An algorithm for blind equalization and synchronization

J. Franklin Inst. Vol. 333(B), No. 3, pp. 339-347, 1996

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Pergamon

S0016--0032(96)00028--2

Copyright © 1996 The Franklin Institute Published by Elsevier Sci. . . . Ltd Printed in Great Britain 00164)032/96 $15.00+0.00

An Algorithmfor Blind Equalization and Synchronization by ABDERRAHMANESSEBBARt,JEAN-MARCBROSSIER,DIDIERMAUUARY C E P H A G ( U R A C N R S 346), Ensieg, BP46, 38402 Saint Martin d'Hbres, France and BENO~T GELLER

LERISS-Universitb Paris XII, ave Gbnkral de Gaulle, 94010 Creteil cedex, France

ABSTRACT: This article concerns the problem of blind equalization. In practical situations, one must synchronize the receiver with the transmitter. The classical use of two separate algorithms, one for synchronization and one for equalization, does not work correctly as the channel interference increases. In such cases, one must prefer a joint estimation of the two unknown parameters (synchronization and equalization). Here, an original algorithm that jointly estimates the synchronism and the equalizer without any training period (blind estimation) is proposed. This algorithm is validated using both synthetic and real data. This study is limited to constant modulation schemes such as PSK. Copyright © 1996 Published by Elsevier Science Ltd L Introduction

In digital communication, one must cope with several problems such as InterSymbol Interference (ISI), carrier recovery and symbol synchronization. Each problem is usually solved with a corresponding algorithm. For instance, one may use an equalizer to reduce ISI, a Phase Locked Loop for carrier recovery and a synchronization device to recover the symbol rate. This approach, a single algorithm for a single problem, is efficient as long as each algorithm is still able to work in spite of the other perturbations. For example, ISI may disturb synchronization or phase recovery. In some channels of practical interest such as multipath channels, ISI is often very large and leads to a weak phase tracking and prevents the receiver from good synchronization. To avoid this practical difficulty, our aim is to build a blind algorithm based on Higher Order Statistics (H.O.S.) that jointly estimates the sampling instant (symbol synchronization) and the equalizer (ISI reduction). The main interest of joint estimation lies in its robustness vs ISI. Some algorithms, based on second order statistics have been proposed in the literature [see for example Kobayashi (1), Stojanovic et al. (2)]. Since these algorithms are only based on second order moments, a training sequence is required. The algorithm we suggest manages to converge without any training sequence. tAuthor for correspondence. 339

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Godard (3) has proposed a criterion for blind equalization. Our aim is to extend his scheme for blind equalization to the problem of equalization and synchronization. In fact, the Godard criterion is considered as a function of the delay (sampling instant) and of the equalizer. An algorithm based on this criterion is given. This procedure is evaluated using some simulations and then applied to real data used in an underwater channel context. It clearly appears to behave correctly in real world situations. The plan of this paper is as follows. First, paragraph 2.1 settles the notations of the transmission scheme. Paragraph 2.2 recalls the Godard algorithms and discusses some problems related to a bad synchronization are introduced in paragraph 2.3. Section III presents the fourth order algorithm for blind equalization and synchronization. This algorithm is finally validated on synthetic (Section IV) and real data (Section V).

II. Problem and Classical Solutions 2.1. Transmission scheme The transmission is modelled using a complex baseband model (see Fig. 1). A complex valued digital signal (ak} is to be transmitted through a multipath channel from a transmitter to a receiver. {ak} is assumed to be an independent identically distributed sequence of random variables. We assume that a linear modulation is used (for example, in the case of BPSK, ak can take the values { - 1 , + 1} with equal probability). The transmitted signal is filtered by the channel. The complex baseband impulse response is denoted by h(t). The complex baseband noise n(t) is assumed to be additive, white, circular and Gaussian. The output of the noisy channel is to be sampled at the symbol rate (symbol duration 7). In practical situations, the clock used at the receiver could be slightly different from the one used at the transmitter. Even if the two frequencies are identical, the sampling instants may differ by a constant delay (0 < z < T). In other cases, if the two frequencies are not exactly the same, there will be a linear drift between the two sampling clocks. Globally speaking, z is time dependent and we note by Zk is optimum value for the kth symbol. The kth discrete sampling time is then tk = k T + rk. After sampling, the complex discrete signal xk = x ( k T + zk) is filtered using a linear complex equalizer 0. We wish to estimate the sampling time z and the equalizer 0 so that the equalizer output is as close as possible to the transmitted signal in some sense. Of course, both 0 and z could be time dependent in real world applications. A Finite Impulse Response (F.I.R.) structure is considered for the equalizer. The notations are nft)

a~ ~

Equalizer

I ........

