Advanced Modulation and Coding : MIMO Communication Systems 1
Multiple-input multiple-output (MIMO)communication systems
Advanced Modulation and Coding : 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
M
)k(r
)k(r
RN
1
M
k : time index (k = 1, ..., K)
(coded) data symbols an(k), E[|an(k)|2] = Es
Advanced Modulation and Coding : MIMO Communication Systems 3
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
Advanced Modulation and Coding : MIMO Communication Systems 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(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)
Advanced Modulation and Coding : MIMO Communication Systems 5
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)
Advanced Modulation and Coding : MIMO Communication Systems 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[P⋅Q] = Tr[Q⋅P], Tr[M] = Tr[MT]
ML detection :
Advanced Modulation and Coding : MIMO Communication Systems 7
Bit error rate (BER)
info-bits
encoder
rMIMO
codedbits
NTK coded symbols
mapper S/P
#1
#NT
#n
M-pointconstellation
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
Advanced Modulation and Coding : MIMO Communication Systems 8
Bit error rate (BER)
)(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
Advanced Modulation and Coding : MIMO Communication Systems 9
Bit error rate (BER)
⋅−≤
⋅=
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 :
Advanced Modulation and Coding : MIMO Communication Systems 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
Advanced Modulation and Coding : MIMO Communication Systems 11
Single-input single-output (SISO) systems
Advanced Modulation and Coding : MIMO Communication Systems 12
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
Advanced Modulation and Coding : MIMO Communication Systems 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
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 ⋅⋅⋅ =
Advanced Modulation and Coding : MIMO Communication Systems 14
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)
Advanced Modulation and Coding : MIMO Communication Systems 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 30Eb/N0 (dB)
BER
AWGNAWGN boundRayleigh fadingRayleigh fading bound
SISOBPSK, uncoded
BER fading channel >> BER AWGN channel
Advanced Modulation and Coding : MIMO Communication Systems 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 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
Advanced Modulation and Coding : MIMO Communication Systems 17
Single-input multiple-output(SIMO) systems
Advanced Modulation and Coding : MIMO Communication 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
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
Advanced Modulation and Coding : MIMO Communication Systems 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
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
Advanced Modulation and Coding : MIMO Communication Systems 20
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(|
Advanced Modulation and Coding : MIMO Communication Systems 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 4x
N_R = 1
N_R = 2
N_R = 3
N_R = 5
N_R = 10
Probability density function of <|h|2>
Advanced Modulation and Coding : MIMO Communication Systems 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 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>
Advanced Modulation and Coding : MIMO Communication Systems 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 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
Advanced Modulation and Coding : MIMO Communication Systems 24
SIMO : numerical results
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
Advanced Modulation and Coding : MIMO Communication Systems 25
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 5 10 15 20 25 30Eb/N0 (dB)
BER
SISOarray gain onlyboundcorrect
SIMO, N_R = 3uncoded BPSK
fading channel : array gain + diversity gain
Advanced Modulation and Coding : MIMO Communication Systems 26
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 5 10 15 20 25 30Eb/N0 (dB)
BER
SISOarray gain onlyboundcorrect
SIMO, N_R = 4uncoded BPSK
fading channel : array gain + diversity gain
Advanced Modulation and Coding : MIMO Communication Systems 27
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 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
Advanced Modulation and Coding : MIMO Communication Systems 28
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 5 10 15 20 25 30Eb/N0 (dB)
BER
SISO uncodeduncodedShannon limitExt. Hamming (8,4)
SIMO, N_R = 3BPSK
fading channel : coding gain increases with NR
Advanced Modulation and Coding : MIMO Communication Systems 29
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 5 10 15 20 25 30Eb/N0 (dB)
BER
SISO uncoded
uncoded
Shannon limit
Ext. Hamming (8,4)
SIMO, N_R = 4BPSK
fading channel : coding gain increases with NR
Advanced Modulation and Coding : MIMO Communication Systems 30
Multiple-input multiple-output(MIMO) systems
Advanced Modulation and Coding : MIMO Communication Systems 31
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
M-pointconstellation
RbMIMO
b
rR
)M(logrR
2MIMO
b
Advanced Modulation and Coding : MIMO Communication Systems 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 ⋅⋅⋅⋅⋅ =
Advanced Modulation and Coding : MIMO Communication Systems 33
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 ∆⇒
Advanced Modulation and Coding : MIMO Communication Systems 35
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)
Advanced Modulation and Coding : MIMO Communication Systems 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
Advanced Modulation and Coding : MIMO Communication Systems 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 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
⇒
Advanced Modulation and Coding : MIMO Communication Systems 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 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)
Advanced Modulation and Coding : MIMO Communication Systems 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)
Advanced Modulation and Coding : MIMO Communication Systems 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 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
Advanced Modulation and Coding : MIMO Communication Systems 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 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
Advanced Modulation and Coding : MIMO Communication Systems 43
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
Advanced Modulation and Coding : MIMO Communication Systems 44
Three classes of space-time coding :
• space-time trellis codes (STTrC)• delay diversity• space-time block codes (STBC)
Advanced Modulation and Coding : MIMO Communication Systems 45
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
Advanced Modulation and Coding : MIMO Communication Systems 46
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)
Advanced Modulation and Coding : MIMO Communication Systems 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, δ1)T, (δ2, 0)T
with nonzero δ1 and δ2⇒ rank = 2, i.e., max. rank
0
1
2
3
Advanced Modulation and Coding : MIMO Communication Systems 48
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
Advanced Modulation and Coding : MIMO Communication Systems 49
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
Advanced Modulation and Coding : MIMO Communication Systems 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
Advanced Modulation and Coding : MIMO Communication Systems 51
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
Advanced Modulation and Coding : MIMO Communication Systems 52
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
−=
Advanced Modulation and Coding : MIMO Communication Systems 53
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
Advanced Modulation and Coding : MIMO Communication Systems 54
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)
Advanced Modulation and Coding : MIMO Communication Systems 55
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
Advanced Modulation and Coding : MIMO Communication Systems 56
STBC : numerical results
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
Space-time block codesBPSKN_R = 2
Advanced Modulation and Coding : MIMO Communication Systems 57
STBC : numerical results
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
Space-time block codesBPSKN_R = 3
Advanced Modulation and Coding : MIMO Communication Systems 58
STBC : numerical results
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
Space-time block codesBPSKN_R = 4
Advanced Modulation and Coding : MIMO Communication Systems 59
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
Advanced Modulation and Coding : MIMO Communication Systems 60
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