Spectrum of a bandlimited signal as a function of frequency

In signal processing, the Nyquist rate is two times the bandwidth of a bandlimited signal or a bandlimited channel. A bandlimited signal is a Deterministic or Stochastic signal whose Fourier transform or Power spectral density is zero above a certain finite Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Bandwidth is the difference between the upper and lower Cutoff frequencies of for example a filter, a Communication channel, or a Signal spectrum A bandlimited signal is a Deterministic or Stochastic signal whose Fourier transform or Power spectral density is zero above a certain finite This term is used to mean two different things under two different circumstances: as a lower bound for the sample rate for alias-free signal sampling,[1] and as an upper bound for the signaling rate across a bandwidth-limited channel such as a telegraph line. [2]

## Nyquist rate relative to sampling

The Nyquist rate is the minimum sampling rate required to avoid aliasing, equal to twice the highest frequency contained within the signal. Sampling theorem The Nyquist–Shannon sampling theorem states that perfect reconstruction This article applies to signal processing including computer graphics

$f_N \ \stackrel{\mathrm{def}}{=}\ 2 B\,$

where $B\,$ is the highest frequency at which the signal can have nonzero energy. Frequency is a measure of the number of occurrences of a repeating event per unit Time.

To avoid aliasing, the sampling rate must exceed the Nyquist rate:

$f_S > f_N\,$.

To avoid aliasing, the bandwidth must be considered to be the upper frequency limit of a baseband signal. In Signal processing, baseband is an adjective that describes signals and systems whose range of Frequencies is measured from zero to a maximum bandwidth Bandpass sampling signals must be sampled at at least twice the frequency of the highest frequency component of the bandpass signal in order to avoid aliasing. In Signal processing, sampling is the reduction of a Continuous signal to a Discrete signal. However, it is typical to use aliasing to advantage, to allow sampling of bandpass signals at rates as low as 2B, where B is the bandwidth of the bandpass signal. An alternative is to mix (heterodyne) the bandpass signals down to baseband, and sample there in the usual way; in this case, the baseband bandwidth can be as low as B/2 in the case of symmetric signals such as amplitude modulation, so the sampling rate can be as low as B in such cases. In Radio and Signal processing, heterodyning is the generation of new frequencies by mixing or multiplying two Oscillating waveforms Amplitude modulation ( AM) is a technique used in electronic communication most commonly for transmitting information via a Radio Carrier wave

## Nyquist rate relative to signaling

Long before Harry Nyquist had his name associated with sampling, the term Nyquist rate was used differently, with a meaning closer to what Nyquist actually studied. Harry Nyquist ( né Harry Theodor Nyqvist pron, not as often pronounced ( February 7, 1889 – April 4, 1976) was an important Quoting Harold S. Black's 1953 book Modulation Theory, in the section Nyquist Interval of the opening chapter Historical Background:

"If the essential frequency range is limited to B cycles per second, 2B was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less half a quantum step. Harold Stephen Black (1898-1983 was an American Electrical engineer, who revolutionized the field of applied electronics by inventing the Negative feedback amplifier This rate is generally referred to as signaling at the Nyquist rate and 1/(2B) has been termed a Nyquist interval. " (bold added for emphasis; italics as in the original)

According to the OED, this may be the origin of the term Nyquist rate. The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English [3]

Nyquist's famous 1928 paper was a study on how many pulses (code elements) could be transmitted per second, and recovered, through a channel of limited bandwidth. Signaling at the Nyquist rate meant putting as many code pulses through a telegraph channel as its bandwidth would allow. Shannon used Nyquist's approach when he proved the sampling theorem in 1948, but Nyquist did not work on sampling per se. The Nyquist–Shannon sampling theorem is a fundamental result in the field of Information theory, in particular Telecommunications and Signal processing

Black's later chapter on "The Sampling Principle" does give Nyquist some of the credit for some relevant math:

"Nyquist (1928) pointed out that, if the function is substantially limited to the time interval T, 2BT values are sufficient to specify the function, basing his conclusions on a Fourier series representation of the function over the time interval T. "

• Harry Nyquist
• Nyquist–Shannon sampling theorem
• Sampling frequency
• Nyquist frequency — The Nyquist rate is defined differently from the Nyquist frequency, which is the frequency equal to half the sampling rate of a sampling system, and is not a property of a signal. Harry Nyquist ( né Harry Theodor Nyqvist pron, not as often pronounced ( February 7, 1889 – April 4, 1976) was an important The Nyquist–Shannon sampling theorem is a fundamental result in the field of Information theory, in particular Telecommunications and Signal processing Sampling theorem The Nyquist–Shannon sampling theorem states that perfect reconstruction The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the Sampling frequency
• Nyquist ISI criterion

## References

1. ^ Yves Geerts, Michiel Steyaert, and Willy Sansen (2002). In communications the Nyquist ISI criterion describes the conditions which when satisfied by a communication channel, result in no Intersymbol interference or ISI Design of multi-bit delta-sigma A/D converters. Springer.
2. ^ Roger L. Freeman (2004). Telecommunication System Engineering. John Wiley & Sons, 399.
3. ^ Black, H. S., Modulation Theory, v. Harold Stephen Black (1898-1983 was an American Electrical engineer, who revolutionized the field of applied electronics by inventing the Negative feedback amplifier 65, 1953, cited in OED
The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English
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