A noncoherent sequential PN code acquisition scheme in DS/SS receivers

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A

NONCOHERENT SEQUENTIAL PN CODE ACQUISITION SCHEME

IN DS/SS RECEIVERS

Jia-Chin

Lm

and Lin-Shan

Lee

Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. Abstract-A noncoherent sequential P N code

acquisition technique is proposed in this paper. Noncoherent detection is employed to handle the seriously noisy environments, while the carrier frequency offset and data modulation effects can be simultaneously taken care of. To realize the sequential acquisition, t h e out-of-phase and in-phase sequences are prop- erly modeled to avoid t h e significantly high error probabiities caused by the conventional widely used zero sequence model. Extensive computer simulation results indicated that t h e proposed PN code acquisition outperforms t h e fixed-dwell-time counterparts by roughly 2

-

6 dB with much lower false alarm and m i s s probabilities, and the superiority of t h e proposed technique becomes more significant at lower SNR conditions.

I.

INTRODUCTION

Substantial darts have been made in the code acquisition problem [l], [2], [3], [4], [5], but most of the analyses for the mean acquisition time were only in the context of ked-sample-size (F- SS, or fixed-dwell-time) techniques, in which the correlation duration is fixed. Although the FSS P N acquisition techniques are the simplest and the most well studied, it is inefficient in terms of the acquisition time, since it always takes a long enough fixed correlation duration before the rejection or acceptance of the synchronization s- tate can be decided. Because the code acquisition usually t&es very long time, it is highly desired to speed up the code acquisition process- es for efficient DS/SS communications. The re- moval of the

FSS

requirement is therefore very attractive. The PN code acquisition in DS/SS can be considered as a detection problem test- ing between two simple hypotheses, i.e., the received and locally generated PN sequences are aligned in phase or not. It is well-known that the sequential probability ratio test (SPRT) [6] is very efficient in obtaining a decision between t-

wo simple hypotheses in the sense that it requires the minimum average detection time for given error probabilities (i.e., false a l a r m / m i s s probabilities) [7]. By modeling the acquisition problem as

testing between two hypotheses, several sequential test techniques under coherent demodulation environments were proposed and analyzed [8], [9] using different out-of-phase sequence model-

s. However, it is practically almost impossible to achieve coherent demodulation, because the signal-to-noise ratio

(SNR)

before despreading is very low. Several sequential detectors or sequential probability ratio tests (SPRT) designed for P N code acquisition under noncoherent demodulation environments have also been discussed

[l], [6],

[lo],

[ll], but under the assumption that the out-of-phase sequence could be modeled as

a zero sequence. However, such assumption may not be very practical as well, because i t is almost impossible to design a front-end filter with narrow enough bandwidth on the one hand to reject completely the residual cross-correlation between the incoming and loca.Uy generated P- N sequences under the out-of-phase condition so that the out-of-phase sequence vanishes; but with wide enough bandwidth on the other hand to accommodate the data modulation and frequency offset effects so that the in-phase se- quence can be kept from a y damage caused by the data modulation and frequency offset effect- s. As a result, the upper bound of the cross- correlation between any two non-synchronized PN sequences was employed in some analyses t o model the out-of-phase sequence t o avoid any performance degradation occumng with the con- ventionalSPRT based on the zero sequence mod-

In t h i s paper, an SPRT-based acquisition technique is derived with the upper bound of the cross-correlation sequence as the out-of-phase sequence model, with the modified in-phase se- quence and under noncoherent demodulation en- el

PI,

[91.

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vironments. In addition, the data modulation and ckrier frequency offset effects can be simultaneously coped with. A sliding correlator is employed in the search of the acquisition process, and so the mean acquisition time is proportional to the average number of the received

PN

code chips required for each synchronization test. Simulation results show that the proposed technique outperforms the FSS counterparts by roughly 2

-

6 dB in terms of SNR in order to achieve given error probabilities. Furthermore, such superiority even improves with lower error probabilities and/or lower SNR’s.

11. CONVENTIONAL SEQUENTIAL ACQUISITION In a M-PSK DS/SS communication system, the information-bearing signal can be expressed

TECHNIQUES

where

d;

is the i-th M-PSK data symbol, T is the symbol duration and the above signal d ( t ) is

then spread by a

PN

waveform,

where c j E {-l,l} is the j-th PN code value,

cj = C ~ + N for all j ,

N

is the period of the

PN

se-

quence, and

T,

is the code chip duration. With- out loss of genera.lity, here we assume that the processing gain is the PN code period,

N.

