Frequency Offset And Symbol Timing Recovery In Flat-fading

400 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 3, MARCH 1998

Frequency Offset and Symbol Timing Recovery inFlat-Fading Channels: A Cyclostationary Approach

Fulvio Gini, Member, IEEE, and Georgios B. Giannakis,Fellow, IEEE

Abstract—Two open-loop algorithms are developed for esti-mating jointly frequency offset and symbol timing of a linearlymodulated waveform transmitted through a frequency-flat fadingchannel. The methods exploit the received signal’s second-ordercyclostationarity and, with respect to existing solutions: 1) theytake into account the presence of time-selective fading effects; 2)they do not need training data; 3) they do not rely on the Gaussianassumption of the complex equivalent low-pass channel process;and 4) they are tolerant to additive stationary noise of any coloror distribution. Performance analysis of the proposed methodsusing Monte Carlo simulations, unifications, and comparisonswith existing approaches are also reported.

Index Terms—Cyclostationarity, fading, synchronization.

I. INTRODUCTION

DEMODULATION in digital communication systems re-quires knowledge of symbol timing and frequency offset.

Mistiming and frequency drifts arise due to propagation,Doppler effects, and mismatch between transmit and receiveoscillators. Both data-aided and nondata-aided feedforward (oropen-loop) estimation structures have been proposed. Block(or batch) schemes include the feedforward data-aided fre-quency estimators proposed in [6], [9], and [11], which ex-ploit the information signal’s autocorrelation sequence toestimate the frequency offset. They do not account for fadingand are simpler than the maximum likelihood, but are notbandwidth-efficient because they rely upon the training dataof a preamble. The alternative is nondata-aided structures thatrecover synchronization parameters from the received data,exploiting only side information concerning the statistics ofthe information signal.

With fading effects, present in mobile cellular terrestrial ra-dio systems [16], or in ionospheric channels [15], the synchro-nization problem is even more challenging. In [18] and [10],nondata-aided open-loop algorithms were proposed for jointfrequency offset and symbol timing estimation in frequency-

Paper approved by E. Panayirci, the Editor for Synchronization andEqualization of the IEEE Communications Society. Manuscript receivedJanuary 7, 1997; revised September 18, 1997. This work was supported bythe National Science Foundation under Grant NSF-MIP 9 424 305. This paperwas presented in part at the 1st IEEE Signal Processing Workshop on WirelessCommunications, Paris, France, April 16–18, 1997.

F. Gini is with the Department of Information Engineering, University ofPisa, I-56126 Pisa, Italy (e-mail: [email protected]).

G. B. Giannakis is with the Department of Electrical Engineering, Uni-versity of Virginia, Charlottesville, VA 22903-2442 USA (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(98)02128-X.

flat fading channels. The performance of [18] and [10] wassimulated for both time- and frequency-selective fading en-vironments, but the analytical results were derived assumingconstant fading over the entire burst.

In this paper we propose an approach for fully digitalnondata-aided joint frequency offset and symbol timing esti-mation of a linearly modulated waveform transmitted througha frequency-flat fading channel. Our approach exploits second-order cyclostationarity of the sampled received sequence andconsiders nonconstant fading over the entire burst. More-over, the proposed algorithms can be used in a data-aidedscenario without any change, if the appropriate preamble isemployed.

A synchronization method relying explicitly on the cyclosta-tionarity of the oversampled data was also proposed in [17],but fading effects were not considered. In continuous-time,[18] formed the instantaneous product of the received signalwith its delayed-conjugated replica, and proved it periodic withperiod equal to the symbol interval. Although included in theoriginal model, fading effects were argued to be small andwere omitted from the instantaneous product prior to extractingits Fourier coefficients that were used for frequency offsetand timing estimation [18]. Despite superficial resemblance,the approach herein relies on periodicity of discrete-timeensemble products of oversampled data, and fading effectsare included in the cyclostationary (CS) statistics used toform consistent estimators of the synchronization parameters.The term CS processes and the role of cyclostationarityin communications dates back to Bennett [1]. Even if notclearly acknowledged, cyclostationarity has been exploitedfor synchronization purposes by many authors (see, e.g.,[13], [10], and [5]). The underlying common idea is touse nonlinear combinations of the data to reveal periodiccomponents containing synchronization parameters. In thispaper, we attempt to unify and improve upon these approacheswithin a discrete-timeCS framework.

