Maravic, Irena

Vetterli, Martin

Infoscience - École polytechnique fédérale de Lausanne

DIGITAL DS-CDMA RECEIVERS WORKING BELOW THE CHIP RATEIrena Maravic´ and Martin Vetterli Audio-Visual Communications LaboratorySwiss Federal Institute of Technology in Lausanne, CH-1015 Lausanne, SwitzerlandDepartment of EECS, University of California at Berkeley, Berkeley CA 94720, USAABSTRACTWe consider the problem of designing low-complexity dig-ital receivers for CDMA systems operating over channelswith single or multiple propagation paths. We extend someof our recent sampling results for certain classes of non-bandlimited signals and develop a method that takes ad-vantage of transform techniques to perform channel esti-mation and signal detection from a low-dimensional sub-space of a received signal, that is, by sampling below thetraditional Nyquist rate. By lowering the sampling rate wecan reduce computational requirements compared to exist-ing solutions, allow for slower A/D converters and signif-icantly reduce power consumption of digital receivers. Ineffect, we use an algorithmic solution to improve hardwarespecifications in terms of complexity and power.1. INTRODUCTIONCode-division multiple access (CDMA) is a widespreadaccess protocol well suited for voice and data transmis-sion, particularly over wireless communication networks.CDMA possesses several advantages over earlier accesstechniques due to a coded signal format and an expandedbandwidth. In particular, selective addressing capability,low-density power spectra and interference rejection areamong properties that have prompted increased interest inCDMA as a flexible and spectrally efficient access strategy[10].Although theoretically sound, implementation ofCDMA receivers is not trivial in practice. On the one hand,in current attempts at low-power integrated radio solutions,a trend is to eliminate as much as possible the necessity foranalog components and perform all processing digitally.On the other, application of proposed high-performancedigital schemes [1] [5] [7] to real time systems is often notfeasible, since most of them require high sampling rates(chip rate sampling or even fractional sampling) and there-fore very fast and expensive analog-to-digital converters.Besides, sampling at the chip rate implies that all opera-tions are performed on an extensive set of samples, whichmakes such schemes computationally intense and often in-efficient.In this paper, we present a new approach toward de-signing low-complexity digital receivers for CDMA sys-tems which uses lower than the Nyquist sampling rateand yet has performances similar to those of chip-sampledschemes. We extend some of our recent sampling re-sults for certain classes of non-bandlimited signals [4] [9]to the problem of channel estimation and signal detec-tion from a lowpass approximation of a received signal.Our approach leads to reduced computational requirementsand lower power consumption compared to existing tech-niques, while providing similar performances.2. SUBSPACE CHANNEL ESTIMATION2.1. Problem FormulationConsider the general case of a direct sequence (DS) CDMAsystem with  users operating over a multipath fadingchannel with at most  propagation paths for each user.We will assume that the channel varies slowly, i.e. it isconsidered constant over a channel estimation window. Acommon model for a received baseband signal 	 is	 ﬀﬂﬁﬃ  ﬃ !#"	$&%(' ﬃ)!%+*&,-.0/(1	$ (1)where "  	$ is a signature sequence of length 2 assigned touser 3 , ' ﬃ)! denotes the delay of the signal of user 3 alongthe 4 -th path, ﬃ)! is a complex propagation coefficient,ﬀdenotes the * -th bit sent by user 3 , 1	$ is additive whiteGaussian noise and ,- denotes a symbol duration.The above model can be interpreted as saying that thereceived signal 	 is given by a sum of attenuated anddelayed copies of transmitted signals and noise. Note that	$ has only 567 degrees of freedom (given knowledgeof bits sent by each user), namely time delays ' ﬃ ! andpropagation coefficients ﬃ ! , which can be efficiently esti-mated if we consider the problem in the frequency domain.Let 8&9 :<;denote DFT coefficients of the received sig-nal 	8 9 :=;>ﬀ?