blind adaptive interference suppression for the near-far ...124 ieee transactions on signal...

124 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 1, JANUARY 1997

Blind Adaptive Interference Suppressionfor the Near-Far Resistant Acquisition and

Demodulation of Direct-Sequence CDMA SignalsUpamanyu Madhow,Senior Member, IEEE

Abstract— Two key operations required of a receiver ina direct-sequence (DS) code division multiple access (CDMA)system are thetiming acquisition of transmissions that are start-ing up or have lost synchronization, and thedemodulation oftransmissions that have been acquired. The reliability of boththese operations is limited by multiple-access interference, es-pecially for conventional matched filter-based methods, whoseperformance displays an interference floor and is vulnerable tothe near-far problem. Recent work has shown that, providedtiming information is available for a given transmission, it canbe demodulated reliably using blind or training-sequence-basedadaptive interference suppression techniques. These techniquesare near-far resistant, unlike the matched filter demodulator, anddo not require explicit knowledge of the interference parameters,unlike nonadaptive multiuser detectors. In this paper, we presenta blind adaptive interference suppression technique forjointacquisition and demodulation, which has the unique feature thatthe output of the acquisition process is not simply the timing of thedesired transmission, but a near-far resistant demodulator thatimplicitly accounts for knowledge of the timing and amplitudesof all transmissions to suppress the multiple-access interference.The only knowledge required by the scheme is that of the desiredtransmission’s signature sequence, so that it is amenable to adecentralized implementation. On the other hand, it can beefficiently implemented as a centralized scheme in which thebulk of the computations for the adaptation are common to alltransmissions that need to be acquired or demodulated.

I. INTRODUCTION

T HE major limitation on the performance and capacityof direct-sequence (DS) code division multiple access

(CDMA) is the multiple-access interference due to simulta-neous transmissions. In particular, the conventional matchedfilter demodulator, which ignores the structure of the in-terference, suffers from aninterference floor(i.e., its errorprobability does not go to zero as the background noise levelvanishes), and from thenear-far problem(i.e., its error proba-bility can deteriorate significantly if an interfering transmissionhas a much higher power than the desired transmission). Inrecent work [1], [8], [10], [14], it has been shown that min-

Manuscript received November 16, 1995; revised August 27, 1996. Thiswork was supported by the Office of Naval Research under Grant N00014-95-1-0647. This paper was presented in part at the 29th Annual Conferenceon Information Sciences and Systems, March 22–24, 1995, Baltimore, MD,and at the IEEE Military Communications Conference, November 5–8, 1995,San Diego, CA.

The author is with the Electrical and Computer Engineering Departmentand the Coordinated Science Laboratory, University of Illinois, Urbana, IL61801 USA (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(97)00540-0.

imum mean squared error (MMSE) receivers can be used tosuppress multiple-access interference. The MMSE receiver canbe implemented adaptively, e.g., by using a training sequenceof symbols for the desired transmission for initial adaptation,followed by decision-directed adaptation. The MMSE detectorcan also be implemented viablind adaptation [3], in whichknowledge of the desired transmission’s timing and spreadingwaveform is used instead of a training sequence. The resultsof [8] imply that the adaptive demodulators in [1], [8], [10],[14], and in [3] do not exhibit an interference floor andare near-far resistant. Further, these demodulators do notrequire explicit knowledge of the interference parameters andhave relatively low complexity, unlike the near-far resistantcentralized multiuser detectors proposed in the past (see [17]for a survey of the latter).

Demodulation of a CDMA signal must, however, be pre-ceded byacquisition, in which the receiver acquires the timingof a transmission that is starting up or has lost synchronization.The adaptive demodulators in [1], [8], [10], [14], and in [3]all assume that some form of timing information regardingthe desired transmission is available. While this informationcould conceivably be obtained using conventional acquisitiontechniques based on matched filters or correlators, the lattertechniques also suffer from the near-far problem, and are atleast as interference-limited [9] as conventional demodulationmethods.

In this paper, we use the blind adaptive demodulator in[3] as a building block for a blind adaptive interferencesuppression scheme forjoint acquisition and demodulation.The only knowledge assumed by the receiver is a knowledgeof the spreading sequence of the desired transmission. The keyidea underlying joint acquisition and demodulation is to choosean observation interval for demodulation that is large enoughso that one complete symbol of the desired transmission fallsinto it, regardless of the timing uncertainty. For a system withmultipath spread that is small compared to the bit interval

, choosing an observation interval of suffices for thispurpose. We then quantize the timing uncertainty into afinite set of hypotheses, run an adaptive demodulator undereach hypothesis, and pick the demodulator that performs thebest according to information derived from receiver statistics.Thus, a unique feature of our method is that it results notonly in an explicit estimate of the desired signal’s timing,but also in a near-far resistant receiver which automaticallyaccounts for the delays and amplitudes of the interfering

1053–587X/97$10.00 1997 IEEE

MADHOW: DIRECT-SEQUENCE CDMA SIGNALS 125

transmissions. The latter can be directly used for subsequentdemodulation as well as for continuing blind or decision-directed adaptation as proposed in [3] or [8]. Our approach isdifferent from more complexsimultaneoustiming estimationand demodulation schemes such as [13], since demodulationin our method occursafter a near-far resistant demodulatorhas been computed based on the timing acquisition algorithm.