I

0

T

Fit. 1. Baseband transmission model.

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• Xk = (xk+u . . . . . xk N)r the observation vector composed of M = 2 N + 1 samples of the channel output • 0 = (O_u,..., 0+u) T the vector of equalizer taps • zk = OrXk = XrO the equalizer output.

2.2. Equalization In this paragraph, we assume that the problem of synchronization is solved. Then a algorithm for blind equalization can be used. Godard algorithms are briefly described in paragraph 2.2.1. Paragraph 2.2.2 points out problems related to weak synchronization. 2.2.1. Godardalgorithms. Godard proposed to estimate an equalizer by minimization of the following cost function. i ( o ) = E(Lz~I ~ - Rp) 2

where zk is the equalizer output: zk = OTXk. The most important case is given setting p = 2. This choice leads to the so-called Constant Modulus Algorithm (CMA). The constant Rp depends on the modulation scheme. It can be calculated according to the following formula

Rp=-

E(lakl 2p)

E(I ak IP) "

Minimizing with respect to 0, a CMA criterion using a stochastic gradient scheme leads to G o d a r d algorithm (or CMA algorithm) 0k : 0~_, -7(I~I ~ - RAX*~zk.

(1)

The step-size 7 is to be chosen: small enough for the algorithm to be stable and to obtain a small variance, large enough to achieve a fast convergence rate. 2.2.2. Influence of a weak synchronization. When the sampling time is not chosen correctly, it becomes difficult for the algorithm to achieve perfect equalization. For example, if no synchronization is done and if the optimum sampling time is linearly drifting because of a misadjustment of the two clocks (transmitter and receiver), some errors bursts will appear periodically (when symbol skipping occurs). This is not acceptable in practice and the sampling time has to be adapted in order to obtain a good equalization. Let us consider some synchronization schemes.

2.3. Classical alyorithms for symbol synchronization For a Nyquist channel, ISI can be cancelled by choosing the right value of v; no equalizer is needed. Some criteria have been proposed in the literature on how to choose v. We recall here the principle of two of them, that will be useful to understand the following paragraphs. 2.3.1. Godard algorithm. Among numerous criteria for time delay estimation, one can use the criterion by Godard. A Godard criterion is used at the sampler output, According to Fig. 1, when no equalizer is needed, the sampler output clearly depends

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on z. Using stochastic gradient scheme, the value zk is adapted in order to minimize the Godard cost function. This can be written as follows J(w) = E(lzkl 2 - R p ) e where Zk = Zk(Z) is the sampler output. Thus, by minimization with respect to z, the adaptive algorithm becomes

/ OZk\ Zk = Zk_ x - P(lz~12 - Rp)Re~z*~z )

where Re(y) denotes the real part of y. As it was mentioned above, the time delay variations can often be approximated by a linear drift (due to clocks misadjustment). Then, one should prefer a second order algorithm like

Zk = Zk-1-- Pl +

Izkl2--Rp Re z*k Oz J"

Such an algorithm should be understood as a couple of equations: the first one estimates the mean slope (linear drift) and the second one estimates the delay itself. The derivative (8Zk/OZ) is to be estimated. In our case, it is replaced by a finite difference approximation. 2.3.2. L M S algorithm. A classical decision directed Least Mean Squares algorithm can also be used to achieve a right synchronism between the transmitter and the receiver. Using a pseudo-error between the transmitted data and its estimate, the criterion is J(z) = E(z~--6k) 2 where

~ik = decision (ak).

A stochastic gradient scheme still leads to an efficient algorithm for sampling time tracking. Of course, a second order algorithm can still be obtained. 2.3.3. Influence o f l S I . Generally, the previous algorithms work correctly if the ISI amount is low enough. For large ISI, they becomes arbitrary and an equalizer is to be used. IlL Joint Estimation

As mentioned previously, classical algorithms do not always lead to efficient algorithms, therefore, we now derive an algorithm that estimates jointly the synchronism and the equalizer.