The resulting signal is then used to modulate a carrier, which is further corrupted by

AWGN.

The complex representation of the signal received at the front-end of the acquisition receiver is

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where E, is the received chip energy, r is the received code phase offset which is assumed to be an integer, O ( t ) is the carrier phase error,

O(t)

= 2.xA

f

t

+

00, introduced by the carrier frequency offset A f , and n‘(t) is a complex white Gaus- sim noise with two-sided power spectral density N0/2. The baseband representation of the conventional noncoherent acquisition method based

Fig. 1. The baseband representation of the conven- tional noncoherent sequential acquisition for D-

S/SS communication.

on SPRT& shown in Fig. 1. The received signal is b s t down-converted by means of a noncoherent local carrier, e j w c t , and then passed through

a lowpass filter

LPFO.

To minimize the noise variance, LPFO has a bandwidth B =

&

(Hz).

The resulting signal is then cross-correlated with the local code sequence p ( t -+Tc), where i is the local PN code phase offset. The cross-correlation signal is passed through a second lowpass filter LPF1, whose bandwidth

W

(Hz)

is assumed to be wide enough to accommodate the data modulation. The output of LPFl is then fed into an envelope detector. The envelope sam- ples S k are obtained by sampling the output of the envelope detector at a rate low enough (say,

A)

such that the samples can be con- sidered to be independent. Finally, the envelope samples enter the likelihood ratio calculator and the sequential detection logic. The detection of the SPRT is then directly derived from the statistics of the envelope samples of the cross-correlation function iiltered by LPF1. In the above structure, the lowpass iilter LPFl may be very diEcult to design. The bandwidth of this filter has to be wide enough to accommodate the frequency offset and data modulation effects, or to prevent the in-phase sequence from any damages caused by such effects. But if its bandwidth is too wide, say W G

&,

the

cross-correlation of the incoming and the local sequences under the out-of-phase condition can not be suppressed at all. This makes it impossible to distinguish the envelopes of the cross- correlation under in-phase and out-of-phase conditions and leads to a significantly high false alar- m probability. As a result, the bandwidth W of

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this filter has to be narrow enough as well, say W 2

&,

t o reject the residual cross-correlation under the out-of-phase conditions, because the likelihood ratio calculator and the sequential de- tection logic are designed based on the assump- tion that the out-of-phase sequence can be mod- eled as a zero sequence here.

O n

the other hand, the SPRT algorithm is derived based on another assumption that the samples entering the likelihood ratio calculator are spaced s a d e n t l y , say

m'T,

where

m

'

>>

1, and can be considered independent, but with that assumption the carrier frequency offset and data modulation effects may possibly change the statistics of the in-phase sequence significantly and destroy the functions of SPRT, which may lead to a high probability of miss. Therefore, it may be practically very difEicult to actually design and implement the conventional noncoherent sequential acquisition scheme.

111. PROPOSED NONCOHERENT SEQUENTIAL

ACQUISITION TECHNIQUE

The noncoherent sequential acquisition method proposed in this paper is shown in Fig. 2. The received signal is also down-converted first by means of a noncoherent local carrier, ejwet,

passed through a lowpass filter (LPFO) and then sampled at the chip rate to generate the baseband received sample stream Tk

Tk = ~ ~ ( ( k - T ) T = ) p ( ( k - T ) T = ) e j e ! - + n k , ( 4 )

where 8k = 27rAf. kT,

+

80, and n k is the noise

sample. To minimize the noise variance, the bandwidth of the lowpass filter LPFO is set at

B

=

&

(Hz), and it can be shown that the noise samples { n k } are independent and iden-

tically distributed 0.i.d.) random variables with zero mean and variance U$ =

3

on each dimen-

sian. Far simplicity and without loss ai genera.. ity, we let

2

= 1. The resulting sequence, { T k } ,

is cross-correlated with the local code sequence p ( ( k

-

?)Tc), where i is the local P N code phase offset and it is an integer, and then the samples

{Xi)

are generated as below,

XL

=

~id((k

-

T ) T , ) e j e k

+

Ni,

_{( 5 )}

5

-Fig. 2. Block diagram of the proposed noncoherent sequential acquisition for direct-sequence spread- spectrum communication.

where yk' = &((k

-

~ ) T = ) p ( ( k

-

i)Te) and

Ni

=

nk. p ( ( k

-

f)Ts).

