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Joint Data Detection and Channel Estimation for Fading Unknown Time-Varying Doppler Environments

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Joint Data Detection and Channel Estimation for Fading Unknown Time-Varying Doppler Environments
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  IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010 2277 Joint Data Detection and Channel Estimation forFading Unknown Time-VaryingDoppler Environments Usa Vilaipornsawai and Harry Leib  Abstract —This work considers a joint channel estimationand data detection technique for Multiple Space-Time TrellisCodes (MSTTCs) operating over unknown time-varying channelswith large Doppler spread. We propose an algorithm, calledDoppler Adaptive Smoothed Data Detection and Kalman Es-timation (DA-SDD-KE), that jointly detects data and estimatesthe channel as well as the time-varying Doppler. In this scheme,an Adaptive Kalman Predictor (AKP) consisting of a KP anda covariance-based Doppler estimator is incorporated into aPer-Survivor Processing (PSP)-based algorithm that utilizes thepast, present and future received symbols for smoothed datadetection. For comparison purposes, we also develop a DopplerAdaptive version of the Delayed Mixture Kalman Filtering(DMKF) technique, referred to as DA-DMKF, where the adaptiveestimations of the channel and the Doppler shift are basedon sequences of importance samples. Moreover, we propose amodel for generating a Rayleigh fading process with time-varyingDoppler using the sum of sinusoids method. The performance of the DA-SDD-KE and DA-DMKF algorithms over channels withconstant, linear and quadratic Doppler functions is evaluatedusing computer simulations, revealing that the DA-SDD-KEalgorithm performs well for all considered Doppler functions,and provides a considerably gain over the DA-DMKF algorithm.  Index Terms —Per-survivor processing, Kalman estimation,decoding, channel estimation, Doppler estimation, space-timecoding. I. I NTRODUCTION S PACE-TIME Trellis Codes (STTCs) [1], while crucial tomodern wireless communications, are designed under theassumption of perfect Channel State Information (CSI) at thereceiver. In practice, this assumption is unrealistic and CSIhas to be estimated. One common method to achieve this taskis to use periodically transmitted pilot symbols for channelestimation [2]. This approach, however, leads to loss in powerand bandwidth ef  Þ ciencies, especially for fast time-varying(high Doppler) fading channels [3]. Paper approved by T. M. Duman, the Editor for Coding Theory andApplications of the IEEE Communications Society. Manuscript receivedAugust 27, 2009; revised January 3, 2010.U. Vilaipornsawai was with the Department of Electrical & ComputerEngineering, McGill University, Montreal, Quebec, Canada, H3A 2A7. Sheis now a research fellow at the Institute for Systems and Robotics, Uni-versity of Algarve, Campus de Gambelas, Faro, Portugal, 8005-139 (e-mail:usa.vilaipornsawai@mail.mcgill.ca).H. Leib is with the Department of Electrical & Computer Engineer-ing, McGill University, Montreal, Quebec, Canada, H3A 2A7 (e-mail:harry.leib@mcgill.ca).Digital Object Identi