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The AWGN channel is represented by a series of outputs at discrete-time event index . is the sum of the input and noise, , where is independent and identically distributed and drawn from a zero-mean normal distribution with variance (the noise).
This model is called a Gaussian white noise signal (or process). In the mathematical field known as white noise analysis, a Gaussian white noise is defined as a stochastic tempered distribution, i.e. a random variable with values in the space ′ of tempered distributions.
For the simple case of the additive white Gaussian noise (AWGN) channel: = +, ... For a zero mean, variance ...
In signal processing theory, Gaussian noise, named after Carl Friedrich Gauss, is a kind of signal noise that has a probability density function (pdf) equal to that of the normal distribution (which is also known as the Gaussian distribution). [1] [2] In other words, the values that the noise can take are Gaussian-distributed.
Here an AWGN channel is assumed. In digital communication or data transmission, / (energy per bit to noise power spectral density ratio) is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit".
Additive white Gaussian noise (AWGN) channel without fading. A worst-case scenario is a completely random channel, where noise totally dominates over the useful signal. This results in a transmission BER of 50% (provided that a Bernoulli binary data source and a binary symmetrical channel are assumed, see below).
Thermal noise is approximately white, meaning that its power spectral density is nearly equal throughout the frequency spectrum. The amplitude of the signal has very nearly a Gaussian probability density function. A communication system affected by thermal noise is often modelled as an additive white Gaussian noise (AWGN) channel.
An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: C = B log 2 ( 1 + S N ) {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)\ }