Part I — Foundations

Chapter 7: The Z-Transform

The Z-transform is the discrete-time counterpart of the Laplace transform. It converts difference equations into algebraic equations, enables transfer-function analysis of digital systems, and provides a clean algebraic framework for studying stability, causality, and frequency response through the geometry of poles and zeros in the complex plane.

7.1 Definition of the Z-Transform

Given a discrete-time signal $x[n]$, its Z-transform is defined as a power series in the complex variable $z$.

The Bilateral Z-Transform

For a sequence $x[n]$, the bilateral (two-sided) Z-transform is

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

where $z \in \mathbb{C}$ is a complex variable. The transform exists for those values of $z$ for which the sum converges absolutely.

The Unilateral Z-Transform

For causal signals and system analysis the unilateral (one-sided) Z-transform is often preferred:

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

The complex variable $z$ can be written in polar form as $z = r\, e^{j\omega}$, where $r = |z|$ is the radius and $\omega$ is the angle. This polar representation is the key to understanding the connection between the Z-transform and the frequency domain.

Region of Convergence (ROC)

The Region of Convergence is the set of all $z$ for which $X(z)$ converges:

$$\text{ROC} = \left\{ z \in \mathbb{C} : \sum_{n=-\infty}^{\infty} |x[n]|\,|z|^{-n} < \infty \right\}$$

The ROC is always an annular region of the form $r_1 < |z| < r_2$. For a causal signal the ROC is the exterior of a circle ($|z| > r_1$); for an anti-causal signal it is the interior ($|z| < r_2$).

Why the ROC matters: Two different sequences can have the same algebraic expression for $X(z)$ but different ROCs. Without specifying the ROC the inverse Z-transform is ambiguous. For example, $X(z) = z/(z-a)$ corresponds to $a^n u[n]$ if $|z|>|a|$, but to $-a^n u[-n-1]$ if $|z|<|a|$.

Example: Z-Transform of a^n u[n]

Let $x[n] = a^n u[n]$ where $|a| < 1$. Then:

$$X(z) = \sum_{n=0}^{\infty} a^n z^{-n} = \sum_{n=0}^{\infty} \left(\frac{a}{z}\right)^n = \frac{1}{1 - az^{-1}} = \frac{z}{z - a}, \quad |z| > |a|$$

The ROC is the exterior of a circle of radius $|a|$, which for $|a| < 1$ includes the unit circle — confirming that the DTFT also exists.

7.2 Relationship to the DTFT and Laplace Transform

The Z-transform unifies the discrete-time Fourier transform and the Laplace transform under one algebraic roof.

Z-Transform on the Unit Circle = DTFT

When we evaluate $X(z)$ on the unit circle $z = e^{j\omega}$, we recover the DTFT:

$$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n} = X_{\text{DTFT}}(\omega)$$

This is valid only when the ROC of $X(z)$ includes the unit circle.

Connection to the Laplace Transform

If a continuous-time signal $x_c(t)$ is sampled with period $T_s$ to produce $x[n] = x_c(nT_s)$, then the Z-transform variable is related to the Laplace variable by:

$$z = e^{sT_s}$$

This mapping transforms the left half of the $s$-plane ($\operatorname{Re}(s) < 0$) to the interior of the unit circle ($|z| < 1$), and the $j\omega$ axis to the unit circle itself. It is a many-to-one mapping due to the periodicity of the exponential in $j\omega$.

Mapping summary: In the $s$-plane, stability requires poles in the left half-plane ($\operatorname{Re}(s)<0$). Via $z = e^{sT_s}$, this maps to $|z|<1$, i.e., poles inside the unit circle. The imaginary axis $s = j\omega$ maps to the unit circle $|z|=1$, which is where we evaluate the frequency response.

7.3 Key Properties of the Z-Transform

The Z-transform inherits many of the algebraic conveniences of the Laplace transform. The following table summarises the most important properties.

