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2_psd.tex
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\documentclass[aspectratio=169]{beamer}
%\documentclass[aspectratio=169,handout]{beamer}
\usepackage{../styles/dspmooc}
\usepackage{pst-3dplot}
%\setbeameroption{show only notes}\def\logoEPFL{}
\setbeameroption{show notes}
\begin{document}
\begin{frame} \frametitle{A simple discrete-time random signal generator}
For each new sample, toss a fair coin:
\[
x[n] = \begin{cases}
+1 & \mbox{if the outcome of the $n$-th toss is head} \\
-1 & \mbox{if the outcome of the $n$-th toss is tail}
\end{cases}
\]
\vspace{1em}
\begin{itemize}
\item each sample is independent from all others
\item each sample value has a 50\% probability
\end{itemize}
\end{frame}
\begin{frame} \frametitle{A simple discrete-time random signal generator}
\centering
every time we turn on the generator we obtain a different {\em realization}\/ of the signal
\vspace{2em}
\begin{figure}
\begin{dspPlot}[height=3cm]{0, 32}{-1.3, 1.3}
\moocStyle
\only<1>{\dspSignal{\dspRand 0 ge {1} {-1} ifelse}}
\only<2>{\dspSignal{\dspRand 0 ge {1} {-1} ifelse}}
\only<3>{\dspSignal{\dspRand 0 ge {1} {-1} ifelse}}
\only<4>{\dspSignal{\dspRand 0 ge {1} {-1} ifelse}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{A simple discrete-time random signal generator}
\begin{itemize}
\item every time we turn on the generator we obtain a different {\em realization}\/ of the signal
\item we know the ``mechanism'' behind each instance
\item but how can we analyze a random signal? What about its frequency content?
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Spectral properties?}
%% Using script flipSpec.m for generating the data
\centering
let's try with the DFT of a finite set of random samples
\vspace{2em}
\setbeamercovered{invisible}
\begin{figure}
\begin{dspPlot}[height=3cm,ylabel={$|X[k]|^2$}]{0, 32}{0, 100}
\moocStyle
\only<1>{\dspTapsAt{0}{64.0000 62.2679 17.8387 37.6703 8.0000 15.0515 53.5022 2.4645 16.0000 29.4352 83.0663 1.0156 8.0000 74.2626 5.5928 33.8324 64.0000 33.8324 5.5928 74.2626 8.0000 1.0156 83.0663 29.4352 16.0000 2.4645 53.5022 15.0515 8.0000 37.6703 17.8387 62.2679}}
\only<2>{\dspTapsAt{0}{4.0000 41.8675 59.1772 1.4475 23.3137 46.9278 52.2640 14.0436 20.0000 48.2468 1.7949 90.3235 0.6863 5.3013 62.7639 39.8422 4.0000 39.8422 62.7639 5.3013 0.6863 90.3235 1.7949 48.2468 20.0000 14.0436 52.2640 46.9278 23.3137 1.4475 59.1772 41.8675}}
\only<3>{\dspTapsAt{0}{64.0000 31.1581 43.0615 0.0013 29.6569 50.1169 32.6090 6.7854 8.0000 65.8737 47.3910 34.7881 18.3431 43.0937 36.9385 24.1828 16.0000 24.1828 36.9385 43.0937 18.3431 34.7881 47.3910 65.8737 8.0000 6.7854 32.6090 50.1169 29.6569 0.0013 43.0615 31.1581}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Spectral properties?}
\centering
every time it's different; maybe with more data?
