Question

Consider a discrete-time causal LTI system with the transfer function H(z) = 1 + 0.5z⁻¹ + (1 - 0.8z⁻¹)/(1 - 1.8cosθz⁻¹ + 0.81z⁻²)/(1 - 0.9cosθz⁻¹), where θ ≠ 0.

a) Identify and illustrate all poles of H(z).

Answer 1

To identify the poles of the transfer function H(z), we need to solve the equations where the denominator is equal to zero. In this case, we need to solve a quadratic equation. The values of z that satisfy the quadratic equation correspond to the poles of H(z).

Step-by-step explanation:

The poles of a transfer function in a discrete-time causal LTI system are the values of z that make the denominator of the transfer function equal to zero. In this case, the transfer function is H(z) = 1 + 0.5z⁻¹ + (1 - 0.8z⁻¹)/(1 - 1.8cosθz⁻¹ + 0.81z⁻²)/(1 - 0.9cosθz⁻¹). To identify the poles, we need to solve the equations where the denominator is equal to zero.

To illustrate the poles, we can first express the transfer function in a factored form, if possible. Then, the poles are the values of z that make each factor equal to zero.

In this case, the transfer function cannot be factored further, so we need to solve the entire denominator for zero. This involves solving a quadratic equation. By setting the denominator equal to zero and solving for z, we can find the values of z that correspond to the poles of H(z).