Question

A time-continuous sinusoidal signal of frequency f0 = 32kHz is sampled with ft = 48kHz without an anti-alias filter preceding the AD converter (no sigma-delta converter).

Is the sampling theorem satisfied or violated? Justify your answer.

Answer 1

The sampling theorem is violated because the signal with a frequency of 32kHz is being sampled at a rate of 48kHz, which is less than the required 64kHz, resulting in the potential for aliasing.

Step-by-step explanation:

The sampling theorem, also known as Nyquist theorem, states that a time-continuous sinusoidal signal must be sampled at a rate that is at least twice the frequency of the signal (the Nyquist rate) to be accurately reconstructed. In this case, the signal frequency, f0, is 32kHz. Therefore, according to the sampling theorem, the sampling frequency, ft, should be at least 64kHz to satisfy the theorem and avoid aliasing.

However, the signal is being sampled at 48kHz, which is less than the required 64kHz. This means that the sampling theorem is violated, and aliasing can occur because the sampler cannot distinguish between the original frequency and another frequency that is an integer multiple of the sampling rate minus the original frequency. Without an anti-alias filter, this can cause distortions in the digitized signal, as higher frequency components of the signal get folded back into the lower frequencies.