In a linear time-invariant system with a single input and output, the relation between input x(t) and output y(t) is given by which expression?

• A. y(t) equals x(t) convolved with h(t).
• B. y(t) equals x(t) times h(t).
• C. y(t) equals the derivative of x(t).
• D. y(t) equals the integral of x(t).

Answer: (A)

In a linear time-invariant system, the output is found by convolving the input with the system’s impulse response. The impulse response h(t) is how the system responds to a unit impulse δ(t). Because the system is linear, any input can be built from scaled and shifted impulses, and by time invariance, each shifted impulse response is just a shifted copy of h(t). The total output is the integral of these shifted responses, giving y(t) = ∫ x(τ) h(t−τ) dτ, the convolution of x with h. This convulsive form captures how the system blends contributions from all past inputs according to its impulse response. The other forms would correspond to specific, particular systems (like a derivative or an integral) and do not describe the general LTI relationship.