Þ er 10.1109/TCOMM.2010.08.090511 Joint channel estimation and data detection schemes, re-quiring only a small number of pilot symbols for an initialchannel estimate, offer another alternative. Iterative channelestimation and data detection techniques for STTCs based onthe Expectation Maximization (EM) algorithm [4], presentedin [5], [6], belong to this category. In [5], [6] knowledge of normalized Doppler    󽠵   󰀽     󽠵   (with      being the maximumDoppler frequencyand  󽠵   being the transmitted symbol period)is required to determine the correlation function of the fadingprocess used in channel estimation. The resulting channelestimate is used in data detection, showing that knowledge of    󽠵   is important for the operation of joint channel estimationand data detection schemes over time-varying fading channels.Per-Survivor Processing (PSP) techniques [7] using multiplehypothetical data sequences (survivors), that also belong to the joint channel estimation and data detection family of schemes,are well suited for high Doppler channels [8]. These PSPalgorithms embed adaptive channel estimation into tree-searchor trellis-search (Viterbi) algorithms [7], [9] where the survivoris used for channel estimation. The development of PSP-based joint channel estimation and data detection algorithms forSTTCs is documented in [8], [10], [11]. Another joint channelestimation and data detection approach is based on particle Þ ltering 1 [12]. In [12], a Delayed Mixture Kalman Filtering(DMKF) technique for single-antenna systems is presented,where importance samples take into account future receivedsymbols. In [13] this approach is applied to STTCs. A KalmanPredictor (KP) is employed for channel estimation in [8],[11], [13], due to its ef  Þ cient recursive implementation andits tracking abilities [14]. Using a KP requires a state-spacemodel that in turns requires    󽠵  . The knowledge of     󽠵   isalso required to determine the step size in the Least MeanSquare (LMS)-based algorithm of [10].In practice      (or   󽠵  ) has to be estimated. For Doppler esti-mation, one can use the Level Crossing Rate (LCR) approachand the covariance approach [15]. Moreover, an asymptoticMaximum Likelihood (ML) estimator based on the Whittleapproximation is proposed in [16] for      estimation. With ashort sequence of channel coef  Þ cients, the covariance methodperforms better than the asymptotic ML approach [16] andthe LCR approach [15]. With a KP, there are several ways toattack the Doppler estimation problem. One way is to includethe unknown parameters into a state space model. However, inthis case the model becomes nonlinear and the extended KP 1 Also known as sequential importance sampling0090-6778/10$25.00 c ⃝ 2010 IEEE  2278 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010 must be used. Initial work in [17] shows that PSP-based jointchannel estimation and data detection, using an Adaptive KP(AKP) for STTC systems, performs well in fading channelswith unknown constant Doppler.In [11], we proposed an algorithm called Smoothed DataDetection and Kalman Estimation (SDD-KE), belonging tothe PSP family, where detection employs the Maximum APosteriori Probability (MAP) criterion with a  Þ xed