Effects of Cell Correlations in a Matched-filter PN Code Acq

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724 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 3, MAY 1999

Effects of Cell Correlations in a Matched-FilterPN Code Acquisition for Direct-Sequence

Spread-Spectrum SystemsWern-Ho Sheen, Member, IEEE, Jiun-Kai Tzeng, and Ching-Kae Tzou

Abstract Rapid pseudonoise (PN) code acquisition withmatched-filter correlators has been very popular in direct-sequence (DS) spread-spectrum systems. Conventionally, theanalysis of this acquisition method is based on the assumptionthat the detections among cells are independent. However, theremay be strong correlations among cell detections for the casethat the cell size is less than a chip duration. In this paper, themean acquisition time performance of the acquisition methodis analyzed with the cell correlations being taken into account.Numerical results show that depending on the threshold valueand other system parameters, the effect of cell correlations maybe over 20% of the mean acquisition time for signal-to-noiseratios (SNRs) of practical interest. The analytical results aresubstantiated by computer simulations.

Index Terms DS spread spectrum, matched filter, PN codeacquisition.

I. INTRODUCTION

IN DIRECT-SEQUENCE (DS) spread-spectrum systems,

pseudonoise (PN) code acquisition aligns the received and

locally generated PN codes to within a small range of timing

error before a code tracking loop can finetune and maintain a

precise timing synchronism of the two PN codes. (The range

of permissible timing error is often determined by the pull-inrange of the code tracking loop.) Rapid PN code acquisition

has been the most challenging work faced by a DS spread-

spectrum receiver [1][3].

PN code acquisition for DS spread-spectrum systems has

been drawing much research attention in the past [1][20].

Acquisition methods such as multiple-dwell serial search ac-

quisition [1][6], serial search acquisition with sequential

detection [1][3], [7][9], serial (or parallel) search acquisition

based on matched-filter (MF) correlators [10][15], and ac-

quisition with recursion-aided sequential estimation [16], [17]

have been investigated extensively in the literature. Of these,

the method based on MF correlators has a faster acquisition

speed at the expense of a larger system complexity for a

large PN code period, as opposed to other types of acquisition

Manuscript received July 7, 1996; revised February 25, 1997. This workwas supported in part by the National Science Council, Taiwan, R.O.C., underNSC Grant 85-2213-E-194-001.

W.-H. Sheen is with the Department of Electrical Engineering, NationalChung Cheng University, Chia-Yi 621, Taiwan, R.O.C.

J.-K. Tzeng is with the National Space Program Office, Science-BasedIndustrial Park, Hsin-Chu 300, Taiwan, R.O.C.

C.-K. Tzou is with the TranSwitch Corporation, Taipei 13F-3, Taiwan,R.O.C.

Publisher Item Identifier S 0018-9545(99)01043-9.

methods. In this study, we are concerned with the performance

analysis of the serial search PN code acquisition based on MF

correlators.

In a serial search PN code acquisition, the time uncertainty

of the incoming code, denoted as , is divided into cells,

and the cells are searched on a cell-by-cell basis until true

code acquisition is accomplished (timing error is within the

permissible range).1 Quite often, the cell size is taken as a

fraction of a chip duration. Therefore, there are more than one

cell at which true acquisition is possible. These types of cellswill be denoted as cells, and the set of all cells will be

denoted as the region. All other cells for which the true

acquisition is not possible will be denoted as the region,

and the cells in the region be denoted as the cells. For the

acquisition method based on MF correlators, since the decision

whether a particular cell is a correct cell (true acquisition) can

be made in a duration of cell size, the acquisition speed is much

faster than other types of acquisition methods, for example, the

method of multiple-dwell acquisition [2], [3].

Conventionally, the performance analysis of the PN code

acquisition based on MF correlators assumes that the detec-

tions between cells are independent. Then, a transfer function

and/or a time domain approach can be employed to obtain

the mean and variance and/or the probability density function

(pdf) of the acquisition time [1][3], [10], [18][20]. However,

as to be shown, the assumption of independency between cell

detections is only valid for the case when the cell size is equal

to the chip duration. For cell sizes less than one chip duration,

the detections between cells may be strongly correlated. In

this paper, the mean acquisition time performance of the

acquisition method is analyzed with the cell correlations being

taken into account. The results show that, depending on the

threshold values and other system parameters, the effect of

cell correlations may be over 20% of the mean acquisition

time for signal-to-noise ratios (SNRs) of practical interest.Computer simulations have been used to substantiate the

analytical results.

