Circular Convolution

When a complex sinusoid is given as an input to an LTI system, the output should be the same complex sinusoid scaled by the magnitude response of the system and phase-shifted by the phase response. This did not happen. Why?

DFT periodicity

Recall that sampling a signal in the time domain gives rise to spectral aliases in the frequency domain. Similarly, when we sample the spectrum in the frequency domain, time-domain aliases arise that actually make the signal periodic. This is why simple convolution, in which the output length is larger than both signals, cannot generate the correct answer.

Consider the process of convolution with a periodic signal as shown in the figure below. As soon as the one period ends, there is a short duration during which the convolving sequence overlaps with both the original signal and its copy in the time domain. These are the points a linear convolution cannot replicate.