ML-oriented NDA carrier synchronization for general rotationally ...

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IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 42, NO. 8, AUGUST 1994

ML-Oriented NDA Carrier Synchronization for General Rotationally Symmetric Signal Constellations Marc Moeneclaey and Geert de Jonghe

Abstract- In this contribution we point out that the nondecision-aided (NDA) carrier synchronizer, maximizing the low E s / N o limit of the likelihood function averaged over a general 2r/N-rotationally symmetric signal constellation, reduces to the familiar timing-aided Nth power synchronizer; this extends a result from [6] where only M-PSK constellations have been considered. Whereas in the case of M-PSK the tracking error variance of this NDA ML synchronizer is known to converge to the Cramer-Rao bound (CRB) with increasing E s / N o [2], we show that for other rotationally symmetric constellations (such as QAM) the tracking error variance is substantially larger than the CRB.

I. INTRODUCTION DERIVATION OF THE NDA ML CARRIER SYNCHRONIZATION ALGORITHM

A

and E,[.] indicates averaging over the signal constellation. In (3), p ( k ) denotes the sample, taken at the decision instant kT,of the output of the matched filter (with impulse response h(-t)) when driven by ~ ( t Note ) . that in [6] the term with I C I ~ has ~ ~ been dropped from the argument of the exponential in (3), because for M-PSK its value is the same for all points of the constellation; for more general constellations, this term should be kept. The maximization of (2) requires knowledge about the operating E,/N,. This problem can be circumvented by replacing L(8’) be either its limit Lo(@) for E , / N , approaching zero, or its limit Lm(8’) for E,/N, approaching infinity. The maximization of L,(B’) is known to yield the decision-directed (DD) feedforward ML algorithm [31, [4]; the corresponding DD ML estimate is given by

SSUMING the symbol timing to be known, the complex envelope of the received noisy PAM signal is given by

s =arg

xF;p(k) (k11

m

)

(DD ML)

(4)

where Ek is the receiver’s decision about the data symbol Ck. Unlike feedforward NDA carrier synchronization algorithms, the feedforward DD ML algorithm (4) needs a preamble to acquire the carrier phase, which makes (4) not suitable for short burst TDMA applications. On the other hand, the maximization of Lo(@’)does yield a truly NDA carrier synchronizer. As the straightforward computation of L,(B’) is quite difficult for arbitrary signal constellations, this approach has hitherto been successfully applied only to M-PSK constellations, for which the resulting NDA ML synchronizer reduces to the timing-aided Mth power synchronizer; this result is well known for BPSK ( M = 2) and QPSK ( M = 4) [l], [4], and has been extended in [6] to arbitrary integer values of M . For more general constellations, we propose an alternative method to compute L o ( @ )which , we have used in [ 5 ] for obtaining ML-oriented symbol synchronization algorithms. First, we expand the exponential function in (3) into a power (2) series, and average each term of this power series over the signal constellation. This yields

where {cm} is a sequence of independent identically distributed equiprobable data symbols with E[Ickl] = 1, 1/T is the symbol rate, 8 is the unknown carrier phase, h ( t ) is a real-valued unit-energy baseband pulse, and n(t) is complexvalued white Gaussian noise, whose real and imaginary parts are statistically independent and have the same power spectral density N 0 / ( 2 E , ) .The nondecision-aided (NDA) maximumlikelihood (ML) carrier synchronizer maximizes, with respect to the trial value 8’ E (0, 27r) of the carrier phase, the log-likelihood function log L( e’). Taking into account the statistical independence of the data symbols and assuming that the baseband pulse g ( t ) , which is defined as the convolution of h ( t ) and h(-t), satisfies the first Nyquist criterion (Le., g(mT) = 0 for m # 0), this log-likelihood function is given by [I], 131, [41, [61 K

logL(k; e’)

logL(0’) = k=l

where

Paper approved by W. R. Brown, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received May 15, 1992; revised August 15, 1992. This work was supported by the Belgium National Fund for Scientific Research (NFWO). The authors are with the Communications Engineering Laboratory, University of Ghent, B-9000 Gent, Belgium. IEEE Log Number 9401936.

where

n=On‘=O

00904778/94$04.00 0 1994 IEEE

~

1

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994

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and I3

1"

1 :true linearized

-1

variance

i

K.Var

2 :approximateilnearlzed

variance

L ( k ; 0') can be written as the sum of two components, which consist of phase-independent terms and phase-dependent terms, respectively. Together with the first term of the righthand side of ( 5 ) , the terms with n = n' in (6) contribute to the phase-independent component of L( IC; e'); for vanishing E,/N,, this component converges to 1. Let us denote by Al(p(lc),m; 0') the summation of the terms with n # n' in (6), which contribute to the phase-dependent component of L ( k ; e'); for vanishing E,/N,, this component is dominated by the term with m = m, where m, is the smallest integer for which E,[Al(p(IC),m; e')] is nonzero. Hence, the low E,/N, limit L0(6")of L(0') is determined by K

m,; 0'11

logL,(O = C l o g k=l

.