The remainder of this paper is organized as follows.

Section II describes the serial search PN code acquisitionbased on MF correlators. Section III describes a state

transition diagram with which the mean acquisition time

can be evaluated. In Section IV, the expression for the mean

acquisition time is derived, and some numerical examples

1 For simplicity, the frequency uncertainty will not be considered.

00189545/99$10.00 1999 IEEE


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SHEEN et al.: EFFECTS OF CELL CORRELATIONS IN PN CODE ACQUISITION 725

Fig. 1. A typical matched-filter detector for DS PN code acquisitions.

are shown in Section V. Finally, conclusions are given in

Section VI.

II. SYSTEM DESCRIPTION

A typical structure for DS PN code acquisition based on

MF correlators is shown in Fig. 1. The received signal is

(1)

where and are the carrier power, frequency and

phase, respectively, is a maximum length sequence with

nonreturn-to-zero shaping function, is the timing information

needed to be acquired, and is additive white Gaussian

noise (AWGN) with one-sided power spectra density (PSD) of

watts/hertz. For simplicity, the effects of data modulation

and frequency shift will not be considered. The received

signal is first downconverted to the inphase and quadrature

components. After integration by s, the inphase and

quadrature signals are sampled and correlated, respectively,

with an MF correlator as shown in Fig. 2, whereand denotes the chip duration. The samples

are taken every s at the instants of

with The MF correlators perform the correlation

between the incoming and locally generated PN codes. Each

MF uses an -stage tapped delay line with a tap

spacing of s, where is the period of the maximum length

sequence (full period correlation). is the locally

generated PN sequence to be defined later. The MF correlator

outputs are squared, and the sum is used to compare

with a threshold For one-dwell systems, if exceeds

the threshold then the cell being tested is considered as

a correct cell and the synchronization process is transferred

to the code tracking. For two-dwell systems, however, theacquisition enters the second dwell (verification mode) after

the excess of .2 As an example, an active correlator with

a second threshold comparison is assumed for the

verification mode, although some other schemes may also be

employed, for example, the coincidence detector [10]. If the

threshold is exceeded again, then the cell is considered

as a correct cell. Otherwise, the cell will be considered as

an incorrect one, and a new cell will be searched and tested

[1][3]. This means that the first sample in the tap delay lines

2 The extension of the proposed method to more than two-dwell systems isstraightforward.

Fig. 2. The matched-filter correlator.

will be shifted out and the new coming one will be moved in,

and the correlation and threshold comparisons repeat. Many

strategies can be used for searching a new cell [2], [3]. Here,

only the straight line search strategy will be used as an examplefor evaluating the mean acquisition time.

The case of returning false alarm will be considered which

means that the tracking loop can always detect the false

alarm after some time, called the penalty time, and return the

synchronization to the code acquisition. The penalty time will

be modeled as a fixed value and denoted by with

This model is a bit simplified. Nevertheless, it is commonly

employed in the study of acquisition systems.

By using the chip synchronous model [8], i.e., and

assume that , the inphase and quadrature outputs at the

MF correlators are given by

(2)

and

(3)

respectively, where

is the maximum length sequence, is the integer


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Fig. 4. The simplified state transition diagram with 1

Fig. 5. The simplified state transition diagram for no correlations from the

to

region with 1

To derive the mean acquisition time, the following defini-

tions are useful.

The detection probability for one passes through theregion. Let denote a sequence of state tran-

sitions with the initial state and the final state and

denote the probability of this particular sequence

of transitions. Then, the overall detection probability is

given by

(22)

where denote the set of all possible sequences

of state transitions with the initial state and final state

TABLE IIGENERALEXPRESSIONS FOR THE

0 AND

0 INFIG. 4

The average dwell time from state to

state given that acquisition is reached during the

next pass of the region. Let be the

dwell time for the particular state transition sequence

Then

(23)

The average dwell time from the starting state

to the state For (the

region)

(24)

where

(25)

and

(26)

with and

In (25) and (26), and

denote the transition probability and dwell time for the

transitions associated with the substate pair

respectively. Recall that there may be two transitions

associated with the substate pair and that

and are independent of with

For

(the region)

(27)


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where

(28)

is the average dwell time from to and

(29)

is used in calculating

The average dwell time

from the state to the state itself (no

acquisition occurs when passing through region).

