DS-CDMA Code Acquisition in the Presence of Correlated Fading ...

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 2004

DS-CDMA Code Acquisition in the Presence of Correlated Fading—Part I: Theoretical Aspects Giovanni E. Corazza, Member, IEEE, Carlo Caini, Member, IEEE, Alessandro Vanelli-Coralli, Member, IEEE, and Andreas Polydoros, Fellow, IEEE

Abstract—This paper deals with the analysis of the code-acquisition process performed by a mobile terminal in a code-division multiple-access cellular system. The fundamental contribution is the setting of a theoretical framework for code-acquisition analysis, accounting for the presence of fading and shadowing with specified coherence time. A multidwell architecture is considered, accounting for different verification strategies. The MAX/TC detection criterion is used, and the acquisition performance is analyzed, following both the direct and flow-graph approaches. Expressions for the acquisition time probability density function under different assumptions are given, as well as closed-form expressions for the mean and variance of the acquisition time. Practical applications and numerical results are reported in the Part II companion paper. Index Terms—Code-division multiple access (CDMA), code acquisition, fading, IMT-2000, IS-95, spread spectrum.

I. INTRODUCTION

T

HE advent of code-division multiple-access (CDMA) networks for cellular mobile communications has shifted the major focus of the application of spread-spectrum (SS) systems from the military to the commercial arena. Most of the cellular third-generation standards adopt direct sequence (DS) as the spreading technique. As known, the most significant peculiarity of an SS transceiver is the need for spreading/despreading circuitry, along with the code-synchronization subsystem. The latter is the critical core of the SS receiver, due to the fact that it must operate in extremely low signal-to-noise ratio (SNR), typically with unknown phase and a frequency offset. In a DS-CDMA network, additional impairments are present, such as multiple-access interference (MAI), multipath fading, and shadowing, which render the code-synchronization procedure even more critical. The most challenging step in code synchronization is the initial acquisition, i.e., the coarse alignment between the received code and the locally generated replica, which is typically accomplished by testing a finite number of code epochs obtained by discretizing the time uncertainty region into cells. Acquisition is Paper approved by S. N. Batalama, the Editor for Spread Spectrum and Estimation of the IEEE Communications Society. Manuscript received March 15, 2002; revised April 24, 2003. This paper was presented in part at the IEEE International Symposium on Spread Spectrum Techniques and Applications, Prague, Czech Republic, September 2002. G. E. Corazza, C. Caini, and A. Vanelli-Coralli are with DEIS/ARCES, University of Bologna, 40136 Bologna, Italy (e-mail: [email protected]; [email protected]; [email protected]). A. Polydoros is with the Department of Physics, University of Athens, Panepistimiopolis 157 87 Athens, Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.831414

followed by fine tracking of the code epoch. In detection-theory terms, acquisition can be formulated as a classical binary testing hypothproblem, which consists of deciding in favor of an hyesis (correct detection), while discarding all incorrect potheses (correct rejection) and trying to avoid missed detection and false alarm events. Several papers have appeared in the recent literature that address one or more aspects of code acquisition in DS-CDMA networks, e.g., see [1]–[11]. Building upon [12], the purpose of this two-part paper is to provide a general framework which encompasses most of the previous efforts and introduces novel theoretical and numerical results. We begin by considering the available options at system level which have a direct impact on code acquisition, discriminating between the forward link (FL), i.e., the link from the base station (BS) to the mobile terminal (MT), and the reverse link (RL). Acquisition in the FL is the first step that must be accomplished at MT power up. This is generally achieved by working on an unmodulated pilot signal, which is spread at the chip rate and is broadcast over the entire cell. The pilot can be continuous [13] or bursty [14]. Evidently, no power control can be enforced during this step, implying that shadowing and multipath fading effects cannot be counteracted by the BS. On the positive side, this procedure usually does not have a definite time out. It takes as long as necessary, although a short acquisition time is an obvious requirement for acceptable quality of service (QoS). Further, multiple pilot tones may be present, due to multipath and to adjacent BSs. The objective of the acquisition procedure is to collect information about the set of all of these hypotheses, which is then fed to the Rake1 manpossible ager. A totally different scenario characterizes the RL acquisition problem. In this case, the MT attempts to be detected by the BS by sending a short random-access burst, generally incorporating a known spread preamble to aid acquisition, followed by data containing a channel-assignment request. The MT can enforce an open-loop power-control procedure by estimating the received power in the FL, and adjusting its transmitted power accordingly. In case no acknowledgment is received from the BS (which is sent on a FL common channel), repeated attempts are performed with increasing transmitted power at pseudorandom time instants. The BS must recognize the MT burst and collect all the multiple paths for raking. A false-alarm or missed-detection event during RL initial acquisition translates into the loss of the random-access burst altogether. Due to the fact that power 1The Rake receiver [15] is a parallel structure with multiple fingers, each dedicated to collecting the energy of a particular multipath component or adjacent BS signal.

