Which expression represents the discrete-time convolution sum?

Which expression represents the discrete-time convolution sum?

• A. y[n] = ∑_{k=−∞}^{∞} x[k] h[n−k].
• B. y[n] = ∑ x[n−k] h[k].
• C. y[n] = x[n] h[n].
• D. y[n] = ∑ x[k] h[k−n].

Answer: (A)

The main idea is that discrete-time convolution builds the output at each time n by sliding one sequence across the other, multiplying overlapping samples, and summing those products. This is captured by y[n] = sum over all k of x[k] times h[n − k]. Here, k indexes every sample of x, while h[n − k] picks the sample of h that aligns with x[k] when the second sequence is shifted by n. Only the overlapping (nonzero) parts contribute, so the infinite sum effectively reduces to a finite sum if the signals have finite support. This form directly expresses the sliding-and-mumming process, which is why it’s the standard representation. Other formulations are equivalent in value in many cases (for example, due to commutativity, h can be slid across x as in x[n − k] h[k]), but they are less straightforward as a concrete, step-by-step description of the convolution. A simple pointwise product x[n] h[n] is just a single sample, not a sum over shifts, so it does not represent the convolution.

The main idea is that discrete-time convolution builds the output at each time n by sliding one sequence across the other, multiplying overlapping samples, and summing those products. This is captured by y[n] = sum over all k of x[k] times h[n − k]. Here, k indexes every sample of x, while h[n − k] picks the sample of h that aligns with x[k] when the second sequence is shifted by n. Only the overlapping (nonzero) parts contribute, so the infinite sum effectively reduces to a finite sum if the signals have finite support.

This form directly expresses the sliding-and-mumming process, which is why it’s the standard representation. Other formulations are equivalent in value in many cases (for example, due to commutativity, h can be slid across x as in x[n − k] h[k]), but they are less straightforward as a concrete, step-by-step description of the convolution. A simple pointwise product x[n] h[n] is just a single sample, not a sum over shifts, so it does not represent the convolution.