### Reconstruction from Samples--The Math

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.*,

*bandlimited*to less than half the sampling rate if and only if for all . In this case, the sampling theorem gives us that can be uniquely reconstructed from the samples by summing up shifted, scaled, sinc functions:

*ideal lowpass filter*. This means its Fourier transform is a rectangular window in the frequency domain. The particular sinc function used here corresponds to the ideal lowpass filter which cuts off at half the sampling rate. In other words, it has a gain of 1 between frequencies 0 and , and a gain of zero at all higher frequencies.

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*
[72].

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Continuous-Time Aliasing Theorem

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The Sinc Function