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Maximum Likelihood Turbo Iterative Channel Estimation for Space-Time Coded Systems and Its Application to Radio Transmission in Subway Tunnels

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Maximum Likelihood Turbo Iterative Channel Estimation for Space-Time Coded Systems and Its Application to Radio Transmission in Subway Tunnels
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  EURASIP Journal on Applied Signal Processing 2004:5, 727–739c  2004 Hindawi Publishing Corporation MaximumLikelihoodTurboIterativeChannelEstimationforSpace-TimeCodedSystemsandItsApplicationtoRadioTransmissioninSubwayTunnels MiguelGonz´alez-L´opez Departamento de Electr´onica y Sistemas, Universidade da Coru˜ na, Campus de Elvi˜ na s/n, 15071 A Coru˜ na, Spain Email: miguelgl@udc.es  Joaqu´ınM´ıguez Departamento de Electr ´onica y Sistemas, Universidade da Coru˜ na, Campus de Elvi˜ na s/n, 15071 A Coru˜ na, SpainEmail: jmiguez@udc.es LuisCastedo Departamento de Electr´onica y Sistemas, Universidade da Coru˜ na, Campus de Elvi˜ na s/n, 15071 A Coru˜ na, Spain Email: luis@udc.esReceived 31 December 2002; Revised 31 July 2003 This paper presents a novel channel estimation technique for space-time coded (STC) systems. It is based on applying the max-imum likelihood (ML) principle not only over a known pilot sequence but also over the unknown symbols in a data frame. Theresulting channel estimator gathers both the deterministic information corresponding to the pilot sequence and the statisticalinformation, in terms of a posteriori probabilities, about the unknown symbols. The method is suitable for Turbo equalizationschemes where those probabilities are computed with more and more precision at each iteration. Since the ML channel estimationproblem does not have a closed-form solution, we employ the expectation-maximization (EM) algorithm in order to iteratively compute the ML estimate. The proposed channel estimator is first derived for a general time-dispersive MIMO channel and thenis particularized to a realistic scenario consisting of a transmission system based on the global system mobile (GSM) standardperforming in a subway tunnel. In this latter case, the channel is nondispersive but there exists controlled ISI introduced by theGaussian minimum shift keying (GMSK) modulation format used in GSM. We demonstrate, using experimentally measuredchannels, that the training sequence length can be reduced from 26 bits as in the GSM standard to only 5 bits, thus achieving a14% improvement in system throughput. Keywords and phrases:  STC, turbo equalization, turbo channel estimation, maximum likelihood channel estimation, GSM, sub-way tunnels. 1. INTRODUCTION Recently, the so-called Turbo codes [1, 2, 3] have revealed themselves as a very powerful coding technique able to ap-proach the Shannon limit in AWGN channels. A Turbo codeis made up of two component codes (block or convolutional)parallely or serially concatenated via an interleaver. This sim-ple coding scheme produces very long codewords, so eachsource information bit is highly spread through the trans-mitted coded sequence. At reception, optimum maximumlikelihood (ML) decoding can be carried out by consideringthe hypertrellis associated with the concatenation of the twocomponent codes. Obviously, such a decoding approach be-comes impractical in most situations. The key idea behindTurbo coding is to overcome this problem by employing asuboptimal, but very powerful, decoding scheme termed  it-erative maximum a posteriori  (MAP) decoding [3, 4]. Basi- cally, the method relies on independently decoding each of the component codes and exchanging in an iterative fashionthe statistical information, that is, the a posteriori probabili-ties about symbols, obtained in each decoding module.The same decoding principle has also been successfully applied, under the term  Turbo equalization  [5], to e ff  ec-tively compensate the ISI induced by the channel and/or the  728 EURASIP Journal on Applied Signal Processingmodulation scheme. This technique exploits the fact that ISIcan be viewed