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4_DTFT.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame}
\frametitle{Overview:}
\begin{itemize}
\item DTFT Existence
\item Properties
\item DTFT as basis expansion
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Discrete-Time Fourier Transform}
\[
X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n}
\]
\begin{itemize}[<+->]
\item when does it exist?
\item is it a change of basis?
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Existence easy for absolutely summable sequences}
\note<1>{say abs summ implies finite energy\\ but not vice versa}
\begin{align*}
|X(e^{j\omega})| &= |\sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n}| \\ \pause
&\leq \sum_{n=-\infty}^{\infty} |x[n]\, e^{-j\omega n}| \\ \pause
&= \sum_{n=-\infty}^{\infty} |x[n]| \\ \pause
&< \infty
\end{align*}
\end{frame}
\begin{frame}
\frametitle{Inversion easy for absolutely summable sequences}
\begin{align*}
\frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega})\,e^{j\omega n}d\omega
&= \frac{1}{2\pi} \int_{-\pi}^{\pi} \left( \sum_{k=-\infty}^{\infty} x[k]\, e^{-j\omega k} \right)e^{j\omega n}d\omega \\ \pause
&= \sum_{k=-\infty}^{\infty} x[k] \int_{-\pi}^{\pi} \frac{e^{j\omega (n-k)}}{2\pi}d\omega \\ \pause
&= x[n]
\end{align*}
\end{frame}
%%%% 1st EDITION SLIDE FLOW %%%%%
%%%%%%%%%%% Intuition by periodization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\intertitle{the DTFT as the limit of the DFS}
\begin{frame} \frametitle{Synopsis}
\begin{itemize}[<+->]
\item $x[n]$ absolutely summable $\Rightarrow X(e^{j\omega})$ exists formally
\item $x[n]$ absolutely summable $\Rightarrow $ we can {\em periodize}\/ it into $\tilde{x}_N[n]$
\item natural Fourier representation for $\tilde{x}_N[n]$ is DFS
\item DFS of $\tilde{x}_N[n]$ turns out to be $X(e^{j\omega})$ at $\omega=(2\pi/N)k$
\item as $N$ grows to infinity $\tilde{x}_N[n]$ becomes $x[n]$
\item as $N$ grows to infinity natural Fourier representation becomes $X(e^{j\omega})$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Some intuition}
\centering
With $x[n]$ absolutely summable we can build arbitrarily ``periodized'' sequences:
\[
\tilde{x}_N[n] = \sum_{p=-\infty}^{\infty} x[n + pN]
\]
\vspace{2em}\pause
clearly $\tilde{x}_N[n] = \tilde{x}_N[n + N]$
\end{frame}
%% absolutely summable signal
\def\sig#1{dup \dspPorkpie{#1}{10} 0.2 mul exch #1 sub 2 div dup mul 1 add 1 exch div sub -1 mul }
%% periodization
\def\perSig#1{ 0 -80 #1.0 80 {/i exch def x \sig{i} add } for }
\begin{frame}
\frametitle{Periodization}
\begin{center}
\vphantom{Let $N$ grow large...}
\begin{figure}
\begin{dspPlot}[xticks=100,yticks=none,sidegap=0]{-50,50}{-.2, 1.2}
\moocStyle
\only<1-4>{\dspFunc{x \sig{0}}}
\only<2-5>{\dspFunc[linecolor=red!50]{x \sig{10}}}
\only<3-5>{\dspFunc[linecolor=red!50]{x \sig{20}}}
\only<4-5>{\dspFunc[linecolor=red!50]{x \sig{30}}}
\only<5-6>{\multido{\n=-80+10}{16}{\dspFunc[linecolor=red!50]{x \sig{\n}}}}
\only<6-7>{\dspFunc[linecolor=blue!70]{\perSig{10} }}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
%# widening periodization
\def\perSigSlide#1#2{\only<#1>{%
\multido{\n=-80+#2}{16}{\dspFunc[linecolor=red!50]{x \sig{\n}}}
\dspFunc[linecolor=blue!70]{0 -80 #2.0 80 {/i exch def x \sig{i} add } for }
\dspText(60,0.5){$N=#2$}}}
\begin{frame}
