16.548 Coding and Information Theory

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16.548 Coding and Information Theory

• Lecture 15 Space Time Coding and MIMO

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Credits
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Wireless Channels
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Signal Level in Wireless Transmission
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Classification of Wireless Channels
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Space time Fading, narrow beam
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Space Time Fading Wide Beam
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Introduction to the MIMO Channel
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Capacity of MIMO Channels
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Single Input- Single Output systems (SISO)
x(t) transmitted signal y(t) received
signal g(t) channel transfer function n(t)
noise (AWGN, ?2)
g
y(t)
x(t)

• y(t) g x(t) n(t)

Signal to noise ratio Capacity
C log2(1?)
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Single Input- Multiple Output (SIMO) Multiple
Input- Single Output (MISO)

• Principle of diversity systems (transmitter/
  receiver)
• Higher average signal to noise ratio
• Robustness
• - Process of diminishing return
• Benefit reduces in the presence of correlation
• Maximal ratio combining gt
• Equal gain combining gt
• Selection combining

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Idea behind diversity systems

• Use more than one copy of the same signal
• If one copy is in a fade, it is unlikely that all
  the others will be too.
• C1xNgtC1x1
• C1xN more robust than C1x1

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Background of Diversity Techniques

• Variety of Diversity techniques are proposed to
  combat Time-Varying Multipath fading channel in
  wireless communication
• Time Diversity
• Frequency Diversity
• Space Diversity (mostly multiple receive
  antennas)
• Main intuitions of Diversity
• Probability of all the signals suffer fading is
  less then probability of single signal suffer
  fading
• Provide the receiver a multiple versions of the
  same Tx signals over independent channels
• Time Diversity
• Use different time slots separated by an interval
  longer than the coherence time of the channel.
• Example Channel coding interleaving
• Short Coming Introduce large delays when the
  channel is in slow fading

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Diversity Techniques

• Improve the performance in a fading environment
• Space Diversity
• Spacing is important! (coherent distance)
• Polarization Diversity
• Using antennas with different polarizations for
  reception/transmission.
• Frequency Diversity
• RAKE receiver, OFDM, equalization, and etc.
• Not effective over frequency-flat channel.
• Time Diversity
• Using channel coding and interleaving.
• Not effective over slow fading channels.

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RX Diversity in Wireless
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Receive Diversity
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Selection and Switch Diversity
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Linear Diversity
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Receive Diversity Performance
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Transmit Diversity
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Transmit Diversity with Feedback
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TX diversity with frequency weighting
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TX Diversity with antenna hopping
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TX Diversity with channel coding
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Transmit diversity via delay diversity
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Transmit Diversity Options
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MIMO Wireless Communications Combining TX and RX
Diversity

• Transmission over Multiple Input Multiple Output
  (MIMO) radio channels
• Advantages Improved Space Diversity and Channel
  Capacity
• Disadvantages More complex, more radio stations
  and required channel estimation

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MIMO Model
T Time index W Noise

• Matrix Representation
• For a fixed T

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Part II Space Time Coding
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Multiple Input- Multiple Output systems (MIMO)
H11
HN1
H1M
HNM

• Average gain
• Average signal to noise ratio

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Shannon capacity

• K rank(H) what is its range of values?
• Parameters that affect the system capacity
• Signal to noise ratio ?
• Distribution of eigenvalues (u) of H

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Interpretation I The parallel channels approach

• Proof of capacity formula
• Singular value decomposition of H H SUVH
• S, V unitary matrices (VHVI, SSH I)
• U diag(uk), uk singular values of H
• V/ S input/output eigenvectors of H
• Any input along vi will be multiplied by ui and
  will appear as an output along si

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Vector analysis of the signals

• 1. The input vector x gets projected onto the
  vis
• 2. Each projection gets multiplied by a different
  gain ui.
• 3. Each appears along a different si.

u1
ltx,v1gt v1
ltx,v1gt u1 s1
u2
ltx,v2gt v2
ltx,v2gt u2 s2
uK
ltx,vKgt uK sK
ltx,vKgt vK
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Capacity sum of capacities

• The channel has been decomposed into K parallel
  subchannels
• Total capacity sum of the subchannel capacities
• All transmitters send the same power
• ExEk

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Interpretation II The directional approach