as a form of rate-1, nonrecursive coding. So,whatever coding scheme is used, if an interleaver is locatedprior to the channel, the overall e ff  ect of coding and ISIcan be treated as a concatenated code and therefore, itera-tive MAP decoding can be applied. Luschi et al. [6] presentan in-depth review of this technique and further improve-ments can be found in [7, 8, 9, 10]. In general, iterative MAP processing can be applied to a variety of situations where theoverall system can be viewed as a concatenation of moduleswhose input/output relationship can be described as a (hid-den) Markov chain. Several works have appeared in the last years exploiting this idea. For instance, G¨ortz [11], Garcia-Frias and Villasenor [12], and Guyader et al. [13] worked on the problem of joint source-channel decoding and Zhangand Burr [14] addressed the problem of symbol timing re-covery.In practical receivers, where the channel impulse re-sponse has to be estimated, it is convenient to have chan-nel estimators capable of benefiting from the high perfor-mance of Turbo equalizers [15, 16, 17]. Moreover, second- and third-generation mobile standards consider the trans-mission of pilot sequences known by the receiver for channelestimation purposes. In the  global system mobile  (GSM) stan-dard, this sequence is 26 bits long, which represents 17 . 6%of the total frame length (148 bits) [18]. Such a long train-ing sequence is necessary if classical estimation techniques,such as least squares (LS), are used. Employing more re-fined channel estimators, such as the one presented in thispaper, we can dramatically decrease the necessary length of the training sequence and therefore increase the overall sys-tem throughput. In [19], an ML-based channel estimator ispresented where the ML principle is applied not only to thepilot sequence, but also to the whole data frame. Since the in-volved optimization problem had no analytical solution, theexpectation-maximization (EM) algorithm [20] was used foriteratively obtaining the solution.Also, wireless communications research has been very in-fluenced by the discovery of the potentials of communicatingthrough multiple-input multiple-output (MIMO) channels,which can be carried out using antenna diversity not only at reception, as classical space-diversity techniques have beendoing, but also at transmission. MIMO techniques have theadvantage to provide high data rate wireless services at noextra bandwidth expansion or power consumption. Telatar[21] calculated the capacity associated with a MIMO chan-nel that in certain cases grows linearly with the number of antennas [22]. More recent progress in information theoret-ical properties of multiantenna channel can be found in [23].Although MIMO channel capacity can be really high,it can only be successfully exploited by proper coding andmodulation schemes. The term space-time Coding (STC)[24, 25] has been adopted for such techniques. Special ef- forts have been made in code design [24, 26] and several de- coding approaches have been developed for these codes. Inboth fields, the Turbo principle has been applied in profu-sion. TurboSTcodesdesigns canbefoundin [27,28,29]and various Turbo decoding schemes are exposed in [30, 31]. As in single-antenna systems, practical ST receivers mustperform the operation of channel estimation. Having e ffi -cient and robust estimators is crucial to guarantee that thesystem performance degradation due to the channel estima-tion error is minimized. In this paper, we present a novelchannel estimation technique that gathers both the deter-ministic information corresponding to the pilot sequenceand the statistical information, in terms of a posteriori prob-abilities, about the unknown symbols. The method is suit-able for Turbo equalization schemes where those probabili-ties are computed with more and more precision at each it-eration. We derive the channel estimator for general MIMOtime-dispersive channels and analyze its performance in amultiple-antenna communication system based on the GSMstandard operating inside subway tunnels.The main motivation for developing a