What is the impulse response of an LTI system and how is the output computed for any input?

• A. h(t) is the impulse response; y(t) = x(t) * h(t) (CT) or y[n] = x[n] * h[n] (DT).
• B. h(t) is the input signal; y(t) = x(t) h(t).
• C. h(t) is the output signal; y(t) = h(t) * h(t).
• D. h(t) is the impulse input to the system only for DT.

Answer: (A)

An LTI system’s impulse response is the output produced when the input is a Dirac delta. Because the system is linear and time-invariant, any input can be built as a sum of shifted impulses, and the total output is the sum of the corresponding impulse responses. That sum is exactly the convolution of the input with the impulse response: in continuous time, y(t) = ∫ x(τ) h(t − τ) dτ; in discrete time, y[n] = Σ x[k] h[n − k]. This explains why h(t) fully characterizes the system and why the output for any input is found by convolving the input with h. The other options mistreat the roles (multiplication instead of summation; mislabeling h; or restricting the impulse input to discrete time).