Sampling Theorem

The Nyquist rate is the minimum sampling frequency required to perfectly reconstruct a band-limited signal. It must be at least twice the highest frequency present in the original signal. Meeting this criterion ensures no information loss during sampling and prevents spectral overlap in the frequency domain.

Multiplying a continuous time signal by an impulse train produces a series of discrete impulses at specific time intervals determined by the sampling interval. This operation converts the continuous signal into discrete samples that can be analyzed and processed digitally using the Fourier transform.

The Fourier transform reveals that the sampled signal's spectrum consists of multiple shifted versions of the original signal's spectrum, spaced apart by the sampling frequency. If the sampling frequency exceeds twice the highest frequency, these shifted spectra do not overlap, enabling perfect signal reconstruction.

Oversampling occurs when the sampling frequency exceeds the Nyquist rate, providing benefits like reduced noise and simpler filter design. Undersampling happens when the sampling rate falls below the Nyquist rate, causing aliasing where different frequency components become indistinguishable, distorting the reconstructed signal.

The Sampling Theorem guarantees that when sampling frequency meets or exceeds twice the highest signal frequency, the spectral copies do not overlap. This non-overlapping condition is crucial for perfect reconstruction of signal using interpolation from discrete samples without information loss.

If sampling frequency is less than twice the highest frequency, shifted spectral versions overlap, causing information loss and aliasing. This overlap makes it impossible to distinguish original frequency components, corrupting the signal. The 2x requirement ensures spectral separation and accurate digital representation.

The impulse train, characterized by the sampling interval and sampling frequency, acts as a mathematical tool to extract discrete samples from a continuous signal. When multiplied with the original signal, it creates discrete impulses at regular intervals, forming the foundation for digital signal processing and frequency domain analysis.