Modeling of Code Acquisition Process in CDMAi

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 18, NO. 1, JANUARY 2000 73

Modeling of Code Acquisition Process in CDMANetworksAsynchronous Systems

Marcos D. Katz and Savo Glisic , Senior Member, IEEE

Abstract In this paper, we discuss modelingof serialcodeacqui-sition process in a code division multiple access (CDMA) network.Due to multiple access interference (MAI), the process is charac-terized by a different probability of signal detection and prob-ability of a false alarm in each cell of the code delay uncer-tainty region. We derive exact expressions for average code acqui-sition time and its variance. In addition to this, we also present sev-eral useful approximations, which enable easy engineering use of these results for practical applications in future universal mobiletelecommunications system (UMTS) solutions. Numerical resultsbased on this analysis are used for decision threshold optimizationin code acquisition process for asynchronous CDMA networks.

Index Terms Code acquisition, code division multiple access

(CDMA), synchronization.

I. INTRODUCTION

T HE THEORY of code acquisition in spread spectrum sys-tems is well established and documented. A recent paperby Polydoros and Glisic [ 1] presents a comprehensive reviewof the problems and possible solutions. The main criteria forthe performance assessment are the average acquisition timeand its variance. Different strategies use different algorithms tosearch the code delay uncertainty region. This uncertainty re-gion is usually divided into a finite number of discrete pointscalled cells. The optimum maximum likelihood approach ex-

amines simultaneously all possible candidates of the code delay(cells) resulting into a considerable complexity, especially if thecode is long. A less complex algorithm, called serialacquisition,would examine one cell in time, resulting into much simpler im-plementation but longer acquisition time. In this approach, onlyone out of these cells represents the synchroposition wherethe correlation between the local and the input sequence is high.Most of the previous work is limited to the analysis of the caseof spread spectrum signal in a Gaussian channel. A commoncharacteristic of that work is an assumption that the probabilityof a false alarm in all nonsynchrocells is the same.

In a general case, a CDMA radio network is characterized bytheso-callednearfar problem where differentusers reach a par-

ticular receiver with different power levels. After despreadingat receiver , the overall received signal will have three compo-nents: the useful signal, the multiple access interference (MAI),and Gaussian noise. The useful signalwill be proportional to theautocorrelation function of the code and will have high

Manuscript received October 1, 1998; revised April 15, 1999.M. D. Katz is with Nokia Networks, Oulu FIN 90651 Finland and the Centre

for Wireless Communications, University of Oulu, Oulu FIN 90014, Finland.S. Glisicis withthe Departmentof ElectricalEngineering, Universityof Oulu,

Telecommunication Laboratory, Oulu FIN 90571, Finland.Publisher Item Identifier S 0733-8716(00)01202-6.

value if the input code and the locally generated replica are syn-chronized; otherwise its value is close to zero. The MAI is pro-portional to the cross correlations between the despreadingcode and the codes of all other users in the network.

In a quasi-synchronous system [ 3], [9] special care is takenthat thecontribution of theMAIterm in synchrocell is negligibleso that is constant for each cell of the code delay uncertaintyregion. It depends only on the signal-to-Gaussian noise powerratio. In such a system, all users are synchronized within therange where mutual cross correlations have values , where

is the length of the code. For a specific class of codes, likeGold, Gold-like, small, and large sets of Kasami codes, crosscorrelation has three or five values, one of which is . Thismeans that all users are synchronized within an error, which isless than the distance between the two adjacent peaks of thecorrelation function. Due to the fact that the synchronization isdefined as a range of errors (a not-zero error is required), theterm quasi-synchronous network is used.

In an asynchronous network, MAItakes on differentvalues inall cells including the synchrocell, so that in general, is dif-ferent. Modeling and analysis of ML-code acquisition for suchcases has been analyzed in a number of papers [ 1][13]. Com-plexity of these algorithms is further increased by the presenceof MAI. For these reasons, in this paper, we analyze modelingof serial code acquisition process in such an environment. In ouropinion, this is a viable solution at this stage of technology. Inthe next stage, we expect that combination of parallel and serialalgorithms will be more feasible and even then, this analysiswill be useful for the system performance evaluation. In Sec-tion II, a short description of the model is given by emphasizingthe differencesbetween the quasi-synchronous and synchronouscase. In Section III, expressions for the average acquisition timeand its variance are presented and discussed. Most of the math-ematical derivations are given in the Appendixes A and B. Inaddition to this, we also present several useful approximationswhich enable easy engineering use of these results for practicalapplications in future UMTS solutions.

