Category: MIMO systems

Maximum Ratio Combining (MRC) architecture simulation

In the previous post on Single Input Multiple Output (SIMO) models for receive diversity, various receiver diversity techniques were outlined. One of them is maximum ratio combining, the focus of the topic here.

Channel model

Assuming flat slow fading channel, the received signal model is given by

where, is the channel impulse response, is the received signal, is the transmitted signal and is the additive Gaussian white noise.

Assuming small scale Rayleigh fading, the channel impulse response is modeled as complex Gaussian random variable with zero mean and variance

Therefore, the instantaneous channel power is exponentially distributed

In the context of AWGN channel, the signal-to-noise ratio (SNR) for a given channel condition, is a constant. But in the case of fading channels, the signal-to-noise ratio is no longer a constant as the signal is fluctuating when passed through a fading channel. Therefore, for fading channel, the SNR has a random variable component built into it. Hence, we just don’t call it SNR, instead it is called instantaneous SNR which depends on the current conditions of the channel (or equivalently, the value of the random variable at that instant). Since the SNR is a random variable, we can also talk about its expected (average) value, which is called average SNR. Denoting the average SNR as and for convenience, let’s assume that the average power of the channel is unity, i.e,

The instantaneous SNR is given by

Therefore, like the channel impulse response, the instantaneous SNR is also exponentially distributed

Maximum Ratio Combining (MRC)

The selection combining technique is the simplest technique, where in, the received signal from the antenna that experiences the highest SNR (i.e, the strongest signal from N received signals) is chosen for processing at the receiver. Therefore this technique throws away of observations. Whereas, in maximum ratio combining (MRC) all observations are used.

MRC works on the signal in spatial domain and is very similar to what a matched filter in frequency domain does to the incoming signal. MRC maximizes the inner product of the weights and the signal vector.

maximum ratio combining
Figure 1: Processing the received samples at the receiver

The maximum ratio combining technique, uses all the received signal elements (Figure 1), it weighs them and combines the weighted signals so that the output SNR is maximized. Requiring the knowledge of the individual channels , the weights are chosen as

With the weights set as , the output of the MRC combiner is

Therefore, the output SNR after MRC processing is

MRC processing results in the weighted average of the received signals and hence the overall output SNR is equal to the sum of the SNRs of all individual receive signals, which yields the maximum possible diversity gain of . This is the maximum achievable SNR for all possible receive diversity schemes (selection combining, equal gain combining, etc..,).

Generally, two figures of merits are used to gauge the performance of the diversity schemes – outage probability and error rate performance for PSK modulation.

Outage probability

As we know, fading channels are characterized by deep fades, i.e, the period when the signal level falls below a certain threshold or certain noise level. During such fades, the user experiences signal outage. We would like to compute the probability, in certain fading channel, that a user will experience signal outage. This is called outage probability. Outage probability can be easily computed if we know the probability distribution characteristics of the fading.

The outage probability with which the instantaneous output SNR of MRC falls below a given SNR target is

For high average SNR conditions , the outage probability can be approximated as

Python code

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import factorial

gamma_ratio_dB = np.arange(start=-10,stop=40,step=2)
Ns = [1,2,3,4,10,20] #number of received signal paths

gamma_ratio = 10**(gamma_ratio_dB/10) #Average SNR/SNR threshold in dB

fig, ax = plt.subplots(1, 1)

for N in Ns:
        n = np.arange(start=0,stop=N,step=1)
        P_outage = 1 - np.exp(-1/gamma_ratio)*np.sum(((1/gamma_ratio)**n[:,None])/factorial(n[:,None]),axis=0)
        ax.semilogy(gamma_ratio_dB,P_outage,label='N='+str(N))

ax.legend()
ax.set_xlim(-10,40);ax.set_ylim(0.0001,1.1)
ax.set_title('MRC outage probability (Rayleigh fading channel)')
ax.set_xlabel(r'$10log_{10}\left(\Gamma/\gamma_t\right)$')
ax.set_ylabel('Outage probability');fig.show()
Figure 2: Outage probability of MRC processing in Rayleigh fading channel

Figure 2, plots the outage probability against (the ratio of average SNR and the SNR threshold) for different values – the number of received signals received over an Rayleigh flat fading channel. For example, the outage probability dramatically improves when going from branch to branches. At outage probability of 0.01% (projected y-value in the graph at ), there is an approximate reduction in the required SNR.

Error rate performance

In the case of receive diversity schemes with antennas, the received signal vector is given by

Considering the QPSK modulated symbols that are transmitted (denoted as ), the maximum likelihood detection criterion for detecting the transmitted symbols by the equalizer block at the receiver is given by,

The solution to this problem can be obtained using the least squares method (refer equation 8.17 given in chapter 8 of this book)

The solution can be re-written as

As an example, the symbol error rate performance of a QPSK modulated transmission over a Rayleigh flat fading SIMO channel, for a range of values for the number of receive antennas () is simulated here. Maximum ratio combining is used in the receiver.

