IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 1, JANUARY 1996 29

A New Approach for Evaluating the Performance of a Symbol Timing Recovery System Employing a General Type of Nonlinearity

Erdal Panayirci and Elisha Yeheskel Bar-Ness

Abstract-A new technique is presented for evaluating the jitter performance of a symbol timing recovery (STR) subsystem for digital data transmission systems. The STR system consists of any even-symmetric zero-memory nonlinear device followed by a narrowband filter tuned to the pulse repetition frequency. Exact analytical expressions are derived for the mean and the mean- square values of the timing wave, based on iterative computations of high-order moments of the input signal. Then, the root mean- square (rms) jitter performance is determined as a function of various system parameters such as the power series expansion of the zero-memory nonlinear device, the rolloff factor of the input pulse shape, and the postfiltering. Finally, the numerical results obtained from some specific examples serve to illustrate several aspects of the timing recovery problem.

I. INTRODUCTION POPULAR method for symbol timing recovery (STR) A consists of passing the incoming signal, either at IF or

at baseband, through a zero-memory device with an even nonlinearity and then feeding the resulting waveform to a phase locked-loop or equivalently to a bandpass filter centered at the pulse repetition frequency, 1/T, so that a discrete tone is generated at the symbol rate frequency. Many forms of nonlinearities may be used for this purpose. The most common are square-law [l], absolute value [2], and fourth law [3]. The jitter performances of STR circuits with second-order nonlinearities have been analyzed in [l], [4], and [5], by adapting a linearized mean-square phase error criterion. Based on this criterion, a complete analytical technique is given in [6] and [7] for evaluation of the jitter performance of a digital STR circuit. Recently, the same performance analysis for the square-law STR circuit employed in a digital sub- scriber loop environment has been done by [8]. When dealing with strongly bandlimited pulses, nonlinearities other than the square-law must be considered. Unfortunately, clock circuits implemented with nonsquare-law devices are hardly tractable mathematically [9], and in fact, their performance has only been evaluated by computer simulation [lo]. In this paper, we describe a new technique for evaluating the performance of an STR circuit, which uses a zero-memory device with any even, high-order nonlinearity. The technique is based on the iterative computation of the high-order cross-moments of the input

Paper approved by M. Moeneclaey, the Editor for Synchronization of the IEEE Communications Society. Manuscript received March 17, 1994; revised January 17, 1995 and May 10, 1995.

E. Panaylrci is with the Faculty of Electrical and Electronics Engineering, Istanbul Technical University, Maslak 80626, Istanbul, Turkey.

E. Y. Bar-Ness was with the Department of Electrical and Computer Engineering, Center for Communications and Signal Processing, New Jersey Institute of Technology, Newark, NJ 07102 USA. He passed away on May 28, 1992.

Publisher Item Identifier S 0090-6778(96)00806-9.

r I

I J I I

Fig. 1. Block diagram of the symbol timing recovery circuit.

signal to the STR circuit. The results are rather general and are applicable to many cases of practical interest. The thermal noise is taken into account as well as bandwidth limitations due to transmit and receive filters. Theoretical results obtained are also confirmed by the computer simulations presented in [lo], using the same STR scheme.

11. PROBLEM STATEMENT AND SYSTEM MODEL The main objective of this section is to calculate the root

mean-square (rms) value of the jitter of the timing wave, extracted from the output, ~ ( t ) , of the STR circuit whose block diagram is shown in Fig. 1, where z ( t ) is the demodulated baseband signal represented by

M

k=-cc

where the data sequence {Q} is assumed to be a statisti- cally independent binary sequence taking the values fl , with equal probabilities and g ( t ) is the pulse shape employed in transmission. The thermal noise, n(t), is modeled by the zero-mean stationary Gaussian process characterized by the power spectral density N ( f ) . In Fig. 1 f ( . ) represents a high order zero-memory nonlinear transformation defined by a finite power series of the form:

N

n=O

where the c,’s, n = 0, 1, . . . , N are given real constants and 2N is the order of the nonlinearity. The nonlinear device is followed by a phase locked-loop or, equivalently, by a narrowband bandpass filter whose transfer function H ( f ) is centered at the symbol rate frequency 1/T and satisfies the bandlimiting condition: H ( f ) = 0 for l l f l - (1/T)I > 1/2T.

