Question

Direction: Given question consists of two statements, one labeled as the 'Assertion (A)' and the other as 'Reason (R)'.

You are to examine these two statements carefully and select the answers to these items using the codes given below.

Assertion (A): The system function \(H(s) = \frac {z^3 - 2z^2 + z}{z^2 + \frac 1 4 z + \frac 18}\) is not causal.

Reason (R): If the numerator of H(s) is of lower order than the denominator, the system may be causal.

1. Both A and R are individually true and R is the correct explanation of A
2. Both A and R are individually true but R is NOT the correct explanation of A
3. A is true but R is false
4. A is false but R is true

Answer 1

Correct Answer - Option 1 : Both A and R are individually true and R is the correct explanation of A

Concept:

Causal – degree (order) of denominator ≥ degree of the numerator

Non-causal – degree of denominator < degree of the numerator

Analysis:

H(z) \( = \frac{{{z^3} - 2{z^2} + z \to numerator}}{{{z^2} + \frac{1}{4}z + \frac{1}{8} \to denominator}}\)

Degree of denominator = 2

Degree of numerator = 3

Since the degree of D^r < degree of N^r. Hence, It is a non-causal system.

Eg.

 \(H\left( z \right) = \frac{{{z^3} + 4{z^2} + 5z + 9}}{{{z^2} + 4z + 5}} = z + k + \ldots {z^{ - 1}} + \ldots {z^{ - 2}}\)

When the degree of N^r > degree of D^r then the positive power of z comes that’s why the system will be non-causal.

\(H\left( z \right) = \frac{{z + 1}}{{{z^2} + 2z + 1}} = {z^{ - 1}} + {z^{ - 2}} \ldots \)^r

In this case degree of D^r  > degree of N^r then only negative power of z appears that’s why the system will be causal.