Multiple-input multiple-output (MIMO) communication systems

Advanced Modulation and Coding : MIMO Communication Systems

1

Multiple-input multiple-output (MIMO)communication systems


2

System model

trans-mitter

receiver

#1

#NT

#1

#NR

#n #minfo-bits

info-bits

NT transmit antennas NR receive antennas

)k(a

)k(a

TN

1

)k(r

)k(r

RN

1

k : time index (k = 1, ..., K)

(coded) data symbols an(k), E[|an(k)|2] = Es


3

System model

)k(w)k(ah)k(r m

N

1nnn,mm

T

)k(

)k(w

)k(w

)k(

)k(a

)k(a

hh

hh

)k(

)k(r

)k(r

RTTRR

T

R N

1

N

1

N,N1,N

N,11,1

N

1

waHr

wm(k) : complex i.i.d. AWGN, E[|wm(k)|2] = N0

hm,n : complex Gaussian zero-mean i.i.d. channel gains (Rayleigh fading), E[|hm,n|2] = 1

(m = 1, ..., NR; k = 1, ..., K)

r(k) = Ha(k) + w(k)

W

www

A

aaaH

R

rrr ))K(,),k(,),1(())K(,),k(,),1(())K(,),k(,),1((

R = HA + W


4

System model

Rb : information bitrateRs : symbol rate per transmit antennaEb : energy per infobitEs : energy per symbol

bits coded#

infobits#rMIMO (code rate)

Es = EbrMIMOlog2(M)

)M(logrN

RR

2MIMOT

bs

(per antenna)

info-bits

encoder

rMIMO

codedbits

NTK coded symbols

mapper S/P

#1

#NT

#n

M-pointconstellation

RbMIMO

b

r

R

)M(logr

R

2MIMO

b

1r0 MIMO

Rb = RsNTrMIMOlog2(M)


5

System model

BRF Rs (necessary condition toavoid ISI between successive symbols)

Spectrum transmitted bandpass signal

f

BRF BRF0

Bandwidth efficiency :

All NT transmitted signals are simultaneously in the same frequency interval

Rb/Rs = NTrMIMOlog2(M) (bit/s/Hz)


6

ML detection

)]()[(Tr||||

)k(ah)k(r),|(plnN

H2

N

1m

K

1k

2N

1nnn,mm0

R T

AHRAHRAHR

HAR

Log-likelihood function ln(p(R |A, H) :

2||||minargˆ AHRAA

(minimization over all symbol matrices that obey the encoding rule)

XH denotes Hermitian transpose (= complex conjugate transpose (X*)T)

][Tr][Tr|x||||| HH

j,i

2j,i

2 XXXXX

i

i,im][Tr M

Frobenius norm of X :

Trace of square matrix M :

Properties : Tr[PQ] = Tr[QP], Tr[M] = Tr[MT]

ML detection :


7

Bit error rate (BER)

info-bits

encoder

rMIMO

codedbits

NTK coded symbols

mapper S/P

#1

#NT

#n


ed transmittinfobits #

errors]infobit #[EBER

symbol matrix A represents NTKrMIMOlog2(M) infobits

MIMO2T

,

)0()i((i,0)info

r)M(logKN

],ˆPr[N

BER)i()0(

AA

AAAA(i,0)infoN : # infobits in which A(i) and A(i)

are different


8


)(PEP],|ˆPr[ )0,i()0()i( HHAAAA

MIMOTMIMO2T rKNr)M(logKN)0( M2]Pr[ AA

],|ˆPr[E]|ˆPr[ )0()0()0()i( HAAAAAAAA H

PEP(i,0)(H) = Pr[||R - HA(i)||2 < ||R - HA(0)||2| A = A(0), H]

PEP(i,0)(H) : pairwise error probability, conditioned on H

(averaging over statistics of H)

