Understand ergodic capacity of a SISO flat-fading system over fading channels. Model and simulate capacity curves in Matlab.
Channel model
In the previous post, derivation of SISO fading channel capacity was discussed. For a flat fading channel (model shown below), with the perfect knowledge of the channel at the receiver, the capacity of a SISO link was derived as
where, is flat fading complex channel impulse response that is held constant for each block of transmitted symbols, is the average input power at the transmit antenna, is the signal-to-noise ratio (SNR) at the receiver input and is the noise power of the channel.
Since the channel impulse response is a random variable, the channel capacity equation shown above is also random. To circumvent this, Ergodic channel capacity was defined along with outage capacity. The Ergodic channel capacity is defined as the statistical average of the mutual information, where the expectation is taken over
Jensen’s inequality [1] states that for any concave function f(x), where x is a random variable,
Applying Jensen’s inequality to Ergodic capacity in equation (2),
This implies that the Ergodic capacity of a fading channel cannot exceed that of an AWGN channel with constant gain. The above equation is simulated in Matlab for a Rayleigh Fading channel with and the plots are shown below.
Matlab code
%This work is licensed under a Creative Commons
%Attribution-NonCommercial-ShareAlike 4.0 International License
%Attribute author : Mathuranathan Viswanathan at gaussianwaves.com
snrdB=-10:0.5:20; %Range of SNRs to simulate
h= (randn(1,100) + 1i*randn(1,100) )/sqrt(2); %Rayleigh flat channel
sigma_z=1; %Noise power - assumed to be unity
snr = 10.^(snrdB/10); %SNRs in linear scale
P=(sigma_z^2)*snr./(mean(abs(h).^2)); %Calculate corresponding values for P
C_erg_awgn= (log2(1+ mean(abs(h).^2).*P/(sigma_z^2))); %AWGN channel capacity (Bound)
C_erg = mean((log2(1+ ((abs(h).^2).')*P/(sigma_z^2)))); %ergodic capacity for Fading channel
plot(snrdB,C_erg_awgn,'b'); hold on;
plot(snrdB,C_erg,'r'); grid on;
legend('AWGN channel capacity','Fading channel Ergodic capacity');
title('SISO fading channel - Ergodic capacity');
xlabel('SNR (dB)');ylabel('Capacity (bps/Hz)');
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As reiterated in the previous article, a MIMO system is used to increase the capacity dramatically and also to improve the quality of a communication link. Increased capacity is obtained by spatial multiplexing and increased quality is obtained by diversity techniques (Space time coding). Capacity equations of a MIMO system over a variety of channels (AWGN, fading channels) is of primary importance. It is desirable to know the capacity improvements offered by a MIMO system over the capacity of SISO system.
To begin with, clarity over few definitions are needed.
Entropy
The average amount of information per symbol (measured in bits/symbol) is called Entropy. Given a set of discrete information symbols – represented as random variable having probabilities denoted by a Probability Mass Function , the entropy of is given by
Entropy is a measure of uncertainty of a random variable , therefore reflects the amount of information required on an average to describe the random variable. In general, it has the following bounds
Entropy hits the lower bound of zero (no uncertainty, therefore no information) for a completely deterministic system (probability of correct transmission ). It reaches the upper bound when the input symbols are equi-probable.
Capacity and mutual information
Following figure represents a discrete memoryless (noise term corrupts the input symbols independently) channel, where the input and output are represented as random variables and respectively. Statistically, such a channel can be expressed by transition or conditional probabilities. That is, given a set of inputs to the channel, the probability of observing the output of the channel is expressed as conditional probability
For such a channel, the mutual information denotes the amount of information that one random variable contains about the other random variable
is the amount of information in before observing and thus the above quantity can be seen as the reduction of uncertainty of from the observation of latex .
The information capacity is obtained by maximizing this mutual information taken over all possible input distributions [1].
SISO fading Channel
A SISO fading channel can be represented as the convolution of the complex channel impulse response (represented as a random variable ) and the input .
Here, is complex baseband additive white Gaussian noise and the above equation is for a single realization of complex output . If the channel is assumed to be flat fading or of block fading type (channel does not vary over a block of symbols), the above equation can be simply written without the convolution operation (refer this article to know how equations (5) & (6) are equivalent for a flat-fading channel).
For different communication fading channels, the channel impulse response can be modeled using various statistical distributions. Some of the common distributions as Rayleigh, Rician, Nakagami-m, etc.,
Capacity with transmit power constraint
Now, we would like to evaluate capacity for the most practical scenario, where the average power, given by , that can be expensed at the transmitter is limited to . Thus, the channel capacity is now constrained by this average transmit power, given as
For the further derivations, we assume that the receiver possesses perfect knowledge about the channel. Furthermore, we assume that the input random variable is independent of the noise and the noise is zero mean Gaussian distributed with variance -i.e, .
Note that both the input symbols and the output symbols take continuous values upon transmission and reception and the values are discrete in time (Continuous input Continuous output discrete Memoryless Channel – CCMC). For such continuous random variable, differential entropy – is considered. Expressing the mutual information in-terms of differential entropy,
Mutual Information and differential entropy
Since it is assumed that the channel is perfectly known at the receiver, the uncertainty of the channel conditioned on is zero, i.e, . Furthermore, it is assumed that the noise is independent of the input , i.e, . Thus, the mutual information is
For a complex Gaussian noise with non-zero mean and variance , the PDF of the noise is given by
The differential entropy for the noise is given by
This shows that the differential entropy is not dependent on the mean of . Therefore, it is immune to translations (shifting of mean value) of the PDF. For the problem of computation of capacity,
and given the differential entropy , the mutual information is maximized by maximizing the differential entropy . The fact is, the Gaussian random variables itself are differential entropy maximizers. Therefore, the mutual information is maximized when the variable is also Gaussian and therefore the differential entropy . Where, the received average power is given by
Thus the capacity is given by
Representing the entire received signal-to-ratio as , the capacity of a SISO system over a fading channel is given by
For the fading channel considered above, the term channel is modeled as a random variable. Thus, the capacity equation above is also a random variable. Thus, for fading channels two different capacities can be defined.
Ergodic Capacity
Ergodic capacity is defined as the statistical average of the mutual information, where the expectation is taken over
Outage Capacity
Defined as the information rate at which the instantaneous mutual information falls below a prescribed value of probability expressed as percentage – .
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