In

the above equation, the sequence

{Yi}

provides the desired inform* tion to detect whether the received and the locally generated P N sequences are synchronized. It is corrupted by the data modulation effec- t, d((k

-

T)T,), and the carrier frequency off-

set &ct,

@!-.

It

can be shown that the noise

term

iVi

is a complex Gaussian random variable with zero mean and the same variance as The cross-correlation samples,

Xi,

are in- tegrated/dumped (I/D-ed) for each m (non- overlapped) samples to generate the I/D-ed cross-correlation samples

Xi

n k .

(i+l)m-l

Xi

=

XL

=

E;.Peo

+

Ni,

(6)

k=im

where E;.

=

Ck=im

(i+lb-l

yl k e j ( 2xAfkT.) and

Ni

= L (i+lh-l i m

Ni.

The acquisition scheme considered here is a sliding correlator structure with serial search. The I/D-ed cross-correlation samples

Xi

enter the likelihood ratio calculator one

by one, and they are then used to test whether the phase of the local P N sequence is synchronized with that of the received sequence. If they are not synchronized, the phase of the local P- N generator is shifted by

6T,

and the above test process is re-started and so on until they are syn- chronized. The step size 6 for each phase update can be usually 1,$, or some other fractional Val- ue, depending on the p d - i n range of the fol- lowing tracking subsystem. Here it actually suf- fices to use 6 = 1 in order to avoid any possibil- ity of lost code phase states caused by updating the code phase, since the pnll-in range is usually

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TJ2,

To test the synchronization of the two s e quences, assume hypothesis Ha is the case when the two sequences are not aligned, i.e., T

#

i,

and hypothesis H Z is the case when T = i . The

test statistic, X(w), to be compared with some thresholds t o decide between

HO

and H I is given by:

U)

X(w) = h(Xi), (7)

i=l

where h(.) is some monotonic function, and w is a random variable, called the dwell time and determined by the detection method. The stop- ping time (or sample number) of a test is defined

W = min{w : X(w) ( A , B ) } , (8)

where A and B are the threshold levels used to decide between HO and

HI.

W

is a random variable which denotes the minimum number w

of the I/D-ed cross-correlation samples, Xi, required to decide between HO or H I . So, the required chip number to make a decision can be also obtained by mW. By contrast with a fixed- samplesize (FSS) test where any decision is al- ways made after some fixed number of samples have entered the synchronization tester, the sequential probability ratio test (SPRT) exploits an "active" correlator, where the correlation and the following synchronization test are performed sequentially on a "sample-by-sample'' basis.

We denote the false alarm probability (i.e.,

H I

claimed while H O is true) by (L and the m i s s prob-

ability (i.e., H o claimed while H1 is true) by

1-p

(i.e., the detection probability is

p).

For calcu- lation of the mean acquisition time and for simplicity, it is assumed that the initial phase offset between the received and the local

PN

sequences is some integer multiple of

T,,

and it is uniform- ly distributed over the range of the whole period (i.e., from 0 to ( N

-

l)Tc). It has been shown that the average acquisition time using a sliding correlator is proportional to the average number of chips, mW, required for each synchronization test [4], [8]. So, to reduce the.code acquisition time is equivalent to reducing the average chip number required for each synchronization test.

Based on the model of the partial correlation of

PN sequences [8], the acquisition problem can be considered as the testing of a composite hypoth- by

esis

Ho

against a simple alternative H1 by means of the envelopes of the I/D-ed cross-correlation samples, Xi. So, the desired information sequence,

{E},

in Eq. (6) can be modeled as a constant being either

P1

for

HI

or

Po

for H o ,

i.e.,

where fl

=

Ijxr=lej(hrAfkTc)ll is a factor in-

troduced by the carrier frequency offset and it approaches to rn as m decreases, and fo

=

0.5

is used here because it corresponds to the worst- case correlation under synchronization (i.e., fo is the PN code cross-correlation value computed at the boundaries of the lock range of the code tracking subsystem) for the case whenever the chip timing is unknown and the local PN generator is updated by

T,

when each HO claimed. Based on the model of yi and after some manipu- lation, the envelope samples, llXill, of the I/D-ed Correlation samples can be described as i.i.d.

