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A Low-Complexity Space-Time RAKE Receiver for DS-CDMA Communications Javier Ramos, Associate Member, IEEE, Michael D. Zoltowski, Senior Member, IEEE, and Hui Liu, Member, IEEE
Abstract—An algorithm is presented for estimating the quantities needed by a space-time RAKE receiver for DS-CDMA to achieve maximum SINR for the desired user. The proposed algorithm, which is applicable in the case of either periodic or aperiodic spreading codes, asymptotically provides the exact time of arrival of each dominant multipath within a bit period, and the optimum beamformer for extracting each multipath. This is achieved by using cyclostationarity to exploit the difference between the respective spectra of the desired user and each multiuser access interferer at the output of the correlator based on the desired user’s code. Estimates of the relative time delay and optimal beamformer for each RAKE finger are used by the spacetime RAKE receiver to optimally combine the desired user’s multipath while simultaneously canceling strong multiuser access interference. Simulations of a near–far scenario are presented demonstrating the efficacy of the proposed algorithm. Index Terms—Array processing, CDMA, RAKE.
I. INTRODUCTION
A
SPACE-TIME RAKE receiver in a DS-CDMA system works to attain maximum SINR for the signal with the desired code (SDC) by optimally combining the SDC’s multipaths to achieve diversity gains, while simultaneously canceling strong multiuser access interference (MUAI) to provide near–far resistance. Several schemes have been proposed in recent years for effecting an optimum space-time RAKE receiver knowing only the spreading waveform of the desired user [1], [2]. An advantageous attribute of the schemes proposed in [1] and [2] is that they are applicable whether the spreading waveform for each user is aperiodic or periodic. In contrast, decorrelating receivers for DS-CDMA are not applicable when the spreading waveforms are aperiodic. Many military and commercial DS-CDMA systems, including that based on the IS-95 standard, employ aperiodic spreading waveforms. However, the schemes proposed in [1] and [2] require the estimation and processing of large space-time correlation matrices. In addition, these schemes require estimation of the Manuscript received January 4, 1997. This work was supported by the Air Force Office of Scientific Research under Grants F49620-95-1-0367 and F49620-97-1-0318, and the Army Research Office under Grant DAAHO495-1-0246. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. K. Buckley. J. Ramos is with the Polytechnic University of Madrid, Madrid 28007, Spain (e-mail:
[email protected]). M. D. Zoltowski is with the School of Electrical Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail:
[email protected]). H. Liu is with the Department of Electrical Engineering, University of Virginia, Charlottesville, VA USA 22903 (e-mail:
[email protected]). Publisher Item Identifier S 1070-9908(97)06699-6.
space-time correlation matrix of the MUAI’s alone. To estimate such, the scheme proposed in [1] requires computation of the output of the correlator matched to the desired user’s code for a large number of delays encompassing roughly a bit period. In contrast, the scheme proposed herein requires that one multiply by a sampled version of the desired user’s spreading waveform and integrate (sum) for each of a number of delays encompassing only the multipath delay spread, which is typically a fraction of a bit period. This is commensurate with a standard RAKE receiver for a DS-CDMA system. The proposed scheme also offers reduced computational complexity relative to the schemes proposed in [1] and [2], since the information needed to form the weights for optimally combining the RAKE fingers across both time and space is obtained from processing matrices of dimension equal to the number of antennas, as opposed to large dimension space-time correlation matrices. The underlying basis for the new algorithm may be gleaned from observing the sample spectral densities plotted in Fig. 1. For this illustrative example, each user was assigned a (unique) maximal length PN code with 127 chips per bit and a spectral . Recall raised cosine chip pulse waveform with that the front end operation at each antenna is correlation with a sampled version of the SDC’s spreading waveform. In the example, the sampling rate was twice per chip. Fig. 1(a) plots the fast Fourier transform (FFT) of the square of the output of the correlator when the input is the SDC’s spreading waveform. Fig. 1(b) plots the FFT of the square of the output of the correlator when the input is the spreading waveform for another user. Again, since the correlator is matched to the SDC’s spreading waveform, Fig. 1(b) displays the FFT of the square of the cross-correlation between the respective spreading waveform for the desired user and an MUAI. It is observed to be negligible except at DC. In contrast, the FFT of the square of the autocorrelation of the spreading waveform for the desired user plotted in Fig. 2(a) is observed to have significant energy at other frequencies as well as a large DC value. Thus, the basic principle underlying the SCORE family of algorithms motivates the formation of a pair of cyclic spatial correlation matrices from samples obtained at the output of the correlator behind each antenna. A cyclic correlation matrix evaluated at zero (DC) cycle frequency will have contributions from both the MUAI’s and the SDC, while a cyclic correlation matrix evaluated at some other cycle frequency will only have contributions from the SDC. It is shown that the generalized
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(a)
Fig. 2. BER versus number of strong MUAI’s.