ﬁﬃ  ﬃ !A@9 :<;CBED6FHG>IKJMLON)P QR/72 9 :<;(2)where SUTVW5EXZY[, - and @ 9 :=;and 2&9 :=;are DFT coef-ficients of signature sequences and noise respectively. Amore compact form of (2) is given by8]\ 9 :<;[ﬀ_^G/#2 9 :<;(3)where^G@9 :=;ﬁﬃ 	 ﬃ)!BD6FHG`I J L NaP QR , or equiva-lently, bdce/gfih+\(4)Note that the coefficients^G are given by a sum ofcomplex exponentials, where the unknown time delays ap-pear as complex frequencies while propagation coefficientsappear as unknown weights. Therefore, the problem ofchannel estimation can be considered as a special case ofclassic spectral estimation problems [2] [6], where the pa-rameters ' ﬃ ! and ﬃ ! can be uniquely found from 5j coef-ficients^G, by applying either subspace methods for har-monic retrieval [6] or the annihilating filter approach [9].A set of 56 adjacent DFT coefficients 8 \ &9 :<;used in (4)can be obtained, for example, from a set of uniform sam-ples taken with the sinc kernel of bandwidth9%klSUTEmn&SUT;[9].The minimum required size of the matrixbis 5jAop ,thus in order to estimateb(giveneand h+\ ) each user3 must send a training sequenceelqeq9q0 qﬀ rtsnsusqﬀ;(5)In the noiseless case, our method leads to a perfect esti-mation of the channel parameters by taking only 5j sam-ples per symbol, while the number of symbols in the train-ing sequences depends on the number of active users  .In the noisy case, the sampling rate should be increased tomake the algorithm robust to noise, however, the requiredsampling rate is typically far below the chip rate.Clearly, we have decomposed the problem of multiuserchannel estimation into a series of single user problems,where parameters of each user are estimated separately byconsidering only one corresponding column of the matrixb. Besides, unlike some high-resolution channel estima-tion methods that use iterative search and require a longtraining phase [5], our algorithm ensures a high channelthroughput since the length of the training interval directlyaffects the channel capacity.3. SUBSPACE SIGNAL DETECTIONThe developed channel estimation algorithm is rather gen-eral and allows for implementation of different detec-tion strategies, such as matched filters, multiuser detectionschemes or RAKE receivers. In the following, we will dis-cuss the application of our sampling results to the case oflinear multiuser detection where all the operations are im-plemented on a reduced set of samples, obtained by sam-pling both the received signal and the reference with thesame kernel and at the same rate.3.1. Linear Subspace DetectorIn quasi-synchronous CDMA systems, where a maximumdelay spread of each user is relatively small compared tothe symbol duration , - , equation (4) can be directly used tosolve for bits sent by each user, that is, without extractingindividual channel parameters. During the training phase,the receiver should only estimate the matrixb, while dur-ing the detection phase, bits sent by users v.m56m s$sws  aregiven byebDch(6)It is clear that in this case the receiver does not requireknowledge of the user’s signature sequences, however, (6)can be successfully applied to systems where approximatetiming of users is a priori known to the receiver. If that isnot the case, the receiver must first estimate the unknownchannel parameters and then compute a new matrixbyxthatwill be used for detection. The elements^G ofbyxaregiven by^G@9 :<;<ﬁﬃ)	 ﬃ)!B.D6FHG`IEJzLN)P QR (7)where {M' ﬃ ! now denotes a delay shift with respect to thenew reference at the receiver, which is assumed to be onthe order of a chip duration. The same approach can be ex-tended to sequence detection, that is, we can solve directlyfor a sequence of bitse9ﬀ 0 rsusnsﬀﬁ;sent byeach user |bk}Dch (8)The developed subspace detector is only one possiblelinear strategy that leads to error-free detection in the ab-sence of noise, while the concept of subspace detection ismore general and can be applied to other linear detectionmethods as well. For example, consider a decorrelating de-tector for a synchronous non-fading channel [8]. Assumethat the received baseband signal 	 is sampled with thesinc sampling kernel and then passed through a bank of matched filters, each matched to the lowpass approxima-tion of the corresponding signature waveform. The outputof such a filter bank is given by~]pe/<1 (9)wheren	rms$swsmeW	rmsws$smwhile  is the normalized cross-correlation matrix of thefiltered signature sequences "  	 and " F 	$ , with elementsF given byFG"Ł	:, -G`"F	:, -G` (10)By multiplying the vector of matched filter outputs (9) byD, we getD~ e(11)for ? . In order to recover transmitted bits, one cantake the sign of each component in (11), that is, sgn 	D&~] sgn 	e (12)which achieves perfect detection for each active user in theabsence of noise. The subspace detection is thus a simpleand natural strategy that can be implemented in virtuallyany detection scheme, thus potentially allowing CDMA re-ceivers to exhibit a good performance/complexity tradeoff.4. COMPUTATIONAL COMPLEXITYIt is of interest to explicitly evaluate computational orderof our developed scheme and compare it with that of theexisting techniques.The major computational requirement of the channelestimation method is dominated by the singular value de-composition part, which in the case of timing synchroniza-tion results in an overall computational order of 	Hi]per user (in one synchronization