Due to the random nature of the data sequence, the band limitations of the channel and thermal noise, a realization of the timing wave ~ ( t ) at the output of the STR circuit of Fig. 1 appears as a nearly sinusoidal waveform with slowly varying amplitude and small fluctuations in phase shift, called “phase jitter.” Therefore, the zero crossings of the random waveform, ~ ( t ) , do not coincide with the zero crossings, t o , of E [ z ( t ) ] . Let At be the difference that represents the “timing error” or

0090-6778/96$05.00 0 1996 IEEE

30 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 1, JANUARY 1996

"jitter." Then the rms value of At/T, where T is the signaling interval, is given in terms of the first and second moments of the timing wave z ( t ) at the mean zero crossing instant t = t o for which E[z(to)] = 0 [ll;

where E[.] denotes expectation and the dot indicates differen- tiation with respect to time. From (3), in order to evaluate the rms value of the jitter, we must compute the mean, E[z( t ) ] and mean-square, E [ z 2 ( t ) ] , values of z ( t ) at the value to. These moments are obtained analytically as a function of the postfiltering characteristics and of the Taylor series expansion coefficients as follows.

A. Evaluation of the Mean Value of the Timing Wave

g ( t ) with h(t) . Hence From Fig. 1, the output z ( t ) is simply the convolution of

00

(4)

From (2), the expected value of y ( a ) can be expressed as

N

n = O

where E[x2"(a)] = M2n(a) is the 2nth moment of the random variable, .(a). Ho and Yeh [ll] obtained a recursive relation for M2, = M2n(a), assuming that the thermal noise, n(t) , is absent. Their results can be extended to include the effect of thermal noise and the following recursive expression is obtained for Mzn

where MO = 1 , and

f 2i-1 =

x g 2 ( a - k T ) + 2a2 i f i = I I k

I = - 00

o2 is the variance of the Gaussian noise, n(t), g( . ) is the pulse shape as defined in (l), and Bi's are the Bemoulli numbers which are defined by the coefficients of the function

Since x ( t ) is a cyclostationary process (CT), the high-order components, xzn(a) , at the output of the nonlinear device are also CT processes. Therefore, their mean functions, Mzn(a) =

E[z2"(a)] , are periodic in time, with period T , and they can be expanded into a Fourier series as

n = 0 , 1 , . . ' ,AT (7)

where the complex Fourier coefficients mfn)s are computed from

Using (5) and (7) in (4) and then taking into account the band limiting condition on H ( f ) we get the following expression for E[z(t )]:

where p1 = H ( l / T ) E,"==, enmy") and 4 is the phase of PI. Equation (8) represents the mean value of the timing waveform from which the zero crossings, t o , of E[z( t ) ] is obtained by setting (27rtolT) +4 = n ( x / 2 ) e t o = n(T/4) - (4T /27r ) , where n is an odd integer.

B. Evaluation of the Mean Squared Value of the Timing Wave

Our main objective is to obtain an exact expression for E[z2( t ) ] that remains manageable even if the degree of pulse overlapping is very large. The derivation is lengthy and tedious. Therefore, we skip the analytical details and point out only the important steps. From Fig. 1 we can write

A where E[y(t - a)y( t - P) ] = R,(t - a, t - P) . From (2), the autocorrelation function R,(t, s) can be

expressed as

"

.R?J(t, s ) = x c,cnE[x2"(t).2n(s)] (10) m=O n = O

which is a linear combination of the higher-order cross mo- ments of ~ ( t ) at time instants t and s, namely

Mzm, 2 n ( t , s) = ~ [ x ~ ~ ( t ) l ~ . ~ ~ ( s ) ] . (11)

We now derive an expression to calculate the high-order cross-moments M2m,2n of the input signal iteratively, in the presence of thermal noise. The derivation is an extension of [ 111 and can be found in [ 121. The final recursive expressions for M2m,2n can be summarized as follows.

