Nyquist theorem

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Nyquist Theorem

A theorem stating that when an analogue waveform is digitised, only the frequencies in the waveform below half the sampling frequency will be recorded. In order to reconstruct (interpolate) a signal from a sequence of samples, sufficient samples must be recorded to capture the peaks and troughs of the original waveform. If a waveform is sampled at less than twice its frequency the reconstructed waveform will effectively contribute only noise. This phenomenon is called "aliasing" (the high frequencies are "under an alias").

This is why the best digital audio is sampled at 44,000 Hz - twice the average upper limit of human hearing.

The Nyquist Theorem is not specific to digitised signals (represented by discrete amplitude levels) but applies to any sampled signal (represented by discrete time values), not just sound.

Nyquist (the man, somewhat inaccurate).
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Nyquist theorem

The concept behind digitizing sound. Working at Bell Labs, Harry Nyquist discovered that it was not necessary to capture the entire analog waveform, and samples of the wave could be taken at various points. He also found that in order to have enough information in the sample pool to reconstruct the original waveform, the sampling rate must be at least twice the signal bandwidth.

The Basis for PCM
These realizations became the foundation for using pulse code modulation (PCM) to convert analog sound to digital in North America and Japan. A typical 4 kHz voice signal is sampled 8,000 times a second, with each sample converted into an 8-bit number, resulting in a 64 Kbps data stream (a single DS0 channel). See sampling and PCM.

The Human Ear
It was believed that people could not hear a frequency greater than 20 kHz (20,000 cycles per second). However, this number was always an approximation and has been challenged since the first music CDs came on the market in the mid-1980s. To deliver 20 kHz to the human ear, analog waves are sampled at 44.1 kHz for a regular music CD. Next-generation audio takes samples more frequently (see SACD and DVD-Audio.
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Thereby, this method provides a new approach of receiving echo data via 2-D random sparse sampling with a significant reduction in the number of sampled data beyond the Nyquist theorem. This will directly impact A/D conversion, and has the potential to reduce the overall data rate and to simplify hardware design.
Interestingly, since undersampling violates the Nyquist theorem, you may wonder if something is lost in the signal information, and if so, what is it?
Nyquist Theorem The requirement to sample a signal at a rate at least twice its maximum frequency.