16.548 Coding and Information Theory
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Transcript of 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)
y(t) = g • x(t) + n(t)
x(t): transmitted signaly(t): received signalg(t): channel transfer functionn(t): noise (AWGN, 2)
Signal to noise ratio :
Capacity : C = log2(1+)
x(t)y(t)
g
2x2
σEρ g
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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 >
Equal gain combining > 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.
• C1xN>C1x1
• C1xN more robust than C1x1
1
N
)N1(log2N1 xC
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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
Space-TimeEncoder
Datasymbolsd
N DataSymbols
Pilo
tsy
mbo
ls
P
L_tTransmitantennas
Wireless Channel(What a Big Cloud!)
Space-TimeDecoder
L_rReceiveantennas
Datasymbols
Pilo
tsy
mbo
ls
P
d_hat
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MIMO Model
• Matrix Representation
– For a fixed T
TNTMMNTN WXHY
T: Time index
W: Noise
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Part II: Space Time Coding
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Multiple Input- Multiple Output systems (MIMO)
H
1
M
1
N
Nx1Mx1NxMNx1nxy H
22
totalP
• Average gain
• Average signal to noise ratio
H11
HN1
H1M
HNM
HH
1,H22 ijE
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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
H
2
H22
T2
H2x
2
Mρdetlog
MσPdetlog
σEdetlogC
HHI
HHIHHI g
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Interpretation I: The parallel channels approach
• “Proof” of capacity formula
• Singular value decomposition of H: H = S·U·VH
• S, V: unitary matrices (VHV=I, 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 vi’s
2. Each projection gets multiplied by a different gain ui.
3. Each appears along a different si.u1
u2
uK
<x,v1> · v1
<x,v2> · v2
<x,vK> · vK<x,vK> uK sK
<x,v1> u1 s1
<x,v2> u2 s2
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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:Ex=Ek
2
k2
k2
k
2k
2k
k σEu
s,nE
v,xEuρ
K
1ik2
K
1ik ρ1logCC
K
1i
222 1logC kk uE
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Interpretation II: The directional approach
• Singular value decomposition of H: H = S·U·VH
• Eigenvectors correspond to spatial directions (beamforming)
1
M
1
N
(si)1
(si)N
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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(N+P) block of information symbols broadcast from transmit antenna: i
Si(d, t)
• 1X(N+P) block of received information symbols taken from antenna: j
Rj = hjiSi(d, t) + nj
• Matrix representation:
Function (mapping) S at antennai defines the Space-Timeencoding process
Assuming a single user andquasi-static and independentfading, n is AWGN
NHS
r
r
R
tL
...
1
trrr
t
t
LLLL
L
L
hhh
hhhhhh
H
...............
...
...
21
22221
11211
),(...
),(),(
2
1
tdS
tdStdS
S
tL
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Related IssuesSpace-Time
EncoderS_i(d,t)
i=1,2,…,K
Datasymbolsd
N DataSymbols
Pilo
tsy
mbo
ls
P
L_tTransmitantennas
• 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?
Wireless Channel(What a Big Cloud!)
Space-TimeDecoder
L_rReceiveantennas
Datasymbols
Pilo
tsy
mbo
ls
P
d_hat
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Specific Example of STBC: Alamouti’s Orthogonal Code
• Let’s consider two antenna i and i+1 at the transmitter side, at two consecutive time instants t and t+T:
• The above Space-Time mapping defines Alamouti’s Code[1].• A general frame design requires concatenation of blocks (each 2X2)
of Alamouti code,
d_0 d_1
-d_1*
Tim
e t
Tim
e
d_0*
Ant i+1
Tim
e t+
T
Ant. i
Space
...||...||
*2
*3
**1
321
dddddddd
Do
o
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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
P 5P P 5D P 2P 3D P
1 7 13 19
5D P
25
2P 3D P
31
2P 3D P
37
5D …... 5D P 3D 2P P 3D 2P P …... P 5P
edge part uniformpart edge part
Frame size = 300
265 271 277 295
L_t = i
A typical slow fadingchannel
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Estimated CSI (cont.d) Block diagram of the receiver
+
n(t)
r(t) A/DConverter Delay
Pilot SymbolExtractor
Channel StateEstimator
MLDecoder
Matched Filteru*(-t)
r_k
r_p_1, r_p_2, …, r_p_N
r_k+1,r_k+2,…,r_k+K
h_hat
D_hat
tL
ltlstlh
1)()()()(
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Channel State Estimation (cont.d)
P 5P P 5D P 2P 3D P
1 7 13 19
5D P
25
2P 3D P
31
2P 3D P
37
5D …... 5D P 3D 2P P 3D 2P P …... P 5P
edge part uniformpart edge part
Frame size = 300
265 271 277 295
L_t = i
