1
A semianalytical method for predictingthe performance and convergence behaviorof a multiuser turboequalizer/demapper
Val´ery Ramon*, C´edric Herzet, Luc VandendorpeCommunications and Remote Sensing Laboratory, Universite catholique de Louvain (UCL)2, place du Levant, B1348 LouvainlaNeuve, Belgium
{
ramon, herzet, vandendorpe
}
@tele.ucl.ac.be
Abstract
—This paper proposes a simple semianalyticalmethod with reduced simulation time for predicting at any iteration the performance of a turboequalization/demapping schemeusing the Wang and Poor softin/softout (SISO) Minimum MeanSquare Error (MMSE) / Interference Cancellation (IC) equalizerand a SISO decoder. The proposed method may be applied tomultilevel/phase data modulations as well as multiuser context.This paper shows that the equalizer behavior may be very reliablypredicted totally by calculations (no simulations are needed)whereas that of the decoder still requires simulations.Comparisonbetween the proposed prediction method and plain simulationsof the overall turbo equalization scheme demonstrates thatour method accurately determines the system performance atany iteration. Static channels, long frames and perfect channelknowledge are assumed throughout the paper.
I. I
NTRODUCTION
Turboequalization/demapping (references in [1]) is a powerful mean to perform joint equalization/demapping and decoding when considering coded data transmission over frequency selective channels. The association of the code and thediscretetime equivalent channel (separated by an interleaver)may be regarded as the serial concatenation of two codes. Theturbo principle may then be used at the receiver : the systemperformance measured in terms of the bit error rate (BER)is improved through the exchange of extrinsic informationbetween a softin/softout (SISO) equalizer/demapper and aSISO decoder.The goal of this paper is to predict the performance andconvergence behavior of such a turboequalization/demappingscheme at any iteration using a simple semianalytical method.The intention is to replace simulations of some part of thereceiver chain, namely the equalizer, by calculations. The proposed prediction method may be applied to multilevel/phasedata modulations as well as multiuser contexts. Nevertheless,for the sake of clarity, only simple cases will be consideredin this paper. Three cases of gradually increasing difﬁculty,namely singleuser Binary PhaseShift Keying (BPSK), singleuser Quadrature PhaseShift Keying (QPSK) and multiuserBPSK, will thus be successively developed. This way to proceed will ease understanding and enable to point out the slightdifferences between these various cases. The SISO Minimum
Val´ery Ramon would like to thank the Belgian NSF for its ﬁnancial support.This research is partly funded by the Federal Ofﬁce for Scientiﬁc, Technicaland Cultural Affairs (OSTC, Belgium) through the IAP contract No P5/11.
Mean Square Error (MMSE) / Interference Cancellation (IC)equalizer/demapper which will be analyzed in this paper isthe one proposed by Wang and Poor [2] for BPSK datamodulation and multiuser context. This multiuser equalizeris suboptimal but it exhibits reasonable complexity whileoffering interesting performance [2]. It was extended notablyin [1] to multilevel/phase data modulation, in [3] to multiple
input multipleoutput (MIMO) systems and simpliﬁed in [4]in order to deal with timeinvariant equalizer coefﬁcients. Itis more particularly this latter version which will be analyzedin the present paper. However, an extension of the proposedprediction method to other linear equalizers using a prioriinformation on data symbols is straightforward. A SISOconvolutional decoder using the BCJR algorithm [5] will beconsidered in the sequel although other SISO decoders (lowdensity parity check (LDPC) decoder, ...) might also be used.Various methods for predicting the convergence behaviorof turbo decoding schemes as well as parallel or serially ([6]and references therein, [7] and references therein) concatenated codes have been previously proposed. They are basede.g. on variance or signaltonoise ratio transfer analysis orextrinsic information transfer chart (EXIT chart). Almost allof them require simulations letting vary at least one inputparameter for each of the two devices involved in the turboprocess. To our knowledge, the only papers which proposecalculations rather than simulations for some parts of thereceiver chain are [8], [9] and [10]. Reference [8] found the
