A blind joint estimator for multipath diversity and PN timing error in direct-sequence spread-spectrum receivers

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BLIND

JOINT ESTIMATOR FOR MULTIPATH DIVERSITY AND P N

TIMING ERROR IN DIRECT-SEQUENCE SPREAD-SPECTRUM RECEIVERS

Jia-Chin Lin

Department of Electrical Engineering, National Taiwan University Taipei, Taiwan

ABSTRACT

A blind joint estimator for multipath diversity and PN code timing error is proposed in this paper for direct-sequence spread-spectrum signaling on a frequency-selective fading channel. In the multipath diversity combiner, a modified known modulus adaptive (KMA)

algorithm is used to cope with time-varyingmultipath effects and to perform multipath diversity combining in the blind mode. In the code timing recovery, the timing error signal is extracted from each propagation path independently and also combined in the same fashion as the multipath diversity combining process. By taking advantage of the inherent diversity based on a known mod- ulus adaptive (KMA) algorithm, this modified code timing recov- ery can avoid the problem due to the drift or flutter effects of the error signals, and provide better performance on frequency selective fading channels. Extensive computer simulation results have verified the analysis and indicated very attractive behavior of the proposed joint estimator for multipath diversity and PN code timing error.

I. INTRODUCTION

Frequency-selective fading can often lead to severe performance degradation in wideband communication systems. Substantial efforts have been made in adap- tive equalization and diversity combining (RAKE) tech- niques in order to improve overall receiver performance

[l]. However, only when both multipath diversity com- bining and code timing recovery are optimized simulta- neously on the frequency-selective fading channel, can overall receiver performance be improved. In this pa- per, a blind joint estimator for multipath diversity and code timing error is proposed. Based on this technique, a KMA-based multipath diversity combiner operates in conjunction with a modified code timing recovery to achieve simultaneous improvement of overall receiver per- formance with lower computation load and lower pro- cessing rate. The modified KMA exploits a nonlinear step-size parameter to achieve better stability and im- proved robustness with noise and time-varying channel to accomplish better blind multipath diversity combin- ing. In the code timing recovery derived from the PN tracted from each propagation path independently and also combined in the same fashion as the multipath di- versity combining process. Due to the inherent diversity

code t i m i n g error estimator, t h e t i m i n g error signal is ex-

based on the modified KMA technique, this modified code timing recovery can avoid the problem caused by the drift or flutter effects of the error signals [2], and provide better code tracking performance. Therefore, the proposed technique can improve the overall receiver performance.

11. DESCRIPTION OF THE PROPOSED BLIND JOINT ESTIMATOR

To describe in detail the operations in the RAKE re- ceiver, which is based on the proposed blind joint esti- mator for multipath diversity and the PN code timing error, a complete block diagram of the RAKE receiver is sketched in Fig. 1. The complex representation of the baseband signal at the output of the chip matched filter is

L-1

F(t)

=

a,(t)s(t

-

nT,)

+

n(t) (1)

n=O

where s ( t ) =

Czo

dj

CE,'

qg(t-IT,-iTb) is the data- modulated PN sequence with raised cosine chip shaping, a,

( t )

are the time-varying complex-valued tap weights with R,ayleigh distributed magnitudes and uniformly dis- tributed phases, L is the number of resolvable paths, di

is the i-th information-bearing symbol, C I is the I-th chip

value of the PN sequence, Ta and T, are the symbol in- terval and chip duration respectively,

M

=

$

is the processing gain, g ( t ) is the overall chip shape. The ef- fect of carrier phase error introduced by the front-end noncoherent down conversion process is absorbed into the time-varying channel weights, a,(t). Without loss

of generality, it is assumed for simplicity here the period

of the PN code is exactly M .

11-A. Multipath Diversity Combining Technique

The integer-instant sample stream L - 1

?k = an(kT')s(kT, - nT,

+

E k T c ) 4- n ( k T , ) (2)

is fed into the KMA-based multipath diversity combiner.

