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?

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 an LTI system, the output is obtained by convolving the input with the impulse response. The impulse response h(t) represents the system’s reaction to a delta input, and due to linearity and time invariance, the overall output is built by summing the scaled and shifted copies of h(t) corresponding to every moment of the input. This yields the convolution integral y(t) = ∫ x(τ) h(t−τ) dτ. The convolution operation is the precise way to combine x(t) with h(t) to produce y(t), capturing how past and present input values influence the current output.

A simple product would ignore how the input affects the output at different times; a sum would just add signals without the system’s weighted, time-shifted response; differentiation doesn’t describe the general input-output relationship for an LTI system.