delay  􍠵 ,and the channel is estimated by a KP assuming that the Þ xed    󽠵   is known. In this paper, we derive the DopplerAdaptive SDD-KE (DA-SDD-KE) algorithm that does notrequire knowledge of     󽠵  . This novel algorithm jointly detectsdata, and estimates the channel as well as the  time-varying Doppler by incorporatingan AKP into the SDD-KE algorithm.We also propose a novel method for generating a Rayleighfading process with time-varying Doppler because of absenceof adequate techniques in the literature.This paper is organized as follows. In Section 2, we discussa general model for Multiple STTC (MSTTC) systems andpresent a technique for generating Rayleigh fading chan-nels with time-varying Doppler. In Section 3 we considercovariance-based    󽠵   estimation, the AKP and the DA-SDD-KE algorithm. Section 4 presents computer simulation resultsof an MSTTC scheme over fading channels, with constant,linear, and quadratic time-varying Doppler functions, usingthe DA-SDD-KE algorithm. Section 5 presents the DA-DMKFalgorithm, a Doppler adaptive extension of the DMKF of [13] that we developed for comparison with the DA-SDD-KEscheme. Finally, Section 6 presents the conclusions.II. S YSTEM AND  C HANNEL  M ODELS  A. System Model Consider a wireless communication system with  󝠵 󽠵   trans-mit and  󝠵 􍠵  receive antennas, employing an MSTTC (e.g thesmart-greedy STTCs of [1] and the STTCs of [13], [18])with    code vectors per trellis branch. The codeword lengthof the MSTTC is   󝠵   󰀽      󰀨   󰀫   󰀩 , where    is theinformation frame length, and    is the memory order of thecode. Let    󰀽 󰀱 ,..., 󝠵   be the transmitted/received vectorindex, and    󰀽 󰀱 ,...,  󰀫    be the trellis stage index. Thecode matrix associated with the   ℎ trellis stage is  󰁃   󰀽󰀨 󰁣   󰀨   󰀱󰀩󰀫󰀱 ,..., 󰁣    󰀩  where 󰁣  , ∈󰁻   󰀨   󰀱󰀩󰀫󰀱 ,...,   󰁽 isthe   ℎ transmitted vector of dimension  󝠵 󽠵  . The   ℎ receivedvector  󰁲  󰀽󰀨  󰀨󰀱󰀩   ,..., 󰀨   󰀩   󰀩 󽠵  is given by 󰁲   󰀽 √       󰁣   󰀫 󰁮     󰀽 󰀱 , 󰀲 ,..., 󝠵   (1)where     is transmit symbol energy per antenna,   󰀽 󰁻  󰀨 , 󰀩   󰁽 is the  󝠵 􍠵 󰃗 󝠵 󽠵   channel matrix affecting  󰁣  , whose elementsare independent identically distributed zero-mean circular-symmetric complex Gaussian random variables of unit vari-ance. The additive noise 󰁮   is a zero-mean circular-symmetriccomplex Gaussian vector with covariance matrix    󰀰    󰁉   where  󰁉    denotes the  󝠵 􍠵 󰃗 󝠵 􍠵  identity matrix.With multiple antenna systems in quasi-static or indepen-dent fading environments,the channel coef  Þ cients are assumedto be fully correlated or independent in time, respectively, andthey are assumed to be spatially independent. Such models areconvenient for design and analysis. However, fading channelswith Doppler spread represent more realistic wireless linkssince they model time-correlated fading with coef  Þ cients thatchange continuously. In this work, we consider Doppler fadingchannels of communication systems where the transmitter,receiver and/or surrounding objects move with time-varyingspeed, resulting in a time-varying Doppler frequency     󰀨  󰀩 [19].The decoder employs the DA-SDD-KE algorithm. This al-gorithm, while utilizing future received symbols for smootheddata detection as in