The code utilizes the modem class discussed in the book here. The modem class incorporates modulation and demodulation techniques for PSK,PAM,QAM and FSK modulation schemes. It uses the object oriented programming method for implementing the various modems.

The addition of Gaussian white noise needs to be multidimensional. The method discussed in this article is extended here for computing and adding the required amount of noise across the branches of signals.

Figure 3: Symbol error rate Vs EbN0 for QPSK over i.i.d Rayleigh flat fading channel with MRC processing at the receiver
"""
Eb/N0 Vs SER for PSK over Rayleigh flat fading with MRC
@author: Mathuranathan Viswanathan
Created on Jan 16, 2020
"""
import numpy as np # for numerical computing
import matplotlib.pyplot as plt # for plotting functions
#from matplotlib import cm # colormap for color palette
from numpy.random import standard_normal

from DigiCommPy.modem import PSKModem
from DigiCommPy.channels import awgn

#---------Input Fields------------------------
nSym = 10**6 # Number of symbols to transmit
EbN0dBs = np.arange(start=-20,stop = 36, step = 2) # Eb/N0 range in dB for simulation
N = [1,2,4,8,10] # [1,2,3,4,10,20] #number of diversity branches
M = 4 #M-ary PSK modulation

k=np.log2(M)
EsN0dBs = 10*np.log10(k)+EbN0dBs # EsN0dB calculation

fig, ax = plt.subplots(nrows=1,ncols = 1) #To plot figure

for nRx in N: #simulate for each # of received branchs
        #Random input symbols to modulator
        inputSyms = np.random.randint(low=0, high = M, size=nSym)
        modem = PSKModem(M)
        s = modem.modulate(inputSyms) #modulated PSK symbols

        #nRx signal branches
        s_diversity = np.kron(np.ones((nRx,1)),s);

        ser_sim = np.zeros(len(EbN0dBs)) # simulated symbol error rates

        for i,EsN0dB in enumerate(EsN0dBs):

                #Rayleigh flat fading channel as channel matrix
                h = np.sqrt(1/2)*(standard_normal((nRx,nSym))+1j*standard_normal((nRx,nSym)))
                signal = h*s_diversity #effect of channel on the modulated signal

                #Computing the signal power and adding noise
                gamma = 10**(EsN0dB/10) #converting EsN0dB to linear scale
                P = np.sum(np.abs(signal)**2,axis=1)/nSym #calculate power in each branch of signal
                N0 = P/gamma #required noise spectral density for each branch
                #Scale each row of noise with the calculated noise spectral density
                noise = (standard_normal(signal.shape)+1j*standard_normal(signal.shape))*np.sqrt(N0/2)[:,None]

                r = signal+noise #received signal branches

                #Receiver processing
                equalized = np.sum(r*np.conj(h),axis=0) #equalized signal

                detectedSyms = modem.demodulate(equalized) #demodulation decisions
                ser_sim[i] = np.sum(detectedSyms != inputSyms)/nSym

        #ax.grid(True,which='both');
        ax.semilogy(EbN0dBs,ser_sim,label='N='+str(nRx))#plot simulated error rates

ax.set_xlim(-20,35);ax.set_ylim(0.0001,1.1);ax.grid(True,which='both');
ax.set_xlabel('Eb/N0(dB)');ax.set_ylabel('Symbol Error Rate($P_s$)')
ax.set_title('SER performance for QPSK over Rayleigh fading channel with MRC')
ax.legend();fig.show()

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Andrea Goldsmith, “Wireless Communications”, ISBN: 978-0521837163, Cambridge University Press, 1 edition.↗

[2] Barry-Lee-Messerschmitt, “Digital Communication”, ISBN: 978-0792375487 , Springer, 3rd edition, September 30, 2003.↗

Articles in this series

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

Selection Combining – architecture simulation

In the previous post on Single Input Multiple Output (SIMO) models for receive diversity, various receiver diversity techniques were outlined. One of them is selection combining, the focus of the topic here.

Channel model

Assuming flat slow fading channel, the received signal model is given by

where, is the channel impulse response, is the received signal, is the transmitted signal and is the additive Gaussian white noise.

Assuming small scale Rayleigh fading, the channel impulse response is modeled as complex Gaussian random variable with zero mean and variance

Therefore, the instantaneous channel power is exponentially distributed

In the context of AWGN channel, the signal-to-noise ratio (SNR) for a given channel condition, is a constant. But in the case of fading channels, the signal-to-noise ratio is no longer a constant as the signal is fluctuating when passed through a fading channel. Therefore, for fading channel, the SNR has a random variable component built into it. Hence, we just don’t call it SNR, instead it is called instantaneous SNR which depends on the current conditions of the channel (or equivalently, the value of the random variable at that instant). Since the SNR is a random variable, we can also talk about its expected (average) value, which is called average SNR.

Denoting the average SNR as and for convenience, let’s assume that the average power of the channel is unity, i.e,

The instantaneous SNR is given by

Therefore, like the channel impulse response, the instantaneous SNR is also exponentially distributed

Selection Combining

In selection combining, the received signal from the antenna that experiences the highest SNR (i.e, the strongest signal from N received signals) is chosen for processing at the receiver (Figure 1).

