Z-Transforms in Discrete-Time Signal Processing

The Z-Transform is a fundamental tool in digital signal processing (DSP) for analyzing discrete-time systems and solving difference equations. It allows engineers to work in the frequency domain for systems that are represented by difference equations, facilitating the design of digital filters and system stability analysis.

Key Concepts

Definition: The Z-Transform converts a discrete-time sequence \( x[n] \) into a complex frequency-domain function \( X(z) \) as follows: \[ X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \] Here, \( z \) is a complex number \( z = r e^{j\omega} \), where \( r \) is the magnitude and \( \omega \) is the phase.
Region of Convergence (ROC): The region in the complex plane where the Z-Transform converges. Understanding the ROC is crucial for determining system stability.
Inverse Z-Transform: The process of converting \( X(z) \) back into the time domain \( x[n] \), typically done using tables, partial fraction decomposition, or contour integration.
Poles and Zeros: Poles are points where \( X(z) \) becomes infinite, and zeros are points where \( X(z) = 0 \). Their locations help determine the frequency response and stability of the system.
Unit Circle: The behavior of the system is typically analyzed on the unit circle (\( |z| = 1 \)). If all poles are inside the unit circle, the system is stable.

Properties of Z-Transforms

Linearity: \[ \mathcal{Z}\{a x_1[n] + b x_2[n]\} = a \mathcal{Z}\{x_1[n]\} + b \mathcal{Z}\{x_2[n]\} \] The Z-Transform of a sum is the sum of the individual Z-Transforms.
Time Shifting: \[ \mathcal{Z}\{x[n - n_0]\} = z^{-n_0} X(z) \] Shifting in the time domain results in multiplication by \( z^{-n_0} \) in the Z-domain.
Convolution: \[ \mathcal{Z}\{x_1[n] * x_2[n]\} = X_1(z) X_2(z) \] Convolution in the time domain is multiplication in the Z-domain.
Differentiation in Z-Domain: \[ \mathcal{Z}\{n x[n]\} = -z \frac{dX(z)}{dz} \] Differentiation in \( z \) corresponds to multiplying by \( n \) in the time domain.
Initial Value Theorem: \[ x[0] = \lim_{z \to \infty} X(z) \] Allows determination of the initial value of \( x[n] \) directly from \( X(z) \).
Final Value Theorem: \[ \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z - 1) X(z) \] Provides the steady-state value of the sequence.

Common Z-Transforms

Below are commonly used Z-Transforms along with their regions of convergence (ROC):

Time-Domain Sequence \( x[n] \)	Z-Transform \( X(z) \)	ROC
\( \delta[n] \) (Unit impulse)	\( 1 \)	All \( z \)
\( u[n] \) (Unit step)	\( \frac{1}{1 - z^{-1}} \)	\( \|z\| > 1 \)
\( -a^n u[-n-1] \)	\( \frac{1}{1 - az^{-1}} \)	\( \|z\| < \|a\| \)
\( a^n u[n] \)	\( \frac{1}{1 - az^{-1}} \)	\( \|z\| > \|a\| \)
\( n a^n u[n] \)	\( \frac{az^{-1}}{(1 - az^{-1})^2} \)	\( \|z\| > \|a\| \)
\( n^2 a^n u[n] \)	\( \frac{2a z^{-1}}{(1 - az^{-1})^3} \)	\( \|z\| > \|a\| \)
\( \cos(\omega_0 n) u[n] \)	\( \frac{1 - z^{-2}}{1 - 2z^{-1} \cos(\omega_0) + z^{-2}} \)	\( \|z\| > 1 \)
\( \sin(\omega_0 n) u[n] \)	\( \frac{z^{-1} \sin(\omega_0)}{1 - 2z^{-1} \cos(\omega_0) + z^{-2}} \)	\( \|z\| > 1 \)
\( e^{\alpha n} u[n] \)	\( \frac{1}{1 - z^{-1} e^{\alpha}} \)	\( \|z\| > e^{\alpha} \)
\( (n + 1)u[n] \)	\( \frac{z^{-1}}{(1 - z^{-1})^2} \)	\( \|z\| > 1 \)

Solving Difference Equations

Z-Transforms are an effective tool for solving linear difference equations in discrete-time systems. The approach mirrors how Laplace Transforms solve continuous differential equations.

Steps to Solve Difference Equations:

Apply the Z-Transform to both sides of the difference equation. Use the properties of Z-Transforms to express the equation algebraically in terms of \( Z \).
Solve for \( X(z) \), the Z-Transform of the unknown sequence \( x[n] \).
Take the inverse Z-Transform to find the solution in the time domain.

Example: Solving a First-Order Difference Equation

Given the equation \( x[n] - \frac{1}{2}x[n-1] = u[n] \), where \( x[0] = 1 \).

Step 1: Apply the Z-Transform:

\[ X(z) - \frac{1}{2} z^{-1} X(z) = \frac{1}{1 - z^{-1}} \]

Step 2: Solve for \( X(z) \):

\[ X(z) \left(1 - \frac{1}{2} z^{-1}\right) = \frac{1}{1 - z^{-1}} \]

Simplifying gives:

\[ X(z) = \frac{2}{2 - z^{-1}} \]

Step 3: Take the inverse Z-Transform to find \( x[n] \):

\[ x[n] = 2 \left(\frac{1}{2}\right)^n u[n] \]

Applications in DSP

Z-Transforms are essential in discrete-time signal processing (DSP) and are commonly used in the following areas:

Digital Filters: Z-Transforms provide the transfer function of digital filters used for processing discrete-time signals. Filters such as low-pass, high-pass, and band-pass are designed using Z-domain analysis.
System Stability: The poles of the Z-domain transfer function determine the system's stability. If all poles are inside the unit circle (\( |z| = 1 \)), the system is stable.
Frequency Response: By evaluating the Z-Transform on the unit circle (\( z = e^{j\omega} \)), we obtain the frequency response of a system, which is critical for understanding its behavior with respect to input signal frequencies.
Solving Difference Equations: Many DSP problems are modeled by linear constant-coefficient difference equations, which can be solved efficiently using Z-Transforms.

Examples

Example 1: Analyzing a Discrete-Time System

Consider the difference equation \( y[n] = 0.5 y[n-1] + x[n] \), where \( x[n] \) is the input, and \( y[n] \) is the output.

Step 1: Apply the Z-Transform to both sides:

\[ Y(z) = 0.5z^{-1}Y(z) + X(z) \]

Step 2: Solve for \( Y(z) \):

\[ Y(z) = \frac{X(z)}{1 - 0.5z^{-1}} \]

Step 3: Interpret the result:

This transfer function represents a low-pass filter, with the pole at \( z = 0.5 \) indicating stability as it is inside the unit circle.

Example 2: Solving a Second-Order Difference Equation

Given the equation \( y[n] - 1.2y[n-1] + 0.32y[n-2] = x[n] \), where \( x[n] \) is the input, solve for the output response \( y[n] \).

Step 1: Apply the Z-Transform to both sides:

\[ Y(z) - 1.2z^{-1}Y(z) + 0.32z^{-2}Y(z) = X(z) \]