Fractionally Sampled Linear Detectors for DS-CDMA

Transcription

1 Fractionally Sampled Linear Detectors for DS-CDMA DR Brown, DL Anair, and CR Johnson, Jr Cornell University Ithaca, NY 4853 Abstract In this paper we analyze the performance of fractionally chip sampled linear multi-user detectors for Direct- Sequence CDMA communication systems We consider a general DS-CDMA system model accounting for user asynchronism and frequency selective propagation channels Analysis shows that FIR linear detectors with chip rate sampling cannot perfectly recover or more users for a system with spreading gain in the presence of frequency selective channel dynamics or user asynchronism Drawing inspiration from fractionally spaced equalization, we propose the fractionally chip sampled receiver and show that a FIR linear detector may be able to perfectly recover users Introduction This paper considers the problem of demodulating digitally modulated signals in the presence of multi-access and multipath interference with a linear multi-user detector Linear multi-user detectors for DS-CDMA systems have received increased attention due to their advantages of relatively low complexity over the optimal MLSE detector and significantly improved performance over the conventional matched filter detector In this paper we investigate the advantages of fractional chip sampling for linear multi-user detectors proposed in [6] where the sampling period of the receiver is some fraction of the chip duration Fractionally sampled equalizers FSEs) for single user linear detection have been studied since the late 96 s and their desirable properties have led to several commercial applications [8] The documented advantages of FSEs include robustness to timing phase errors [7], the ability to effectively compensate for more severe delay distortion than a baud spaced equalizer [], and the existence of FIR zero-forcing ZF) solutions for data distorted by a FIR channel [3] The necessary and sufficient Supported in part by NSF grants MIP-959, ECS , and ECS and Applied Signal Technology conditions for the existence of FIR ZF solutions for a single user) FSE can be summarized as sufficient equalizer length, subchannel disparity, and no additive channel noise This paper examines similar conditions for multi-user DS- CDMA communication systems In particular, we focus on deriving a length condition necessary for the existence of ZF solutions which, in the absence of noise, perfectly recover the transmitted symbols by completely canceling all multi-access and multipath interference The derived length condition indicates that FIR linear detectors with chip rate sampling are unable to perfectly recover users in the presence of multipath channel dynamics or user asynchronism Our necessary length condition also implies that FIR fractionally chip sampled multi-user detectors may be able to perfectly recover transmitted symbols from users Simulation results verify the analysis and additionally demonstrate that fractionally chip sampled multi-user detectors may also provide significant performance advantages in the presence of additive channel noise 2 Discrete Time System Model Consider the baseband asynchronous DS-CDMA system model shown in Figure Let represent the spreading gain common to all users and assume the spreading codes are periodic with period Let represent the combined system impulse response of the pulse shaping filter, the user s fixed or slowly time varying) channel, and the receiver input filter The baseband signal at the input of the sampler may be written as!#"%$ ) "+ '-, )/ 3254# : =< &%' '?> A@BC3D2E@BF@ )

2 , P Ä Û ÚÙÙÙÙÙ Û Œ Symbol Sequence User Symbol Sequence User K where, ` Z [a\ b YZ [\ ] Chip Sequence User ` Z ^_\ b YZ ^_\ ] Chip Sequence User K Chip Pulse Shaping Filter Chip Pulse Shaping Filter Receiver Input Filter GIH J Channel Channel K STL RUVIWX9O FIR Linear Filter KML NO Aggregate Channel Noise PSfrag replacements cd egfmhiikjml monpqmsrutwvxyvzi {}{~ Mi9 al )/7 a represents the spread and oversampled) discrete information bearing sequence of the user, is the chip duration, : +ƒe g is an arbitrary delay with respect to some reference, > is the aggregate additive channel noise, and C3 is the impulse response of the receiver input filter Note that )/ for 4 6_86 ƒeˆ where ˆ, denotes the set of all integers The output of the sampler may then be expressed as Š 6_8 o ) "+ '-, Q L RO )/ Š 6_8 254# Š 6_8 25: a > Š 6_8 2) > Š 6_8 represents the sampled receiver filtered Denoting the 6_8 where and