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7/31/2019 Space Time Codes for MIMO
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Space Time code for MIMOSystems
S72.333-Postgraduate Course
Fernando [email protected]
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
OutlineIntroductionSpace Time codesAlamouti codes
Space-Time block codesSpace-time Trellis CodesDifferential Space-Time block codes
Space-Time for OFDM systemsConclusionsReferences
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
IntroductionDepending on surrounding environment, a transmittedradio signal propagates through several differentpaths multipath propagation.
The signal received by the receiver antenna consistsof the superposition of various multipaths.The attenuation coefficients corresponding to differentpaths are assumed to be independent and identicallydistributed
The path gain can be modeled as a complex Gaussianrandom variable Rayleigh fading channel
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Diversity GainSignal power in a wireless system fluctuates. When this signalpower drops significantly, the channel is said to be in fade.Diversity is used in wireless channels to combat the fading.Receive diversity and transmit diversity mitigate fading and
significantly improve link quality .The receive antennas see independently faded versions of thesame signals. The receiver combines these signals so that theresultant signal exhibits considerably reduced amplitude fading.
Diversity order M R x MTMIMO turns multipath propagation into a benefit for the user
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Diversity gain
Variance Diversity
Mean ArrayGain
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
An introductory example
One transmit antenna and two receive antenna
TX
RX
RX
Diversity Gain=2
S- Transmitted signal
h1
h2
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
An introductory exampleTwo transmit antenna and one receive antenna
Channel Unknown: If the weights are not allowed to depend on h 1and h 2 it is impossible to achieve diversity of order 2.Channel known : The error probability of detecting s decay asSNR -2.
RX
TX
TX
Diversity gain
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
St+1
St
A2A1
Using two time intervals
An introductory example
Equal to 1x2 systemDiversity gain=2
Data rate is reduced !!
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
An introductory exampleWithout channel knowledge at the transmitter,diversity can not be achieved.Using more than one time interval for the
transmission, diversity gain is achievedTransmit diversity is easy to achieve if a sacrifice ininformation rate is acceptable.Space Time coding is concerned with
Maximize the transmitted information rate Minimize the error probability
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1/31/2005 Fernando GregorioS72-333 Space time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Transmit diversityChannel information at the transmitter
Beamforming methods
Channel unknown at the transmitter
Space time coding:Coding techniques designed for multiple antenna
transmission.Coding is performed by adding properly designed
redundancy in both spatial and temporal domainswhich introduces correlation into the transmittedsignal.
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Space Time coding
r
InformationSource
S-TEncoder S/P
R e c ei v er C t Xt n t
1
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Space Time coding
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Alamouti code2 by 1 orthogonalspace time blockcode2 TX antenna 1 RX
antenna
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HELSINKI UNIVERSITY OF TECHNOLOGY
Signal Processing LaboratorySMARAD Centre of Excellence
Alamouti codeEncoding and Transmission Sequence
At a given symbol period, two signals are simultaneouslytransmitted from the two antennas. The signal transmitted fromantenna zero is denoted by S 0
and from antenna one by S 1. During the next symbol period signal -S 1* is transmitted from antenna zero, and S 0* signal is transmittedfrom antenna one where is the complex conjugate operation.
S 0*- S 1*t+T
S 1S 0t
A1A0Time
Space
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Alamouti codeEncoding and Transmission Sequence
Assuming that fading is constant across two consecutivesymbols
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Alamouti codeThe Combining Scheme
The Maximum Likelihood Decision Rule
Low complexityThese combinedsignals areequivalent to thatobtained from twobranch MRRC.
Diversity gain is equal totwo branch MMRC !!
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Properties of Alamouti codeUnitary
The product of its transmission matrix with its Hermitian transpose is equal to the 2 x 2 identity matrix.
Full-rate complex code Is the only complex S-T block code with a code rate of unity.
Linearity The Alamouti code is linear in the transmitted symbols.
