## 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*:

### 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

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 .
### 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.*,

*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