• Singular value decomposition of H H SUVH
• Eigenvectors correspond to spatial directions
  (beamforming)

1 M
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Example of directional interpretation
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Space-Time Coding

• What is Space-Time Coding?
• Space diversity at antenna
• Time diversity to introduce redundant data
• Alamouti-Scheme
• Simple yet very effective
• Space diversity at transmitter end
• Orthogonal block code design

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Space Time Coded Modulation
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Space Time Channel Model
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STC Error Analysis
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STC Error Analysis
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STC Design Criteria
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STC 4-PSK Example
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STC 8-PSK Example
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STC 16-QAM Example
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STC Maximum Likelihood Decoder
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STC Performance with perfect CSI
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Delay Diversity
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Delay Diversity ST code
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Space Time Block Codes (STBC)
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Decoding STBC
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Block and Data Model

• 1X(NP) block of information symbols broadcast
  from transmit antenna i
• Si(d, t)
• 1X(NP) block of received information symbols
  taken from antenna j
• Rj hjiSi(d, t) nj
• Matrix representation

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Related Issues

• How to define Space-Time mapping Si(d,t) for
  diversity/channel capacity trade-off?
• What is the optimum sequence for pilot symbols?
• How to get best estimated Channel State
  Information (CSI) from the pilot symbols P?
• How to design frame structure for Data symbols
  (Payload) and Pilot symbols such that most
  optimum for FER and BER?

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Specific Example of STBC Alamoutis Orthogonal
Code

• Lets consider two antenna i and i1 at the
  transmitter side, at two consecutive time
  instants t and tT
• The above Space-Time mapping defines Alamoutis
  Code1.
• A general frame design requires concatenation of
  blocks (each 2X2) of Alamouti code,

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Estimated Channel State Information (CSI)

• Pilot Symbol Assisted Modulation (PSAM) 3 is
  used to obtain estimated Channel State
  Information (CSI)
• PSAM simply samples the channel at a rate greater
  than Nyquist rate,so that reconstruction is
  possible
• Here is how it works

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Channel State Estimation
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Estimated CSI (cont.d) Block diagram of the
receiver
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Channel State Estimation (cont.d)

• Pilot symbol insertion length, Pins6.
• The receiver uses N12, nearest pilots to obtain
  estimated CSI

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Channel State Estimation Cont.d

• Pilot Symbols could be think of as redundant data
  symbols
• Pilot symbol insertion length will not change the
  performance much, as long as we sample faster
  than fading rate of the channel
• If the channel is in higher fading rate, more
  pilots are expected to be inserted

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Estimated CSI, Space-time PSAM frame design

• The orthogonal pilot symbol (pilots chosen from
  QPSK constellation) matrix is, 4
• Pilot symbol insertion length, Pins6.
• The receiver uses N12, nearest pilots to obtain
  estimated CSI
• Data 228, Pilots 72

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Channel State Estimation (cont.d)MMSE estimation

• Use Wiener filtering, since it is a Minimum Mean
  Square Error (MMSE) estimator
• All random variables involved are jointly
  Gaussian, MMSE estimator becomes a linear minimum
  mean square estimator 2
• Wiener filter is defined as,
  .
• Note, and

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Block diagram for MRRC scheme with two Tx and one
Rx
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Block diagram for MRRC scheme with two Tx and one
Rx

• The received signals can then be expressed as,
• The combiner shown in the above graph builds the
  following two estimated signal

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Maximum Likelihood Decoding Under QPSK
Constellation

• Output of the combiner could be further
  simplified and could be expressed as follows
• For example, under QPSK constellation decision
  are made according to the axis.

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Space-Time Alamouti Codes with Perfect CSI,BPSK
Constellation
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Space-Time Alamouti Codes with PSAM under QPSK
Constellation
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Space-Time Alamouti Codes with PSAM under QPSK
Constellation
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Performance metrics

• Measures of comparison
• Gaussian iid channel
• ideal channel

• Eigenvalue distribution
• Shannon capacity
• for constant SNR or
• for constant transmitted power
• Effective degrees of freedom(EDOF)
• Condition number

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Measures of comparison

• Gaussian Channel
• Hij xijjyij x,y i.i.d. Gaussian random
  variables
• Problem poutage

• Ideal channel (max C)
• rank(H) min(M, N)
• u1 u2 uK

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Eigenvalue distribution
Limits Power constraints System size Correlation

