What is the impulse response of a pure delay system h[n] = δ[n−N]?

• A. Output is x[n−N] but scaled by 2.
• B. Output is x[n−N] with a phase shift.
• C. Output is x[n+N].
• D. Output is x[n−N] with magnitude unchanged.

Answer: (D)

A pure delay shifts a signal in time without changing its shape or size. If the impulse response is h[n] = δ[n−N], then the system simply delays the input by N samples. In an LTI system, the output is the convolution y[n] = x[n] * h[n]. Convolving with δ[n−N] gives y[n] = x[n−N], i.e., the entire signal appears N samples later but with the same amplitude.

So the correct description is that the output is x[n−N] with magnitude unchanged. The other options would imply scaling, advancing the signal, or a different kind of effect, which a pure delay does not produce.