multiple-antennaGSM-based communication system is the following. GSMis, by far, the most widely deployed radio-communicationsystem. Since 1993, its radio interface (GSM-R) has beenadopted by the European railway digital radio-communic-ation systems. Due to the conservative nature of its market,it is expected that railway radio-communication systems willemploy GSM-R for the long-term future. For this reason,when subway operators wish to deploy advanced, high datarate, digital services for security or entertainment purposes,it is very likely that they will prefer to increase the capac-ity of the existing GSM-R system rather than switch to an-other radio standard. STC and Turbo equalization are very promising ways of achieving this capacity growth [32]. Inthis specific application, we will show that the proposed it-erative MLMIMO channel estimation method has large ben-efits over traditional channel estimation approaches.The rest of the paper is organized as follows. Section 2presents the signal model and Section 3 describes the Turboequalization scheme for STC systems. Next, in Section 4, wederivetheMLchannelestimatorforageneraltime-dispersiveMIMO channel. Since direct application of the ML principleleads to an optimization problem without closed-form solu-tion, the EM algorithm is applied for computing the actualvalue of the solution, resulting in the so-called ML-EM es-timator. The application of the proposed channel estimatorto a STC GSM-based system operating in subway tunnels isdetailed in Section 5. Section 6 presents the results of com- puter experiments for both the general case and experimen-tal measurements of subway tunnel MIMO channels. Finally,Section 7 is devoted to the conclusions. 2. SIGNALMODEL We consider the transmitter signal model corresponding toan STC system shown in Figure 1. The srcinal bit sequence u ( k ) feeds an ST encoder whose output is a sequence of vectors  c ( k )  =  [ c 1 ( k )  c 2 ( k )  ···  c N  ( k )] T  , with  N   beingthe number of transmitting antennas. The specific spatio-temporal structure of the sequence of vectors  c ( k ) dependsontheparticularSTCtechniqueemployed.AnyoftheseveralSTC methods that have been proposed in the literature couldbe used in our scheme. However, we have focused on ST  ML Turbo Iterative Channel Estimation for STC Systems 729 s N  ( t  ; b N  )Mod. b N  ( k ) π c N  ( k )STcoder u ( k ) ...... s 2 ( t  ; b 2 )Mod. b 2 ( k ) π c 2 ( k ) s 1 ( t  ; b 1 )Mod. b 1 ( k ) π c 1 ( k ) Figure  1: Transmitter model. trellis codes [24, 25] to elaborate our simulation results. Each component of   c ( k ) is independently interleaved to produce anew symbol vector  b ( k )  =  [ b 1 ( k )  b 2 ( k )  ···  b N  ( k )] T  andthese are the symbols that are afterwards modulated (wave-form encoded) to yield the signals  s i ( t  ; b i )  i  =  1,2, ... , N  that will be transmitted along the radio channel. Withoutloss of generality, we will assume that the modulation formatis linear and that the channel su ff  ers from time-dispersivemultipath fading with memory length  m . It is well knownthat at reception, matched-filtering and symbol-rate sam-pling can be used to obtain a set of su ffi cient statistics forthe detection of the transmitted symbols. Using vector nota-tion, this set of statistics will be grouped in vectors  x  ( k )  = [ x  1 ( k )  x  2 ( k )  ···  x  L ( k )] T  ,  k  =  0,1, ... , K   − 1, where  L  isthe number of receiving antennas and  K   is the number of to-tal transmitted symbol vectors in a data frame. Elaboratingthe signal model, it can be easily shown that the su ffi cientstatistics  x  ( k ) can be expressed as x  ( k ) = Hz ( k ) +  v  ( k ), (1)where matrix   H  =  [ H  ( m − 1)  H  ( m − 2)  ···  H  (0)] rep-resents the overall dispersive MIMO channel with memory length  m . Each submatrix  H  ( i ) =  h 11 ( i )  h 12 ( i )  ···  h 1 N  ( i ) h 21 ( i )  h 22 ( i )  ···  h 2 N  ( i )...... ... ... h L 1 ( i )  h L 2 ( i )  ···  h LN  ( i )  (2)contains the fading coe ffi cients that a ff  ect the symbol vector b ( k − i ). Vector  z ( k ) results from stacking the source vectors b ( k ), that is, z ( k ) = [ b T  ( k − m  + 1)  b T  ( k − m  + 2)  ···  b T  ( k ) T  ] .  (3)Finally, the noise component  v  ( k ) is a vector of mutually in-dependent complex-valued, circularly symmetric Gaussianrandom processes, that is, the real and imaginary parts arezero-mean, mutually independent Gaussian random pro-cesses having the same variance. We will also assume that thenoise is temporally white with variance  σ  2 v  . 3. STTURBODETECTION Figure 2 shows the block diagram of an ST Turbo de-tector. The MAP equalizer [4] computes  L [ b ( k ) | ˜ x  ] whichare the  a posteriori  log-probabilities of the input sym-bols  b ( k ) based on the available observations ˜ x   = [ x  T  (0)  x  T  (1)  ···  x  ( K   − 1)] T  . Due to its time-dispersivenature, it is convenient to represent our MIMO channel by means of a finite-state machine (FSM) having 2 N  ( m − 1) states.This FSM has 2 N  transitions per state which implies thatthere is a total number of 2 Nm transitions between two timeinstants. Let  e k  = ( s k − 1 , b ( k ), s ( k ), s k ) be one of the 2 Nm pos-sible transitions at time  k  of this FSM. This transition de-pends on four parameters: the incoming state  s k − 1 , the out-going state  s k , the input symbol vector  b ( k ), and the outputsymbol vector without noise  s ( k ) = Hz ( k ). It is important topoint out that the incoming state is determined by the  m − 1previous symbol vectors, that is,  s k − 1  = ( b ( k − m  + 1), b ( k − m  + 2), ... , b ( k − 1)). On the other hand, the outgoing stateis a function of the previous state and the current input sym-bols, that is,  s k  =  f  next ( s k − 1 , b ( k )). For a better description of the MAP equalizer, we are going to introduce the notation b ( k ) = L in ( e k ) and  s ( k ) = L out ( e k ) to represent the input andoutput symbol vectors associated to the transition  e k , respec-tively. Note that the output vector does not depend on theoutgoing state  s k , so we will slightly change our notation andwrite s ( k ) = L out  e k  = L out  s k − 1 , b ( k )  = L out  z ( k )  = Hz ( k ) . (4)The aposteriori log-probabilities L [ b ( k ) | ˜ x  ]canberecursively computed by means of the Bahl-Cocke-Jelinek-Raviv (BCJR)algorithm [3, 4] which is summarized in the sequel. The first stage when computing the  a posteriori  log-probabilities isnoting that L  b ( k ) | ˜ x   = L  b ( k ), ˜ x   +  h b , (5)where  h b  is the constant that makes  P  [ b ( k ) | ˜ x  ] a probability mass function and L  b ( k ), ˜ x   = log  e k : L in ( e k ) = b ( k ) exp L  e k , ˜ x    (6)is the joint log-probability of the transition  e k  and the setof available observations ˜ x  . This joint log-probability can beexpressed as L  e k , ˜ x   = α k − 1  s k − 1  +  γ k  e k  +  β k  s k  , (7)where α k [ s ] = L  s k − 1 , ˜ x  − k  , γ k  e k  = L  b ( k )  +  L  x  ( k ) | s ( k )  ,  β k [ s ] = L  ˜ x  + k | s k  ,(8)  730 EURASIP Journal on Applied Signal Processing Decision L [ u ( k ); I  ] L [ u ( k ); O ] L [ c ( k ); O ]MAPSTDEC − L [ u ( k ); I  ] L [ c ( k ); I  ] π π  − 1 − L [ b ( k ) | ˜ x  ] L [ b ( k )]Channelestimator L [ z ( k ) | ˜ x  ] ˆH MAPSTEQ x  ( k )MF Figure  2: Receiver model. with L  x  ( k ) | s ( k )  =−  1 σ  2 v   x  ( k ) − Hz ( k )  2 , (9)˜ x  − k  =  x  T  (0)  x  T  (1)  ···  x  T  ( k − 1)  , (10)˜ x  + k  =  x  T  ( k  + 1)  x  T  ( k  + 2)  ···  x  T  ( K   − 1)  .  (11)Note that the noise variance  σ  2 v   is needed in (9). Our simu-lation results assume this parameter as known. However, itcould be estimated and, in particular, it can be consideredas another parameter to be estimated by the ML estimatordescribed in Section 4, as shown in [33], for the case of a de- cision feedback-equalizer (DFE) instead of a MAP detector.The computation of the quantities  α k [ s ],  γ k [ e k ], and  β k [ s ]can be carried out recursively by first performing a forwardrecursion α k − 1  s k − 1  = log  b ( k ), s k − 2 :  f  next ( s k − 2 , b ( k − 1)) = s k − 1 exp  α k − 2  s k − 2  +  L  b ( k − 1)  +  L  x  ( k ) | s ( k )   (12)with initial values  α 0 [ s  =  0]  =  0 and  α 0 [ s  =  0]  = −∞ , andthen proceeding with a backward recursion  β k  s k  = log  b ( k +1), s k +1 :  f  next ( s k , b ( n +1)) = s k +1 exp   β k +1  s k +1  +  L  b ( k  + 1)  +  L  x  ( k  + 1) | s ( k  + 1)  (13)using as initial values  β K  − 1 [ s  =  s K  − 1 ]  =  0 and  β K  − 1 [ s  = s K  − 1 ] =−∞ .Similarly, thedecoder hasto computethe  a posteriori  log-probabilities of the srcinal symbols  L [ u ( k ); O ] from their apriori log-probabilities  L [ u ( k ); I  ]  =  log(0 . 