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A Semi-Analytical Method for Predicting the Performance and Convergence Behavior of a Multiuser Turbo-Equalizer/Demapper

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A Semi-Analytical Method for Predicting the Performance and Convergence Behavior of a Multiuser Turbo-Equalizer/Demapper
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  1 A semi-analytical method for predictingthe performance and convergence behaviorof a multi-user turbo-equalizer/demapper Val´ery Ramon*, C´edric Herzet, Luc VandendorpeCommunications and Remote Sensing Laboratory, Universite catholique de Louvain (UCL)2, place du Levant, B-1348 Louvain-la-Neuve, Belgium { ramon, herzet, vandendorpe } @tele.ucl.ac.be  Abstract —This paper proposes a simple semi-analyticalmethod with reduced simulation time for predicting at any itera-tion the performance of a turbo-equalization/demapping schemeusing the Wang and Poor soft-in/soft-out (SISO) Minimum MeanSquare Error (MMSE) / Interference Cancellation (IC) equalizerand a SISO decoder. The proposed method may be applied tomulti-level/phase data modulations as well as multi-user 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 Turbo-equalization/demapping (references in [1]) is a pow-erful mean to perform joint equalization/demapping and de-coding when considering coded data transmission over fre-quency selective channels. The association of the code and thediscrete-time 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 soft-in/soft-out (SISO) equalizer/demapper and aSISO decoder.The goal of this paper is to predict the performance andconvergence behavior of such a turbo-equalization/demappingscheme at any iteration using a simple semi-analytical method.The intention is to replace simulations of some part of thereceiver chain, namely the equalizer, by calculations. The pro-posed prediction method may be applied to multi-level/phasedata modulations as well as multi-user contexts. Nevertheless,for the sake of clarity, only simple cases will be consideredin this paper. Three cases of gradually increasing difficulty,namely single-user Binary Phase-Shift Keying (BPSK), single-user Quadrature Phase-Shift Keying (QPSK) and multi-userBPSK, will thus be successively developed. This way to pro-ceed 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 multi-user context. This multiuser equalizeris suboptimal but it exhibits reasonable complexity whileoffering interesting performance [2]. It was extended notablyin [1] to multi-level/phase data modulation, in [3] to multiple- input multiple-output (MIMO) systems and simplified in [4]in order to deal with time-invariant 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 (low-density 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) concate-nated codes have been previously proposed. They are basede.g. on variance or signal-to-noise 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 non-cancelled multi-user interference. Reference [9] used the so-called unscented transformation for calculating the mean of the Log-Likelihood Ratios (LLRs) output by the MMSE/ICequalizer. Nevertheless, this approach has been developed onlyin the single-user 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 cross-entropy minimization techniques as well as a prac-tical implementation of this algorithm. A theoretical analysisshowed that, with this approach, multiuser detection/decodingis capable of achieving optimal single-user performance for  2 user correlations approaching one. For our part, we alreadypresented in [11] an overview of the prediction method devel-oped hereafter but only in the single-user 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 simula-tion) 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 sum-product decoding algorithmsin the general setup of decoding of graph-based codes. Refer-ences [4] and [6], when drawing their EXIT charts respectively for turbo-equalization 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 Gaus-sian noise (AWGN) channel or with perfect knowledge of apriori information at the equalizer input may then also bedirectly obtained.In the single-user 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. Sec-tion II will introduce the system model. Our method for pre-dicting the performance of the turbo-equalization/demappingscheme will be explained in section III. This section willsuccessively consider single-user BPSK, single-user QPSKand multi-user BPSK cases. Section IV will compare fordifferent channels the performance obtained with our pre-diction method to that obtained by plain simulations of theturbo-equalizer/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 II-A), theoverall iterative receiver (subsection II-B) and the equationsof the considered MMSE/IC equalizer (subsection II-C) willbe successively presented for the most general case :  K   userspropagating through frequency-selective channels and usingmulti-level/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 Direct-Sequence Code Division Multiple Access (DS-CDMA) com-munication system with spreading factor  M  . The block di-agram of the transmitter-end 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 frequency-selective 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, chip-matched filteringand chip-rate sampling are successively performed. Each  k -th user composite channel may thus be represented by itsequivalent discrete-time model. This one includes the effectof the  k -th user spreading, the  k -th user channel itself, thechip-matched filtering and the chip-rate sampling. It resultsin a causal discrete-time filter with chip-rate 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 illus-trated in Fig. 2 and is more described in [1], [2] or [4]. It consists of two stages : a SISO multi-user equalizer/demapperfollowed by  K   parallel single-user SISO channel decoders.The two stages are separated by bit-deinterleavers and in-terleavers. 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 well-known 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 : convolutionally-coded DS-CDMA system with K users transmitting through a frequency-selective channel. . multi−userinterleaverdeinterleaverdeinterleaverinterleaveruser 1user Kequalizer/demapperchip−ratesamplingchip−matchedfilteringdecoderdecoder r ( t )  r ( iM   +  m ) . Fig. 2. Iterative receiver : soft-in soft-out (SISO) multi-user equalizer/demapper followed by K parallel single-user 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 sliding-window 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 single-user 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 so-called 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 )  calcu-lated from the a priori LLRs at the equalizer input. Nota-tion “ 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 perfor-mance of the multi-user turbo-equalization/demapping schemewill be developed.The idea is to predict the behavior of the turbo-equalizer/demapper by solely looking separately at theinput/output relations of each constituent stage (equal-izer/demapper, decoder). More precisely, the mean and vari-ance 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 III-A will bedevoted to the equalizer/demapper and subsection III-B to thedecoder. Interleaver and de-interleaver 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 single-user BPSK, single-user QPSK and multi-user BPSK cases.Both of them will also conclude with a few words on how toextend the methodology to high-order constellations.  A. Predicting the equalizer/demapper behavior  In the remainder of this paper, we will refer to the “equal-izer/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. Single-user 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) Single-user BPSK case:  In the single-user 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) (leav-ing 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) Single-user QPSK case:  In the single-user 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 ex-plained 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.
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