Question

Using the modified Program P3.1 compute and plot the frequency response of a causal LTI discrete-time system with a transfer function given by

H(z) = 0.15(1-z⁻²)/1-0.5z ⁻¹+0.7z⁻²

for 0 ≤ ω ≤ π. What type of filter does it represent?

Answer 1

To compute the frequency response of the given discrete-time system with the transfer function H(z), we substitute the complex exponential form of z into the transfer function and evaluate it for different frequencies. This transfer function represents a low-pass filter, allowing lower frequencies to pass through while attenuating higher frequencies.

Step-by-step explanation:

To compute the frequency response of a discrete-time system with the given transfer function H(z) = 0.15(1-z⁻²)/(1-0.5z⁻¹+0.7z⁻²), we need to evaluate the transfer function for various frequencies ω in the range 0 ≤ ω ≤ π. The frequency response of a system represents how the system processes different frequencies. The given transfer function represents a type of filter known as a low-pass filter, as it allows lower frequencies to pass through while attenuating higher frequencies.

1. Substitute the complex exponential form of z, z = e^(jω), into the transfer function H(z).

2. Simplify the transfer function by expressing it in terms of real and imaginary components.

3. Compute the magnitude of the transfer function, given by |H(z)|, for different frequencies ω in the range 0 ≤ ω ≤ π.

4. Plot the magnitude of the transfer function against frequency ω to visualize the frequency response of the system.