For n = I, 2, . . . a n d m = 0, 1, 2 , . . . , n-1

~ 2 m , 2 n == - [ ( 2 n - 12i p 2 z + (an - 122 + 1 S Z + ~ ]

i=O (12)

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 1, JANUARY 1996

where

. fZm-Zj, 2n-1-Zi Mzj, 2i

Zm-23+1,2n-2-2i . f 2 M23-1,22+1.

Similarly, for n = 0, 1, 2, . .. and m = 0, 1, 2, ... , M2m+1,2n+l =

- [ 2 (2) Qzi + ( 2:T 1) Qzi+i] (13) i=O i=O

where

. fim-zj, 2n-1-2i M2j, 2i

Zm-Z3,2n--22-1 . f2 Mz3+1,22+1

(2m - 1 2 j - 1) ( - 1 ) ~ f ~ - 1 ~ 0 ~ ~ ( m - 3 ) , 0 .

(14)

and, initially, for n = 0 and m = 0, 1, 2, . . . , m

~ 2 m , 0 = - 3=1

Note that f,">' and f l ) ' are defined as follows:

f y =

+ 2, i f p = l , q = O

U k V k + a 2 d 4 i f p = O , q = l k

k

C(P, dlB,+q+ll u;+lv,& if p + 4 E odd N k

0, i f p + q E evenN.

x u ; + 2, i f p = O , q = l

u k v k + a2p(r), i f p = l , q = O k

k

f;,' =

C(P, 4)IB,+,+ll u:ll;+l, if P + 4 E odd N k

0, if p + q E even N.

A

A ~k = g ( t - kT) V k = g ( s - k T ) a 2,+'+1(2,+4+1 - 1)

p + q + l *

I where p ( ~ ) is the autocorrelation function of n(t), N denotes the set of natural numbers and

C(P, 4 ) =

As previously stated, since ~ ( t ) is a CT process so is y(t) and thus its autocorrelation function, R,(t - a, t - p) is

l 0 l L I I t I

t

\ E b/N,, I 20 dB

Eb/No= md6 \

31

0 25 0 50 075 100

ROLL OFF F A C T 0 R . a

Fig. 2. Jitter variance using a raised-cosine pulse shape for SL-STR circuit.

periodic with period T. Its Fourier series expansion and its complex Fourier coefficients rk ( r ) , are given by

R,(t - a , t - p) =

Using these relations, and then after some algebraic manip- ulations lead to expressing (9) in the frequency domain

where H ( f ) is the transfer function of the postfilter and Rk(f) is the Fourier Transform of T ~ ( T ) . It is easy to see that by using the bandlimiting condition on H ( f ) , the product H ( k / T - v ) H ( v ) is identically zero except for the case where IC = 0 and IC = f 2 . One can show by using the symmetry

32 JEEE TRIWSACI'IONS ON COMMUNICATIONS, VOL. 44, NO. 1, JANUARY 1996

10' I I

Q =lo0

w -

10'

lo6 I 0 2 5 0 50 0 75

ROLL OFF FACTOR.^

Fig 3 Jitter variance using a raised-come pulse shape for E-STR cmut .

properties of H ( f ) and &( f ) that the terms in (16) for k = f 2 are complex conjugates. Hence

where

Vz = 1: H ($ - u ) H(u)R2 (v - $) du

and I V2 I and Q are the magnitude and phase of V,. The expression for rms relative jitter (3) can be written

in terms of the coefficients VO and V2 for the mean and mean-square value of the timing wave by substituting the value of t o , obtained earlier, in (17) and in the expression for E[i( to)] = (47r/T)lpl I sin (27rtlT + $), obtained from (8). As a result of the postfiltering characteristics used in real applications, it can be seen that B M 24 and the minimum mean-square value occurs approximately at t o , (91. The rms