II. MODEL OF CODE ACQUISITION PROCESS IN ASYNCHRONOUSNETWORKS

We will analyze a serial search [ 2] of the uncertainty regionpartitioned into cells. Most of the time in practice, the numberof cells is equal to the code length . In order to increase thechance to detect the synchrocell, sometimes is chosen to be

. A careful optimization is required to simultaneously mini-mize and keep high the probability of synchrostate detection.Thus, each cell represents one of phase positions between re-ceived and local sequences. In general, there are three different

07338716/00$10.00 2000 IEEE

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KATZ AND GLISIC: MODELING OF CODE ACQUISITION PROCESS IN CDMA NETWORKS 75

Fig. 3. The reduced decision process graph.

the expression for generating function is further simplified, re-sulting in

(9)

Substituting from (5) gives

(10)

Denoting gives the expression for the gener-ating function in a compact form as

Generating function (11)

III. THE MEAN ACQUISITION TIME AND ITS VARIANCE

The mean acquisition time is obtained by differentiating thegenerating function and evaluating the result at .

Thus(12)

The details are given in Appendix A, and the result is

(13)

where

and

(14)

Note that the first two terms of (13) preserve the form of pre-viously known results for the mean acquisition time, namelywhen and are constant [ 2] or when is -valuedand is constant [ 9]. If is considered constant, we havethat , , and the third term in (13) be-comes zero. Under these conditions, the mean acquisition timeagrees with the corresponding results obtained in [ 1]. The vari-ance of the acquisition time can be calculated by any of the fol-lowing equivalent expressions [ 2]:

(15)

(7)

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and by using (11), we have

(16)

From (16), we have

(17)

(18)

where

(19)

Combining the obtained results gives the following expressionfor the variance

(20)

(21)

Since and are related, the variance can bewritten as a function of the mean acquisition time. The expres-sion for the mean acquisition time has already been derived, andit is related to and by

and

(22)

Thus

Substituting these expressions into the equation for the vari-ance yields

(23)

Using the results derived in Appendix B, for ,can be expressed as

(24)

where the following notation is used:

(24a)

The variance of the acquisition time is obtained by substi-tuting (24) and (13) into (23). It is interesting to compare ex-pressions for the mean acquisition time with previous results.Table I summarizes the results obtained for case 1, constant

, and [2]; case 2, -valued , and a constant [ 9]in quasi-synchronous networks; and case 3, -valued and

-valued in asynchronous networks.

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TABLE IMEAN ACQUISITION TIME FOR DIFFERENT DISTRIBUTIONS OF AND

The form of the three expressions provides an easy insightinto the major differences in average acquisition times for thethree cases. In the expression for case 2, when compared withcase 1, should be replaced by , and in the numer-ator shouldbe modifiedby a factor givenby (14). Thefirstfactortakes into account the average , and the second modifica-tion takes into account the position of the initial search cell withrespect to the distribution of . In the expression for case 3,when compared with case 2, should be replaced by [see(14)] in addition to a new term that should be added to thenumerator. This term can be expressed as

(25)

A first observation is that a sufficient condition forto be zero is that or or both of them have aconstant distribution; that is, at least one of the followingconditions is met: , or

. A proof for it is straightforwardfrom the definitions of , , and . Sinceand , the sign of cannot be determined withoutknowledge of the particular distributions of and .

From the definition of , one can see that aslong as . However, it is enough thatat least one is small to cause a considerable reduction of the final value of . The variation of also depends on thenumber of cells .

IV. APPROXIMATIONS

To get an additional insight into the system behavior, we willpresent some approximations. A block diagram of the acquisi-

tion scheme, implementing the algorithm described in SectionII, is given in Fig. 4. In the figure, is the assumption thatthe decision variable is larger than the threshold, and is theopposite assumption. For a CDMA network with users, thereceived overall signal can be represented as

(26)

where is the bit of the signal transmitted by user with index ; is the code of the signal transmitted by user with index

; is the phase of the signal transmitted by user with index

.For simplicity, a real signal is assumed. The extension to com-plex signal (I&Q) is straightforward.

In the receiver with index after frequency downconversionand correlation with code , we have

(27)

where is the cross correlation between codes with indexes

and ; .

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Fig. 4. Generic model for the serial-search code acquisition receiver.