\end{frame}
\begin{frame} \frametitle{Spectral properties?}
%% Using script flipSpec.m for generating the data
\centering
DFT of an increasing number of samples
\vspace{2em}
\begin{figure}
\only<1>{%
\begin{dspPlot}[height=3cm,ylabel={$|X[k]|^2$}]{0, 32}{0, 100}
\moocStyle
\dspTapsAt{0}{64.0000 62.2679 17.8387 37.6703 8.0000 15.0515 53.5022 2.4645 16.0000 29.4352 83.0663 1.0156 8.0000 74.2626 5.5928 33.8324 64.0000 33.8324 5.5928 74.2626 8.0000 1.0156 83.0663 29.4352 16.0000 2.4645 53.5022 15.0515 8.0000 37.6703 17.8387 62.2679}
\end{dspPlot}}
\only<2>{%
\begin{dspPlot}[height=3cm,ylabel={$|X[k]|^2$}]{0, 64}{0, 400}
\moocStyle
\dspTapsAt{0}{16.0000 3.6797 81.3613 139.6348 85.5641 165.5380 67.2710 5.2560 54.6274 76.8998 18.8899 47.6250 68.2459 20.5414 39.1286 72.1031 16.0000 341.1534 26.9212 79.4058 43.7541 3.8631 20.8038 50.1676 9.3726 248.6221 139.6628 0.2183 26.4359 2.1542 53.9615 23.1377 16.0000 23.1377 53.9615 2.1542 26.4359 0.2183 139.6628 248.6221 9.3726 50.1676 20.8038 3.8631 43.7541 79.4058 26.9212 341.1534 16.0000 72.1031 39.1286 20.5414 68.2459 47.6250 18.8899 76.8998 54.6274 5.2560 67.2710 165.5380 85.5641 139.6348 81.3613 3.6797}
\end{dspPlot}}
\only<3->{%
\begin{dspPlot}[height=3cm,ylabel={$|X[k]|^2$}]{0, 128}{0, 800}
\moocStyle
\dspTapsAt{0}{196.0000 103.9152 12.0329 5.2858 399.1202 238.1588 19.9432 72.8450 140.4164 44.2936 149.4802 145.6537 117.6417 1.4158 82.1600 0.2776 78.7452 131.5514 130.5073 50.9976 123.7012 99.8657 101.1329 218.0622 499.4595 32.9301 31.5729 16.3724 54.9893 521.6938 230.2043 501.9915 292.0000 64.4835 49.7240 16.2963 103.3038 114.0325 100.3522 171.6879 32.9913 129.9932 49.0327 11.4079 6.0399 260.6908 223.8790 87.1789 169.2548 127.5767 140.2200 17.7706 331.3623 225.6281 26.8281 45.1971 143.1328 75.1143 76.6386 126.5341 111.8415 132.1963 48.2919 176.9017 100.0000 176.9017 48.2919 132.1963 111.8415 126.5341 76.6386 75.1143 143.1328 45.1971 26.8281 225.6281 331.3623 17.7706 140.2200 127.5767 169.2548 87.1789 223.8790 260.6908 6.0399 11.4079 49.0327 129.9932 32.9913 171.6879 100.3522 114.0325 103.3038 16.2963 49.7240 64.4835 292.0000 501.9915 230.2043 521.6938 54.9893 16.3724 31.5729 32.9301 499.4595 218.0622 101.1329 99.8657 123.7012 50.9976 130.5073 131.5514 78.7452 0.2776 82.1600 1.4158 117.6417 145.6537 149.4802 44.2936 140.4164 72.8450 19.9432 238.1588 399.1202 5.2858 12.0329 103.9152}
\end{dspPlot}}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Averaging}
\begin{itemize}[<+->]
\item DFTs of realizations show no clear pattern... we need a new strategy
\item when faced with random data an intuitive response is to take ``averages'' (i.e. expectation)
\item for the coin-toss signal:
\[
\expt{x[n]} = -1\cdot P[\mbox{n-th toss is tail}] + 1\cdot P[\mbox{n-th toss is head}] = 0
\]
\item so the average value for each sample is zero...
