Which expression describes how to obtain the output y(t) from the input x(t) and the impulse response h(t) in an LTI system?

• A. Convolution of x(t) with h(t).
• B. Product of x(t) and h(t).
• C. Sum of x(t) and h(t).
• D. Differentiation of x(t) with respect to h(t).

Answer: (A)

In linear time-invariant systems, the output is obtained by convolving the input with the impulse response. The impulse response h(t) is the system’s reaction to a delta impulse, so the input x(t) can be seen as a continuous sum of shifted impulses weighted by x at each time τ. The response to an impulse at time τ is h(t−τ), and summing all these shifted responses over all τ gives y(t) = ∫ x(τ) h(t−τ) dτ. This convolution process embodies superposition and the way a past input influence spans time, exactly describing how the output arises from the input and the system’s impulse response. The other operations don’t capture this spreading and accumulation: multiplying x(t) and h(t) would not reflect the time-spread effect, adding them would ignore the system’s memory, and differentiating with respect to h(t) isn’t the mechanism by which the input maps to the output.