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CORAZZA et al.: DS-CDMA CODE ACQUISITION IN THE PRESENCE OF CORRELATED FADING—PART I: THEORETICAL ASPECTS

control can be used and that BS complexity is, in general, not much of an issue, RL acquisition can be considered to be less critical. Here we will focus on FL acquisition (see, for example, [6] for an approach to the RL problem). A very large number of options are available for the definition of the synchronization subsystem architecture. Here we update the classification provided in [16] to fit the present scenario. First, a search strategy for scanning the cells in the uncertainty region must be selected. The classic choices are parallel search (with dedicated hardware per cell, hence, maximum complexity) and serial search, with possible hybrid serial/parallel solutions. Due to the harsh complexity limitations in the MT, for FL acquisition, only serial strategies (or hybrid with low/moderate parallelism) are feasible. Then, the stage is open for a host of possible serial-search strategies (straight, expanding window, broken Z, etc.). In particular, the straight search is reasonable for acquiring the first path (initial acquisition), while an expanding window centered upon this first detected epoch can be adopted for acquiring all successive paths (postinitial acquisition). Reference [1] provides an in-depth consideration of the postinitial-acquisition problem. An interesting proposal for a serial-search strategy exploiting the possible knowledge on the multipath delay spread is reported in [8], where consecutive tests are performed on cells which are separated by the number of resolvable paths. In this paper, we will focus on initial acquisition in an unknown multipath delay spread, because it corresponds to the maximum uncertainty, and hence, to the most critical case. Once a search strategy is selected, it is necessary to adopt a criterion for testing cells. The discretization of the uncertainty region reduces the maximum-likelihood (ML) criterion to the MAX criterion [17] in additive white Gaussian noise (AWGN) channels,2 or to a slightly more complex decision rule in fading channels [7]. Interestingly, in [7], it is shown that the gain in using this more sophisticated rule is generally very small, implying that the MAX criterion can be safely adopted, even in fading channels. The MAX criterion requires the collection of all test variables in the uncertainty region, which may be a lengthy process for long codes. In order to anticipate decisions, the threshold-crossing (TC) criterion can be adopted, which, in general, is the most popular choice.3 Threshold design can be performed according to many possible criteria: Bayes, minimax, Neyman–Pearson, etc. The latter turns out to be particularly useful in fading channels, leading to a constant false-alarm rate (CFAR) design. Careful consideration of the threshold design problem in fading channels can be found in [4]. To improve testing performance, it is possible to combine the test variables pertaining to multiple cells before making a TC decision (cell bundling). Double-cell and triple-cell bundling are, respectively, considered in [11] and [8], while [7] considers the general -cell bundling case. In this paper, a hybrid MAX/TC criterion [17] is considered, 2The MAX criterion consists of choosing the cell corresponding to the maximum test variable value. It should not be confused with parallel search. 3The TC criterion consists of comparing a single test variable with a threshold.