That is, the weight of the path that has the highest is chosen.

Therefore, the output SNR (at the combiner output) is the maximum SNR of all the received signals

Figure 1: Processing the received samples at the receiver

As we know, fading channels are characterized by fades, i.e, the period when the signal level falls below a certain threshold or certain noise level. During such fades, the user experiences signal outage. We would like to compute the probability, in certain fading channel, that a user will experience signal outage. This is called outage probability. Outage probability can be easily computed if we know the probability distribution characteristics of the fading.

For a selection combining scheme, for an user to experience outage, the SNR of all the received branches should fall below the given threshold $\gamma_{t}$. In otherwords, the output SNR at the combiner is below the threshold . The outage probability of selection combining receiver is given by

For high average SNR conditions , the outage probability can be approximated as

Python code

import numpy as np
import matplotlib.pyplot as plt

gamma_ratio_dB = np.arange(start=-10,stop=41,step=2)
Ns = [1,2,3,4,10,20] #number of received signal paths

gamma_ratio = 10**(gamma_ratio_dB/10) #Average SNR/SNR threshold in dB

fig, ax = plt.subplots(1, 1)
for N in Ns:
        P_outage = (1 - np.exp(-1/gamma_ratio))**N
        ax.semilogy(gamma_ratio_dB,P_outage,label='N='+str(N))

ax.legend()
ax.set_xlim(-10,40);ax.set_ylim(0.0001,1.1)
ax.set_title('Selection combining outage probability (Rayleigh fading channel)')
ax.set_xlabel(r'$10log_{10}\left(\Gamma/\gamma_t\right)$')
ax.set_ylabel('Outage probability');fig.show()
Figure 2: Outage probability of selection combining in Rayleigh fading channel

Figure 2, plots the outage probability against (the ratio of average SNR and the SNR threshold) for different values – the number of received signals received over an Rayleigh flat fading channel. For example, the outage probability dramatically improves when going from branch to branches. At outage probability of 0.01% (y-value in the graph at ), there is an approximate reduction in the required SNR.

Error rate performance

As an example, the symbol error rate performance of a QPSK modulated transmission over a Rayleigh flat fading SIMO channel, for a range of values for the number of receive antennas () is simulated here. Selection combining is used in the receiver. After the selection combining the received signal is equalized by multiplying the selected branch with the conjugate of the corresponding channel sample.

The code utilizes the modem class discussed in the book here. The modem class incorporates modulation and demodulation techniques for PSK,PAM,QAM and FSK modulation schemes. It uses the object oriented programming method for implementing the various modems.

The addition of Gaussian white noise needs to be multidimensional. The method discussed in this article is extended here for computing and adding the required amount of noise across the branches of signals.

Figure 3: Symbol error rate Vs EbN0 for QPSK over i.i.d Rayleigh flat fading channel with Selection Combining at the receiver
"""
Eb/N0 Vs SER for PSK over Rayleigh flat fading with Selection Combining
@author: Mathuranathan Viswanathan
Created on Dec 10, 2019
"""
import numpy as np # for numerical computing
import matplotlib.pyplot as plt # for plotting functions
#from matplotlib import cm # colormap for color palette
from numpy.random import standard_normal

from DigiCommPy.modem import PSKModem

#---------Input Fields------------------------
nSym = 10**6 # Number of symbols to transmit
EbN0dBs = np.arange(start=0,stop = 36, step = 2) # Eb/N0 range in dB for simulation
N = [1,2,4,8,10] # [1,2,3,4,10,20] #number of diversity branches
M = 4 #M-ary PSK modulation

k=np.log2(M)
EsN0dBs = 10*np.log10(k)+EbN0dBs # EsN0dB calculation

fig, ax = plt.subplots(nrows=1,ncols = 1) #To plot figure

for nRx in N: #simulate for each # of received branchs
        #Random input symbols to modulator
        inputSyms = np.random.randint(low=0, high = M, size=nSym)
        modem = PSKModem(M)
        s = modem.modulate(inputSyms) #modulated PSK symbols

        #nRx signal branches
        s_diversity = np.kron(np.ones((nRx,1)),s);

        ser_sim = np.zeros(len(EbN0dBs)) # simulated symbol error rates

        for i,EsN0dB in enumerate(EsN0dBs):

                #Rayleigh flat fading channel as channel matrix
                h = np.sqrt(1/2)*(standard_normal((nRx,nSym))+1j*standard_normal((nRx,nSym)))
                signal = h*s_diversity #effect of channel on the modulated signal

