multiple-input multiple-output (mimo) communication systems
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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
)k(r
)k(r
RN
1
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
)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
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(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)
Advanced Modulation and Coding : MIMO Communication Systems
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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)
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[PQ] = Tr[QP], 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[N
BER)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 - 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
Advanced Modulation and Coding : MIMO Communication Systems
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Bit error rate (BER)
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 :
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
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
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
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
Advanced Modulation and Coding : MIMO Communication Systems
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)
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 30
Eb/N0 (dB)
BE
R
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 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
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
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
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
*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(|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(|
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 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>
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 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>
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 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
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 30
Eb/N0 (dB)
BE
R
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 30
Eb/N0 (dB)
BE
R
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 30
Eb/N0 (dB)
BE
R
SISO
array gain only
bound
correct
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 30
Eb/N0 (dB)
BE
R
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 30
Eb/N0 (dB)
BE
R
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 30
Eb/N0 (dB)
BE
R
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 = 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
M-pointconstellation
RbMIMO
b
r
R
)M(logr
R
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
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
Advanced Modulation and Coding : MIMO Communication Systems
34
MIMO : spatial multiplexing
Advanced Modulation and Coding : MIMO Communication Systems
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)
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 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
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 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)
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 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
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 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
Advanced Modulation and Coding : MIMO Communication Systems
42
MIMO : space-time coding
Advanced Modulation and Coding : MIMO Communication Systems
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
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
)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
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 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)
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, 1)T, (2, 0)T
with nonzero 1 and 2
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(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
Advanced Modulation and Coding : MIMO Communication Systems
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
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 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
Advanced Modulation and Coding : MIMO Communication Systems
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
Advanced Modulation and Coding : MIMO Communication Systems
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
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
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)
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 30
Eb/N0 (dB)
BE
R (
bo
un
d)
SISO
N_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 30
Eb/N0 (dB)
BE
R (
bo
un
d)
SISO
SIMO
N_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 30
Eb/N0 (dB)
BE
R (
bo
un
d)
SISO
SIMO
N_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 30
Eb/N0 (dB)
BE
R (
bo
un
d) SISO
SIMO
N_T = 2 (Alamouti)
N_T = 3
Space-time block codesBPSKN_R = 4
Advanced Modulation and Coding : MIMO Communication Systems
59
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
Advanced Modulation and Coding : MIMO Communication Systems
60
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
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