Multicarrier Code Division Multiplex with iterative MAP Symbol by Symbol Estimation

Transcription

1 D Multicarrier Code Division Multiplex iterative MA Symbol by Symbol stimation Frieder Sanzi Alexander Slama Joachim Speidel Institute of Telecommunications University of Stuttgart faffenwaldring Stuttgart Germany Abstract In this paper we consider a Multi Carrier Code Division Multiplex MC CDM) scheme At the receiver side the Maximum A osteriori Symbol by Symbol stimator MASS) is used for the detection of the CDM signal Therefore the influence of the spreading factor on the overall it rror Rate R) is investigated We concatenate the MASS the channel which allows for iterative decoding This system can be considered as a serially concatenated iterative decoding scheme the inner is replaced by the MASS Therefore the bit error rate can be further reduced by means of iterative decoding A combination an inner Recursive Systematic Convolutional RSC) component code rate is suggested to further improve performance The applications are broadcasting and some extensions two way communications eywordsfdm CDMA iterative decoding I INTRDUCTIN A Transmitter source conv frequency II SYSTM MDL multicarrier modulation ifft mapper time- frequency selective channel CDM AWGN For mobile communication systems rthogonal Frequency Division Multiplexing FDM) has received a lot of attention in the recent years Therefore FDM has become an important modulation scheme for several applications like Digital Subscriber Line DSL) Digital Audio roadcasting DA) Terrestrial Digital Video roadcasting DV-T) and Wireless LAN eg YRLAN) ne advantage of FDM is that it turns a frequency selective channel into a flat fading channel for each sub carrier So the frequency domain equalizer is just a one tap equalizer for each sub carrier To achieve frequency diversity FDM can be combined Code Division Multiplex CDM) where the signal is spread over several sub carriers This concept was introduced as FDM CDMA or multicarrier CDMA MC CDMA) by [] [] [] and is also called MC spread spectrum MC SS) In this paper the authors consider a MC-CDM system in which each receiver can decode the total bitrate by using all orthogonal codes For the detection the maximum a posteriori symbol by symbol estimator MASS) is proposed With this scheme we investigate the impact of the spreading factor on the overall bitrate In recent time since the invention of Turbo Codes by [4] iterative decoding algorithms for spectrally efficient modulation have become a vital field of research in digital communication In this paper we suggest an application of the Turbo rinciple for iterative MASS concatenated the channel This system can be regarded as a serially concatenated iterative decoding scheme the inner is replaced by the MASS ften the reliability information of the channel is used for soft interference cancellation eg [5] We propose to feed back this information directly to the MASS stage To further improve the performance of the system a combination an inner RSC component code of rate is suggested The performance of the proposed scheme is evaluated on the basis of bit error rate R) charts In addition the convergence of the iterative decoding loop is studied the xtrinsic Information Transfer Chart XIT Chart) recently introduced in [6] [7] Fig Transmitter and channel model As shown in Fig the signal from the source is convolutionally encoded interleaved mapped S alphabet ) and spread by the CDM block which takes consecutive S symbols creates the vector the! )+ -- coefficients "$#&% and ) and multiplies it the Walsh matrix 4 The resulting vector the coefficients is con- 87:; ;<>= is integer division opera- verted to the composite multicode CMD sequence 56 5 where?c tor Ṫhe set of orthogonal Walsh codes can be calculated recursively by using GFI GFI : GFI GFJL NM4 Q and is the SRT adamard Matrix [8] and is the spreading factor The serial order of 5 is interchanged by the subsequent which results in the sequence 5 So the input 5 to the multicarrier modulator is frequency interleaved 5 is modulated onto U sub carriers using ifft For the following mathematical description it is convenient to separate the discrete time axis into intervals of length U V! X W JZ 5Y #[%]\I^ and similarly for V: and )+ -- ) ) U_ ) The transmission is done on a block by block basis blocks of U sub carriers in frequency and a FDM symbols in time direction $7 C) I