                #Computing the signal power and adding noise
                gamma = 10**(EsN0dB/10) #converting EsN0dB to linear scale
                P = np.sum(np.abs(signal)**2,axis=1)/nSym #calculate power in each branch of signal
                N0 = P/gamma #required noise spectral density for each branch
                #Scale each row of noise with the calculated noise spectral density
                noise = (standard_normal(signal.shape)+1j*standard_normal(signal.shape))*np.sqrt(N0/2)[:,None]

                r = signal+noise #received signal branches

                #Receiver processing
                idx = np.abs(h).argmax(axis=0) #indices of max |h| values along all branches
                
                hSelected = h[idx,np.arange(h.shape[1])] #branches with max |h| values
                ySelected = r[idx,np.arange(r.shape[1])] #output of selection combiner
                equalized = ySelected*np.conj(hSelected) #equalized signal

                detectedSyms = modem.demodulate(equalized) #demodulation decisions
                ser_sim[i] = np.sum(detectedSyms != inputSyms)/nSym

        print(ser_sim)
        #ax.grid(True,which='both');
        ax.semilogy(EbN0dBs,ser_sim,label='N='+str(nRx))#plot simulated error rates

ax.set_xlim(0,35);ax.set_ylim(0.00001,1.1);ax.grid(True,which='both');
ax.set_xlabel('Eb/N0(dB)');ax.set_ylabel('Symbol Error Rate($P_s$)')
ax.set_title('SER performance for QPSK over Rayleigh fading channel with Selection Diversity')
ax.legend();fig.show()

Rate this post: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Andrea Goldsmith, “Wireless Communications”, ISBN: 978-0521837163, Cambridge University Press, 1 edition.↗

[2] Barry-Lee-Messerschmitt, “Digital Communication”, ISBN: 978-0792375487 , Springer, 3rd edition, September 30, 2003.↗

Articles in this series

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

Receive diversity schemes – channel models

SIMO channel configuration is characterized by 1 transmit antenna and multiple receiver antennas (Figure 1). SIMO configuration is used to provide receive diversity, where the same information is received across independent fading channels to combat fading. When multiple copies of the same data are received across independently fading channels, the amount of fade suffered by each copy of the data will be different. This guarantees that at-least one of the copy will suffer less fading compared to rest of the copies. Thus, the chance of properly receiving the transmitted data increases. In effect, this improves the reliability of the entire system. 

Figure 1: Single Input Multiple Output (SIMO) channel model

The channels are independent and identically distributed (i.i.d) and hence the error event across those independent channels are also independent. The signal to noise ratios (SNR) of the channels are also i.i.d and the every channel has the same average SNR. The is the fundamental assumption of the SIMO model incorporating receive diversity.

For the case of receive antennas, independent received signal copies are

Writing in compact form, the received signal

where,

Figure 2: Processing the received samples at the receiver

The receiver, takes in these received copies, applies weights according to the chosen receive diversity technique (see below) and outputs a single copy (Figure 2). The combined output signal is given by

where, the weight vector is

How are the weights chosen ? The weights are chosen based on the following receive diversity techniques.

Selection Combining (SC)

In selection combining, the received signal from the antenna that experiences the highest SNR is chosen for further processing. That is, the path that has the highest is chosen.

This is the simplest of all the receive diversity techniques. It simply chooses one branch that experiences the highest SNR. It seems to waste the remaining branches in the selection processes. Also, channel phase information is not needed for this technique.

This technique provides SNR gain in the order of .

Maximum Ratio Combining (MRC)

As we saw earlier, the selection combining is the simplest algorithm but it wastes receive elements in the computations. Maximum ratio combining technique uses all the received signals in order to maximize the output SNR. It represents maximum likelihood estimation. The output signal is the weighted sum of all the received branches. The weights for MRC technique is chosen as

Therefore, it requires the knowledge of the channel at the receiver and matching of both magnitude and phase. The output signal, with weight set as , is given by

The SNR gain achieved by this technique is .

Equal Gain Combining (EGC):

The MRC technique requires matching of both magnitude and the phase of the channel. The magnitudes can fluctuate significantly. In equal gain combining, only the phase of the channel is cancelled out. The weights are chosen as

where is the channel phase information. The output signal is given by

The technique suffers small SNR loss compared to MRC, but it is a good alternative for implementation.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Andrea Goldsmith, “Wireless Communications”, ISBN: 978-0521837163, Cambridge University Press, 1 edition.↗

[2] Barry-Lee-Messerschmitt, “Digital Communication”, ISBN: 978-0792375487 , Springer, 3rd edition, September 30, 2003.↗

Articles in this series

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

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.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

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

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

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

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
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

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…

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

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

Model and characterize MIMO channels

Two flavors of MIMO implementation – spatial multiplexing and spatial diversity – were discussed in the previous article. In that, it was mentioned that the reliability of a MIMO system is governed by diversity and the capacity of the link is governed by degrees of freedom.

Channel State Information (CSI)

Multiple data streams can be spatially multiplexed over the M transmit antennas and are received by the N receiver antennas. Spatially multiplexing increases the capacity of the link, since multiple data streams are transmitted over the same available frequency band. On the other hand, antenna diversity systems (dubbed as MIMO using diversity) merely improve the reliability of the link.

But the question of whether the transmission of multiple streams of data over multiple antenna really works, depends on the actual geometry of the antenna systems. Transmission of independent data streams over multiple antennas depends on the correlation factor that measures the influence of the spatially separated signals over each other. One way to eliminate correlation (there by mutual influence of spatially separated signals) is to use orthogonally polarized antennas (one antenna is horizontally polarized and the other is vertically polarized) that sufficiently separate the signals in spatial dimension.

