aliasing

Assume that a signal to be digitized is well behaved, changing relatively slowly as a function of time.
Consider the cosine signals shown in figure below, where the x-axis shows sample intervals.
The top waveform is a cosine at the fraction 0.35 of the sampling rate \(f_S\); the middle waveform is at \(0.65f_S\).
The bottom row shows that identical samples result from sampling either of these waveforms: Either of the waveforms can masquerade as the same sample sequence.
If the middle waveform is sampled, then reconstructed conventionally, the top waveform will result.
This is the phenomenon of aliasing.

\[\omega=2\pi f_S \]

\[t_S=\frac{1}{f_S} \]