Ri-

cian random variables with parameter either Y1 for

HI

or

Lo

for

Bo.

The log-likelihood ratio for the i-th envelope sample, IlXill, is then

Zi

-

( 9 0 ) 2 4 p 1 ) *

ZnW0

+ w ~ a ( + ) )

Y'

xi

-

WrJ(+9)>

PO

x.

(10) and we will consider the sequential test using the statistic below,

UI

X(w) =

c

zj.

i=l

i.e., h(Xi) in Eq. (7) is taken as Zj here. The SPRT is then described by:

2 B j H i ( T = + )

X(w) < A + H a ( T # ? ) E ( A , B ) take another sample,

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i

where the threshold levels, A and B , should be

chosen so that the test can achieve the false alar- m probability a and the miss probability 1

-

p

[6]. The sample number W of the test has been defined in Eq. (8), and its mean value has been used for performance evaluation.

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IV. SIMULATION

RESULTS

Computer simulation results illustrating the performance of the proposed noncoherent se-

quential P N code acquisition technique are pre- sented in this section. The false alarm probability, miss probability and average chip numbers required for each synchronization test are evalu- ated by Monte Carlo methods.

For several reasons, the baseband received signal may be afTected by a finite carrier frequency offset

A

f comparable t o the symbol rate

+.

A

f

is in general much smaller than the chip rate

$-,

but may turn out to be of the same order of mag- nitude as

*

in some voice transmission networks and even a few times larger for low-bit-rate applications [13].

Shown in Fig. 3 are the false alarm and mis- s probabilities of the proposed and conventional sequential acquisition techniques with vazious values of the frequency offset and LPFl bandwidth for

$

=

0.025 in case the desired error probability (false alarm and miss probabilities) is The false alarm and miss probabilities with m'

=

4 for the conventional SPRT and m = 4 for the proposed SPRT algorithm are plotted in Fig. 3(la) & (lb), while those for

m' = 8 and m

=

8 are shown in Fig. 3(2a) & (2b). From Fig. 3(la), the false alarm probabilities of the conventional SPRT are significantly higher than the desired false alarm probability for a wide range of signal-to-noise ratios, because when m' = 4 the bandwidth, W

=

-,

of LPFl is too wide to reject the residual cross- correlation under the out-of-phase conditions, but the out-of-phase sequences were incorrectly modeled as a zero sequence. The miss probabilities in Fig. 3(lb) are also relatively high for some range of signal-to-noise ratios. For a larger

m

'

value or a narrower LPFl bandwidth, say

m' = 8 as shown in Fig. 3(2a), the false alarm probability of the conventional SPRT algorithm is reduced, because the narrower bandwidth of LPFl can reject more residual cross-correlation. However, such narrow bandwidth may lead to the accumulation of the canier frequency offset effects and change the statistics of the in-phase sequence. This is why much higher miss prob- abilities were obtained as shown in Fig. 3(2b).

1

On

the other hand, as cas be seen from Fig. 3, the false alarm and miss probabilities of the proposed SPRT algorithm are well under or around the desired value of It is clear in Fig. 3 that for the conventional SPRT algorithm when m' is smaller, the false alarm probabilities are very high and they decrease as m' increases; while the miss probabilities are very high when m' is larger, and they decrease as

m'

decreases. These results indicate that no matter how to choose a

m' value or the LPFl bandwidth, either a high false alarm probability or a high miss probability will be inextable with the conventional sequen- tial acquisition technique. But such a problem is w d taken care of in the proposed technique due to proper modeling of the in-phase and out- of-phase sequences.

Finally, the average required chip numbers ver-

sus SNR is shown in Fig. 4 in the case of a carrier frequency offset

$-

=

10-3 for different error probabilities t o and m values.

From

Fig. 4, the proposed technique is shown to yield significant improvement (up to roughly 2

-

6 dB) in SNR. Furthermore, such improvement increases as the input SNR decreases. It should be remarked that the optimum threshold setting for

FSS

must be incorporated in the e- valuation of the average required chip numbers for each SNR value, while the thresholds, A &

B,

of SPRT are not functions of SNR values. In other words, for a given

SNR

d u e and a given chip number, there exists an optimal threshold to achieve the desired error probabilities for

FSS.