2 $ W ^ a ^ a l n Channel Model For the mobile channel we use the wide sense stationary uncorrelated scattering WSSUS) channel model introduced in [] The frequency response of the channel can be expressed as the denotes the number of are randomly chosen depending on the corresponding joint probability density func- of the considered channel model We assume a channel model where the phase $ is uniformly distributed the delay is exponentially distributed probability density function DF) % and the Doppler shift is distributed according to Jakes power spectral density function In this case the auto correlation function in time is D! #" where $ &% is the phase the Doppler frequency and delay of the th path The variable propagation paths The $ )% and tion + % - X ) %5467 Z 8: 8 is the essel function is the duration of one FDM symbol useful %;4<6 7 part plus guard interval) Z is the discrete time index and is the maximal Doppler shift The complex auto correlation function in frequency direction writes as 46 7 >= - W?@ 4A W C "? 4<6 7?@ 4A DGF )! JI 6) LM is the channel delay spread is the sub carrier spacing and ^ is the discrete frequency index I NM is chosen such that C Receiver After multicarrier demodulation of the U sub carriers and frequency deinterleaving the received signal is fed into the MASS stage Fig ) multicarrier frequency demodulation de FFT \ X]Y+Z [ Q RUT XJYZ [ MASS V 6 de QSRUT Q A coded bits Fig Receiver iterative MASS and decoding hard decision sink The MASS stage s a log likelihood ratio value L value see []) for each coded bit After deinterleaving and soft insoft out decoding an A osteriori robability calculator A or MA calculator see []) the estimates on the transmitted bits are available at the of the hard decision device This can be accomplished by just considering the sign of the A soft values To allow for turbo processing iterative MASS and decoding the extrinsic information on the coded bits is fed back from the A and after interleaving it becomes a priori knowledge a for the MASS stage 4) 5) III MASS WIT A RIRI NWLDG For our further considerations we assume the channel characteristic to be approximately unchanged during the duration of one FDM symbol Under this assumption and provided that the guard interval is longer than the delay spread of the channel the cyclic prefix avoids inter carrier interference ICI) and also inter symbol interference ISI) In this case we can compute the received composite multicode chip after multicarrier demodulation as ^ V! D Z is the FDM symbol index ^ is the sub carrier V are the transmitted composite multicode chips and index are independent and identically distributed complex Gaussian noise variables component wise noise power _ The are sample values of the channel frequency response a IZ 8 After the frequency de the signal writes as ^ b V! D c fg e = W " X and i 7) 8) dc5e fhg= W " X ) j g denotes the inverse operation of the frequency Note that this operates only in frequency direction permutes the composite multicode chips on the sub carriers) and not in time direction The MASS operates blockwise and takes received ^ composite multicode chips which are grouped in a vector k and s L values on the S symbols which are also called coded bits in the following For simplification we just write for the coded bits for the channel state information V! for the composite multicode chips #ml -- n ) and k ^o - ^ First we consider a simple example to get an idea of the principles on the MASS algorithm In this case the MASS block needs to calculate L values on the coded bits for each incoming vector k The L-value of bit conditioned on k can be calculated as follows see []) V qp5rjstl" V 6 vu wyx+z qp5r" {; } r~ = Z } = ~ e = q x+z {< } r+~ = Z } = ~ = ƒ # ˆŠ 6 x+z {; } r ~ z } = e = Z } = ~ e = q x+z {< } r ~ e = Z } = ~ = ƒ # ˆŠ 6 z } = ) The a priori L-values write as h Œ yž ŽC respectively ) The conditional probability density function in ) is given by see []) t s p;r p = " z J = ; š) = V V s œ!r ž r Ÿ rjs s œ = ž Ÿ = = s " ) ) utting ) into ) we obtain the L value on the coded bit and respectively $7 C) I

3 F \ F M - ~ ~ Š Š - A U For a given spreading factor we obtain the L value of the coded bit as follows see []) V qp st " V 6 qp " e = e = u w ~ r x {< } ~ = Z } Z ~ r Z Z e = Z ~ z ƒ # ˆ e = 6 ~ r Z z } z Z ~ = e = e = ~ r x {; } ~ e = Z } Z ~ r Z Z e = Z ~ z ƒ# # ˆ e = 6 ~ r Z z } z Z ~ = 4) < denotes the joint event of the variables! having the values according to! ) and "ž being the decomposition of The function takes on the value # # " if bit number is set in the decomposition of otherwise it is # # " %$ F F& \ 5) According to ) the conditional probability density function writes as t s p;r p e = " z J = š) = V --- V -- e = ~ r s œ ž Ÿ s + IV INSRTING AN INNR RSC CMNNT CD 6) 7) We can further improve the performance of the iterative decoding loop by inserting an inner RSC component code of rate at the transmitter Fig shows the necessary changes at the transmitter source outer inner mapper Fig Modified transmitter inner RSC code We introduce a special puncturing scheme to make use of RSC component codes small memory and low decoding complexity Fig 4 illustrates the