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Averaging the DFT}
\begin{itemize}[<+->]
\item but the DFT is linear so $\expt{\DFT{x[n]}} = \DFT{\expt{x[n]}} = 0$
\item however the signal ``moves'', so its energy or power must be nonzero
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Energy and power}
\begin{itemize}
\item the coin-toss process produces realizations with infinite energy:
\[
E_{x} = \lim_{N\rightarrow\infty}\sum_{n=-N}^{N}|\breve{x}[n]|^2 = \lim_{N\rightarrow\infty} (2N+1) = \infty
\]
\item which, however, have has finite \textit{power}:
\[
P_{x} = \lim_{N\rightarrow\infty} \frac{1}{2N+1}\sum_{n=-N}^{N} |\breve{x}[n]|^2 = 1
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Averaging}
let's try to average the DFT's square magnitude, normalized:
\begin{itemize}[<+->]
\item pick an interval length $N$
\item pick a number of iterations $M$
\item run the signal generator $M$ times and obtain $M$ $N$-point realizations
\item compute the DFT of each realization
\item average their square magnitude divided by $N$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Averaged DFT square magnitude}
\begin{center}
\begin{figure}
\begin{dspPlot}{0,32}{0,2.6}
\moocStyle
\only<1>{%
\dspText(28,2.3){$M=1$}
\dspTapsAt{0}{2.0000 1.2263 1.3266 0.0859 0.4393 0.0259 0.8647 1.4264 1.0000 0.1274 1.1353 1.0376 2.5607 0.4364 0.6734 1.6340 2.0000 1.6340 0.6734 0.4364 2.5607 1.0376 1.1353 0.1274 1.0000 1.4264 0.8647 0.0259 0.4393 0.0859 1.3266 1.2263}}
\only<2>{%
\dspText(28,2.3){$M=10$}
\dspTapsAt{0}{0.8250 1.6433 1.1168 0.9961 0.5452 0.9240 0.8540 1.4178 0.5500 1.0745 0.5399 1.3725 1.0048 1.1538 0.4892 1.0179 1.7750 1.0179 0.4892 1.1538 1.0048 1.3725 0.5399 1.0745 0.5500 1.4178 0.8540 0.9240 0.5452 0.9961 1.1168 1.6433}}
\only<3>{%
\dspText(28,2.3){$M=1000$}
\dspTapsAt{0}{0.9394 0.9929 0.9606 0.9819 0.9769 0.9358 0.9567 1.0096 1.0049 0.9951 1.0083 0.9810 1.0639 1.0595 1.0539 1.0512 0.9964 1.0512 1.0539 1.0595 1.0639 0.9810 1.0083 0.9951 1.0049 1.0096 0.9567 0.9358 0.9769 0.9819 0.9606 0.9929}}
\only<4->{%
\dspText(28,2.3){$M=5000$}
\dspTapsAt{0}{0.9673 1.0021 0.9745 1.0184 0.9908 0.9739 0.9863 0.9771 0.9960 0.9855 1.0135 1.0083 1.0203 1.0277 1.0172 1.0165 1.0166 1.0165 1.0172 1.0277 1.0203 1.0083 1.0135 0.9855 0.9960 0.9771 0.9863 0.9739 0.9908 1.0184 0.9745 1.0021}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Power spectral density}
\[
P[k] = \expt{|X_N[k]|^2/N}
\]
\vspace{1em}
\begin{itemize}[<+->]
\item it looks very much as if $P[k] = 1$
\item if $|X_N[k]|^2$ tends to the {\em energy} distribution in frequency...
\item ...$|X_N[k]|^2/N$ tends to the {\em power} distribution (aka \emph{density}) in frequency
\item the frequency-domain representation for stochastic processes is the power spectral density
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Power spectral density: intuition}
\begin{itemize}[<+->]
\item $P[k] = 1$ means that the power is equally distributed over all frequencies
\item i.e., we cannot predict if the signal moves ``slowly'' or ``super-fast''
\item this is because each sample is independent of each other: we could have a realization of all ones or a realization in which the sign changes every other sample or anything in between
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Filtering a random process}
\begin{itemize}[<+->]
\item let's filter the random process with a 2-point Moving Average filter
\item $y[n] = (x[n] + x[n-1])/2$
\item what is the power spectral density?
