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16.548 Coding and Information Theory

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(coherent distance) ... MMSE estimation Use Wiener filtering, since it is a Minimum Mean ... Space-Time Alamouti Codes with Perfect CSI,BPSK Constellation Space ... – PowerPoint PPT presentation

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Title: 16.548 Coding and Information Theory


1
16.548 Coding and Information Theory
  • Lecture 15 Space Time Coding and MIMO

2
Credits
3
Wireless Channels
4
Signal Level in Wireless Transmission
5
Classification of Wireless Channels
6
Space time Fading, narrow beam
7
Space Time Fading Wide Beam
8
Introduction to the MIMO Channel
9
Capacity of MIMO Channels
10
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11
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?)
12
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

13
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

14
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

15
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.

16
RX Diversity in Wireless
17
Receive Diversity
18
Selection and Switch Diversity
19
Linear Diversity
20
Receive Diversity Performance
21
Transmit Diversity
22
Transmit Diversity with Feedback
23
TX diversity with frequency weighting
24
TX Diversity with antenna hopping
25
TX Diversity with channel coding
26
Transmit diversity via delay diversity
27
Transmit Diversity Options
28
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

29
MIMO Model
T Time index W Noise
  • Matrix Representation
  • For a fixed T

30
Part II Space Time Coding
31
Multiple Input- Multiple Output systems (MIMO)
H11
HN1
H1M
HNM
  • Average gain
  • Average signal to noise ratio

32
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

33
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

34
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
35
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

36
Interpretation II The directional approach
  • Singular value decomposition of H H SUVH
  • Eigenvectors correspond to spatial directions
    (beamforming)

1 M
37
Example of directional interpretation
38
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39
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

40
Space Time Coded Modulation
41
Space Time Channel Model
42
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43
STC Error Analysis
44
STC Error Analysis
45
(No Transcript)
46
(No Transcript)
47
STC Design Criteria
48
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49
STC 4-PSK Example
50
STC 8-PSK Example
51
STC 16-QAM Example
52
STC Maximum Likelihood Decoder
53
STC Performance with perfect CSI
54
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55
(No Transcript)
56
Delay Diversity
57
Delay Diversity ST code
58
(No Transcript)
59
Space Time Block Codes (STBC)
60
Decoding STBC
61
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62
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63
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



64
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?

65
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,

66
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

67
Channel State Estimation
68
Estimated CSI (cont.d) Block diagram of the
receiver
69
Channel State Estimation (cont.d)
  • Pilot symbol insertion length, Pins6.
  • The receiver uses N12, nearest pilots to obtain
    estimated CSI

70
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

71
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

72
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



73
Block diagram for MRRC scheme with two Tx and one
Rx
74
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

75
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.

76
Space-Time Alamouti Codes with Perfect CSI,BPSK
Constellation
77
Space-Time Alamouti Codes with PSAM under QPSK
Constellation
78
Space-Time Alamouti Codes with PSAM under QPSK
Constellation
79
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

80
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

81
Eigenvalue distribution
Limits Power constraints System size Correlation
  • Ideally
  • As high gain as possible
  • As many eigenvectors as possible
  • As orthogonal as possible

82
Example Uncorrelated correlated channels
83
Shannon capacity
  • Capacity for a reference SNR (only channel info)
  • Capacity for constant transmitted power (channel
    power roll-off info)

84
Building layout
RCVR (hall)
85
LOS conditions Higher average SNR, High
correlationNon-LOS conditions Lower average
SNR,More scattering
86
Example C for reference SNR
87
Example C for constant transmit pwr
88
Other metrics
89
From narrowband to wideband
  • Wideband delay spread gtgt symbol time
  • - Intersymbol interference
  • Frequency diversity
  • SISO channel impulse response
  • SISO capacity

90
Matrix formulation of wideband case
91
Equivalent treatment in the frequency domain
  • Wideband channel Many narrowband channels
  • H(t) ? H(f)

Noise level
f
92
Extensions
  • Optimal power allocation
  • Optimal rate allocation
  • Space-time codes
  • Distributed antenna systems
  • Many, many, many more!

93
Optimal power allocation
  • IF the transmitter knows the channel, it can
    allocate power so as to maximize capacity
  • Solution Waterfilling

94
Illustration of waterfilling algorithm
?
Stronger subchannels get the most power
95
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

96
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

97
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

98
Distributed antenna systems
  • Idea put your antennas in different places
  • lower correlation
  • - power imbalance, synchronization,
    coordination

99
Practical considerations
  • Coding
  • Detection algorithms
  • Channel estimation
  • Interference

100
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!!!

101
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

102
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

103
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!!!!

104
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????

105
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

106
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
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