]Pr[]|ˆPr[],ˆPr[ )0()0()i()0()i( AAAAAAAAAA

PEP(i,0) = EH[PEP(i,0)(H)] : pairwise error probability, averaged over H


9


0

2)0,i(

0

2)0,i()0,i(

N4

||||exp

2

1

N2

||||Q)(PEP

ΔHΔHH

R

TR

T

NN

1n 0

)0,i(n

N

0

H)0,i()0,i(

N)0,i()0,i(

N41

2

1

N4

)(det

2

1)(PEPEPEP

ΔΔ

IHH

Define error matrix : (i,0) = A(0) - A(i)

set of eigenvalues of NTxNT matrix (i,0)((i,0))H

(these eigenvalues are real-valued non-negative)}N,...,0n,{ T

)0,i(n

MIMO2T

0i

)0,i((i,0)info

)0(

r)M(logKN

PEPN]Pr[

BER 0

A

AA

Averaging over Rayleigh fading channel statistics :


10

To be further considered

Single-input single-output (SISO) systems : NT = NR = 1

Single-input multiple-output (SIMO) systems : NT = 1, NR 1

Multiple-input multiple-output (MIMO) systems : NT 1, NR 1

•Spatial multiplexing : ML detection, ZF detection•Space-time coding : trellis coding, delay diversity, block coding


11

Single-input single-output (SISO) systems


12

SISO : observation model

info-bits

Rb

encoder

rSISO

coded bits

coded symbols

mapper

SISO

b

r

R

)M(logr

RR

2SISO

bs

Bandwidth efficiency : Rb/Rs = rSISOlog2(M) (bit/s/Hz)

Observation model : r(k) = ha(k) + w(k) k = 1, ..., K


13

SISO : ML detection

k

2

)}k(a{k

2

)}k(a{|)k(a)k(u|minarg|)k(ah)k(r|minarg)}k(a{

h

)k(r

|h|

h)k(r)k(u 2

*

xr(k) u(k)

2

*

|h|

h

h

1

ML detector looks for codeword that is closest (in Euclidean distance) to {u(k)}

In general : decoding complexity exponential in K (trellis codes, convolutional codes : complexity linear in K by using Viterbi algorithm)

# codewords = SISOSISO2 rKr)M(logK M2


14

SISO : PEP computation

0

2)0,i(,E

2)0,i(

N4

d|h|exp

2

1)h(PEP

1

0

2)0,i(,E)0,i(

N4

d1

2

1PEP

K

1k

2)i()0(2)0,i(,E |)k(a)k(a|d

0

2)0,i(,E)0,i(

N4

dexp

2

1PEP

Conditional PEP :

squared Euclidean distance between codewords {a(0)(k)} and {a(i)(k)}

Average PEP, AWGN (|h| = 1) :

Average PEP, Rayleigh fading :

decreases exponentially with Eb/N0

inversely proportional

to Eb/N0

b2SISOs2

)0,i(,E E)M(logrEd

(occasionally, |h| becomes very small large PEP)


15

SISO : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

AWGNAWGN boundRayleigh fadingRayleigh fading bound

SISOBPSK, uncoded

BER fading channel >> BER AWGN channel


16

SISO : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

-5 0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

AWGN uncoded

AWGN Ext. Hamming (8,4)

AWGN Shannon limit

Rayleigh fading uncoded

Rayleigh fading, ext. Hamming (8,4)

Rayleigh fading, Shannon limit

SISOBPSK, coded

coding gain on fading channel << coding gain on AWGN channel


17

Single-input multiple-output (SIMO) systems


18

SIMO : observation model

Same transmitter as for SISO; NR antennas at receiver

info-bits

Rb

encoder

rSIMO

coded bits

coded symbols

mapper

SIMO

b

r

R

)M(logr

RR

2SIMO

bs

Bandwidth efficiency : Rb/Rs = rSIMOlog2(M) (bit/s/Hz)

Observation model : rm(k) = hma(k) + wm(k) m = 1, ..., NR; k = 1, ..., K


19

SIMO : ML detection

k

2

)}k(a{k

N

1m

2mm

)}k(a{|)k(a)k(u|minarg|)k(ah)k(r|minarg)}k(a{

R

R

R

N

1m

2m

N

1mm

*m

|h|

)k(rh)k(u

xr1(k)

u(k)

*mh

x

x

rm(k)

