Diversity and MIMO System for Fading Channels

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Diversity and MIMO System for Fading Channels
Dr. Essam Sourour Alexandria University, Faculty
of Engineering, Dept. Of Electrical Engineering
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Overview

• Diversity and its types
• MIMO systems

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Why Diversity?

• BER in fading channel is calculated by averaging
  the AWGN BER over the fading pdf
• In Rayleigh fading channel the pdf of the SNR, g
  is given by the exponential function
• This pdf causes large performance loss
• If we get a better pdf of the SNR, performance
  will increase
• This is done by diversity techniques

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What is Diversity ?

• Diversity is achieved by creating several
  independent paths between the transmitter and
  receiver
• Each path fades independently, hence, there is a
  low chance they fade together
• Receiver combines the received signal for the
  several paths using some method
• Diversity is used in all mobile communication
  systems

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Types of Diversity

• Space diversity multiple transmit and multiple
  receive antennas
• Multiple Tx split power over several Tx
  antennas. More antennas more power split
• Multiple Rx collect signal by several Rx
  antennas. More antennas more collected power
• Antennas separation about l/2 is required
• If directional antennas (typically) larger
  separation is required

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Types of Diversity

• Polarization diversity
• Transmit and/or receive with both vertical and
  horizontal polarization
• Scattering is independent for each polarization,
  giving independent paths
• Limited to 2 transmit and 2 receive diversity
• Tx polarization diversity half power for each
  polarization

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Types of Diversity

• Frequency diversity
• Transmit same signal with several frequencies
• Frequencies separated by gt coherence bandwidth
• Also wideband signals achieve frequency
  diversity, like OFDM techniques over wideband
  (WLAN, WiMax, LTE)
• Multipath diversity
• In Direct-Sequence-Spread-Spectrum signals we can
  receive from multipath separately using Rake
  receiver
• Used in all CDMA systems (IS-95, CDMA2000, WCDMA)

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Types of Diversity

• Time diversity
• Signal is re-transmitted (repeated) after gt
  coherence time
• Also achieved using coding and interleaving
• Reduces overall transmission data rates
• Coding and interleaving used in all mobile
  communication systems
• Also combined with repeat-diversity in what is
  called Hybrid Automatic Repeat Request (H-ARQ)

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Types of Diversity

• Beam-forming
• Transmit with antenna array
• Each antenna is fed with different phase
• Forms a directional beam towards the receiver, or
  group of receivers
• Antenna beam tracks the intended receiver
• Requires knowledge of the fading channel at Tx
• Optional is 4G mobile communication systems
  (WiMax and LTE)

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Types of Diversity

• In all types of diversity we achieve several
  independent paths between Tx and Rx
• If not independent there is some performance
  degradation
• If correlation coefficient lt 0.5 degradation is
  not noticeable
• Space diversity will be studied, but approach
  applies to all methods

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Diversity Order

• In many cases the average probability of error
  looks like
• M is the diversity order
• If the number of paths between Tx and Rx is N,
  then diversity order ? N
• For example if Nt is the number of Tx antennas
  and Nr is the number of Rx antennas
• Number of branches N Nt Nr
• Diversity order M ? N
• Full diversity order if M number of branches

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Receiver Diversity

• Signal from multiple receiver antennas are
  linearly combined
• Weighted sum of each branch
• Each branch weight is complex
• The phase of each received path are aligned
• Signals from each branch are coherently combined

• Receiver must estimate the phase (and some times
  gain) of each branch
• Estimation through pilot signals

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Receiver Diversity

• Receiver diversity achieves two types of gain
  Array gain and Diversity gain
• Array gain due to coherent combining of multiple
  branches
• Defined average combined SNR /average branch SNR
  
• Achieved even if there is no fading
• Diversity gain due to the better (than
  exponential) pdf p(g) of the combined SNR
• Better BER results of the integration

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Receiver Diversity

• Several methods to combine the receiver branches
• Selection combining
• Threshold combining
• Maximal ratio combining
• Equal gain combining
• Mix of the above
• Tradeoff between performance and complexity

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Selection Combining

• Fading channel path gain to each branch is
• Noise PSD at each branch is Ni
• Receiver selects the branch with largest
  instantaneous SNR
• If the noise PSD No is the same for all branches,
  this is equivalent to selecting largest
• Easier to implement
• Combiner output SNR best branch SNR

