Capacity of SISO system over a fading channel

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, we will be looking into the capacity equations for a conventional SISO system over AWGN and fading channels followed by capacity equations for a MIMO systems. As a pre-requisite, readers are encouraged to go through the detailed discussion on channel capacity and Shannon’s Theorem.

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 – .

Continue reading on simulating ergodic capacity of a SISO channel…

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References:

[1] Andrea J. Goldsmith & Pravin P. Varaiya, Capacity, mutual information, and coding for finite-state Markov channels,IEEE Transactions on Information Theory, Vol 42, No.3, May 1996.↗

Articles in this series
[1] Introduction to Multiple Antenna Systems
[2] MIMO - Diversity and Spatial Multiplexing
[3] Characterizing a MIMO channel - Channel State Information (CSI) and Condition number
[4] Capacity of a SISO system over a fading channel
[5] Ergodic Capacity of a SISO system over a Rayleigh Fading channel - Simulation in Matlab
[6] Capacity of a MIMO system over Fading Channels
[7] Single Input Multiple Output (SIMO) models for receive diversity
[8] Receiver diversity - Selection Combining
[9] Receiver diversity – Maximum Ratio Combining (MRC)

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