Optimality of capacity For 2 transmit antennas and a single receive antenna, the
Alamouti code is the only optimal S-T block code in terms of capacity
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Alamouti code- Performance
From: ALAMOUTI, Simple transmit diversity technique for wirelesscommunications
Coherent BPSK withMRRC and two-branch transmitdiversity in Rayleigh
fading.
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Alamouti code- PerformanceThe performance of Alamouti code with twotransmitters and a single receiver is 3 dB worse thantwo-branch MRRC.
The 3-dB penalty is incurred because is assumedthat each transmit antenna radiates half the energy inorder to ensure the same total radiated power as withone transmit antenna.
If each transmit antenna was to radiate the sameenergy as the single transmit antenna for MRRC, , theperformance would be identical.
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Space Time Block Codes (STBC)Parameters
1/2844G4
1/2843G3
1222Alamouti
RateR=l/m
Number oftime slots
m
Number oftransmitted
symboll
Number oftransmit
antennasNt
Space Timecode
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
STBC - Performance
G4
G3
G2
Code
1/216 QAM4
1/216 QAM3
14 BPSK2
CodeRate
Modulationantennas
5 dB
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
STBC Performance [7]
4 DPSK44 DPSK3
BPSK2
Modulationantennas
3.5 dB7.5 dB
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Error probability in slow-fading channelThe fading channel coefficients are constant within each frame.Codeword difference matrix B
Codeword distance matrix A (Nt x Nt)
The rows of V are theeigenvectors of A matrix
The diagonal elements ofare the eigenvalues of
A matrix
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Error probability in slow-fading channelThe Euclidean distance
The upper bound of the error probability is given by [2]
rnR
s
nRr
i
i N
E x xP
01
^
4),(
Diversity
gain
Eigenvalues ofmatrix ARank(A)
Inner
product
Gd
cerror SNRGP
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Example 1 - A Time-Switched ST codeOnly one antenna is active in each time slot
xt is transmittedfor antenna 1 attime 2t
xt is transmittedfor antenna 2 attime 2t+1.
R=1/2
Diversity gain = 2
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Example 2 - Repetition code
The same modulated symbols are transmitted from twoantennas
The repetition code has the same performance as a nodiversity scheme (1x1 system) !!
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
S-T Code Design criteria
The design criteria for slow Rayleigh fading channeldepend on the value of
The maximum possible value of isThe error probability at high SNR is dominated by theminimum rank r of the matrix A over the all possiblecodewords pairs
rnR
s
nRr
i
i N
E x xP
01
^
4),(
rnR
rnR nRnT .
Maximize the minimum rank r of matrix A over allpairs of distinct codewords
Maximize the minimum product of matrix A
r
i
i
1
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Space-Time Trellis Codes (STTC)
A stream of data is encoded via Nt convolutionalencoders to obtain N t streams x1xntThe design of STTC codes is a relatively hard problem.Advantages
Coding gain !! Similar Diversity gain than STBC
Disadvantages Viterbi decoder.
The complexity of decoding algorithm grows exponentiallywith the memory length of the trellis code.
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Space-Time Trellis Codes (STTC)
TX 2
TX 1C C
C Cs(n)
x1(n)
x2(n)
Example
4-state STTC
Two transmit antennas
CC g 1=[(0 2),(2 0)]
g2
=[(0 1),(1 0)]The encoder takes m=2 bits as its input at eachtime.