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whereby the uncertainty region is divided into sectors; the MAX criterion is applied within a sector, while different sectors are tested serially in a TC fashion. The MAX/TC criterion contains the MAX and TC criteria as particular cases, giving an added value of generality to the results. Performing the above-described test requires a decision variable, obtained through a correlator which can be active or passive; the latter is referred to as code matched filter. Unfortunately, due to possible frequency offsets, it is not possible, in general, to correlate coherently for a sufficiently long time span. Therefore, noncoherent postdetection integration (PDI) is generally adopted, or an alternative differential PDI [18]. The analytical framework described in this paper encompasses both active and passive correlation, and arbitrary PDI. In order to reduce false-alarm probability, multiple tests on a hypothesis can be performed before moving on to tracking, adopting an -stage multidwell strategy. The first tests (verification dwell (search mode) is followed by mode). There are different strategies for verification. In the case of immediate rejection, the procedure is stopped as soon as a single test is failed, and the search mode is reinitiated. successful tests In the case of nonimmediate rejection, dwells are sufficient for passing control to tracking. over The difficulty for the designer lies in the fact that a host of particular cases are possible, and a unitary vision is lacking. In this paper, we introduce a novel analytical formulation, which is able to accommodate as particular cases both immediate and nonimmediate rejection strategies. The achievable performance is strongly affected by the propagation channel conditions. As a matter of fact, considering the code-acquisition problem in mobile fading channels, the main theoretical issue to be solved is the correct account for the influence that the fading correlation has on the search through the uncertainty region. Unfortunately, the memory introduced by the fading random process prevents a straightforward use of the classical tools for the analysis of the performance of the acquisition subsystem. In this paper, we show how the applicability of these tools, namely, the direct approach and the flow-graph approach (see [16], [17], and references therein), can be extended to account for the presence of fading with given spectral characteristics. This is done considering a Rice-lognormal (RLN) [19] fading channel, which contains as particular cases Rayleigh-lognormal, Rice, Rayleigh, and lognormal distributions, with variable channel dynamics. The analysis of some particular cases has been addressed in [9] and [10]. This paper aims at the definition of a general theoretical framework, and as such, it does not refer to any particular implementation. A detailed numerical analysis of a possible practical application is referred to the companion Part II paper [20]. The remainder of the paper is organized as follows. The fading channel is described in Section II; the multidwell MAX/TC criterion, along with the derivation of the testing probabilities, are introduced in Section III. Section IV deals with the multidwell acquisition time characterization through either the direct or the flow-graph approaches. Finally, conclusions are drawn in Section V.

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II. FADING-CHANNEL MODELING The pilot signal is transmitted through a linear time-variant communication channel, with its lowpass equivalent impulse response described by the following model: (1) where is the number of paths, and, for the th path, is is the phase shift, is the Doppler the fading envelope, frequency shift, is the delay, and indicates Dirac’s delta function. The effect of multipath components is twofold: 1) they contribute additional interference in the despreading process (self-noise); 2) they provide replicas of the pilot which are all valid candidates for acquisition. In other words, multipath has both negative and positive effects in the acquisition procedure. The negative effect can be equivalently modeled as an increase in the additive background noise. The positive effect can be neglected in order to perform a conservative analysis.4 Therefore, , accounting for we limit the analysis to a single path the self-noise in the equivalent background noise. The fading envelope is assumed to follow a RLN distribution and can be written as

is obtained through active or passive correlation, possibly followed by PDI. In order to achieve the CFAR objective, these test variables are appropriately normalized to the background [2], [20], yielding the sufficient noise variance estimate . statistics Considering the multidwell testing architecture, let and be, respectively, the duration and threshold of the th dwell, . Although the simplest solution consists of a series of identical tests, better performance may be achieved by varying the dwell correlation lengths, as well as the threshold values, from dwell to dwell. The design of dwell number, length, and threshold values is identified as the problem of multidwell optimization, which has not yet been given sufficient consideration in the literature. An applied optimization example is reported in Part II. In the search mode, the uncertainty region is divided into sectors consisting of cells. The following rule is applied: In a sector, select cell

if else, go to the next sector. (4)

where , depending on the tracker indicate a complementary hypothesis. pull-in range. Let It is common practice to discretize the estimate domain , uniformly spaced), in which case, the ( uncertainty region becomes a collection of cells of , centered upon the aforelength , a test variable mentioned discrete values. In each cell

Note that for and , the MAX/TC criterion , the MAX/TC reduces to the MAX criterion, while for criterion reduces to the TC criterion. Assume for simplicity that there is a single cell satisfying .5 Starting from a generic position, all sectors are serially cell, identified as tested to find the sector that contains the sector. Testing the sector, in the presence of fading, the three disjoint events are possible: cell test variable is above 1) Correct detection: The ; threshold and is the maximum, with probability test variables are below 2) Missed detection: All of the threshold, with probability ; sector): At least one test vari3) Error (false alarm in the cell is above threshold, and able associated with an cell test variable, with probability is greater than the . sector, with CFAR, two disjoint Conversely, testing an events are possible: 1) False alarm: At least one test variable is above threshold, with probability ; test variables are below 2) Correct rejection: All the . threshold, with probability In the verification mode, samples of the test variable of the candidate cell are sequentially compared with specified . thresholds, cell in the th dwell, two disjoint events are Verifying the possible: 1) Correct detection: The test is passed with probability ; 2) Missed detection: The test is failed with probability . cell in the th dwell, we have: Conversely, verifying an 1) False alarm: The test is passed with probability ;

4The acquisition subsystem must be able to operate also in environments where the delay spread is very small, i.e., less than the chip time.