3.1. Second order solution Such an algorithm was first suggested by Kobayashi (1). This solution based on a classical Minimum Mean Square Error criterion leads to an efficient algorithm [tested

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on real data from underwater acoustics by Stojanovic et al. (2)]. But its main drawback is to require a training period. 3.2. Fourth order solution: a blind algorithm In order to derive a blind algorithm, let us start with the G o d a r d cost function. According to Fig. 1, the equalizer output can be seen as a function of z and 0. We have J(O, z) = E(lzkl 2 - Rp)2

where z, = z~(O, z) = 0~Xk. We use a stochastic gradient to estimate both z and 0. Thus, minimizing J(O, z) by differentiating with respect to 0 and to z, one obtains

IOk=Ok-I--i('Z '2--RP)XiZ i ) 1-:

1

9.

(2)

Iz~12-Rp Re z z ~ z

In this algorithm, three step-sizes are to be chosen • V which is related to the rate of variation of the optimum equalizer, the value M and the Signal to Noise Ratio (SNR) •/& and #2 which are related to the SNR and to the time delay variations. When no a priori knowledge on the channel is available, the choice of step-sizes becomes a difficult problem. This difficulty is encountered with all adaptive algorithms. Some optimization schemes can be found in (4-6); for instance example, an algorithm for adapting the step-size using the same system measurement which are used for the tracking was suggested by Benveniste et al. (4), a practical study is made by Geller et al. in (5), a theoretical study is given by Kushner et al. (6). In our particular case, those optimization schemes do not appear to be very robust (because of the fourth-order statistics). A practical is to keep ~2 ((/11. Also, ~: must be small enough for the gradient scheme to be stable but large enough to achieve an efficient convergence rate. IV. Simulation Results

The previous algorithm (2) is tested over two kinds of simulations. The channel impulse response is given by h(t) = t e x p ( - t) (cf. Fig. 2). 0.50 0.40 0.30 e~

E 0.20 < 0. I0 0.00 0

............................. I0 20 30 T i m e (I=I/16

40

symbol)

FIG. 2. Simulated channel impulse response.

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FIG. 4. Linear delay drift: 7 = 4 e - 3 , M = 5, zo = 8,/~] = 0.1 and/~2 = l e - 3 . Estimated delay (....... ) and true delay ( ). The first (and simplest) one considers a constant delay ~, this delay is known and we just wish to check for the convergence of the adaptive algorithm (see Fig. 3). As it is readily seen, the ISI at the output of the equalizer becomes neglectable and the estimated delay converges towards the true delay. The second simulation is more realistic and considers a linear delay drift. Figure 4 shows the true and the estimated delay variation vs time. This simulation clearly points out the good tracking ability of the adaptive scheme. Also, once again, the ISI is minimized by the algorithm. Remark 1: I f the channel's z transform has one or several zeros on (or very close to) the unit circle of the complex plane, the performance is limited by the finite impulse response structure of the equalization filter. Remark 2: Algorithm 2 requires around 3000 symbols to adjust correctly the equalizer and 5000 symbols for an appropriate estimation of z. The convergence duration is almost the same as the one provided by a G o d a r d algorithm (it is the same one for #1 =/t2 = 0). Since estimations of r and 0 are coupled, a bad estimation of 0 does not enable the algorithm to achieve a good estimation of ~. This problem remains when two independent algorithms are used (one for symbol synchronization and the other one for equalization).

Algorithm for Blind Equalization

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0.0 iF_~ = ::~ 7700 7750

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Time (=1/16 symbol) FIG. 5. Channel impulse response. V. A Real Transmission Test

We now consider some data from an underwater linkt. Let us first briefly describe the experiment.