concept of heavy puncturing : Most of the parity bits sometimes also referred to as coded bits) of the rate RSC mother code are discarded such that from the total of - parity bits only - remain The parity bits are periodically inserted into the systematic bit stream in such a way that they replace the at these positions Therefore we obtain a rate code - - and - parity bits The insertion period of the parity bits is - - wing to the heavy puncturing - & code structure almost destroyed) the error correcting capabilities of the inner are very poor owever we noticed that particularly those properties which are crucial for good iterative decoding performance are still preserved as will be further detailed in Section V- In Fig 5 the necessary changes at the receiver are shown input A input A FI rate recursive systematic convolutional code FI D rate encoding parity bits added) puncturing puncture x only keep parity bits and replace x A bits A x and x parity bits Fig 4 uncturing scheme of inner rate T mother code; :;6 MASS <>= of 4?5 R 4: inner inner A <!= of 4: inner de outer A Fig 5 Modified receiver inner A outer coded bits The extrinsic information on the coded bits is fed back from the outer A and after interleaving it becomes a priori knowledge a on the inner - bits) The a priori knowledge a for the MASS stage should be inner and on - inner parity bits ecause there is no a priori knowledge on the inner parity bits we can only use the a priori knowledge on the - - inner information bits for the MASS stage The of the MASS stage becomes the input of the inner A - - L values on the inner and - L values on the parity bits Additionally we can use the remaining a priori knowledge on the - inner for the inner A which is not fed to the MASS stage Therefore the whole a priori knowledge is used which is fed back from the outer A V SIMULATIN RSULTS We use the following channel and multicarrier system pa- DC sub rameters: duration 8 of FG JI one FDM symbol A carrier spacing channel delay spread NM DC % 46 7 and maximal Doppler shift JI With U adjacent sub carriers a bols in time the interleaving depth is a consecutive FDM sym- outer) coded bits and the frequency interleaving depth is U R charts are used for performance analysis of different system configurations In addition to that we apply the xtrinsic Information Transfer Chart XIT chart) to better compare the different system configurations and to gain more insight into the convergence behavior of iterative decoding The outer) convolutional code is recursive systematic feedforward polynomial and code rate -NL + We further assume that feedback polynomial I LAM $7 C) I

4 the channel state information is perfectly known at the receiver side Channel estimation for a MC CDMA system can be accomplished by inserting pilot symbols into the transmitted data stream see [4]) A Without inner code Fig 6 shows mutual information transfer characteristics of the MASS stage The a priori input to the MASS is on the abscissa mutual information = in bit per symbol) The a posteriori is on the ordinate mutual information = ) Mutual information transfer characteristics describe the input relations of the MASS and are calculated by applying a Gaussian distributed random variable as a priori input and quantifying the a posteriori in terms of mutual information [6] [7] of MASS Fig out a priori input to MASS Mutual information transfer characteristics of MASS stage for different spreading factor and or out frequency 6 ) As can be seen the curves are straight lines Increasing the spreading factor results in a higher ascending slope and also the curves start for no a priori knowledge! very left side of chart) at a higher a posteriori! for a system frequency For the ascending slope of the curve is zero Therefore this system can not be improved by means of iterative decoding A similar result is achieved if there is no frequency " As can be seen from Fig 6 for the case the ascending slope is nearly zero We can summarize the results as follows: A frequency demolishes the orthogonality of the Walsh Codes but this results in a better diversity gain and the system can be improved by means of iterative decoding Whereas in a system out frequency the orthogonality nearly remains but there is almost no possibility of system improvement by means of iterative decoding In addition the diversity gain is smaller The trajectory of iterative decoding shows the exchange of channel and extrinsic information between MASS stage and In Fig 7 the XIT Chart is depicted The trajectory is a simulation result of the iterative scheme whereas the transfer characteristics are computed individually for the inner MASS stage and the outer applying independent Gaussian distributed random variables as a priori inputs The achieved trajectory matches the characteristics fairly well After about iterations the trajectory gets stuck owing to the intersection of both characteristics 5 8 becomes outer a priori knowledge 5 67 inner MASS information trajectory of iterative decoding at MASS 6 6 transfer characteristic of outer rate T memory 4 #%$&)$+!6#%-! outer extrinsic information becomes inner a priori knowledge 4 = 6 ) and outer memory 4 simulated trajectory of iterative decoding at 6 Fig 7 XIT chart MASS The R chart of: Fig 8 shows the system performance for spreading factor or out frequency and for different numbers of iterations R Fig 8 j j j j j j g g g out iterations out g g g iterations iterations iteration iterations iterations e ; < =?