5The formulation can be extended to multiple more cumbersome.

(2) where is a stationary Rice random process accounting for small-scale diffuse multipath fading plus a direct signal compois a stationary lognormal random process modnent, and eling large-scale shadowing effects. In general, the coherence time of multipath fading is much shorter than that of shadowing. The two processes are assumed to be statistically independent. be the vectors conFor a digital receiver, let taining the samples of the overall fading, multipath, and shadowing which are experienced at the th step of the acquisition procedure. These vectors will be used in the following to summarize in a short notation the effects of the fading channel. For simplicity, the superscript will be omitted whenever its value can be easily deduced from the context. III. MULTIDWELL- MAX/TC TESTING Let be the delay in the received sequence, being the sequence length and the chip time, and its estimate, . Synchronism is achieved if the following holds: hypothesis or (3)

H

cells, but notation becomes


2) Correct rejection: The test is failed with probability . Evidently, the above probabilities depend on the particular implementation of the acquisition subsystem. Therefore, we refer to the Part II paper for their analytical definition considering a particular architecture. Here, it is appropriate to leave these basic elements in symbolic form to carry out a general analytical characterization of the verification process.

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. The expression for equal to results from a false alarm, both at the first and at the th failed tests in the dwell, and exactly intermediate dwells. Equations (5) and (6) are involved, but can be simplified for the , yielding “immediate rejection” policy, by setting (7)

A. Verification Strategy We describe a unifying verification strategy which encompasses various rejection policies. Letting be the number of tolerated test failures, the verification process has to be terminated whenever the remaining dwells are inconsequential for the verification result. This happens either when the hypothesis is th failed test), or, conrejected (i.e., immediately after the versely, when the hypothesis is verified (i.e., immediately after tests). It is evident having passed the threshold of that setting or yields the immediate or nonimmediate rejection rule, respectively. Therefore, the overall multidwell test duration is not fixed, rather, it is a random variable (r.v.). In a way, this can be considered as a time-discrete approximation of sequential testing. In the following, we derive the probabilities of the overall sector acquisition process, beginning with those related to tests, since under CFAR, these probabilities do not depend on fading, and notation is less involved. • :6 Probability of no false alarm after the th dwell (i.e., false alarm not confirmed, with exit from the verification process at the th dwell, as shown in (5) at the bottom of the page), where is the set containing the indexes of which all binary vectors . Note that the first have Hamming weight equal to equality results from the necessity of having a false alarm at the first dwell, a correct rejection (i.e., a test failed) at the th dwell, and tests failed in the intermediate dwells.7 •

: Probability of false alarm verified, with verification process terminated at the th dwell, as is shown in (6) at the bottom of the page, where the set containing the indexes of all binary vectors having Hamming weight

6Capital letters in the subscripts are intended to distinguish probabilities that refer to the overall acquisition process from probabilities referring to single dwells. 7It is intended that x = 1; 8x.

(8) Testing the sector, dependence on fading must be considered. To this end, the following probabilities are conditional on the fading vector . • : Probability of detection not confirmed, with exit from the verification process at the th dwell. It can be formally obtained from (5), substituting with . : Probability of detection confirmed, with exit • from the verification process at the th dwell. It can be formally obtained from (6), substituting with . • : Probability of detection confirmed at any dwell. : Error not verified, with exit from the verifica• tion process at the th dwell. It can be formally obtained with . from (6), substituting • : Error verified, with exit from the verification process at the th dwell. It can be formally obtained from with . (6), substituting Having defined all the necessary probabilities, it is possible to derive the acquisition-system performance in terms of acquisition time characterization following two alternative methods, the direct approach and the flow-graph approach. In both cases, we will refer to the following time parameters: • : Test duration up to the th dwell (9) •

: Penalty time due to nonabsorbing false alarm. Whenever a false alarm, or an error, is erroneously confirmed at the end of the verification process, the system enters a state of nonabsorbing false alarm. The penalty time represents the time spent in this state (large compared with the dwell times), which is modeled as a fixed value.