frame), where  denotesthe number of samples per symbol. It is worth noting atthis point that reducing the sampling rate y- times notonly reduces the computational complexity, but also powerconsumption of A/D converters and the required processorspeed, and that approximately by a factor of r-.Implementation of the simplest timing synchronizationmethod based on matched filters requires 	H2r opera-tions, where 2 denotes the spreading factor. As we will seein the next section, the subspace estimator has essentiallythe same computational order, yet the technique based onmatched filters does not have the satisfactory resolutionperformance and is suitable only for initial synchroniza-tion. High resolution methods [1] typically involve twocomputationally intense steps, namely, estimation of the2o2 signal correlation matrix and the singular-value-decomposition step, both requiring on the order of 	H2]computations.We next focus on the computational complexity of de-veloped subspace detectors. In the case of the subspacedetection (6), the receiver must first compute a pseudoin-verse of the matrixb, which requires 	r opera-tions, while the detection phase itself has computationalorder of 	H4	H per symbol (associated to comput-ing  DFT coefficients of the received signal). Similarly,implementation of the subspace decorrelating or MMSElinear detector involves inversion of a  o matrix, astep that requires 	HpO computations, whereas the com-putational order during the detection phase is 	H . Thisis clearly much lower than the order of the channel estima-tion part, yet the detection phase requires higher samplingrates to achieve performances comparable to those of chip-sampled schemes, as we will see in the next section. Notethat equivalent chip-sampled multiuser detection schemesrequire on the order of 	¡¢/g2£ operations per one de-tection frame.5. NUMERICAL RESULTSIn this section, we show simulation results that illustratethe performance of the developed scheme. Figures 1-4illustrate the timing synchronization performance of ourmethod for an AWGN channel with BPSK modulation,where the spreading is achieved with pseudo-random se-quences of length 511. Figure 1 shows an average tim-ing estimation error (normalized to the chip duration ,¤ )versus the sampling rate in non-fading channels with 10users. Our algorithm clearly yields high resolution esti-mates1 even with a modest number of samples. For ex-ample, by sampling the received signal at one fourth of thechip rate, the average estimation error is less than one tenthof the chip duration (for ¥¦uYu2§TW¨ ©Eﬀª ). Comparisonof the timing estimation errors for systems with 5 and 10users is illustrated in Figure 2, indicating that the numberof active users doesn’t affect the estimator performance,since we have decomposed the multiuser estimation prob-lem into a series of single-user equivalents. Due to thesame reason, our scheme is near-far resistant, as shownin Figure 3. The performance of our estimator in multi-path fading channels is illustrated in Figure 4, where weassumed 3 propagation paths for each user. The error issomewhat higher than in the case of non-fading channels,however, it is interesting to note that if there is only onedominant path, the estimation error is basically the sameas in the non-fading channels.1Within a fraction of the chip duration.We next consider the bit-error-rate performance ofthe developed subspace detection scheme in quasi-synchronous systems, where timings of all users are as-sumed to be perfectly estimated. Figure 5 illustrates theperformance in the system with 10 users versus the sam-pling rate, while dotted lines correspond to the decorre-lating detector implemented on the same set of samples.The two methods have essentially the same performance,however, the BER exhibits a relatively slow decay rate.Namely, if the sampling rate is below the chip rate «¢¤ , welose a part of the signal energy and effectively reduce theSNR, which in turn increases the average BER. Note thatthe same argument holds for the timing estimation perfor-mance, however, in this case a dependence of the estima-tion error on the SNR is not the same. In Figure 6 wecompare the detection performance of chip-sampled andsubsampled schemes for different values of SNR. Roughlyspeaking, sampling at half the chip rate is sufficient to at-tain the average BER close to that of the chip-sampledscheme. Whether or not it is advisable to detect the sig-nal from its subspace is obviously a matter of tradeoff be-tween the complexity of the receiver on the one hand andits performance on the other, which is particularly relevantin the case when computational requirements and powerconsumption are the main constraints in the system design.6. CONCLUSIONWe have presented a new approach