Question

(Please work on this problem using Matlab. Attach the Matlab code to the end). Suppose we have the transfer function of a causal LTI system as

H(z)= 1−2.9z¹ + z² / 1−0.21z¹ −0.3666² − 0.039z³ + 0.0077z⁴
(a) (10%) Draw the pole-zero plot of H(z).

Answer 1

To draw the pole-zero plot of the given transfer function in Matlab, identify the values of z where the numerator and denominator of H(z) equal zero, and plot these on the complex plane.

Step-by-step explanation:

To draw the pole-zero plot of the transfer function H(z), we need to identify the values of z where the numerator and denominator of H(z) equal zero.

For the numerator, z² - 2.9z + 1 = 0, we can find the roots as z = 1 and z = 1.9.

For the denominator, z⁴ - 0.039z³ - 0.3666z² - 0.21z + 1 = 0, we can use software like MATLAB to find the roots: z = 0.4812, z = 0.4336 + 0.5793i, z = 0.4336 - 0.5793i, and z = -0.3487.

Plotting these values on the complex plane will give us the pole-zero plot of H(z).

Answer 2

Here's the MATLAB code to generate the pole-zero plot:

```matlab

% Define the numerator and denominator coefficients

num = [1, -2.9, 1]; % Coefficients of the numerator

den = [1, -0.21, -0.3666, -0.039, 0.0077]; % Coefficients of the denominator

% Create a transfer function using the coefficients

H = t.f.(num, den);

% Plot the pole-zero plot

figure;

zplane(num, den);

title('Pole-Zero Plot of H(z)');

xlabel('Real');

ylabel('Imaginary');

```

Step-by-step explanation:

The pole-zero plot visualizes the behavior of an LTI system by showing the locations of its poles (denoted by 'x') and zeros (denoted by 'o') in the complex plane. In this case, the plot displays the poles and zeros of the transfer function H(z). Poles represent system stability and dictate its response, while zeros affect the system's frequency response.

The plot aids in understanding system characteristics: stable systems have poles within the unit circle, zeros influence frequency response, and pole-zero patterns indicate system behavior like resonances or dampening. This visual representation assists in analyzing system stability and frequency behavior.