# Digital Audio: Part 3 - The Mathematics Of Sampling

Digital audio relies completely on sampling and no treatment of the subject can be complete without looking at how it works.

Fig.1. Sampling can be thought of as the audio waveform amplitude modulating a series of narrow pulses or needles, to produce a pulse-amplitude-modulated (PAM) signal.

Fig.2. Here, two different waveforms are represented by the same set of samples. Which is correct depends on the associated filters.

Fig.3. All sampling processes must be preceded and followed by appropriate filters. It is not meaningful to discuss sampling without considering the action of the filters.

Fig.4. When there is contra-rotation, as shown here, the horizontal processes are in opposition and cancel out, whereas the vertical process results in a sine wave having equal parts of positive and negative frequency. Modulation causes one sine wave to carry another so that negative frequencies subtract to produce lower side bands.

Fig.5. a) An un-modulated set of sampling needles has a spectrum like this. b) In conventional sampling, the base band is repeated as upper and lower sidebands around multiples of the sampling rate. c) The limit is reached at Fs/2, the Nyquist Frequency, when the lower sideband touches the baseband. d) An alternative sampling system in which the base band is band-limited. The lower side band is now below the baseband in the spectrum. The baseband can still be recovered with a band-pass filter.

Fig.6. In a reconstruction filter the zero crossings in the filter impulses align with adjacent samples so the output waveform effectively joins the samples into a continuous waveform identical within the available bandwidth to the original.

Fig.7. Transform duality holds that a square in either the frequency or time domains results in a sin/x in the other domain.

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