5.1. Experiment description The transmitter and the receiver are about 150 m below sea level. The distance between them is about 750 m. The carrier frequency is v0 = 400 Hz. A BPSK modulation is used. The estimated SNR is 10 dB. 5.2. Channel characterization In this case, the transmitted data is a pseudo-noise sequence composed of plus or minus one (with the same probability). Thus, the channel impulse response can be estimated by computing the cross-correlation function of the channel output. The estimated impulse response is plotted on Fig, 5. As one can see, the impulse response duration is very long with respect to the symbol duration. The two peaks that appear on this response are related to two different propagation channel paths. A long impulse response duration is somehow characteristic of underwater communications. Of course, it is one of the main difficulties of this kind of transmission. 5.3. Data processin9 The previous channel output is processed by a classical Godard equalizer (algorithm 1: without synchronization). The corresponding constellations are displayed in Fig. 6. The dots on the right half of this figure around the transmitted symbols plus and minus one are not well separated; good equalization is thus not achieved and synchronization is necessary. The joint algorithm (2) is then used to equalize the channel. The received and equalized constellations (using one point per symbol) are plotted in Fig. 7 as well as the estimated delay. It should be clear now that as the two dots are well separated the data recovery is easily achieved with a simple phase locked loop. The algorithm made no error after the initial convergence. "tThe experiment has been supported by the IFREMER (THETIS I-MAST project).

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R e c e i v e d signal

Equalizer output

3

3

2

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1

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--]

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FIG. 6. Constellations at the input and at the output using Algorithm 1: y = 6 e - 3 , M = 10, z0 = 12,/~ = 0. and/~2 = 0 e - 4 .

._=

0 0

10219 Symbol Equalizer output

R e c e i v e d signal

0 -2 -2

0

2

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-3 -2

0

2

FIG. 7. Fourth order joint algorithm: Estimated z and constellations at the input and at the outpatient: ? = 6 e - 3 , M = 10, z0 = 12,/~1 = 0.2 and/z: = 0. In the same situation, Fig. 8 illustrates the b e h a v i o u r o f the classical second o r d e r a l g o r i t h m b y K o b a y a s h i (1). In fact, the first 500 p o i n t s are used to e n a b l e the convergence (since the t r a n s m i t t e d sequence is k n o w n in this p a r t i c u l a r case), a n d then we switch to a decision d i r e c t e d m o d e . In p r a c t i c a l situations, the j o i n t blind a l g o r i t h m c o u l d be used to achieve convergence, a n d after convergence, one w o u l d then switch to a decision scheme in o r d e r to reduce the variance.

VI. Conclusion A new a l g o r i t h m for b l i n d e q u a l i z a t i o n a n d s y n c h r o n i z a t i o n was p r e s e n t e d in this article. T h e a l g o r i t h m was tested o v e r synthetic, as well as real, d a t a f r o m a n u n d e r w a t e r

Algorithm for Blold Equalization

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=_ E 0 10219

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Equalizer output 3

0

x

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-2

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0

2

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2

FIO. 8. Second order joint algorithm: 7 = 5 e - 3 , M = 30, z0 = 10,/~, = le-2,/~2 = l e - 4 and switching to decision directed mode after 500 symbols. transmission. In both cases, the results allow a correct data recovery and are thus promising. These results were obtained without any training sequence which is a highly desirable property for real transmission systems.

Acknowledgement Part of this work was funded by the European Community under the PARACOM-MAST contract (MAS2-CT91-0005).

References (1) H. Kobayashi, "Simultaneous adaptive estimation and decision algorithms for carrier modulated data transmission systems", IEEE Trans. Commun., Vol. COM-19, pp. 268 280, June 1971. (2) M. Stojanovic, J. Catipovic and J. Proakis, "Phase coherent digital communications for underwater acoustic channels", IEEEJ. Ocean. Enyng., Vol. OE-19, pp. 100-111, January 1994. (3) D. N. Godard, "Self-recovering equalization and carrier tracking in two dimensional data communication systems", IEEE Trans. Commun. Vol. 28, No. 11, pp. 1867-1876, November 1980. (4) A. Benveniste, M. M6tivier and P. Priouret, "Adaptive Algorithms and Stochastic Approximations, Theory and Applications", Springer, NY, 1990. (5) B. Geller, V. Capellano, J. M. Brossier, A. Essebbar and G. Jourdain, "Equalizer for video rate transmission in multipath underwater communications", IEEE J. Ocean. Enqn 9. Vol. OE-21, No. 2, April 1996. (6) H. J. Kushner and J. Yang, "Analysis of adaptive step-size SA algorithms for parameter tracking", IEEE Trans. Automat. Contr., Vol. 40, No. 8, pp. 1403-1410, August 1995.