the SDD-KE algorithm [11], does notassume perfect knowledge of the Doppler as in [11]. Inthe DA-SDD-KE algorithm, we incorporate an AKP into theSDD-KE scheme where the survivor sequences are used by theAKP for channel tracking and    󽠵   estimation. Hence, unlikethe SDD-KE, the new algorithm not only estimates jointly thechannel matrices and transmitted data sequence, but also    󽠵  .The following notations will be used throughout this pa-per. Let  󰀨  󰀩 󽠵  , 󰀨  󰀩   , 󰀨  󰀩 ∗ , denote the transpose, Hermitian, andconjugate operations on a vector or matrix. For an    -statetrellis diagram, a branch at stage    connects a state     ∈    to a state    󰀫󰀱  ∈    󰀫󰀱  where      ⊆ 󰁻 󰀰 ,...,     󰀱 󰁽  with   󰀱  󰀽     󰀫  󰀫󰀱  󰀽  󰁻 󰀰 󰁽 , and     󰀨   󰀫󰀱 󰀩  ⊆     is the set of states in      for which there are branches connecting to thesestates terminating at    󰀫󰀱  ∈    󰀫󰀱 . Denote by  󽠵   and  ℂ  the set of all possible  󰁣   and  󰁃  , respectively. Furthermore, ℂ  󰀨   󰀫󰀱 󰀩  ⊂  ℂ   denotes a set of code matrices associatedwith branches at stage    whose terminating state is    󰀫󰀱 , and 󰁃  󰀨   ,  󰀫󰀱 󰀩 󰀽  󰁣   󰀨   󰀱󰀩󰀫󰀱 󰀨   ,  󰀫󰀱 󰀩 ,..., 󰁣    󰀨   ,  󰀫󰀱 󰀩  denotes a code matrix at stage    associated with a branchconnecting states     and    󰀫󰀱 , where  󰁣  󰀨   ,  󰀫󰀱 󰀩 ,   󰀽     󰀨     󰀱󰀩 󰀫 󰀱 ,...,     , is the code vector at discretetime   . Let  󰁃  󰀱  be a code matrix sequence,  󰁣  󰀱  be a codevector sequence,  ℂ  󰀱  and  󽠵  󰀱  be the set of possible  󰁃  󰀱  and 󰁣  󰀱 , respectively. Furthermore,  󰁒   󰀽 󰀨 󰁲   󰀨   󰀱󰀩󰀫󰀱 ,..., 󰁲    󰀩 is the received matrix corresponding to trellis stage   , and 󰁒  󰀱 ,  󰁲  󰀱  denote the sequence of received matrices and vectors,respectively. Let  ℂ  󰀫  󰀱  󰀨 󰁃  󰀨   ,  󰀫󰀱 󰀩󰀩 ⊂ ℂ  󰀫  󰀱  be the set of sequences  󰁃  󰀫  󰀱  for  􍠵  ≥  󰀰  such that  󰁃   󰀽  󰁃  󰀨   ,  󰀫󰀱 󰀩 .Furthermore,  󰁃 , 󰀨   󰀫󰀱 󰀩  is a code matrix at trellis stage   associated with a survivor at    󰀫󰀱 , 󰁃  󰀱 , 󰀨   󰀫󰀱 󰀩  is the sequenceof code matrices corresponding to the survivor at    󰀫󰀱 , and 󰁃  󰀫  󰀱 ,  󰀨   󰀫  󰀫󰀱 󰀩  is the detected sequence.  B. A Technique for Generating Time-Varying Doppler FadingChannels In [20], Jakes proposed a model (that became very popular)for generating a time-correlated Rayleigh fading process basedon a Sum of Sinusoids (SoS) technique. In [21], differentgeneration methods for fading channels with constant Dopplershifts, using the SoS technique, are compared in terms of complexity and performance. One of the results of [21]recommends the model from [22] for Rayleigh fading channelsimulation because of its superior performance (while slightlymore complex) over other models. Note that all techniques inthe literature for Rayleigh fading generation assume a constant    . However, channels with time-variant Doppler that modelsituations where the transmitter, receiver and/or surroundingobjects move with time-varying speed are also of interest  VILAIPORNSAWAI and LEIB: JOINT DATA DETECTION AND CHANNEL ESTIMATION FOR FADING UNKNOWN TIME-VARYING DOPPLER  ...  