Additive white Gaussian noise 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 across the frequency band for the information system. It is an
analogy to the color white which has 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.
Wideband noise comes from many natural sources, such as the thermal vibrations
of atoms in conductors, 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 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 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 are further assumed to not be correlated with the .
The capacity of the channel is infinite unless the noise n is nonzero, and the
are sufficiently constrained. The most common constraint on the input is the
so-called "power" constraint, requiring that for a codeword transmitted through
the channel, we have: where represents the maximum channel
power. Therefore, the channel capacity for the power-constrained channel is
given by: Where is the distribution of . Expand ,
writing it in terms of the differential entropy:
But and are independent, therefore: Evaluating the differential entropy of a
Gaussian gives: Because and are independent and their
sum gives : From this bound, we infer from a
property of the differential entropy that
Therefore the channel capacity is given by the highest achievable bound on the
mutual information: Where is maximized when:
Thus the channel capacity for the AWGN channel is given by:
= Channel capacity and sphere packing= Suppose that we are sending messages
through the channel with index ranging from to , the number of distinct
possible messages. If we encode the messages to bits, then we define the
rate 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 approaches infinity. The capacity is the highest achievable
rate. Consider a codeword of length sent
through the AWGN channel with noise level . When received, the codeword
vector variance is now , and its mean is the codeword sent. The vector is very
likely to be contained in a sphere of radius around the codeword sent. 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. Each codeword vector has an associated
sphere of received codeword vectors which are decoded to it and each such
sphere must map uniquely onto a codeword. Because these spheres
therefore must not intersect, we are faced with the problem of sphere
packing. How many distinct codewords can we pack into our -bit codeword vector?
The received vectors have a maximum energy of and therefore must occupy a
sphere of radius . Each codeword sphere has radius . The volume of an
n-dimensional sphere is directly proportional to , so the maximum number
of uniquely decodeable spheres that can be packed into our sphere with
transmission power P is: By this argument, the rate R can be no
more than . = Achievability=
In this section, we show achievability of the upper bound on the rate from the
last section. A codebook, known to both encoder and
decoder, is generated by selecting codewords of length n, i.i.d. Gaussian
with variance and mean zero. For large n, the empirical variance of the
codebook will be very close to the variance of its distribution, thereby
avoiding violation of the power constraint probabilistically.
Received messages are decoded to a message in the codebook which is
uniquely jointly typical. If there is no such message or if the power constraint
is violated, a decoding error is declared.
Let denote the codeword for message , while is, as before the received
vector. Define the following three events:
Event :the power of the received message is larger than .
Event : the transmitted and received codewords are not jointly typical.
Event : is in , the typical set where , which is to say that the incorrect
codeword is jointly typical with the received vector.
An error therefore occurs if , or any of the occur. By the law of large
numbers, goes to zero as n approaches infinity, and by the joint Asymptotic
Equipartition Property the same applies to . Therefore, for a sufficiently large
, both and are each less than . Since and are independent for , we have that
and are also independent. Therefore, by the joint AEP, . This allows us to
calculate , the probability of error as follows:
Therefore, as n approaches infinity, goes to zero and . Therefore there is a
code of rate R arbitrarily close to the capacity derived earlier.
= Coding theorem converse= Here we show that rates above the
capacity are not achievable. Suppose that the power constraint is
satisfied for a codebook, and further suppose that the messages follow a
uniform distribution. Let be the input messages and the output messages. Thus
the information flows as: Making use of Fano's inequality gives:
where as Let be the encoded message of codeword
index i. Then: Let be the average power of the
codeword of index i: Where the sum is over all input messages
. and are independent, thus the expectation of the power of is, for
noise level : And, if is normally distributed, we
have that Therefore,
We may apply Jensen's equality to , a concave function of x, to get:
Because each codeword individually satisfies the power constraint, the
average also satisfies the power constraint. Therefore
Which we may apply to simplify the inequality above and get:
Therefore, it must be that . Therefore, R must be less than a value arbitrarily
close to the capacity derived earlier, as .
Effects in time domain In serial data communications, the AWGN
mathematical model is used to model the timing error caused by random jitter.
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. This
results in increased uncertainty Δt. 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 f0 = the center frequency of the filter
B = the filter bandwidth SNR = the signal-to-noise power ratio in
linear terms Effects in phasor domain
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. The instantaneous response of the Noise
Vector cannot be precisely predicted, however its time-averaged response can
be statistically predicted. As shown in the graph, we confidently predict that
the noise phasor will reside inside the 1σ circle about 38% of the time; the
noise phasor will reside inside the 2σ circle about 86% of the time; and the
noise phasor will reside inside the 3σ circle about 98% of the time.
See also Ground bounce
Noisy-channel coding theorem Gaussian process
References