# 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*:

### 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].

## 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

## 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

sinc where sinc

The ``sinc function'' is defined with in its argument so that it
has zero crossings on the nonzero integers, and its peak magnitude is
1. Figure D.2 illustrates the appearance of the sinc function.
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

*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
*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

*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

*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''.

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Taylor Series Expansions

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Selected Continuous-Time Fourier Theorems