Near-far resistant estimation of the timing of the desiredtransmission using the eigendecomposition-based MUSIC al-gorithm [11] has been proposed in [2], and [15]. However,since the observation interval used is of lengthand isnot necessarily aligned with the bit interval for the desiredtransmission, this algorithm only yields the timing of the de-sired transmission, which is not sufficient for near-far resistantdemodulation. This method can be extended to obtain a near-far resistant demodulator in the following two different ways.(a) Use an observation interval of length, estimate the delaysfor all transmissions, and then compute a near-far resistantdemodulator based on these estimates. Delay estimation forall transmissions would require knowledge of all the spreadingsequences in the method proposed in [2], [15]. (b) Apply theMUSIC method with a observation interval, in which casecomputation of a near-far resistant demodulator based on theestimated timing can be done exactly as in this paper (see (12)).Method (b) would therefore yield a blind joint acquisitionand demodulation scheme, albeit with a complexity that issomewhat higher than that of the scheme presented here. TheMUSIC algorithm as presented in [2] and [15] also requiresknowledge of the number of users, although this requirementcould be removed by estimating the number of significantusers using a number of methods (e.g., see [18]). A detailedcomparison of suitably optimized versions of the MUSICmethod and the scheme presented here, especially recursiveversions for time-varying channels, is an important topic forfurther investigation.

A simpler approach to acquisition, based on MMSE adap-tation using an all-one training sequence, has been proposedin [12]. Again, since the observation interval is of lengthin [12], this method does not yield information sufficient tocompute a near-far resistant demodulator. Modification of themethod to use a observation interval removes this problem.However, the use of an all-one training sequence means thatdifferent transmissions in acquisition mode cannot be distin-guished in a near-far resistant manner. If the timing uncertaintyis finite, this second problem can be addressed by usingdifferent, and “sufficiently random” training sequences fordifferent users in acquisition mode, and by running adaptivedemodulators for each of a finite number of timing hypotheses,as in this paper. Since our purpose in this paper is to developand understand the blind method in detail, we refer the readerto [7] for a training based method for joint acquisition anddemodulation that incorporates the preceding modifications,and to [6] for a numerical comparison of early versions of ourblind and training based schemes.

Section II contains background material, including the sys-tem model and a review of the blind demodulator. Section IIIprovides a description of our joint acquisition and demodu-lation scheme. Section IV contains performance analysis of

the scheme. Numerical results are given in Section V, andSection VI contains our conclusions.

II. BACKGROUND

A. Asynchronous CDMA

We consider an asynchronous DS CDMA system withsimultaneous antipodal transmissions over an additive whiteGaussian noise (AWGN) real baseband channel. In principle,the scheme proposed here extends trivially to a complexbaseband model that encompasses two-dimensional signalingand multipath fading. However, theperformanceof adaptivemethods over time-varying complex baseband channels is anopen issue that is currently under study. The received signaldue to the th transmission ( ) is given by

(1)

where is the bit interval, is the th bit ofthe th transmission, is its amplitude, is its relativedelay with respect to the receiver, and is its spreadingwaveform, given by

(2)

Here, is the th element of the spreadingsequence for theth transmission, is the chip waveform(typically assumed to be of duration ), and isthe processing gain.

The net received signal is given by

(3)

where is AWGN. The bits are assumed to beuncorrelated for all and . Taking the first transmission tobe the desired transmission, our objective is to demodulateits bit sequence . The only knowledge assumed is thatof the desired signature sequence. The delays and amplitudesfor all transmissions are unknown, as are the signature se-quences for the interfering transmissions. The adaptivealgorithm for joint acquisition and demodulation will result ina linear receiver that implicitly accounts for these unknownparameters.

B. The Equivalent Synchronous Discrete Time Model

Since the digital signal processing required for interferencesuppression occurs in discrete time, we restrict attention to anequivalent synchronous discrete timemodel obtained by chipmatched filtering the received signal, sampling at a multipleof the chip rate, and limiting attention to a finite observationinterval for each bit decision. All results in this paper arefor a rectangular chip waveform and chip rate sampling.Generalizations to other chip waveforms and sampling rates


is straightforward.1 The length of the observation interval ischosen to be , which is the minimum length such that onecomplete bit of the desired transmission falls in the intervalregardless of the relative delay .2 The latter property iscrucial to the design of our acquisition algorithm. If morethan one transmission is being acquired or demodulated bythe receiver, this property also enables the use of commonobservation intervals for all transmissions. For least squaresor recursive least squares (RLS) adaptation, this will implythat the computation of the inverse of the empirical cross-correlation matrix can be used for all transmissions, so thatan efficient centralized implementation of the decentralizedadaptive method proposed here is possible.

The th discrete time sample can be written as

(4)

Each observation interval corresponds to samples, and thevector of received samples for theth observation intervalis . Thenumber of samples used for each bit decision is, where

is theprocessing gain, or the number of chips per bit, andwe may henceforth consider anequivalent synchronous systemwith received vector of samples for the thobservation interval.

We will now express the observation vector in terms ofthe parameters of the asynchronous CDMA model (1)–(3).Without loss of generality, let denote the bit of the

th transmission that falls completely in theth observationinterval, and let denote the delay of this bit relative to theleft edge of the th observation interval. Sincewe may write it as a multiple of the chip interval asfollows: , where is an integer between

and , and . Let denote a vector oflength consisting of the elements of the signaturesequence of the th transmission followed by zeroes,i.e., . Let denotethe acyclic left shift operator, and denote the acyclicright shift operator, both operating on vectors of length.Thus, for a vector , we have

and .For each asynchronous transmission, three consecutive bit

intervals overlap with a given observation interval of length. Furthermore, since the system is chip-asynchronous, two

adjacent chips contribute to each chip sample. The contributionof the th transmission to the received vector of

samples for the th observation is therefore given by

where

(5)

1While sampling at twice the chip rate would preserve most of theinformation in the continuous-time signal, it would also lead to a larger numberof adaptive taps for a fixed observation interval.

2Longer observation intervals result in receivers with better steady stateperformance, but with higher complexity and slower adaptation speed.

Thus, the contribution due to theth transmission consists ofthree signal vectors, each modulating a different bit. Each ofthese signal vectors are linear combinations of two adjacentshifts of the signature sequence for theth transmission. Thenet received vector is given by

(6)

where is white Gaussian noise with covariance . Ourtask is to demodulate bit based on the observation vector

. The observation vectors are identically distributed,which means that adaptive mechanisms that exploit the struc-ture of for demodulation can be devised. Note that theobservation vectors are not independent, since a given bitappears in three consecutive observation intervals, and sincethe first noise samples for are the same as the last

noise samples of . This is irrelevant in determining thestructure of the adaptive algorithm, however.

Since the model (5)–(6) is notationally cumbersome, itis convenient to consider the followinggeneric equivalentsynchronous model:

(7)

where is thedesired bitthat we wish to demodulate,is the vector modulating it, and, for , areinterfering bits due to intersymbol interference and multiple-access interference, and are interference vectors modulatingthese bits. Recalling that the first transmission is the desiredtransmission, the correspondence between (7) and (5)–(6) isas follows: , the desired signal vector ,and the interfering vectors are the setof vectors due to multiple-access interference together with the vectors dueto intersymbol interference. Thus, the number of interferencevectors is given by , three for each interferingtransmission and two for the adjacent bits of the desiredtransmission. The interfering bits are simply the bitsmodulating the interfering vectors, as specified in (5). The bits

are uncorrelated by virtue of our assumptionthat a given bit is uncorrelated with different bits of the sametransmission, as well as with bits of other transmissions.

For the remainder of this paper, we find it notationally con-venient to hide the fine structure of the equivalent synchronousmodel and work with (7).

C. Blind Demodulation

In this section, we supply background material adapted from[3] together with some additional definitions and formulas thatwill be required in our acquisition algorithm. Lettingdenote inner product, the blind minimum output energy (MOE)demodulator [3] for the equivalent synchronous system (7)corresponds to an estimate ,where the correlator is chosen to minimize the outputenergy , subject to the constraint

(8)


Here is a nominal signal vector, which is the receiver’sestimate of thedirection of the desired signal vector . Wewill assume that without loss of generality.

The norm squared of will be referred to as thedetector energy , and is a measure of the amount of noiseenhancement at the output. If has a nonzero component

orthogonal to the space spanned by the interferencevectors , complete cancellation of the interferenceis obtained by setting , where is chosen tosatisfy (8), and the resulting choice ofsatisfies

(9)

where is the energy of the component of thenominal orthogonal to the interference space (this equals therelative energy of the orthogonal component by virtue of thenormalization ). Smaller values of correspond tolarger and, hence, to more noise enhancement at the outputof the detector.

Mismatchbetween the nominal and the signal vectorcan occur due to errors in the timing estimate or due to thepresence of multipath components not accounted for in thenominal. This can result in signal loss. In particular, completesignal cancellation is possible by setting ,where denotes the projection of the nominal orthogonalto the space spanned by the desired signal vector, and where

is chosen to satisfy (8). The resulting choice ofsatisfies

(10)

where is the energy of the component of the nominalorthogonal to the space spanned by the signal vector, i.e.,

is the near-far resistance of a hypothetical system in whichthe nominal is the desired signal vector and the interferenceconsists solely of the desired signal vector.

Equations (9) and (10) imply that, if the nominal is closerto the space spanned by the desired signal vectorthan to theinterference subspace (i.e., if ),then interference suppression can be achieved while avoidingexcessive signal cancellation by constraining the norm of

, i.e., by constraining the detector energy. Ideally, theconstraint should be such that is allowed to exceed

but not . Using Lagrange multipliers to reflect the normconstraint and (8), the cost function to be minimized becomes

(11)

While must be chosen to satisfy (8), it is convenientto define anunscaled MOE solution,which corresponds to

as follows:3

(12)

where is the statistical correlation matrix forthe observation vector. This solution must be scaled downby in order to satisfy the constraint (8), and the

3Note that demodulation performance does not depend on scaling for aconstant modulus constellation.

resulting MOE is given by

(13)

Similar, the detector energy for the appropriately scaledMOE solution is given by

(14)

Recalling that the bits in (7) are uncorrelated, thecorrelation matrix is given by

(15)

From (12) and (15), it is clear that the Lagrange multiplierforthe constraint on plays the same role as additional noisevariance in determining the MOE solution. We will henceforthcall thefictitious noise variance.Large leads to less signalloss, and hence to greater tolerance to mismatch at the cost ofless interference suppression.

Our acquisition scheme coarsely quantizes the delay intoa number of hypotheses, computes the MOE solution foreach corresponding nominal, and attempts to choose the besthypothesis based on the resulting MOE’s. If all shifts ofthe desired spreading sequence have reasonably low crosscorrelations with the interference, the amount of interferencesuppression under different hypotheses should be comparable.However, it should be harder to suppress the desired signal fornominals that are close to the direction of the desired signalvector. For the same value of detector energy, therefore,one would expect a larger MOE for the better hypotheses.However, constraining to be the same under differenthypotheses requires, in general, that different values ofareused. This is computationally cumbersome, since the inverseof must be computed for each. In order to avoid this,we assume that the same fictitious noise varianceis used forfinding the MOE solutions under all delay hypotheses. Wethen compare the MOE in (13) for different hypotheses, andchoose the one that is the largest.

The MOE computed in (13) is normalized such that. A different normalization that will be useful

in our acquisition algorithm is , which yields thenormalized MOE defined by

(16)

The utility of the normalized MOE is as follows: Once acoarse estimate of the delay has been obtained, a local searchfor refining this estimate can be performed by maximizing.This is because, if there is a nominal with no mismatch, thesignal is not suppressed, leading to a large output energy.


Further, in the absence of mismatch, the detector energyisexpended only on suppressing interference, so thatshouldbe smaller at the correct delay. Thus, should be largeat the true delay. We will use this idea to interpolate betweentwo coarse delay hypotheses to estimate the true delay.