Finally, the transmission matrix (also called Channel State Information (CSI) ) determines the suitability of MIMO techniques and influences the capacity to a great extent. In a SISO channel, the channel state information is constant and does not change from bit to bit. Thus the knowledge of CSI in a SISO link is often not needed as it is characterized by steady state SNR. In the case of rapid fading channels, the channel state information varies rapidly and we may think of employing MIMO to break the channel variations into spatially separated sub channels. Thus, the knowledge channel state information (at transmitter or receiver) will open up the possibility of incorporating this information in intelligent system design.

In a MIMO configuration, a typical CSI matrix is formed by transmitting a symbol (say value ‘1’)  from each of the transmitting antenna and its response on the multiple receiving antennas are noted. For example, in a configuration, at some time instant, we transmit the voltage ‘1’ from the first antenna and record its response on the three receiving antennas. Lets say the three receiver antennas picks up the following voltage values –  [0.8, 0.7, 0.9 ].

At the same time instant, the procedure is repeated for other transmit antennas and the response of multiple receive antennas are recorded. A complete CSI matrix is shown below

In this method, the transmitter transmits the data blindly and the receiver constructs the CSI matrix. This method of transmission is called open loop transmission scheme and are not generally effective. From the sample CSI matrix above, it can be noted that the transmission through antenna 2 is not effective (note the low voltage values recorded at the receiver antennas (second column on the right) ) the receiver may feed back the CSI matrix to the transmitter and the transmitter may decide not to transmit on antenna 2, there by saving power. This is an example for closed loop diversity scheme. In this way the knowledge of CSI opens up the possibility for intelligent communication.

The CSI matrix shown above contain only real numbers that describe the amplitude variations. In reality the CSI matrix contains elements that are complex and they describe both the amplitude and phase variations of the link.

MIMO channel Model

A channel model is needed to properly assess a MIMO channel. In MIMO, the system configuration typically contains M antennas at the transmitter and N antennas at the receiver front end as illustrated in the following figure.

Here, each receiver antenna receives not only the direct signal intended for it, but also receives a fraction of signal from other propagation paths. Thus, the channel response is expressed as a transmission matrix H. The direct path formed between antenna 1 at the transmitter and the antenna 1 at the receiver is represented by the channel response . The channel response of te path formed between antenna 1 in the transmitter and antenna 2 in the receiver is expressed as  and so on. Thus, the channel matrix is of dimension .

The received vector  is expressed in terms of the channel transmission matrix , the input vector and noise vector as

where the various symbols are

Note that the response of the MIMO link is expressed as a set of linear equations. For a simple MIMO configuration, the received signal vector is expressed as

The receiver has to solve this set of equations to find out what was transmitted (  ). The stability of the solution depends on the condition number of the transmission matrix  (CSI).

Condition Number

Solving a set of linear equation has its own challenges – rounding effects and how bad a matrix is. Obviously an on-board computer will be solving those equations. Storage of co-efficients in computer memory is prone to fixed point effects or rounding. Pivoting is method that address the problems with rounding effects when Gauss Jordan elimination procedure is used. It makes sure that the Gaussian elimination procedure proceeds as intended. Problems do occur even without rounding effects. A small change in input can cause drastic difference in the solution. In the set of linear equations mentioned above, the variations to the solutions can be effected by the noise term. The solution should be robust against variations in the noise (at-least to certain extent). The sensitivity of the solution to small changes in the input data is measured by condition number of the transmission matrix ( ). It indicates the stability of the solution ( ) to small change in incoming data ( ).

At the receiver, the received data is known and is often corrupted by noise. Let’s consider the received vector  that is corrupted by noise . Thus the system of linear equations is given as

Also, the channel transmission matrix is usually estimated approximately. The solution is obtained as

The solution to the above equation may or may not exist and may or may not be unique. Let’s consider a symmetric transmission matrix . From matrix and linear algebra[1][2], if the input is arbitrary (as is the case here), an unique solution is possible only if the matrix is non-singular. The condition number () of a non-singular matrix is given as

where  denotes the matrix norm[1]. The condition number measures the relative sensitivity of the solutions to the changes in the input data (  ). The changes to the solution can be expressed as

where represents the change in the solution, represent a change in the observed or received samples and denotes the condition number of the transmission matrix.

In other words, a small change in the input data gets multiplied by the condition number and produces changes in the output (solution). Thus high condition number is bad and is regarded as ill-conditioned matrix. An ill-conditioned matrix will behave similar to a singular matrix which will not render any solution or will give infinite non-unique solutions (see the table below).

Translating to the problem of transmission by MIMO, the ability to transmit multiple data streams across a MIMO channel – relies on the ability of the receiver to solve the system of linear equations in an unambiguous and stable way. Thus the condition number of the transmission matrix affects the suitability of spatial multiplexing in a MIMO link. A well-conditioned matrix (low condition number) allows reliable transmission of spatially multiplexed signal, whereas an ill-conditioned matrix makes it difficult to do so.