(b) Fig. 1.
j
Sk` (fTc )j for (a) k = 1 (SDC term) and (b) k 6= 1 (MUAI term).
eigenvalues and eigenvectors of this cyclic correlation matrix pencil are one-to-one related to the relative time-delay and optimal beamformer for each dominant multipath of the SDC. Choosing the nonzero cycle frequency equal to the reciprocal of the multipath delay spread insures a one-to-one mapping between the phase of each generalized eigenvalue and the relative time delay of the associated SDC multipath. II. SIGNAL MODEL
where is the th symbol value, is the code assigned to the th user, and is the bit period. Each code is composed of chips of duration . is the array manifold that describes the relative gain and phase at each antenna induced by a planewave arriving from , assuming the propagation delays to be small compared to the chip pulse duration as is the case in current cellular DS-CDMA systems. is the noise vector. Equation (1) assumes the code remains the same from bit to bit. However the IS-95 standard and other CDMA systems use different parts of a longer code for each bit. We refer to such systems as aperiodic. Besides substituting by , where now the code utilized by the user depends on the bit index , the rest of the signal model is still valid for aperiodic systems. III. BLIND SPACE-TIME RAKE RECEIVER The vector of outputs obtained after correlating with a sampled version of the SDC’s spreading waveform at each antenna may be expressed as
(2) where the SDC is enumerated as
An -element antenna array is employed at a base station to collect superimposed uplink signals from cochannel users in a DS-CDMA system. Each user is assigned a different spreading waveform typically based on a pseudonoise (PN) code. The signal transmitted from the th user, , arrives at the antenna array via paths, each of which is attenuated and phase shifted by a complex gain . The th path of the th user arrives with a delay of at the two-dimensional (2-D) direction . The baseband array output can be expressed in vector form as
and (3)
The operator denotes convolution, and . For ease of explanation, the algorithm for estimating the relative time delays and optimal beamformer for each dominant multipath of the SDC will first be developed in a continuous time framework. Estimators of the cyclic spatial correlation matrices in the practical case of sampled data will be prescribed subsequently. A. Second-Order Cyclic Statistical Properties Consider the
time-varying spatial correlation matrix rect
(1)
(4)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 9, SEPTEMBER 1997
where is the expectation operator, denotes conjugate transpose, is the multipath delay spread, and rect is unity over an interval of width centered at and zero elsewhere. Approximate synchronization is assumed so that the time window encompassed by rect encompasses the dominant RAKE fingers for a given bit period. Approximate synchronization may be obtained blindly (knowing only the SDC code) by setting equal to the bit interval, , as described in [3]. Exploiting the cyclostationarity of the DS-CDMA data described by (2), is estimated as
rect
B. Multipath Delay Estimation and Beamforming Assuming the number of dominant SDC multipaths to be less than the number of antennas, , (9) dictates that has rank . As a result, zero is a generalized eigenvalue of the matrix pencil of multiplicity . Theorem 2 below was proved in [3]. Theorem 2: Let denote the nonzero generalized eigenvalues of . The argument of is one-to-one related to the relative time-delay of the th SDC multipath as . Further, the corresponding generalized eigenvectors defined may be expressed as by
(5)
is defined as , where represents the Fourier transform operating elementwise on . Theorem 1 below was proved in [3] under the following assumptions. 1) The symbol sequences are independent and identically distributed (i.i.d.) for a given user and among users, i.e., , where is the average energy per symbol. 2) The noise is i.i.d. among antennas with variance . 3) The noise is uncorrelated with all CDMA signals. 4) The number of chips per bit is large such that the Fourier transform of is a circular complex Gaussian random process. Theorem 1:
(10) above, we first prescribe To understand the significance of how the cyclic correlation matrices and are estimated from the sampled data. Again, approximate synchronization is assumed so that a tapped-delay line is set up at the correlator output behind each antenna encompassing the multipath delay spread. For the sake of simplicity, assume the time spacing between taps to be ; the number of taps for each antenna is . For the th delay tap, , form the following matrix by averaging over bits, for which the multipath parameters are constant, as follows: (11) .