The reason that the MOE is not used to interpolatebetween delay hypotheses is that maximizing it locally isdifficult. On the other hand, the normalized MOEis notused for deciding among the coarse delay hypotheses because,in the presence of mismatch, the detector energyunder thegood delay hypotheses can be large at low noise levels (sincethe MOE criterion is trying to take advantage of the mismatchto suppress the signal as well as the interference, especiallywhen the interference is weak). Thus, even thoughis largerfor the good hypotheses, so is, which may causeto be smaller than for some incorrect delay hypothesis. Thus,testing the hypotheses based onbreaks down in a low noise,single-user regime, where we would like to obtain the bestperformance. This phenomenon has been verified numerically,although we do not include those results here. Note that theproblem occurs due to the mismatch due to coarse delayquantization, and is not an issue when we are trying to chooseamong a set of nominals such that one corresponds to the truedelay. We will therefore use the MOE for the coarse delayestimate and the normalized MOEfor refining the estimate.

III. B LIND ACQUISITION AND DEMODULATION

In the acquisition phase, the receiver does not know either ofthe parameters or specifying the delayfor the first transmission, and therefore does not know thedesired signal vector . However, we do know that is alinear combination of two shifts of the desired transmission’sspreading sequence, given by

(17)

We must now determine the level of delay discretizationneeded to translate the uncertainty in and into a finitenumber of hypotheses.

A. Number of Delay Hypotheses Needed

The delay discretization should be such that the mismatch,and hence the signal cancellation, for the best hypothesis isnot excessive, regardless of the true delay. This is true(see the previous section) if, for each , there isa hypothesis for which the distance of the nominal from thesignal space is smaller than its distance from the interferencespace, i.e., if , where the inequality is preferablysatisfied by a wide margin. For signature sequences with goodcross correlation properties, any given shift of the desiredsignature sequence will have a nonzero component orthogonalto the interference space, so that the near-far resistanceof the nominal corresponding to each delay hypothesis can beexpected to satisfy some designed lower bound, e.g., .For the best hypothesis, must be smaller than this lowerbound.

Consider given by (17). Suppose the hypothesized delaysare integer multiples of the chip interval, so that the nomi-

nals corresponding to the two closest hypotheses are (scalarmultiples of) and , respectively.To obtain a rough idea of the number of delay hypothesesneeded, it suffices to assume that these two nominals areorthogonal, since shifts of typical signature sequences arenearly orthogonal, in practice. We then obtain that the near-farresistances of these nominals relative to the signal vectorin (17) are, respectively,

(18)

(19)

For , we have , which is comparableto typical values of the near-far resistance relative to theinterference space. Thus, if the delay hypotheses are spacedby the chip interval, the amount of signal loss could be largeeven under the closest hypothesis.

Consider now an intermediate delay hypothesis with nomi-nal (a scalar multiple of) , whichcorresponds to a hypothesized delay of . The near-far resistance of this nominal relative to the signal space canbe shown to be

(20)

For a given , the best hypothesis is the one with thesmallest value of . From (18)–(20), it can be shown thatthe worst-case value of for the best hypothesis is

. The value of should therefore be larger than thisvalue for near-far resistant timing acquisition. In contrast,when the delay is perfectly known, the value of is, intheory, only required to be nonzero. The penalty in terms ofnear-far resistance for not knowing the delay can be reduced byusing a finer delay discretization, which in turn implies a largernumber of delay hypotheses and thus greater complexity.

B. Acquisition Algorithm

For , the delay hypotheses and thecorresponding nominals are given by: Hypothesis: Delay

, Nominal Hypothesis: Delay , Nominal

. All nominals are normalized tounit energy to enable a fair comparison of the MOE’sfordifferent hypotheses.

Step 1: Main ComputationsFor the th hypothesis, , compute the

MOE solution, as follows, as in (12):

(21)

and the MOE and normalized MOE as in (13) and (16),respectively, as follows:

(22)


(23)

Step 2: Finding the Best HypothesesLet denote the index of the hypothesis with the largest

MOE, i.e.,

Similarly, let denote the index of the best adjacenthypothesis, i.e.,

except for , for which we set ,and , for which . This is because it isassumed (without loss of generality) that the delay .

Step 3: Combining RuleDefine theinterpolated nominal

(24)

where , and let denote the normalized MOEcorresponding to this nominal. Let

denote the value of that maximizes thenormalized MOE. As shown in Appendix A, the computationof is simple, involving solution of a quadratic equationfor and comparison of and withthe values of evaluated at the solutions to the quadraticequation.

Step 4: Algorithm OutputsThe demodulator produced by the acquisition algorithm is

given by

(25)

which is the (unscaled) MOE solution for themaximizinginterpolated nominal

(26)For the delay estimate, applying the definitions of the andusing (24), the maximizing interpolated nominal is rewritten as

where and have a straightforward dependence onand (see Appendix A). Referring to (5), the

delay estimate is given by

(27)

We will illustrate the operation of our algorithm via a leastsquares implementation, which follows the steps 1 through 4described previously, except that the cross correlation matrix

is replaced by the empirical crosscorrelation matrixcomputed over bit intervals

(28)

The MOE corresponding to hypothesis is denoted by ,in order to distinguish it from its steady state equivalent.

In practice, an RLS implementation might be preferred inorder to track time variations. It is also possible to use astochastic gradient algorithm, but the convergence of suchalgorithms for blind adaptation has been found to be slow [3].Furthermore, the complexity of running stochastic gradientalgorithms corresponding to the hypotheses is comparableto that of RLS algorithms that share the most significantpart of the computation, i.e., the recursive computation of theinverse of the empirical crosscorrelation matrix.

Choice of : The fictitious noise variance must bechosen to optimize performance: a small value allows moreinterference suppression, but also more signal loss. Numer-ical results in [3] and [5] indicate that afictitious SNR of

of 10 dB appears to work well over mostranges of relative amplitudes.4 However, since such a choiceof cannot be implemented without knowing the amplitude ofthe desired transmission, we consider a more practical choiceof , which scales according to the net power in the receivedsignal, estimated as trace . Thus, we set trace .As discussed in Section V, the performance is sensitiveto the choice of . While one fairly complex method forautomatically choosing is provided in Appendix B, findingmore satisfactory solutions is left as an open problem.

