In a linear time-invariant system, the output response to a unit step input is obtained by convolving the step input with the system's impulse response h(t).

• A. The impulse response h(t) alone
• B. The derivative of the step input
• C. The convolution of the step input with the impulse response h(t)
• D. There is no relation to h(t)

Answer: (C)

In linear time-invariant systems, the output for any input is found by convolving the input with the system’s impulse response. For a unit step input, the output is the convolution of the unit step with the impulse response h(t): y(t) = (u * h)(t). This is the standard way to obtain the step response from the impulse response.

If the system is causal, this convolution reduces to y(t) = ∫₀^t h(τ) dτ, meaning the step response is the time integral of h(t). Intuitively, feeding a step turns the impulse response into a running sum (an accumulation) of the system’s response over time.

The other options don’t fit: the impulse response alone would be the output to a delta input, not a step; the derivative of the step input is a delta function and doesn’t directly give the step output; and there is indeed a direct relationship to h(t) for a step input, so claiming no relation is incorrect.