In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.

Multiplying the continuous-time signal by the impulse train results in a series of discrete impulses. This operation produces a sampled signal, which can be analyzed in the frequency domain using the Fourier transform. The Fourier transform reveals that the spectrum of the sampled signal consists of multiple shifted versions of the original signal's spectrum. These spectral copies are spaced apart by the sampling frequency.

A fundamental principle in sampling theory is that to avoid overlap between these shifted spectra, the sampling frequency must be sufficiently high. Specifically, the sampling frequency fsf_sfs​ must be greater than twice the highest frequency present in the original signal, a condition known as the Nyquist rate. When fsf_sfs​ meets or exceeds this rate, the spectra do not overlap, ensuring the original signal can be perfectly reconstructed from its samples. This requirement is encapsulated in the Sampling Theorem, which states that for a band-limited signal, the sampling frequency must be at least twice the highest frequency component of the signal.

When a signal is sampled at a frequency higher than the Nyquist rate, it is considered oversampled. Oversampling can provide benefits such as reduced noise and more straightforward digital filter design. Conversely, if the sampling rate is lower than the Nyquist rate, the signal is under-sampled, leading to a phenomenon known as aliasing. Aliasing causes different frequency components to become indistinguishable from each other, distorting the reconstructed signal.

In practical applications, adherence to the Nyquist rate is crucial for accurate digital representation and reconstruction of analog signals. This principle underpins various technologies, including digital audio, telecommunications, and medical imaging, ensuring that signals can be sampled, processed, and reconstructed without loss of critical information.