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.1As 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 . 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 positionD.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.
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.,
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 .
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)
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
Let denote any continuous-time signal having a continuous Fourier transform
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
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
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
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''.
Taylor Series Expansions
Selected Continuous-Time Fourier Theorems