For a single-input, single-output time-invariant linear system, the impulse response h(t) is the system's output when the input is which signal?

For a single-input, single-output time-invariant linear system, the impulse response h(t) is the system's output when the input is which signal?

• A. A delta function.
• B. A unit step.
• C. A sine wave.
• D. A ramp.

Answer: (A)

In an LTI (linear time-invariant) system, the impulse response h(t) is defined as the output when the input is a Dirac delta signal δ(t). This is because, for any input x(t), the output is the convolution x(t) * h(t). The Dirac delta acts as the identity for convolution, so feeding δ(t) directly reveals h(t) itself: y(t) = δ(t) * h(t) = h(t). The other inputs produce different responses: a unit step gives the step response (the integral of h(t) if the system is causal), a sine wave produces a steady-state response related to the system’s frequency response, and a ramp yields a response tied to the system’s integration properties. None of these equal the impulse response itself.

In an LTI (linear time-invariant) system, the impulse response h(t) is defined as the output when the input is a Dirac delta signal δ(t). This is because, for any input x(t), the output is the convolution x(t) * h(t). The Dirac delta acts as the identity for convolution, so feeding δ(t) directly reveals h(t) itself: y(t) = δ(t) * h(t) = h(t).

The other inputs produce different responses: a unit step gives the step response (the integral of h(t) if the system is causal), a sine wave produces a steady-state response related to the system’s frequency response, and a ramp yields a response tied to the system’s integration properties. None of these equal the impulse response itself.