IV. PERFORMANCE ANALYSIS

Steady state benchmarks for the performance of the al-gorithm are established by running the algorithm using thestatistical cross correlation matrix. The error probabilityfor the resulting demodulator , and the cosine of theangle between and the ideal MMSE solution,

, are computed.5 The error probability forthe matched filter is computed as a benchmarkthat any adaptive scheme should be able to beat when theinterference powers are significant. This comparison is biasedin favor of the matched filter, since we assume a perfectdelay estimate in this case. The latter would require a separateacquisition scheme in practice. The steady state performancemeasures are compared with corresponding results obtainedfor each simulation run of a least squares implementation.The bias and variance of the delay estimate are otherperformance measures of interest associated with the leastsquares implementation.

In addition to the preceding, a key performance measureis the acquisition error probability, defined as the event thatthe best delay hypothesis is not one of the two selectedby an adaptive implementation (in particular, by the leastsquares implementation considered here). The best hypothesisis defined to be the one with the largest normalized MOEin steady state, provided that the delay it corresponds to iswithin (the delay quantization used) of the true delay.Otherwise, it is defined as the hypothesis that corresponds toa delay closest to the true delay. In the latter instance, the

4In this case, ifjju0jj2=�2 is smaller than 10 dB, then the noise level ishigh enough that no fictitious noise would be needed, so that� = 0 wouldsuffice.

5For the model 7, the error probability for any linear receiver is computedanalytically by averaging over the bitsfbj [n]g modulating the signal andinterference vectors; see [8], for instance.


algorithm would choose an incorrect hypothesis even in steadystate, so that acquisition with an adaptive implementationwould necessarily be unreliable. In most cases of interest,as the number of least squares iterations increases, the ac-quisition error probability quickly becomes too small to beestimated directly by simulation. On the other hand, an exactanalytical estimate of the acquisition error probability for theleast squares implementation appears intractable. We thereforedevelop a simple approximation as follows.

Let denote the index of the best hypothesis. Letdenote the estimated MOE under hypothesis for a

least squares implementation. We will approximate theas jointly Gaussian. Under this approximation, the randomvariable , is Gaussian. Denoting itsmean by and its variance by , we obtain the followingapproximation for the probability of choosing over thecorrect hypothesis (this would be exact if the jointlyGaussian assumption were exact):

(29)

Analytical computation of and is difficult; hence, wesimply estimate these using the empirical statistics of theover multiple simulation runs.

Assuming that (29) provides an accurate estimate of theprobabilities , we can obtain a union bound on the prob-ability of acquisition error as follows. Let andindex the best hypothesis and the best adjacent hypothesisin steady state. Acquisition error occurs if is notchosen by the adaptive implementation of the algorithm. If

is chosen but is not, this is not considered anacquisition error. This is because, if is truly a significanthypothesis, the probability of this event is very small, whileif is not significant, then it does not matter if thewrong adjacent hypothesis is chosen, since the combining rulewould give a low weight to whichever adjacent hypothesisis chosen. While it is possible, using the jointly Gaussianapproximation, to carry out a detailed analysis of the choiceof adjacent hypothesis taking into account the combining rule,little additional insight would be gained by doing so.

The event that neither nor is chosen lies in theunion of the events for . On theother hand, if is chosen, the event that is not thebest adjacent hypothesis also lies in the preceding union. Thus,the probability that acquisition error occurs is bounded by

(30)

V. NUMERICAL RESULTS

We consider a symbol- and chip-asynchronous system withprocessing gain and number of transmissions .No attempt is made to optimize the set of signature sequences,so that all numerical results are for a fixed, but random, choiceof the set of signature sequences. Steady state analysisand simulations for other choices of signature sequences yieldqualitatively similar results. The delays of theinterfering transmissions are chosen randomly in and

then kept fixed. Two different values of the delay of the first(desired) transmission are considered: (which isperfectly matched to hypothesis ), and (whichfalls in the “middle” of hypotheses and , so that thereis mismatch under either of these hypotheses).

The SNR for the equivalent synchronous model (7) is. Since this depends on the chip delay, we define

SNR to be that corresponding to a chip-synchronous system( ), so that SNR . The numerator, , issimply the bit energy , and , so that SNR(dB) . We fix the amplitude of the desiredtransmission, and assume thateach interfering transmissionhas power relative to that of the desired transmission. Threedifferent values of are considered: 20 dB (to check for near-far resistance), 0 dB (to examine the performance with perfectpower control), and 20 dB (to check that signal cancellationis not excessive, and that the performance is not degraded toomuch relative to the matched filter receiver). The least squaresimplementation is run for a value of fictitious noise variance

trace , where unless specified otherwise.

A. Steady State Benchmarks

We consider two values of , 7 dB (which is moderatefor single-user applications, but low for linear multiuser de-tection, which causes noise enhancement) and 17 dB (whichis high for single-user systems, but not uncommon for CDMAsystems whose performance is limited by multiple-accessinterference). These values correspond to (chip-synchronous)SNR’s of 10 dB and 20 dB, respectively.

We define to be the largest MOE among thetiming hypotheses within at most of the true delay, and

to be the largest MOE among all other hypotheses.The difference between these two quantities is a measure ofhow well the algorithm can distinguish between good andbad hypotheses (the bias and variance in estimates of thesein a least squares implementation will ultimately determineacquisition performance). These and other relevant steady statequantities are displayed in Tables I and II.

For of 17 dB, the delay estimates based on interpo-lating between good hypotheses are excellent, and the cosineof the angle between the resulting demodulator and the MMSEsolution is close to one. The demodulator is near-far resistant,and performs much better than the matched filter even withperfect power control ( dB). The separation between

and is substantial, so that the scheme isexpected to be robust to least squares estimation errors. For asmaller of 7 dB, the acquisition scheme can go wrongfor high interference levels (see the entry forin Table II) in the presence of mismatch.6 In every othercase for of 7 dB, the acquisition algorithm selects thecorrect hypotheses and the demodulator and delay estimatebased on interpolating the hypotheses is again very close tothe MMSE solution. However, in this low SNR regime, theperformance gains over the matched filter receiver are not asdramatic. The separation between and is

6In this case, increasingEb=N0 by a further 2 dB leads to correctacquisition in steady state.