)k(rRN

*1h

*NR

h

x

RN

1m

2m|h|

1ML detector looks for codeword that is closest to {u(k)}

Same decoding complexity as for SISO


20

SIMO : PEP computation

0

avg22

)0,i(,ER)0,i(

N4

)|h(|dNexp

2

1)(PEP h

RN

0

2)0,i(,E)0,i(

N4

d1

2

1PEP

0

2)0,i(,ER)i,0(

N4

dNexp

2

1PEP

Conditional PEP :

PEP, AWGN (|hm| = 1) :

PEP, Rayleigh fading :

decreases exponentially with Eb/N0

inversely proportional

to (Eb/N0)^NR

total received energy per symbol increases by factor NR as compared to SISO same PEP as with SISO, but with Eb replaced by NREb

(“array gain” of 10log(NR) dB)

- “array gain” of 10log(NR) dB - additional “diversity gain” (diversity order NR) : probability that (|h|2)avg is small, decreases with NR

PEP is smaller than in SISO setup with Eb replaced by NREb

RN

1m

2m

Ravg

2 |h|N

1)|h(|


21

SISO : statistical properties of <|h|2>

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4

x

pd

f

N_R = 1

N_R = 2

N_R = 3

N_R = 5

N_R = 10

Probability density function of <|h|2>


22

SISO : statistical properties of <|h|2>

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

x

Pr[

<|h

|2>

< x

]

N_R = 1

N_R = 2

N_R = 3

N_R = 5

N_R = 10

Cumulative distribution function function of <|h|2>


23

SIMO : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Eb/N0 (dB)

BE

R

N_R =1N_R =2N_R =3N_R = 4

SIMO AWGNUncoded BPSK

AWGN channel : array gain of 10log(NR) dB


24


fading channel : array gain + diversity gain

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISOarray gain onlyboundcorrect

SIMO, N_R = 2uncoded BPSK array

gain

diversity gain


25


1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISOarray gain onlyboundcorrect

SIMO, N_R = 3uncoded BPSK



26


1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISO

array gain only

bound

correct

SIMO, N_R = 4uncoded BPSK



27


1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISO uncodeduncodedShannon limitExt. Hamming (8,4)

SIMO, N_R = 2BPSK

fading channel : coding gain increases with NR


28


1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISO uncodeduncodedShannon limitExt. Hamming (8,4)

SIMO, N_R = 3BPSK



29


1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R

SISO uncoded

uncoded

Shannon limit

Ext. Hamming (8,4)

SIMO, N_R = 4BPSK



30

Multiple-input multiple-output (MIMO) systems


31

MIMO : observation model

Bandwidth efficiency : Rb/Rs = NTrMIMOlog2(M) (bit/s/Hz)

)k(w)k(ah)k(r m

N

1nnn,mm

T

(m = 1, ..., NR; k = 1, ..., K)

R = HA + W at any receive antenna, symbols from different transmit antennas interfere

)M(logrN

RR

2MIMOT

bs

(per antenna)

info-bits

encoder

rMIMO

codedbits

NTK coded symbols

mapper S/P

#1

#NT

#n


RbMIMO

b

r

R

)M(logr

R

2MIMO

b


32

MIMO : ML detection

2||||minargˆ AHRAA

ML detector :

detector complexity in general increases exponentially with NTrMIMO and K

# different symbol matrices A =

KrNK)M(logrN MIMOT2MIMOT M2


33

MIMO : PEP computation

RT

R

T

NN

1n 0

)0,i(n

N

0

H)0,i()0,i(

N)0,i(

N41

2

1

N4det

2

1PEP

I

PEP, Rayleigh fading :

# nonzero eigenvalues equals rank((i,0)), with 1 rank((i,0)) NT

PEP(i,0) inversely proportional to (Eb/N0)^(NRrank((i,0)))

PEPs that dominate BER are those for which (i,0) = A(0)-A(i) has minimum rank.

