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Iterative detection and decodingto approach MIMO capacity
Jun Won Choi
Introduction MIMO capacity (CSI only at receiver) Fast fading scenario – Ergodic I.I.D. Rayleigh fading
Channel
Under fast fading assumption, transmission of independent data stream with same power is sufficient to achieve capacity (V-BLAST).
Capacity achieving Gaussian codes are used at each antenna as outer code.
Slow fading scenario Coding across transmit antennas is needed space-time
coding, advanced layering
log det Hr
PE I
t
HH Teletar, 1999
Introduction Optimal transmitter structure
AWGN coder
(outer code)
AWGN coder
(outer code)
Signal processing operation
(V-BLAST, D-BLAST,
space-time coding)
H
noise
Coding across transmit antennas is needed in slow fading
y = Hx+n
Inner code
Introduction Optimal receiver structure Maximum a posteriori (MAP) decoder
Model a received signal as Markov process whose trellis is formed to include AWGN code, space-time coding, and MIMO channel.
Map decoding rule is optimal.
Computationally infeasible ! Iterative detection and decoding (IDD)
Divide decoding job into MIMO detection (inner code) and AWGN channel decoding (outer code).
Approximately approach to optimal performance via information exchange between two constitutional blocks.
1arg max | , ,i i Tb P b y y
Transmitter design example 1 (IDD) Turbo-Blast (Haykin 2002) Random layered space time coding
AWGNcoder
AWGNcoder
Diagonal
Layering
Interleaver
Interleaver
M-arymodul
M-arymodul
Space-time interleaver
Transmitter design example 2 Space-time bit interleaved coded modulation (Tonello,
2000)
AWGNcoder
Interleaver
S/P
M-arymappe
r
M-arymappe
r
Principle of IDD Iterative (MIMO) detection and (channel) decoding
MIMO detector
MIMO detector
Deinterleaver
SISOdemapper
SISO channel decoder
SISO channel decoder
Interleaver
Information exchange
y
Soft information is expressed as L-value
1pL
1eL
2pL
2eL
1
1p
p xL
p x
eL
Priori LLR
Extrinsic LLR
IDD SISO Channel decoder BCJR algorithm – based on trellis-based search Low-complexity APP decoder - LOG-MAX algorithm, Soft
output viterbi algorithm (SOVA) MIMO detector Complexity and performance trade-off
MAP versus Sub-optimal detector with linear structure
Definition (Space-time bit interleaved coded modulation)
,
: information bit
c : coded bit
: interleaved coded bit
: symbol
i
i
n m
n c
b
c
x M ary
,1 ,, ,tn n M nc c x
1
tN
x
x
x
Map detector Map detector A posteriori L-value of the bit
,
1 ,
,
1ln
1n m
a n m
n m
P cL c
P c
yy
y
, , 1
. , 1
1 , 1 , 1 ,
, 1 ,1( ) 1( )
, 1 ,1( ) 1( )
1| exp
2ln
1| exp
2
t c
n m
t c
n m
e n m a n m p n m
N M
k l p k lX k n l m
N M
k l p k lX k n l m
L c L c L c
p c L c
p c L c
x
x
y y
y x
y x
Extrinsic information (output)
2
22
1exp
tNww
p
y -Hxy | x
Map decoder Map detection rule Log-Max approximation
Complexity Complexity of MAP decoder is exponential in
modulation size, antenna size.
1 2ln max( 1, 2)a ae e a a
, , 1
, , 1
1 , , 1 ,21( ) 1( )
, 1 ,21( ) 1( )
1 1max
2
1 1max
2
t c
n m
t c
n m
N M
e n m k l p k lX
k n l mw
N M
k l p k lX
k n l mw
L c c L c
c L c
x
x
y y -Hx
y -Hx
There are combinations for each hypothesis. 2 1t tM N
, , 1 ,: 1n m n mX c x
List sphere decoding Idea (Hochwald, 2003) Find the combinations of symbol vector that are highly
likely to be transmitted. It is called candidate list.
Define the candidate list, L as Then, extrinsic L-value can be find over such candidate
list, i.e.,
2p y | x y -Hx
:L B x y -Hx
, , 1
, , 1
1 , , 1 ,21( ) 1( )
, 1 ,21( ) 1( )
1 1max
2
1 1max
2
t c
n m
t c
n m
N M
e n m k l p k lX L
k n l mw
N M
k l p k lX L
k n l mw
L c c L c
c L c
x
x
y y -Hx
y -Hx
List sphere decoding List sphere decoding Efficient tree pruning problem
y = Hx+w
y HxLattice
Form skewed lattice
Number of points tNtM
Sphere constraint2By -Hx
2
2 2
1 ,1
' 't t
t
t
n nn
n i i k ki k i
d y r x
y Hx y Rx x
2
,'t t
t
n nn
q q i i k ki q k i
d y r x
x
2
1 1 1 ,1
't
t t
nn n
q q q q q i k kk q
d d y r x
x x
Define the cost metric
Stage 1
Stage 2
Stage 3
Stage 4
root
0 1
0 1
10
0 1
ML path
43 3d Bx
Sphere constraint is violated.
Prune sub-tree.
List sphere decoding Procedure 1. Find the points inside sphere by tree search. 2. Select closest points. (when number of points
found is larger than predefined list size) 3. Increase radius and restart the search. (when
number of points found is less than list size) 4. If candidate list has no common entry with or
, the extrinsic L-value is set to –inf or inf depending on the sign of entries.
How to choose B?, For true x
candN
, , 1n mX , , 1n mX
2 2 2 22 tW N y -Hx w 2
=CDF , t Bp B B N P w
0.9BP
Turbo-Blast detector Turbo-Blast detector Sub-optimal detector with linear structure Derive based on linear MMSE criterion
y = Hx+n
Assume that are available.1 [1: ],[1: ]( )t tp N ML c
1ˆ Cov , Covn n n nx E x x y E
y y,y H x
Let 1 ,[1: ]( ) 0tp n ML c 0nE x
var 1nx
Interference cancellation stepInterference nulling step
Turbo-Blast detector Interference cancellation step
Interference nulling step
1: 1
1:
0
t
n
n
n N
E
E
E
x
z y H x y H
x
, 1 ,1
11 tanh |
2
tM
k n m p n mx m
E x x c L c
y
12n n w n
a HΛ H I h
ˆ Hn n nx a z
1 1 1diag var( ), , var( ),1, var( ), , var( )tn n n Nx x x x Λ
22
, 1 ,1
1var 1 tanh |
2
tM
n n m p n m nx m
x x c L c E x
y
Turbo-Blast detector Gaussian approximation
1: 1 1: 1
1: 1:
ˆ
t t
n nH H
n n n n n n n n
n N n N
E
x H x x
E
x x
a z a w
x x Interference + noise term
2 2n n
, 1
, 1
, 1 , 1
2,
1 ,
,2
2 2
ˆexp
ˆ 1ˆ| ln ln
ˆˆ 1exp
ˆ ˆmax max
m
m
m m
n n
n n m x X
e n m nn nn n m
x X
n n n n
x X x X
x xp x c
L c xx xp x c
x x x x
There are only combinations for each hypothesis. 2 1tM
, 1 ,ˆ: 1m n n mX x c
Conclusions Capacity achieving MIMO architecture Transmitter architecture
V-BLAST + AWGN code for fast fading Coding across transmit antenna for slow fading space time
coding, D-BLAST, Treaded space time coding Receiver architecture
Global MAP decoding Iterative detection and decoding
Map decoding List sphere decoder Linear MMSE detector