Property	Time Domain	Z-Domain
Linearity	$a\, x_1[n] + b\, x_2[n]$	$a\, X_1(z) + b\, X_2(z)$
Time Delay	$x[n - k]$	$z^{-k}\, X(z)$
Time Advance	$x[n + k]$	$z^{k}\, X(z)$
Convolution	$x[n] * h[n]$	$X(z)\, H(z)$
Scaling	$a^n\, x[n]$	$X(z/a)$
Differentiation in z	$n\, x[n]$	$-z \,\frac{dX(z)}{dz}$
Accumulation	$\displaystyle\sum_{k=-\infty}^{n} x[k]$	$\frac{X(z)}{1 - z^{-1}}$
Initial Value	$x[0]$	$\displaystyle\lim_{z \to \infty} X(z)$
Final Value	$\displaystyle\lim_{n \to \infty} x[n]$	$\displaystyle\lim_{z \to 1} (1 - z^{-1})\, X(z)$

Convolution Theorem

If $y[n] = x[n] * h[n]$, then

$$Y(z) = X(z) \cdot H(z)$$

with $\text{ROC}_Y \supseteq \text{ROC}_X \cap \text{ROC}_H$. This is the fundamental reason that the Z-transform is so useful: convolution in the time domain becomes multiplication in the Z-domain.

Example: Time Delay Property

Consider a unit delay system: $y[n] = x[n-1]$. Taking the Z-transform of both sides:

$$Y(z) = z^{-1} X(z)$$

Thus the transfer function of a single-sample delay is simply $H(z) = z^{-1}$. A cascade of $k$ delays gives $H(z) = z^{-k}$.

7.4 Common Z-Transform Pairs

The following table collects the most frequently encountered Z-transform pairs. Memorising these accelerates both analysis and design work.

Signal $x[n]$	Z-Transform $X(z)$	ROC
$\delta[n]$	$1$	All $z$
$u[n]$	$\dfrac{z}{z - 1}$	$\|z\| > 1$
$a^n u[n]$	$\dfrac{z}{z - a}$	$\|z\| > \|a\|$
$n\, a^n u[n]$	$\dfrac{az}{(z - a)^2}$	$\|z\| > \|a\|$
$\cos(\omega_0 n)\, u[n]$	$\dfrac{z(z - \cos\omega_0)}{z^2 - 2z\cos\omega_0 + 1}$	$\|z\| > 1$
$\sin(\omega_0 n)\, u[n]$	$\dfrac{z\sin\omega_0}{z^2 - 2z\cos\omega_0 + 1}$	$\|z\| > 1$
$r^n \cos(\omega_0 n)\, u[n]$	$\dfrac{z(z - r\cos\omega_0)}{z^2 - 2rz\cos\omega_0 + r^2}$	$\|z\| > r$
$r^n \sin(\omega_0 n)\, u[n]$	$\dfrac{rz\sin\omega_0}{z^2 - 2rz\cos\omega_0 + r^2}$	$\|z\| > r$

Pattern to notice: Every causal exponential $a^n u[n]$ has ROC $|z|>|a|$. The damped sinusoids ($r^n \cos$, $r^n \sin$) have poles at $z = r\, e^{\pm j\omega_0}$ and ROC $|z| > r$. When $r < 1$ the ROC includes the unit circle, so the DTFT exists and the system is stable.

7.5 Transfer Function of Discrete Systems

A causal, linear, time-invariant discrete system is characterised by a constant-coefficient difference equation:

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

Taking the Z-transform of both sides (assuming zero initial conditions) and using the time-delay property:

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

Transfer Function H(z)

The transfer function is the ratio of the output to input Z-transforms:

$$H(z) = \frac{Y(z)}{X(z)} = \frac{\displaystyle\sum_{k=0}^{M} b_k\, z^{-k}}{\displaystyle\sum_{k=0}^{N} a_k\, z^{-k}} = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}$$

Multiplying numerator and denominator by $z^{\max(M,N)}$ gives the equivalent polynomial form in positive powers of $z$.

Example: First-Order IIR Filter

The difference equation $y[n] = x[n] + 0.8\, y[n-1]$ has transfer function:

$$H(z) = \frac{1}{1 - 0.8\, z^{-1}} = \frac{z}{z - 0.8}$$

This system has a single pole at $z = 0.8$ and a single zero at $z = 0$. Since $|0.8| < 1$, the system is stable.