Additionally, the rank of the transmission matrix –  indicates how many data streams can be spatially multiplexed on a MIMO link. Thus the rank and the condition number of the transmission matrix play an important role in a MIMO system design.

Some useful prepositions

Existence and uniqueness

Given a system of linear equations , existence and uniqueness of the solution depends on whether the matrix is singular or non-singular. It also depends on the input vector for the singular case.

Matrix norm[1]

Matrix norm (the maximum absolute row sum) is calculated as

Non-Singular Matrix

An matrix  is non-singular if it has any of the following properties

● Inverse exists
●
●
● For any vector ,

Rate this article : Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Stephen Boyd, Symmetric matrices, quadratic forms, matrixnorm, and SVD, Stanford university, EE263 Autumn 2007-08.↗
[2] Review of Linear Algebra.↗

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

Diversity techniques and spatial multiplexing

The wireless communication environment is very hostile. The signal transmitted over a wireless communication link is susceptible to fading (severe fluctuations in signal level), co-channel interference, dispersion effects in time and frequency, path loss effect, etc. On top of these woes, the limited availability of bandwidth posses a significant challenge to a designer in designing a system that provides higher spectral efficiency and higher quality of link availability at low cost.

Previous article in this series : Introduction to Multiple Antenna Systems

Multiple antenna systems are the current trend in many of the wireless technologies that is essential for their performance (you will even see it in your future hard disk drives as Two Dimensional Magnetic Recording (TDMR) technology). Multiple Input Multiple Output systems (MIMO) improve the spectral efficiency and offers high quality links when compared to  traditional Single Input Single Output (SISO) systems. Many theoretical studies [1-2] and communication system design experimentations [3-5] on MIMO systems demonstrated a great improvement in performance of such systems.

Techniques for improving performance

Spatial Multiplexing techniques [6], example – BLAST[7] yields increased data rates in wireless communication links. Fading can be mitigated by employing receiver and transmit diversity (Alamouti Scheme [8] , Tarokh et. al[9]) , there by improving the reliability of the transmission link. Improved coverage can be effected by employing coherent combining techniques – which gives array gain and increases the signal to noise ratio of the system. The goals of a wireless communication system are conflicting and a clear balance of the goals is needed for maximizing the performance of the system.

The following text concentrates on two of the above mentioned techniques – diversity and spatial multiplexing.

MIMO classification with respect to antenna configuration

In MIMO jargon, communication systems are broadly categorized into four categories with respect to number of antennas in the transmitter and the receiver, as listed below.

● SISO – Single Input Single Output system – 1 Tx antenna , 1 Rx antenna
● SIMO – Single Input Multiple Output system – 1 Tx antenna, Rx antennas ()
● MISO – Multiple Input Single Output system – Tx antennas, 1 Rx antenna ()
● MIMO – Multiple Input Multiple Output system – Tx antennas, Rx antennas ()

Diversity and Spatial-Multiplexing

Apart from the antenna configurations, there are two flavors of MIMO with respect to how data is transmitted across the given channel. Existence of multiple antennas in a system, means existence of different propagation paths. Aiming at improving the reliability of the system, we may choose to send same data across the different propagation (spatial) paths. This is called spatial diversity or simply diversity. Aiming at improving the data rate of the system, we may choose to place different portions of the data on different propagation paths (spatial-multiplexing). These two systems are listed below.

● MIMO – implemented using diversity techniques – provides diversity gain – Aimed at improving the reliability
● MIMO – implemented using spatial-multiplexing techniques – provides degrees of freedom or multiplexing gain – Aimed at improving the data rate of the system.

Diversity:

As indicated, two fundamental resources available for a MIMO system are diversity and degrees of freedom. Let’s see what these terms mean

In diversity techniques, same information is sent across independent fading channels to combat fading. When multiple copies of the same data are sent across independently fading channels, the amount of fade suffered by each copy of the data will be different. This guarantees that at-least one of the copy will suffer less fading compared to rest of the copies. Thus, the chance of properly receiving the transmitted data increases. In effect, this improves the reliability of the entire system. This also reduces the co-channel interference significantly. This technique is referred as inducing a “spatial diversity” in the communication system.

Consider a SISO system where a data stream [1, 0, 1, 1, 1] is transmitted through a channel with deep fades. Due to the variations in the channel quality, the data stream may get lost or severely corrupted that the receiver cannot recover.The solution to combat the rapid channel variations is to add independent fading channel by increasing the number of transmitter antennas or receiver antennas or the both.

The SISO antenna configuration will not provide any diversity as there is no parallel link. Thus the diversity is indicated as (0).

