Question

Consider the statements given below: A. Discrete-time LTI system with impulse response $h[n] = \left(\frac{1}{2}\right)^n u[n-1]$ is causal B. Response $(-2)^n u[n]$ may or may not be stable C. Response $(-2)^n u[n]$ is unstable D. Response $(-2)^n u[-n-1]$ is unstable

• A. A, B and D only
• B. A and B only
• C. C and D only
• D. A and C only

Hint:

Stability → absolute summability.

Answer 1

The Correct Option is D

Concept:
• Causality: $h[n]=0$ for $n<0$
• Stability: $\sum |h[n]| < \infty$

Step 1: Statement A \[ h[n] = \left(\frac{1}{2}\right)^n u[n-1] \] Since $u[n-1]=0$ for $n<1$, hence for $n<0$ → zero. \[ \Rightarrow \text{System is causal} \]

Step 2: Statement B \[ (-2)^n u[n] \] Magnitude grows exponentially → not bounded. So cannot be stable. \[ \Rightarrow \text{B is FALSE} \]

Step 3: Statement C \[ (-2)^n u[n] \Rightarrow \sum |(-2)^n| = \infty \] \[ \Rightarrow \text{Unstable} \]

Step 4: Statement D \[ (-2)^n u[-n-1] \] Left-sided exponential but still diverging. \[ \Rightarrow \text{Unstable} \] Final: Correct statements: A and C \[ \therefore \text{Correct answer is (D)} \]