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