Find z-transform and plot it's pole-zero map with MATLAB

2
votes

I have this function:

x[n] = (1/2) ^ n * u[n] + (-1/3) ^ n * u[n]

I need to do two things with this using MATLAB:

  1. Find it's z-transform.

  2. Plot it's poles and zeros.

I am using the following code:

syms n;
f = (1/2)^n + (-1/3)^n;
F = ztrans(f);

I get the z-transform in the F variable, but I can't see how to create it's pole-zero plot. I am using the built-in function pzmap (pzmap(F);), but it doesn't seem to work with the output of ztrans(f).

What am I doing wrong? Do I need to change the z-transform into some other form like like a transfer function model or a zero-pole gain model? If so, can someone explain how that can be done using the output of ztrans(f)?

1
matlabplot

1 Answers

1
votes

The first bit of code you gave uses symbolic math to solve for the z-transform. You'll need to convert the output to a discrete-time model supported by the Control System toolbox.

syms n;
f = (1/2)^n + (-1/3)^n;
F = ztrans(f)

returns z/(z - 1/2) + z/(z + 1/3). You can optionally use collect to convert this

F2 = collect(F)

to (12*z^2 - z)/(6*z^2 - z - 1). Then you'll want to find the coefficients of the polynomials in the numerator and denominator and create a discrete-time transfer function object with tf for a particular sampling period:

[num,den] = numden(F2);
Ts = 0.1; % Sampling period
H = tf(sym2poly(num),sym2poly(den),Ts)

Then pzmap(H) will produce a plot like this:

Pole-Zero Map plot