(5)

(6)

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IV. MULTIDWELL ACQUISITION TIME CHARACTERIZATION A. Direct Approach The direct approach has been introduced for the case of unfaded TC criterion (e.g., [21]), and extended to the MAX/TC criterion for a two-dwell detector in AWGN in [17]. Here we further extend it according to the more general description of the MAX/TC criterion provided in the previous section (multidwell architecture, presence of fading and shadowing, generalized verification strategy). and indicate that, at the th step of a serial Let is being tested for the search, the th sector th time . For a straight serial search, we have and , where stands for the largest integer not greater than . The time needed for correct acquisition is given by the sum of the time intervals pertaining to the following random events: sector with correct detection; • one test over the tests over the sector with missed detection; • tests over sectors. • Dealing with a sum of r.v.’s, it is convenient to use characteristic functions (cf’s). To this end, let us express the conditional cf of sector as the time needed for detection testing the (10) where

be the cf of the time needed to reject any Further, let sector. This includes the nonabsorbing false-alarm event as sector the least efficient way to reject an

(16) Following the same procedure which led to (15), we assign to all verified false-alarm events the same duration . By defining as the probability of rejection at the th dwell (17) we can write (18) Although (11), (15), and (18) have been derived consid, ering the general case of a multidwell system it can be verified that they hold also in the particular case , no verification), by setting of a single dwell ( . Finally, the cf of the conditional acquisition time can be calculated as

(11) Similarly, the conditional cf of the time spent for a missed detection is (12)

(19) where is the probability that the sector labeled sector, and the

is

where (20) (13) To achieve analytical tractability, it is expedient to introduce a more compact expression by assigning to all verified error events the same duration . This corresponds to a worst-case analysis (because the error could be verified at an earlier stage), but the excess time introduced is negligible, since . By defining the probability of missing the sector at as the th dwell (14) it is possible to rewrite (13) in the following compact form: (15)

Substituting and simplifying, we obtain the final expression

(21) Equation (21) is very general, needing only the CFAR assumption; therefore, it holds for arbitrary correlation characteristics in the fading process, although eliminating the conditional dependence from can be very complex, or altogether analytically intractable. In order to consider some significant applications of (21), it is useful to classify the fading process on the basis of the memory time intervals8 of its multipath and shadowing components, 8Defined as the temporal span that, if exceeded, ensures the statistical independence of the fading samples.


and (with ). Let be the minimum time interval sectors. As far as shadbetween two consecutive tests of owing is concerned, we distinguish between the following alternatives: ; shadowing is • Static shadowing, whenever constant during the overall acquisition process, involving sector tests; several ; shadowing • Slow shadowing, whenever sector test to the changes (with correlation) from an following; sectors), when• Fast shadowing (between consecutive ; shadowing is considered indepenever sector test to the following, but constant dent from an sector.9 in testing a single Considering the multipath component, we distinguish among the following cases: • Static multipath, whenever ; multipath is constant during the overall acquisition process; • Slow multipath, whenever ; multipath may sector test to the change (with correlation) from an following [10]; ; multipath is consid• Fast multipath, whenever ered independent from an sector test to the following. Note that as far as Part I is concerned, the above classification of multipath dynamics would be sufficient. However, a finer classification for the fast multipath case is necessary to the end of Part II: • Full correlation (inside a single sector test), whenever ; multipath is independent from an sector test to the following, but is constant during a single test; ; multi• Partial correlation, whenever sector test, but it is conpath changes during a single stant over chip times (accumulation length); ; it is the extreme con• Uncorrelated, whenever dition of the previous case. Multipath is independent over two consecutive accumulated samples, but it is practically constant over chip times. The analysis of all the possible channel conditions would lead to an excessively long treatise. Therefore, in the following, we will focus on three significant cases. 1) Fast Shadowing and Fast Multipath (Very High MT Speed): In this case, it is correct to average the acquisition time over fading, as this average value can be considered representative of the system performance. Being independent from sector test to sector test, (21) reduces to

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lowing the derivations reported in Appendix A, the probability is obtained as density function (pdf) of the acquisition time

(23)

2) Static Shadowing and Fast Multipath (Medium-High MT Speed): In this case, it is correct to average the acquisition time over multipath fading only, as the effects of a constant shadowing can be more efficiently treated as an SNR fluctuation. Then, averaging over multipath fading and letting be the unique value for shadowing, (21) reduces to

(24) where . To achieve the conditional average acquisition-time pdf, let us define , and . Then, following a procedure similar to that reported in Appendix A, we obtain

(25)

(22) where Let us define

. , and

9The

case < T

can be neglected for all practical purposes.

, . Fol-

3) Static Shadowing and Static Multipath (Static-Slow Moving MT): In this case, it is not correct to carry out any average on (21), as the effects of a constant fading should be considered as an SNR fluctuation. Letting be the unique value for the overall fading, the final result, i.e., the conditional average acquisition-time pdf, can be formally derived from (25) by simply substituting with .

