In a discrete-time linear time-invariant system, the output y[n] equals the convolution of the input x[n] with the impulse response h[n].

• A. True
• B. False
• C. Only for stable systems
• D. Only for causal systems

Answer: (A)

The main idea tested is that a discrete-time linear time-invariant (LTI) system outputs the input convolved with its impulse response. Because the system is linear, you can break any input x[n] into a sum of scaled shifted impulses: x[n] = sum_k x[k] δ[n − k]. Time invariance means the response to each shifted impulse δ[n − k] is simply h[n − k]. By superposition, the overall output becomes y[n] = sum_k x[k] h[n − k], which is exactly the discrete convolution x[n] * h[n].

This relationship holds for any input x[n] and any impulse response h[n], independent of stability or causality. Stability (BIBO) concerns whether the output remains bounded for bounded inputs and ties to the convergence of the convolution sum (usually requiring h[n] to be absolutely summable). Causality is about whether the impulse response is zero for n < 0. Neither condition changes the fact that the output is given by the convolution with the impulse response.