0 25 0 50 0 75 100

ROLL OFF FACTOR , a

Fig. 4. Jitter vanance using a raised-cosine pulse shape for AV-STR circuit

jitter is then characterized by

m. NUMERICAL RESULTS We calculated the jitter variance [square of the rms value in

(25)], evaluated exactly at the mean zero crossings, for several STR circuits employing such zero-memory nonlinear devices as: a) square law (SL), b) absolute value (AV), c) fourth-law (EX), and d) In cosh (.). By curve fitting, the absolute value nonlinearity was approximated by the following finite power series

1x1 M x2 - x4 + & x6 for 1x1 5 3 (19)

and similarly, the In cos (.) function was expanded in Taylor series around x = 0 and approximated by the power series

In cosh (x) M x2 - - ;2 x4 + 2. (20)

The pulse shape, g( t ) , entering the nonlinear device was chosen to have a raised-cosine characteristic, with rolloff factor a. The noise level is expressed by the ratio of &/No of the energy per bit to the noise spectral density at the receiver input. The numerical computations were carried out under the assumption that the post filtering characteristic looks like a

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO 1, JANUARY 1996 33

_ - - a Maximum Likelihood type of STR scheme is employed. At istanbul Technical UniGersity, Tech. Rep. ITU-01-93, Apr. 1993.

106 1_ 0 25 0 5 0 75

ROLL OFF F A C T 0 R . a

Fig. 5. circuit.

Jitter variance using a raised-cosine pulse shape for In cosb-STR

single-tuned resonator with resonant frequency 1/T and the quality factor Q

(21) H ( f ) = HLP(f - 1/T) + H L P ( f + 1/T)

with HLP (f ), the lowpass equivalent, given by 1 1

(22) I f 1 elsewhere.

The jitter variances are computed as a function of rolloff factor a, and plotted in Figs. 2 4 , corresponding to the STR circuits employing a SL, FL, and AV types of nonlinearities for some values of signal-to-noise ratios and for the post filter quality factor Q = 100. It can be observed that there is an excellent agreement between these theoretical jitter curves and the computer simulation results obtained by D’Andrea and Mengali [ 121.

Finally, Fig. 5 illustrates the jitter performance of the STR circuit employing the In cosh (.) type of nonlinearity. It is well known that this type of nonlinearity is an optimal choice when

high SNR, this nonlinearity can be approximated by a fullwave rectifier. At low SNR In cosh (z) M !jxz and the optimal nonlinearity is an SL device. The jitter curves presented in Fig. 5 agree with these observations.

IV. CONCLUSION In this paper, a new analytical method has been presented to

evaluate the jitter performance of a popular type of STR circuit employing any even-symmetric, zero memory nonlinear device followed by a narrow band filter tuned to the pulse repetition frequency. The method is based on the iterative computations of the high-order moments of the input signal to the STR circuit. The results are rather general and are applicable to many cases of practical interest. The thermal noise was taken into account as well as bandwidth limitations due to the channel. It has been concluded that the theoretical results obtained were in an excellent agreement with the computer simulations presented earlier in the literature [lo].

DEDICATION

E. Panayirci would like to dedicate this work to Elisha Yegal Bar-Ness, the coauthor of this paper and the late son of Prof. and Mrs. Yeheskel Bar-Ness, who passed away suddenly on May 28, 1992, at a very young age. His genuine and close friendship and the memories will always be treasured.

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[4] A. N. D’Andrea and U. Mengali, “Performance analysis of the delay-line clock regenerator,” IEEE Trans. Commun., vol. COM-34, pp. 321-328, Apr. 1986.

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IEEE Trans. Commun., vol. 39, pp. 133-140, Jan. 1991. [lo] N. A. D’Andrea and U. Mengali, “A simulation study of clock recovery

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r l l l E. Y. Ho and Y. S. Yeh, “A new approach for evaluating the error - - probability in the presence of intersymbol interference and additive Gaussian noise,” Bell Syst. Tech. J., pp. 2249-2265, Nov. 1970.

[12] E. Panayirci and E. Y. Bar-Ness, “A new approach for evaluating the oerformance of a STR system employing a general type of nonlineanty,”