Multiple-input multiple-output (MIMO) communication systems · Advanced Modulation and Coding :...

Advanced Modulation and Coding : MIMO Communication Systems 1

Multiple-input multiple-output (MIMO)communication systems


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

M

)k(r

)k(r

RN

1

M

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

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


System model

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

N

1nnn,mm

T

+=∑=

43421

M

43421

M

444 3444 21L

MOM

L

43421

M

)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) = H⋅a(k) + w(k)

4444 34444 21LL

4444 34444 21LL

4444 34444 21LL

Wwww

AaaaH

Rrrr ))K(,),k(,),1(())K(,),k(,),1(())K(,),k(,),1(( +⋅=

R = H⋅A + W


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(logrNRR

2MIMOT

bs =

(per antenna)

info-bits

encoder

rMIMO

codedbits

NTK coded symbols

mapper S/P

#1

#NT

#n

M-pointconstellation

RbMIMO

b

rR

)M(logrR

2MIMO

b

1r0 MIMO ≤<

Rb = RsNTrMIMOlog2(M)


System model

BRF ≥ Rs(necessarycondition toavoid ISIbetweensuccessivesymbols)

Spectrum transmitted bandpass signal

f

BRF BRF0

Bandwidth efficiency :

All NT transmitted signals are simultaneously in the same frequency interval

Rb/Rs = NT⋅rMIMO⋅log2(M) (bit/s/Hz)


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[P⋅Q] = Tr[Q⋅P], Tr[M] = Tr[MT]

ML detection :


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[NBER

)i()0(∑ ==

= AA

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

are different



)(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 - H⋅A(i)||2 < ||R - H⋅A(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



⋅−≤

⋅=

0

2)0,i(

0

2)0,i()0,i(

N4||||exp

21

N2||||Q)(PEP ∆H∆HH

[ ]R

TR

T

NN

1n 0

)0,i(n

N

0

H)0,i()0,i(

N)0,i()0,i(

N41

21

N4)(det

21)(PEPEPEP

−

=

−

λ+=

⋅+≤= ∏∆∆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 =λ

MIMO2T

0i

)0,i((i,0)info

)0(

r)M(logKN

PEPN]Pr[BER 0

∑ ∑

=

≤ ≠AAA

Averaging over Rayleigh fading channel statistics :


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


Single-input single-output (SISO) systems


SISO : observation model

info-bits

Rb

encoder

rSISO

coded bits

coded symbols

mapper

SISO

b

rR

)M(logrRR

2SISO

bs =

Bandwidth efficiency : Rb/Rs = rSISO⋅log2(M) (bit/s/Hz)

Observation model : r(k) = h⋅a(k) + w(k) k = 1, ..., K


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

h1=

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 ⋅⋅⋅ =


SISO : PEP computation

−≤

0

2)0,i(,E

2)0,i(

N4d|h|

exp21)h(PEP

1

0

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

N4d

121PEP

−

+≤

∑=

−=K

1k

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

−≤

0

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

N4d

exp21PEP

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 proportionalto Eb/N0

b2SISOs2

)0,i(,E E)M(logrEd =∝

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


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 30Eb/N0 (dB)

BER

AWGNAWGN boundRayleigh fadingRayleigh fading bound

SISOBPSK, uncoded

BER fading channel >> BER AWGN channel


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 30Eb/N0 (dB)

BER

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


Single-input multiple-output(SIMO) systems


SIMO : observation model

Same transmitter as for SISO; NR antennas at receiver

info-bits

Rb

encoder

rSIMO

coded bits

coded symbols

mapper

SIMO

b

rR

)M(logrRR

2SIMO

bs =

Bandwidth efficiency : Rb/Rs = rSIMO⋅log2(M) (bit/s/Hz)

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


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

Σ

*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


SIMO : PEP computation

−≤

0

avg22

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

N4)|h(|dN

exp21)(PEP h

RN

0

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

N4d

121PEP

−

+≤

−≤

0

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

N4dN

exp21PEP

Conditional PEP :

PEP, AWGN (|hm| = 1) :

PEP, Rayleigh fading :

decreases exponentially with Eb/N0

inversely proportionalto (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|N1)|h(|


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 4x

pdf

N_R = 1

N_R = 2

N_R = 3

N_R = 5

N_R = 10

Probability density function of <|h|2>


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 3x

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>


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 15Eb/N0 (dB)

BER

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

SIMO AWGNUncoded BPSK

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



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 30Eb/N0 (dB)

BER

SISOarray gain onlyboundcorrect

SIMO, N_R = 2uncoded BPSK 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 30Eb/N0 (dB)

BER


SIMO, N_R = 3uncoded BPSK




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 30Eb/N0 (dB)

BER


SIMO, N_R = 4uncoded BPSK




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 30Eb/N0 (dB)

BER

SISO uncodeduncodedShannon limitExt. Hamming (8,4)

SIMO, N_R = 2BPSK

fading channel : coding gain increases with NR



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 30Eb/N0 (dB)

BER

SISO uncodeduncodedShannon limitExt. Hamming (8,4)

SIMO, N_R = 3BPSK




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 30Eb/N0 (dB)

BER

SISO uncoded

uncoded

Shannon limit

Ext. Hamming (8,4)

SIMO, N_R = 4BPSK



Multiple-input multiple-output(MIMO) systems


MIMO : observation model

Bandwidth efficiency : Rb/Rs = NTrMIMO⋅log2(M) (bit/s/Hz)

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

N

1nnn,mm

T

+=∑=

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

R = H⋅A + W at any receive antenna, symbols from different transmit antennas interfere