error variance on the symbol estimates at the output of aparallel interference canceller assuming gaussianity of the noncancelled multiuser interference. Reference [9] used the socalled unscented transformation for calculating the mean of the LogLikelihood Ratios (LLRs) output by the MMSE/ICequalizer. Nevertheless, this approach has been developed onlyin the singleuser BPSK case and is a little more complex thanthe one described in this paper. Reference [9] also calculatedthe probability density function (pdf) of the extrinsic LLRsdelivered by a LDPC decoder assuming a mixture of Gaussianpdf for the decoder input LLRs distribution. Reference [10]presented an optimal iterative multiuser detector/decoder basedon crossentropy minimization techniques as well as a practical implementation of this algorithm. A theoretical analysisshowed that, with this approach, multiuser detection/decodingis capable of achieving optimal singleuser performance for
2
user correlations approaching one. For our part, we alreadypresented in [11] an overview of the prediction method developed hereafter but only in the singleuser BPSK case.The prediction method proposed in this paper requiressimulations only of the decoder. Indeed, as we will show inthe sequel, the equalizer/demapper behavior may be accuratelyand totally predicted by calculations (so without any simulation) using simplifying approximations. In other words, theinput/output relations of the equalizer/demapper may be givenanalytically. One of these approximations is the gaussianity of the extrinsic LLRs output by the equalizer/demapper as wellas the decoder. The impact of these Gaussian assumptions onthe performance of our prediction method will be illustrated.The gaussianity of the LLRs computed by the constituentstages of the receiver is a hypothesis made in all the hereabovecited papers. Reference [7] provides theoretical justiﬁcation forthe Gaussian approximation in turbo decoding when the framelength becomes very large. Reference [12] shows that thisassumption is accurate for sumproduct decoding algorithmsin the general setup of decoding of graphbased codes. References [4] and [6], when drawing their EXIT charts respectively
for turboequalization and parallel concatenated codes, usedthis assumption which turned out to be accurate enough forpredicting the system convergence behavior. The reliability of this hypothesis may also be illustrated by simulation results.If the decoder may be simulated in advance for someﬁxed parameters (constraint length, code rate,...), our methodenables to foretell the performance and convergence behaviorof the equalized system given that decoder, without furthersimulations, for any frequency selective channel and at any
E
b
/N
0
. Performance with the reference additive white Gaussian noise (AWGN) channel or with perfect knowledge of apriori information at the equalizer input may then also bedirectly obtained.In the singleuser BPSK case, the proposed method forcalculating the equalizer/demapper behavior will signiﬁcantlyreduce the number of simulations needed to establish theEXIT chart [4]. Indeed, we are now able to calculate (ratherthan simulate) the equalizer characteristic curves for anychannel and any
E
b
/N
0
. Simulations remain necessary onlyfor drawing the decoder characteristic curve.Static channels, long frames and perfect channel knowledgeat the receiver will be assumed throughout the paper.The sequel of this paper will be organized as follows. Section II will introduce the system model. Our method for predicting the performance of the turboequalization/demappingscheme will be explained in section III. This section willsuccessively consider singleuser BPSK, singleuser QPSKand multiuser BPSK cases. Section IV will compare fordifferent channels the performance obtained with our prediction method to that obtained by plain simulations of theturboequalizer/demapper. It will be observed that our methodaccurately determines the system performance at any iteration.II. S
YSTEM
M
ODEL
In this section, the transmitter model (subsection IIA), theoverall iterative receiver (subsection IIB) and the equationsof the considered MMSE/IC equalizer (subsection IIC) willbe successively presented for the most general case :
K
userspropagating through frequencyselective channels and usingmultilevel/phase data modulation. As no novelty is introducedwith respect to the equalizer of [1], [2] and [4], only the
equations which will be useful for section III will be given.