a represent the integer and fractional number of samples in the delay, respectively, we can rewrite 2) as Œ where aggregate channel noise user s symbol delay as : Ž Š 6_8 o ) "+ '5, T M a 254 š6_8 2 š6_8 > Š 6_8 # 3) Assuming that is FIR with support 7œo 6_8 for ƒ-žmÿm 7 we can rewrite 3) as Š 6_8 o a + M a ª > Š 6_8 4) where «a ž ª ª ƒ ˆ and ª 2 a ƒ g œ±šo Note that there are exactly œ terms in the inner summa- of these terms are zero since Š$ tion and approximately /š³ for µ 6_ 6 ƒ ˆ, ² Given a linear detector ¹»º½¼ ¾ with taps, 4) enables us to directly express the output of the linear detector in terms of finite dimensional linear operators on a vector of source symbols The linear detector output may be expressed as ÀŠ Á ¹yÂ+à where à Á ÅŠ Å $ Å BÆDÇ & $šè  and ÉÅÊË Š #6_8 In light of the linear convolution in 4), let ÌÍ»ºÏÎ denote the»º œ 2=Ÿ dimensional convolution matrix such that ÀŠ o ¹ ÂÑÐ Ì aóòd Œ Ô ÖÕ 5) where Ò is the user s -sample delayed vector of oversampled) chip rate information constructed from 4) In general, Ì has the Toeplitz form a $ Ü Ý $ Ì a Ø ÙÚ á a $ Þ Ý $ ßaà Recognizing that Ò a may be formed from a linear combination source symbols, we can rewrite 5) as À oâ¹ Â½Ð Ì Öãkåäšæ_ Œ Ô Õ 6) where ã is the map from the user s symbol sequence æ a oèçié_a Å šé Å Š$ šé Å a Bê a & $7ë  to Ò In general, ã has the form ì ì a ãk ßaà ì á ì àààà where ì a is a column vector of length 8 representing the user s oversampled) spreading code The expressions ì a and ì denote spreading codes which may be upper or lower-truncated, respectively, due to user asynchronism and the finite observation interval Finally, constructing a user-ordered stacked source vector æ æ aíaïî3 æ ïíðî3 æ ñšðî3 ò Â, we can rewrite 6) as ÀŠ o ¹ Â Ì æ Ô Tò 7) where ÌË ÌÍïí ã ïíøìíóö ã åót ÌÍañŠ ã ñšóò

3 ê 3 Common Linear Multi-user Detectors Given the discrete time linear model in 7) it is straightforward to write expressions for zero-forcing ZF) and minimum mean square error MMSE) detectors under arbitrary asynchronism and multipath channels We have intentionally avoided deriving the decorrelating detector since it is conventionally defined without consideration for multipath interference The ZF detector may be viewed as a generalized decorrelating detector that completely cancels the effects of both multi-user and multipath interference [2] Given a desired user and symbol delay ô, a ZF detector is defined as any element of the set žy¹þ õ¹  à ö éya 2Eô9š Let ø ù u represent the total number of symbols in the observation interval or, equivalently, the number of columns in Ì The ZF detector exists for all and ô if and only ifædç ü ê the span of the columns of Ìú is all of û Since Ì ƒ û then this condition is equivalent to requiring that rankaìúý or Ì must have full column rank Under this assumption, the unique minimum norm ZF linear detector may be written as ¹ þ ÿöaì Â! where! is a column vector with all elements equal to zero except a value of one in the position corresponding to the é_ Í2Êô9 position in æ and denotes the Moore- Penrose pseudoinverse Given a desired user and symbol delay ô, the MMSE detector is defined as ¹ö arg ¹  à +2Eé 2 ô9 Assuming that the source symbols are BPSK and iid, 7) leads to a straightforward closed form expression for the MMSE detector: where! % # # ¹ ÉÌÏÌ Â "!$# % # %'& Ì! % is the autocorrelation matrix of the receiver filtered noise 4 System Matrix Dimensional Analysis This section applies the previous development of the discrete time º model to analyze the effects of linear detector length ) on the existence of ZF solutions for ¹ To be specific, we will require that ZF solutions exist for all ô g ŸM 7 ï ð ð 7 uab2âÿ for each úƒwžmÿm 7 Since these ZF detectors exist if and only if Ì has full column rank, we recognize that a necessary condition for full column rank is that Ì must be tall in the sense that it must have at least as many rows than columns We note that the tall condition is not sufficient for full column rank in the same sense that length conditions are not sufficient for the existence of zero-forcing FSEs for single user equalization Inspection of the individual users sub-system matrices Ì ã allows us to express the number of columns per user as a ) º œ 2wŸ 8 Ÿ 8) where Ÿy ƒ žé Ÿ9 is a term representing the effects of user asynchronism on Ì To be precise, Ÿ_oõŸ is equivalent to the condition that the user contributes an additional column to the system matrix by having an additional symbol in the observation interval when compared to a synchronized user As an example, Figure 2 shows a two user asynchronous scenario where Ÿyïí± Ÿ and ŸyåóT } Note that the overlap in the bits is due to multipath channel effects when œ,+ñÿ User, bit User, bit User 2, bit PSfrag replacements User, bit 2 User 2, bit Observation Interval User, bit 3 User 2, bit 2 User, bit 4 User 2, bit 3 cd egfmhii-gl/243 {65 i ~4783yvx29:< hi~=9y~ fyvfyvi hðvél In the synchronous case Ÿyuè?