# Sampling Theory

In this appendix, sampling theory is derived as an application of the
DTFT and the Fourier theorems developed in Appendix C. First, we
must derive a formula for *aliasing* due to uniformly sampling a
continuous-time signal. Next, the *sampling theorem* is proved.
The sampling theorem provides that a properly bandlimited
continuous-time signal can be sampled and reconstructed from its
samples without error, in principle.

An early derivation of the sampling theorem is often cited as a 1928
paper by Harold Nyquist, and Claude Shannon is credited with reviving
interest in the sampling theorem after World War II when computers
became public.^{D.1}As a result, the sampling theorem is often called
``Nyquist's sampling theorem,'' ``Shannon's sampling theorem,'' or the
like. Also, the sampling rate has been called the
*Nyquist rate* in honor of Nyquist's contributions
[48].
In the author's experience, however, modern usage of the term
``Nyquist rate'' refers instead to *half* the sampling rate. To
resolve this clash between historical and current usage, the term
*Nyquist limit* will always mean *half* the sampling rate in this
book series, and the term ``Nyquist rate'' will not be used at all.

## 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].

## Aliasing of Sampled Signals

This section quantifies aliasing in the general case. This result is then used in the proof of the sampling theorem in the next section.

It is well known that when a continuous-time signal contains energy at
a frequency higher than half the sampling rate , sampling
at samples per second causes that energy to *alias* to a
lower frequency. If we write the original frequency as
, then the new aliased frequency is
,
for
. This phenomenon is also called ``folding'',
since is a ``mirror image'' of about . As we will
see, however, this is not a complete description of aliasing, as it
only applies to real signals. For general (complex) signals, it is
better to regard the aliasing due to sampling as a summation over all
spectral ``blocks'' of width .

### Continuous-Time Aliasing Theorem

Let denote any continuous-time signal having a Fourier Transform (FT)

*aliasing terms*. They are said to

*alias*into the

*base band*. Note that the summation of a spectrum with aliasing components involves addition of complex numbers; therefore, aliasing components can be removed only if both their

*amplitude and phase*are known.

*Proof: *
Writing as an inverse FT gives

The inverse FT can be broken up into a sum of finite integrals, each of length , as follows:

Let us now sample this representation for at to obtain , and we have

since and are integers. Normalizing frequency as yields

## Sampling Theorem

Let denote any continuous-time signal having a *continuous* Fourier transform

^{D.3}

*Proof: *From the continuous-time aliasing theorem (§D.2), we
have that the discrete-time spectrum
can be written in
terms of the continuous-time spectrum
as

To reconstruct from its samples , we may simply take the inverse Fourier transform of the zero-extended DTFT, because

By expanding as the DTFT of the samples , the formula for reconstructing as a superposition of weighted sinc functions is obtained (depicted in Fig.D.1):

where we defined

or

We have shown that when is bandlimited to less than half the
sampling rate, the IFT of the zero-extended DTFT of its samples
gives back the original continuous-time signal .
This completes the proof of the
sampling theorem.

Conversely, if can be reconstructed from its samples , it must be true that is bandlimited to , since a sampled signal only supports frequencies up to (see §D.4 below). While a real digital signal may have energy at half the sampling rate (frequency ), the phase is constrained to be either 0 or there, which is why this frequency had to be excluded from the sampling theorem.

A one-line summary of the essence of the sampling-theorem proof is

The sampling theorem is easier to show when applied to sampling-rate
conversion in discrete-time, *i.e.*, when simple downsampling of a
discrete time signal is being used to reduce the sampling rate by an
integer factor. In analogy with the continuous-time aliasing theorem
of §D.2, the downsampling theorem (§7.4.11)
states that downsampling a digital signal by an integer factor
produces a digital signal whose spectrum can be calculated by
partitioning the original spectrum into equal blocks and then
summing (aliasing) those blocks. If only one of the blocks is
nonzero, then the original signal at the higher sampling rate is
exactly recoverable.

##
Appendix: Frequencies Representable

by a Geometric Sequence

Consider , with . Then we can write in polar form as

Forming a geometric sequence based on yields the sequence

*sampled complex sinusoid*with unit amplitude, and zero phase. Defining the

*sampling interval*as in seconds provides that is the

*radian frequency*in radians per second.

A natural question to investigate is what frequencies are possible. The angular step of the point along the unit circle in the complex plane is . Since , an angular step is indistinguishable from the angular step . Therefore, we must restrict the angular step to a length range in order to avoid ambiguity.

Recall from §4.3.3 that we need support for both positive and negative frequencies since equal magnitudes of each must be summed to produce real sinusoids, as indicated by the identities

The length range which is symmetric about zero is

However, there is a problem with the point at
: Both
and correspond to the same point in the
-plane. Technically, this can be viewed as *aliasing* of the
frequency on top of , or vice versa. The practical
impact is that it is not possible in general to reconstruct a sinusoid
from its samples at this frequency. For an obvious example, consider
the sinusoid at half the sampling-rate sampled on its zero-crossings:
. We cannot be expected to
reconstruct a nonzero signal from a sequence of zeros! For the signal
, on the other hand, we sample
the positive and negative peaks, and everything looks fine. In
general, we either do not know the amplitude, or we do not know phase
of a sinusoid sampled at exactly twice its frequency, and if we hit the
zero crossings, we lose it altogether.

In view of the foregoing, we may define the valid range of ``digital frequencies'' to be

While one might have expected the open interval , we are free to give the point either the largest positive or largest negative representable frequency. Here, we chose the largest negative frequency since it corresponds to the assignment of numbers in two's complement arithmetic, which is often used to implement phase registers in sinusoidal oscillators. Since there is no corresponding positive-frequency component, samples at must be interpreted as samples of clockwise circular motion around the unit circle at two points per revolution. Such signals appear as an alternating sequence of the form , where can be complex. The amplitude at is then defined as , and the phase is .

We have seen that uniformly spaced samples can represent frequencies only in the range , where denotes the sampling rate. Frequencies outside this range yield sampled sinusoids indistinguishable from frequencies inside the range.

Suppose we henceforth agree to sample at *higher* than twice the
highest frequency in our continuous-time signal. This is normally
ensured in practice by lowpass-filtering the input signal to remove
all signal energy at and above. Such a filter is called an
*anti-aliasing filter*, and it is a standard first stage in an
*Analog-to-Digital (A/D) Converter (ADC)*. Nowadays, ADCs are normally
implemented within a single integrated circuit chip, such as a CODEC
(for ``coder/decoder'') or ``multimedia chip''.

**Next Section:**

Taylor Series Expansions

**Previous Section:**

Selected Continuous-Time Fourier Theorems