> R curves of system or out frequency for As can be seen for the system out frequency the improvement of the iterative decoding loop is low whereas for the system frequency a remarkable improvement is achieved At R C the gain between the DGF curve for iterations and the curve for iterations is about We can also conclude that for the system frequency almost iterations are enough In Fig the system performance frequency is shown after iterations for different spreading factors except for In the case no improvement is achieved $7 C) I

5 the iterative decoding loop R iterations iterations iterations iterations The R C curves of Fig show a steep turbo cliff At a R of the advantage to the system out inner code is about A+ D F for and also A+ D F for R out inner code iterations out inner code iterations inner code 7 iterations inner code 7 iterations e Fig spreading factors ; < =?> R curves of system frequency for different The R chart of Fig illustrates the performance improvement by increasing the spreading factor The gain between spreading A D F factor and no spreading) is about at R C ut we have to note that the complexity of the MASS stage increases exponentially for increasing the spreading factor With inner code 5 8 becomes trajectory at ) ) - MASS inner rate RSC outer rate T memory 4 # $ & $+!6# -! becomes 4 = Fig XIT chart of MASS 6 ) and inner rate one RSC 6 8 ) in combination outer memory 4 ; decoding trajectory at 6) - The iterative MASS and decoding loop of the system inner can be studied in the XIT chart of Fig for one particular block U a ) We choose for A+ heavy D F puncturing The decoding trajectory at spreading factor can converge towards which directly relates to reaching a very low R e ; < =?> Fig R chart of the system inner ; for better comparison two reference curves are copied from Fig VI CNCLUSIN We have shown that the use of the MASS concatenated an outer channel can reduce the R by A D means F of iterative decoding A performance gain of about is achieved by increasing the CDM spreading factor from to 8 The system performance of the iterative decoding loop can be further improved by about A+ D F an additional inner rate one RSC component code RFRNCS [] N Yee J Linnartz and G Fettweis Multi Carrier CDMA in Indoor Wireless Radio Networks roc I Int Symp on ersonal Indoor and Mobile Radio Commun IMRC) pp D D5 September [] G Fettweis A Shaikh abai and Anvari n Multi Carrier Code Division Multiple Access MC CDMA) Modem Design roc I Vehicular Technology Conference VTC) pp June 4 [] Fazel erformance of CDMAFDM for Mobile Communication System roc I Int Conf on Universal ersonal Commun ICUC) pp 75 7 ctober [4] C errou A Glavieux and Thitimajshima Near Shannon limit error correcting coding and decoding: Turbo codes roc I Int Conf on Commun ICC) Geneva Switzerland pp 64 7 May [5] S aiser and J agenauer Multi carrier CDMA Iterative Decoding and Soft Interference Cancellation roc I Global Telecommun Conf Globecom) hoenix USA pp 6 November 7 [6] S ten rink Iterative Decoding Trajectories of arallel Concatenated Codes roc rd IITG Conf on Source and Channel Coding Munich Germany pp 75 8 January [7] S ten rink Design of Serially Concatenated Codes based on Iterative Decoding Convergence nd International Symposium on Turbo Codes rest France pp September [8] J G roakis Digital Communication McGraw ill rd edition 5 [] oeher A Statistical Discrete-time model for the WSSUS multipath channel I Trans on Veh Tech vol 4 pp Nov [] J agenauer ffer L apke Iterative Decoding of inary lock and Convolutional Codes I Trans Inform Theory vol 4 no pp March 6 [] L ahl J Cocke F Jelinek J Raviv ptimal decoding of linear codes for minimizing symbol error rate I Trans Inform Theory vol pp Mar 74 [] S ten rink J Speidel R- Yan Iterative Demapping and Decoding for Multilevel Modulation roc I Global Telecommun Conf Globecom) Sydney Australia pp November 8 [] S ten rink J Speidel R- Yan Iterative demapping for QS modulation I lectronic Letters vol 4 no 5 pp July 8 [4] oeher S aiser Robertson Two dimensional pilot symbol aided channel estimation by Wiener filtering roc ICASS pp April $7 C) I