>Š and in the asynchronous case Ÿ_ is a function of : a,, 8, œ, and º The exact expression for Ÿya is not required to develop the necessary length condition It follows directly from 8) that the tall condition necessary for full column rank may be stated as º = º œ 2=Ÿ 8 Ÿ a 9) We consider the implications of this result in the following examples 5 Examples Consider the conventional case with no multipath œ Ÿ ), chip-rate sampling 8 ½Ÿ ), and synchronous users ) If the number of users equals the spreading gain ) then 9) reduces to»º º

4 F A Î œ ] l or any positive integer multiple of ) This agrees with the well known result for decorrelating detectors [4] where the decorrelating detector is realized as a bank of code matched filters A followed by the inverse of the matrix! of cross-correlations between spreading codes, ie, B 5! $ A à Since! Š$ has dimensions Î, it follows that the decorrelating detector is equivalent to an -tap, chip-spaced, linear filter for each user If the spreading codes are linearly independent then the decorrelating detector achieves perfect symbol recovery and is equivalent to a ZF detector which is satisfied when»º 2 Consider the previous case with the addition of multipath channels œ+ Ÿ ) When the number of users equals the spreading gain 9) reduces to º º œ 2=Ÿ º œ 2=ŸC+ which implies that no finite value for º can cause Ì to be tall The failure to satisfy this necessary condition implies that FIR ZF detectors do not exist for synchronous users in the presence of multipath interference with chip rate sampling 3 Consider the first case with the addition of user asynchronism such that Ÿya øÿ for at least one user When the number of users equals the spreading gain 9) reduces to º º!#"%$ Ÿ º Ÿ + which implies that no finite value for º can cause Ì to be tall Again, the failure to satisfy this necessary condition implies that FIR ZF detectors do not exist for asynchronous users with chip rate sampling 4 Consider the case of an oversampling receiver with 8 è and Ë º º 2 Ÿ Note that the number of users is greater than the spreading gain Assume further that the users are asynchronous In this case, we can manipulate 9) to write»º ÑA 2=ŸÉœ 2=Ÿ ED Ÿ Æ Ç & Ý $ Æ Ç & ÝŠ$ where Æ G Æ Æ H and D ƒ žé g Ÿ 8 2wŸ º Observe that ¼â¾ even for worst case asynchronism, is finite for and œ ¼ ¾ The next section verifies these results with simulations and demonstrates that, in addition to the aforementioned desirable properties in a noiseless scenario, the fractionally sampled receiver may provide improved performance in the presence of additive channel noise as well 6 Simulations Figure 3 shows a comparison between MMSE solutions for and 76 sampled linear detectors in a noiseless, synchronous communication scenario with multipath interference To provide a fair comparison, the channel delay ŸI and spread is fixed at g ŸIM which implies that œ èkj in the Š and Š 6 sampled simulations, respectively The transmit filter is assumed to have square root raised cosine spectrum with excess bandwidth L5 å and the propagation channel coefficients are random The receiver input filter is an ideal low pass filter with cutoff The length J spreading codes are also random with NMO elements in ž 2 Ÿ Ÿ ) The linear detector length in both cases is»º QP which satisfies the length condition in 9) for the 76 sampled receiver MMSE, Tc spaced MMSE, Tc/2 spaced user user user user user user user 2 user 3 user 4 user 5 user 6 user 7 user user user user 8 cd e f hiisr lbr rknt/vu#xkfévi9h3w9_ Mi9 3xS7~h 3W9_ vn3 {65 ĩ ý dx9éi 39h Mi#zi:#z~hðvÉl}n da{ fm 3Éz7d ~Y9Z539h[3 { izi9hiv\ š6 :<3W929yi9 vf25w59~hïz v59hði 3_ dx9yeíew3 dx9 ŸIM i`gf3 Y59~ aoi9hdfyvi hðv ^] º _JY] èqj bpm Observe that the 6 sampled linear detector is able to perfectly recover the symbols of all users at all delays implying that the linear detector completely canceled both multi-user and multipath interference Inspection of the system matrix confirmed that full column rank was achieved for the fractionally sampled receiver and that ZF solutions exist for all users at all possible delays The Š sampled detector exhibits considerably worse performance since the system matrix is not full column rank for any finite choice»º of when and œc+ýÿ Note that the MMSE solutions may be indexed by user and delay, ie, MMSE! d é_ 2-ô9D25ÀŠ where ÀŠ is the output of the appropriate MMSE linear detector We will use this notation in the next simulation

