The convolution sum formula y[n] = sum_k x[k] h[n-k] is used to compute the output of a discrete-time system; which property must hold for this to be valid?

The convolution sum formula y[n] = sum_k x[k] h[n-k] is used to compute the output of a discrete-time system; which property must hold for this to be valid?

• A. The system must be linear and time-invariant
• B. The system must be nonlinear
• C. The input must be zero
• D. The sampling rate must be infinite

Answer: (A)

Convolution sum expresses how a discrete-time system maps input x[n] into output y[n] by sliding the impulse response h[n] across time, weighting each shifted copy by the corresponding input sample, and summing. For this to work, the system must be linear and time-invariant. Linearity ensures that the response to a sum of inputs is the sum of responses, so the effect of each input sample x[k] is a scaled and shifted version of h[n], and time invariance ensures that shifting the input in time produces a proportional shift in the output without changing shape. Under these conditions, the impulse response h[n] fully characterizes the system, and the output is exactly y[n] = sum_k x[k] h[n-k]. If the system is nonlinear or time-varying, this representation breaks down because responses to scaled or shifted inputs are not predictable solely from h[n]. The input need not be zero, and the sampling rate being infinite is not a requirement for the validity of the convolution expression; discrete-time convolution assumes samples exist, and the rate affects interpretation, not the fundamental relation.