MIMO Systems: Principles, Iterative Techniques, and Advanced Polarization - Adaptive Signal Processing in Wireless Communications - page 107

1

2

APP

MIMO
Soft−input
Soft−output
detector

Soft−input
Soft−output
decoding
of C

P Det

ext ( c )

Π −1

M R

Extrinsic

P Dec
ext

( c )

Π

CSI

FIgure 4.8 Block diagram of the receiver.

decoding, we can use the optimal maximum a posteriori (MAP) algorithm, or a sub-
optimal solution based on sphere decoding or interference cancelling, for example. In
fact, the optimal MAP detector becomes too complex to implement in practice, espe-
cially for large Q or large signal constellations.
Soft channel decoding, on the other hand, can be done using the well-known forward-
backward algorithm [57], the soft-output Viterbi algorithm (SOVA) [58], or a simplifica-
tion of them [59]. For the final decision making on the transmitted data bits, we use the
a posteriori probabilities at the decoder output. Like in any other turbo processing case,
extrinsic information is exchanged between the two blocks of MIMO detector and chan-
nel decoder. In what follows, we explain the principle of MIMO detection based on
MAP, sphere decoding, and soft interference cancelling, while focusing on the third
approach.

4.4.3.1 MAP Signal Detection
We present here the formulation of the MAP detector based on probabilities. It can also
be implemented using logarithmic likelihood ratios (LLRs). Remember the expression
of Y from (4.16). The MIMO detector provides at its output extrinsic probabilities on the
coded bits c . Let Q be the cardinality of S of size q  |Q| = 2 BQ . Let also c i , i = 1, … , BQ
be the bits corresponding to a vector of symbols S ∈ Q. The extrinsic probability on the
bit c j at the MIMO detector output, P ex Det ( c j ), is calculated as follows [60]:

 

2 BQ

YHS −

∑


 


 

∏ ()

( ) =

eq

Det Dec

Pc 1 K exp − Pc i ,

(4.17)

ext j ext

2

σ

n

 

SQ ∈ i

ij

=
≠

1

c = 1

j

where K is the normalization factor satisfying P ex Det ( c j = 1) + P ex Det ( c j = 0) = 1. he prob-
ability P ex Dec ( c i ) is in fact the a priori information on bit c i , fed back from the channel
decoder. At the first iteration, where no a priori information is available on bits c i , P ex Dec
are set to 1/2. The summation in (4.17) is taken over the product of the conditional