2279 [23]–[25]. For example, [26], [27] consider non-stationarityissues associated with channels for vehicle-to-vehicle com-munications where each vehicle can move with time-varyingspeeds and/or directions, and hence can experience time-variant Doppler effects. From the literature it is not clear howsuch time-varying Doppler channels can be generated. In thispaper we show that substituting a time-varying     󰀨  󰀩  in placeof a constant      in a technique for generating a fading processwith constant Doppler is not proper. The key point that veryoften is missed is that the Doppler shift should be treated asan instantaneous frequency. In [28], the authors point out toinaccurate methods of generating time-varying carrier offsetsin the literature where     󰀨  󰀩  is applied in place of a constant    . However, in [28], a single-carrier frequency-offset modelis considered, while in our work we consider a technique forgenerating a Rayleigh fading process with a prescribed timevarying Doppler shift based on the SoS method.Extending the method of [22] to time-varying Doppler fre-quency (instantaneousfrequency)    󰀨  󰀩 , results in a continuoustime fading process   󰀨 , 󰀩   given by (2)-(6), with  󰀨 , 󰀩   󰀽 󰀲 󝠵    󰀫  󰀨 , 󰀩 󰀴    (7)where   󰀨 , 󰀩   ,  󰀨 , 󰀩   and   󰀨 , 󰀩 are uniformly distributed on 󰁛  , 󰀩  and independent for all  󝠵  󰀽 󰀱 ,...,   ,  󰀽 󰀱 ,...,󝠵 󽠵  and    󰀽 󰀱 ,...,󝠵 􍠵 . Note that (3) reduces to (4) since the phaseat    󰀽 󰀰  associated with ∫  󰀰 ∞    󰀨  󰀩   can be absorbed inthe random phase   󰀨 , 󰀩   . Similarly, we obtain (6) from (5).For a constant Doppler frequency, i.e.     󰀨  󰀩 󰀽     , we have ∫   󰀰     󰀨  󰀩   󰀽       and the model (2)-(6) is reduced to the onepresented in [22]. Theorem 1:  If       → ∞ , then the process   󰀨 , 󰀩   tends tobe Gaussian.Proof of this theorem can be found in Appendix A. Theorem 2:  The fading process   󰀨 , 󰀩   speci Þ ed by (2)-(6) has the properties shown in (8)-(12), where  󰋜    󰀨 ,  󰀩 󰀽 ∫   󰀫 󰀯 󰀲   󰀯 󰀲     󰀨  󰀩   and    󰀰 󰀨  󰀩  is the zero-order Bessel function of the  Þ rst kind.Proof of this theorem can be found in Appendix B.From Theorem 2, if the Doppler shift is time-invariant, i.e.    󰀨  󰀩 󰀽      and ∫   󰀰     󰀨  󰀩   󰀽       is used in the generationof    󰀨 , 󰀩   (2), then we have  󰋜    󰀨 ,  󰀩 󰀽       , and (11) reducesto    󰁛  󰀨 , 󰀩  󰀫 󰀯 󰀲  󰀨 , 󰀩 ∗   󰀯 󰀲 󰁝  󰀽    󰀰 󰀨󰀲      󰀩  which is the well knownresult for constant Doppler. For a linear time-varying Doppler    󰀨  󰀩 󰀽   󰀫  , we have  󰋜    󰀨 ,  󰀩 󰀽 󰀨  󰀫  󰀩   , and (11) reducesto    󰁛  󰀨 , 󰀩  󰀫 󰀯 󰀲  󰀨 , 󰀩 ∗   󰀯 󰀲 󰁝  󰀽    󰀰 󰀨󰀲  󰀨   󰀫   󰀩   󰀩 . Note that if this    󰀨  󰀩  is applied in place of       in the model for generatingthe fading process with constant Doppler [22], or equivalently    󰀨  󰀩   is used in place of  ∫   󰀰     󰀨  󰀩   in (2), then it can beshown (using the approach in the Proof of Theorem 2) thatthe autocorrelation function of the process is    󰀰 󰀨󰀲  󰀨󰀲   󰀫  󰀩   󰀩 , rather than    󰀰 󰀨󰀲  󰀨   󰀫   󰀩   󰀩 . This shows that a directsubstitution of      󰀨  󰀩  for      in the model for constant Doppleris not a proper way to generate time-varyingDoppler channels.Next, we consider the generation of discrete-time fadingcoef  Þ cients using (2)-(6). First, consider 󲈫    󰀰    󰀨  󰀩    󰀽 󽠵  󰀽 󲈫   󽠵  󰀰    󰀨  󰀩   󰀽  󽠵  󲈫    󰀰    󰀨 󽠵  󰀩  󰀽 󲈫    󰀰   󽠵  󰀨  󰀩   (13)where    󰀽  󽠵   , and    󽠵  󰀨  󰀩 󰀽  󽠵   󰀨 󽠵  󰀩  is the normalizedDoppler shift function. For a polynomial Doppler function of