\frametitle{Periodization}
\begin{center}
\only<1-4>{Let $N$ grow large...}
\only<5->{... as $N$ grows, $\tilde{x}_N[n] \rightarrow x[n]$}
\begin{figure}
\begin{dspPlot}[xticks=100,yticks=none,sidegap=0]{-50,50}{-.2, 1.2}
\moocStyle
\perSigSlide{1}{10}
\perSigSlide{2}{15}
\perSigSlide{3}{20}
\perSigSlide{4}{40}
\perSigSlide{5}{80}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{From DFS to DTFT}
\note<2>{ remark that we can always add full multiples of $2\pi$\\
to a complex exponential in the last line \\ \vspace{2em}
also, remark on the exchange of the summations}
%
Natural spectral representation for $\tilde{x}_N[n]$ is the DFS:
\begin{align*}
\tilde{X}[k] &= \sum_{n=0}^{N-1} \tilde{x}_N[n] e^{-j\frac{2\pi}{N}nk} \\ \pause
&= \sum_{n=0}^{N-1} \sum_{p = -\infty}^{\infty} x[n +pN] e^{-j\frac{2\pi}{N}nk} \\ \pause
&= \sum_{p = -\infty}^{\infty} \sum_{n=0}^{N-1} x[n +pN] e^{-j\frac{2\pi}{N}(n+pN)k}
\end{align*}
\hspace{24em}\small (remember $e^{j\alpha} = e^{j(\alpha + 2K\pi)} \quad \forall K$)
\end{frame}
\begin{frame} \frametitle{Same trick we used before:}
\[
\sum_{p = -\infty}^{\infty} \sum_{n=0}^{N-1} y[n + pN] = \sum_{i = -\infty}^{\infty} f[i]
\]
\end{frame}
\begin{frame} \frametitle{Example (N=4)}
\centering
\Large
$p$~~
\begin{tabular}{c||c|c|c|c|}
\multicolumn{5}{c}{$n$} \\
& 0 & 1 & 2 & 3 \\ \hline\hline
& & & & \\ \hline
$\ldots$ & & & & \\ \hline
-1 & -4 & -3 & -2 & -1 \\ \hline
0 & 0 & 1 & 2 & 3 \\ \hline
1 & 4 & 5 & 6 & 7 \\ \hline
2 & 8 & 9 & 10 & 11 \\ \hline
$\ldots$ & & & & \\ \hline
& & & &
\end{tabular}
\hspace{3em}$i=4p+n$
\end{frame}
\begin{frame}
\frametitle{From DFS to DTFT}
\begin{align*}
\tilde{X}[k] &= \sum_{p = -\infty}^{\infty} \sum_{n=0}^{N-1} x[n +pN] e^{-j\frac{2\pi}{N}(n+pN)k} \\ \pause
&= \sum_{i = -\infty}^{\infty} x[i] e^{-j\frac{2\pi}{N}k\,i} \\ \pause
&= X(e^{j\omega})|_{\omega = \frac{2\pi}{N}k}
\end{align*}
\end{frame}
\begin{frame} \frametitle{From DFS to DTFT}
\begin{itemize}[<+->]
\item we're comfortable with DFS: change of basis, energy conservation, etc.
\item as $N$ grows, $\tilde{x}_N[n] \rightarrow x[n]$ and the spectral representation ``becomes'' the DTFT
\item we can retain the ``change of basis'' paradigm for the DTFT
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Review: DFT}
\note<1>{Idea now is to contextualize the next step\\ i.e. extending the basis-expansion paradigm\\ at least formally to the DTFT. To do so \\ we first recall the vector spaces involved \\ in the DFT and DFS and associated analysis \\
and synthesis formulas}
\begin{center}
\begin{figure}
\psset{linewidth=1pt}
\begin{dspBlocks}{4}{0.2}
\circlenode{A}{~~$\mathbb{C}^{N}$~~} & & \circlenode{B}{~~$\mathbb{C}^{N}$~~}
\end{dspBlocks}
\end{figure}
\ncarc[nodesep=3pt,arcangle=45]{->}{A}{B}\only<2->{\naput{$X[k] = \langle e^{j\frac{2\pi}{N}nk}, x[n] \rangle$}}
\ncarc[nodesep=3pt,arcangle=45]{->}{B}{A}\only<3->{\naput{$x[n] = (1/N)\sum X[k]\,e^{j\frac{2\pi}{N}nk}$}}
\ncline[linestyle=none]{A}{B}\only<4->{\ncput{basis: $\{e^{j\frac{2\pi}{N}nk}\}_k$}}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Review: DFS}
\begin{center}
\begin{figure}
\psset{linewidth=1pt}
\begin{dspBlocks}{4}{0.2}
\circlenode{A}{~~$\tilde{\mathbb{C}}^{N}$~~} & & \circlenode{B}{~~$\mathbb{C}^{N}$~~}
\end{dspBlocks}
\end{figure}