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Selection Combining

• For M branch diversity, the CDF of the combined
  SNR is given by
• Probability of best SNR lt g Probability all SNR
  lt g
• In Rayleigh fading each SNR is exponential
• CDF of M branches is given by
• If the target SNR is go, outage is given by
• If the average SNR per branch are equal

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Selection Combining

• Most benefit from M1 (no diversity) to M2
• Benefit reduces as M increases

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Selection Combining

• From basic probability, the PDF is the
  differential of CDF
• If the average SNR per branch are equal the CDF
  is given by
• The PDF of the combined SNR is given by
  differentiating w.r.t g
• The average SNR after combining is
• Increases with M, but not linear

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Selection Combining

• Probability of error is calculated by averaging
  the AWGN formula over the PDF of the SNR
• Now we use
• For BPSK there is no closed form, needs numerical
  integration (or simulation)
• For DPSK we get

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Selection Combining

• Similarly, more gain when moving from M1 to M2
• Gains reduces as M increases

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Selection Combining

• Same observation for D-BPSK (equation 7.11)
• Gains reduces as M increases

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Threshold Combining

• A simpler approach is to stay with a branch till
  its SNR falls below a threshold
• If the SNR falls below threshold, switch to
  another branch according to
• Random selection, or
• Best branch
• Example 2 branches

Bad switch
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Threshold Combining

• Assume M2 and switching happens when the SNR on
  one antenna falls below gT (even if to lower
  SNR)
• The CDF of selected SNR is the probability that
  SNR lt g
• Two possibilities for overall SNR, SNR lt gT , or
  gt gT
• If SNR lt gT
• Prob(SNRltg)Prob(SNR1ltgT) x Prob(SNR2ltg)
• If SNR gt gT
• Prob(SNRltg)Prob(gT ltSNR1ltg)
• Prob(SNR1ltgT) x Prob(SNR2ltg)
• For Rayleigh fading and equal average SNR

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Threshold Combining

• Hence, the CDF is given by
• The outage probability is given by
• If the threshold gT ?o, the outage probability
  is similar to selection diversity with M2

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Threshold Combining

• The PDF is found by differentiating the CDF
• Again, for BPSK there is no closed form for BER.
  Requires numerical integration or simulation
• For DPSK the BER is given by

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Threshold Combining

• M2
• DBPSK with threshold combining
• Slightly worse than selection combining

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Maximal Ratio Combining

• Instead of selecting one branch, all branches are
  added with weight ai exp(-j?i)
• If a transmitted symbol is s with unity power
  s21
• The received symbol at branch i is s ri
  exp(j?i) ni
• The combined symbol is
• Assume all ni have equal variance No, the SNR is
• We need to select ai to maximize SNR

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Maximal Ratio Combining

• Using Swartz inequality we find the optimum
  weights
• The resulting SNR is
• Combined SNR is the sum of all branches SNR
• Each gi is exponential with mean
• Assuming all branch SNR have equal mean, g? is
  chi-square with 2M degrees of freedom (see
  Proakis, chapter 2) and mean
• The PDF is given by

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Maximal Ratio Combining

• The outage probability is found from PDF
• The average BER is calculated by averaging the
  AWGN BER over the random SNR
• For BPSK

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Maximal Ratio Combining

• BER performance for BPSK
• Best diversity method

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Comparison
BPSK with Maximal Ratio Combining
BPSK with Selection Combining
Maximal Ratio Combining provides better
performance (lower BER)
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Equal Gain Combining

• MRC required knowledge of the SNR on each branch
• Simpler approach is equal weight for all branches
  (all ai1)
• The combined SNR is
• There is no closed form solution for the CDF or
  PDF except for M2

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Equal Gain Combining

• From the CDF we find the outage probability
• Also the BER for BPSK is

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Equal Gain Combining

• BER for BPSK

BPSK with Maximal Ratio Combining
BPSK with Equal Gain Combining
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MIMO

• Traditional diversity is based on multiple
  receiver antennas
• Multiple-In Multiple-Out (MIMO) is based on both
  transmit and receive diversity
• Also known as Space Time Coding (STC)
• With Mt transmission antennas and Mr receiver
  antennas we have Mt Mr branches
• Tx and Rx processing is performed over space
  (antennas) and time (successive symbols)