5) and the a pri-orilog-probabilities L [ c ( k ); I  ]whichcomefromthedetector.Again, the BCJR algorithm applies [3, 4]. It also computes the  a posteriori  log-probabilities of the transmitted symbols L [ c ( k ); O ] using L  c ( k ); O  = log  e k : L out ( e k ) = c ( k ) exp  α k − 1  s k − 1  +  γ k  s k  +  β k  s k  , (14)where L [ c ( k ); I  ]isutilizedasbranchmetric.Thesecomputedlog-probabilities are then fed back to the detector to act asthe  a priori  log-probabilities  L [ b ( k )]. As reflected in Figure 2,notethatitisalwaysnecessarytosubtractthe apriori compo-nent from the computed log-probabilities before forwardingthem to the other module in order to avoid statistical depen-dence with the results of the previous iteration. 4. MAXIMUMLIKELIHOODCHANNELESTIMATION Channel estimation is often mandatory when practically im-plementing ST detection strategies, unless we deal with somekind of blind processing techniques. In this section, we willpresent a novel channel estimation method that will enableus to take full advantage from the Turbo detection schemepresented in the Section 3.When developing our channel estimation approach,we will exploit the fact that transmitted data frames inmost practical systems contain a deterministic known pi-lot sequence of length  M   for the purpose of estimatingthe channel at reception. For instance, in GSM, this se-quence is  M   =  26 bits long [18]. Let ˜ b  f   =  [˜ b T t   ˜ b T  ] T  denote the overall data frame, which includes ˜ b t   = [ b T t   (0)  b T t   (1)  ···  b T t   (  M  − 1)] T  as the training sequenceand ˜ b  =  [ b T  (  M  )  b T  (  M   + 1)  ···  b T  ( K   − 1)] T  as the in-formation sequence. Analogously, ˜ x   f   =  [˜ x  T t   ˜ x  T  ] T  are theobservations corresponding to one data frame, where ˜ x  t   = [ x  T t   (0)  x  T t   (1)  ···  x  T t   (  M  − 1)] T  represents the pilot se-quence and ˜ x   =  [ x  (  M  )  x  (  M   + 1)  ···  x  ( K   − 1)] T  corre-sponds to the information sequence. The ML estimator isthus given by   H = argmax  H  f  ˜ x  | ˜ b t  ; H (˜ x  ), (15)where  f  ˜ x  t  | ˜ b t  ; H  is the probability density function (pdf) of theobservations conditioned on the available information (thetraining sequence  b t  ) and the parameters to be estimated  ML Turbo Iterative Channel Estimation for STC Systems 731(the channel matrix   H ). Although, this is a problem with-out closed-form solution, the EM algorithm [20] can be em-ployed to iteratively solve (15). The EM algorithm relies on defining a so-called “complete data” set formed by the ob-servable variables and by additional unobservable variables.At each iteration of the algorithm, a more refined estimateis computed by averaging the log-likelihood of the completedata set with respect to the pdf of the unobservable vari-ables conditioned on the available set of observations. Us-ing the EM terminology, we define the union of the observa-tions (which are the observable variables) and the transmit-ted bit sequence (which are the unobservable variables) ˜ x  e  = [˜ b T  f   ˜ x  T  f   ] T  as the complete data set, whereas the observations˜ x   f   are the incomplete data set. The relationship between ˜ x  e and ˜ x   f   must be given by a noninvertible linear transforma-tion, that is, ˜ x   f   =  T ˜ x  e . It can be easily seen that in our case,thistransformationisgivenby  T = [ 0 L (  M  + K  ) × N  (  M  + K  ) I L (  M  + K  ) ].With these definitions in mind, the estimate of the channel atthe  i  + 1th iteration is obtained by solving  H i +1  = argmax  H E ˜ x  e | ˜ x   f   ,˜ b t  ;  H i  log  f  ˜ x  e | ˜ b t  ; H  ˜ x  e  , (16)where  E  f  {·}  denotes the expectation operator with respectto the pdf   f   ( x  ). Expanding the previous expression, we have  H i +1  = argmax  H E ˜ b | ˜ x  ;  H i  log   f  ˜ x   f  | ˜ b  f   ; H  ˜ x   f    f  ˜ b (˜ b )  = argmax  H E ˜ b | ˜ x  ;  H i  log   f  ˜ x  t  | ˜ b t  ; H  ˜ x  t    f  ˜ x  | ˜ b ; H (˜ x  )  = argmax  H log  f  ˜ x  t  | ˜ b t  ; H  ˜ x  t   +  E ˜ b | ˜ x  ;  H i  log  f  ˜ x  | ˜ b ;  H (˜ x  )  = argmin H  M  − 1  k = 0  x  t  ( k ) − Hz t  ( k )  2 +  E ˜ b | ˜ x  ;  H i  K  − 1  k =  M   x  ( k ) − Hz ( k )  2  ,(17)where the last equality follows from the fact that, as far as weassume AWGN, the pdf of the observations conditioned onthe transmitted symbols  f  ˜ x  | ˜ b ;  H i is Gaussian. This leads to thefollowing quadratic optimization problem:  H i +1  = argmin H  M  − 1  k = 0  x  t  ( k ) − Hz t  ( k )  2 + K  − 1  k =  M  E z ( k ) | ˜ x  ;  H i  x  ( k ) − Hz ( k )  2   (18)with the closed-form solution 1  H i +1  =  R xz  , t   +  R xz   ×  R z  , t   +  R z   − 1 , (19) 1 Since the expectation operator is linear, the derivation leading to (19)follows, step by step, the usual optimization procedure to find the LS es-timate of a linear system given a set of noisy observations (see, e.g., [34]).Such a procedure includes the calculation of the gradient with respect to thesystem coe ffi cients and then solving for the points where the gradient van-ishes. Hence, solving (17) is tedious, since derivatives have to be computed for the coe ffi cients in matrix   H , but conceptually straightforward. where R xz  , t   =  M  − 1  k = 0 x  t  ( k ) z H t   ( k ), (20) R z  , t   =  M  − 1  k = 0 z t  ( k ) z H t   ( k ), (21) R xz   = K  − 1  k =  M  E z ( k ) | ˜ x  ;  H i  x  ( k ) z H  ( k )  , (22) R z   = K  − 1  k =  M  E z ( k ) | ˜ x  ;  H i  z ( k ) z H  ( k )  .  (23)Note that for computing (22) and (23), it is necessary to know the probability mass function  p z ( k ) | ˜ x  ;  H i . Towards thisaim, we take benefit from the Turbo equalization process be-cause L  z ( k ) | ˜ x  ;   H i  = L  z ( k ), ˜ x  ;   H i  +  h z   = L  e k , ˜ x   +  h z  , (24)where  h z   is the constant that makes  p z ( k ) | ˜ x  ;  H i a probability mass function and  L [ e k , ˜ x  ] is the joint log-probability of thetransition e k  andthesetofavailableobservations.Noticethatthis quantity has already been computed in the Turbo equal-ization process (see (7)). This fact makes the proposed chan-nel estimator very suitable to be used within a Turbo equal-ization structure. 5. APPLICATIONTOANSTCSYSTEMFORSUBWAYENVIRONMENTS We focus now on the application of the ML-EM channel esti-mator described in Section 4 to an STC GSM-like system forunderground railway transportation systems. Some practicalconsiderationsfollow.Insubwaytunnelenvironments,prop-agation conditions result in flat multipath fading because itsdelay spread is small when compared to the GSM symbolperiod [35]. Nevertheless, the modulation employed by theGSM standard, Gaussian minimum shift keying (GMSK),induces controlled ISI and thus Turbo ST Equalization canbe employed for the purpose of joint demodulating and de-coding. In addition, experimental measurements [36] haverevealed that in this environment, there exist strong spatialcorrelations between subchannels. These spatial correlationswill be taken into account when evaluating the receivers’performance in the following section because we will use,in the computer simulations, experimental measurements of MIMO channel impulse responses obtained in subway tun-nels. These field measurements have been carried out in theframework of the European project “ESCORT” [37]. We willshow how the proposed channel estimator allows to reducethe necessary length of the training sequence from 26 bits inthe GSM standard up to only 5 bits, while performance ismaintained very close to the optimum (i.e., the bit error rate(BER) obtained when the channel is perfectly known at re-ception) which clearly implies a very high gain in the overallsystem throughput.
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