Definition of Nyquist’s Theorem

In order to fully understand Nyquist's Theorem, one needs to understand the basics of waves and signals. The frequency of a wave is the rate at which waves are observed; that is, it is the number of complete oscillations of the wave per second. The period of a wave is the inverse of the frequency: it is the number of seconds required for one complete oscillation. The frequency of a wave is measured in Hertz, which is defined as 1/seconds, or cycles per second. The period is thus measured in seconds, sometimes phrased as seconds per cycle. Simple signals are comprised of just one frequency, but signals can actually have many frequencies contained within them. Multiple-frequency-containing signals are often called composite signals.

Nyquist's Theorem states that if a signal contains no frequencies higher than a certain value B, then the all of the necessary information in the signal can be captured with a sampling frequency of 2B or higher. [5] This means that to obtain an accurate understanding of a signal, the sampling period must be at most half the length of the period of oscillation of the signal. But why is this the case?

The proof for Nyquist's Theorem requires use of a concept known as the Fourier Transform, which converts a signal or other sinusoid from the time domain (that is, the graph is a graph of amplitude of the wave versus time) to the frequency domain (where the graph is of amplitude versus frequency). One method of implementing the Fourier Transform, called the Discrete Time Fourier Transform, essentially repeats the process of computing the Fourier Transform every sampling period, resulting in a "shift" of the graph, as shown in green below. [6] More information about how the Fourier Transform works can be found here. As a result of the Fourier Transform, the graph of a waveform can be converted into one similar to the image below, where the frequency is graphed on the x-axis. In order for Nyquist's Theorem to be valid, one particular condition must be satisfied. The function must be of finite bandwidth, meaning that there should be no positive or negative frequencies beyond a certain constant value. Waveforms that satisfy this criterion are referred to as bandlimited signals. In practical cases, bandwidth is limited by filters, such as the one in a car radio, that filters out frequencies that are not very close to, say, 104.5 Megahertz. In the image below, there is no frequency beyond +B or -B, meaning that the sampling frequency, according to the theorem, should be at least +2B.

Aliasing results from sampling frequency that is too low. [1]

Once a signal is bandlimited, it is fairly simple to understand why Nyquist's Theorem is true. Shown above is the Discrete Time Fourier Transform of two different signals, X(f) and X_A(f). The two signals, which are not equivalent at frequencies close to B, have Discrete Time Fourier Transforms that are the same. This situation results because there is an overlap between samples due to the low sampling frequency, and the signals are identical when outside of overlapping sections. The parts of the signals within the overlap are essentially lost, and so the two signals appear to be the same. This result from under-sampling is called aliasing, because the two different signals are "aliases" of one another. This explains why Nyquist's Theorem works: the sampling frequency must be more than two times the greatest frequency contained within a signal in order to completely capture all of the information in the signal. Shown below is an example of the use of an appropriate sampling frequency.

Appropriate Sampling Frequency results in exact representation of signal. [1]

This minimum sampling rate of two times the maximum signal frequency is often referred to as the Nyquist Rate. [7]
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