This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Additive white Gaussian noise" – news · newspapers · books · scholar · JSTOR (February 2020)

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:

• Additive because it is added to any noise that might be intrinsic to the information system.
• White refers to the idea that it has uniform power spectral density across the frequency band for the information system. It is an analogy to the color white which may be realized by uniform emissions at all frequencies in the visible spectrum.
• Gaussian because it has a normal distribution in the time domain with an average time domain value of zero (Gaussian process).

Wideband noise comes from many natural noise sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson–Nyquist noise), shot noise, black-body radiation from the earth and other warm objects, and from celestial sources such as the Sun. The central limit theorem of probability theory indicates that the summation of many random processes will tend to have distribution called Gaussian or Normal.

AWGN is often used as a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of bandwidth) and a Gaussian distribution of amplitude. The model does not account for fading, frequency selectivity, interference, nonlinearity or dispersion. However, it produces simple and tractable mathematical models which are useful for gaining insight into the underlying behavior of a system before these other phenomena are considered.

The AWGN channel is a good model for many satellite and deep space communication links. It is not a good model for most terrestrial links because of multipath, terrain blocking, interference, etc. However, for terrestrial path modeling, AWGN is commonly used to simulate background noise of the channel under study, in addition to multipath, terrain blocking, interference, ground clutter and self interference that modern radio systems encounter in terrestrial operation.

Channel capacity

The AWGN channel is represented by a series of outputs ${\displaystyle Y_{i}}$ at discrete-time event index ${\displaystyle i}$. ${\displaystyle Y_{i}}$ is the sum of the input ${\displaystyle X_{i}}$ and noise, ${\displaystyle Z_{i}}$, where ${\displaystyle Z_{i}}$ is independent and identically distributed and drawn from a zero-mean normal distribution with variance ${\displaystyle N}$ (the noise). The ${\displaystyle Z_{i}}$ are further assumed to not be correlated with the ${\displaystyle X_{i}}$.

${\displaystyle Z_{i}\sim {\mathcal {N}}(0,N)\,\!}$

${\displaystyle Y_{i}=X_{i}+Z_{i}.\,\!}$

The capacity of the channel is infinite unless the noise ${\displaystyle N}$ is nonzero, and the ${\displaystyle X_{i}}$ are sufficiently constrained. The most common constraint on the input is the so-called "power" constraint, requiring that for a codeword ${\displaystyle (x_{1},x_{2},\dots ,x_{k})}$ transmitted through the channel, we have:

${\displaystyle {\frac {1}{k}}\sum _{i=1}^{k}x_{i}^{2}\leq P,}$

where ${\displaystyle P}$ represents the maximum channel power. Therefore, the channel capacity for the power-constrained channel is given by:^{[clarification needed]}

${\displaystyle C=\max \left\{I(X;Y):f{\text{ s.t. }}E\left(X^{2}\right)\leq P\right\}\,\!}$

where ${\displaystyle f}$ is the distribution of ${\displaystyle X}$. Expand ${\displaystyle I(X;Y)}$, writing it in terms of the differential entropy:

${\displaystyle {\begin{aligned}&I(X;Y)=h(Y)-h(Y\mid X)\\={}&h(Y)-h(X+Z\mid X)\\={}&h(Y)-h(Z\mid X)\end{aligned}}\,\!}$

But ${\displaystyle X}$ and ${\displaystyle Z}$ are independent, therefore:

${\displaystyle I(X;Y)=h(Y)-h(Z)\,\!}$

Evaluating the differential entropy of a Gaussian gives:

${\displaystyle h(Z)={\frac {1}{2}}\log(2\pi eN)\,\!}$

Because ${\displaystyle X}$ and ${\displaystyle Z}$ are independent and their sum gives ${\displaystyle Y}$:

${\displaystyle E(Y^{2})=E((X+Z)^{2})=E(X^{2})+2E(X)E(Z)+E(Z^{2})\leq P+N\,\!}$

From this bound, we infer from a property of the differential entropy that

${\displaystyle h(Y)\leq {\frac {1}{2}}\log(2\pi e(P+N))\,\!}$

Therefore, the channel capacity is given by the highest achievable bound on the mutual information:

${\displaystyle I(X;Y)\leq {\frac {1}{2}}\log(2\pi e(P+N))-{\frac {1}{2}}\log(2\pi eN)\,\!}$

Where ${\displaystyle I(X;Y)}$ is maximized when:

${\displaystyle X\sim {\mathcal {N}}(0,P)\,\!}$

Thus the channel capacity ${\displaystyle C}$ for the AWGN channel is given by:

$$Zdroj:https://en.wikipedia.org?pojem=Additive_white_Gaussian_noise
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---