The rest of the paper is organized as follows. In SectionII we derive discrete-time models of the fading channel andthe received signal, and introduce our modeling assumptions.The relationships between synchronization parameters, cycliccorrelation, and cyclic spectrum of the observed sequenceare derived in Sections III and IV, respectively. In SectionV we show how to estimate the cyclic correlation and thecyclic spectrum consistently. The estimation algorithms andtheir performance can be found in Section VI along with uni-fications and comparisons with existing approaches. Finally,conclusions are drawn in Section VII.

0090–6778/98$10.00 1998 IEEE

GINI AND GIANNAKIS: FLAT-FADING CHANNELS: A CYCLOSTATIONARY APPROACH 401

II. M ODELING AND PROBLEM STATEMENT

The complex envelope of a linearly modulated signal,transmitted through a frequency-flat fading channel [8], [15,Ch. 14], [18], is received as

(1)

where is the fading-induced multiplicative (or pattern)noise, is the transmitter’s signaling pulse, is thesymbol period, ’s are the complex information sym-bols, is the frequency offset, is the initial phase, and

is the propagation delay within a symbol periodComplex additive noise is assumed stationary

but not necessarily white and/or Gaussian (subscriptdenotescontinuous-time signals).

After the receiving matched filter , the signaldenotes convolution) is

(over)sampled at a rate , where is an integer. Wethus obtain the following discrete-time data:

(2)

with , ,

, and ,where is the combined impulse response of the knowntransmitter and receiver filters in cascade

For the moment, we do not invoke any specificassumption on the shape of

Models (1) and (2) are valid as long as:

1) the fading distortion is approximately constantover a pulse duration or, equivalently, the Dopplerspread is small [15, Ch. 14], where denotesthe bandwidth of Typical values for practicalsystems range from (very slow fading)to (very fast fading) [12];

2) the frequency offset is small compared to the symbolrate, so that the mismatch of the receive-filterdue to can be neglected. In [11], wastypically assumed (see also [9], [10], and [18]).

The following assumptions are imposed on (2).

(AS1): is a zero-mean independently identically dis-tributed (i.i.d.) sequence with values drawn from afinite-alphabet complex constellation, with variance

(AS2): is stationary complex process with autocorre-lation ; the Fouriertransform (FT) of is calledDoppler spec-trum [15, Ch. 14].

(AS3): is a wide-sense stationary complex process,independent of

(AS4): satisfies the so-calledmixing conditions(see e.g., [3] and [4]), which state that theth-ordercumulant of at lag ,denoted by , is absolutely summable:

Similarly, we assume thatis mixing as well.

Mixing requires that sufficiently separated samples are ap-proximately independent and is satisfied by all finite memorysignals in practice. (AS4) will prove useful in establishingconsistency of our estimation algorithms.

The goal here is to derive estimates of and in (1)based on consecutive samplesfrom (2), corresponding to transmitted symbolsdenotes integer part of If is given (data-aidedscenario), and the distributions of and are known(e.g., is Gaussian or i.i.d. non-Gaussian with knownparameters), then maximum-likelihood (ML) estimation ofand is possible although computationally demanding. Weseek consistent, albeit computationally efficient, estimates thatcannot only initialize the nonlinear search required by MLestimates in the data-aided case, but, most importantly, remainoperational in the blind (nondata-aided) scenario with realisticfading and colored noise processes of unknown distributions.

In the ensuing section we establish that the fractionally-sampled discrete-time process is CS with period , andshow how this property leads to consistent estimators ofand

III. CYCLIC CORRELATION APPROACH

The time-varying correlation of a general nonstationaryprocess is defined as ,where is an integer lag. Signal is termed second-orderCS with period iff there exists an integer such that

(see, e.g., [7] and [3]).With given by (2), and under assumptions (AS1)–(AS3),we have

(3)

To establish that in (2) is CS with period , consider (3)and replace with

(4)

where in deriving the second equality we setFor a fixed , (4) shows that is periodic

in with period Thus, it has discrete Fourierseries coefficients given by

, which are periodicwith respect to (w.r.t.) with period ; is termedcyclic correlation andare called cyclic frequencies or cycles. From (3), the cyclic


correlation turns out to be1

(5)

We wish to reveal the dependence of on thetime epoch which was absorbed in when we passedfrom (1) to (2). Let us denote the FT of as

and use Parseval’s relation to rewritethe sum in (5) as

(6)

Now recall that and suppose thatthe bandwidth of is less than ; thus, samplingat a rate does not introduce aliasing. In the absence ofaliasing, we have (e.g., [14, Ch. 3])

for (7)

where denotesthe continuous-time frequency variable and is thesampling period. Inserting (7) in (6) and then (6) in (5), weobtain

(8)

where we defined

We observe from (8) that frequency offset and symbol tim-ing appear as separable one-dimensional complex exponentialsin the two-dimensional sequence w.r.t. the lag

and w.r.t. the cycle We will use this observation toestimate and separately.

