Signals & Systems Lecture 15

ZTransform

Z-transform

The Z-transform is a powerful tool used in signal processing and control theory, primarily for analyzing discrete-time systems. It is a generalization of the discrete-time Fourier transform (DTFT) and allows us to work with discrete signals in the frequency domain.

Definition

For a discrete-time signal ( x[n] ), the Z-transform ( X(z) ) is defined as:

$$X(z)=\sum _{n=-\infty }^{\infty }x[n]z^{-n}$$

where:

• ( x[n] ) is the discrete-time signal (sequence),
• ( z ) is a complex variable.

Region of Convergence (ROC)

The Z-transform only exists if the sum converges. The set of values of ( z ) for which the Z-transform converges is called the Region of Convergence (ROC).

Inverse Z-transform

The inverse Z-transform is used to convert back from the Z-domain to the time domain. It is given by:

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

where ( C ) is a contour in the complex plane that encircles the origin and lies within the ROC.

Properties of Z-transform

1. Linearity: $Z\{a{x}_1[n]+b{x}_2[n]\}=a{X}_1(z)+b{X}_2(z)$

2. Time Shifting: $Z\{x[n-k]\}=z^{-k}X(z)$

3. Scaling in the Z-domain: $Z\{a^{n}x[n]\}=X\left(\frac{z}{a}\right)$

4. Convolution: $Z\{x_1[n]\ast x_2[n]\}=X_1(z)X_2(z)$

5. Differentiation in the Z-domain: $Z\{nx[n]\}=-z\frac{dX(z)}{dz}$

Applications of Z-transform

• Analysis and design of discrete-time control systems.
• Solving difference equations.
• Digital signal processing (DSP).

Questions

Question 1

$X(N)=u(n)$ The Z-transform of the unit step function ( $u(n)$ ) is given by:

$$Z\{u(n)\}=\sum _{n=0}^{\infty }u(n)z^{-n}=\sum _{n=0}^{\infty }{z}^{-n}$$

This is a geometric series, and for ( |z| > 1 ), it converges to:

$$Z\{u(n)\}=\frac{1}{1-z^{-1}}=\frac{z}{z-1}$$

Thus, the Z-transform of ( u(n) ) is:

$$X(z)=\frac{z}{z-1},\text{for}|z|>1$$

Question 2

The given sequence is ( $x(n)=(0.5z^{-1}{)}^n$ ). To find its Z-transform, we use the standard formula for the Z-transform of ( a^n u(n) ), where ( u(n) ) is the unit step function.

For ( $x(n)=(0.5z^{-1}{)}^n={\left(\frac{0.5}{z}\right)}^n$ ), the Z-transform is:

$$Z\{(0.5z^{-1}{)}^n\}=\sum _{n=0}^{\infty }{\left(\frac{0.5}{z}\right)}^n$$

This is a geometric series and converges for ( $\left|\frac{0.5}{z}\right|<1$), or ( $|z|>0.5$ ). The sum of the geometric series is:

$$X(z)=\frac{1}{1-\frac{0.5}{z}}=\frac{z}{z-0.5}$$

Thus, the Z-transform of ( $(0.5z^{-1}{)}^n$) is:

$$X(z)=\frac{z}{z-0.5},\text{for}|z|>0.5$$

Question 3