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R2653/teaching/COM303/2021/lectures/week6/ztrans.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\intertitle{the $z$-transform}
\begin{frame} \frametitle{The most general LTI systems}
\begin{itemize}
\item ideal filters cannot be implemented
\item what is the most general, realizable LTI transformation?
\begin{itemize}
\item linearity: only sums and multiplications
\item time-invariance: only multiplications by constants
\item realizability: only finite number of past and future samples
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Constant-Coefficient Difference Equation}
\[
\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]
\]
\vspace{2em}
\begin{itemize}
\item uses $M+1$ input and $N$ output values
\item $a_0 = 1$ (otherwise renormalize)
\item completely specified by $M + N + 1$ scalar coefficients
\item causal formulation:
\[
y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The $z$-transform}
\[
X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}, \quad z \in \mathbb{C}
\]
\vspace{2em}
\begin{itemize}
\item linearity: $\mathcal{Z}\{\alpha x[n] + \beta y[n] \} = \alpha X(z) + \beta Y(z)$
\item time shift: $\mathcal{Z}\{x[n - N] \} = z^{-N} X(z)$
\item convolution: $\mathcal{Z}\{h[n] \ast x[n] \} = H(z) X(z)$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Rational Transfer Function}
\[
H(z) =
\frac{\hlBox{b}{green!30}{\displaystyle\sum_{k=0}^{M} b_k z^{-k}}}{\hlBox{a}{red!30}{\displaystyle 1 + \sum_{k=1}^{N} a_k z^{-k}}}
\]
\vspace{2em}
{
\begin{itemize}
\item feedforward part \rnode{B}{}
\item feedback part \rnode{A}{}
\end{itemize}
\nccurve[linecolor=darkred,angleA=0,angleB=180]{->}{B}{b}
\nccurve[linecolor=darkred,angleA=0,angleB=-90]{->}{A}{a}
}
\end{frame}
\begin{frame} \frametitle{Existence and region of convergence (ROC)}
\begin{center}
ROC is defined by the absolute convergence of the power series:
\end{center}
\[
z \in \mbox{ROC}\bigl\{H(z)\bigr\} \Longleftrightarrow \sum_{n=-\infty}^{\infty} \big| h[n]z^{-n} \big| < \infty
\]
\vspace{1em}
where are the convergence problems? \\
\vspace{1em}
in general, difficult question; but we're only interested in rational transfer functions!
\end{frame}
\begin{frame} \frametitle{ROC for rational transfer functions}
Consider the transfer function for an LTI system:
\[
H(z) = \frac{b_0 + b_1 z^{-1} + \ldots + b_{M}z^{-M}}{1 + a_1 z^{-1} + \ldots + a_{N}z^{-N}}
\]
It can always be factored as:
\[
H(z) = b_0\, \frac{\displaystyle \prod_{n = 1}^{M}(1 - z_n z^{-1})}{\displaystyle \prod_{n = 1}^{N}(1 - p_n z^{-1})}
\]
\end{frame}
\begin{frame} \frametitle{ROC for causal LTI systems}
\begin{itemize}
\item $z_n$'s: {\em zeros} of the transfer function
\item $p_n$'s: {\em poles} of the transfer function
\item only trouble spots for ROC are the poles
\end{itemize}
\end{frame}
\begin{frame} \frametitle{ROC for causal LTI systems}
We know:
\begin{itemize}
\item ROC has circular symmetry (see video lecture)
\item ROC of causal sequences extends outwards (see video lecture)
\item ROC cannot include poles
\end{itemize}
\vspace{2em}
\centering
{ROC extends outwards from a circle touching the largest-magnitude pole}
\end{frame}
\begin{frame}
\frametitle{ROC for causal systems}
\begin{figure}
\center
\begin{dspPZPlot}[width=5cm,circle=0,xticks=none,yticks=none]{1.8}
\dspPZ[label=none]{0.5,0.5}
\dspPZ[label=none]{0.5,-0.5}
\dspPZ[label=none]{0.2,0}
\dspPZ[type=zero,label=none]{0,1}
\dspPZ[type=zero,label=none]{0,-1}
\dspPZ[type=zero,label=none]{-.5,0}
\dspPZ[type=zero,label=none]{-1.25, 0.78}
\dspPZ[type=zero,label=none]{-1.25, -0.78}
\end{dspPZPlot}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{ROC for causal systems}
\begin{figure}
\center
\begin{dspPZPlot}[width=5cm,circle=0,xticks=none,yticks=none,roc=0.705]{1.8}
\moocStyle
\dspPZ[label=none]{0.5,0.5}
\dspPZ[label=none]{0.5,-0.5}
\dspPZ[label=none]{0.2,0}
\dspPZ[type=zero,label=none]{0,1}
\dspPZ[type=zero,label=none]{0,-1}
\dspPZ[type=zero,label=none]{-.5,0}
\dspPZ[type=zero,label=none]{-1.25, 0.78}
\dspPZ[type=zero,label=none]{-1.25, -0.78}
\end{dspPZPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{System stability criterion}
Consider a filter with impulse response $h[n]$
\begin{itemize}
\item BIBO stability $\Longleftrightarrow$ $\displaystyle \sum_{n=-\infty}^{\infty}|h[n]| < \infty$
\item $1\in\mbox{ROC} \Longleftrightarrow$ $\displaystyle \sum_{n=-\infty}^{\infty}|h[n]| < \infty \qquad$ because $H(z)$ converges absolutely in $z=1$
\end{itemize}
\vspace{2em}
\centering
{\bf an LTI system is stable if and only if the ROC includes the unit circle!}
\vspace{2em}
{\bf a \textit{causal} system is stable if and only if all poles are inside the unit circle}
\end{frame}
\begin{frame}
\frametitle{Stable system}
\begin{figure}
\center
\begin{dspPZPlot}[width=5cm,clabel={$1$},roc=0.705]{1.8}
\moocStyle
\dspPZ[label=none]{0.5,0.5}
\dspPZ[label=none]{0.5,-0.5}
\dspPZ[label=none]{0.2,0}
\dspPZ[type=zero,label=none]{0,1}
\dspPZ[type=zero,label=none]{0,-1}
\dspPZ[type=zero,label=none]{-.5,0}
\dspPZ[type=zero,label=none]{-1.25, 0.78}
\dspPZ[type=zero,label=none]{-1.25, -0.78}
\end{dspPZPlot}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Unstable system}
\begin{figure}
\center
\begin{dspPZPlot}[width=5cm,clabel={$1$},roc=1.2]{1.8}
\moocStyle
\dspPZ[label=none]{-1.2,0}
\dspPZ[label=none]{0.5,0.5}
\dspPZ[label=none]{0.5,-0.5}
\dspPZ[label=none]{0.2,0}
\dspPZ[type=zero,label=none]{0,1}
\dspPZ[type=zero,label=none]{0,-1}
\dspPZ[type=zero,label=none]{-.5,0}
\end{dspPZPlot}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{System stability}
given a causal filter with impulse response $h[n]$
\begin{itemize}
\item if FIR: always stable
\item if IIR: verify that poles are inside unit circle
\end{itemize}
\end{frame}
\end{document}
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