Example: Second-Order System

Consider $y[n] - 1.2\, y[n-1] + 0.81\, y[n-2] = x[n] - 0.5\, x[n-1]$:

$$H(z) = \frac{1 - 0.5\, z^{-1}}{1 - 1.2\, z^{-1} + 0.81\, z^{-2}} = \frac{z(z - 0.5)}{z^2 - 1.2z + 0.81}$$

The poles are at $z = 0.6 \pm 0.6j$ with magnitude $|p| = \sqrt{0.36 + 0.36} \approx 0.849 < 1$, so the system is stable. The zero at $z = 0.5$ shapes the frequency response.

7.6 Stability: The Unit Circle Criterion

For a causal LTI system with rational transfer function $H(z)$, stability is determined entirely by the locations of the poles.

BIBO Stability Criterion

A causal discrete-time LTI system is bounded-input bounded-output (BIBO) stable if and only if all poles of $H(z)$ lie strictly inside the unit circle:

$$|p_i| < 1 \quad \text{for all poles } p_i$$

Equivalently, a causal system is BIBO stable if and only if its impulse response is absolutely summable: $\sum_{n=0}^{\infty} |h[n]| < \infty$.

The stability criterion connects directly to the ROC:

ROC and Stability Connection

For a causal system, the ROC is $|z| > r_{\max}$, where $r_{\max}$ is the magnitude of the outermost pole. The system is stable if and only if the ROC includes the unit circle, which requires $r_{\max} < 1$.

For DTFT analysis we need to evaluate $H(z)$ on $|z| = 1$. If a pole sits on or outside the unit circle, the frequency response either diverges or the system is unstable.

Stability Classification

Stable: All poles satisfy $|p_i| < 1$. Impulse response decays to zero.
Marginally stable: At least one pole on the unit circle ($|p_i| = 1$), none outside. Impulse response neither grows nor decays (e.g., pure oscillation).
Unstable: At least one pole outside the unit circle ($|p_i| > 1$). Impulse response grows without bound.

Continuous vs. discrete stability: In the Laplace domain stability requires poles in the left half-plane ($\operatorname{Re}(s) < 0$). The mapping $z = e^{sT}$ transforms the left half-plane to the interior of the unit circle, so the Z-domain criterion $|z| < 1$ is the discrete-time analogue of the continuous-time condition.

7.7 Inverse Z-Transform

Recovering $x[n]$ from $X(z)$ is called the inverse Z-transform. There are three principal methods.

Formal Inversion Formula

The inverse Z-transform is given by the contour integral:

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

where $C$ is a closed contour in the ROC encircling the origin counterclockwise. In practice, we rarely evaluate this integral directly.

Method 1: Partial Fraction Expansion

The most common approach. We decompose $X(z)/z$ (or $X(z)$ directly) into partial fractions, then use the known transform pairs from the table.

Example: Partial Fractions

Given $X(z) = \dfrac{2z^2 - 1.5z}{z^2 - 1.5z + 0.5}$ with ROC $|z| > 1$:

Factor the denominator: $(z-1)(z-0.5)$. Compute $X(z)/z$:

$$\frac{X(z)}{z} = \frac{2z - 1.5}{(z-1)(z-0.5)} = \frac{A}{z-1} + \frac{B}{z-0.5}$$

Solving: $A = 1$, $B = 1$. Therefore $X(z) = \dfrac{z}{z-1} + \dfrac{z}{z-0.5}$, giving:

$$x[n] = u[n] + (0.5)^n u[n] = \left(1 + 0.5^n\right) u[n]$$

Method 2: Long Division (Power Series Expansion)

Perform polynomial long division of the numerator by the denominator to obtain a power series in $z^{-1}$. The coefficients are the signal values: $X(z) = \sum_n x[n] z^{-n}$, so the coefficient of $z^{-n}$ is $x[n]$.

When to use each method: Partial fractions give a closed-form expression and are preferred for rational transforms. Long division is useful when you need only the first few values of $x[n]$ or when the partial fraction decomposition is unwieldy. Contour integration is mostly of theoretical interest.

