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It connects Hartley's result with Shannon's channel capacity theorem in a form that is equivalent to specifying the M in Hartley's line rate formula in terms of a signal-to-noise ratio, but achieving reliability through error-correction coding rather than through reliably distinguishable pulse levels.
The term Nyquist Sampling Theorem (capitalized thus) appeared as early as 1959 in a book from his former employer, Bell Labs, [22] and appeared again in 1963, [23] and not capitalized in 1965. [24] It had been called the Shannon Sampling Theorem as early as 1954, [25] but also just the sampling theorem by several other books in the early 1950s.
The channel capacity can be calculated from the physical properties of a channel; for a band-limited channel with Gaussian noise, using the Shannon–Hartley theorem. Simple schemes such as "send the message 3 times and use a best 2 out of 3 voting scheme if the copies differ" are inefficient error-correction methods, unable to asymptotically ...
the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem; the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as; the bit—a new way of seeing the most fundamental unit of information.
This result is known as the Shannon–Hartley theorem. [11] When the SNR is large (SNR ≫ 0 dB), the capacity ¯ is logarithmic in power and approximately linear in bandwidth. This is called the bandwidth-limited regime.
This relationship is described by the Shannon–Hartley theorem, which is a fundamental law of information theory. SNR can be calculated using different formulas depending on how the signal and noise are measured and defined.
For this calculation, it is conventional to define a normalized rate = / (), a bandwidth utilization parameter of bits per second per half hertz, or bits per dimension (a signal of bandwidth B can be encoded with dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is:
Nyquist–Shannon sampling theorem; S. Schwartz–Zippel lemma; Shannon–Hartley theorem; Shannon's source coding theorem This page was ...