5 l Figure 4 is a Monte Carlo comparison between and Š 6 sampled linear detectors in the presence of AWGN with synchronous and asynchronous equal power users The asynchronous users each have uniformly distributed delay over As in the first simulation, MMSE is calculated for all users at all possible delays but in this case the delay-optimal MMSE, given by MMSE!e f MMSE!, is selected for each user and then averaged over all users and 5 experiments The noise variance g % ðÿ The linear detector length º is varied from 2 to All other parameters are as in the first simulation Experimentally averaged, delay optimal MMSE db) Synchronous Users K=2 users K=4 users Length, N f Experimentally averaged, delay optimal MMSE db) Asynchronous Users K=2 users K=4 users Length, N f cd e f hðiihbl t'jyi9h[3ye ĩ i 3#xÉpÖ~Y5 z7da{s3»r rknt/k7~h l v~g d = dx9éĩ vmf3w9y vn3 6 {n5m ié l W3yv^<yĩ = dx9yivmö dx9ép i 39h izi:#z~hðva d zd<5vxw9:< hð~y9é~gfévo3w9_ Z3yvxW9:< hð~y9é~gfév i ` f3 Y59~ aoi h+févi9hiṽ longda{ f 3Éz7d ~=9539h[3 { izi9hiv\ :<3W9W9Éi vf25w59~hðz v^5 hði 3_ dx9yeke23 dx9 ŸI ] QJ This simulation provides evidence indicating that 6 sampled receivers may achieve better MMSE performance than sampled receivers in the presence of additive channel noise, but only for larger values of»º The relatively poor performance of the fractionally sampled receiver for small values of»º is due to the fact that, for a fixed value of»º, the observation interval of the 6 sampled receiver is half that of the sampled receiver In this simulation, values of º ¼ I cause the š6 sampled receiver s observation interval to not contain even one full bit including multipath) from a synchronous user Also note that this simulation suggests that the effects of asynchronism on averaged delay-optimal MMSE performance are almost negligible for both and 6 sampled receivers 7 Conclusions In this paper we have explored the concept of fractional sampling to improve the performance of linear multiuser detectors for DS-CDMA communication systems The analysis has shown that fractionally sampled receivers of sufficient length satisfy a necessary condition for the existence of zero-forcing solutions under conditions where no FIR solution exists for chip rate sampled receivers Furthermore, simulations in the presence of additive channel noise indicate that fractionally sampled receivers may also offer improved MMSE performance over chip rate receivers The robustness to timing phase errors property of single user FSEs was not explored in this paper for fractionally sampled multi-user detectors and remains an open research topic Necessary and sufficient conditions for the existence of FIR zero-forcing solutions in DS-CDMA communication systems with arbitrary asynchronism and multipath channels are also a topic of current research References [] RD Gitlin and SB Weinstein, Fractionally-spaced equalisation: An improved digital transversal equaliser, Bell System Technical Journal, vol 6, pp , Feb 98 [2] S Gollamudi, et al, On the dimensional limitations of linear multiuser detection, in Proc Thirty-Fifth Annual Allerton Conference on Communication, Control, and Computing Monticello, IL), pp tbd, Sep 25-29, 998 [3] CR Johnson Jr, et al, Blind equalization using the constant modulus criterion: A review, Proceedings of the IEEE, vol 86, no, pp 927-5, Oct 998 [4] R Lupas and S Verdu, Linear multi-user detectors for synchronous code-division multiple-access channels, IEEE Trans on Information Theory, vol 35, pp 23-36, Jan 989 [5] U Madhow and M L Honig, MMSE interference suppression for direct-sequence spread-spectrum CDMA, IEEE Trans on Communications, vol 42, no 2, pp , Dec 994 [6] U Madhow, Signal processing for interference suppression in direct-sequence CDMA systems, in Proc International Conference on Acoustics, Speech and Signal Processing Atlanta, GA), pp 65-8, May 996 [7] M Shafi and DJ Moore, Further results on adaptive equalizer improvements for 6 QAM and 64 QAM digital radio, IEEE Trans on Communications, vol 34, no, pp 59-66, Jan 986 [8] JR Treichler, I Fijalkow, and CR Johnson, Jr, Fractionally spaced equalizers: How long should they really be?, Signal Processing Magazine, vol 3, pp 65-8, May 996