Note now that in (8) is known because the sig-naling pulse is known. Hence, we can “compensate” forit by multiplying (8) with , provided that

, where Upon definingwe can rewrite (8) as

(9)

Noise affects the cyclic correlation in (9) at cycleTo avoid it, we henceforth consider Moreover, (9)suggests cancelling by multiplying terms with cyclefrequencies and

1To derive (5) we used the identity�P�1p=0 �

+1

l=�1f(p � lP ) =

�+1

n=�1 f(n), with f(n) := g(n)g�(n+ �) exp(�j(2�=P )kn):

Thus, denoting with the unwrapped phase, we can re-trieve from the phase of a “compensated” (or “normalized”)cyclic correlation product as follows:

for

(10)

where denotes the maximum inGiven the frequency offset , the time epoch can be

theoretically derived as [c.f. (9)]

for

(11)

Relationships (10) and (11) form the basis for the estimationalgorithms developed in this paper. Estimatingand basedon (10) and (11) may require phase unwrapping. In thenext section we will avoid phase unwrapping by workingin the cyclic spectrum domain (see also [13] for a thoroughdiscussion on phase unwrapping issues).

Remark 1: Having retrieved and , we can obtainthe correlation of the fading from (9) as

for andBased on , parametric (e.g., autoregressive)

models can be fit to Such models may be potentiallyuseful for parsimonious characterization, classification, ortracking of possible variations in the fading process (see also[19]).

IV. CYCLIC SPECTRUM APPROACH

By definition, the cyclic spectrum of is theFT of w.r.t. [7], [3], i.e.,

From(9), we obtain

(12)

where andIt is now customary to introduce a further

assumption on the fading channel characteristics.

(AS5): is a complex low-pass process with powerspectral density peaking at a known fre-quency; w.l.o.g. we assume that peaks at

(see [15, Ch. 14] and also [2], [12], and[20]).

Under (AS5), can be recovered by peak-picking themagnitude of (12) as

(13)

and the time epoch can be found from the phase at the peak

(14)


It is worth observing that ambiguity due to spectral foldingdoes not occur in (13) if , while the estimator in(10) requires with

In addition, note that (AS5) is not strictly necessary; forexample, the Doppler spectrum could also have two peaks aslong as they are known, but in this case the structure of thefrequency estimator (13) should be modified accordingly. InSection VI, we will make use of (10) and (11) or (13) and(14) to derive consistent estimators of and But first, weneed sample estimates of and

V. CYCLIC STATISTICS

Since we do not have access to ensemble cyclic quan-tities, we should estimate them from finite samples. Weestimate from a single record , usingthe (normalized) fast Fourier transform (FFT) of the product

as follows:

(15)

Negative lags are obtained by symmetryUnder (AS4), is asymptot-

ically unbiased and mean square sense (m.s.s.) consistent; i.e.,[4]. As a consequence,

the estimators of and , obtained by replacing byin (10) and (11), are asymptotically unbiased and consistent.

Similar reasoning applies to (13) and (14) when samplecyclic spectra are used in practice. Consistent estimates ofthe cyclic spectra can be obtained by windowingwith having support andFourier transforming to obtain [3]

(16)

Alternatively, it has been shown in [3] that can beestimated directly from the data, relying upon the so-calledcyclic periodogram defined as

(17)

Note that denotes the conventional periodogram forstationary processes. Although the cyclic periodogram is anunbiased estimator of the cyclic spectrum, it is inconsistent.Under (AS4) it has been proven that smoothingwith an appropriate spectral window , the cyclicspectral estimate

(18)

is consistent and asymptotically normal with computable vari-ance [3]. As with stationary processes, windowing controls

the bias-variance tradeoff encountered with cyclic spectralestimates. Without it, the variance of in (16) and(18) would not decrease as (see [3] for rigorousstatements and guidelines on the choice of windows).

