Z-Transform in Digital Signal Processing using MATLAB

Z-TRANSFORM

 THEORY: As discussed in the class, Z-transform is an extremely useful mathematical tool for the analysis and design of discrete time systems. Z-transform is defined as.

MATLAB Symbolic Toolbox gives the z-transform of a function

Example1:

To find out Z transform of the following and enter the following commands. 

>>syms z n

>> ztrans(1/4^n)

ans =

z/(z - 1/4)

 POLES AND ZEROS: H (z) is defined as the ratio of the z-transform of the output to the z-transform of the input when all initial conditions are zero.

 Mathematically,

 The roots of the denominator of H(z) are called “poles” and those of the numerator are called “zeros”. The above equation has M zeros and N poles. A system is said to be stable if all of its poles lie inside of a unit circle on the z-plane. Following example illustrates the use of MATLAB to compute zero and poles of a transfer function and to plot them onto the z-plane.

Example2: Find poles and zeros of the following pulse transfer function and plot them onto the z-plane.

num = [2 -1]; % numerator of H(z)

den = [1 -0.1 -0.02];% denominator of H(z)

zplane(num,den)

 On the z-plane, “x” indicates the poles and “0” indicate the zeros.

 Is this system stable? Why?

The given system is a stable system, because its poles are inside the unit circle.

Exercise

Q 1.                      Find Z transform of following

>>syms z n

>>ztrans(1+n)

>>syms z n

>>a =4;

>>ztrans(a^n+a^-n)

Q 2.                      Find poles and zeros plot of following

>>num = [2.25 -2.1 -3.95 -1.6 -0.2]; % numerator of H(z)

>>den = [4 -2.96 0.8 -0.1184 -0.0064];% denominator of H(z)

>>zplane(num,den)

>>num = [0 1 0.5]; % numerator of H(z)

>>den = [1 3/5 2/25];% denominator of H(z)

>>zplane(num,den)