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Averaged DFT magnitude of filtered process}
\begin{center}
\begin{figure}
\begin{dspPlot}{0,32}{0,1.5}
\moocStyle
\only<1>{%
\dspText(28,1.3){$M=1$}
\dspTapsAt{0}{0.0000 1.0030 0.7074 1.0655 0.6402 0.0828 0.7157 0.5397 1.2500 0.5401 0.0343 0.2211 0.1098 0.0270 0.0426 0.0207 0.0000 0.0207 0.0426 0.0270 0.1098 0.2211 0.0343 0.5401 1.2500 0.5397 0.7157 0.0828 0.6402 1.0655 0.7074 1.0030}}
\only<2>{%
\dspText(28,1.4){$M=10$}
\dspTapsAt{0}{0.7562 1.3015 0.5172 0.9269 1.0265 0.8844 0.8881 0.6273 0.4625 0.5450 0.3602 0.2185 0.0985 0.0757 0.0594 0.0207 0.0187 0.0207 0.0594 0.0757 0.0985 0.2185 0.3602 0.5450 0.4625 0.6273 0.8881 0.8844 1.0265 0.9269 0.5172 1.3015}}
\only<4->{%
\dspFunc[linestyle=dashed,linewidth=4pt,linecolor=blue!50,xmax=31]{x 32 div 360 mul cos 1 add abs 0.5 mul}
\dspLegend(0,1.3){blue!50 $|(1+e^{-j(2\pi/N)k})/2|^2$}}
\only<3->{%
\dspText(28,1.3){$M=5000$}
\dspTapsAt{0}{0.9921 0.9656 0.9443 0.8946 0.8352 0.7845 0.6834 0.5931 0.5119 0.4102 0.3159 0.2307 0.1545 0.0987 0.0523 0.0255 0.0160 0.0255 0.0523 0.0987 0.1545 0.2307 0.3159 0.4102 0.5119 0.5931 0.6834 0.7845 0.8352 0.8946 0.9443 0.9656}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Filtering a random process}
\begin{itemize}[<+->]
\item it looks like $P_y[k] = P_x[k]\,|H[k]|^2$, where $H[k] = \DFT{h[n]}$
\item can we generalize these results beyond a finite set of samples?
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Energy and Power Signals}
\begin{itemize}
\item energy signals: $\displaystyle \sum_{n=-\infty}^{\infty}|x[n]|^2 < \infty$
\item power signals: $\displaystyle \lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2 < \infty$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Energy Signals}
\begin{itemize}
\item finite support, $\sinc(n)$, $\alpha^n\,u[n]$ for $|\alpha| < 1$, ...
\item DTFT is well defined
\item DTFT square magnitude is \textit{energy} distribution in frequency
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Power Signals}
\begin{itemize}
\item $x[n] = 1$, $u[n]$, $e^{j\omega n}$, $\sin, \cos$, ...
\item DTFT uses the Dirac delta formalism
\item ``DTFT square magnitude'' doesn't make sense!
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Power Spectral Density}
Consider a truncated DTFT
\[
X_N(e^{j\omega}) = \sum_{n = -N}^{N} x[n]e^{-j\omega n}
\]
\vspace{1em}
\pause
define the power spectral density of a signal as:
\[
P(e^{j\omega}) = \lim_{N\rightarrow\infty}\,\frac{1}{2N+1}|X_N(e^{j\omega})|^2
\]
\end{frame}
\begin{frame} \frametitle{Power Spectral Density}
Examples:
\begin{itemize}[<+->]
\item $x[n] = a$, $P_x(e^{j\omega}) = a^2\tilde{\delta}(\omega)$
\item $x[n] = ae^{j\sigma n}$, $P_x(e^{j\omega}) = a^2\tilde{\delta}(\omega - \sigma)$
\item $x[n] = au[n]$, $P_x(e^{j\omega}) = (a^2/2)\tilde{\delta}(\omega)$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\centering
For a random process
\[
P_x(e^{j\omega}) = \lim_{N\rightarrow\infty}\displaystyle\frac{1}{2N+1}\expt{|X_N(e^{j\omega})|^2 }
\]
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\begin{align*}
\expt{\left|\sum_{n = -N}^{N} x[n]e^{-j\omega n}\right|^2} &= \expt{\displaystyle \sum_{n = -N}^{N} x[n]e^{j\omega n}\sum_{m = -N}^{N} x[m]e^{-j\omega m}} \\
\only<2->{&= \sum_{n = -N}^{N}\sum_{m = -N}^{N}\expt{x[n]x[m]}e^{-j\omega (m-n)} \\ }
\only<3->{&= \sum_{n = -N}^{N}\sum_{m = -N}^{N}\hlBox[\sum_0^0]{wss}{blue!30}{r_x[m-n]}\ e^{-j\omega (m-n)}}
