18.4 : 混叠

Consider a time-domain signal and its Fourier transform to reveal the spectrum.

Sampling the signal at a specific frequency creates multiple scaled replicas of the original spectrum.

If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the recovery of the original signal using a low-pass filter.

This overlapping effect, known as aliasing, distorts the reconstructed signal.

Consider a sinusoidal signal and its spectrum, analyzing the sampled signal spectrum involves considering various values of the fundamental frequency with a fixed sampling frequency.

When the fundamental frequency is less than half the sampling frequency, increasing the fundamental frequency leads to a higher output frequency. 

Conversely, when the fundamental frequency is between half of the sampling frequency and the sampling frequency, increasing the fundamental frequency decreases the output frequency.

Because of aliasing, the reconstructed signal cannot return to its original form.

Accurate reconstruction of the original signal is only possible when the sampling frequency exceeds the Nyquist rate, thereby avoiding aliasing.