A. Transmitter model
Let us consider a
K
user convolutionally coded DirectSequence Code Division Multiple Access (DSCDMA) communication system with spreading factor
M
. The block diagram of the transmitterend is similar to [1] and [2] and
is shown in Fig. 1. For any user
k
(
k
= 1
,...,K
), aframe of information bits
u
k
(
p
)
is encoded by a same rate
r
convolutional encoder. The resulting encoded bits
b
k
(
n
)
areinterleaved using a random permutation function to give theinterleaved coded bits
b
k
(
l
)
. These bits are then mapped ontosymbols
s
k
(
i
)
(
i
= 1
,...,I
) belonging to some constellationalphabet
A
. These symbols are modulated by a spreadingwaveform and transmitted over a frequencyselective channel,which is assumed to be static and perfectly known. Both thespreading waveform and the channel are speciﬁc to each user.At the receiver, after the summation of all the users signals andaddition of the white gaussian noise, chipmatched ﬁlteringand chiprate sampling are successively performed. Each
k
th user composite channel may thus be represented by itsequivalent discretetime model. This one includes the effectof the
k
th user spreading, the
k
th user channel itself, thechipmatched ﬁltering and the chiprate sampling. It resultsin a causal discretetime ﬁlter with chiprate coefﬁcients
h
k
(
jM
+
m
)
where
j
= 0
,...,L
and
m
= 0
,...,M
−
1
.
L
+1
thus denotes the length of the channel impulse responseexpressed in number of symbols. The observations
r
(
iM
+
m
)
at the output of the channel may thus be expressed as
r
(
iM
+
m
) =
K
k
=1
L
j
=0
h
k
(
jM
+
m
)
s
k
(
i
−
j
) +
n
(
iM
+
m
)
,
(1)where
n
(
iM
+
m
)
are Gaussian noise samples of variance
σ
2
n
.
B. Overall iterative receiver
The block scheme of the overall iterative receiver is illustrated in Fig. 2 and is more described in [1], [2] or [4]. It
consists of two stages : a SISO multiuser equalizer/demapperfollowed by
K
parallel singleuser SISO channel decoders.The two stages are separated by bitdeinterleavers and interleavers. They exchange extrinsic information, on iterativefashion, in order to improve the system performance. Thechannel code considered in the sequel is a convolutional codeand decoding is implemented using the wellknown BCJRalgorithm. Nevertheless, other codes (e.g. LDPC code) andSISO decoding algorithms might be used.
C. MMSE/IC equalizer equations
Now that the overall iterative receiver has been quicklypresented, let us focus more deeply on the MMSE/IC equalizerequations. These will be useful for understanding section III.
3
.
channel encoder interleaverinterleaver spreadingspreading channel 1channel Kchannel encodersymbol mappersymbol mapper
user 1user K
n
(
t
)
r
(
t
)
b
K
(
n
)
b
1
(
n
)
b
1
(
l
)
b
K
(
l
)
s
K
(
i
)
s
1
(
i
)
s
1
(
t
)
s
K
(
t
)
u
K
(
p
)
u
1
(
p
)
.
Fig. 1. Transmitter : convolutionallycoded DSCDMA system with K users transmitting through a frequencyselective channel.
.
multi−userinterleaverdeinterleaverdeinterleaverinterleaveruser 1user Kequalizer/demapperchip−ratesamplingchip−matchedfilteringdecoderdecoder
r
(
t
)
r
(
iM
+
m
)
.
Fig. 2. Iterative receiver : softin softout (SISO) multiuser equalizer/demapper followed by K parallel singleuser SISO channel decoders.
For
0
≤
j
≤
L
, deﬁning a
M
×
K
matrix
H
(
j
) =
h
1
(
jM
)
... h
K
(
jM
)
.........
h
1
(
jM
+
M
−
1)
... h
K
(
jM
+
M
−
1)
and the following vectors
r
(
i
)
[
r
(
iM
)
r
(
iM
+ 1)
... r
(
iM
+
M
−
1)]
T
,s
(
i
)
[
s
1
(
i
)
s
2
(
i
)
... s
K
(
i
)]
T
,n
(
i
)
[
n
(
iM
)
n
(
iM
+ 1)
... n
(
iM
+
M
−
1)]
T
,
we may write
r
(
i
) =
L
j
=0
H
(
j
)
s
(
i
−
j
) +
n
(
i
)
.
Deﬁning the equalizer length as
N
N
1
+
N
2
+ 1
, we mayintroduce a slidingwindow model using the following vectors
r
(
i
)
[
r
(
i
−
N
1
)
... r
(
i
)
... r
(
i
+
N
2
)]
T NM
×
1
,
s
(
i
)
[
s
(
i
−
N
1
−
L
)
... s
(
i
)
... s
(
i
+
N
2
)]
T
(
N
+
L
)
K
×
1
,
n
(
i
)
[
n
(
i
−
N
1
)
... n
(
i
)
... n
(
i
+
N
2
)]
T NM
×
1
,
and the
(
MN
×
(
N
+
L
)
K
)
channel matrix
H
=
H
(
L
)
... H
(0) 0
... ...