Method 3: Contour Integration (Residue Method)

By the residue theorem, the inversion integral evaluates to:

$$x[n] = \sum_{k} \operatorname{Res}\left[ X(z)\, z^{n-1},\, z = z_k \right]$$

where the sum is over all poles $z_k$ of $X(z)\, z^{n-1}$ inside the contour $C$. For a simple pole at $z = z_k$:

$$\operatorname{Res}[X(z)\, z^{n-1}, z_k] = \lim_{z \to z_k} (z - z_k)\, X(z)\, z^{n-1}$$

7.8 Pole-Zero Analysis

A rational transfer function can always be factored as:

$$H(z) = G \cdot \frac{\displaystyle\prod_{k=1}^{M}(z - z_k)}{\displaystyle\prod_{k=1}^{N}(z - p_k)}$$

where $z_k$ are the zeros and $p_k$ are the poles. The pole-zero plot is the most compact representation of an LTI system.

Geometric Interpretation of Frequency Response

On the unit circle ($z = e^{j\omega}$), the magnitude response is:

$$|H(e^{j\omega})| = |G| \cdot \frac{\displaystyle\prod_{k=1}^{M} |e^{j\omega} - z_k|}{\displaystyle\prod_{k=1}^{N} |e^{j\omega} - p_k|}$$

Each factor $|e^{j\omega} - z_k|$ or $|e^{j\omega} - p_k|$ is the Euclidean distance from the point $e^{j\omega}$ on the unit circle to the zero or pole in the complex plane. When $e^{j\omega}$ passes near a pole, the denominator term becomes small and $|H|$ peaks; when it passes near a zero, $|H|$ dips.

This geometric interpretation is extraordinarily useful for intuitive filter design:

Poles near the unit circle create peaks (resonances) in the magnitude response at the corresponding frequency.
Zeros on the unit circle create nulls (perfect rejection) at that frequency.
Poles and zeros far from the unit circle have a mild, broadband effect on the response.
A notch filter places zeros on the unit circle at the frequency to be rejected and nearby poles just inside the circle to keep the rest of the response flat.
A resonator places poles close to the unit circle at the desired frequency, creating a sharp peak.

Example: Designing a Simple Notch Filter

To reject frequency $\omega_0 = \pi/4$, place zeros at $z = e^{\pm j\pi/4}$ and poles at $z = r\, e^{\pm j\pi/4}$ with $r = 0.9$ (close to 1 for a narrow notch):

$$H(z) = \frac{(z - e^{j\pi/4})(z - e^{-j\pi/4})}{(z - 0.9e^{j\pi/4})(z - 0.9e^{-j\pi/4})} = \frac{z^2 - \sqrt{2}\,z + 1}{z^2 - 0.9\sqrt{2}\,z + 0.81}$$

At $\omega = \pi/4$, the numerator vanishes (the zero is on the unit circle), giving $|H| = 0$. The nearby poles keep the response close to unity at other frequencies.

Python Lab: Pole-Zero Map and Frequency Response

We analyse a second-order system $H(z) = \frac{1 - 0.5z^{-1}}{1 - 1.2z^{-1} + 0.81z^{-2}}$ by computing its poles, zeros, and frequency response. The pole-zero map gives immediate geometric insight into the shape of the magnitude and phase responses.

import numpy as np
import matplotlib.pyplot as plt

# H(z) = (1 - 0.5z^{-1}) / (1 - 1.2z^{-1} + 0.81z^{-2})
# Zeros: z = 0.5, Poles: z = 0.6 +/- 0.6j (|p| = 0.849)

b = [1, -0.5]        # numerator coefficients
a = [1, -1.2, 0.81]  # denominator coefficients

# Find poles and zeros
zeros = np.roots(b)
poles = np.roots(a)

print("Zeros:", zeros)
print("Poles:", poles)
print("Pole magnitudes:", np.abs(poles))
print("Stable:", all(np.abs(poles) < 1))

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(15, 5))
fig.suptitle('Z-Transform: Pole-Zero Analysis', color='white', fontsize=14)

# Pole-zero map
theta = np.linspace(0, 2*np.pi, 100)
ax1.plot(np.cos(theta), np.sin(theta), '--', color='#475569', alpha=0.5)
ax1.plot(zeros.real, zeros.imag, 'o', color='#818cf8', markersize=12, markeredgewidth=2, fillstyle='none', label='Zeros')
ax1.plot(poles.real, poles.imag, 'x', color='#ef4444', markersize=12, markeredgewidth=2.5, label='Poles')
ax1.axhline(y=0, color='#475569', linewidth=0.5)
ax1.axvline(x=0, color='#475569', linewidth=0.5)
ax1.set_aspect('equal')
ax1.set_title('Pole-Zero Map', color='#a5b4fc')
ax1.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#94a3b8')