10.

0.

for E,/No

( $ ) m O )

(8)

+0

m,

(9) where we made use of log (1+x) + z for x + 0. The specific value of m, is a function of the signal constellation. For signal constellations that are unvariant under a rotation of 27rfN, it is easily shown that

E[cYc;"] = 0 when m - n

(0, N , 2N, ...}.

(10)

The result (10) is valid for practical constellations such as M-PSK (invariant under rotation of 2.rr/M, i.e., N = M ) and QAM with square and cross constellations (invariant under rotation Of T/2, i.e., N = 4). For these COnStellatiOnS, E[cF]# 0, SO it follows from (10) that m, = N , and that Only the terms With ( n = 0, 71' = N ) and ( n = N ; 72' = 0) in (6) contribute to E[Al(p(IC),N ; e')]. Hence, within a factor not depending on e', (9) reduces to

30.

20.

Es/No[dB] Fig. 1. Tracking performance of the NDA ML synchronizer for M2-QAM constellations.

synchronizer results from the maximization of Lo(#), in the case of general 2.rr/N-rotationally symmetric constellations (with E[,-;] # 0). 11.

PERFORMANCE CONSIDERATIONS

TRACKING

Following the approach from [2], the linearized tracking error resulting from the NDA algorithm (12) is given by

e~

=

1

NIEk

11

[

1

, I m E [ ~ ; "l] ~ Kx p " ( k ) e - j .~ '(13) k=l

For moderate and high E,/N,, quadratic and higher order noise terms in (13) can be ignored; in this case the tracking error variance of the NDA algorithm is approximated by

Var[O - e]

1

--B1

K

(2,) iBz -

(14)

+

logLo(8') = 2Re ~ E [ C ~ ~ ] ~ P ~ ( I C ) ~ - (11) " ~ ' . with ].

L

k=l

J

~ [ I ~ : I I ~ [ I ~ f l " - ' >l 1

The resulting NDA feedforward carrier phase estimate is given bv

B1 =

which corresponds to the familiar timing-aided Nth power synchronizer [ l l , 121, [41, [61, which belongs to the class of NDA feedforward carrier-synchronizers introduced in [2]. Hence, we have shown that the timing-aided Nth power

The approximation in (14) consists of omitting terms containing the square and higher powers of NO/(2E,).The first and second term in (14) are caused by additive noise and self-noise, respectively.

lE[CF1l2

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In the case of the DD algorithm (4), decision errors can be ignored for moderate and high E s / N o . This also gives rise to (14), but with B1 = 1 and Bz = 0: the additive noise contribution equals the Cramer-Rao bound (CRB), which is a lower bound on the tracking error variance [l], [4], and self-noise is absent. This indicates that the DD ML algorithm is essentially optimum, irrespective of the specific signal constellation. Because of the inequality (15), the NDA algorithm (14) has a constellation dependent noise penalty as compared to the CRB. Also, the NDA algorithm suffers from self-noise when (16) yields B2 > 0. For M-PSK constellations, (15) and (16) yield B1 = 1 and B2 = 0, in which case the tracking performance of the NDA algorithm is basically the same as the CFtB for moderate and high Es/No;a similar observation has been made in [ 2 ] , [6]. Fig. 1 shows the tracking performance of the NDA ML algorithm for M2-QAM constellations. Displayed are the true variance of the linearized tracking error (13), the approximate variance (14) for moderate and high E,/No, and the CRB. Whereas for 4-QAM (which is

equivalent with 4-PSK) the NDA ML algorithm is essentially optimum for moderate and large E,/N,. this is no longer the case for the larger QAM constellations.

REFERENCES M. H. Meyers and L. E. Franks, “Joint carrier phase and symbol timing recovery for PAM systems,” IEEE Trans. Commun., vol. COM-28, pp. 1121-1129, Aug. 1980. A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated Carrier phase with application to burst digital transmission,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 543-551, July 1983. P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun., vol. COM-34, pp. 522-527, June 1986. F. M. Gardner, “Demodulator reference recovery techniques suited for digital implementation,” Final Rep., ESA Contract no. 6847/86/NL/DG, Aug. 1988. M. Moenclaey and G. Jonghe, “Tracking performance comparison of two fmdfonvard ML-oriented carrier-independent NDA symbol synchonizrs,” IEEE Trans. Commun., vol. 40,pp. ln3-1425, SePl. 1093 _._-. [6] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “Carrier phase recovery for narrow-band polyphase shift keyed signab,” Al/a Frequenm, vol. LVII, pp. 575-581, Dec. 1988.