Discrete-time convolution has the same fundamental relationship as continuous-time convolution. Which statement is true?

Discrete-time convolution has the same fundamental relationship as continuous-time convolution. Which statement is true?

• A. It yields a time-domain product of inputs
• B. It is commutative, associative, and linear, yielding y[n] = x[n] * h[n].
• C. It does not relate to impulse response
• D. It cannot be used with non-periodic signals

Answer: (B)

Discrete-time convolution describes how an LTI system combines an input sequence with its impulse response to produce an output sequence. If the system has impulse response h[n], the output is y[n] = x[n] * h[n] = sum over k of x[k] h[n − k]. This shows the process as a weighted sum of shifted copies of the impulse response, with each input sample x[k] acting as the weight for the impulse response shifted by k. Because the system is linear and time-invariant, convolution has key properties: it is commutative (x * h = h * x), so the order doesn’t matter; it is associative, so you can chain convolutions without changing the result; and it is linear, so superpositions of inputs produce proportional superpositions of outputs. This combination is what makes y[n] = x[n] * h[n] the correct general description of how the output relates to the input and the impulse response. The other statements don’t fit. Convolution is not simply a time-domain product of the inputs; it’s a sum of products that blends the input with shifted versions of the impulse response. It clearly relates to the impulse response, since h[n] defines the system’s behavior and appears in the convolution. And convolution is not restricted to periodic signals; it applies to non-periodic (finite or infinite) sequences as well.

Discrete-time convolution describes how an LTI system combines an input sequence with its impulse response to produce an output sequence. If the system has impulse response h[n], the output is y[n] = x[n] * h[n] = sum over k of x[k] h[n − k]. This shows the process as a weighted sum of shifted copies of the impulse response, with each input sample x[k] acting as the weight for the impulse response shifted by k.

Because the system is linear and time-invariant, convolution has key properties: it is commutative (x * h = h * x), so the order doesn’t matter; it is associative, so you can chain convolutions without changing the result; and it is linear, so superpositions of inputs produce proportional superpositions of outputs. This combination is what makes y[n] = x[n] * h[n] the correct general description of how the output relates to the input and the impulse response.

The other statements don’t fit. Convolution is not simply a time-domain product of the inputs; it’s a sum of products that blends the input with shifted versions of the impulse response. It clearly relates to the impulse response, since h[n] defines the system’s behavior and appears in the convolution. And convolution is not restricted to periodic signals; it applies to non-periodic (finite or infinite) sequences as well.