Key points you have learned

A discrete time sinusoid has discrete-time frequency w, where w has the interpretation as the fraction of a full period of the sinusoid between adjacent values.

The only interesting range of discrete-time frequency is 0<=w<0.5. In particuar, whenever we increase w by unity, the discrete-time sinusoid is not changed.

When a continuous-time sinusoid with frequency f is sampled with sampling interval T (sampling frequency 1/T), the result is a discrete-time sinusoid with discrete-time frequency w = f T.

If frequency of a continuous-time sinusoid f is increased by the sampling rate 1/T, the samples are unchanged. This is called aliasing.

Aliasing does not occur if 2f < 1/T; that is, the sampling rate is greater than twice the frequency of a sinusoid. In this case, the sinusoid can be reconstructed from its samples using a superposition of Sinc[] pulses.

A continuous-time signal consisting of a superposition of sinusoids is bandlimited to f0, where f0 is the highest-frequency-sinusoid. A signal bandlimited to f0 can be recontructed from its samples if 2 f0 < 1/T; that is, the sampling rate is greater than twice the bandwidth.