Question

The question consists of two statements, one labeled as ‘Statement (I)’ and the other labeled as ‘Statement (II)’. You are to examine these two statements carefully and select the answers to these items using the codes given below:

Statement (I): ROC of the z-transform of the signal x(n)= aⁿ u(n) + bⁿ u(-n-1) is  |a| < |z| < |b|

Statement (II): ROC of a causal infinite length sequence is |z| < r₁

1. Both Statement (I) and Statement (II) are individually true, but Statement (II) is the correct explanation of Statement (I)
2. Both Statement (I) and Statement (II) are individually true, but Statement (II) is not the correct explanation of Statement (I)
3. Statement (I) is true, but Statement (II) is false
4. Statement (I) is false, but Statement (II) is true
5.

Answer 1

Correct Answer - Option 3 : Statement (I) is true, but Statement (II) is false

We know that,

ROC of z-transform of aⁿ u(n) is |z| > |a|.

ROC of z-transform of bⁿ u(-n-1) is |z| < |b|.

By combining both the ROC’s we get the ROC of z-transform of the signal x(n) as |a| < |z| < |b|.

The ROC of the causal infinite sequence is of form |z| > r₁ where r₁ is the largest magnitude of poles

Properties of ROC: 

1. Linearity:

If Laplace Transform of x₁(t) is X₁(s) with ROC R₁ and Laplace transform of x₂(t) is X₂(s) with ROC R₂. Then 

Laplace transform of a x1(t + b x₂(t) ⇒ a X₁(s) + b X₂(s) with > ROC: R₁ ∩ R₂

2. Time-shifting:

 x(t) ↔ X(s) with > ROC : R

Then, x(t - t0) ↔ e-st0 X(s) with > ROC: R

3. Shift in S-domain:

 x(t) ↔ X(s) with ROC : R

Then, x(t) e^s₀t ↔ X (s - s₀) with > ROC: R + Re (s₀).

4. Time-reversal:

 x(t) ↔ X(s) with > ROC : R

Then, x(-t) ↔ X(-s) with ROC : - R

5. Differentiation in time:

 x(t) ↔ X(s) with ROC : R

Then, dx(t) / dt ↔ s X(s) with > ROC : R