The MAI term can be repre-

sented in each cell of the delay uncertainty region as a Gaussianzero mean variable with variance . In

order to average out the impact of MAI on the code acquisi-tion process, we approximate with the average value

. Furthermore, we will use discrete approx-imation of the correlation function ,where and is the code length. One should beaware that if the number of cells , then represents theindex of the cell . For , . Therefore, the previousequation can be expressed as

(28)

where and are two arbitrary sequences from the same setof sequences. In the sequel, we will evaluate this expression fordifferent classes of codes, which are considered for applicationsin UMTS system. For sequences, we will use the well-knownrelation [ 10] which is based on Cauchy ineqality

(29)and

(29a)

For sequence, and for .Therefore, we have for the upper and lower bound on thefollowing expressions:

(30)

which for becomes . For short codes, thisshould be further elaborated. For a preferred pair of sequences[10], [11] obtained by decimation by factor andsequence length , and divides ,the three-valued cross correlation is given as

times

times

times(31)

For a specific choice of these parameters, we have1) Gold Sequences:

integer of odd

even. (32)

2) Small Set of Kasami Sequences:

even (33)

By using (28), we have in general case

(34)

V. NUMERICAL EXAMPLES

For the generation of MAI, we use eight Gold codes of length31, generated by a preferred pair of polynomials

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KATZ AND GLISIC: MODELING OF CODE ACQUISITION PROCESS IN CDMA NETWORKS 79

Fig. 5. Gaussian probability density functions (pdfs) at the integrator output in each cell. Pdf of the decision variable at the square-law detector output.Signal-to-noise ratio (SNR) dB, relative delay factor and 0 .

and . Vectordefines initial delays (in chips) of the

sequences. By using known results for a square law detector fol-lowed by an integrator (integration interval and signal band-

width ), the detector output signal (decision variable) proba-bility density function (pdf) is shown in Fig. 5. In this analysis,theequivalent noise varianceis given by (28). Thecurves areob-tained for 16 different cells of the code delay uncertainty region.From this figure, one can expect that parameterswill depend on the decision threshold .

Three parallel results are presented for the normalized av-erage acquisition time is obtained by usingexact resultgivenby (13)(case3 inTable I), istheapprox-imation where the standard expression for is used (case 1in Table I) with and and is theapproximation where MAI is approximated by Gaussian noisewith variance given by (28), (30), and (35).

In Fig. 6, is presented versus the threshold withsignal-to-noise ratio (SNR) being a parameter. One can see thatfor large (> 10 dB) and small (

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Fig. 6 Normalized mean acquisition time ( ), eight users, SNR 0 0 . , . Phase vector ,Solid line: . Dotted line: . Dasheddotted line: .

(a)

Fig. 7. Upper and lower bounds of the mean acquisition time for 30 realizations of a random phase shift vector. Normalized mean acquisition time ( ),eight users, SNR . (a) , .

created nearfar effect by base station power control. The basestation will send more power to the users at the edge of the cellwhere the impact of the interference coming from the adjacentcell is more severe. In the uplink, unequal signal power levelsare due to both power control imperfections and unequal signallevel due to differentprocessing gain for different data rates. Forcontinuous transmission in the paging channel, the model is di-rectly applicable. For discontinuous transmission, the model isthe same with a modification , where is thevoice activity factor.

APPENDIX ADERIVATION OF EXPRESSION FOR THE MEAN ACQUISITION

TIME

In order to determine the mean acquisition time, a similarprocedure to the one that was used in [ 2] is followed. Only themost important differences will be pointed out here. The meanacquisition time is defined as

(a1)

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KATZ AND GLISIC: MODELING OF CODE ACQUISITION PROCESS IN CDMA NETWORKS 81

(b)

(c)

Fig. 7. (Continued.) Upper and lower bounds of the mean acquisition time for 30 realizations of a random phase shift vector. Normalized mean acquisition time( ), eight users, SNR . (b) , . Solid line: . Dotted line: . Dasheddotted line: . (c) , . Solidline: . Dotted line: . Dasheddotted line: .

where is the generating function defined by (11). Equation(11) can be written as

(a2)

where , and the generic th termis

(a3)

The derivative of the generating function at is

(a4)

The first term of (a4) is simply . The second term of (a4) canbe determined taking into account that is a sum of expres-sions and then its derivative will be the sum of derivatives of the

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(a)

(b)

(c)

Fig. 8. Normalized mean acquisition time for different thresholds. Number of users 0 , , phase vector Solidline: . (a) SNR dB. (b) SNR dB. (c) SNR dB.

corresponding expressions. The derivative of the generic termgiven by (a3) is computed by induction, resulting in

(a5)

Substituting (a5) into (a4) results in

(a6)

After some straightforward manipulations, the terms in-volving double summation are reduced as

(a7)

and

(a8)

Combining (a7) and (a8) according to (a6) and arrangingterms results in the following expression for the mean acqui-sition time:

(a9)

Denoting

(a10)

(a11)

(a12)

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and substituting these expressions into (a9) leads to (after a te-dious algebraic manipulation) the final equation for the meanacquisition time, namely

(a13)

where ; ; ; ; is defined in (14).