# Magnitude response
w = np.linspace(0, np.pi, 500)
z_eval = np.exp(1j * w)
H = np.polyval(b, z_eval) / np.polyval(a, z_eval)

ax2.plot(w/np.pi, 20*np.log10(np.abs(H)), color='#34d399', linewidth=2)
ax2.set_title('Magnitude Response', color='#a5b4fc')
ax2.set_xlabel('Normalized Frequency (xpi rad)', color='#94a3b8')
ax2.set_ylabel('|H| (dB)', color='#94a3b8')

# Phase response
ax3.plot(w/np.pi, np.degrees(np.unwrap(np.angle(H))), color='#c084fc', linewidth=2)
ax3.set_title('Phase Response', color='#a5b4fc')
ax3.set_xlabel('Normalized Frequency (xpi rad)', color='#94a3b8')
ax3.set_ylabel('Phase (degrees)', color='#94a3b8')

for ax in [ax1, ax2, ax3]:
    ax.grid(True, alpha=0.3)
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

Python Lab: Impulse Response from Pole Locations

The character of a system's impulse response is dictated by its pole locations. We compare three systems: a stable first-order system with $|p| = 0.8$, a marginally stable system with $|p| = 1$, and a resonant system with complex conjugate poles.

import numpy as np
import matplotlib.pyplot as plt

# Three different systems
systems = [
    {'name': 'Stable (|p|=0.8)', 'b': [1], 'a': [1, -0.8]},
    {'name': 'Marginally Stable (|p|=1)', 'b': [1], 'a': [1, -1.0]},
    {'name': 'Resonant (complex poles)', 'b': [1], 'a': [1, -1.6, 0.81]},
]

fig, axes = plt.subplots(1, 3, figsize=(14, 5))
fig.suptitle('Impulse Response h[n] from Pole Locations', color='white', fontsize=14)

colors = ['#34d399', '#f59e0b', '#c084fc']

for ax, sys, c in zip(axes, systems, colors):
    # Compute impulse response by filtering
    N = 50
    h = np.zeros(N)
    x = np.zeros(N)
    x[0] = 1  # impulse

b, a = sys['b'], sys['a']
    # Direct form I
    y = np.zeros(N)
    for n in range(N):
        for k in range(len(b)):
            if n-k >= 0:
                y[n] += b[k] * x[n-k]
        for k in range(1, len(a)):
            if n-k >= 0:
                y[n] -= a[k] * y[n-k]

ax.stem(range(N), y, linefmt=c, markerfmt='o', basefmt='#334155')
    ax.set_title(sys['name'], color='#a5b4fc')
    ax.set_xlabel('n', color='#94a3b8')

poles = np.roots(a)
    pole_str = ', '.join([f'{p:.2f}' for p in poles])
    ax.text(0.5, 0.95, f'Poles: {pole_str}', transform=ax.transAxes,
            color='#94a3b8', fontsize=9, va='top', ha='center')

ax.grid(True, alpha=0.3)
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

Python Lab: Geometric Interpretation of Frequency Response

The frequency response at any angle $\omega$ is determined by the distances from the point $e^{j\omega}$ on the unit circle to each pole and zero. When the evaluation point passes close to a pole, the denominator distance shrinks and the magnitude peaks. This visualisation makes the connection concrete.

import numpy as np
import matplotlib.pyplot as plt

# H(z) = z / (z - 0.9e^{j*pi/4})*(z - 0.9e^{-j*pi/4})
# Resonant at omega = pi/4

r = 0.9
omega_p = np.pi/4
poles = [r * np.exp(1j*omega_p), r * np.exp(-1j*omega_p)]

w = np.linspace(0, np.pi, 500)
z_on_circle = np.exp(1j * w)

# H(w) = product of (z - zeros) / product of (z - poles)
H = 1.0 / ((z_on_circle - poles[0]) * (z_on_circle - poles[1]))
H_mag = np.abs(H)
H_mag_norm = H_mag / np.max(H_mag)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
fig.suptitle('Geometric View: Closer pole -> larger response', color='white', fontsize=14)

