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?

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)

For a linear time-invariant system, the output is obtained by convolving the input with the system’s impulse response. The relationship is y(t) = ∫ x(τ) h(t − τ) dτ. This means every value of the input acts like a scaled copy of the impulse response shifted in time, and the overall output is the sum of all those shifted responses. This convolution reflects both linearity (superposition of scaled responses) and time invariance (shifts in time of the input produce corresponding shifts in the response).

In short, the output is not just a simple product or a derivative or an integral of the input in general; it’s the convolution with the impulse response, which generalizes all linear, time-invariant behavior. (Special cases exist: a differentiator corresponds to h(t) = δ′(t), an integrator to h(t) = u(t), etc., but the general relation is the convolution.)