Assume first the c o d e acquisition process has been achieved. In the each arm of the multipath diversity combiner, the input samples, Fk-m, m = 0, l , . . . , L

-

1, are cross- correlated with the local PN sequence c k - ( ~ - ~ ) , which

n=O

(2)

has been code-acquired, and then pass through the arm filter, h k , which is lowpass filter with the bandwidth

comparable with the symbol rate and its transfer func- tion, H ( z ) =

-,

to produce the KMA input sam- ples,

X r = F k - m x C k - ( L - 1 )

*

h k

= a ( L - l ) - m g ( E k ) d i

.

D k - i ~

+

S.N.

+

ni-m

m = O , l , . . . , L - 1. (3)

where x denotes correlation o erator,

*

denotes the con- volution operator, Dk =

Ckil

s ( k

-

/>

*

h k

= (1

-

a k ) [ ~ k - U k - ~ ] + O 1 for each one symbol duration, S.N

is caused by self-noise, U k denotes a discrete unit step

function, 01 is an undesired term which is the response outside one symbol duration and it can be eliminated by reseting the arm filter output at the beginning of each symbol period. To achieve the multipath diversity com- bining, a multipath diversity combiner with coefficient vector, W k =

[WE,

wk,.

. .

,

w f 1 I T , is employed. The multipath diversity combiner output is

Y k

=

W F X k (4)

where x k

=

[ x " , z : , ~..,x~-']' is the input vector of

the KMA algorithm. In order to achieve blind RAKE combining together with carrier phase recovery, the cost function of the conventional KMA algorithm given as below

must be written as J k =

J F

+

Jj!, where

J F

and JL are

the cost functions for real and imaginary parts of the equalizer output, y k

=

y f

+

J'&, and are defined as

J k = E ( l l Y k 1 2 -

&I2),

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Assuming the transmitted data symbols, d k = d F + j d f ,

are i.i.d. and sampled at the time instant t = kTc, then

RF and

Ri

are determined by

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for the real and imaginary parts of the lowpass filtering transmitted data symbols respectively. Because in the receiver the despreading process, i.e., cross-correlation with local code-acquired P N sequence, is followed by the lowpass arm filter, h k , the desired multipath diversity combiner output is lowpassed and thus Rf and Ri de- fined above is determined by the lowpass data symbols,

d;* h k and

d i *

h k . With this modification, the modified

KMA can not only combine the signals from each prop- agation path, but also track the carrier frequency offset

to avoid random spinning of the output signal constel- lation. In accordance with [6], we rewrite the update procedure as follows,

,

the error signal, [ E f + j E f ] , could thus be approximated

and CR and Cl are the confidence zones of real and

imaginary parts respectively defined in [7];

fip

and yf

are the real and imaginary parts of the decision result of the multipath diversity combiner output. Initially, the combining process begins when the error is large, the modified KMA operates in the blind mode described as

where a is a small positive number included to avoid any possible divergence caused by llXk1121yk12 = 0. When-

ever the output error is low enough, the decision-directed algorithm with normalized LMS (NLMS) adaptation takes place in the tracking mode. The update equation is de- scribed as

where 6 is a small positive value again and y k is the de- cision result of the multipath diversity combiner output

at time k. These two modes can be switched over auto-

matically and smoothly according to the magnitude of

the multipath diversity combiner output error without any specific detection mechanism.

In addition, if we define the step-size parameter as

we m a y view t h e modified K M A a l g o r i t h m proposed here

as a KMA algorithm with a time-varying step-size pa- rameter, @. This time-varying step-size parameter can overcome the gradient noise amplification problem oc- curring with conventional estimation algorithms when- ever the input vector XI, increases suddenly. Obviously,

@ is indeed identical to the Kalman filter gain with a

fixed positive-definite matrix as the error covariance ma- trix to guarantee local stability in the blind-mode of the modified KMA algorithm.

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11-B.

PN Code Chip Timing Recovery

In the previous subsection, the operation of the multi- path diversity combiner is described as yk =

If the cost function of the conventional KMA algorithm,

w r x r

J k = E { l l Y k 1 2 -

&I2},

where

Rk

=

~ { ~ [ ~ ~ ~ ~ ~ ~ ~ ~ { ,

is em- ployed for the timing error estimator, the PN code tim-

ing error, E k , can be estimated by the following equa-

where 7; is the step-size parameter and

i p

is the differ- entiate value of

~ ( t )

at the timing instant t =

( k

- m)T,.

