Aliasing

Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.

If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal from being accurately recovered using a low-pass filter. This overlapping effect, known as aliasing, distorts the reconstructed signal and makes it impossible to recover the original signal.

To analyze the spectrum of the sampled signal, one must consider the fundamental frequency and how it interacts with a fixed sampling frequency. When the fundamental frequency of the signal is between half the sampling frequency and the sampling frequency itself, any increase in the fundamental frequency will paradoxically result in a decrease in the perceived output frequency. This counterintuitive effect is due to aliasing, where higher frequencies are indistinguishable from lower frequencies after sampling. Consequently, the reconstructed signal is significantly distorted and cannot return to its original form.

Conversely, if the fundamental frequency is less than half the sampling frequency, increasing the fundamental frequency results in an increase in the output frequency. This behavior aligns with expectations and allows for a clearer reconstruction of the original signal. Therefore, for accurate reconstruction, the sampling frequency must exceed the Nyquist rate, which is twice the highest frequency present in the original signal. By meeting or exceeding this rate, aliasing is avoided, and the replicas in the frequency domain do not overlap.

Adhering to the Nyquist criterion ensures the sampling frequency is high enough to capture the necessary information from the original signal, making accurate signal reconstruction possible. This principle is critical in various applications such as audio processing, telecommunications, and data acquisition, where maintaining signal integrity is paramount. Avoiding aliasing by using an appropriate sampling frequency allows for the faithful recovery of the original time-domain signal, preserving its quality and fidelity.