degree      , we have     󰀨  󰀩 󰀽 ∑   󽠵  󰀽󰀰     where     takes realvalue. Then,    󽠵  󰀨  󰀩  in this case is given by   󽠵  󰀨  󰀩 󰀽  󽠵     󽠵 󲈑  󰀽󰀰    󰀨 󽠵  󰀩  󰀽   󽠵 󲈑  󰀽󰀰    󽠵   󰀫󰀱    󰀽   󽠵 󲈑  󰀽󰀰  󽠵    (14)where   󽠵  󰀽       󽠵   󰀫󰀱 . From (13) and (14), we have ∫   󰀰    󽠵  󰀨  󰀩   󰀽 ∑   󽠵  󰀽󰀰  􍠵󝠵  󰀫󰀱   󰀫󰀱 . To generate the discrete-timefading process  󰀨 , 󰀩   󰀽  󰀨 , 󰀩   ∣  󰀽 󽠵   with    󽠵  󰀨  󰀩  given by (14),we use ℜ 󰀨  󰀨 , 󰀩   󰀩 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󰀨   󽠵 󲈑  󰀽󰀰  󽠵    󰀫 󰀱   󰀫󰀱 󰀩󰁣󰁯󰁳  󰀨 , 󰀩  󰀫  󰀨 , 󰀩   󰀩  (15) ℑ 󰀨  󰀨 , 󰀩   󰀩 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󰀨   󽠵 󲈑  󰀽󰀰  󽠵    󰀫 󰀱   󰀫󰀱 󰀩󰁳󰁩󰁮  󰀨 , 󰀩  󰀫   󰀨 , 󰀩   󰀩  (16)Non-polynomial forms for     󰀨  󰀩  can be approximated bypolynomials to synthesize a large variety of normalizedDoppler functions    󽠵  󰀨  󰀩 .The validity of the discrete-time model obtained by sam-pling the continuous-time model is investigated in [29] forconstant Doppler, and shown to depend on BER  ß oor      ,the roll-off factor     of the transmitted square-root raisedcosine pulse shape, and the constant    󽠵  . A fading processwith autocorrelationfunction   󰁛    ∗     ′ 󰁝 󰀽    󰀰 󰀨󰀲   󽠵    ′ 󰀩  andJakes U-shape power spectrum is considered. With a given      ,it is shown in [29] that as    󽠵   increases, a larger     is required.In this paper we use the same approach as in [29] toinvestigate the validity of the discrete-time model for time-varying Doppler channels with QPSK and coherent detection.Appendix C shows that the BER  ß oor associated with the   ℎ information symbols      󰀨  󰀩  as a function of the instantaneousDoppler    󽠵  󰀨  󰀩  and     is     󰀨  󰀩 ≈ 󰀰 . 󰀲󰀷   󽠵  󰀨  󰀩 󰀴   󰀳  (17)Note that (17) is approximately the same for linear andquadratic Doppler functions    󽠵  󰀨  󰀩 󰀽 ∑   󽠵  󰀽󰀰  󽠵     consid-ered in Section IV and the Doppler function with       󰀽 󰀵 considered in [30, Case 8]. Moreover, it is also approximatelythe same as [29, Eqn. (63)] for constant Doppler. Hence,for time-varying Doppler channels and a given   , we canuse (17) or equivalently the result of [29] for a constant   󽠵   󰀽    󽠵  󰀨  󰀩  to determine the roll-off factor     for aspeci Þ c       󰀽      󰀨  󰀩 . For time-varying Doppler channels,pulse shaping and the receiver front-end should be designedfor the maximum instantaneous normalized Doppler (the worst  2280 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010  󰀨 , 󰀩   󰀽  ℜ 󰀨  󰀨 , 󰀩   󰀩 󰀫   ℑ 󰀨  󰀨 , 󰀩   󰀩  (2) ℜ 󰀨  󰀨 , 󰀩   󰀩 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󲈫    ∞    󰀨  󰀩  󰁣󰁯󰁳  󰀨 , 󰀩   󰀫  󰀨 , 󰀩   󰀩  (3) 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󲈫    󰀰    󰀨  󰀩  󰁣󰁯󰁳  󰀨 , 󰀩   󰀫  󰀨 , 󰀩   󰀩 ,  󰀰 ≤  󰀼 ∞  (4) ℑ 󰀨  󰀨 , 󰀩   󰀩 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󲈫    ∞    󰀨  󰀩  󰁳󰁩󰁮  󰀨 , 󰀩   󰀫   󰀨 , 󰀩   󰀩  (5) 󰀽 󰀱 √       󲈑  󰀽󰀱 󰁣󰁯󰁳󰀨󰀲  󲈫    󰀰    󰀨  󰀩  󰁳󰁩󰁮  󰀨 , 󰀩   󰀫   󰀨 , 󰀩   󰀩 ,  󰀰 ≤  󰀼 ∞  (6)   󰁛  󰀨 , 󰀩  󰁝  󰀽 󰀰  (8)   󰁛 ℜ 󰀨  󰀨 , 󰀩  󰀫 󰀯 󰀲 󰀩 ℜ 󰀨  󰀨 , 󰀩   󰀯 󰀲 󰀩 󰁝  󰀽    󰁛 ℑ 󰀨  󰀨 , 󰀩  󰀫 󰀯 󰀲 󰀩 ℑ 󰀨  󰀨 , 󰀩   󰀯 󰀲 󰀩 󰁝 󰀽 󰀱󰀲   󰀰 󰀨󰀲   󰋜    󰀨 ,  󰀩󰀩  (9)   󰁛 ℜ 󰀨  󰀨 , 󰀩  󰀫 󰀯 󰀲 󰀩 ℑ 󰀨  󰀨 , 󰀩   󰀯 󰀲 󰀩 󰁝  󰀽    󰁛 ℑ 󰀨  󰀨 , 󰀩  󰀫 󰀯 󰀲 󰀩 ℜ 󰀨  󰀨 , 󰀩   󰀯 󰀲 󰀩 󰁝 󰀽 󰀰  (10)   󰁛  󰀨 , 󰀩  󰀫 󰀯 󰀲  󰀨 , 󰀩 ∗   󰀯 󰀲 󰁝  󰀽    󰀰 󰀨󰀲   󰋜    󰀨 ,  󰀩󰀩  (11)   󰁛  󰀨 , 󰀩  󰀫 󰀯 󰀲  󰀨 , 󰀩   󰀯 󰀲 󰁝  󰀽 󰀰  (12)case) since it requires the largest    for a given     . The discretetime fading model in our work assumes such proper pulseshaping and receiver front-end.III. T HE  DA-SDD-KE A LGORITHM This section presents the DA-SDD-KE algorithm that in-cludes an AKP for channel and Doppler estimations in theSDD-KE scheme of [11]. We consider the use of covariance-based    󽠵   estimation in the AKP, and its integration in SDD-KE algorithm.  