Additive white Gaussian noise
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 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.
The AWGN channel is represented by a series of outputs
are independent, therefore: Evaluating the differential entropy of a Gaussian gives: Because
as: A rate is said to be achievable if there is a sequence of codes so that the maximum probability of error tends to zero as
If we decode by mapping every message received onto the codeword at the center of this sphere, then an error occurs only when the received vector is outside of this sphere, which is very unlikely.
The volume of an n-dimensional sphere is directly proportional to
A codebook, known to both encoder and decoder, is generated by selecting codewords of length n, i.i.d.
If there is no such message or if the power constraint is violated, a decoding error is declared.
Define the following three events: An error therefore occurs if
goes to zero as n approaches infinity, and by the joint Asymptotic Equipartition Property the same applies to
, the probability of error as follows: Therefore, as n approaches infinity,
Therefore, there is a code of rate R arbitrarily close to the capacity derived earlier.
be the average power of the codeword of index i: where the sum is over all input messages
, a concave (downward) function of x, to get: Because each codeword individually satisfies the power constraint, the average also satisfies the power constraint.
Therefore, R must be less than a value arbitrarily close to the capacity derived earlier, as
In serial data communications, the AWGN mathematical model is used to model the timing error caused by random jitter (RJ).
The graph to the right shows an example of timing errors associated with AWGN.
The variable Δt represents the uncertainty in the zero crossing.
As the amplitude of the AWGN is increased, the signal-to-noise ratio decreases.
[1] When affected by AWGN, the average number of either positive-going or negative-going zero crossings per second at the output of a narrow bandpass filter when the input is a sine wave is where In modern communication systems, bandlimited AWGN cannot be ignored.
When modeling bandlimited AWGN in the phasor domain, statistical analysis reveals that the amplitudes of the real and imaginary contributions are independent variables which follow the Gaussian distribution model.
When combined, the resultant phasor's magnitude is a Rayleigh-distributed random variable, while the phase is uniformly distributed from 0 to 2π.
The graph to the right shows an example of how bandlimited AWGN can affect a coherent carrier signal.