)rank(minNorderdiversity (i,0)

)0,i(R Δ


34

MIMO : spatial multiplexing


35

Spatial multiplexing

info-bits

encoder

r

mapper

rN

R

T

b

)M(logrN

RR

2T

bs

#1

T

b

N

R

encoder

r

mapper

rN

R

T

b

#NT

T

b

N

R

S/P

Rb

MIMO encoder : rMIMO = r(per antenna)

Aim : to increase bandwidth efficiency by a factor NT as compared to SISO/SIMO

TN

T1

Ta

a

A

))N(a,),k(a,),1(a( nnnTn a codeword transmitted

by antenna #n

codewords transmitted by different antennas are statistically independent

Bandwidth efficiency : Rb/Rs = NTrlog2(M)


36

Spatial multiplexing :ML detection

TT N

1nn

Hn

N

1'n,n'n

Hn'n,n

H

2

Re2)()(minarg

||||minargˆ

uaaaHH

AHRA

A

AML detector :

As HHH is nondiagonal, codewords on different rows must be detected jointly instead of individually

ML detector complexity increases exponentially with NT

as compared to SISO/SIMO

# different symbol matrices A =KrNK)M(logrN T2T M2

(un)T : n-th row of HHR


37

Spatial multiplexing : PEP computation

1)(rankmin )0,i(

0i

dominating PEPs are inversely proportional to (Eb/N0)^NR

diversity order NR (same as for SIMO)

rank((i,0)) = 1 when all rows of (i,0) are proportional to a common row vector T = ((1), ..., (k), ..., (K)).

Example : (i,0) has only one nonzero row; this corresponds to a detection error in only one row of A

RN

n

2n

0

2)0,i( ||

N4

||1

2

1PEP

δ

(i,0) nonzero rank((i,0)) 1

Denoting the n-th row of a rank-1 error matrix (i,0) by n T, the bound on the corresponding PEP is


38

Spatial multiplexing : zero-forcing detection

Complexity of ML detection is exponential in NT

Simpler sub-optimum detection : zero-forcing (ZF) detection

r(k) = Ha(k) + w(k)

z(k) = (HHH)-1HHr(k) = a(k) + n(k) n(k) = (HHH)-1HHw(k)

rank(H) = NT (HHH)-1 exists

E[n(k)nH(k)] = N0(HHH)-1 : ni(k) and nj(k) correlated when ij

z(k) : no interference between symbols from different transmit antennas

linear combination of received antenna signals rm(k) to eliminatemutual interference between symbols an(k) from differenttransmit antennas and to minimize resulting noise variance

Condition for ZF solution to exist : rank(H) = NT (this requires NR NT)


39

Spatial multiplexing : zero-forcing detection

Sequences {an(k)} are detected independently instead of jointly, ignoring the correlation of the noise on different outputs :

(HHH)-1HH

r1(k)

rm(k)

)k(rRN

z1(k)

zn(k)

)k(zTN

detector

detector

detector

)}k(a{ 1

)}k(a{ n

)}k(a{TN

k

2nn

)k(an |)k(a)k(z|minarg)}k(a{

n

Complexity of ZF detector is linear (instead of exponential) in NT

Performance penalty : diversity equals NR-NT+1 (instead of NT)


40

Spatial multiplexing : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R (

bo

un

d)

N_T = 1, N_R = 1

N_T = 5, N_R = 1

N_T = 1, N_R = 2

N_T = 5, N_R = 2

N_T = 1, N_R = 3

N_T = 5, N_R = 3

N_T = 1, N_R = 4

N_T = 5, N_R = 4

Spatial multiplexinguncoded BPSKML detection

ML detection : penalty w.r.t. SIMO decreases with increasing NR


41

Spatial multiplexing : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R(N_R, N_T) = (1,1), (2,2), (3,3) ,...

(N_R, N_T) = (2,1), (3,2), (4,3) ,...

(N_R, N_T) = (3,1), (4,2), (5,3) ,...

(N_R, N_T) = (4,1), (5,2), (6,3) ,...

