Unit 6 - Notes

ECE220

Unit 6: The Z-transform

1. Introduction

The Z-transform is a mathematical tool used for the analysis and design of discrete-time signals and systems. It serves as the discrete-time counterpart to the Laplace transform used in continuous-time systems.

While the Discrete-Time Fourier Transform (DTFT) provides a frequency-domain representation of a discrete signal, it only exists if the signal is absolutely summable (stable). The Z-transform generalizes the DTFT, allowing for the analysis of a broader class of signals (including unstable ones) and the characterization of transient behaviors in Discrete-Time Linear Time-Invariant (DT-LTI) systems.

Key Analogies

Continuous Time	Discrete Time
Differential Equations	Difference Equations
Laplace Transform ( $s$ -domain)	Z-transform ( $z$ -domain)
Stability: Left Half of $s$ -plane	Stability: Inside Unit Circle of $z$ -plane

2. The Z-transform

Definition

The bilateral (two-sided) Z-transform of a discrete-time signal $x[n]$ is defined as a power series:

X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}

Where:

$z$ is a complex variable defined as $z = r e^{j\Omega}$ .
$r$ is the magnitude of $z$ .
$\Omega$ is the angle (frequency) in radians.

Relation to DTFT

If we substitute $z = e^{j\Omega}$ (i.e., restricting $z$ to the unit circle where $r=1$ ), the Z-transform reduces to the DTFT:

X(z)\big|_{z=e^{j\Omega}} = \sum_{n=-\infty}^{\infty} x[n] e^{-j\Omega n} = X(e^{j\Omega})

Note: The Z-transform exists on the complex z-plane, while the DTFT exists specifically on the Unit Circle of that plane.

The Unilateral Z-transform

Used primarily for solving difference equations with initial conditions (causal systems), defined as:

X^+(z) = \sum_{n=0}^{\infty} x[n] z^{-n}

3. The Region of Convergence (ROC)

The Region of Convergence is the set of values of $z$ in the complex plane for which the infinite summation $\sum x[n]z^{-n}$ converges to a finite value.

\text{ROC} = \{ z : |X(z)| < \infty \}

A Z-transform expression is uniquely defined only when its ROC is specified. For example, $a^n u[n]$ and $-a^n u[-n-1]$ result in the same algebraic expression $\frac{z}{z-a}$ , distinguished only by their ROCs.

Properties of the ROC

Shape: The ROC consists of a ring or disk in the z-plane centered at the origin.
No Poles: The ROC cannot contain any poles (values of $z$ where $X(z) \to \infty$ ).
Finite Duration Signals: The ROC is the entire z-plane, except possibly $z=0$ or $z=\infty$ .
Right-Sided Signals (Causal): If $x[n]$ is right-sided (zero for $n < N_1$ ), the ROC is the region outside the outermost pole:
$|z| > |p_{\text{max}}|$
Left-Sided Signals (Anti-Causal): If $x[n]$ is left-sided (zero for $n > N_2$ ), the ROC is the region inside the innermost pole:
$|z| < |p_{\text{min}}|$
Two-Sided Signals: The ROC is an annular ring bounded by two poles (does not include the poles themselves):
$r_1 < |z| < r_2$
Stability: For a system to be stable, the ROC must include the Unit Circle ( $|z|=1$ ).

4. Properties of the Z-transform

Assuming $x[n] \leftrightarrow X(z)$ with ROC $R_x$ and $y[n] \leftrightarrow Y(z)$ with ROC $R_y$ .

1. Linearity

a x[n] + b y[n] \leftrightarrow a X(z) + b Y(z)

ROC: Contains

R_x \cap R_y

2. Time Shifting

x[n-k] \leftrightarrow z^{-k} X(z)

ROC: Same as

R_x

(except possibly

z=0

\infty

3. Scaling in the z-domain

a^n x[n] \leftrightarrow X(z/a)

ROC:

|a|R_x

(scaled expansion/contraction of the ROC).

4. Time Reversal

x[-n] \leftrightarrow X(1/z)

ROC: Inverted (

1/R_x

5. Differentiation in the z-domain

n x[n] \leftrightarrow -z \frac{d}{dz} X(z)

6. Convolution

x[n] * y[n] \leftrightarrow X(z) Y(z)

ROC: Contains

R_x \cap R_y

. This is the foundation of system analysis (Output = Input

\times

Transfer Function).

7. Initial Value Theorem (Causal Signals)

If $x[n] = 0$ for $n < 0$ :

x[0] = \lim_{z \to \infty} X(z)

8. Final Value Theorem (Causal Signals)

If the poles of $(z-1)X(z)$ are inside the unit circle:

\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1) X(z)

5. The Inverse Z-transform

The process of recovering the discrete-time signal $x[n]$ from $X(z)$ .

Method 1: Power Series Expansion (Long Division)

Useful when $X(z)$ is a rational function and we need the first few samples of $x[n]$ .

Divide the numerator polynomial by the denominator.
The coefficients of the resulting series correspond to the sequence values.

Method 2: Partial Fraction Expansion (Most Common)

Used for rational functions of the form $X(z) = \frac{N(z)}{D(z)}$ .

