Question

A causal, discrete time system is described by the difference equation \[ y[n] = 0.5 \, y[n-1] + x[n], \text{for all n, \] where \(y[n]\) denotes the output sequence and \(x[n]\) denotes the input sequence. Which of the following statements is/are TRUE?}

• A. The system has an impulse response described by \(0.5^n u[-n]\) where \(u[n]\) is the unit step sequence.
• B. The system is stable in the bounded input, bounded output sense.
• C. The system has an infinite number of non-zero samples in its impulse response.
• D. The system has a finite number of non-zero samples in its impulse response.

Hint:

- Impulse responses of IIR systems typically have infinite duration.
- BIBO stability requires that poles lie inside the unit circle in the \(z\)-plane.

Answer 1

The Correct Option is B, C

Step 1: The system equation is: \[ y[n] = 0.5 y[n-1] + x[n]. \] This is a first-order linear constant coefficient difference equation (an IIR system).
Step 2: To find the impulse response, set \(x[n] = \delta[n]\). Then the system recursion gives: \[ h[n] = 0.5 h[n-1] + \delta[n]. \] For \(n = 0\): \(h[0] = 1\).
For \(n = 1\): \(h[1] = 0.5 h[0] = 0.5\).
For \(n = 2\): \(h[2] = 0.5 h[1] = 0.25\).
Continuing: \[ h[n] = (0.5)^n u[n]. \] Step 3: Now check statements:
(A) Wrong, because impulse response is \(0.5^n u[n]\), not \(u[-n]\).
(B) True, because the pole of the system is at \(z = 0.5\), inside the unit circle. Therefore, the system is BIBO stable.
(C) True, because the impulse response \((0.5)^n u[n]\) has infinite length (non-zero for all \(n \geq 0\)).
(D) False, since it has infinite length, not finite.
\[ \boxed{\text{Correct statements: (B) and (C)}} \]