00
H
(
L
)
... H
(0) 0
...
0
...... ... ... ... ......
0
... ...
0
H
(
L
)
... H
(0)
.
(2)This enables to write for each symbol time index
i
r
(
i
) =
Hs
(
i
) +
n
(
i
)
,
(3)where
n
(
i
)
∼ N
c
(0
,σ
2
n
I
)
,
I
being the
MN
×
MN
identitymatrix. In the singleuser case, the dimensionality of thepreceding vectors and matrices signiﬁcantly reduces since
K
and
M
are then both equal to 1 (spreading is not needed).Let
˜
s
k
(
i
)
E
{
s
k
(
i
)
}
and var
{
s
k
(
i
)
}
E
{
s
k
(
i
)

2
} −
˜
s
k
(
i
)

2
denote respectively the socalled a priori mean (
E
stands for expectation) and variance of symbol
s
k
(
i
)
whichare computed as explained in [1], [2] and [4] from the
a priori LLRs at the equalizer input. For normalized PSKconstellations, we have of course
E
{
s
k
(
i
)

2
}
= 1
∀
k,i
.
4
Using the following deﬁnitions
˜
s
(
i
)
[˜
s
1
(
i
)
...
˜
s
K
(
i
)]
T
,
˜
s
(
i
)
[˜
s
(
i
−
N
1
−
L
)
T
...
˜
s
(
i
)
T
...
˜
s
(
i
+
N
2
)
T
]
T
,
˜
s
k
(
i
)
˜
s
(
i
)
−
˜
s
k
(
i
)
e
k
,r
ss,k
(
i
+
l
)
[
var
{
s
1
(
i
+
l
)
}
...
var
{
s
K
(
i
+
l
)
}
]
T
for
l
=
−
N
1
−
L,...,
−
1
,
1
,...,N
2
,r
ss,k
(
i
)
[
var
{
s
1
(
i
)
}
...
var
{
s
k
−
1
(
i
)
}
,
1
,
var
{
s
k
+1
(
i
)
}
,...,
var
{
s
K
(
i
)
}
]
T
,
R
ss
,k
(
i
)
diag
[
r
ss,k
(
i
−
N
1
−
L
)
T
... r
ss,k
(
i
)
T
... r
ss,k
(
i
+
N
2
)
T
]
,
w
k
(
i
)
[
HR
ss
,k
(
i
)
H
H
+
σ
2
n
I
]
−
1
He
k
,
(4)the symbol estimate
ˆ
s
k
(
i
)
is given by
ˆ
s
k
(
i
) =
w
H k
(
i
) [
r
(
i
)
−
H
˜
s
k
(
i
)]
,
(5)where
e
k
denotes a length
(
N
+
L
)
K
vector of zeros exceptfor the
((
N
1
+
L
)
K
+
k
)
th element, which is 1.An efﬁcient approximation made in [4]  which is not usedin [1] and [2]  leading to a complexity reduction of the
equalizer and weak performance degradation may be obtainedby computing for each user
k
the mean a priori variance overthe
I
transmitted symbols in the
k
th user frame
v
k
1
I
I
i
=1
var
{
s
k
(
i
)
}
.
(6)Letting
v
= [
v
1
,v
2
,...,v
K
]
T
and
v
k
=[
v
1
,...,v
k
−
1
,
1
,v
k
+1
,...,v
K
]
T
, a matrix
R
ss
,k
=
diag
[
v
T
... v
T
v
T k
v
T
... v
T
]
which is independent of
i
(but still dependent on
k
) may thus be deﬁned. Consequently
w
k
(
i
)
calculated with
R
ss
,k
instead of
R
ss
,k
(
i
)
does notdepend on
i
either and will be denoted by
w
k
. The equalizercomplexity is thus reduced since the calculation of theequalizer coefﬁcients
w
k
by (4)  which requires a matrixinversion  has to be performed only once for each user andno longer for each symbol. We will use this approximationthroughout the remainder of this paper.Exactly as in [1], [2] and [4], we assume that the estimate
ˆ
s
k
(
i
)
provided by the equalizer is the output of an equivalentAWGN channel having
s
k
(
i
)
as its input i.e.