It can be shown that

(30)

where

(31)

is the average dwell time for missing the region and

(32)

is used in calculating

With these definitions, the mean acquisition time for the

case that the starting cell is is given as follows. For

(33)

From (33), it is easy to see that

(34)

For

(35)with and , where

the last term is due to the fact that it is possible to acquire

the code phase without searching through the region when

Consequently, the overall mean

acquisition time is given by

(36)

A simpler example with is given in the Appendix to

explain the derivation of (33) more clearly.

The final step to complete the analysis is to obtain the

probabilities and FromTables II and III, we know that and take the

general forms of

(37)

or

(38)

where is given in (13). In (37), it is well

known that is chi-square distributed with the pdf given by

(39)

where is the zeroth-order modified Bessel function of

the first kind, , and is defined

in (9). Therefore, (37) can be evaluated as

(40)

where

(41)

is the Marcum generalized function, which can be evaluatedvery efficiently by the saddle point integration method [21].

For then (40) becomes

(42)

For active correlators, the probabilities and take the

form of (37) and can be evaluated similarly as in (40). Un-

fortunately, there is no efficient method to calculate (38) due

to that are correlated central or noncentral

chi-square random variables. To the best of our knowledge,

for central chi-square cases, one way to do it is to express the

joint pdf of in a convergent infinite series


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Fig. 6. Effects of the cell correlations on the performance of mean acqui-sition time.

and then the probability of the form

can be evaluated numerically [22], [23]. However, for non-

central chi-square cases, no such convergent infinite series

have been found in the open literature. Moreover, since a

numerical integration is still needed in the above infinite series

method, the method would not be easier than the Monte Carlo

integration employed in this study. Note that all conditional

probabilities for the thresholds of interest can be found by

using one integration. This is important in the process of

finding the optimum threshold. Let

(43)

where or and Since

we have

(44)

By using (44), (4) and (5) become

(45)

It can be shown that and are approximatelyuncorrelated, if for a large and, hence,

they are independent Gaussian variables. Equation (45) can be

used to efficiently generate the random variables

Taking as an example, the Monte-Carlo

integration can be used to calculate the probability

(46)

by generating the random variables and accord-

ing to (45) and (13).

Fig. 7. Comparisons of analytical and simulation results for the cases of1 and

dB.

V. NUMERICAL EXAMPLES

In this section, numerical examples are used to show the

effects of cell correlations. For simplicity, only one-dwell

systems will be considered.3 is employed for all

numerical examples in this section. Monte-Carlo integration

with 10 samples have been used to obtain the probability

(38). The mean acquisition time is evaluated only for the case

of , i.e., the worst case.

Fig. 6 shows typical examples of the effects of cell corre-

lations. Recall that is the time uncertainty of the incoming

code. In the figures, the SNR is defined by

(47)

As expected, the effects of the cell correlations are generally

more significant for a smaller Also, at the expense of a

larger system complexity, a better performance is obtained

with a smaller (larger ) due to the fact that there are more

cells in the region. However, as evident in the figure, the

performance improvement becomes smaller with less than

1/3. Similar results are observed for the cases of In

Fig. 7, computer simulations are used to verify the analytical

results for the case of As can be seen, analytical

results agree very well with the simulation ones, and, therefore,

it is verified that the correlations from the to regions

can be safely neglected. In our simulations, the sequence is

generated by the primitive polynomial , and

500 acquisitions are averaged to obtain the mean acquisition

time.

Fig. 8 shows the relative acquisition time error for the case

of and with various SNRs and s. The

relative acquisition time error is defined as

(48)

3 For two-dwell systems, since the verification process provides independentsamples, the effects of cell correlations may be smaller than those presentedhere.


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Fig. 8. Example relative acquisition time errors.

TABLE IIITHEOPTIMUMTHRESHOLD, THE MINIMUMMEAN ACQUISITIONTIME, AND THE

ASSOCIATED RELATIVE ERROR FOR VARIOUS SYSTEM PARAMETERS

where is the mean acquisition time without

considering the cell correlations. As can be seen, the larger

the SNR, the larger the relative acquisition time error. Also,

over 20% of errors are observed. Note that the largest relative

error does not necessarily occur at the optimum threshold.