\end{align*}
\only<3->{\begin{flushright} \rnode{B}{\color{darkred} \parbox{4cm}{WSS}} \end{flushright}
\nccurve[linecolor=darkred,angleA=180,angleB=-90]{->}{B}{wss}}
\end{frame}
\begin{frame} \frametitle{A clever manipulation}
\[
S = \sum_{m = -N}^{N} \sum_{n = -N}^{N} f(m-n)
\]
\vspace{1em}
\pause
\[
-2N \leq (m-n) \leq 2N
\]
\vspace{1em}
\pause
\[
S = \sum_{k = -2N}^{2N} c(k)f(k)
\]
\end{frame}
\def\sp{ }
\def\btick#1{\psline[linewidth=0.6pt](#1,-.1)(#1,.1)}
\def\qtick#1#2{\btick{#1}\uput{10pt}[-90](#1,0){#2}}
\def\qseg#1#2{%
\color{darkred}
\psline[linecolor=darkred]{|-|}(#1,0.2)(! #1\sp #2\sp add 0.2)%
\uput{10pt}[90](#1,0.2){$n$}\uput{10pt}[90](! #1\sp #2\sp add 0.2){$m$}}%
\begin{frame} \frametitle{A clever manipulation}
\centering
\only<1>{$c(k)$: number of ways we can pick $n, m$ in $[-N,N]$ so that $(m-n) = k$}
\only<2->{geometrically: $c(k) = $ number of ways we can fit a segment of length $k$ over $[-N,N]$:
\begin{figure}
\center
\psset{xunit=2.4cm,yunit=2cm}
\begin{pspicture}(0.5,-1)(5.5,1)
\psline[linewidth=1pt,tickwidth=2pt,](1,0)(5,0)%
\qtick{1}{$-N$}\btick{2}\btick{3}\btick{4}\qtick{5}{$N$}%
\only<3>{\qseg{1}{4}}%
\only<4>{\qseg{1}{3}}%
\only<5>{\qseg{2}{3}}%
\only<6>{\qseg{1}{2}}%
\only<7>{\qseg{2}{2}}%
\only<8>{\qseg{3}{2}}%
\only<9>{\qseg{1}{1}}%
\end{pspicture}
\end{figure}}
\end{frame}
\begin{frame} \frametitle{A clever manipulation}
\centering
$c(k)$: number of ways we can pick $n, m$ in $[-N,N]$ so that $(m-n) = k$
\vspace{2em}
\[
c(k) = 2N+1 - |k|
\]
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\begin{align*}
\expt{\left|\sum_{n = -N}^{N} x[n]e^{-j\omega n}\right|^2} &= \sum_{k = -2N}^{2N}(2N+1-|k|)\ r_x[k]e^{-j\omega k}
\end{align*}
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\begin{align*}
P_x(e^{j\omega}) &= \lim_{N\rightarrow\infty} \sum_{k = -2N}^{2N} \left(\frac{2N+1-|k|}{2N + 1}\right)\ \left( r_X[k]e^{-j\omega k}\right) \nonumber \\
&= \lim_{N\rightarrow\infty} \sum_{k = -2N}^{2N} w_N[k] r_x[k]e^{-j\omega k}
\end{align*}
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\center
$w_N[k]$
\begin{figure}
\begin{dspPlot}[yticks=1,sidegap=0,height=5cm]{-40,40}{0,1.2}
\psclip{\psframe[linestyle=none](-40,0)(40,1.2)}
\only<1>{\dspFunc[linecolor=red,linewidth=2pt]{1 x abs 2 5 mul 1 add div sub }%
\uput[0](10,0.2){$N=5$}}%
\only<2->{\dspFunc[linecolor=gray,linewidth=2pt]{1 x abs 2 5 mul 1 add div sub }%
\uput[0](10,0.2){$N=5$}}%
\only<2>{\dspFunc[linecolor=red,linewidth=2pt]{1 x abs 2 20 mul 1 add div sub }%
\uput[0](22,0.5){$N=20$}}%
\only<3->{\dspFunc[linecolor=gray,linewidth=2pt]{1 x abs 2 20 mul 1 add div sub }%
\uput[0](22,0.5){$N=20$}}%
\only<3>{\dspFunc[linecolor=red,linewidth=2pt]{1 x abs 2 500 mul 1 add div sub }%
\uput[0](22,0.85){$N=500$}}%
\endpsclip
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\[
\lim_{N\rightarrow\infty} w_N[k] = 1
\]
\end{frame}
\begin{frame} \frametitle{Power Spectral Density for WSS Processes}
\begin{align*}
P_x(e^{j\omega}) &= \lim_{N\rightarrow\infty} \sum_{k = -2N}^{2N} w_N[k] r_x[k]e^{-j\omega k} \\
&= \sum_{k = -\infty}^{\infty}r_X[k]e^{-j\omega k} \\
&= \mbox{DTFT}\{r_x[k]\}
\end{align*}
\end{frame}
\begin{frame} \frametitle{Noise}
\begin{itemize}[<+->]
\item noise is everywhere:
\begin{itemize}
\item thermal noise
\item sum of extraneous interferences
\item quantization and numerical errors
\item ...