For simplicity, yk[lykI2

-

Rk] is replaced by the deci- sion result of the multipath diversity combiner output,

& ,

and then the update e uation can be rewritten as

E ~ + I = ~k

-

r;’Re{yz

C;zo

wrkr}. Because of full-

digital implementation, the differentiate operation must be replaced by the difference operation. If we define

xr

= &{g?-$ -xr+,}, we can implement the PN code

timing recovery based on the timing error estimator de- scribed above in the same structure as that of the mod- ified code tracking loop [5].

tion ~ k + l = - ~ , ‘ R ~ { Y ~ [ I Y ~ I ~ - R ~ I .

[

E

;

:

,

w r i r l } ,

P

111. SIMULATION RESULTS

Computer simulation results illustrating the perfor- mance of the proposed blind joint estimator for multi- path diversity and PN code timing error are presented

in this section. The residual inter-chip interference (ICI) after the modified KMA proposed here, the timing er- ror signals, mean square timing error and mean time to lose lock (MTLL) for the code timing recovery derived from the proposed PN code chip timing error estimator

are evaluated by Monte Carlo methods. The simulation parameters are given below.

Modulation: 4-QAM modulation

PN code: m-sequence Chip Shaping: Chip Rate: Symbol Rate: Sampling Rate: Channel :

Square root raised cosine

&

= 1.27M chips/sec

&

= lOIl symbols/sec 16 samples per chip period 7 tap weights are taken from [7], and each tap weight is corruptted by an independent Jakes’ fading maximal Doppler frequencies 8, 80, 10, 100, 25, 30, 35 Hz

7 arms with filters: Bb =

&

RAKE Structure:

Shown in Fig. 2-(a) is the signal constellation obtained after the multipath diversity combiner with the modified KMA algorithm proposed in this paper. It is obvious the proposed KMA-based combiner is qualified to perform

multipath diversity combining and cluster the signal con- stellations at the right position very well. The residual inter-chip interference (ICI) at the output of the multi- path diversity combiner is defined in the similar fashion

of residual inter-symbol interference (ISI) [8] as follows:

where I h k * w k l m Q + is the maximumabsolute value among all values of h k

*

wk. Fig. 2-(b) shows the ensemble av- erage residual IC1 after the KMA-based combiner pro- posed here from 100 Monte Carlo runs. Fig. 3 presents several short-period error signals of the proposed code timing recovery, the modified code tracking loop with

the maximum ratio combining (MCTL/MRC) technique

[5], and the conventional DLL. It is obvious from Figs. 3-(a) & (b) that the error signals of the proposed code timing recovery and MCTL/MRC [5] with stable locked points at E = 0, there exist no phenomena of drift or

flutter of the locked point which are observed in Fig. 3-(c) for conventional DLLs [2]. Because of the time- varying channel, the received signals don’t have con- stant strength. Thus, the strength of the error signal of the MCTL/MRC [5] shown in Fig. 3-(b) is time- varying. In performing KMA, the adaptive algorithm acts as an automatic gain control (AGC) in addition to its desired task of spectral shaping. The effect of AGC on the multipath diversity combiner is rejection of mul- tiple access interference, while it also makes the error signals more stable and have the same strength shown in Fig. 3-(a). The mean square timing errors of the pro- posed code timing recovery, MCTL/MRC [5], and the conventional DLL under different SNR conditions are

shown in Fig. 4-(a). We can see that the conventional DLL is vulnerable to multipath channel effect and suffers from an irreducible error floor. On the other hand, the proposed code timing recovery and MCTL/MRC [5] are with much less mean square timing errors than the con- ventional DLL. Moreover, the mean square timing errors of the proposed code timing recovery are less than that of MCTL/MRC [5]. It is believed that this is because the effect of AGC makes the error signals more steady (with the same strength) to avoid the misadjustment caused by variation of the error signals. Fig. 4-(b) presents the

simulated results for the MTLL of the proposed code timing recovery, MCTL/MRC [5] and the conventional DLL under different SNR conditions. We can see that

the proposed code timing recovery and MCTL/MRC [5] have much longer MTLL than the conventional DLL. Furthermore, the proposed code timing recovery has a

longer MTLL than MCTL/MRC [ 5 ] . This is because the larger area, in average, under the error signals of the proposed code timing recovery for positive E implies higher escape energy, which can support longer MTLL.