ˆ
s
k
(
i
) =
µ
k
s
k
(
i
) +
ν
k
(
i
)
(7)where
ν
k
(
i
)
is a complex noise and
ν
k
(
i
)
∼ N
c
(0
,σ
2
ν
k
)
.Parameters
µ
k
and
σ
2
ν
k
may be easily calculated as follows[2] :
µ
k
=
w
H k
He
k
,
(8)
σ
2
ν
k
=
µ
k
−
µ
2
k
.
(9)Finally, the equalizer outputs extrinsic LLRs on the interleavedcoded bits
b
k
(
l
)
denoted by
L
EQe
(
b
k
(
l
))
. Assuming that thebits characterizing symbol
s
k
(
i
)
are denoted by
b
pk
(
i
) =
b
k
((
i
−
1)
q
+
p
)
(
p
= 1
,...,q
where
q
is the number of bits per symbol for the considered constellation
A
) and thatthey are independent from each other thanks to interleaving,
L
EQe
(
b
pk
(
i
))
is given [1] by
L
EQe
(
b
pk
(
i
)) =ln
s
k
(
i
):
b
pk
(
i
)=1
p
(ˆ
s
k
(
i
)

s
k
(
i
))
r
=1
,...,qr
=
p
P
a
(
b
rk
(
i
))
s
k
(
i
):
b
pk
(
i
)=0
p
(ˆ
s
k
(
i
)

s
k
(
i
))
r
=1
,...,qr
=
p
P
a
(
b
rk
(
i
))
,
(10)where
p
(ˆ
s
k
(
i
)

s
k
(
i
)) = 1
π σ
2
ν
k
exp
−
ˆ
s
k
(
i
)
−
µ
k
s
k
(
i
)

2
σ
2
ν
k
.
(11)
P
a
(
b
rk
(
i
))
are the a priori probabilities on bits
b
rk
(
i
)
calculated from the a priori LLRs at the equalizer input. Notation “
s
k
(
i
) :
b
pk
(
i
)
” represents the subset of symbols
s
k
(
i
)
in constellation
A
with a given value (0 or 1) for
b
pk
(
i
)
.III. M
ETHOD FOR PREDICTING THE MULTI

USERTURBO

EQUALIZER
/
DEMAPPER PERFORMANCE
In this section, a simple method for predicting the performance of the multiuser turboequalization/demapping schemewill be developed.The idea is to predict the behavior of the turboequalizer/demapper by solely looking separately at theinput/output relations of each constituent stage (equalizer/demapper, decoder). More precisely, the mean and variance of the extrinsic LLRs output by each of the two stageswill be determined as a function of the mean and the varianceof their input LLRs, called a priori LLRs. This will be donein the two following subsections : subsection IIIA will bedevoted to the equalizer/demapper and subsection IIIB to thedecoder. Interleaver and deinterleaver do not of course affectthose input/output relations. Gaussianity of the extrinsic LLRsoutput by the two stages will be assumed like in [4]. Its impacton the performance of our prediction method will be illustratedin section IV. Nevertheless, this Gaussian hypothesis is notneeded for calculating the equalizer behavior as it will beexplained later on in this section. It only reduces the numberof equalizer input parameters to be considered (mean andvariance of the distribution rather than the pdf itself). Bothof the following subsections will successively consider singleuser BPSK, singleuser QPSK and multiuser BPSK cases.Both of them will also conclude with a few words on how toextend the methodology to highorder constellations.
A. Predicting the equalizer/demapper behavior
In the remainder of this paper, we will refer to the “equalizer/demapper” simply as the “equalizer” for the sake of conciseness. Before going through the different cases, let
g
(
x
;
µ,σ
2
) = 1
√
2
π σ
2
exp
−
(
x
−
µ
)
2
2
σ
2
(12)denote the pdf of a Gaussian random variable with mean
µ
and variance
σ
2
. Let also
g
c
(
z
;
µ,σ
2
) = 1
π σ
2
exp
−
z
−
µ

2
σ
2
(13)
5
.