\end{itemize}
\item we can model noise as a stochastic signal
\item the most important noise is white noise
\end{itemize}
\end{frame}
\begin{frame} \frametitle{PSD of white noise}
White noise:
\begin{itemize}
\item $m = 0$
\item $r[k] = \sigma^2 \delta[k]$
\end{itemize}
\pause
\begin{align*}
P(e^{j\omega}) &= \sigma^2
\end{align*}
\end{frame}
\begin{frame} \frametitle{PSD of white noise}
\begin{figure}
\begin{dspPlot}[xtype=freq,yticks=custom,ylabel={$P_w(e^{j\omega})$}]{-1,1}{0,1}
\moocStyle
\dspFunc{0.7}
\dspCustomTicks[axis=y]{0.7 $\sigma^2$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{White noise}
\begin{itemize}[<+->]
\item the PSD is independent of the probability distribution of the single samples (depends only on the variance)
\item distribution is important to estimate bounds for the signal
\item very often a Gaussian distribution models the experimental data the best
\item AWGN: additive white Gaussian noise
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Filtering a Random Process}
\center
\begin{figure}
\begin{dspBlocks}{1cm}{1cm}
$x[n]$~ & \BDfilter{$h[n]$} & ~$y[n]$
\BDConnH{1}{1}{2}{}
\BDConnH{1}{2}{3}{}
\end{dspBlocks}
\end{figure}
\vspace{2em}
\begin{itemize}[<+->]
\item is $y[n]$ a random process?
\item if $x[n]$ WSS, is $y[n]$ WSS?
\item what are $m_y$ and $r_y[n]$ ?
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Mean of the Filtered Process}
\begin{align*}
m_{y[n]} & = \expt{y[n]} = \expt{\displaystyle \sum_{k = -\infty}^{\infty} h[k] x[n - k]} \\
& = \sum_{k = -\infty}^{\infty} h[k] \expt{x[n - k]} \\
& = \sum_{k = -\infty}^{\infty} h[k] m_x \qquad \mbox{($x[n]$ is WSS)}\\
& = m_x \sum_{k =-\infty}^{\infty}h[k] \\
& = m_x H(e^{j0})
\end{align*}
\end{frame}
\begin{frame} \frametitle{Autocorrelation of the Filtered Process}
\begin{align*}
\expt{y[n]y[m]} &= \expt{\displaystyle \sum_{k = -\infty}^{\infty} h[k] x[n - k] \, \sum_{i = -\infty}^{\infty} h[i] x[m - i] }\\
&= \sum_{k = -\infty}^{\infty}\sum_{i = -\infty}^{\infty} h[k]h[i] \expt{x[n - k]x[m - i]} \\
&= \sum_{k = -\infty}^{\infty}\sum_{i = -\infty}^{\infty} h[k]h[i] r_x[(n - m) - k + i]
\end{align*}
\centering
output depends only on lag $(n-m) \longrightarrow y[n]$ is WSS
\end{frame}
\begin{frame} \frametitle{Fundamental Result}
\centering
with a change of variable in the double sum:
\[
r_y[n] = h[n] \ast h[-n] \ast r_x[n]
\]
\vspace{2em}
so that:
\[
P_y (e^{j\omega})\ =\ \left| H(e^{j\omega}) \right|^2 P_x(e^{j\omega})
\]
\vspace{2em}
Deterministic filters can be used to shape the power distribution of WSS random processes
\end{frame}
\begin{frame} \frametitle{Stochastic signal processing}
key points:
\begin{itemize}[<+->]
\item filters designed for deterministic signals still work (in magnitude) in the stochastic case
\item we lose the concept of phase since we don't know the shape of a realization in advance
\end{itemize}
\end{frame}
\end{document}

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