Frequency Shifting and Modulation

Given a signal, a shift in time results in a product between the original signal spectrum and a frequency-domain complex sinusoid. This has a counterpart in the frequency domain, as follows.

Complex signals

If we multiply a signal $x[n]$ with a complex sinusoid in the time domain, we get a shift in the signal spectrum. Let’s call this new signal $y[n]$.

$$y[n]=x[n]\cdot e^{j2\pi\frac{k_0}{N}n}$$ ...