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Under all cases 1–3, it is readily seen that the averaged cf’s for both missed detection and correct detection do not depend on the index of the sector under test. This is an essential condition to find manageable expressions for the acquisition time distribution using the direct approach. It is also the necessary condition to be able to apply the flow-graph approach at all.

from which expressions for the average and variance of the acquisition time can be obtained through derivation, see [17, eqs. (30) and (32)]. Inserting in [17, eq. (30)] the new expressions and , and denoting by for

(30) B. Flow-Graph Approach The drawback of the direct approach is that the computation of the acquisition-time pdf can be very lengthy, especially when the number of sectors is large. A much faster way to analyze performance is given by the flow-graph approach, which leads to closed-form expressions for the moments of the acquisition time. This approach exploits the duality existing between the state transition diagram of a discrete-time Markov chain and the flow graph of electrical systems. When the MAX/TC criterion is used, the nodes in the flow graph correspond to sectors in the uncertainty region. Here we extend the use of the flow-graph approach to a MAX/TC multidwell system working in a fading environment. The fundamental assumption for modeling the acquisition procedure as a discrete Markov chain is that decision statistics in different sectors must be statistically independent. As shown in Section IV-A, this is true in the study case 1, and is conditionally true in cases 2 and 3, where the conditioning is intended on shadowing or on overall fading, respectively. Under either of these assumptions, the treatment of the flow-graph approach for the MAX/TC criterion presented in [17] applies completely, as long as the original expressions for the branch gains are properly modified according to the fading correlation characteristics. In particular, in the following, we will focus on case 2, which is the reference scenario for numerical evaluation in the Part II paper. On the other hand, whenever fading correlation introduces statistical dependence, then the flow-graph approach fails, and the direct approach is the only resource. Let the branch gains be

(26) (27) (28)

Once the flow graph is completely defined, the generating function can be computed and used to derive the moments of the acquisition time. Assuming a uniform a priori probability , we have

(29)

we obtain the final expression for the average acquisition time as

(31) where (32)], and introducing

. Similarly, working on [17, eq. and , we have for the acquisition

time variance

(32) As far as cases 1 and 3 are concerned, expressions for the average and variance of the acquisition time can be obtained from (31) and (32) by formally eliminating the fading dependence in case 1 and by substituting with in case 3. Note that by means of the flow-graph approach, it is not possible to derive the exact expression for the acquisition-time pdf, but an accurate chi-square approximation can be found using and , as described in detail in [17]. This approximation is useful for calculating the percentiles of the acquisition time. V. CONCLUSION The adoption of the CFAR criterion proves to be essential in the design of a spread-spectrum acquisition system which is robust against fading. In fact, by designing the dwell tests to achieve CFAR, the effect of fading on the acquisition procesector. Starting dure can be confined to tests involving the from this essential observation, a theoretical framework for the performance evaluation of a spread-spectrum acquisition subsystem in the presence of arbitrary fading conditions has been described. This framework is derived considering a general and quite complex architecture (multidwell MAX/TC criterion, with different dwell lengths and thresholds, general rejection policy), and following both the direct and the flow-graph approaches. The general expressions of the acquisition-time pdf, as well as of its statistical average and variance, represent the main results of the analysis. They lay the foundations for the analytical (or semianalytical) investigation of the impact of fading on a practical acquisition subsystem, as reported in Part II.


APPENDIX A DERIVATION OF ACQUISITION-TIME PDF IN PRESENCE OF FAST SHADOWING AND FAST MULTIPATH

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which leads to the following:

With the definitions introduced in the text, we have

(33) from which, expanding the last term

(39) Finally, averaging on , we obtain the desired result (23). REFERENCES (34)

Then, noting that (35) we obtain

(36) Similarly, it can be shown that

(37) Now, applying (36) and (37), we have

(38)