Spatial multiplexinguncoded BPSKZF detection

ZF detection : same BER as SIMO with NR-NT+1 receive antennas


42

MIMO : space-time coding


43

Space-time coding

in fo -b its

R b

encoder

rM IM O

codedb its

coded sym bols

m apper

MIMO

b

r

R

)M(logr

R

2MIMO

b

S /P

#1

#N T

#n

)M(logrN

RR

2MIMOT

bs

(per an tenna)Aim : to achieve higher diversity order than with SIMO any (i,0) must have rank larger than 1 (max. rank is NT, max. diversity order is NTNR) coded symbols must be distributed over different transmit antennas and different symbol intervals (“space-time” coding)

Property : rMIMO 1/NT is necessary condition to have rank NT for all (i,0)

at max. diversity, Rb/Rs log2(M) Rb/Rs limited by bandwidth efficiency of uncoded SIMO/SISO


44

Three classes of space-time coding :

•space-time trellis codes (STTrC)•delay diversity•space-time block codes (STBC)


45

STTrC : trellis representation

)K(,),k(,),1( aaaA

)k(b,),k(b)k( L1 b L input bits at beginning of k-th symbol interval

S(k) : state at beginning of k-th symbol interval

)k(a,),k(a,),k(a)k(TNn1

T a

state transitions : (S(k+1), a(k)) determined by (S(k), b(k))

trellis diagram :(L=1)

k k+1

S(k)

S(k+1)

b(k), a(k)

0

1

2

3

2L branches leaving from each state

2L branches entering each state


46

STTrC : bandwidth efficiency

ML decoding by means of Viterbi algorithm :

decoding complexity linear (instead of exponential) in K

but still exponential in NT (at max. diversity)

rMIMO = L/(NTlog2(M)) Rb/Rs = L (bit/s/Hz)

at max. diversity : rMIMO 1/NT L log2(M)

Bandwidth efficiency :

#states 1NTM (exponential in NT)


47

STTrC : example

Example for NT = 2, 4 states, 4-PSK, L = 2 infobits per coded symbol pair

0 1 2 30 0,0 0,1 0,2 0,31 1,0 1,1 1,2 1,32 2,0 2,1 2,2 2,33 3,0 3,1 3,2 3,3

entryi,j : 2 data symbols corresponding to transition from state i to state j

max. diversity (NTNR = 2NR), max. bandwidth efficiency (L = log2(M)), corresponding to max. diversity

any (i,0) has columns (0, 1)T, (2, 0)T

with nonzero 1 and 2

rank = 2, i.e., max. rank

0

1

2

3


48

Delay diversity : transmitter

)2K(a)3K(a)4K(a)5K(a)2(a)1(a00

0)2K(a)3K(a)4K(a)3(a)2(a)1(a0

00)2K(a)3K(a)4(a)3(a)2(a)1(a

A

Example : NT = 3

rank((i,0)) = NT diversity order = NTNR maximum possible diversity order !

info-bits encoder

r

mapper

r

Rb

)M(logr

RR

2

bs

#1

delay (NT -1)T#NT

Rb

(per antenna)

delay (n-1)T#n


49

Delay diversity : bandwidth efficiency

Equivalent configuration :

Bandwidth efficiency (for K >> 1) : Rb/Rs = NTrMIMOlog2(M) = rlog2(M)

same bandwidth efficiency as SIMO/SISO with code rate r

Max. bandwidth efficiency (at max. diversity) of log2(M) achieved for r = 1 (uncoded delay diversity)

info-bits encoder

r

mapper

)M(logr

RR

2

bs

#1

delay (NT -1)T#NT

(per antenna)

delay (n-1)T#n

mapper

mapper

MIMO encoder : rMIMO = r/NT

Rb


50

Delay diversity : trellis representation

Uncoded delay diversity (r = 1) can be interpreted as space-time trellis code :

state S(k) = (a(k-1), a(k-2), ..., a(k-NT+1))

input at time k : a(k) output during k-th symbolinterval : (a(k), a(k-1), ..., a(k-NT+1))

#states = 1NTM

decoding complexity : linear in K, exponential in NT


51

Space-time block codes

Space-time block codes can be designed to yield maximum diversity and to reduce ML decoding to simple symbol-by-symbol decisions

decoding complexity linear (instead of exponential) in NTK

ML decoding of space-time code : minimizing ||R - HA||2 over all possible codewords A

A contains NTK coded symbols, which corresponds to NTKrMIMOlog2(M) infobits

In general, ML decoding complexity is exponential in NTK


52

STBC : example 1

*12

*21

aa

aaA

Example 1 : NT = 2 (Alamouti code )