Input Sequence c=(10, 01,11,00,01,)
Output sequence x=(02, 21,13,30,01,)
Antenna 1 x1=(0,2,1,3,0,)
Antenna 2 x2=(2,1,3,0,1,)
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Differential space time block codes
Channel acknowledge in the receiver is necessary forSTBC Overhead Channel estimation Problems in high mobility channels
DSTBC eliminates the need for channel estimationVery simple Maximum Likelihood decoding
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Differential space time block codes
S 2tS 2t-12t-1
S 1*- S 2*
S 2t-1 *- S 2t*2t
S 2S 1Antenna 2Antenna 1Time
No information
At time 2t+1 a block of 2b bits B (2t+1) arrives at the encoder.Using the mapping M computes S 2t+1 and S 2t+2
S 2t+1 *- S 2t+2 *2t+2
S2t+2
S2t+1
2t+1
The process is inductively repeated until the end of the frame
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Differential space time block codes
Example BPSK symbolsMapping function M
(-1 0)M(1 1)
(0 1)M(1 0)
(0 -1)M(0 1)
(1 0)M(0 0)
0.7070.7072t
-0.7070.7072t-1
M
M(1 0)=(0 1)
A=0
B=1
0.7070.7072t+2
-0.7070.7072t+1(S 2t+1 , S 2t+2 )=A (S 2t+1 , S 2t+2 )+B(S 2t*,S2t-1 *)
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Z-2 I
Mapper
Differential encoder
Z-2 IComputeclosestmatrix toA(t+2,t+3 )
InverseMapper
Differential decoder
Compute the matrix A that isclosest to Yin terms Euclideandistance
Incoming pairs of bits
t+2, t+3
At+2,t+3
St,t+1
S t+2,t+3
xt+2,t+3
xt,t+1
yt+2,t+3Estimate of At+2,t+3
Estimate of original pairs of data bits transmitted at t+2 and t+3
[1]
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
DSTBC-Performance [6]
The DSTBC detectionscheme is 3 dB worse thanthat of the transmit diversityscheme of employs coherentdetection at high SNR.
BPSK 2 x 1 system
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Spatial Diversity coding for MIMO-OFDM [4]
The time index is replaced by the tone index in OFDMAlamouti code requires that the channel remains constantover consecutive symbols periods.In OFDM context, the channel must remain constant over
consecutive tones.Problems in frequency selective channels !!!
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Spatial Diversity coding for MIMO-OFDM
A
IFFT
IFFT
B
C
A l a m o u
t i
E n c o
d e r
D
FFT
A l a m o u
t i
d e c o
d e r
E
H[k]2 s 2H[k]2 s 1E
-h1s 2*+h2s 1*h1s 1+h 2s 2D
S 1*s 2C
-S 2*S 1B
S 2S 1A
Tone K+1Tone k
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Spatial Diversity coding for MIMO-OFDMThe receiver detected the transmitted symbols from thereceived signals on the two tones using the Alamouti detectiontechnique.The use of consecutive tones is not strictly necessary, any pairof tones can be used as long as the associated channels are
equal.The technique can be generalized over a large number ofantennas to extract spatial diversity using STBC The blocksize is T N T .The channel must be identical over the T tones
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
ConclusionsAlamouti code is the best option when 2 Transmissionantennas is considered.Low complexity receiver is a good characteristic forSTBCSTTC provides coding gain. But Viterbi decodermust be implemented in the receiver.DSTBC can be considered in high mobility channels.
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
References[1] Simon Haykin and Michael Moher, Modern Wireless Communications.
[2] B. Vucetic and J. Yuan, Space-Time Coding , John Wiley, 2003[3] P. Stoica and E. Larson, Space-Time Block Coding for WirelessCommunications, Cambridge University Press, 2003[4] A. Paulraj, R. Nabar, D. , Introduction to Space-Time WirelessCommunications, Cambridge University Press, 2003[5] D. Gespert, et. Al. , From theory to practice: An overview of MIMO Space-Time Coded Wireless Systems , IEEE JSAC, Vol. 21, April 2003[6] Tarokh, V.; Jafarkhani, H., A differential detection scheme for transmitdiversity, Selected Areas in Communications, IEEE JSAC ,Vol. 18 ,July 2000 .[7] Tarokh, V.; Jafarkhani, H.; Calderbank, A.R.; Space-time block coding forwireless communications: performance results , IEEE JSAC ,Vol. 17 , March
1999
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HELSINKI UNIVERSITY OF TECHNOLOGYSignal Processing LaboratorySMARAD Centre of Excellence
Homework
Alamouti code dont provide coding gain .
Justify