[1] S. Glisic and M. D. Katz, “Modeling of the code acquisition process for Rake receivers in CDMA wireless networks with multipath and transmitter diversity,” IEEE J. Select. Areas Commun., vol. 19, pp. 21–32, Jan. 2001. [2] B. B. Ibrahim and A. H. Aghvami, “Direct sequence spread spectrum matched filter acquisition in frequency-selective Rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 12, pp. 885–890, Jan. 1994. [3] J. Iinatti, “Performance of DS code acquisition in static and fading multipath channels communications,” Inst. Elect. Eng. Proc., vol. 147, pp. 355–360, Dec. 2000. , “On the threshold setting principles in code acquisition of DS-SS [4] signals,” IEEE J. Select. Areas Commun., vol. 18, pp. 62–72, Jan. 2000. [5] C. Kim, T. Hwang, H. Lee, and H. Lee, “Acquisition of PN code with adaptive threshold for DS/SS communications,” Electron. Lett., vol. 33, no. 16, pp. 1352–1354, July 1997. [6] W. A. Krzymien, A. Jalali, and P. Mermelstein, “Rapid acquisition algorithms for synchronization of bursty transmissions in CDMA microcellular and personal wireless systems,” IEEE J. Select. Areas Commun., vol. 14, pp. 570–579, Apr. 1996. [7] R. R. Rick and L. B. Milstein, “Optimal decision strategies for acquisition of spread-spectrum signals in frequency-selective fading channels,” IEEE Trans. Commun., vol. 46, pp. 686–694, May 1998. [8] O. Shin and K. B. Lee, “Utilization of multipaths for spread-spectrum code acquisition in frequency-selective Rayleigh fading channels,” IEEE Trans. Commun., vol. 49, pp. 734–743, Apr. 2001. [9] W. Sheen, M. Chang, and H. Wang, “A new analysis of direct-sequence pseudo-noise code acquisition on Rayleigh fading channels,” in Proc. IEEE Int. Symp. Spread-Spectrum Theory and Applications, 2000, pp. 40–44. [10] H. Wang and W. Sheen, “Variable dwell-time code acquisition for direct-sequence spread-spectrum systems on time-variant Rayleigh fading channels,” IEEE Trans. Commun., vol. 48, pp. 1037–1046, June 2000. [11] L. Yang and L. Hanzo, “Serial acquisition of DS-CDMA signals in multipath fading mobile channels,” IEEE Trans. Veh. Technol., vol. 50, pp. 617–628, Mar. 2001. [12] G. E. Corazza and A. Polydoros, “Code acquisition in CDMA cellular mobile networks. I. Theory,” in Proc. IEEE Int. Symp. Spread-Spectrum Theory and Applications, 1998, pp. 454–458. [13] Qualcomm Inc., San Diego, CA, “System and Method for Generating Signal Waveforms in a CDMA Cellular Telephone System,” Patent PCT/US91/04 400—WO 92/00 639, Jan. 9, 1992. [14] H. Holma and A. Toskala, WCDMA for UMTS. New York: Wiley, 2001. [15] A. J. Viterbi, Principles of Spread Spectrum Communications. Reading, MA: Addison-Wesley, 1995. [16] A. Polydoros and C. L. Weber, “A unified approach to serial search spread-spectrum code acquisition—Part I: General theory,” IEEE Trans. Commun., vol. COM-32, pp. 542–549, May 1984. [17] G. E. Corazza, “On the MAX/TC criterion for code acquisition and its application to DS-SSMA systems,” IEEE Trans. Commun., vol. 44, pp. 1173–1182, Sept. 1996. [18] M. H. Zarrabizadeh and E. S. Sousa, “A differentially coherent PN code acquisition receiver for CDMA systems,” IEEE Trans. Commun., vol. 45, pp. 1456–1465, Nov. 1997.

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[19] G. E. Corazza and F. Vatalaro, “A statistical model for land mobile satellite channels and its application to nongeostationary orbit systems,” IEEE Trans. Veh. Technol., vol. 43, pp. 738–742, Aug. 1994. [20] G. E. Corazza, C. Caini, and A. V. Coralli, “DS-CDMA code acquisition in presence of correlated fading—Part II: Application to cellular networks,” IEEE Trans. Commun., to be published. [21] V. M. Jovanovic, “Analysis of strategies for serial-search spread-spectrum code acquisition—Direct approach,” IEEE Trans. Commun., vol. 36, pp. 1208–1220, Nov. 1988.