AAH = (|a1|2 + |a2|2)I2

(A has orthogonal rows, irrespective of the values of a1 and a2)

ML decoding (H = (h1, h2), R = (r1, r2), a1 and a2 i.i.d. symbols)

)|||)(||a||a(|)](a)(aRe[2minarg

)])([(Tr]][TrRe[2minarg||||minarg)a,a(

22

21

22

21

*2

T12

H2

*2

*2

T11

H1

*1

a,a

HHHH

a,a

2

a,a11

21

2121

hhrhrhrhrh

AAHHRHAHAR

22

21

*2

T21

H1

1 ||||u

hh

rhrh

22

21

*2

T11

H2

2 ||||u

hh

rhrh

no cross-terms involving both a1 and a2 symbol-by-symbol detection

211

a1 |au|minarga

1

222

a2 |au|minarga

2


53

STBC : example 1

*12

*21

aa

aaA

*)i(

1)0(

1)i(

2)0(

2

*)i(2

)0(2

)i(1

)0(1)0,i(

)aa(aa

)aa(aaΔ

orthogonal rows rank = 2(= max. rank for NT = 2)

Assume M-point constellation A contains 4 symbols, i.e. 4log2(M)rMIMO = 2log2(M) infobits rMIMO = 1/2 (= 1/NT) : highest possible rate for max. diversity

Alamouti space-time block code yields - maximum diversity (i.e., 2NR)- maximum corresponding bandwidth efficiency (i.e., Rb/Rs = log2(M))- small ML decoding complexity


54

STBC : example 2

Example 2 : NT = 3

*2

*1

*4

*32143

*3

*4

*1

*23412

*4

*3

*2

*14321

aaaaaaaa

aaaaaaaa

aaaaaaaa

A AAH = 2(|a1|2 + |a2|2 + |a3|2 + |a4|2)I3

rows of A are orthogonal any (i,0) has rank = 3 max. diversity (i.e., 3NR)

ML detection of (a1, a2, a3, a4) : symbol-by-symbol detection (no cross-terms)

Bandwidth efficiency : Rb/Rs = 4log2(M)/8 = log2(M)/2 i.e., only half of the max. possible value for max. diversity

max. bandwidth efficiency (under restriction of max. diversity and symbol-by-symbol detection) cannot be achieved for all values of NT

rMIMO = 1/6 (1/NT = 1/3)


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STBC : numerical results

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R (

bo

un

d)

SISO

N_T = 2 (Alamouti)

N_T = 3

Space-time block codesBPSKN_R = 1


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1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R (

bo

un

d)

SISO

SIMO

N_T = 2 (Alamouti)

N_T = 3



57


1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R (

bo

un

d)

SISO

SIMO

N_T = 2 (Alamouti)

N_T = 3



58


1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30

Eb/N0 (dB)

BE

R (

bo

un

d) SISO

SIMO

N_T = 2 (Alamouti)

N_T = 3



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Summary : SISO/SIMO/MIMO

B a n d w id the ffic ie n cy

D iv e r s ity o r d e r(M L d e tec tio n )

M L d e tec to rc o m p le x ity

S IS O rS IS O lo g 2(M ) 1 in g e n e ra l : e x p o n . in K(tre llis : lin e a r in K )

S IM O rS IM O lo g 2(M ) N R in g e n e ra l : e x p o n . in K(tre llis : lin e a r in K )

M IM O N T rM IM O lo g 2(M )

)(rankmaxN )0,i(

)0,i(R

in g e n e ra l :e x p o n . in N T K


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Summary : MIMO

Bandwidthefficiency

Diversity order(ML detection)

ML detectorcomplexity

spatialmultiplexing

NTrlog2(M) NRin general : expon. in NTK(trellis : linear in K, expon. in NT)

STTrC log2(M) (*) NTNR expon. in NT, linear in Kdelay

diversityrlog2(M) NTNR

if uncoded (r = 1) :expon. in NT, linear in K

STBC log2(M) (*) NTNR symbol-by-symbol decision

(*) assuming max. diversity NTNR is achieved

Remark : spatial multiplexing with ZF detection- complexity linear in NT

- diversity order is NR-NT+1

Multiple-input multiple-output (MIMO) communication systems

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