Nyquist-Shannon Sampling Theorem Nyquist Shannon Sampling

The theorem states that, when converting from an analog signal to digital (or otherwise sampling a signal at discrete intervals), the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version.

If the sampling frequency is less than this limit, then frequencies in the original signal that are above half the sampling rate will be " aliased" and will appear in the resulting signal as lower frequencies, therefore audible. If the sampling frequency is exactly twice the highest frequency of the input signal, then phase mismatches between the sampler and the signal will distort the signal. For example, sampling at t=0,1,2... will give you the discrete signal , as desired. However, sampling the same signal at t=0.5,1.5,2.5... will give you a constant zero signal - these samplers, which differ only in phase, not frequency, give dramatically different results because they sample at exactly the critical frequency.

Therefore, an analog low-pass filter is typically applied before sampling to ensure that no components with frequencies greater than half the sample frequency remain. This is called an " anti-aliasing filter". The quality of analog to digital converters depends critically upon that filter, which is also one of the most expensive components to build, since a poor filter causes phase distortion and other difficulties.

The theorem also applies when reducing the sampling frequency of an existing digital signal.

The theorem was first formulated by Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), but was only formally proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). Mathematically, the theorem is formulated as a statement about the Fourier transformation.

If a function s(x) has a Fourier transform F[s(x)] = S(f) = 0 for |f| ≥ W, then it is completely determined by giving the value of the function at a series of points spaced 1/(2W) apart. The values s_n = s(n/(2W)) are called the samples of s(x).

The minimum sample frequency that allows reconstruction of the original signal, that is 2W samples per unit distance, is known as the Nyquist frequencyThe Nyquist rate named after the Nyquist-Shannon sampling theorem, is immediately below the minimum theoretical sampling rate that will fully describe a given band-limited signal, enabling its faithful reconstruction from the samples. If the signal's larg, (or Nyquist rate). The time inbetween samples is called the Nyquist intervalIn telecommunication, the Nyquist interval is the maximum time interval between equally spaced samples of a signal that will enable the signal waveform to be completely determined. The Nyquist interval is equal to the reciprocal of twice the highest frequ.

If S(f) = 0 for |f| > W, then s(x) can be recovered from its samples by the Nyquist-Shannon interpolation formulaThe Nyquist-Shannon sampling theorem states that if a function has a Fourier transform for , then can be recovered from its samples by the formula: :..

A well-known consequence of the sampling theorem is that a signal cannot be both bandlimitedA bandlimited signal is a deterministic or stochastic signal (e. function of time) whose Fourier transform, or power spectrum are zero after a certain frequency. This has the consequence that the signal can be fully reconstructed from its samples, provide and time-limited. To see why, assume that such a signal exists, and sample it faster than the Nyquist frequency. These finitely many time-domain coefficients should define the entire signal. Equivalently, the entire spectrum of the bandlimited signal should be expressible in terms of the finitely many time-domain coefficients obtained from sampling the signal. Mathematically this is equivalent to requiring that a (trigonometric) polynomial can have infinitely many zeros since the bandlimited signal must be zero on an interval beyond a critical frequency which has infinitely many points. However, it is well-known that polynomials do not have more zeros than their orders due to the fundamental theorem of algebraThe fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if : (where the coefficients a . a can be r. This contradiction shows that our original assumption that a time-limited and bandlimited signal exists is incorrect.

1 Undersampling

It has to be noted that even if the concept of "twice the highest frequency" is the more commonly used idea, it is not absolute. In fact the theorem stands for "twice the bandwidth", which is totally different. BandwidthAnalog Bandwidth is the width, usually measured in hertz, of a frequency band f f. It can also be used to describe a signal, in which case the meaning is the width of the smallest frequency band within which the signal can fit. It is usually notated B, W, is related with the range between the first frequency and the last frequency that represent the signal. Bandwidth and highest frequency are identical only in basebandIn telecommunication, the term baseband has the following meanings: 1. The original band of frequencies produced by a transducer, such as a microphone, telegraph key, or other signal-initiating device, prior to initial modulation. In transmission systems, signals, that is, those that go very nearly down to DCDirect current DC or continuous current ) is the continuous flow of electricity through a conductor such as a wire from high to low potential. In direct current, the electric charges flow always in the same direction, which distinguishes it from alternati. This concept led to what is called undersampling, that is very used in software-defined radio.

Imagine that you want to sample all the FM commercial radio stations that broadcast in a given area. They broadcast in channels that span from 88 MHz to 108 MHz, giving a signal with bandwidth of 20 MHz. In the baseband interpretation of the theorem, this would require a sampling frequency more than 216 MHz. In fact, doing undersampling one is only required to sample at more than 40 MHz, as long as the antenna signal is passed by a bandpass filter to keep the signal in the 88-108 MHz range. Sampling at 44 MHz, the frequency 100 MHz will be reflected as a 12 MHz digital frequency.

In certain problems, the frequencies of interest are not an interval of frequencies, but perhaps some more interesting set F of frequencies. Again, the sampling frequency must be proportional to the size of F. For instance, certain domain decomposition methods fail to converge for the 0th frequency (the constant mode) and some medium frequencies. Then the set of interesting frequencies would be something like 10 Hz to 100 Hz, and 110 Hz to 200 Hz. In this case, one would need to sample at 360 Hz, not 400 Hz, to fully capture these signals.

2 References

H. Nyquist, "Certain topics in telegraph transmission theory," Trans. AIEE, vol. 47, pp. 617-644, Apr. 1928.
C. E. Shannon, "Communication in the presence of noise," Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10-21, Jan. 1949.

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1 Undersampling

2 References

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