and
are then estimated
as sinc (12) (6) where (13) if
(7)
if and sinc and
is the Fourier transform of the chip pulse,
. Consider the matrix pencil formed by evaluating and , respectively. It follows from (7)
at
(8)
is the standard Substituting (11) into (12), it follows that sample spatial correlation matrix formed from array snapshots recorded in the vicinity of the RAKE fingers, averaged over bits. Thus, the weight vectors in (10) are the well-known MVDR beamformers that maximize the SINR for each of the SDC multipath rays. C. Optimum Combination of Beamspace-Time Samples The tap index, , of the th RAKE finger, associated with the th SDC multipath, is estimated as the closest integer to . Define , where is the time series obtained by applying the generalized eigenvector to the vector of samples obtained at the th tap at each antenna, and then sampling once per bit.
(9) Thus, asymptotically, neither the MUAI’s or the receiver noise contribute to .
The decision statistic for the th bit period, denoted is constructed as a weighted sum of
, :
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. The vector is determined as that which maximizes the SINR in . Recall that is asymptotically the MVDR beamformer that extracts the th SDC multipath and, thus, ideally steers spatial nulls toward strong MUAI’s. Thus, there is ideally negligible MUAI contribution to . The goal then is to maximize the signalto-noise ratio (SNR) in . If the noise spectral density is flat across the front-end bandpass filter, then it easy to show that chip rate sampling leads to temporally uncorrelated noise assuming use of a chip pulse waveform either having a rectangular shape or a raised cosine spectrum. Since the beams are formed at tap times separated by an integer multiple of , the following criterion defines that vector , which maximizes the SNR in : (14) is the “largest” eigenvector of is estimated as
. .
IV. SIMULATIONS A scenario involving an SDC with multipath parameters listed in Table I and strong MUAI’s emulating a near–far problem was simulated to assess the efficacy of a space-time RAKE receiver using the proposed scheme for estimating the time-delay and MVDR beamformer for each SDC path. A linear array of eight omni antennas equispaced by a halfwavelength was used; 0 represents broadside. The chip waveform had a raised cosine spectrum with . The number of MUAI’s was varied from one to ten; each MUAI was 20 dB stronger than the SDC path arriving at
TABLE I SDC MULTIPATH PARAMETERS
. The th MUAI arrived via a single path at 90 20 . The code for each MUAI was a different cyclic shift of the same maximal length sequence of 127. The code for the SDC was a different maximal length sequence of 127. For all users, the code was periodic. bits were averaged to estimate and . In the Monte Carlo simulations, the output of the reduced dimension space-time RAKE receiver using the time delays and MVDR beamformers estimated according to Section II-B is fed to a slicer. Fig. 2 shows the bit error rate (BER) as a function of the number of strong MUAI’s. This is compared with that obtained with a standard RAKE receiver at a single antenna. The presence of even a single strong MUAI causes the latter to provide totally unreliable bit decisions.
REFERENCES [1] M. D. Zoltowski, Y.-F. Chen, and J. Ramos, “Blind 2D RAKE receivers based on space-time adaptive MVDR processing for the IS-95 CDMA system,” in Proc. Milcom’96, pp. 618-622. [2] H. Liu and M. D. Zoltowski, “Blind equalization in antenna array CDMA systems,” IEEE Trans. Signal Processing, vol. 45, pp. 161–172, Jan. 1997. [3] J. Ramos and M. D. Zoltowski, “Blind 2D RAKE receiver for CDMA incorporating code synchronization and multipath time delay estimation,” in Proc. ICASSP’97, pp. 4025–4029.
