## Introduction to Sampling

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.2}of 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.

### Reconstruction from Samples--Pictorial Version

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*:

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

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

### 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].

**Next Section:**

Aliasing of Sampled Signals

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The Uncertainty Principle