Ensure the order of $N(z)$ is less than $D(z)$ . If not, use long division first.
Factor the denominator to find poles.
Express $\frac{X(z)}{z}$ as a sum of partial fractions:
$\frac{X(z)}{z} = \frac{A}{z-p_1} + \frac{B}{z-p_2} + \dots$
Multiply by $z$ to isolate $X(z)$ :
$X(z) = A \frac{z}{z-p_1} + B \frac{z}{z-p_2} + \dots$
Apply the inverse transform based on the ROC:
- If ROC is $|z| > |p_k|$ (Right-sided): $\mathcal{Z}^{-1} \to p_k^n u[n]$
- If ROC is $|z| < |p_k|$ (Left-sided): $\mathcal{Z}^{-1} \to -p_k^n u[-n-1]$

Method 3: Contour Integration (Residue Method)

Based on Cauchy's Integral Theorem:

x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz

Where

C

is a counter-clockwise contour in the ROC encircling the origin.

x[n] = \sum [\text{Residues of } X(z)z^{n-1} \text{ at the poles inside } C]

6. Analysis and Characterization of LTI Systems

An LTI system is characterized by its impulse response $h[n]$ . The Z-transform of $h[n]$ is the System Function or Transfer Function, $H(z)$ .

Y(z) = H(z) X(z) \implies H(z) = \frac{Y(z)}{X(z)}

System Function from Difference Equations

Given a Linear Constant Coefficient Difference Equation (LCCDE):

\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]

Taking the Z-transform (assuming zero initial conditions):

Y(z) \sum_{k=0}^{N} a_k z^{-k} = X(z) \sum_{k=0}^{M} b_k z^{-k}

H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}}

Causality

A system is Causal if $h[n] = 0$ for $n < 0$ .

ROC Condition: The ROC is the exterior of a circle $|z| > R$ , including $z = \infty$ .
Rational Function Condition: The order of the numerator polynomial must be less than or equal to the order of the denominator polynomial (proper fraction).

Stability

A system is BIBO Stable (Bounded Input, Bounded Output) if $\sum |h[n]| < \infty$ .

ROC Condition: The ROC must include the Unit Circle ( $|z|=1$ ).
Pole Location: For a causal system to be stable, all poles must lie strictly inside the unit circle.

7. Software Simulation and Pole-Zero Analysis

Modern analysis relies on software tools (MATLAB, Python/SciPy) to visualize the relationship between the algebraic $H(z)$ and system behavior.

Pole-Zero Plot

Zeros ( $O$ ): Roots of the numerator $N(z)$ . Values of $z$ where $H(z) = 0$ .
Poles ( $X$ ): Roots of the denominator $D(z)$ . Values of $z$ where $H(z) = \infty$ .

Effect of Pole Locations (Causal Systems)

Real Pole inside unit circle ( $0 < p < 1$ ): Decaying exponential response.
Real Pole outside unit circle ( $p > 1$ ): Growing exponential (Unstable).
Pole at origin ( $p=0$ ): Simple delay.
Complex Conjugate Poles ( $re^{\pm j\theta}$ ):
- $r < 1$ : Damped sinusoidal oscillation (Stable).
- $r = 1$ : Constant amplitude oscillation (Marginally Stable/Oscillator).
- $r > 1$ : Growing sinusoidal oscillation (Unstable).

Python Simulation Example

Using scipy.signal to analyze a system described by: $y[n] - 0.5y[n-1] = x[n]$ .

H(z) = \frac{1}{1 - 0.5z^{-1}} = \frac{z}{z - 0.5}

PYTHON

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

# Define system coefficients
# H(z) = B(z) / A(z)
# Numerator: 1 (which is 1*z^0)
# Denominator: 1 - 0.5*z^-1
b = [1]          # Numerator coefficients
a = [1, -0.5]    # Denominator coefficients

# 1. Pole-Zero Analysis
z, p, k = signal.tf2zpk(b, a)

print(f"Zeros: {z}")
print(f"Poles: {p}") 
# Result: Pole at 0.5. Since |0.5| < 1, system is stable.

# 2. Frequency Response
w, h = signal.freqz(b, a)

plt.figure()
plt.title('Digital Filter Frequency Response')
plt.plot(w, 20 * np.log10(abs(h)), 'b')
plt.ylabel('Amplitude [dB]')
plt.xlabel('Frequency [rad/sample]')
plt.grid()
plt.show()

# 3. Impulse Response (Inverse Z-transform simulation)
# Create an impulse signal
impulse = np.zeros(20)
impulse[0] = 1
response = signal.lfilter(b, a, impulse)

plt.figure()
plt.stem(response)
plt.title('Impulse Response h[n]')
plt.xlabel('n')
plt.grid()
plt.show()
# Result will show a decaying exponential (0.5)^n

Interpretation of Simulation

zplane: Visualizes stability immediately. If x markers are inside the circle drawn at radius 1, the filter is stable.
freqz: Evaluates $H(z)$ along the unit circle ( $z = e^{j\Omega}$ ) to show how the system amplifies or attenuates specific frequencies.
lfilter: Implements the difference equation numerically to simulate time-domain behavior.