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].

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)

In a discrete-time LTI system, the output is the convolution of the input with the impulse response because linearity lets us break any input into a sum of scaled, shifted impulses, and time invariance makes the response to each shifted impulse just a shifted version of the system’s impulse response. Specifically, any input x[n] can be written as a sum x[n] = sum_k x[k] δ[n − k]. By linearity, the response to δ[n − k] is h[n − k], so the total output is y[n] = sum_k x[k] h[n − k], which is precisely the convolution x[n] * h[n].

The impulse response h[n] is defined as the system’s output to δ[n]. This relationship holds regardless of stability or causality. Stability affects whether the convolution sum converges for a given input, and causality affects the support of h[n], but the basic equation y[n] = x[n] * h[n] remains valid for an LTI system.