Iv.

CONCLUSION

A blind joint estimator for multipath diversity and PN

code timing error is proposed in this paper. It has been shown that this technique can cope with time-varying

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multipath effects to accomplish multipath diversity com- 1 6

bining in blind mode. By exploiting the multipath di- versity, the code timing recovery rovides better per- t

formance on frequency-selective fazing channels in the

V. ACKNOWLEDGEMENT

sense of mean square timing error and MTLL. O S

I am deeply indebted to my advisor, Prof. Lin-Shan Lee, for E

making so many valuable suggestions to improve this paper. 4 5

1

VI. REFERENCES

[I] G. J. R. Povey, P. M. Grant and R. D. Pringle, “A Decision- Directed Spread-Spectrum RAKE Receiver for Fast-Fading Mobile Channels,” IEEE Trans. Vehicular Tech., Vol. 45, No. . j t 3, pp. 491-502, August 1996.

[2] W.-H. Sheen and Gordon L. Stuber, “Effects of Multi- path Fading on Delay-Locked Loops for Spread Spectrum Systems,” IEEE Trans. Commun., Vol. 42, pp. 1947-1956 Feb./March/April1994.

[3] J. R. ’Beichler and M. G. Larimore, “New Processing Tech- niques Based on the Constant Modulus Adaptive Algorithm,”

LEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-33, No. 2, pp. 420-431, April, 1985.

[4] John G. Proakis, Digital Communications, NY: MacGraw Hill, 1989.

[5] Jia-Chin Lin and Lin-Shan Lee, “A Modified Code Track- ing Loop for Direct-Sequence Spread-Spectrum Systems on Frequency-Selective Fading Channels,” IEICE Trans. Com- mum, Vol. E80-B, No. 7, pp. 1055-1063, July 1997.

[SI Jia-Chin Lin and Lin-Shan Lee, “A Modified Blind Equaliza- tion Technique Based on a Constant Modulus Algorithm,” sub- mitted for review.

[7] K. N. Oh and Y. 0. Chin, “New Blind Equalization Techniques Based on Constant Modulus Algorithm,” Proc. Glo becomm ’95,1995.

[8] 0. Shalvi and E. Weinstein, “New Criteria for Blind Decon- volution of Nonminimum Phase Systems (Channels),” IEEE

Trans. Inform. Theory, Vol. IT-36, pp, 312-321, Mar. 1990.

Rate:

mc

Fig. 1. The RAKE receiver based on the proposed blind joint estimator for multipath diversity and PN code timing error.

Fig. 2. (a)Constellation of the output of the KMA-based multi- path diversity combiner with PN code timing recovery.(b)The residual inter-chip interference after KMA with proposed code timing recovery. U o d i . d C h b T h C y e- (.> 0 1 D I -0 * 4 D -3 -.-E, - 0 . .”* - 0 , 0 o 1 o 1 “I ”. 0 ,

Fig. 3. (&)several short-period error signals of the proposed code timing recovery, (b)several short-period error signals of MCTL/MRC [ 5 ] , (c)severd short-period error signals of the conventional DLL. , e . , . -I I (a> (b)

Fig. 4. (a)Mean squared timing jitter and (b)Mean time to lose lock with (1) x: conventional DLL, ( 2 )

+:

MCTL/MRC [ 5 ] ,

and ( 3 ) *: the proposed technique.

數據

Fig.  1.  The  RAKE  receiver based  on  the proposed blind joint  estimator for multipath diversity and PN code timing error
Fig. 1. The RAKE receiver based on the proposed blind joint estimator for multipath diversity and PN code timing error p.4

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