Nyquist criterion
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| Nyquist criterion | |
|---|---|
| Name | Nyquist criterion |
| Field | Signal processing, Communication systems |
| Description | A fundamental concept in Harry Nyquist's work on Telegraphy |
Nyquist criterion. The Nyquist criterion is a fundamental concept in Signal processing and Communication systems, developed by Harry Nyquist while working at Bell Labs. It is closely related to the work of Claude Shannon on Information theory and Ralph Hartley on Hartley oscillator. The criterion is essential in understanding the relationship between sampling and reconstruction of analog signals.
Introduction
The Nyquist criterion states that a continuous-time signal can be perfectly reconstructed from its samples if the sampling rate is greater than twice the bandwidth of the signal. This concept is crucial in Digital signal processing and has been influential in the development of Modulation techniques, such as Amplitude-shift keying and Frequency-shift keying, used in Radio communication systems designed by Guglielmo Marconi and Nikola Tesla. The work of Vladimir Kotelnikov and Emile Borel also contributed to the understanding of the Nyquist criterion, which is closely related to the Shannon-Hartley theorem and the work of Norbert Wiener on Cybernetics.
Historical Background
The Nyquist criterion was first proposed by Harry Nyquist in his 1928 paper, "Certain Topics in Telegraph Transmission Theory", published in the Bell System Technical Journal. Nyquist's work built upon the earlier research of James Clerk Maxwell and Oliver Heaviside on Telegraphy and Electrical engineering. The concept was later developed further by Claude Shannon in his 1948 paper, "A Mathematical Theory of Communication", which laid the foundation for Information theory and its applications in Computer science and Cryptography, as seen in the work of Alan Turing and William Friedman. The Nyquist criterion has since become a cornerstone of Signal processing and Communication systems, influencing the work of Andrew Viterbi and Irwin Jacobs on Error-correcting codes.
Mathematical Formulation
The Nyquist criterion can be mathematically formulated as follows: if a continuous-time signal x(t) has a bandwidth B, then it can be perfectly reconstructed from its samples if the sampling rate fs is greater than 2B. This can be expressed as fs > 2B, which is known as the Nyquist rate. The mathematical formulation of the Nyquist criterion is closely related to the work of David Hilbert on Hilbert spaces and the Fourier transform, developed by Joseph Fourier and Carl Friedrich Gauss. The concept is also connected to the Z-transform and the work of Pierre-Simon Laplace on Laplace transforms.
Sampling and Reconstruction
The Nyquist criterion has significant implications for sampling and reconstruction of analog signals. If the sampling rate is less than the Nyquist rate, then the reconstructed signal will be aliased, resulting in a distorted version of the original signal. This is known as aliasing distortion, which can be mitigated using Anti-aliasing filters, such as the Butterworth filter and the Chebyshev filter, developed by Stephen Butterworth and Pafnuty Chebyshev. The work of Bernhard Rieman and Eliahu Jury on Sampling theory has also contributed to the understanding of the Nyquist criterion and its applications in Digital signal processing.
Applications and Limitations
The Nyquist criterion has numerous applications in Signal processing and Communication systems, including Audio signal processing, Image processing, and Telecommunication systems designed by Alexander Graham Bell and Johann Philipp Reis. However, the criterion also has limitations, such as the requirement for a bandlimited signal and the assumption of an ideal sampling process. The work of Dennis Gabor and Yuri Kochiyama on Time-frequency analysis has also explored the limitations of the Nyquist criterion and its applications in Non-stationary signal processing. The concept is closely related to the Heisenberg uncertainty principle and the work of Werner Heisenberg on Quantum mechanics.
Related Concepts
The Nyquist criterion is closely related to other concepts in Signal processing and Communication systems, such as the Shannon-Hartley theorem, the Sampling theorem, and the Reconstruction theorem. The work of Andrey Kolmogorov and Robert Fano on Information theory has also contributed to the understanding of the Nyquist criterion and its applications in Data compression and Error-correcting codes. The concept is also connected to the work of John von Neumann on Computer science and the development of Digital computers, such as the ENIAC and the UNIVAC I, designed by John Mauchly and J. Presper Eckert.