If a discrete-time LTI system has impulse response h[n] = δ[n−N], the output to x[n] is:

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Multiple Choice

If a discrete-time LTI system has impulse response h[n] = δ[n−N], the output to x[n] is:

Explanation:
Time-shifting through convolution with a delayed impulse. In a discrete-time LTI system, the output is the convolution of the input with the impulse response: y[n] = x[n] * h[n]. If h[n] = δ[n − N], then y[n] = sum_k x[k] δ[n − k − N]. The delta function selects k = n − N, giving y[n] = x[n − N]. This shows the system simply delays the input by N samples. So the correct output is the input shifted to the right by N samples, i.e., x[n − N].

Time-shifting through convolution with a delayed impulse. In a discrete-time LTI system, the output is the convolution of the input with the impulse response: y[n] = x[n] * h[n]. If h[n] = δ[n − N], then y[n] = sum_k x[k] δ[n − k − N]. The delta function selects k = n − N, giving y[n] = x[n − N]. This shows the system simply delays the input by N samples. So the correct output is the input shifted to the right by N samples, i.e., x[n − N].

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