• Ideally
• As high gain as possible
• As many eigenvectors as possible
• As orthogonal as possible

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Example Uncorrelated correlated channels
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Shannon capacity

• Capacity for a reference SNR (only channel info)
• Capacity for constant transmitted power (channel
  power roll-off info)

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Building layout
RCVR (hall)
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LOS conditions Higher average SNR, High
correlationNon-LOS conditions Lower average
SNR,More scattering
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Example C for reference SNR
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Example C for constant transmit pwr
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Other metrics
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From narrowband to wideband

• Wideband delay spread gtgt symbol time
• - Intersymbol interference
• Frequency diversity
• SISO channel impulse response
• SISO capacity

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Matrix formulation of wideband case
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Equivalent treatment in the frequency domain

• Wideband channel Many narrowband channels
• H(t) ? H(f)

Noise level
f
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Extensions

• Optimal power allocation
• Optimal rate allocation
• Space-time codes
• Distributed antenna systems
• Many, many, many more!

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Optimal power allocation

• IF the transmitter knows the channel, it can
  allocate power so as to maximize capacity
• Solution Waterfilling

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Illustration of waterfilling algorithm
?
Stronger subchannels get the most power
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Discussion on waterfilling

• Criterion Shannon capacity maximization
• (All the SISO discussion on coding,
  constellation limitations etc is pertinent)
• Benefit depends on the channel, available power
  etc.
• Correlation, available power ?? Benefit ?
• Limitations
• Waterfilling requires feedback link
• FDD/ TDD
• Channel state changes

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Optimal rate allocation

• Similar to optimal power allocation
• Criterion throughput (T) maximization
• Bk bits per symbol (depends on constellation
  size)
• Idea for a given ?k, find maximum Bk for a
  target probability of error Pe

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Discussion on optimal rate allocation

• Possible limits on constellation sizes!
• Constellation sizes are quantized!!!
• The answer is different for different target
  probabilities of error
• Optimal power AND rate allocation schemes
  possible, but complex

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Distributed antenna systems

• Idea put your antennas in different places
• lower correlation
• - power imbalance, synchronization,
  coordination

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Practical considerations

• Coding
• Detection algorithms
• Channel estimation
• Interference

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Detection algorithms

• Maximum likelihood linear detector
• y H x n ? xest Hy
• H (HH H)-1 HH Pseudo inverse of H
• Problem find nearest neighbor among QM points
• (Q constellation size, M number of
  transmitters)
• VERY high complexity!!!

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Solution BLAST algorithm

• BLAST Bell Labs lAyered Space Time
• Idea NON-LINEAR DETECTOR
• Step 1 H (HH H)-1 HH
• Step 2 Find the strongest signal
• (Strongest the one with the highest post
  detection SNR)
• Step 3 Detect it (Nearest neighbor among Q)
• Step 4 Subtract it
• Step 5 if not all yet detected, go to step 2

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Discussion on the BLAST algorithm

• Its a non-linear detector!!!
• Two flavors
• V-BLAST (easier)
• D-BLAST (introduces space-time coding)
• Achieves 50-60 of Shannon capacity
• Error propagation possible
• Very complicated for wideband case

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Coding limitations

• Capacity Maximum achievable data rate that can
  be achieved over the channel with arbitrarily low
  probability of error
• SISO case
• Constellation limitations
• Turbo- coding can get you close to Shannon!!!
• MIMO case
• Constellation limitations as well
• Higher complexity
• Space-time codes very few!!!!

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Channel estimation

• The channel is not perfectly estimated because
• it is changing (environment, user movement)
• there is noise DURING the estimation
• An error in the channel transfer characteristics
  can hurt you
• in the decoding
• in the water-filling
• Trade-off Throughput vs. Estimation accuracy
• What if interference (as noise) is not white????

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Interference

• Generalization of other/ same cell interference
  for SISO case
• Example cellular deployment of MIMO systems
• Interference level depends on
• frequency/ code re-use scheme
• cell size
• uplink/ downlink perspective
• deployment geometry
• propagation conditions
• antenna types

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Summary and conclusions

• MIMO systems are a promising technique for high
  data rates
• Their efficiency depends on the channel between
  the transmitters and the receivers (power and
  correlation)
• Practical issues need to be resolved
• Open research questions need to be answered