Equalizer
σ
2
n
H
L
EQa
(
b
(
i
))
∼N
(
µ
EQa
¯
b
(
i
)
,σ
2
,EQa
)
L
EQe
(
b
(
i
))
∼N
(
µ
EQe
¯
b
(
i
)
,σ
2
,EQe
)
.
Fig. 3. Inputs and output of the equalizer. Singleuser BPSK case.
denote the pdf of a complex Gaussian random variable, withmean
µ
and variance
σ
2
, whose real and imaginary partsare independent and have the same variance
σ
2
/
2
. Thesenotations will be often used in the sequel.
1) Singleuser BPSK case:
In the singleuser BPSK case,
K
= 1
,
q
= 1
and
M
= 1
(no spreading is needed). Inthe sequel we will thus leave out indexes
k
and
p
in all thenotations introduced in section II. Deﬁning
¯
b
(
i
)
2
b
(
i
)
−
1
(
¯
b
(
i
)
∈ {−
1
,
1
}
and
b
(
i
)
∈ {
0
,
1
}
)
; for BPSK,
¯
b
(
i
) =
s
(
i
)
),the a priori equalizer input LLRs
L
EQa
(
b
(
i
))
as well as theextrinsic equalizer output LLRs
L
EQe
(
b
(
i
))
on the interleavedcoded bits
b
(
i
)
are assumed to be Gaussian distributed
L
EQa
(
b
(
i
)) =
µ
EQa
¯
b
(
i
) +
n
EQa
(
i
)
,
(14)
L
EQe
(
b
(
i
)) =
µ
EQe
¯
b
(
i
) +
n
EQe
(
i
)
,
(15)with respective means
µ
EQa
and
µ
EQe
and respective variances
n
EQa
(
i
)
∼N
(0
,σ
2
,EQa
)
and
n
EQe
(
i
)
∼N
(0
,σ
2
,EQe
)
.As illustrated in Fig. 3,
the goal is to express the mean
µ
EQe
and the variance
σ
2
,
EQe
of the extrinsic LLRs asfunctions of the mean
µ
EQa
and the variance
σ
2
,
EQa
of the a priori LLRs as well as the noise variance
σ
2n
andthe channel.
The reasoning will assume long frames andnecessitate the calculation of an intermediary variable whichwill be named
˜
v
(
µ
EQa
,σ
2
,EQa
)
hereafter.It may be easily shown from the deﬁnition of LLRthat
˜
s
(
i
) = tanh(0
.
5
L
EQa
(
b
(
i
)))
and var
{
s
(
i
)
}
= 1
−
tanh
2
(0
.
5
L
EQa
(
b
(
i
)))
. Given this latter expression, (6) (leaving out index
k
) and (12), the mean of
v
calculated overthe Gaussian distribution of the equalizer input LLRs maybe expressed as (see appendix)
˜
v
(
µ
EQa
,σ
2
,EQa
) = 1
−
+
∞−∞
tanh
2
y
2
g
(
y
;
µ
EQa
,σ
2
,EQa
)
dy
(16)where
y
is an integration variable. This integral may becomputed numerically for any values of
µ
EQa
and
σ
2
,EQa
. If the pdf of the equalizer a priori LLRs is known and may notbe satisfactorily regarded as Gaussian,
˜
v
(
µ
EQa
,σ
2
,EQa
)
maybe calculated by incorporating this pdf in (16) rather than theGaussian density.Assuming long enough frames, the variance of
v
around itsmean may be neglected. Indeed,
v
is then by (6) (without index
k
) the sum of a great number of random variables var
{
s
(
i
)
}
(
i
= 1
,...,I
) which may be assumed to be independent thanksto interleaving
1
. Variable
v
may then be approximated by adeterministic value, namely its mean
˜
v
(
µ
EQa
,σ
2
,EQa
)
. Let now
1
var
{
s
(
i
)
}
= 1
−
tanh
2
(0
.
5
L
EQa
(
b
(
i
)))
are random variables since
L
EQa
(
b
(
i
))
are too.