Single Input Single Output (SISO) system – no diversity

Instead of transmitting with single antenna and receiving with single antenna (as in SISO), let’s increase the number of receiving antennas by one more count. In this Single Input Multiple Output (SIMO) antenna system, two copies of the same data are put on two different channels having independent fading characteristics. Even if one of the link fails to deliver the data, the chances of proper delivery of the data across the other link is very high. Thus, additional fading channels increase the reliability of the overall transmission – this improvement in reliability translates into performance improvement – measured as diversity gain. For a system with transmitter antennas and receiver antennas, the maximum number of diversity paths is . In the following configuration, the total number of diversity path created is .

Single Input Multiple Output Channel with diversity

In this way, more diversity paths can be created by adding multiple antennas at transmitter or receiver or both. The following figure illustrates a MIMO system with number of diversity paths equal to .

MIMO system with diversity

Spatial Multiplexing:

In spatial multiplexing, each spatial channel carries independent information, there by increasing the data rate of the system. This can be compared to Orthogonal Frequency Division Multiplexing (OFDM) technique, where, different frequency subchannels carry different parts of the modulated data. But in spatial multiplexing, if the scattering by the environment is rich enough, several independent subchannels are created in the same allocated bandwidth. Thus the multiplexing gain comes at no additional cost on bandwidth or power. The multiplexing gain is also referred as degrees of freedom with reference to signal space constellation [2]. The number of degrees of freedom in a multiple antenna configuration is equal to , where is the number of transmit antennas and is the number of receive antennas. The degrees of freedom in a MIMO configuration governs the overall capacity of the system.

Following figure illustrates the difference between diversity and spatial multiplexing. In the transmit diversity technique shown below, same information is sent across different independent spatial channels by placing them on three different transmit antennas. Here, the diversity gain is 3  (assuming MISO configuration) and multiplexing gain is 0.

In the spatial multiplexing technique, each bit of the data stream (independent information) is multiplexed on three different spatial channels thereby increasing the data rate. Here, the diversity gain is 0 and the multiplexing gain is 3 (assuming MIMO configuration).

MIMO system – Diversity Vs spatial multiplexing

Exploiting diversity and degree of freedom:

As seen above, in a MIMO system with rich scattering environment (independent uncorrelated spatial paths), space time codes are designed to exploit following two resources.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] I. E. Telatar, “Capacity of multi-antenna gaussian channels,” European Transactions on Telecommunication, vol. 10, pp. 585–595, Nov./Dec. 1999.
[2] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., vol. 1, no. 2, pp. 41–59, 1996.
[3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block code from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999.
[4] G. Foschini, G. Golden, R. Valenzuela, and P. Wolniansky, “Simplified processing for high spectal efficiency wireless communication employing multi-element arrays,” IEEE J. Select. Areas Commun., vol. 17, pp. 1841–1852, Nov. 1999.
[5] R. Heath, Jr. and A. Paulraj, “Switching between multiplexing and diversity based on constellation distance,” in Proc. Allerton Conf. Communication, Control and Computing, Oct. 2000.
[6] A. Paulraj and T. Kailath, Increasing capacity in wireless broadcast Systems using distributed  transmission/directional reception (DTDR), US Patent No. 5,345,599, Issued 1993
[7] Gerard. J. Foschini, Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multi-Element Antennas”.Bell Laboratories Technical Journal: 41–59,(October 1996)
[8] S.M. Alamouti (October 1998). “A simple transmit diversity technique for wireless communications”. IEEE Journal on Selected Areas in Communications 16 (8): 1451–1458
[9] V. Tarokh, N. Seshadri, A. Calderbank, ‘Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction,’ IEEE Trans. on. Information Theory, Vol. 44, No.2, pp.744-765, March 1998

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

Introduction to Multiple Antenna Systems

Regarded as a breakthrough in wireless communication system design, multiple antenna systems fuel the ever increasing data rate requirements of advanced technologies like UMTS, LTE, WLAN etc. Multiple antenna systems come in different flavors and are generally referred as Multiple Input Multiple Output systems (MIMO). In a series of articles, I intend to cover various aspects of multiple antenna systems – specifically the diversity techniques.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Diversity techniques:

Diversity techniques are employed to make a communication system robust and reliable even over varying channel conditions. Diversity techniques exploit the channel variations rather than mitigating it. Diversity techniques combat fading and interference by presenting the receiver with multiple uncorrelated copies of the same information bearing signal. Essentially, diversity techniques are aimed at creating uncorrelated random channels -uncorrelated copies of the same signal (may also be in combined form) at the receiver front end. Combining techniques are employed at the receiver to exploit multipath propagation characteristics of a channel.