It is worth noting at this point that if the record lengthis small, then the estimates or

should be zero-padded to sufficiently large length priorto the FFT so that the frequency binsin (27) are small enough to allow accurate estimation of

In [11], it is reported that the Cramer–Rao lower bound(CRLB) for frequency estimation of a pure sinusoid in additivewhite Gaussian noise (AWGN) is given by

Therefore, for the FFT bin sizesto be compatible with this limit, must be large enough tosatisfy Alternatively, high-resolutionalgorithms (such as MUSIC, Kumaresan–Tufts, or matrixpencil) can be used, but these directions are beyond the scopeof this work.

VI. ESTIMATORS AND COMPARISONS

The effects of pattern and additive noise on our estimationalgorithms can be potentially reduced by averaging (10) and(11) over to obtain

(19)

(20)

Correspondingly, the cyclic-spectrum-based estimators are de-rived from (13) and (14) as

(21)

(22)

A remark is now in order on complexity issues.Remark 2: From (19) to (22) we observe that complexity

increases with , but there is also a tradeoff in selectingeven from an estimation viewpoint. Although more samplesare collected as increases, their correlation increases too,which is known to increase the estimators’ variance. Hence, asin [2], moderate values of are recommendedand the corresponding moderate increase in complexity isjustified by the improvement evidenced in our simulations (seeFig. 3). Note that if one wishes to average only a subset ofall possible cycles, complexity may be reduced by replacingthe FFT in (15) with the chirp-FFT [14, p. 623]. In addition,performance improves at the expense of increased complexity


as the number of lags increases. But, as with , thereis a tradeoff also with —one should not average overlags that are far away from the origin (say ) becausethe accuracy of the corresponding sample cyclic correlationsdecreases due to reduced averaging [c.f. (15)].

Expressions (19)–(22) are valid for arbitrary pulse shapes.Next, we will specialize to the commonly used raisedcosine pulse2 (see, e.g., [11] and [18]). Such a choice willnot only reduce complexity but also will allow us to as-sess performance and compare our estimators with existingmethods.

(AS6): is a raised cosine pulse, truncated and delayedby to assure causality

, where is the unit step function andis the noncausal (even w.r.t. raised cosine

pulse [15, Ch. 9]. Let denote itsbandwidth, where is the so-called rolloff factor

In the following, we will show that when is a raisedcosine, we can easily “compensate” for the presence ofwithout normalizing and to obtain and

A. Estimators with Raised Cosine Pulse

Under (AS6) and denoting by the FT of , wehave that in (8) can be written as

(23)

Thanks to the particular shape of , the integral is realand even w.r.t. [15, p. 546]. Recall also thatfor , and since , we have

for ; hence, the productin (23) will be

nonzero only for As a result, in (23) andin (8) will be nonzero only for the cycles ,

, andTo obviate the additive noise, we will thus rely only on

and to estimate and as follows[c.f. (19) and (20)]:

(24)

(25)

where denotes the known delay ofAs concerning the estimators based on the cyclic spec-

trum, they can be derived observing that in this case

2This means that bothg(tr)c (t) andg(rec)c (t) are square-root raised cosine

pulses.

, where andRecalling that in (23) for , andexcluding to suppress noise, we find the cyclic spectrumas

(26)

Plugging in the square bracketed term of (26) the raised cosinespectrum [15, p. 547], it turns out (by setting derivativesequal to zero) that the maxima occur at for

, and for These maximaare shifted by due to the convolution by in(26); hence, and

, from which weobtain easily

(27)Similarly, subtracting the phase at, we estimate the timeepoch as

(28)

Expressions (24)–(25) and (27)–(28) are the main resultsof this paper specialized for raised cosine pulses. Their per-formance will be compared next with existing algorithms.Summarizing, under (AS1)–(AS6) the estimators (24)–(25)and (27)–(28) are asymptotically unbiased and m.s.s. consis-tent, independent of the color and distribution of the additivenoise and the fading distortion.

B. Comparison with Scott–Olasz (S–O)

For comparison purposes we have derived the discrete-timeversion of the estimators proposed by Scott and Olasz (S–O)in [18]. With the notation adopted so far, they are expressed as

(29)

(30)

The estimator originally proposed in [18] contained theterm in order to compensate for thecontribution of the frequency offset. However, the terms

and that are necessary tocompensate for the effects of were not included (see[18, eq. (2.18)]).

