MIMO Systems: Principles, Iterative Techniques, and Advanced Polarization - Adaptive Signal Processing in Wireless Communications - page 109

( ) −

1

ˆ

† † † 2

γ = WY = h HH + σ I Y , (4.18)

p p p eq q n

where W p denotes the filter, h p is the p th column of H eq , and (.) † stands for transpose
complex conjugate. Note that the entries of H eq are real values, and hence (.) † is equiva-
lent to transposition. From the second iteration, we can calculate soft estimates S ˜ of
the transmitted data using the soft decoder outputs. Using these estimates, we perform
interference cancelling followed by a simple zero forcing (ZF) or MMSE filtering:

ˆ  ˆ

, ˆ

†

YYHS =− γ = WY , (4.19)

p p p p pp

1 1

ZF: W =

hh h

, MMSE: W = h p , (4.20)

( )

p p p

†

hh

†

σ

2

pp

pp n

where S ˜ p of dimension ((2 Q - 1) × 1) is S ˜ with its p th entry removed, and H p of dimension
(2 M R T × (2 Q - 1)) is the matrix H with its p th column removed. Notice that, compared
to the exact MMSE filtering proposed in [48], (4.20) are simplified solutions that assume
almost perfect estimation of data symbols and permit a considerable reduction of the
computational complexity. Thanks to iterative processing, the performance loss due to
this simplification would be negligible. In the results that we present later, we will con-
sider the simplified ZF solution.

4.4.3.3.2 Conversion to LLR
For QAM modulation with B (an even number) bits per symbol, we can attribute m =
B /2 bits to the real and imaginary parts of each symbol. Let, for instance, the bit c i cor-
respond to the real (imaginary) part of the symbol s q . Let also a 1, j and a 0, j , j = 1, … , B /2
denote the real (imaginary) part of the signal constellation points, corresponding to
c i = 1 and c i = 0, respectively. Remember that the signal constellation points have nor-
malized average power. The LLR corresponding to c i is calculated as follows [62]:

 

m − 1

∑

2

1
2σ γα

− ( )

2

ˆ

   

exp −

p 1 , j

2

j = 1

 

p

= … LLR i = log , i 1 , , m ,

(4.21)

10

 

m − 1

∑ −

2

1

− ( )

2

  ˆ  

exp

σ γα
p

p 0 , j

2

2

j = 1

 

where σ p 2 is the variance of noise plus the residual interference (RI) that intervenes in the
detection of γ ˆ p , and is assumed to be Gaussian. Note that as the detection is performed
on blocks of Q complex symbols, or in other words, on blocks of 2Q real symbols in our
model, the RI comes in fact from (2 Q - 1) other real symbols in the corresponding chan-
nel use [64]. In LLR calculation, we need the variances σ p 2 . These variances can be cal-
culated analytically as shown in [46], or estimated at each iteration and for each one of