Space Time Codes for MIMO

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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OutlineIntroductionSpace Time codesAlamouti codes

Space-Time block codesSpace-time Trellis CodesDifferential Space-Time block codes

Space-Time for OFDM systemsConclusionsReferences


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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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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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Diversity gain

Variance Diversity

Mean ArrayGain


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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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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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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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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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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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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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Space Time coding


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Alamouti code2 by 1 orthogonalspace time blockcode2 TX antenna 1 RX

antenna


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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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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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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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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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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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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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STBC - Performance

G4

G3

G2

Code

1/216 QAM4

1/216 QAM3

14 BPSK2

CodeRate

Modulationantennas

5 dB


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STBC Performance [7]

4 DPSK44 DPSK3

BPSK2

Modulationantennas

3.5 dB7.5 dB


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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Homework

Alamouti code dont provide coding gain .

Justify

Space Time Codes for MIMO

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