1.17: z-Transform
- Page ID
- 126943
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives: z-transform
- Understand how the DT Fourier transform and z-transform are related.
- Compute the z-transform and its Region-of-Convergence (ROC) for signals such as decaying exponentials, finite rectangular pulses, and simple finite-length signals.
- Determine the poles and zeros using the z-transform of a signal
- Explain how time-domain properties relate to the ROC properties.
YouTube Videos
Big Picture for z-Transform
z-Transform Definition
z-Transform Example 1
z-Transform Example 2
Poles and Zeros
Region of Convergence Properties
Summary
Why the z-transform?
The z-transform is used when a signal does not have a Fourier transform. Whereas the DTFT (i.e., the Discrete-Time Fourier Transform) exists for signals with finite energy, the z-transform is a generalization of the DTFT, working for almost all DT signals.
\[ X(z)=\sum_{n=-\infty}^{\infty}x[n]\,z^{-n}. \]
The system function is defined as the z-transform of the impulse response in an LTI DT system
\[ H(z)=\sum_{n=-\infty}^{\infty}h[n]\,z^{-n}. \]
Connection to the DTFT: if \(z=e^{j\omega}\) (i.e., \(|z|=1\)), then \[ X(e^{j\omega})=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}=X(z)\Big|_{z=e^{j\omega}}. \] In words: the DTFT is the z-transform evaluated on the unit circle.
Complex-plane reminder: \(z=re^{j\theta}\). The unit circle is \(|z|=1\).
Signal: \(\;x_1[n]=\left(\dfrac{1}{3}\right)^n u[n]\)
\[ X_1(z)=\sum_{n=0}^{\infty}\left(\frac{1}{3}\right)^n z^{-n} =\sum_{n=0}^{\infty}\left(\frac{1}{3}z^{-1}\right)^n =\frac{1}{1-\frac{1}{3}z^{-1}}, \] using the geometric-series result \(\sum_{n=0}^{\infty}\alpha^n=\dfrac{1}{1-\alpha}\) for \(|\alpha|<1\).
ROC: \[ \left|\frac{1}{3}z^{-1}\right|<1 \quad\Longleftrightarrow\quad |z|>\frac{1}{3}. \]
Unit-circle evaluation (DTFT): \[ X_1(e^{j\omega})=X_1(z)\big|_{z=e^{j\omega}}=\frac{1}{1-\frac{1}{3}e^{-j\omega}}. \]
Signal: \(\;x_2[n]=-\left(\dfrac{1}{3}\right)^n u[-n-1]\)
(Left-sided signals typically produce an ROC inside a circle.)
\[ X_2(z)=\sum_{n=-\infty}^{-1}-\left(\frac{1}{3}\right)^n z^{-n} =-\frac{3z}{1-3z}. \]
ROC: \[ |3z|<1\quad\Longleftrightarrow\quad |z|<\frac{1}{3}. \]
Figure 1.17.1: Illustration of the Region of Convergence for the signals in Example1 1 (i.e., \(z>\frac{1}{3}\), see purple shadowed circle), and Example 2 (i.e., \(z<\frac{1}{3}\), see red shadowed circle).
How to estimate the poles and zeros using the ROC?
To find the poles and zeros, one should write the z-transform as a rational function: \[ X(z)=\frac{P(z)}{Q(z)}. \]
- Zeros: values of \(z\) where \(P(z)=0\) so \(X(z)=0\).
- Poles: values of \(z\) where \(Q(z)=0\) so \(X(z)\to\infty\).
Key ROC facts
- The ROC is bounded by poles and cannot include them (i.e., poles).
- If the ROC includes the unit circle, the DTFT exists, and the system is stable.
- If the ROC is outside the outermost pole, the system is right-sided, and it is causal.