)M(logrNRR

2MIMOT

bs =

(per antenna)

info-bits

encoder

rMIMO

codedbits

NTK coded symbols

mapper S/P

#1

#NT

#n


RbMIMO

b

rR

)M(logrR

2MIMO

b


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 ⋅⋅⋅⋅⋅ =


MIMO : PEP computation

RT

R

T

NN

1n 0

)0,i(n

N

0

H)0,i()0,i(

N)0,i(

N41

21

N4det

21PEP

−

=

−

λ+=

∆⋅∆+≤ ∏I

PEP, Rayleigh fading :

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

PEP(i,0) inversely proportional to (Eb/N0)^(NR⋅rank(∆(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 ∆⇒


MIMO : spatial multiplexing


Spatial multiplexing

info-bits

encoder

r

mapper

rNR

T

b

)M(logrNRR

2T

bs =

#1

T

b

NR

encoder

r

mapper

rNR

T

b

#NT

T

b

NR

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

aA M

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

by antenna #n

codewords transmitted by different antennas are statistically independent

Bandwidth efficiency : Rb/Rs = NT⋅r⋅log2(M)


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


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 vectorδT = (δ(1), ..., δ(k), ..., δ(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

21PEP

−

α+≤ ∑δ

∆(i,0) nonzero ⇒ rank(∆(i,0)) ≥ 1

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

⇒


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 i≠j

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)


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)


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 30Eb/N0 (dB)

BER

(bou

nd)

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


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 30Eb/N0 (dB)

BER

(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


MIMO : space-time coding


Space-time coding

info-bits

Rb

encoder

rMIMO

codedbits

coded symbols

mapper

MIMO

b

rR

)M(logrR

2MIMO

b

S/P

#1

#NT

#n

)M(logrNRR

2MIMOT

bs =

(per antenna)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


Three classes of space-time coding :

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


STTrC : trellis representation

( ))K(,),k(,),1( aaaA LL=

( ))k(b,),k(b)k( L1 L=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 LL=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


STTrC : bandwidth efficiency

ML decoding by means of Viterbi algorithm :

decoding complexity linear (instead of exponential) in Kbut 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)


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, δ1)T, (δ2, 0)T

with nonzero δ1 and δ2⇒ rank = 2, i.e., max. rank

0

1

2

3


Delay diversity : transmitter

−−−−−−−

−−=

)2K(a)3K(a)4K(a)5K(a)2(a)1(a000)2K(a)3K(a)4K(a)3(a)2(a)1(a000)2K(a)3K(a)4(a)3(a)2(a)1(a

L

L

L

A

Example : NT = 3

⇒ rank(∆(i,0)) = NT ⇒ diversity order = NTNRmaximum possiblediversity order !

info-bits encoder

r

mapper

rR b

)M(logrRR

2

bs =

#1

delay (NT -1)T#NT

Rb

(per antenna)

delay (n-1)T#n


Delay diversity : bandwidth efficiencyEquivalent configuration :

Bandwidth efficiency (for K >> 1) : Rb/Rs = NTrMIMOlog2(M) = r⋅log2(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(logrRR

2

bs =

#1

delay (NT -1)T#NT

(per antenna)

delay (n-1)T#n

mapper

mapper

MIMO encoder : rMIMO = r/NT

Rb


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


Space-time block codes

Space-time block codes can be designed to yield maximum diversityand 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


STBC : example 1

−= *

12

*21

aaaa

A

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

*1a,a

HHHH

a,a

2

a,a11

21

2121

hhrhrhrhrh

AAHHRHAHAR

+++−++−=

+−=−=

22

21

*2

T21

H1

1 ||||u

hhrhrh

++

= 22

21

*2

T11

H2

2 ||||u

hhrhrh

+−

=

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

211a1 |au|minarga

1

−= 222a2 |au|minarga

2

−=


STBC : example 1

−= *

12

*21

aaaa

A

−−−−−

= *)i(1

)0(1

)i(2

)0(2

*)i(2

)0(2

)i(1

)0(1)0,i(

)aa(aa)aa(aa

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

Assume M-point constellation⇒ A contains 4 symbols, i.e. 4⋅log2(M)rMIMO = 2⋅log2(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


STBC : example 2

Example 2 : NT = 3

−−−−−−−−−−

=*2

*1

*4

*32143

*3

*4

*1

*23412

*4

*3

*2

*14321

aaaaaaaaaaaaaaaaaaaaaaaa

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 = 4⋅log2(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 andsymbol-by-symbol detection) cannot be achieved for all values of NT

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


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 30Eb/N0 (dB)

BER

(bou

nd)

SISON_T = 2 (Alamouti)N_T = 3

Space-time block codesBPSKN_R = 1



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 30Eb/N0 (dB)

BER

(bou

nd)

SISOSIMON_T = 2 (Alamouti)N_T = 3




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 30Eb/N0 (dB)

BER

(bou

nd) SISO

SIMON_T = 2 (Alamouti)N_T = 3



Summary : SISO/SIMO/MIMO

Bandwidthefficiency

Diversity order(ML detection)

ML detectorcomplexity

SISO rSISOlog2(M) 1 in general : expon. in K(trellis : linear in K)

SIMO rSIMOlog2(M) NR in general : expon. in K(trellis : linear in K)

MIMO NTrMIMOlog2(M)

∆⋅ )(rankmaxN )0,i(

)0,i(Rin general :expon. in NTK


Summary : MIMO

Bandwidthefficiency

Diversity order(ML detection)

ML detectorcomplexity

spatialmultiplexing NTr⋅log2(M) NR

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

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

diversity r⋅log2(M) NTNRif 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 · Advanced Modulation and Coding :...

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