\ncarc[nodesep=3pt,arcangle=45]{->}{A}{B}\naput{$\tilde{X}[k] = \langle e^{j\frac{2\pi}{N}nk}, \tilde{x}[n] \rangle$}
\ncarc[nodesep=3pt,arcangle=45]{->}{B}{A}\naput{$\tilde{x}[n] = (1/N)\sum \tilde{X}[k]\,e^{j\frac{2\pi}{N}nk}$}
\ncline[linestyle=none]{A}{B}\ncput{basis: $\{e^{j\frac{2\pi}{N}nk}\}_k$}
\end{center}
\end{frame}
\begin{frame}
\frametitle{What about the DTFT?}
\begin{itemize}[<+->]
\item formally DTFT is an inner product in $\mathbb{C}^\infty$:
\[
\sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n} = \langle e^{j\omega n}, x[n] \rangle
\]
\item ``basis'' is an infinite, uncountable basis: $\{e^{j\omega n}\}_{\omega \in \mathbb{R}}$
\item something ``breaks down'': we start with sequences but the transform is a function
\item we used absolutely summable sequences but DTFT exists for all square-summable sequences (proof is rather technical)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{DTFT}
\begin{center}
\begin{figure}
\psset{linewidth=1pt}
\begin{dspBlocks}{4}{0.2}
\circlenode{A}{~~$\ell_2(\mathbb{Z})$~~} & & \circlenode{B}{$L_2([-\pi,\pi])$}
\end{dspBlocks}
\end{figure}
\ncarc[nodesep=3pt,arcangle=45]{->}{A}{B}\naput{$X(e^{j\omega}) = \langle e^{j\omega n}, x[n] \rangle$}
\ncarc[nodesep=3pt,arcangle=45]{->}{B}{A}\naput{$x[n] = (1/2\pi)\int X(e^{j\omega})e^{j\omega n}d\omega$}
\ncline[linestyle=none]{A}{B}\ncput{``basis'': $\{e^{j\omega n}\}_\omega$}
\end{center}
\end{frame}
\begin{frame}
\frametitle{DTFT properties}
\note<1>{if we look at the DTFT as a formal change\\ of basis, the following properties \\ are naturally derived}
\begin{itemize}[<+->]
\item linearity
\[
\mbox{DTFT}\{\alpha x[n] + \beta y[n]\} = \alpha X(e^{j\omega}) + \beta Y(e^{j\omega})
\]
\item time shift
\[
\mbox{DTFT}\{x[n-M]\} = e^{-j\omega M} X(e^{j\omega})
\]
\item modulation (dual)
\[
\mbox{DTFT}\{e^{j\omega_0 n}\,x[n]\} = X(e^{j(\omega-\omega_0)})
\]
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{DTFT properties}
\begin{itemize}[<+->]
\item time reversal
\[
\mbox{DTFT}\{x[-n]\} = X(e^{-j\omega})
\]
\item conjugation
\[
\mbox{DTFT}\{ x^*[n]\} = X^*(e^{-j\omega})
\]
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Some particular cases:}
\begin{itemize}[<+->]
\item if $x[n]$ is symmetric, the DTFT is symmetric:
\[
x[n] = x[-n] \Longleftrightarrow X(e^{j\omega}) = X(e^{-j\omega})
\]
\item if $x[n]$ is real, the DTFT is Hermitian-symmetric:
\[
x[n] = x^*[n] \Longleftrightarrow X(e^{j\omega}) = X^*(e^{-j\omega})
\]
\item in other words: if $x[n]$ is real, the magnitude of the DTFT is symmetric:
\[
x[n] \in \mathbb{R} \Longleftrightarrow |X(e^{j\omega})| = |X(e^{-j\omega})|
\]
\item finally, if $x[n]$ is real and symmetric, the DTFT is also real and symmetric!
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{DTFT as basis expansion}
Some things are OK:
\begin{itemize}[<+->]
\item $\DFT{\delta[n]} = 1$
\item $\DTFT{\delta[n]} = \langle e^{j\omega n}, \delta[n] \rangle = 1$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{DTFT as basis expansion}
Some things aren't:
\begin{itemize}[<+->]
\item $\DFT{1} = \delta[n]$
\item $\DTFT{1} = \sum_{n=-\infty}^{\infty} e^{-j\omega n} = ?$
\vspace{2em}
\item problem: too many interesting sequences are {\em not} square summable!
\end{itemize}
\end{frame}
\end{document}

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