Sampling Theorem

Consider the continuous-time signal 𝑥(𝑡), and a train of impulses where 𝑇_𝑠, is the sampling interval, and 𝑓_𝑠, is the sampling frequency.

Multiplying both signals results in a series of discrete impulses.

The Fourier transform shows that the spectrum of the sampled signal is a sum of shifted versions of the original signal's spectrum.

The spacing of these shifted versions is determined by the sampling frequency.

If the sampling frequency is greater than twice the highest frequency present in the original signal, these shifted spectra will not overlap.

This non-overlapping condition is crucial for the perfect reconstruction of the original signal from its samples.

The Sampling Theorem states that for a band-limited signal, the sampling frequency must be greater than twice the highest frequency in the signal.

This threshold of twice the highest frequency is known as the Nyquist rate, and exceeding this frequency ensures no loss of information during the sampling process.

A signal is considered oversampled if sampled at a rate greater than its Nyquist rate and undersampled if at a rate lower than the Nyquist rate.