Broadly speaking, the term diversity broadly categorized as

● Time diversity:
—-Multiple copies of information sent on different time slots. The time slots are designed in such a way that the delay between the signal replicas should be greater than the Coherence Time () of the channel. This condition will create uncorrelated channels over those time slots. Coding and interleaving are also equivalent techniques that break the channel memory into multiple chunks there by spreading and minimizing the effect of deep fades. This technique consumes extra transmission time.
● Frequency diversity:
—-Signal replicas are sent across different frequency bands that are separated by at-least the Coherence Bandwidth () of the channel – thereby creating uncorrelated channels for the transmission. By this method, the level of fading experienced by each frequency bands are different and at-least one of the frequency bands may experience the lowest level of fading – strongest signal in this band. This technique requires extra bandwidth. Example : Orthogonal Frequency Division Multiplexing (OFDM) and spread spectrum.
● Multiuser diversity [1-6]:
—-Employing adaptive modulation and user scheduling techniques to improve the performance of a multiuser system. In such systems, the channel quality information for each user is utilized by a scheduler to select the number of users, coding and modulation such that an objective function (throughput and fairness of scheduling) is optimized. Example: OFDMA and access scheme used in latest wireless systems such as IEEE 802.16e (Mobile WiMAX)
● Spatial diversity (antenna diversity):
—Aimed at creating uncorrelated propagation paths for a signal, spatial diversity is effected by usage of multiple antennas in the transmitter and/or the receiver. Employing multiple antennas at the transmitter is called “Transmit Diversity” and multiple antennas at the receiver is called “Reception Diversity“. Diversity combining techniques like Selection Combining (SC), Feedback or Scanning Combining (FC or SC), Maximum Ratio Combining (MRC) can be employed by the receiver to exploit the multiplath effects. Spatial diversity techniques can also be used to increase the data rates (spatial multiplexing) rather than improving the reliability of the channel. Example: MIMO, beamforming and Space-Time Coding (STC)
● Polarization diversity (antenna diversity):
—-Multiple copies of the same signal are transmitter and received by antennas of different polarization. Used the mitigate polarization mismatches of transmit and receiver antennas.
● Pattern diversity (antenna diversity):
—-Multiple versions of the signal are transmitted via two or more antenna with different radiation patterns. The antennas are spaced such that they collectively act as a single entity aimed at providing more directional gain compared to an omni-directional antenna.
● Adaptive arrays (antenna diversity):
—-Ability to control the radiation pattern of a single antenna with active elements or an array of antennas. The radiation pattern is adapted based on the existing channel conditions.

The above is just a broad classification. There exists more advanced diversity techniques (example : cooperative diversity↗) that can be a combination of one or more techniques listed above.

The following text concentrates on Spatial Diversity or the MIMO systems.

Single Input Single Output (conventional system)

The conventional radio communication systems contain one antenna at the transmitter and one at the receiver.  In MIMO jargon, this is called Single Input and Single Output system (SISO). The channel capacity (C)) is constrained by Signal to Noise Ratio (S/N)) and the bandwidth (B) as given by the well-known Shannon Hartley theorem for a Continuous Input Continuous Output Memoryless AWGN channel (CCMC) [7].

Figure 1: Single input single output channel

Multiple Antenna Systems or MIMO:

Here, the system configuration typically contains M antennas at the transmitter and N antennas at the receiver front end as illustrated in the next figure. Here, each receiver antenna receives not only the direct signal intended for it, but also receives a fraction of signal from other propagation paths. Thus, the channel response is expressed as a transmission matrix . The direct path formed between antenna 1 at the transmitter and the antenna 1 at the receiver is represented by the channel response . The channel response of the path formed between antenna 1 in the transmitter and antenna 2 in the receiver is expressed as and so on. Thus, the channel matrix is of dimension .

Figure 2: Multiple input multiple output (MIMO) system

The received vector is expressed in terms of the channel transmission matrix , the input vector and noise vector  as

where the various symbols are

For asymmetrical antenna configuration (), the number of data streams (or the number of uncoupled equivalent channels) in the system is always less than equal to the minimum of the number of transmitter and receiver antennas – .

For a single user system, the capacity scales linearly with    relative to a SISO system[8].

More on the classification of multiple antenna systems and their capacity in subsequent posts.

Jump to next article in this series: MIMO – Diversity and Spatial Multiplexing

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References:

[1] M Torabi, D Haccoun, Performance analysis of joint user scheduling and antenna selection over MIMO fading channels. IEEE Signal Process Lett 18(4), 235–238 (2011).↗
[2] L Jin, X Gu, Z Hu, Low-complexity scheduling strategy for wireless multiuser multiple-input multiple-output downlink system, IET Commun 5(7), 990–995 (2011).↗
[3] B Makki, T Eriksson, Efficient channel quality feedback signaling using transform coding and bit allocation, Vehicular Technology Conference, VTC (Ottawa, ON, 2010), pp. 1–5.↗
[4] T Eriksson, T Ottosson, Compression of feedback for adaptive transmission and scheduling. Proc IEEE 95(12), 2314–2321 (2007).↗
[5] G Dimic, ND Sidiropoulos, On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm, IEEE Trans Signal Process 53(10), 3857–3868 (2005).↗
[6] VKN Lau, Proportional fair space-time scheduling for wireless communications. IEEE Trans Commun 53(8), 1353–1360 (2005).↗
[7] J. G. Proakis, Digital Communications, Mc-Graw Hill International, Editions, 3rd ed., 1995.↗
[8] Torlak, M.; Duman, T.M., MIMO communication theory, algorithms, and prototyping, Signal Processing and Communications Applications Conference (SIU), 2012 20th , vol., no., pp.1,2, 18-20 April 2012.↗

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