(The optimum thresholds in the figure can be read from

Table III.) Once again, the result shows that the effects of

the cell correlations are more significant for a smaller

Table III shows the optimum threshold, minimum mean

acquisition time, and the associated for various system

Fig. 9. The state transition diagram with 1

region .

parameters. As can be seen, for an SNR larger than 0 dB,

over 10% and 20% of errors are observed for and

respectively. For 5%8% of errors are

observed for the same range of SNRs. In addition, the effects

of cell correlation are not sensitive to the values of

VI. CONCLUSIONS

In this paper, a new method has been proposed to analyze

the matched-filter PN code acquisition for DS spread-spectrum

systems. By using a time-domain approach along with a state

transition diagram, the mean acquisition time performance

can be evaluated with the cell correlations being taken into

account (the effects of the cell correlations have been neglected

in the previous analyses). Numerical results show that: 1)

the effects of the cell correlations are more significant for

a smaller cell size and/or a larger SNR; 2) for SNRs of

practical interest, over 10% and 20% acquisition time error

(at the optimum thresholds) are observed for and, respectively; and 3) the effects of cell correlations

at the optimum thresholds are quite insensitive to the value

of penalty time.

APPENDIX

DERIVATION OF (33)

In this Appendix, a simple example with is used

to explain the derivation of (33) more clearly. Fig. 9 depicts

the state transition diagram, where is used to denote

the transition gain for easy presentation. Using our notation,

for the first pass, the average acquisition time (if acquired) isgiven by ( in the region)

where and

For the second pass, the


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average acquisition time (if acquired) is

Similarly, the average acquisition time for the third pass is

and so on. Summing all passes is exactly (33).

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[9] S. Tantaratana, A. W. Lam, and P. J. Vincent, Noncoherent sequentialacquisition of PN sequences for DS/SS communication with/withoutchannel fading, IEEE Trans. Commun., vol. 43, pp. 17381746,Feb./Mar./Apr. 1995

[10] A. Polydoros and C. L. Weber, A unified approach to serial searchspread-spectrum code acquisitionPart II: A matched filter receiver,

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Wern-Ho Sheen(S89M91) received the B.S.E.E.degree from the National Taiwan Institute of Tech-

nology, Taiwan, R.O.C., in 1982, the M.S.E.E. de-gree from the National Chiao Tung University,Taiwan, in 1984, and the Ph.D. degree from theGeorgia Institute of Technology, Atlanta, in 1991.

From 1984 to 1993, he was with Telecommuni-cation Laboratories, Taiwan, where he was mainlyinvolved in the projects of personal communicationsand basic rate ISDN. Since 1993, he has been anAssociate Professor in the Department of Electrical

Engineering, National Chung Cheng University, Taiwan. His research interestsinclude adaptive signal processing, spread-spectrum communications, andpersonal and mobile radio systems.

Jiun-Kai Tzeng was born in Taipei, Taiwan,R.O.C., on August 12, 1972. He received theB.S. degree in 1994 from the National Chiao-Tung University, Taiwan, and the M.S. degree in1996 from the National Chung Cheng University,Chia-Yi, Taiwan.

Since October 1996, he has been with the Na-tional Space Program Office, Hsin-Chu, Taiwan. Hiscurrent research interests include spread-spectrumcommunications and satellite communications.

Ching-Kae Tzou was born in Hsin-Chu, Taiwan,R.O.C. He received the B.S.E.E. and M.S.E.E. de-grees from the National Chiao-Tung University,Taiwan, in 1984 and 1986, respectively, and the

Ph.D. degree in system electrical engineering fromthe University of South California, Los Angeles.

During 19931996, he was with the Departmentof Communication Technology at Computer andCommunication Laboratories, Industry of Technol-ogy Research Institute, Taiwan, where he workedon personal wireless communication system design.

In 1997, he became a Senior Engineer in the VLSI Group, TranSwitchCorporation, Taipei, Taiwan, and was promoted to Principle Engineer inMay of the same year, working on ASIC design for telecommunications. Hisresearch interests include wireless communication system design, digital signalprocessing, and ASIC design for applications in communication systems.

Dr. Tzou received the Phi Tau Phi Award from the Phi Tau Phi ScholasticSociety, Taiwan, in 1986.

Effects of Cell Correlations in a Matched-filter PN Code Acq

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