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
where
The sinc function is the impulse response of the 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
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].