• Pilot symbol insertion length, Pins=6. • The receiver uses N=12, 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
P 5P P 5D P 2P 3D P
1 7 13 19
5D P
25
2P 3D P
31
2P 3D P
37
5D …... 5D P 3D 2P P 3D 2P P …... P 5P
edge part uniformpart edge part
Frame size = 300
265 271 277 295
L_t = i
• The orthogonal pilot symbol (pilots chosen from QPSK constellation) matrix is, [4]
• Pilot symbol insertion length, Pins=6. • The receiver uses N=12, nearest pilots to obtain estimated CSI• Data = 228, Pilots = 72
P 5P P 5D P 2P 3D P
1 7 13 19
5D P
25
2P 3D P
31
2P 3D P
37
5D …... 5D P 3D 2P P 3D 2P P …... P 5P
edge part uniformpart edge part
Frame size = 300
265 271 277 295
L_t = i+1
11
11P
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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
1)(][ p
H
prCovhrEW
INhCovrCovo
H
pppp )()(
][)( H
ppphhEhCov
]|[ˆp
rhEh
ppp
H
pWrrrCovhrEh 1)(][ˆ
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Block diagram for MRRC scheme with two Tx and one Rx
+
channelestimator combiner
h_hat_0
h_hat_1
h_hat_0 h_hat_1 d_hat_0 d_hat_1
maximum likelihood detector
n_0n_1
Interference& noise
rx antenna
h_0(t)h_0(t+T)
h_1(t)h_1(t+T)
tx antenna 0 tx antenna 1
d_0-d_1*
d_1d_0*
d_0 d_1
-d_1*
Tim
e t
Tim
e
d_0*
Ant i+1
Tim
e t+
T
Ant. i
Space
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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
oooo ndthdthtrr 11 )()()(
1
*
01
*
11 )()()( ndTthdTthTtrr o
*1
*
1
*
000 ])()([ˆ rthrTthd
*1
*
0
*
011 ])()([ˆ rthrTthd
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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.
*
110
*
00
*
11
*
000 )()()]()()()([ˆ nthnTthdTththTththd *
100
*
11
*
11
*
001 )()()]()()()([ˆ nthnTthdTththTththd
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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• Eigenvalue distribution
• Shannon capacity– for constant SNR or– for constant transmitted power
• Effective degrees of freedom(EDOF)
• Condition number
• Measures of comparison– Gaussian iid channel– “ideal” channel
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Measures of comparisonGaussian ChannelHij =xij+jyij : x,y i.i.d. Gaussian random variablesProblem: poutage
“Ideal” channel (max C)rank(H) = min(M, N)|u1 | = |u2 | = … = |uK |
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Eigenvalue distribution
Ideally: As high gain as possibleAs many eigenvectors as possibleAs orthogonal as possible
H
1
M
1
N
LimitsPower constraintsSystem sizeCorrelation
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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)
Href2 M
ρdetlogC HHI
H
2x
2 σEdetlogC HHI
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Building layout
RCVR(hall)
XMTR
RCVR(lab)
4m
6m
3.3m
3.3m
2m
0o
90o
270o
180o
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LOS conditions: Higher average SNR, High correlationNon-LOS conditions: Lower average SNR,More scattering
XMTR
RCVR(lab)
4m
6m
3.3m
3.3m
2m
0o
90o
270o
180o
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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 EDOF (Effective degrees of freedom)
Condition number
Definition UMAX/ umin
+ Intuition Simplicity
- Dependence on reference SNR
No information on intermediate eigenvalue distribution
ref
outageref
ρ2
ρ2Clim δ
p
δ
0δ
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From narrowband to wideband
• Wideband: delay spread >> symbol time
• -: Intersymbol interference+: Frequency diversity
• SISO channel impulse response:
SISO capacity:
L
1lll τtδgg(t)
L
1l
2l
22
x2
2 gg,σ
Eg1logC
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Matrix formulation of wideband case
)()(
)(
)(
)(
)(
τtδhth
1
1
1111
L
1llij,lij
tntx
tx
ty
ty
tntxty
MNMN
M
N
HH
HH
H
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Equivalent treatment in the frequency domain
• Wideband channel = Many narrowband channels
H(t) H(f)
bandwidthbandwidthNB fffC H
0
x2WB
02H
2x
2NB
)()(N
)(EdetlogC
(BW)Nσ,σEdetlogC
HHI
HHIf
Noise level
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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
total
K
1ik
K
1i
2k2
k2 PE,u
σE1logC
2
2k
k
kk
σu
λ
)λ1(νE
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Illustration of waterfilling algorithm
2
2k
k
kk
σu
λ
)λ1(νE
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
(b/Hz)B T
(bps/Hz) ρ1logC
K
1kk
K
1kk2
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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 = H+yH+ = (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• It’s 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