br /> Bandwidth. Hence the NEB is always greater than the SEB. Therefore, there is<br /> a crucial trade off for windowing functions. The narrower your window is, the<br /> greater your NEB is and hences the lower your signal to noise is going to be.<br /> Depending on your needs, you need to pick your windowing function wisely.<br /> <br /> \end{document}<br /> <br /> &lt;/latex&gt;</div> Zaki Homemade Interferometer 2012-05-24T00:32:16Z <p>Zaki: </p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{baoant.jpg} <br /> \includegraphics[width=12in]{baoant2.jpg} <br /> \caption{The dipole. Copper strips on the board are used for the dipole.}<br /> \end{figure}<br /> <br /> <br /> %\begin{figure}[!h]<br /> %\centering<br /> %\includegraphics[width=10in]{baoant2.jpg} <br /> % \caption{The basic two-element interferometer.}<br /> %\end{figure}<br /> <br /> <br /> \subsection*{The Signal Path}<br /> <br /> <br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> [[Image:SIGNALPATH2.jpeg|thumb|left|The signal path for one antenna.]]<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{signalspath.jpg} <br /> \caption{The backend amplifier and filter chain.}<br /> \end{figure}<br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{corr.jpg} <br /> \caption{The CASPER ROACH board with iADC's used as the correlator hardware.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{hfa2.jpg} <br /> \caption{HFA deployment!}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had an east-west baseline of 575 inches = 36.52 wave lengths at 750MHz. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...<br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=10in]{baseline.jpg} <br /> \includegraphics[width=10in]{blofringes.png}<br /> \caption{A baseline at Leuschner Observatory and our first fringes!}<br /> \end{figure}<br /> <br /> \subsection*{Fifth Trial-Leuschner}<br /> Fifth deployment of BAOBAB-4 at Leuschner on May 24th 2012. <br /> New active Baluns with Noise Temperature ~40K.</div> Zaki Homemade Interferometer 2011-10-20T23:41:42Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{baoant.jpg} <br /> \includegraphics[width=12in]{baoant2.jpg} <br /> \caption{The dipole. Copper strips on the board are used for the dipole.}<br /> \end{figure}<br /> <br /> <br /> %\begin{figure}[!h]<br /> %\centering<br /> %\includegraphics[width=10in]{baoant2.jpg} <br /> % \caption{The basic two-element interferometer.}<br /> %\end{figure}<br /> <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{signalspath.jpg} <br /> \caption{The backend amplifier and filter chain.}<br /> \end{figure}<br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{corr.jpg} <br /> \caption{The CASPER ROACH board with iADC's used as the correlator hardware.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{hfa2.jpg} <br /> \caption{HFA deployment!}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had an east-west baseline of 575 inches = 36.52 wave lengths at 750MHz. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...<br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=10in]{baseline.jpg} <br /> \caption{A baseline at Leuschner Observatory.}<br /> \end{figure}</div> Zaki File:Baseline.jpg 2011-10-20T06:20:40Z <p>Zaki: </p> <hr /> <div></div> Zaki Homemade Interferometer 2011-10-20T06:20:08Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{baoant.jpg} <br /> \includegraphics[width=12in]{baoant2.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> %\begin{figure}[!h]<br /> %\centering<br /> %\includegraphics[width=10in]{baoant2.jpg} <br /> % \caption{The basic two-element interferometer.}<br /> %\end{figure}<br /> <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{signalspath.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{corr.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{hfa2.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had an east-west baseline of 575 inches = 36.52 wave lengths at 750MHz. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...<br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=10in]{baseline.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}</div> Zaki Homemade Interferometer 2011-10-19T23:14:29Z <p>Zaki: </p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{baoant.jpg} <br /> \includegraphics[width=12in]{baoant2.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> %\begin{figure}[!h]<br /> %\centering<br /> %\includegraphics[width=10in]{baoant2.jpg} <br /> % \caption{The basic two-element interferometer.}<br /> %\end{figure}<br /> <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{signalspath.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{corr.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \begin{figure}[!h]<br /> \centering<br /> \includegraphics[width=12in]{hfa2.jpg} <br /> \caption{The basic two-element interferometer.}<br /> \end{figure}<br /> <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had an east-west baseline of 575 inches = 36.52 wave lengths at 750MHz. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...</div> Zaki File:Hfa2.jpg 2011-10-19T23:12:29Z <p>Zaki: </p> <hr /> <div></div> Zaki File:Corr.jpg 2011-10-19T23:11:57Z <p>Zaki: </p> <hr /> <div></div> Zaki Homemade Interferometer 2011-10-19T01:44:18Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had an east-west baseline of 575 inches = 36.52 wave lengths at 750MHz. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...</div> Zaki Homemade Interferometer 2011-10-19T01:41:44Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. We had a baseline of 575 inches. The signal path of one antenna was as follows:<br /> dipole antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...</div> Zaki Homemade Interferometer 2011-10-19T01:18:46Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011 during the afternoon. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. The signal path of one antenna was as follows:<br /> antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds). The power levels into the ADCs were ~20.5 dbm with this set up (measured with the HP power meter). Signal generator clock was set to 800MHz +6.00 dbm. <br /> <br /> This deployment went great! We saw fringes! Although Lots of RFI is still present. More to come...</div> Zaki Homemade Interferometer 2011-10-19T01:15:59Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}<br /> The third deployment was done at Leuschner observatory on October 18th 2011. Again the signal path for each antenna was changed, as well as the correlator design from the previous deployments. The signal path of one antenna was as follows:<br /> antenna-&gt;3ft sma with ~90db shielding-&gt; AP1309C Cougar amp (12.5 db gain NF=2.5) -&gt; AC3064 Cougar amp (18.5 db gain NF~3.5) -&gt; 40ft LMR400 cable -&gt; 530-730 MHz bandpass filter -&gt; AC1309 Cougar amp (13.5db gain NF~4or 5) -&gt; AC1309 Cougar amp -&gt; 6ft sma (90db shielding) cable into iADC's. Power was supplied to antennas in the usual fashion (CAT5e cable). <br /> Correlator details: running baopoco design from october14th. equalization coefficients set to 500. int time set to default (2**28 --&gt; 1.3 seconds).<br /> This deployment went great! We saw fringes! More to come...</div> Zaki Homemade Interferometer 2011-10-17T23:45:03Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> All of the amplifiers are powered by a 15V-5A linear power supply. Power was supplied to the amplifiers at the antennas with 50ft Belden CAT5e cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{Test Equipment}<br /> The test equipment we used to verify power levels and spectra were:<br /> <br /> HP Power Meter with Low Frequency power head.<br /> HP Spectrum Analyzer<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T23:05:11Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Currently, ADT4-1WT 2-775MHz 50 ohm 1:4 balun SMT are being used. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T22:58:41Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> The output from the amplifier chassis is connected to the [https://casper.berkeley.edu/wiki/IADC iADCs] inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T22:52:41Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> The output from the amplifier chassis is connected to the iADCs inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection*{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T22:52:00Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. The baluns are placed on a baun test board. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR400 cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> The output from the amplifier chassis is connected to the iADCs inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> Note: This is the default setup. Many changes were made in the first few deployments to get the levels right. <br /> <br /> \subsection{The Correlator}<br /> The correlator was built with [https://casper.berkeley.edu/ CASPER] architecture. Specifically, we used a 4 input ROACH correlator, known as a Pocket Correlator. This correlator took advantage of iADC's which can be sampled up to 1GHz with two inputs or 2 GHz with one input (interleave mode). The digitized signal then goes through a Polyphase Filter Banks (PFB) which consists of an FIR filter and an FFT. Then the signals are quantized to 8bits (4bits real and 4 bits imaginary). Finally, all the signals are cross multiplied to give the cross correlations. <br /> <br /> The scripts used to collect the cross correlations and initialize the correlator can be found [here].<br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB (Note that this doesn't take into account RFI and other foreign signals). This was due to all the RFI present and the Sun. We underestimated the RFI environment. We removed all but one 13.5 dB gain amplifier for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands. <br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T22:38:18Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards (Prototyping Products 521.<br /> 12x12&quot; single sided copper board;[http://www.mgchemicals.com/products/500.html 590-521]). The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are used to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 rms-counts into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline (however they can be set up any way you want, this is the easiest). Each antenna is propped up onto a ground screen ( a giant metal mesh plane), which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> The output from the amplifier chassis is connected to the iADCs inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB. This was likely due to more RFI present during the day, and possibly the Sun(??). We removed all but one 13.5 dB gain amplifier (I THINK WE TRIED LEAVING THE 1309 BY ITSELF TOO?), for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands.<br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Homemade Interferometer 2011-10-17T22:26:20Z <p>Zaki: /* Reference Material */</p> <hr /> <div>===Reference Material===<br /> &lt;latex&gt;<br /> \documentclass[11pt]{article}<br /> \usepackage{graphicx}<br /> \usepackage{amsmath}<br /> \usepackage{fullpage}<br /> \begin{document}<br /> <br /> \subsection*{The antenna}<br /> The interferometer consists of two dipole antennas. The dipoles are made out of single sided copper circuit boards. The dielectric on these boards was FR4 with a dielectric constant of K=4.3. They were milled so that only 4 strips of copper remained (each being ~5''). These formed two perpendicular dipoles (one for each polarization). <br /> The dipoles are tuned to pick up 750MHz (in the middle of the ultra high frequency (UHF) band). <br /> Above and below the dipole plane are two circular metal sheets or radius ~2.5''. These sheets are use to increase the frequency response of the dipoles by a few hundred megahertz. <br /> <br /> \subsection*{The Signal Path}<br /> The interferometer consists of two dipole antennas; here we describe the signal path for one of the two polarizations available to each antenna.<br /> <br /> The signal path begins at the sky, whose temperature at 750 MHz is approximately 10 K (obtained through models of the sky), which corresponds to a power of approximately -95dBm (P=k*T*BW). This initial -95 dBm needs to be increased to approximately -22 dBm, which is the input power that the analog-to-digital converters (ADC) need to output 16 counts rms into the correlator. This is the optimum value we want going into the correlator. <br /> <br /> The antennas are set up with an East-West baseline of 259 inches (approx. 16.5 wavelengths DOUBLE CHECK THIS NUMBER) . Each antenna is propped up onto a ground screen, which has a circle cut-out in the middle to allow the signal-path cables to go through. The ground screen is used to reflect away anything coming up from the ground, as well as the sidelobes characteristic to dipole antennas. <br /> <br /> The signal is first run through a [http://en.wikipedia.org/wiki/Balun balun], which turns a differential signal into a single-ended signal. Next comes a series of amplifiers. The first is a 12.5 dB gain amp, with a [http://en.wikipedia.org/wiki/Noise_figure noise figure] of 2.5, connected directly to the balun. Following this first amp are two 18.5 dB gain amps (NF = 3.5 each), which are connected by a 40-foot LMR cable to a chassy which contains a 530-730 MHz filter, and three 13.5 dB gain amps (NF = 4.5 each). The three amps and the filter are connected in series by hand conformable SMA cables. <br /> <br /> <br /> The output of the chassis is connected to the iADCs inside the ROACH board. Lastly, the iADC is also connected to a signal generator that outputs a 800 MHz sine wave. This is needed to sample the signal coming in at 800MHz, which corresponds to a bandwidth of 400 MHz. Note that since our bandpass filters go from 530-730 MHz, we are looking at the second Nyquist band (400 - 800MHz). <br /> <br /> <br /> <br /> \subsection*{First Trial-HFA}<br /> The first trial was run at 6pm on 10/11/2011. Unfortunately, we did not observe any fringes. After double-checking all connections, the auto-correlations did not show any convincing passbands.<br /> <br /> \subsection*{Secong Trial-HFA}<br /> The second trial was run at approximately 2pm on 10/12/2011. The signal path had to be changed from that described above, since we found that the sky signal power had significantly increased from the standard -95dB. This was likely due to more RFI present during the day, and possibly the Sun(??). We removed all but one 13.5 dB gain amplifier (I THINK WE TRIED LEAVING THE 1309 BY ITSELF TOO?), for each antenna. Although this increased the power to an appropriate level for our iADCs, we still did not detect any fringes, and the auto-correlations for each antenna again did not show convincing passbands.<br /> <br /> \subsection*{Third Trial-Leuschner}</div> Zaki Coordinates 2011-10-05T22:21:44Z <p>Zaki: Created page with '===Short Topical Videos=== * [http://www.youtube.com/watch?v=dxZy_PR4_zM Coordinate Systems (by Zaki Ali)]'</p> <hr /> <div>===Short Topical Videos===<br /> * [http://www.youtube.com/watch?v=dxZy_PR4_zM Coordinate Systems (by Zaki Ali)]</div> Zaki Units of radiation 2011-10-05T21:54:25Z <p>Zaki: </p> <hr /> <div>===Short Topical Videos===<br /> * [http://www.youtube.com/watch?v=n2wFsybPIes Units of Radiation (by Zaki Ali)]<br /> <br /> ==Main Questions==<br /> * What is the point of talking about specific intensity?<br /> * Why use the Jansky? What does flux density mean?<br /> * Which quantities depend upon observing conditions and which are intrinsic to the source?<br /> <br /> ==Resources==<br /> The NRAO Essential Radio Astronomy course explains this pretty well in their chapter on [http://www.cv.nrao.edu/course/astr534/Brightness.html Brightness].</div> Zaki