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.*,

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*
[70].

[How to cite this work] [Order a printed hardcopy]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]