Let
denote the
th sample of the original
sound
, where
is time in seconds. Thus,
ranges over the
integers, and
is the *sampling interval* in seconds. The
*sampling rate* in Hertz (Hz) is just the reciprocal of the
sampling period,
*i.e.*,

To avoid losing any information as a result of sampling, we must
assume
is *bandlimited* to less than half the sampling
rate. This means there can be no energy in
at frequency
or above. We will prove this mathematically when we prove
the *sampling theorem* in §D.3 below.

Let
denote the Fourier transform of
, *i.e.*,

Then we can say is

where

The sinc function is the impulse response of the

The reconstruction of a sound from its samples can thus be interpreted
as follows: convert the sample stream into a *weighted impulse
train*, and pass that signal through an ideal lowpass filter which
cuts off at half the sampling rate. These are the fundamental steps
of
*digital to analog conversion* (DAC). In practice,
neither the impulses nor the lowpass filter are ideal, but they are
usually close enough to ideal that one cannot hear any difference.
Practical lowpass-filter design is discussed in the context of
*bandlimited interpolation*
[75].

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