Inside computers and modern ``digital'' synthesizers, (as well as
music CDs), sound is
sampled into a stream of
numbers.
Each
sample can be thought of as a number which specifies the
position
D.2of a loudspeaker at a particular instant. When sound is sampled, we
call it
digital audio. The
sampling rate used for CDs nowadays
is 44,100 samples per second. That means when you play a CD, the
speakers in your stereo system are moved to a new position 44,100
times per second, or once every 23 microseconds. Controlling a
speaker this fast enables it to generate any sound in the human
hearing range because we cannot hear frequencies higher than around
20,000 cycles per second, and a
sampling rate more than twice the
highest frequency in the sound guarantees that exact reconstruction is
possible from the samples.

Figure
D.1 shows how a sound is reconstructed from its
samples. Each sample can be considered as specifying the
scaling and
location of a
sinc function. The
discrete-time
signal being interpolated in the figure is
a
digital rectangular pulse:
The
sinc functions are drawn with dashed lines, and they sum to
produce the solid curve. An isolated sinc function is shown in
Fig.
D.2. Note the ``Gibb's overshoot'' near the corners of the
continuous rectangular pulse in
Fig.
D.1 due to bandlimiting. (A true continuous rectangular
pulse has infinite
bandwidth.)
Figure D.1:
Summation of weighted sinc
functions to create a continuous waveform from discrete-time samples.
![\includegraphics[width=\twidth]{eps/SincSum}](http://www.dsprelated.com/josimages_new/mdft/img1765.png) |
Notice that each sinc function passes through zero at every sample
instant but the one it is centered on, where it passes through 1.
The Sinc Function
Figure:
The sinc function
sinc
.
![\includegraphics[width=\twidth]{eps/Sinc}](http://www.dsprelated.com/josimages_new/mdft/img1768.png) |
The sinc function, or
cardinal sine function, is the famous
``sine x over x'' curve, and is illustrated in Fig.
D.2. For
bandlimited
interpolation of discrete-time
signals, the ideal
interpolation kernel
is proportional to the sinc function
sinc
where

denotes the
sampling rate in samples-per-second (Hz), and

denotes time in seconds. Note that the sinc function has zeros at
all the integers except 0, where it is 1. For precise scaling, the
desired interpolation kernel is

sinc

, which has a
algebraic area (time integral) that is independent of the
sampling
rate

.
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
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:
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

, 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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