The impulse response h(t) is the system's output when the input is which signal?

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 linear time-invariant systems, the impulse response is the output that results when the input is an impulse δ(t). This is because the convolution y(t) = ∫ x(τ) h(t−τ) dτ, with x(t) = δ(t), becomes y(t) = δ(t) * h(t) = h(t). The delta input contains all frequencies equally, so the system’s response to it fully reveals its behavior. Therefore h(t) is exactly the system’s output to a delta input, and knowing h(t) lets you predict the output to any other input by convolving that input with h(t). The other signals produce different specific responses (step response, sine-wave response, ramp response), but they are not the impulse response itself.

In linear time-invariant systems, the impulse response is the output that results when the input is an impulse δ(t). This is because the convolution y(t) = ∫ x(τ) h(t−τ) dτ, with x(t) = δ(t), becomes y(t) = δ(t) * h(t) = h(t). The delta input contains all frequencies equally, so the system’s response to it fully reveals its behavior. Therefore h(t) is exactly the system’s output to a delta input, and knowing h(t) lets you predict the output to any other input by convolving that input with h(t). The other signals produce different specific responses (step response, sine-wave response, ramp response), but they are not the impulse response itself.