to the problem of de-signing low-complexity digital receivers for DS-CDMAsystems, where all the necessary steps are carried out ona sampled lowpass version of the received signal. Weextended some of our recent sampling results for certainclasses of non-bandlimited signals to the multiuser chan-nel estimation problem and developed a method that esti-mates the unknown channel parameters of all users simul-taneously, and that from the uniform set of samples of thereceived signal taken well below the chip rate. Once thechannel parameters of each user have been estimated, theyare used in the detection process, carried out on a lowpassversion of the received signal as well. Our approach leadsto lower computational requirements and reduced powerconsumption compared to existing solutions, thus allowingfor a practical hardware implementation and all the bene-fits of a digital design.7. REFERENCES[1] S. E. Bensley and B. Aazhang, “Maximum-Likelihood Synchronization of a Single User Code-Division Multiple-Access Communication Systems”,0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7Sampling Rate / Chip Ratedelay estimation error / TcNormalized delay estimation error, Nu=10, N=511SNR=5dBSNR=7dBSNR=10dBFig. 1. Average timing synchronization error (normalized to ¬j­ ) in themultiuser case (10 users) vs. sampling rate. We assumed a non-fadingchannel. Signature sequences of all users are of length 511.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7Sampling Rate / Chip Ratedelay estimation error / TcNormalized delay estimation error, Nu=5,10, N=511SNR=5dB Nu=10SNR=7dB Nu=10SNR=10dB Nu=10SNR=5 Nu=5SNR=7 Nu=5SNR=10 Nu=5Fig. 2. Comparison of the timing synchronization errors in non-fadingchannels for systems with 5 and 10 users.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7Sampling Rate / Chip Ratedelay estimation error / TcNormalized delay estimation error, Nu=10, N=511, near−far scenario SNR=5dB P2−P10=P1SNR=7dB P2−P10=P1SNR=10dB P2−P10=P1SNR=5dB P2−P10=10P1SNR=7dB P2−P10=10P1SNR=10dB P2−P10=10P1Fig. 3. Timing synchronization errors for a system with 10 users in thenear-far case.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7Sampling Rate / Chip Ratedelay estimation error / TcNormalized delay estimation error, Nu=10, N=511SNR=7dB, one propagating pathSNR=7dB, 3 paths, 1 dominant (estimate only that one)SNR=7dB, 3 paths, equal power (estimate all)Fig. 4. Comparison of timing estimation errors for the system with 10users is single-path and multipath channels (3 propagation paths for eachuser).0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.020.040.060.080.10.12Sampling Rate / Chip RateBERBER performance of the subspace detector, Nu=10, N=511SNR=5dBSNR=7dBSNR=10dBFig. 5. BER performance of the subspace detector in non-fading syn-chronous CDMA systems with 10 users. Dotted lines illustrate the BERperformance of the decorrelating detector.0 2 4 6 8 10 1210−910−810−710−610−510−410−310−210−1SNRBERBER performance of the subspace detector vs. SNRRc0.6 Rc0.5 Rc0.33 RcFig. 6. BER performance of the subspace detector vs. SNR for differentvalues of the sampling rate.IEEE Transactions on Communications, Vol. 46, No.3., p.p. 392-399, March 1998.[2] S. Y. Kung, K. S. Arun and D. V. Bhaskar Rao,“State-space and singular-value decomposition-basedapproximation methods for the harmonic retrievalproblem”, Journal of the Optical Society of America,Vol. 73, No. 12, pp. 1799-1811, December 1983.[3] Z. S. Liu, J. Li and S. L. Miller, “An efficient code-timing estimator for receiver diversity DS-CDMAsystems”, IEEE Transactions on Communications,Vol. 46, pp 826-835, June 1998.[4] I. Maravic and M. Vetterli, “Sampling Results forClasses of Non-Bandlimited 2-D Signals”, IEEETransactions on Signal Processing, submitted De-cember 2001.[5] J. Ramos and M. Zoltowski, “Low-ComplexitySpace-Time Processor for DS-CDMA Communica-tions”, IEEE Transactions on Signal Processing, Vol.48, No. 1, January 2000.[6] B. D. Rao and K. S. Arun, “Model Based processingof Signals: A State Space Approach”, Proceedingsof the IEEE, Vol. 80, No. 2, pp. 283-309, February1992.[7] E. G. Strom, S. Parkvall, S. L. Miller and B. E. Ot-tersten, “DS-CDMA synchronization in time-varyingfading channels”, IEEE J. Selected Areas in Commu-nications, vol. 14, pp. 1636-1642, October 1996.[8] S. Verdu, Multiuser Detection, Cambridge UniversityPress, 1998.[9] M. Vetterli, P. Marziliano and T. Blu, “Sampling sig-nals with finite rate of innovation”, IEEE Transac-tions on Signal Processing, June 2002.[10] A. J. Viterbi, CDMA - Principles of Spread Spec-trum Communication Reading, MA: Adison-Wesley,1995.

Digital DS-CDMA receivers working below the chip rate

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Infoscience - École polytechnique fédérale de Lausanne