Tag: SIMO

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()

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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)

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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()

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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)
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Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
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Checkout Added to cart

Digital Modulations using Matlab
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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.

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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

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.

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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)

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