# Unit circle with poles and vectors
theta = np.linspace(0, 2*np.pi, 100)
ax1.plot(np.cos(theta), np.sin(theta), '--', color='#475569', alpha=0.5)
ax1.plot([p.real for p in poles], [p.imag for p in poles], 'x',
         color='#ef4444', markersize=14, markeredgewidth=3, label='Poles')

# Show vectors at a few frequencies
for w_show in [0.2, np.pi/4, np.pi/2, np.pi]:
    z_pt = np.exp(1j * w_show)
    ax1.plot(z_pt.real, z_pt.imag, 'o', color='#34d399', markersize=6)
    for p in poles:
        ax1.annotate('', xy=(z_pt.real, z_pt.imag), xytext=(p.real, p.imag),
                     arrowprops=dict(arrowstyle='->', color='#818cf8', alpha=0.4))

ax1.set_aspect('equal')
ax1.set_title('Pole-to-Circle Vectors', color='#a5b4fc')
ax1.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#94a3b8')

# Frequency response
ax2.plot(w/np.pi, 20*np.log10(H_mag_norm + 1e-10), color='#818cf8', linewidth=2)
ax2.axvline(x=omega_p/np.pi, color='#ef4444', linestyle='--', alpha=0.7, label=f'Pole angle = pi/4')
ax2.set_title('Magnitude Response', color='#a5b4fc')
ax2.set_xlabel('Normalized Frequency (xpi)', color='#94a3b8')
ax2.set_ylabel('|H| (dB)', color='#94a3b8')
ax2.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#94a3b8')

for ax in [ax1, ax2]:
    ax.grid(True, alpha=0.3)
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

Chapter Summary

The Z-transform maps a discrete-time sequence $x[n]$ to a function of a complex variable: $X(z) = \sum x[n]\, z^{-n}$.
The Region of Convergence (ROC) determines which sequence corresponds to a given algebraic expression. For causal signals the ROC is $|z| > r$.
Evaluating $X(z)$ on the unit circle $z = e^{j\omega}$ recovers the DTFT. The relationship $z = e^{sT}$ connects to the Laplace transform.
Key properties — linearity, time delay ($z^{-k}$), convolution ($X \cdot H$), scaling, accumulation — turn difference equations into algebraic equations.
The transfer function $H(z) = Y(z)/X(z)$ is a rational function whose poles and zeros determine all system behaviour.
A causal system is BIBO stable if and only if all poles lie strictly inside the unit circle ($|p_i| < 1$).
The inverse Z-transform can be found via partial fractions, long division, or contour integration.
Pole-zero analysis provides a geometric interpretation: poles near the unit circle create magnitude peaks; zeros create nulls.

← Ch 6: DFT & FFT Ch 8: Digital Filters →

Signal Theory Course

Chapter 7: The Z-Transform

7.1 Definition of the Z-Transform

The Bilateral Z-Transform

The Unilateral Z-Transform

Region of Convergence (ROC)

Example: Z-Transform of a^n u[n]

7.2 Relationship to the DTFT and Laplace Transform

Z-Transform on the Unit Circle = DTFT

Connection to the Laplace Transform

7.3 Key Properties of the Z-Transform

Convolution Theorem

Example: Time Delay Property

7.4 Common Z-Transform Pairs

7.5 Transfer Function of Discrete Systems

Transfer Function H(z)

Example: First-Order IIR Filter

Example: Second-Order System

7.6 Stability: The Unit Circle Criterion

BIBO Stability Criterion

ROC and Stability Connection

Stability Classification

7.7 Inverse Z-Transform

Formal Inversion Formula

Method 1: Partial Fraction Expansion

Example: Partial Fractions

Method 2: Long Division (Power Series Expansion)

Method 3: Contour Integration (Residue Method)

7.8 Pole-Zero Analysis

Geometric Interpretation of Frequency Response

Example: Designing a Simple Notch Filter

Python Lab: Pole-Zero Map and Frequency Response

Python Lab: Impulse Response from Pole Locations

Python Lab: Geometric Interpretation of Frequency Response

Chapter Summary