TABLE ISTEADY STATE BENCHMARKS FOR THE PERFORMANCE OF THEBLIND ACQUISITION ALGORITHM (CASE 1: �1 = 3:5Tc)

TABLE IISTEADY STATE BENCHMARKS FOR THE PERFORMANCE OF THEBLIND ACQUISITION ALGORITHM (CASE 2: �1 = 3:25Tc)

also smaller here, implying that a large number of least squaresiterations might be required to provide reliable acquisition.For the remainder of the paper, therefore, we restrict attentionto an interference-limited regime more typical of CDMAapplications, taking dB.

While 0.0001 is used for the preceding results, it isimportant to choose according to signal and interferencepowers for rapid acquisition using a least squares implemen-tation. See Section V-B, where we illustrate the advantageof choosing larger when is smaller. Conversely, if weincrease to 50 dB while keeping 0.0001, acquisitionerror occurs in steady state even for of 17 dB. Thisis because, for fixed trace , and hence , increases with

, permitting less interference suppression. Using a smaller0.000 01 restores reliable steady state performance in this

setting.Finally, it is worth noting that even for very weak in-

terference, the error probability performance for all threedemodulators considered is substantially worse than the stan-dard error probability formula for binary phaseshift keying (BPSK). This occurs because the receiver is notchip-synchronous with the desired transmission, causing bothSNR loss and intersymbol interference.

B. Acquisition Error Probability

In view of the poor performance even in steady state forof 7 dB, we restrict attention to of 17 dB

in order to evaluate the performance of the least squaresimplementation of the algorithm in an interference-limitedregime. Our objective is to explore the dependence of theacquisition error probability on the number of leastsquares iterations used. We use 10 000 simulation runs to

estimate the acquisition probability in each case considered,and compare it with the analytical approximation (29)–(30)obtained using the jointly Gaussian assumption for the outputenergies obtained over the simulation runs (the second-orderstatistics of the can be well estimated using many fewerruns than required to estimate the acquisition error probabilityaccurately).

The over independent runs will be identically dis-tributed only if the value of used for the runs is thesame. Since the empirical crosscorrelation matrixdiffersover different runs, so does the value of the fictitious noisevariance trace . However, for all the simulation runsconsidered, trace (which is a rapidly converging estimateof the average received power) is close to the statisticalaverage trace . Thus, from the point of view of analyzingacquisition performance, it suffices to consider a fixed valuefor the fictitious noise variance, trace . For this valueof , 1000 simulation runs of the least squares implementationof the acquisition algorithm are used to compute the empiricalmean and variance of . This is usedto estimate the probability of choosing a given wronghypothesis via (29), and then to compute the union bound(30) on the probability .

In comparing the results of the analysis with direct estimatesof the acquisition error probability, we list the followingquantities.

1) The acquisition error probability obtained via10 000 simulation runs and the (simulation-aided) an-alytical approximation.

2) Let denotethe index of the incorrect hypothesis with the largestprobability of being chosen over the best hypothesis


TABLE IIIACQUISITION PERFORMANCE AS A FUNCTION OF THE RELATIVE INTERFERENCEPOWER PI , THE NUMBER

MLS OF LEAST SQUARES ITERATIONS, AND � = �=trace(R) (�1 = 3:5Tc; Eb=N0 = 17 dB)

TABLE IVACQUISITION PERFORMANCE AS A FUNCTION OF THE RELATIVE INTERFERENCEPOWER PI , THE NUMBER

MLS OF LEAST SQUARES ITERATIONS, AND � = �=trace(R) (�1 = 3:25Tc; Eb=N0 = 17 DB)

, as estimated by simulations. In order to evaluatethe performance of the jointly Gaussian approximationfor the with simulations, we compare ob-tained using the two methods, rather than listing all the

(if the latter are all zero asestimated from simulations, we list the largest of thesevalues according to our analytical approximation).

3) The bias and standard deviation of the delay estimate.

While all the results from the steady state analysis in SectionV-A are for , here we illustrate the effect of varying

on the performance of the least squares implementation.Tables III and IV list the preceding performance measures asa function of the relative interference power, the numberof iterations , and .

For high SNR, is small enough to permitsuppression of very strong interference ( dB), andgives good performance for dB as well, especiallywhen there is no mismatch under the best hypothesis, as inTable III. Recall that the steady state results in Tables I andII show that gives acceptable performance for allvalues of considered. For the least squares implementation,


TABLE VERROR PROBABILITY PERFORMANCE OVER100 SIMULATION RUNS OF

THE LEAST SQUARES IMPLEMENTATION OF THE BLIND ACQUISITION

ALGORITHM (Eb=N0 = 17 dB, �1 = 3:5Tc; MLS = 40)

however, Tables III and IV show that, for smaller , theacquisition error probability and the delay bias and standarddeviation are better for larger values of, such as .001 and.01. This might be because, for a fixed number of least squaresiterations, decreasing makes the empirical cross correlationmatrix more ill conditioned. Further, for fixed, decreasing

amounts to decreasing. Thus, the noise enhancement dueto inverting for small appears to be excessivefor 0.0001. If the least squares implementation is run forlong enough with .0001, reliable acquisition is attainedfor all values of (see the entries for ), whichis consistent with the steady state analysis. However, largervalues of of .001 and .01 produce reliable acquisition muchmore quickly for of 0 and 20 dB. Of course, thesevalues of are not universally good either, since they canbe shown to cause acquisition errors even in steady state for

of 20 dB. If is chosen appropriately, reliable acquisitionis obtained within iterations for andwithin for .

