Question

determine the impulse response and the step response of the following causal systems plot the pole-zero patterns and determine which are stable

y(n) = 3/4y(n - 1)- 1/8y(n - 2) + x(n)

Answer 1

SOLUTION-

The system is,

The Z transform of is,

The transfer function is,

Since the Z transform of is, ,

Hence the impulse response of the system is,

Thus, the impulse response of the system is .

The poles of the system are located at,

and

Plot the pole-zero as shown in Figure 1.

Since the system is causal and the poles of are inside the unit circle, the ROC of the system includes the unit circle. Hence the system is stable.

Find the step response of the system.

The Z transform of is, .

The output function of the system is,

Consider the partial fraction expansion,

The step response of the system is,

Thus, the step response of the system is .

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