Giovanni E. Corazza (M’92) was born in 1964. He received the Dr.Ing. degree (summa cum laude) in electronic engineering in 1988 from the University of Bologna, Bologna, Italy, and the Ph.D. degree in 1995 from the University of Rome “Tor Vergata,” Rome, Italy. He is currently a Full Professor with the DEIS Department, University of Bologna, where he held the Chair for Telecommunications during 2000–2003, and is responsible for Wireless Communications inside the the Advanced Research Center for Electronic Systems (ARCES). He is Chairman for the Advanced Satellite Mobile Systems Task Force (ASMS-TF), a European forum with more than 60 industrial partners. In 1989–1990, he was with COM DEV, Ontario, Canada, as an Advanced Member of Technical Staff. In 1991–1998, he was with the DIE Department, University of Rome “Tor Vergata” as a Researcher. During 1995, he visited ESA/ESTEC, Noordwjik, The Netherlands, as a Research Fellow. During 1996, he was a Visiting Scientist at CSI, University of Southern California, Los Angeles. CSI invited him as a Visiting Professor to hold a graduate course on spread-spectrum systems in the fall of 2000. During 1999, he was a Principal Engineer at Qualcomm, San Diego, CA. His research interests are in communication and information theory, wireless communications (cellular, satellite mobile/fixed), spread-spectrum techniques, CDMA, synchronization and parameter estimation, MAC layer and multicast protocols. He is author or coauthor of more than 80 papers published in international journals and conference proceedings, and author of a patent on the cdma2000 system. Dr. Corazza has been an Associate Editor for Spread Spectrum for the IEEE TRANSACTIONS ON COMMUNICATIONS since 1997. He received the Marconi International Fellowship Young Scientist Award in 1995. He was corecipient of the Best Paper Award at ISSSTA’98 and at ICT 2001, and was corecipient of the 2002 IEEE VTS Best System Paper Award.

Carlo Caini (M’93) was born in Bologna, Italy, in 1960. He received the Dr.Ing. degree (cum laude) in electronic engineering from the University of Bologna, Bologna, Italy, in 1986. Since 1990, he has been with the Department of Electronics Computer Science and Systems of the same University, as a Researcher Associate. His main scientific interests are in the field of terrestrial and satellite cellular mobile radio systems, with a special emphasis on spectrum efficiency, multiple-access techniques and spread spectrum systems. He has participated in several international research projects and he is author of many international publications on these topics. Dr. Caini is a member of IEEE Communications Society.


Alessandro Vanelli-Coralli (S’93–M’97) was born in Bologna, Italy, in 1967. He received the Dr.Ing. degree (cum laude) in electronics engineering and the Ph.D. degree in electronics and computer science, both from the University of Bologna, Bologna, Italy, in 1991 and 1996, respectively. In 1996, he joined the Department of Electronics, Computer Science and Systems (DEIS) at the University of Bologna, where he is currently a Research Associate. Since 2001, he has been a staff member of the Advanced Research Center for Electronic Systems (ARCES) of the University of Bologna. During 2003, he was a Visiting Scientist at Qualcomm, Inc., San Diego, CA. He participates in national and international research projects on satellite mobile communication systems, and he is responsible for the Research and Development Group of the Advanced Satellite Mobile Systems Task Force (ASMS-TF). His research interests are in the area of spread-spectrum communications, synchronization techniques, and digital signal processing. Dr. Vanelli-Coralli was corecipient of the Best Paper Award at the IEEE ICT 2001 Conference, June 4–7, 2001, Bucharest, Romania.

Andreas Polydoros (S’76–M’78–SM’92–F’95) was born in Athens, Greece, in 1954. He received the Diploma in electrical engineering in 1997 from the National Technical University of Athens, Athens, Greece, the M.S.E.E. degree in 1979 from the State University of New York at Buffalo, and the Ph.D. degree in electrical engineering in 1982 from the University of Southern California (USC), Los Angeles. He was a Faculty Member in the Electrical Engineering Department, USC, and the Communication Sciences Institute (CSI) from 1982 to 1997, and was a Professor since 1992. He codirected CSI from 1991 to 1993. Since 1997, he has been a Professor and Director of the Electronics and Systems Laboratory, Division of Applied Physics, Department of Physics, University of Athens, Athens, Greece. His research interests include statistical communication theory with applications to wireless and wireline transmission (including spread spectrum, cellular, and satellite, as well as high-throughput multicarrier systems), signal detection and classification, data detection in uncertain environments, and multiuser radio networks. He is a co-inventor (with R. Raheli) of a U.S. patent on per-survivor processing. He served as Guest Editor of the July 1993 special issue for Digital Signal Processing: A Review Journal, a designated Area Editor for the international journal Wireless Personal Communications, and as a Co-Guest Editor of the March/April 1998 special issue of the European Transactions on Telecommunications. Dr. Polydoros was the recipient of a 1986 U.S. National Science Foundation Presidential Young Investigator’s Award. He served as Associate Editor for Communications of the IEEE TRANSACTIONS ON INFORMATION THEORY from 1987 to 1988.