Comparing Tables XIII and IX in Appendix B with TablesIII and IV, it is interesting to note that the automatic choice of

via the algorithm presented in Appendix B does eliminatevalues of that provide poor performance for a given valueof . However, because of the bias of the algorithm towardsmaller values of (in order to permit more interferencesuppression), it need not select the value ofthat provides thebest performance for a given , especially for small or mod-erate (which requires a large to optimize performance).Thus, we see from Table IV that works better for

dB, but the results in Table IX show that .001 isselected much more often in this case.

Comparing Tables III and IV, note that, for the same numberof least squares iterations, the fact that there is mismatch evenunder the best hypothesis for leads to a largeracquisition error probability and to a larger bias and standarddeviation for the delay estimate, even though the steady stateperformance for this delay is better (see Tables I and II). The

TABLE VIERROR PROBABILITY PERFORMANCE OVER100 SIMULATION RUNS OF

THE LEAST SQUARES IMPLEMENTATION OF THE BLIND ACQUISITION


performance quickly improves as increases, especiallyfor values of that are well matched to .

The analytical prediction based on the jointly Gaussianapproximation is seen to be always larger than the acquisitionerror probability directly estimated from simulations. Thematch is better for . For , the analyticalestimate is sometimes larger by several orders of magnitude.However, even in this case, because the acquisition errorprobability decreases so rapidly with , the analyticalapproximation is still a good tool for conservative design;e.g., for dB, .0001, and a desired of ,simulations show that should suffice (see Table III),while the analytical approximation would lead to .Of course, if very low acquisition error probabilities aredesired, then the analytical approximation provides the onlypossible design approach, since direct simulations would betoo time consuming.

C. Error Probability Performance

The preceding results show that, for high SNR and anappropriate choice of , our method quickly provides a gooddelay estimate. We now evaluate the average error probability

of the demodulator produced by the algorithm afterleast squares iterations. The error probability is evaluated

analytically after each simulation run. For each run, we alsoevaluate the cosine of the angle between and .The range and median of (the mean would be weighted tooheavily by outliers), and the range and mean ofare presentedin Tables V and VI, which should be compared with the steadystate results in Section V-A. As in the previous section, werestrict attention to of 17 dB, and vary , keeping

. For each value of , we choose the valuesof determined by results in the previous subsection (andautomatically chosen by the algorithm described in AppendixB) to give better acquisition performance. Fewer (100) simu-lation runs are used here, due to the complexity of the errorprobability computation.


TABLE VIIERROR PROBABILITY PERFORMANCE FORSELECTED SETTINGS, COMPUTED

USING 100 SIMULATION RUNS OF THE LEAST SQUARES IMPLEMENTATION OF

THE BLIND ACQUISITION ALGORITHM (Eb=N0 = 17 dB, MLS = 80)

TABLE VIIIFRACTION OF TIMES EACH VALUE OF � IS CHOSEN BY THE MODIFIED


The results for of 0 dB and 20 dB show that itis important to choose large enough to prevent excessivesignal suppression and noise enhancement in these situations.In particular, the largest bit error probability for .001 and

dB is unacceptable. Even when is chosen tooptimize performance, there is a large variation in bit errorprobabilities over different runs. Nevertheless, for of atleast 0 dB (which is of most interest in a CDMA system, wherethere are at least a few interferers with strength comparable tothat of the desired transmission), the scheme quickly provides adetector with performance far superior to that of the matchedfilter.

For of 0 dB and 20 dB, since the performance for.001 is unsatisfactory for 0.001, we have tried a largernumber of least squares iterations, , in these cases.The results are shown in Table VII. While the performancefor dB improves significantly, the performance for

dB remains poor. Running the algorithm forselecting in this setting, we have found that the likelihoodof choosing 0.001 is small (a few percent) but nonzero.One possibility for rectifying this (if some estimate of receivedpower level relative to background noise were available) mightbe to bias the algorithm for choosingtoward higher valueswhen the received power level is lower.

VI. CONCLUSIONS

We have presented a blind interference suppression schemefor joint acquisition and demodulation, which requires knowl-edge only of the spreading sequence of the desired trans-

TABLE IXFRACTION OF TIMES EACH VALUE OF � IS CHOSEN BY THE MODIFIED


mission. The performance of the scheme is illustrated viaa steady state analysis and simulations of a least squaresimplementation. Based on our numerical results, we make thefollowing observations.

1) Given that , and hence the fictitious noise variance, is chosen appropriately, the algorithm chooses the

correct timing hypotheses with a fairly small number ofiterations for a wide range of interference powers. Theacquisition algorithm is near-far resistant, so that it canoperate in situations in which conventional acquisitionmethods are useless.

2) If the correct (coarsely quantized) delay hypotheses arechosen, the combining rule for interpolating betweenthese hypotheses produces an excellent delay estimate,which eliminates the need for “pull-in” using a codetracking loop. The interpolation also produces a near-farresistant demodulator which is close to the MMSE so-lution, and outperforms the conventional matched filter(which assumes perfect knowledge of timing) when theinterference levels are significant, including the situationof perfect power control.

3) A Gaussian approximation used to estimate the ac-quisition error probability is found to be consistentlyconservative in regimes where acquisition error prob-abilities are large enough to be directly estimated bysimulation. Application of this approximation shows thatthe acquisition error probability drops rapidly with thenumber of least squares iterations when the SNR is highenough. For low SNR’s and high relative interferencepowers, noise enhancement due to linear interferencesuppression causes acquisition errors regardless of thenumber of iterations. These errors can be predicted usingthe steady state analysis.

4) The biggest disadvantage of the algorithm presented hereis its sensitivity to the choice of. One possible methodfor automatically choosing is given, but it is notcompletely satisfactory because of its high complexity.An analogous training based method for joint acquisitionand demodulation [6], [7] does not suffer from suchsensitivity to the choice of algorithm parameters.

In future work, it is of interest to seek lower complexitymethods for automating the choice of. Extending the range ofoperation of the algorithm to lower SNR’s is another problemthat needs to be addressed. Finally, while there are now anumber of low-complexity methods for interference suppres-sion in CDMA systems, extensive performance evaluation ofthese schemes in a typical time-varying wireless environment


is required to quantify the trade-offs involved in designing afunctioning system based on these ideas.

APPENDIX ACOMBINING RULE FORINTERPOLATINGBETWEENHYPOTHESES

We supply here the details of obtaining the best interpolationbetween the nominals for the best two hypotheses. Simplifyingthe notation, consider two nominals and , andconsider an interpolated nominal ,where . The normalized MOE for this nominal isgiven by

where denotes the statistical correlation matrix, and isreplaced by for a least squares implementation. We want tomaximize over .

The numerator and denominator of are quadratics in ,whose coefficients already need to be evaluated when runningthe acquisition algorithm. Differentiating with respect toand simplifying, we find that the unconstrained extrema of

solve a quadratic equation. Denoting by and thesolutions to the quadratic equation, choose to be thevalue that maximizes over and either of ,that fall in . Thus, evaluating involves a comparisonof at most four real numbers. The demodulator can thenbe obtained using (25).

Obtaining the delay estimate requires expressing the nomi-nals and as linear combinations of shifts of the spreadingsequence. Suppose, without loss of generality, that

and . Then the interpolated

nominal is written as , where

and the delay estimate is given by

APPENDIX BCHOOSING AUTOMATICALLY

We describe here a modified algorithm in which the MOEdetectors under each hypothesis are computed as in (21) fordifferent values of , so that there are parallel algorithmsusing values of given by trace . Thisis admittedly complex, since it requires matrix inversions(or RLS algorithms running in parallel). If is large, itmight even be easier to compute theinverses based on aneigendecomposition of , which would make the complexitycomparable to that of the MUSIC method in [2], [15].

The algorithm chooses between the values ofbased onthe following reasoning. In general, it is preferable to usethe smallest possible value of, since this yields betterinterference suppression and a better approximation to theMMSE detector. However, smallpermits a large detector en-ergy and more noise enhancement, which becomes especially

significant when the interference is weak and the detectorenergy is expended mainly in suppressing the desired signal.Since the detector energy required to suppress the interferenceshould be of the order of (see Section II-C),near-far resistance should still be obtained if the maximumallowable value of is of the order of . This leads to thefollowing modification to the algorithm in Section III-B, basedon constraining the maximum detector energy to be at most

, where is chosen large enough to allow adequateinterference suppression.

For each value of , compute thecorrelators using (21). For , define

as the largest detector energy among all

hypotheses. Choose to be thesmallest such that(set if the latter condition

does not hold for any). Now follow Steps 1 through 4 of thealgorithm using the correlators obtained for .

In an RLS implementation, it might be possible to use asingle value of instead of running parallel RLS algorithms,and to adapt this value based on the detector energies producedby the algorithm. We leave the study of such modificationsfor future work.

While we did not perform extensive simulations with thismodified algorithm due to its complexity, its effectiveness isillustrated by the following example. Consider , with

, , and . Forand (corresponding to 10 dB noise

enhancement), we present in Tables VIII and IX the fractionof 1000 simulation runs in which is chosen, as a function of

. Note that the values of chosen decrease with increasing, as they should. Referring back to Tables V and VI, we see

that the values of chosen for a given value of are the onesthat give good performance for that interference power level.When the algorithm for choosing is applied in steady state,however, the choice is for all values of .It appears, therefore, that least squares estimation errors causea larger value of detector energythan would be obtained insteady state, causing larger values ofto be chosen by thealgorithm.

REFERENCES

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[2] S. E. Bensley and B. Aazhang, “Subspace-based channel estimation forcode division multiple access communication systems,”IEEE Trans.Commun,to be published.

[3] M. L. Honig, U. Madhow, and S. Verdu, “Blind adaptive multiuserdetection,”IEEE Trans. Inform. Theory,vol. 41, pp. 944–960, July 1995.

[4] R. Lupas, S. Verdu, “Near-far resistance of multiuser detectors inasynchronous channels,”IEEE Trans. Commun.,vol. 38, pp. 496–508,Apr. 1990.

[5] U. Madhow, “Blind adaptive interference suppression for acquisition anddemodulation of direct-sequence CDMA signals,”in Proc. Conf. Inform.Sci. Syst. (CISS ’95),pp. 180–185, Johns Hopkins Univ., Baltimore,MD, Mar. 1995.

[6] , “Adaptive interference suppression for joint acquisition anddemodulation of direct-sequence CDMA signals,” inProc. Milcom ’95,pp. 1200–1204.

[7] , “MMSE interference suppression for joint acquisition and de-modulation of direct-sequence CDMA signals,” submitted for publica-tion.


[8] U. Madhow and M. L. Honig, “MMSE interference suppression fordirect-sequence spread-spectrum CDMA,”IEEE Trans. Commun., vol.42, pp. 3178–3188, Dec. 1994.

[9] U. Madhow and M. B. Pursley, “Acquisition in direct-sequence spread-spectrum communication networks: an asymptotic analysis,”IEEETrans. Inform. Theory, vol. 39, pp. 903–912, May 1993.

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Upamanyu Madhow (S’86–M’90–SM’96) re-ceived the B.S. degree in electrical engineeringfrom the Indian Institute of Technology, Kanpur, in1985 and the M.S. and Ph.D. degrees in electricalengineering from the University of Illinois, Urbana-Champaign, in 1987 and 1990, respectively.

From August 1990 to July 1991, he was aVisiting Assistant Professor at the University ofIllinois. From August 1991 to July 1994, he was aresearch scientist at Bell Communications Research,Morristown, NJ. Since August 1994, he has been

as Assistant Professor, Department of Electrical and Computer Engineering,University of Illinois. His current research interests are in communicationsystems and networking, with current emphasis in wireless communicationsand high-speed networks.

Dr. Madhow is a recipient of the NSF Career award. He was awarded thePresident of India Gold Medal for graduating at the top of his undergraduateclass.

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