The $z$-Transform¶

This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Comunications Engineering, Universität Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.

Properties¶

The $z$-transform has a number of specific properties which can be concluded directly from its definition. The most important ones in the context of signals and systems are reviewed in the following.

Linearity¶

As can be concluded from its definition, the $z$-transform is a linear operation. For two signals $x_1[k]$ and $x_2[k]$ with $z$-transforms $X_1(z) = \mathcal{Z} \{ x_1[k] \}$ and $X_2(z) = \mathcal{Z} \{ x_2[k] \}$ the following holds

\begin{equation} \mathcal{Z} \{ A \cdot x_1[k] + B \cdot x_2[k] \} = A \cdot X_1(z) + B \cdot X_2(z) \end{equation}

with $A, B \in \mathbb{C}$. The region of convergence (ROC) of the superposition $A x_1[k] + B x_2[k]$ is as least as large as the intersection of the ROCs of $x_1[k]$ and $x_2[k]$

\begin{equation} \text{ROC} \{ A \cdot x_1[k] + B \cdot x_2[k] \} \supseteq \text{ROC} \{ x_1[k] \} \cap \text{ROC} \{ x_2[k] \} \end{equation}

The $z$-transform of a weighted superposition of signals is equal to the weighted superposition of the individual $z$-transforms. This property is useful to derive the transform of signals that can be expressed as superposition of other signals for which the $z$-transform is known or can be calculated easier.

Transformation of the cosine and sine signal¶

The Laplace transform of the causal cosine $\epsilon[k] \cdot \cos(\Omega_0 k)$ and sine $\epsilon[k] \cdot \sin(\Omega_0 k)$ signal with $\Omega_0 \in \mathbb{R}$ is derived by expressing the cosine/sine as a superposition of harmonic exponential signals using Euler's formula

\begin{align} \cos(\Omega_0 k) &= \frac{1}{2} \left( e^{j \Omega_0 k} + e^{-j \Omega_0 k} \right) \\ \sin(\Omega_0 k) &= \frac{1}{2j} \left( e^{j \Omega_0 k} - e^{-j \Omega_0 k} \right) \end{align}

The $z$-transform of the harmonic exponential signal can be derived from the spectrum of the causal complex exponential signal as

\begin{equation} \mathcal{Z} \{ \epsilon[k] \cdot e^{j \Omega_0 k} \} = \frac{z}{z - e^{j \Omega_0}} \qquad \text{for } |z| > 1 \end{equation}

Exploiting the linearity of the $z$-transform yields

\begin{equation} \mathcal{Z} \{ \epsilon[k] \cdot \cos(\Omega_0 k) \} = \frac{1}{2} \left( \frac{z}{z - e^{j \Omega_0}} + \frac{z}{z - e^{-j \Omega_0}} \right) = \frac{z ( z - \cos(\Omega_0))}{z^2 - 2 z \cos(\Omega_0) + 1} \qquad \text{for } |z| > 1 \end{equation}

and

\begin{equation} \mathcal{Z} \{ \epsilon[k] \cdot \sin(\Omega_0 k) \} = \frac{1}{2j} \left( \frac{z}{z - e^{j \Omega_0}} - \frac{z}{z - e^{-j \Omega_0}} \right) = \frac{z \sin(\Omega_0)}{z^2 - 2 z \cos(\Omega_0) + 1} \qquad \text{for } |z| > 1 \end{equation}

Exercise

• Determine the zeros and poles of above $z$-transforms. What specific symmetries do the zeros and poles show in the $z$-plane?

Symmetry for Real-Valued Signals¶

The $z$-transform of a conjugated signal $x^*[k]$ is given as

\begin{equation} \mathcal{Z} \{ x^*[k] \} = \sum_{k = -\infty}^{\infty} x^*[k] \, z^{-k} = \left( \sum_{k = -\infty}^{\infty} x[k] \, (z^*)^{-k} \right)^* = X^*(z^*) \end{equation}

where $X(z) = \mathcal{Z} \{ x[k] \}$. The ROC of $x^*[k]$ is equal to the ROC of $x[k]$. Above result is used to derive the $z$-transform of the real part of the signal $x[k]$

\begin{equation} \mathcal{Z} \left\{ \Re \{ x[k] \} \right\} = \mathcal{Z} \left\{ \frac{1}{2} ( x[k] + x^*[k] ) \right\} = \frac{1}{2} \left[ X(z) + X^*(z^*) \right] \end{equation}

and imaginary part

\begin{equation} \mathcal{Z} \left\{ \Im \{ x[k] \} \right\} = \mathcal{Z} \left\{ \frac{1}{2j} ( x[k] - x^*[k] ) \right\} = \frac{1}{2j} \left[ X(z) - X^*(z^*) \right] \end{equation}

For a real-valued signal $x[k] \in \mathbb{R}$ with $\Im \{ x[k] \} = 0$ it follows that

\begin{equation} X(z) = X^*(z^*) \end{equation}

It can be concluded, that the $z$-transform of a real valued signal shows complex conjugate symmetry.

The poles and zeros of a $z$-transform which can be expressed as a rational function in $z$

\begin{equation} X(s) = \frac{\sum_{m=0}^{M} \beta_m z^{-m}}{\sum_{n=0}^{N} \alpha_n z^{-n}} = K \cdot \frac{\prod_{\mu=0}^{Q} (z - z_{0 \mu})}{\prod_{\nu=0}^{P} (z - z_{\infty \nu})} \end{equation}

with $Q=M-1$ and $P=N-1$ show also specific symmetries. From above condition for the $z$-transform of a real-valued signal $x[k] \in \mathbb{R}$, it can be concluded that the coefficients $\alpha_n, \beta_m$ of the numerator and denominator are real-valued. The complex conjugate root theorem states that the roots of a polynomial with real valued coefficients are either real-valued or complex conjugate pairs. Hence, the zeros and poles of the $z$-transform $X(z)$ of a real-valued signal $x[k]$ are either real-valued or complex conjugate pairs.

Example

As example for a real-valued signal, the poles and zeros of the $z$-transform $X(z) = \mathcal{Z} \{ \epsilon[k] \cos(\Omega_0 k) \}$ are investigated. First the transform, as derived above, is defined in SymPy

The roots of $X(z)$ are computed

as well as its zeros

Finally the pole-zero plot is plotted for $\Omega = \frac{\pi}{4}$

Copyright

The notebooks are provided as Open Educational Resource. Feel free to use the notebooks for your own educational purposes. The text is licensed under Creative Commons Attribution 4.0, the code of the IPython examples under the MIT license. Please attribute the work as follows: Lecture Notes on Signals and Systems by Sascha Spors.