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MIMO or STC

• In Mobile communication systems it may be
  difficult to put many antennas in the mobile unit
• Diversity in the downlink (from base station to
  mobile station) can be achieved by Multiple-In
  Single-Out (MISO) (i.e., Mr1)
• In the uplink (from mobile station to base
  station) diversity is achieved my conventional
  diversity (SIMO)
• Hence, all diversity cost is moved to the base
  station
• All 3G and 4G mobile communication system employ
  MIMO in their standard

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Type of MIMO

• Two major types of space time coding
• Space time block coding (STBC)
• Space time trellis coding (STTC)
• STBC is simpler by STTC can provide better
  performance
• STBC is used in mobile communications. STTC is
  not used in any systems yet
• We will talk only about STBC

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Space Time Block Codes

• There are few major types
• Transmit diversity main goal is diversity gain
• Spatial multiplexing main goal is increase data
  rate
• Eigen steering main goal is both. Requires
  knowledge of the channel at the transmitter side
• Mix of the above Lots of research
• Transmit diversity, spatial multiplexing and
  simplified version of Eigen steering are used in
  3G and 4G standards
• While in 3G standards MIMO was an enhancement, in
  4G MIMO is a main part

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Transmit Diversity

• Take Mt2 and Mr1
• Two symbols so and s1 are transmitted over two
  transmission periods
• No change in data rate (denoted as rate 1 STBC)
• Channel is known at receiver only

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Transmit Diversity

• Transmission matrix
• Transmission matrix columns are orthogonal to
  guarantee simple linear processing at the
  receiver
• Other transmission matrices are defined in
  literature
• Received signal is
• Performance is same as MRC with M2
• However, if Tx Power is the same, then transmit
  diversity (2x1) is 3 dB worse than (1x2)

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Transmit Diversity

• Take Mt2 and Mr2
• Performance is the same as MRC with M4
• However, if Tx Power is the same, then transmit
  diversity (2x2) is 3 dB worse than (1x4)

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Performance

• MRRCMaximal Ratio Receiver Combining
• Note 3 dB difference in favor of Rx MRC diversity
• Reference S. Alamouti, a simple transmit
  diversity technique for wireless communications,
• IEEE JSAC, October 98

No diversity
Order 2
Order 4
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Spatial Multiplexing

• Purpose is to increase data rate (2x2 gives twice
  data rate)
• The 4 gains must be known at receiver
• Simplest way zero forcing algorithm

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Spatial Multiplexing

• Optimum method Maximum Likelihood
• Try all combinations of s1 and s2
• Find the combination that minimizes the squared
  error
• Complexity increases with high order modulation

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Performance

• Equal rate comparison
• Reference David Gesbert, Mansoor Shafi, Da-shan
  Shiu, Peter J. Smith, and Ayman Naguib, From
  theory to practice an overview of MIMO
  spacetime coded wireless systems, IEEE JSAC,
  April 2003

Zero forcing
ML
Alamouti
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Eigenvalue Steering

• Assume a MIMO system

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Eigenvalue Steering

• Example with Mt 2 and Mr4
• Any matrix H can be represented using Singular
  Value Decomposition as
• U is Mr by Mr and V is Mt by Mt unitary matrices
• ? is Mr by Mt diagonal matrix, elements si

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Eigenvalue Steering

• Using transmit pre-coding and receiver shaping

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Eigenvalue Steering

• This way we created r paths between the Tx and
  specific Rx without any cross interference
• The channel (i.e., Channel State Information)
  must be known to both transmitter and receiver
• The value of r rank of matrix H, r ?min(Mt,
  Mr)
• Not all r paths have good SNR
• Data rate can increase by factor r
• See Appendix C for Singular Value Decomposition
• See Matlab function U,S,V svd(X)

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Example

• Reference Sanjiv Nanda, Rod Walton, John
  Ketchum, Mark Wallace, and Steven Howard, A
  high-performance MIMO OFDM wireless LAN, IEEE
  Communication Magazine, February 2005

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SVD Matlab Example

• X is 4 by 2 matrix
• X1 2 3 4 5 6 7 8
• U,S,V svd(X)
• U
• -0.1525 -0.8226 -0.3945 -0.3800
• -0.3499 -0.4214 0.2428 0.8007
• -0.5474 -0.0201 0.6979 -0.4614
• -0.7448 0.3812 -0.5462 0.0407

• V
• -0.6414 0.7672
• -0.7672 -0.6414
• S
• 14.2691 0
• 0 0.6268
• 0 0
• 0 0
• Then s1 14.2691 and s2 0.6268