˜R
ss
denote matrix
R
ss
when replacing
v
by
˜
v
(
µ
EQa
,σ
2
,EQa
)
,
˜w
denote vector
w
when replacing
R
ss
by
˜R
ss
,
˜
µ
and
˜
σ
2
ν
respectively denote
µ
and
σ
2
ν
when replacing
R
ss
and
w
by
˜R
ss
and
˜w
. Given
ˆ
s
(
i
) =
µs
(
i
) +
ν
(
i
)
, (10)  which gives
L
EQe
(
b
(
i
)) = 2
µσ
2
ν
ˆ
s
(
i
)
(17)in BPSK  and the previous deﬁnitions which consider
µ
and
σ
2
ν
as constants
˜
µ
and
˜
σ
2
ν
, the mean and variance of theextrinsic LLRs output by the equalizer are given by
µ
EQe
E
{
L
EQe
(
b
(
i
))

b
(
i
) = 1
}
= 2 ˜
µ
˜
σ
2
ν
E
{
ˆ
s
(
i
)

b
(
i
) =
s
(
i
) = 1
}
= 2 ˜
µ
2
˜
σ
2
ν
,
(18)
σ
2
,EQe
var
{
L
EQe
(
b
(
i
))
}
=
2 ˜
µ
˜
σ
2
ν
2
var
{
ˆ
s
(
i
)
}
= 4 ˜
µ
2
˜
σ
2
ν
= 2
µ
EQe
.
(19)As
˜
µ
and
˜
σ
2
ν
depend on
˜
v
(
µ
EQa
,σ
2
,EQa
)
and on
σ
2
n
via
˜w
,
µ
EQe
and
σ
2
,EQe
are functions of
µ
EQa
,
σ
2
,EQa
and
σ
2
n
. Theequalizer behavior may thus totally be predicted by thepreceding calculations since we have expressed the mean andthe variance of its output LLRs as functions of the mean andvariance of its input LLRs, the noise variance and the channelmatrix. The computational complexity of these calculations isvery low since the sizes of vectors and matrices involved incalculations of
µ
EQe
and
σ
2
,EQe
are small and independent of the frame length.
2) Singleuser QPSK case:
In the singleuser QPSK case,
K
= 1
,
q
= 2
and
M
= 1
(no spreading is needed). In thesequel we will thus leave out indexes
k
in the notations butnot index
p
which can take on values 1 or 2. Deﬁning
¯
b
p
(
i
)
2
b
p
(
i
)
−
1
(
p
= 1
,
2
,
¯
b
p
(
i
)
∈ {−
1
,
1
}
and
b
p
(
i
)
∈ {
0
,
1
}
)
,the a priori equalizer input LLRs
L
EQa
(
b
1
(
i
))
and
L
EQa
(
b
2
(
i
))
as well as the extrinsic equalizer output LLRs
L
EQe
(
b
1
(
i
))
and
L
EQe
(
b
2
(
i
))
on the interleaved coded bits
b
1
(
i
)
and
b
2
(
i
)
are assumed to be Gaussian distributed. Whereas it may beassumed that both
L
EQa
(
b
1
(
i
))
and
L
EQa
(
b
2
(
i
))
have the samemean and variance
L
EQa
(
b
p
(
i
)) =
µ
EQa
¯
b
p
(
i
) +
n
EQa,p
(
i
)
p
= 1
,
2
(20)with
n
EQa,p
(
i
)
∼ N
(0
,σ
2
,EQa
)
, simulations reveal that
L
EQe
(
b
1
(
i
))
and
L
EQe
(
b
2
(
i
))
calculated from (10) have ingeneral
2
different means and variances
L
EQe
(
b
p
(
i
)) =
µ
EQe,p
¯
b
p
(
i
) +
n
EQe,p
(
i
)
p
= 1
,
2
with
n
EQe,p
(
i
)
∼ N
(0
,σ
2
,EQe,p
)
. This will be intuitively explained in the appendix and illustrated in section IV.As illustrated in Fig. 4,
the goal is to express the means
µ
EQe
,
1
,
µ
EQe
,
2
and the variances
σ
2
,
EQe
,
1
,
σ
2
,
EQe
,
1
of the extrinsicLLRs as functions of the mean
µ
EQa
and the variance
σ
2
,
EQa
of the a priori LLRs as well as the noise variance
σ
2n
and the channel.
Exactly as for BPSK, the reasoningwill assume long frames and necessitate the calculation of
2
It depends on the choice of constellation mapping.