This optimal threshold has been employed in all the above simulation results.

V. CONCLUSION

In this paper, a noncoherent sequential PN code acquisition technique is proposed. The out- of-phase sequence has been modeled as the upper bound of the cross-correlation between any two non-synchronized sequences, and the in-phase sequence has been modified by a factor caused by the carrier frequency offset effect to avoid high error probabilities occurring with the conventional SPRT-based acquisition techniques. The proposed technique significantly outperforms its FSS counterparts and simultaneously overcomes the carrier frequency offset and data modulation

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effects.

REFERENCES

[l] M. Simon, J. Omura, R. Scholtz, and X. Livitt, Spread Spectrum Communication, vol. III, Rockville, MD: Compnter Sdence Press, 1985.

[2] J. Holmes, Coherent Spread Spectrum Systems, New York Wiley, 1982.

[3] J. Holmes and C. Chen, 'Acquisition Time Perfor- mance of PN Spread-Spectrum Systems,' IEEE 5- s. Common., pp. 778-783, Aug. 1977.

[4] R. Pickholte, D. S c h i l l i q and L. Miktdn, 'Thec- ry of Spread-Spectrum Ccmmunications: a Tutorial,'

B E E Trans. Commun., pp. 855-884, May 1982. [5]

D.

Hall and C. L. Weber, "Noncoherent Sequential

Acquisition of DS Waveforms,' Milcom'86 Con€ Rec.,

pp. 13.3.1-13.3.5, Oct 1986.

[6] A. Wald, Sequential Analysis. New York: Wiley, 1947. [7] A. Wald and J. Wolfowits, 'Optimum Charader of the Sequential Probability Ratio Test," Ann. Math. Statist., Vol. 19, p. 326, 1948.

[SI Y.-H. Lee and S. Tantarstana, *Sequential Acquisi- tion of PN Sequences for DS/SS Communications: Design and Performance," B E E Journ. Select. Area

Commun., pp. 750-759, May 1992.

[9] K. K. Chawla and D. V. Sarwate, 'Acquisition of PN Sequences in Chip Synchronous DS/SS Systems U 5 ing a Random Sequence Model and the SPRT," IEEE

Trans. Commnn., Vol. 42, No. 6, pp. 2325-2334, Jun. 1994.

[lo] Y.-T. Su and C. L. Weber, "A Class of Sequential Tests and Its Applications," B E E Trans. Comm~n.,

Vol. 38, No. 2, pp. 165-171, Feb. 1990.

[I11 R. L. Peterson and R. E. Ziemer and D. E. Borth, Introduction to Spread Spectrum Commm'cation, Prentice Hall, 1995.

[12] A. D. Whalen, Detection of Sign& in Noise, Aca- demic Press, 1971.

[13] A. Cavallini, F. Giannetti, M. Luke and R. Reggian- nini, "Chip-Level Differential Encoding/Detection of Spread-Spectrum Signals for CDMA Radio Transmis- sion over Fading Channels," IEEE Trans. Commun.,

Vol. 45, No. 4, pp. 456-463, April, 1997.

[14] M. E. Zarrabizadeh and E. S. Sousa, "A Differential- ly Coherent P N Code Acquisition Receiver for CDMA Systems," E E E Bans. Commun., Vol. 45, No. 11, pp. 1456-1465, Nov., 1997.

n w -

.,, .,I ." _)I ." .U .* I .* .m .,, Y 4, .I 4, 1 .U ." _I< .*

-)9 se,* (la) m' = 4, m = 4 (lb) m'

=

4, m = 4 I -... ". ... ... =e 19 (2a) m'

=

8, m = 8 (2b) m'

=

8, m

=

8

Flg. 3. The false alarm and miss probabilities when

=

0.025 and desired error probability

lo-'

:

(la) M s e alarm probabilities and ( l b ) miss pro& abilities for m'

=

4, (2a) false alarm probabilities and (2b) miss probabilities for m'

=

8.

( c )

(4

Fig. 4. The average required chip numbers for a s p - (c) chronisation test versus SNR when