A. Doppler Shift Estimation Let   󰀨 , 󰀩   󰀽   󰁛  󰀨 , 󰀩    󰀨 , 󰀩 ∗     󰁝  denote the autocorrelation func-tion for lag    of the channel coef  Þ cients associated withreceive antenna    and transmit antenna   , and  󰋆  󰀨 , 󰀩   󰀨  󰀩  denotethe estimate of    󰀨 , 󰀩   based on  󰀨  󰀨 , 󰀩   , 󰀨 , 󰀩   󰀱 ... 󰀨 , 󰀩󰀱  󰀩 . Sincea covariance-based    󽠵   estimator operates well over shortsequences of channel coef  Þ cients [16], only a small numberof pilot symbols are required to provide an initial estimatefor    󽠵  . Hence, we employ this method as in [24]. The    󽠵  estimate is given by 󰋆   󽠵   󰀽 󰁡󰁲󰁧 󰁭󰁩󰁮 􍠵  􍠵  ∈ℱ  󰀱 󽠵 󝠵   󽠵 󽠵  􍠵  ∑  󰀽󰀱   ∑  󰀽󰀱   ∑  󰀽󰀱  󰋆 􍠵 󰀨 󰀬 󰀩   󰀨 󝠵 󰀩󰋆 􍠵 󰀨 󰀬 󰀩󰀰  󰀨 󝠵 󰀩    󰀰 󰀨󰀲   󽠵   󰀩  󰀲 (18) where  ℱ   is a set of     󽠵   candidates, and for    󰀽 󰀰 , 󰀱 ,...,  with   󰀾   󰋆  󰀨 , 󰀩   󰀨  󰀩 󰀽 󰀱 ∑     󰀱  󰀽󰀰       󰀱 󲈑  󰀽󰀰   ℜ 󰀨󰋆  󰀨 , 󰀩    󰋆  󰀨 , 󰀩 ∗      󰀩 ,  (19)where  󰋆  󰀨 , 󰀩   ,  󰋆  󰀨 , 󰀩   󰀱 ,   ,  󰋆  󰀨 , 󰀩󰀱   are channel estimates avail-able from the AKP in the DA-SDD-KE algorithm. Comparedto [24], (19) introduces an exponential forgetting factor  󰀰  󰀼 ≤ 󰀱  to allow for tracking in time-varying    󽠵   estimation.With    󰀽 󰀱  (19) is a conventional sample autocovariancefunction [31, pp. 321-323].A forgettingfactor has been widelyused in Recursive Least Square (RLS) problems [32, Chapter9], allowing past data to be weighted less and hence enhancingtracking capabilities. From (19),  󰋆  󰀨 , 󰀩   󰀨  󰀩  can be updated from 󰋆  󰀨 , 󰀩   󰀨    󰀩  using the equation at the top of the next page,where later we use    󰀽 󰀲  because a phase ambiguity resistantMSTTC with    󰀽 󰀲  is employed, and in the DA-SDD-KEalgorithm the  󰋆   󽠵   is updated at trellis states rather than withintrellis branches.  B. Adaptive Kalman Predictor (AKP) In this section, we present the KP structure embedded in theAKP. We use a state space Auto-Regressive model of order 3,AR(3), matched to the Jakes U-shape power spectrum. Hencethe AR(3) model must depend on    󽠵   and it is given by [8], 󰁘  󰀫󰀱  󰀽 󰁆 󰀨 󰋆   󽠵  󰀩 󰁘   󰀫 󰁇󰁗  󰀨 󰋆   󽠵  󰀩  (20)where 󰁘   󰀽 󰀨       󰀱     󰀲 󰀩 󽠵  , 󰁗  󰀨 󰋆   󽠵  󰀩 󰀽   󰀲󰀨󰀳󰀩 󰀨 󰋆   󽠵  󰀩  󰁉  􍠵  , 󰁇 󰀽 󰀨 󰁉 󽠵  􍠵   󰀰 󽠵  􍠵   󰀰 󽠵  􍠵  󰀩 󽠵  , and 󰁆 󰀨 󰋆   󽠵  󰀩 󰀽 ⎛⎝  󰀱 󰀨 󰋆   󽠵  󰀩 󰁉  􍠵    󰀲 󰀨 󰋆   󽠵  󰀩 󰁉  􍠵    󰀳 󰀨 󰋆   󽠵  󰀩 󰁉  􍠵  󰁉  􍠵   󰀰  􍠵   󰀰  􍠵  󰀰  􍠵   󰁉  􍠵   󰀰  􍠵  ⎞⎠ , (21)with  󰋆   󽠵   provided by the Doppler estimator, and AR(3)coef  Þ cients    󰀨 󰋆   󽠵  󰀩  obtained as in [11].  VILAIPORNSAWAI and LEIB: JOINT DATA DETECTION AND CHANNEL ESTIMATION FOR FADING UNKNOWN TIME-VARYING DOPPLER  ...  