Note that, with exception of the last term on the right-hand sideof (a13), the expression for the mean acquisition time is quitesimilar to that obtained in [ 9] for quasi-synchronous systems.

APPENDIX BDERIVATION OF THE EXPRESSION FOR THE VARIANCE OF THE

MEAN ACQUISITION TIMEEquation (23) shows the variance of the mean acquisition

time as a function of both the mean acquisition time alreadydetermined and the second derivative of . In the remainderof this Appendix, the determination of is outlined.Many of the algebraic manipulations required here are similarto those used in [ 9]. We start from , which is given by

(b1)

Its first derivative results are shown in (b2), at the bottom of the page, while the second derivative is (note that for simplicity,the dependence of H with z has been omitted)

(b3)

which, for = 1, it is reduced to

(b4)In order to facilitate the derivation of , wewill split

(b4) as(b5)

where

(b6)

and

(b7)A few manipulations to (b6) leads to

(b8)

(b2)

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The first term of (b8) was already computed in [ 9], the secondterm is reduced according to

(b9)

and the third term of (b8) can be written as

(b10)

Thus, the expression for results in

(b11)

, given by (b7) can be in turn split into

(b12)

and

(b13)

Splitting (b12) as

(b14)

The first term of (14) can be developed as

(b15)

Distributing the sums on both terms of (b15) and working ongives

(b16)

The second term of (14) can be developed as

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[2] J. K. Holmes and C. C. Chen, Acquisition time performances of PNspread-spectrum systems, IEEE Trans. Commun., , vol. COM-25, pp.778784, Aug. 1977.

[3] R. DeGaudenzi , C. Elia, andR. Viola, Bandlimited quasi-synchronousCDMA; Novel satellite access technique for mobile and personal com-munications systems, IEEE J. Select. Areas Commun. , vol. 10, Feb.1992.

[4] A. Polydoros and M. Simon, Generalized serial search code acqui-sition: The equivalent circular state diagram approach, IEEE Trans.

Commun. , vol. 32, pp. 12601268, Dec. 1984.[5] A. Polydoros and C. L. Weber, A unified approach to serial searchspread-spectrum code acquisitionPart I: General theory, IEEE Trans.Commun. , vol. COM-32, pp. 542549, May 1984.

[6] , A unified approach to serial search spread-spectrum code acqui-sitionPart II: A matched-filter receiver, IEEE Trans. Commun. , vol.32, pp. 550560, May 1984.

[7] D. M. Di Carlo and C. L. Weber, Statistical performance of singledwell serial synchronization systems, IEEE Trans. Commun. , vol. 28,pp. 13821388, Aug. 1980.

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[10] D. Sarwatre and M. Pursley, Crosscorrelation properties of pseudo-

random and related sequences, Proc. IEEE , vol. 68, pp. 593619, May1980.[11] S. Bensley and B. Aazhang, Maximum likelihood synchronization of a

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Marcos D. Katz was born in Salta, Argentina,on January 29, 1962. He received the B.S.E.E.degree from the Universidad de Tucumn, Tucumn,Argentina, in 1987 and the M.S.E.E. degree fromthe University of Oulu, Oulu, Finland, in 1994. Heis currently working toward the Ph.D. degree inelectrical engineering at the Centre for WirelessCommunications, Telecommunication Laboratory,University of Oulu.

From 1987 to 1995, he worked as a Research En-gineer at Nokia Telecommunications, Fixed AccessSystems. In 1995, he joined Nokia Telecommunications (presently Nokia Net-works), Radio Access Systems, where he is currently a part-time Senior Re-search Engineer. His current research interests are in synchronization and arrayprocessing techniques.

Savo Glisic (M90SM94) is a Professor of elec-trical engineering at the University of Oulu, Oulu,Finland and the Director of Globalcomm Institute forTelecommunications. He was a visiting Scientist atCranfield Institute of Technology, Cranfield,Englandin 19761977and at theUniversityof California, San

Diego, in 19861987. He has been active in the fieldof spread spectrum and wireless communications for25 years and has published a number of papers andfive books. He is the Coauthor of the book Spread Spectrum CDMA Systems for Wireless Communica-

tions , (Norwood, MA: Artech, 1997) and Coeditor of the books Code Division Multiple Access Communications (Norwell, MA: Kluwer, 1995) and WirelessCommunications: TDMA Versus CDMA (Norwell, MA: Kluwer, 1997. He isdoing consulting in this field for industry and government.

Dr. Glisic has served as Technical Program Chairman of the Third IEEEISSSTA94, the Eighth IEEE PIMRC97, and IEEE ICC01. He is Director of IEEE ComSoc MD programs.