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 follows the same fundamental idea as in continuous time: the output of an LTI system is found by combining the input with the system’s impulse response through a defined operation. In discrete time, this operation is y[n] = sum over k of x[k] h[n - k], written compactly as y = x * h. This process is linear, and it is both commutative and associative, so swapping the order of signals or chaining multiple convolutions yields the same result. The impulse response fully characterizes the system, and the convolution formula shows exactly how the input is blended with shifted copies of h. It’s not a simple time-domain product, and it works with non-periodic signals as well (periodic cases use a periodic version of convolution). The statement that best captures this behavior is that convolution is commutative, associative, and linear, yielding y[n] = x[n] * h[n].