2281 󰋆  󰀨 , 󰀩   󰀨  󰀩 󰀽 󰀱 ∑     󰀱  󰀽󰀰    􀀨 󰀨       󰀱 󲈑  󰀽󰀰   󰀩   󰋆  󰀨 , 󰀩   󰀨    󰀩 󰀫   󰀱 󲈑  󰀽󰀰   ℜ 󰀨󰋆  󰀨 , 󰀩    󰋆  󰀨 , 󰀩 ∗      󰀩 􀀩 󰋆 󰁘  󰀫󰀱 ∣   󰀽 󰁆 󰀨 󰋆   󽠵  󰀩󰋆 󰁘  ∣   󰀱  󰀫 󰁆 󰀨 󰋆   󽠵  󰀩 󰁍 ′  󰀨󲈚  󰁤 󽠵  Σ 󰁘  ∣   󰀱 󰁤 ∗   󰀫   󰀰 󰀩  󰀱  󰀨 󰁲 󽠵    󰁤 󽠵   󰋆 󰁘  ∣   󰀱 󰀩  (22) Σ 󰁘  󰀫󰀱 ∣   󰀽 󲈚  Σ 󰁘  󰀫󰀱 ∣  󲈚  Σ 󰁘  󰀫󰀱 ∣    (23) ⎡⎢⎢⎣󲈚  󰁤 󽠵  Σ 󰁘  ∣   󰀱 󰁤 ∗   󰀫   󰀰 󽠵  󰀨 󰁆 󰀨 󰋆   󽠵  󰀩 󰁍 ′  󰀩 󽠵  󰀰 √  Σ 󰁘  󰀫󰀱 ∣  󽠵  󰀰 󰀰 ⎤⎥⎥⎦ 󰀽  􍠵  ⎛⎜⎝⎡⎢⎣ √    󰀰  󰀰 󰀨 󰁤 󽠵  √  Σ 󰁘  ∣   󰀱 󰀩 󽠵   󰀨 󰁆 󰀨 󰋆   󽠵  󰀩 √  Σ 󰁘  ∣   󰀱 󰀩 󽠵  󰀰   󰀨󰀳󰀩 󰀨 󰋆   󽠵  󰀩 󰁇 󽠵  ⎤⎥⎦⎞⎟⎠ (24) 󰋆 󰁘 󰀱 ∣ 󰀰 ,   󰀽 󰁛 󰀰 󰁝 󰀳  􍠵  󰃗    (25) Σ 󰁘 󰀱 ∣ 󰀰 󰀬   󰀽 ⎛⎝  󰀰 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀱 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀲 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀱 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀰 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀱 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀲 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵     󰀱 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵    󰀰 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  󰁉  􍠵  ⎞⎠ ,  (26)  , 󰀨󰀳󰀩 󰀨   󽠵 󰀩 󰀽 ⎧⎨⎩∑ 󰀳  󰀽󰀱   󰀨   󽠵 󰀩    , 󰀨󰀳󰀩 󰀨   󽠵 󰀩   󰀾  󰀰 ∑ 󰀳  󰀽󰀱   󰀨   󽠵 󰀩    , 󰀨󰀳󰀩 󰀨   󽠵 󰀩 󰀫  󰀲󰀨󰀳󰀩 󰀨   󽠵 󰀩    󰀽 󰀰  ∗ , 󰀨󰀳󰀩 󰀨   󽠵 󰀩   󰀼  󰀰 .  (27)We use the square root Kalman  Þ lter [14, pp. 150] to avoidnumerical errors, as shown in (22)-(24), where  􍠵  󰀨  󰀩  denotesa triangularization by means of the    decomposition. Theinitial conditions for the AKP are obtained using the initialvalue    󽠵  as shown in (25) and (26), where   , 󰀨󰀳󰀩 󰀨   󽠵 󰀩 is the autocorrelation function of the AR( 󰀳 ) process, as shownin (27) [33, pp.838]. For a given    󰀨   󽠵 󰀩 ,󝠵  󰀽 󰀱 , 󰀲 , 󰀳 ,we solve for   , 󰀨󰀳󰀩 󰀨   󽠵 󰀩 ,  󰀽 󰀱 , 󰀲 , 󰀳  using (27) with  󰀰 , 󰀨󰀳󰀩 󰀨   󽠵 󰀩  = 1, and process noise power   󰀲󰀨󰀳󰀩 󰀨   󽠵 󰀩 󰀽󰀱  ∑ 󰀳  󰀽󰀱   󰀨   󽠵 󰀩  , 󰀨󰀳󰀩 󰀨   󽠵 󰀩 .Subsequently, we use  󰋆  󰀨 , 󰀩 ,   and  󰋆   󽠵   to denote  󰋆  󰀨 , 󰀩   and 󰋆   󽠵   estimated from the sequence of channel estimates asso-ciated with the pilot sequence of length   , respectively. Also,we use  󰋆  󰀨 , 󰀩 ,  󰀨  󰀩  and  󰋆   󽠵  󰀨  󰀩  to denote  󰋆  󰀨 , 󰀩   and  󰋆   󽠵   estimatedfrom the sequence of channel estimates associated with codedsequence  󰁣  󰀱  concatenated with the pilot sequence. Fig. 1illustrates the structure of the AKP considered in this work,where dash lines are used to underline the quantities associatedwith the training mode, while solid lines are used for thedata-aided mode. During the training mode, the algorithmstarts by calculating the AR(3) coef  Þ cients    󰀨   󽠵 󰀩  basedon    󽠵 . Using the AR(3) model, the KP estimates thechannel matrices  󰋆    󰀫󰀱󰀲 ,   based on the pilot sequence of length   .Then,  󰋆    󰀫󰀱󰀲 ,   are used to calculate the autocorrelation function 󰋆  󰀨 , 󰀩 ,   for lag    󰀽 󰀰 , 󰀱 ,...,   󰀽      󰀱  which are fed tothe    󽠵   estimator of Section III-A to provide  󰋆   󽠵  . During 󰁲    󰀱 󰁲   󰀱 ,  󰁣    󰀱 󰁣   󰀱 ,    󽠵 󰋆     󰀫󰀱   󰀨   󰀱󰀩󰀫󰀲 󰋆    󰀫󰀱󰀲 ,  󰋆   󽠵  󰀨   󰀨   󰀱󰀩󰀩󰋆   󽠵    󰀨 󰋆   󽠵  󰀨   󰀨   󰀱󰀩󰀩󰀩   󰀨 󰋆   󽠵 󰀩 󝠵 󰀽󰀱 , 󰀲 , 󰀳 󝠵 󰀽󰀱 , 󰀲 , 󰀳 AR(3)Coef  Þ cientsCalculationKPOperation   󽠵   Estimation 111222Training ModeData-aided Mode Fig. 1. Structure of an adaptive Kalman predictor for Doppler shiftestimation. the data-aided mode, we set  󰋆  󰀨 , 󰀩 ,  󰀨󰀰󰀩 󰀽 󰋆  󰀨 , 󰀩 ,   ,  󰋆   󽠵  󰀨󰀰󰀩 󰀽󰋆   󽠵  ,  󰋆  󰀱  󰀽 󰋆    󰀫󰀱 ,  , and using  󰋆   󽠵  󰀨󰀰󰀩 , the AR(3) model isupdated for the  Þ rst time based on the coef  Þ cients    󰀨 󰋆   󽠵  󰀨󰀰󰀩󰀩 .
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