Tag: outage capacity

Capacity of MIMO system over fading Channels

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 of 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 conventional SISO system. The capacity equations for a conventional SISO system over AWGN and fading channels were discussed in the earlier articles. As a per-requisite, readers are encouraged to go through the detailed discussion on channel capacity and Shannon’s Theorem too.

For those who are directly jumping here (without reading the article on SISO channel capacity), a few definitions are given below.

Entropy

The average amount of information per symbol (measured in bits/symbol) is called Entropy. Given a set of N discrete information symbols – represented as random variable having probabilities denoted by a Probability Mass Function , the entropy of X 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 memory less (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 .

The information capacity C is obtained by maximizing this mutual information taken over all possible input distributions p(x) [1].

MIMO flat fading Channel model

A MIMO system over a flat fading channel can be represented in the complex baseband notation.

where,
– received response from the channel – dimension
– the complex channel matrix of dimension
– vector representing transmitted signal – dimension . Assuming Gaussian signals i.e, , where is the covariance matrix of the transmit vector
– the number of transmit antennas
– the number of receive antennas
– complex baseband additive white Gaussian noise vector of dimension . It is assumed that the noise is spatially white  where is the covariance matrix of noise.

Note: The trace of the covariance matrix of the transmit vector gives the average transmit power, , where is the transmit power constraint applied at the transmitter.

Signal Covariance Matrices

It was assumed that the input signal vector and the noise vector  are uncorrelated. Therefore, the covariance matrix of the received signal vector is given by

In the above equation, the operator on the matrices denote Hermitian transpose operation. Thus, there are three covariance matrix involved here

  – Covariance matrix of input signal vector
– Covariance matrix of channel response vector
– Covariance matrix of noise vector

Channel State Information

The knowledge of the channel matrix , at the transmitter is called Channel State Information at the Transmitter (CSIT). If the receiver knows about the present state of the channel matrix, that knowledge is called Channel State Information at the Receiver (CSIR). Click here for more information on CSI and another related concept called condition number.

MIMO capacity discussion for CSIT known and unknown cases at the transmitter will be discussed later.

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 X is independent of the noise N and the noise vector is zero mean Gaussian distributed with covariance matrix   -i.e, .

Note that both the input symbols in the vector and the output symbols in the vector 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,

Since it is assumed that the channel is perfectly known at the receiver, the uncertainty of the channel h conditioned on X is zero, i.e, . Furthermore, it is assumed that the noise is independent of the input , i.e, . Thus, the mutual information is

Following the procedure laid out here, the differential entropy is calculated as

Using \((6)\) and the similar procedure for calculating above , The differential entropy is given by

Substituting equations (10) and (11) in (9), the capacity is given by

For the case, where the noise is uncorrelated (spatially white) between the antenna branches, , where  is the identity matrix of dimension .

Thus the capacity for MIMO flat fading channel can be written as

The capacity equation (13) contains random variables, and therefore the capacity will also be random. For obtaining meaningful result, for fading channels two different capacities can be defined.

If the CSIT is **UNKNOWN** at the transmitter, it is optimal to evenly distribute the available transmit power at the transmit antennas. That is, , where is the identity matrix of dimension .

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 below which the instantaneous mutual information falls below a prescribed value of probability expressed as percentage – q.

A word on capacity of a MIMO system over AWGN Channels

The capacity of MIMO system over AWGN channel can be derived in a very similar manner. The only difference will be the channel matrix. For the AWGN channel, the channel matrix will be a constant. The final equation for capacity will be very similar and will follow the lines of capacity of SISO over AWGN 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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Ergodic Capacity of SISO flat fading channel

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.

Figure 1: A frequency-flat channel model

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)');
Figure 2: Simulated capacity curves for SISO flat-fading channel

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References

[1] Konstantinos G. Derpanis, Jensen’s Inequality, Version 1.0,March 12, 2005.↗

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)

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
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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)

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing