IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997 1595

PN Acquisition and TrackingPerformance in DS/CDMA Systems with

Symbol-Length Spreading SequencesWalter R. Braun,Member, IEEE

Abstract— Direct sequence spread spectrum code-divisionmultiple-access (DS/CDMA) is receiving increasing attentionfor cellular communications systems. When the users aresynchronized and special symbol-length sequences, such as Goldcodes, are used, the mutual interference can be substantiallyreduced relative to a system with very long or purely randomspreading sequences. It is shown that this approach degradesthe performance of the code phase acquisition and tracking,however. This effect prevents the system from acquiring andtracking long before the data detection is affected by themultiuser interference.

Index Terms—Code division multiaccess, delay lock loops,mobile communication, pseudonoise coded communication, syn-chronization, tracking loops.

I. INTRODUCTION

DIRECT SEQUENCE spread spectrum transmission is at-tracting growing interest for indoor and outdoor multiple-

access wireless communication. This is based on its inherentfrequency diversity, which provides protection against mul-tipath fading, on its simplification of the channel allocationproblem, and on its potential for higher traffic capacity [1], [2].

In such a system, there are two approaches to the problem ofselecting spreading codes. When very long sequences are used(period much longer than the symbol time) the interferencebetween users varies randomly from symbol to symbol andcan, effectively, be modeled as random noise. On the otherhand, if the sequence period is equal to the symbol time and thesymbols of all transmitters are aligned, the interference is thesame in every symbol interval. In this situation, it is possibleto limit the interference between any two users to a smallvalue by selecting sequence sets with known good correlationproperties, thus increasing the system capacity. In principle,it is possible to eliminate interference altogether by usingorthogonal sequences, such as the rows of a Hadamard matrix.As an alternative, Gold codes can achieve almost the samelevel of performance. Note, however, that the interference iseliminated only if the despreader is exactly synchronized withall of the received sequences.

Paper approved by R. Peterson, the Editor for Spread-Spectrum Systems ofthe IEEE Communications Society. Manuscript received February 11, 1994;revised March 25, 1997 and May 11, 1997. This work was performed at ABBAsea Brown Boveri and supported by the European Space Agency underContract 8728/90/NL/RE.

The author is with Ascom Systec Ltd., Gewerbepark, CH-5506M̈agenwil,Switzerland.

Publisher Item Identifier S 0090-6778(97)09081-8.

The type of spreading sequence also affects the synchroniza-tion performance. With long sequences, the analytical resultson the acquisition and tracking performance in additive whiteGaussian noise are applicable. However, with synchronizedshort sequences this analysis is no longer valid—in a serialsearch acquisition system [3] the decision whether synchro-nism has been achieved is based on the power measured atthe output of the despreader. A high output level is takenas a lock indication. While for long codes the interferenceis independent of the lock state, for orthogonal codes theinterference is large in the out-of-sync case and vanishes inthe in-sync state.

The same effect manifests itself in a noncoherent delay-locked loop (DLL) [3]. When a tracking error is present,the arm with the larger offset relative to the true phasereceives more interference power, thus offsetting the reductionin despread signal power to some extent. This reduces theslope of the S-curve of the control loop and results in poorertracking performance. There is a second effect, however—thecross correlation between the codes is not symmetric. Hence,depending on the correlation properties of the codes in use ata particular time, the S-curve of the loop may be biased in onedirection, resulting in a tracking offset.

The above effects are assessed in this contribution. InSection II, the second-order statistics of the output of a digitaldespreader is computed. Based on this, the effect on thetracking performance is analyzed in Section III. The theoryfor the acquisition performance is developed in Section IV.Section V summarizes the results.

II. A NALYTICAL MODEL FOR PSEUDONOISE

(PN) ACQUISITION AND TRACKING

The PN acquisition is assumed to be performed by a sequen-tial search. The acquisition detector is shown in Fig. 1—thereceived signal is demodulated into baseband quadrature com-ponents and filtered by chip matched filters. It is then sampledat the chip rate and multiplied with the local PN sequence.The samples are accumulated over a symbol interval (symbolmatched filter) and squared. The sum of the two quadraturecomponents is accumulated further and, finally, compared toa threshold.

The PN tracking is assumed to be based on a DLL tracker.The corresponding phase detector is shown in Fig. 2. Thereare two parallel paths, each with the same processing as inthe acquisition detector. The only difference is in phase of

0090–6778/97$10.00 1997 IEEE

1596 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997

Fig. 1. Acquisition detector model.

Fig. 2. DLL for PN tracking.

the code reference. For this reason, the signal models will bedefined such that they can be used in both, the acquisition andthe tracking analysis.

Let represent the received signal after demodulationinto quadrature baseband signals and chip matched filtering

(2.1)

where is the desired signal, , are

interfering signals, and is the thermal noise

(2.2)

, are independent Gaussian baseband processes.Here, the following notations were used:

symbol time;chip time; ;

BRAUN: PN ACQUISITION AND TRACKING PERFORMANCE IN DS/CDMA SYSTEMS 1597

timing error of th signal relative to some referenceclock;amplitude of th user signal;carrier phase of th user signal;binary data symbol of th user in th bit interval;th chip in the binary spreading sequence of user;

chip pulseshape after matched filtering.

Note that all signals are modeled as biphase modulated. If thesignals consist of quadrature carriers modulated by differentspreading codes, the same model can be used by letting

,.

The received signal is sampled at the chip rate inaccordance with the local PN clock reference. The samplesare multiplied with the PN sequence

(2.3)

where is the local estimate of the PN chip phase andisan offset which will be used to model the timing offset in thearms of the DLL in the tracking mode and to represent thetiming error in the acquisition mode. Inserting (2.1) and (2.2)into (2.3) yields

(2.4)

Initially, the timing offset will be assumed smallrelative to the symbol time . This corresponds to the track-ing case. For reasonably large values ofthe intersymbolinterference may then be neglected (for the acquisition modeit will be added in again)

(2.5)

The noise samples are independent complex Gaussian randomvariables.

Accumulation of such samples yields, where

(2.6)

(2.7)

For a more concise notation, we introduce the correlationfunction of the signal waveform

(2.8)

where is the aperiodic cross correlation function ofthe spreading sequence

(2.9)

With this, the signal component (2.6) can be rewritten as

(2.10)

As shown in Figs. 1 and 2, the next processing step is to takethe square of the absolute value of the correlation values .This is followed by narrow-band filtering, hence the centrallimit theorem may be invoked and a second-order statisticalcharacterization of the signal is adequate.

The first moment is given by

(2.11)

This expression can be evaluated after substituting (2.7) and(2.10) into it. The cross-product signal times noiseaverages to zero because the noise is zero-mean.

(2.12)

All cross products of signal terms , dropout because of the random phase difference and random data.All cross products of noise terms , dropout because the correlation function of the noise has zeros atinteger multiples of . Hence,

(2.13)

(2.14)

The covariance function of is given by

(2.15)

where

(2.16)

(2.17)

(2.18)

1598 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997

where is the correlation function of the basebandprocesses and

(2.19)

III. STATISTICAL ANALYSIS OF THE DLL PHASE DETECTOR

In the DLL phase detector, the samples are sub-tracted from the samples

(3.1)

The S-curve of a tracking loop is commonly defined as theexpected value of the phase detector output, conditioned onthe phase error. Based on (2.8)–(2.14), this can be expressedas

(3.2)

where .The following comments on the above result are in order.

1) is nonrandom in this model. This is due to thefact that intersymbol interference has been neglected.

2) In a symbol-synchronous and chip-synchronous CDMAsystem may be random and time-varying due to the random variation in user activity(different codes being used at different times, e.g., due tocall initiation/termination or voice activation). However,this is a very slow variation in comparison to the timescale of interest, i.e., the inverse of the bandwidth ofthe tracking loop. Hence, it is more appropriate to study“typical” and “worst case” characteristics of than tomodel this variation as an additional noise term.

3) The expression for the S-curve should really containas its only parameter. is a random processof the same bandwidth as and, hence, should beaveraged over. However, there is usually a large positivecorrelation between these processes since typically thelocal tracking error is the major contribution. Inparticular, for a base-to-mobile link the values ofwill all be the same. This is the model to be used in thenumerical examples.

To illustrate the effect of the interference on the S-curveit is assumed that all received signals are of equal powerand synchronized, i.e.,

. This condition is typicallysatisfied on the base-to-mobile link. The spreading sequencesare selected from Gold sequence sets. These sequences oflength have a three-valued cross correlation

Fig. 3. S-curve of DLLN = 31, M = 14 (a) -��-��-��- ideal; (b) -�-�-�-�-self-interference; (c)� � � � � � cross interference; (d)—— total.

function

(Note: this is actually the periodic cross correlation function.In the vicinity of the lock point this is equivalent to the auto-correlation function, however.) A thorough discussion of Goldsequences and their correlation properties can be found, e.g.,in [4].

The S-curve can be split into three components—the desirederror signal, the self-interference, and the interference fromother users. The desired error signal is

(3.3)

which also corresponds to the S-curve of a system usingpurely random spreading sequences. This is shown as curve(a) in Fig. 3. The following parameters were used—processinggain , number of interferers , raised cosinepulseshape with roll-off and offset .

The self-interference of the desired user is given by

(3.4)

This function, normalized to the same scale as , is plottedas curve (b) in Fig. 3. Its contribution to the error signal isquite negligible.

The interference from the other users is shown by curve(c) in Fig. 3. It cancels a considerable part of the desirederror signal. The sequences selected are those that have across correlation at or

or both. Hence, this is a worst-case example but itis nevertheless not unlikely to occur in practice. The relativestrength of the self-interference decreases with the processinggain since . For fixed theinterference from other users also decreases with increasingprocessing gain. However, for a fair comparison of systems,the number of users should grow with the processing gain,so that the relative interference level remains constant. Fig. 4shows results for with interferers. Note thatthe S-curve becomes virtually flat around the lock point, i.e.,the DLL will be unable to track.

The tracking jitter is determined by the relation, where is the slope of the S-curve at

BRAUN: PN ACQUISITION AND TRACKING PERFORMANCE IN DS/CDMA SYSTEMS 1599

Fig. 4. S-curve of DLLN = 63; M = 27 (a) -��-��-��- ideal; (b) -�-�-�-�-self-interference; (c)� � � � � � cross interference; (d)—— total.

the origin. The noise sequence is white because adjacentsamples depend on different data symbols and contain noisesamples taken apart, which makes them independent.Hence, it is sufficient to compute the variance of the samples

(3.5)

The terms in the above expression are given by (2.15)

(3.6)

Note that the variance is zero for in the absence ofthermal noise; i.e., there is no self-noise. This is due to thefact that the signal is deterministic after squaring.

If the cross correlation function is asymmetric the varianceis also asymmetric, which increases the tracking bias.

Figs. 5 and 6 show the variance of the tracking variableas a function of the offset for the same cases as in Figs. 3and 4, respectively.

IV. STATISTICAL ANALYSIS OF THE ACQUISITION DETECTOR

The signal at the despreader output is the definedin Section II. For tracking, only values of (and )smaller than the chip duration were of interest, hence inter-symbol interference could be neglected. During acquisition,the integration window of the despreader overlaps two symbols

(4.1)

where the offset was assumed to be positive .

Fig. 5. Variance of DLLN = 31, M = 14.

Fig. 6. Variance of DLLN = 63, M = 27.

The further processing of these samples can be described by

(4.2)

where is the number of symbols integrated over before alock decision is made (dwell time ). For the performanceanalysis, the mean value and the variance of are required.

The mean value is given by

(4.3)

The first three terms are readily derived from the resultsof Section II. The fourth term is zero because and

contain independent data symbols, while the lasttwo terms are zero because signal and noise are zero-meanand independent. Hence,

(4.4)

Note that should be determined such that

1600 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997

Fig. 7. Mean and RMS value of acquisition detector outputN = 31,M = 14 (a) —— mean and (b) ------rms value.

The variance is

(4.5)

where we have taken advantage of the fact that noise samplestaken apart are independent. After some algebra this reducesto

(4.6)

Fig. 7 shows the mean and the rms value of the detectionvariable around the correct phase 0 for a system with ,

. The mean value shows a secondary peak atthat is almost as high as the main peak. The rms value tracksthe mean quite closely, except at , where it becomes quitesmall. A peak such as Fig. 7 shows at will almost surelyresult in a false alarm. Since this correlation peak is stable aslong as the same set of codes is used, the DLL will also lockon it. Hence, some other means will be required to detect sucha false lock condition. The variance of the acquisition detectoroutput may be a suitable indicator.

The false alarm and detection performance of the acquisitionsystem is determined by the difference of the mean value of

in sync and out of sync and by the variance ofin the twocases. A convenient performance indicator is the normalized

Fig. 8. Distance measure� versus number of interferersM (N = 63).

distance

(4.7)

This parameter is plotted against the numberof interferersin Fig. 8, again for . Also shown is the same perfor-mance parameter for a system with long spreading sequences.The “worst case” curve corresponds to the secondary peak at

. Note that the distance measure is much lower thanfor long sequences, even for . The “typical cases”correspond to arbitrarily selected code phases. The distancemeasure tends to be slightly higher than for long sequences inthese cases. The thermal noise was neglected in this figuresince spread spectrum systems in CDMA applications aretypically interference-limited.

V. CONCLUSION

The above analyses show that with specially selected shortspreading sequences it is possible to reduce the co-channel(CDMA) interference, but this improvement does not carryover to the PN acquisition and tracking function. Here, the re-duced randomness in the interference can result in an increasedtracking jitter and possibly an offset in the tracking mode. Inthe acquisition mode the ability to detect the synchronizedstate may be severely degraded.

In the downlink (base-to-mobile) direction this problem maybe alleviated by adding a pilot to the signal constellation.This pilot should have at least one unmodulated quadraturecomponent (i.e., modulated by the spreading sequence, but notby random data). The mobile can then perform its acquisitionand tracking—for the PN code, as well as the carrier and clockphase—on the pilot signal. Since there is no data modulationon this signal the coherent integration time can be much longerthan a symbol interval. Such a scheme was shown elsewhere[5] to result in highly acceptable performance.

For the uplink, there is no directly comparable solution.The acquisition problem could be solved by performing theinitial access with an unmodulated signal (spreading sequenceonly—possibly random data on quadrature carrier). This phasecould even use a specially designated sequence, so that theacquisition subsystem could search for only one sequence.There is no way to ease the tracking problem; however, herethe synchronization subsystem has to accept the increasedinterference.

BRAUN: PN ACQUISITION AND TRACKING PERFORMANCE IN DS/CDMA SYSTEMS 1601

REFERENCES

[1] U. Grob, A. L. Welti, E. Zollinger, R. K̈ung, H. Kaufmann, “Microcellu-lar direct-sequence spread-spectrum radio system usingN -path RAKEreceiver,” IEEE J. Select. Areas Commun., vol. 8, pp. 772–780, June1990.

[2] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver,C. E. Wheatley III, “On the capacity of a cellular CDMA system,”IEEETrans. Veh. Technol., vol. 40, pp. 303–312, May 1991.

[3] M. K. Simon, J. K. Omura, R. A. Scholtz, B. K. Levitt,Spread SpectrumCommunications, vol. III. Rockville, MD: Computer Science, 1985,pp. 154–165.

[4] D. Sarwate and M. B. Pursley, “Cross correlation properties of pseu-dorandom and related sequences,”Proc. IEEE, vol. 68, pp. 593–619,May 1980.

[5] R. Gaudenzi, C. Elia, and R. Viola, “Bandlimited quasisynchronousCDMA: A novel satellite access technique for mobile and personalcommunication systems,”IEEE J. Select. Areas Commun., vol. 10, pp.328–343, Feb. 1992.

Walter R. Braun (S’71–M’76) received thediploma in electrical engineering from the SwissFederal Institute of Technology, Zurich, Switzer-land, in 1972, and the M.S.E.E. and Ph.D. degreesfrom the University of Southern California, LosAngeles, in 1973 and 1976, respectively.

From 1976 to 1982, he was with LinComCorporation, Los Angeles, CA. In 1982, he joinedABB Asea Brown Boveri, where he initiated acommunications research program at the corporateresearch center. This group was transferred to

Ascom Corporate Research in 1991. From 1992 to 1997, he was withRadiocom Inc., Switzerland, where he was active in the standardizationand development of land mobile radio systems. He is currently Head ofResearch and Development at Ascom Systec Inc., Gewerbepark,M̈agenwil,Switzerland.