Computer simulations were run to verify the efficacy ofthe proposed algorithms in frequency-flat fading channel en-vironments. The linear modulation format was 4-quadrature


(a) (b)

Fig. 1. (a) Bias and (b) MSE offeT versus SNR (dB). Estimator (24):feT = 0:0 (solid line), feT = 0:1 (dashed line), andfeT = 0:2 (dash–dotline). S–O: feT = 0:0 (circles), feT = 0:1 (�-marks), andfeT = 0:2 (stars).

(a) (b)

Fig. 2. (a) Bias and (b) MSE offeT versusfeT: Estimator (24):SNR = 0 dB (solid line), SNR = 8 dB (dashed line),SNR = 16 dB (dash–dotline). S–O:SNR = 0 dB (circles),SNR = 8 dB (�-marks),SNR = 16 dB (stars).

amplitude modulation (QAM) and i.i.d. symbols weregenerated with The transmit and receive filters werethe square-root raised cosine pulses with 50% rolloff

and the receive filter was simulated as a finite-impulseresponse (FIR) filter with taps; thus, andthe equivalent time duration of the FIR impulse response was

The additive noise was generated by passingzero-mean complex Gaussian deviates through the square-rootraised cosine filter to yield autocorrelation sequence

The land–mobile channel is generally modeled bythe Clarke’s model [16]. However, as pointed out in [2], itis difficult to efficiently simulate the exact mobile channelspectrum with conventional filters. So, we followed the usualpractice of using a low-pass filter with sharp 3-dB cutoff. In[2], an FIR filter with 256 taps was adopted, while [20] useda Butterworth filter with three poles to simulate the fadingdistortion. In our simulations, was modeled as a low-pass autoregressive process of order , with real multiplepoles at It can be shown that the 3-dB bandwidth of such afilter (i.e., the so-called Doppler spread that in Section II wedenoted by ) is related to the multiple pole through theequation

Note that our algorithms put no constraint on the distributionof the fading distortion. Under (AS1)–(AS5), consistency ofthe proposed and estimators is guaranteed irrespectiveof the distribution of Only for simulation purposes,

we assumed a Rayleigh channel, with zero-mean com-plex circular Gaussian (w.l.o.g. we assumed also that

). The signal-to-noise ratio (SNR) was defined as

1) Frequency Offset Estimation—(24) Versus (29):An esti-mation interval of symbols was assumed. Estimatesof the cyclic correlation were obtained as in (15), withsamples per symbol interval, for a total ofsamples. The other parameters were chosen as(fast fading), , and The results reportedin Figs. 1 and 2 were obtained by averaging over 400 MonteCarlo trials. Fig. 1(a) shows the bias and Fig. 1(b) shows themean square error (MSE) of the frequency offset estimator

(normalized to the symbol rate) versus SNR for differentvalues of Fig. 2(a) shows the bias and Fig. 2(b) showsthe MSE of versus at three different SNR’s. Exceptfor , the proposed estimator in (24) outperforms theone in (29), especially at lower SNR’s and greater frequencydrifts. From the same set of simulations we derived the plotsin Fig. 3 that depict MSE of versus SNR, for ,

, and [Fig. 3(a)] different values of or [Fig. 3(b)]different values of the oversampling factor Note that theS–O estimator (stars) does not depend on, so only oneplot for S–O is reported in Fig. 3(b). Larger impliesmore averaging in (25) but also more noise in estimating

, which justifies the tradeoff observed in Fig. 3(a).


(a) (b)

Fig. 3. (a) MSE offeT versus SNR (dB). Estimator (24):Lg = 8 (solid line),Lg = 16 (dashed line),Lg = 24 (dash–dot line). S–O: stars. (b) MSE offeT

versus SNR (dB). Estimator (25):P = 4 (solid line),P = 8 (dashed line),P = 12 (dash–dot line). S–O:P = 4 (circles),P = 8 (�-marks),P = 12 (stars).

Fig. 4. (a) MSE of� versus SNR (dB) for different values of�: Estimator (25):� = 0:125 (solid line), � = 0:5 (dashed line),� = 0:875 (dash–dot line).S–O: � = 0:125 (stars),� = 0:5 (circles), and� = 0:875 (�-marks). (b) MSE of� versus� at SNR = 8 and16 dB. Estimator (25):SNR = 0 dB (solidline), SNR = 12 dB (dashed line),SNR = 12 dB (dash–dot line). S–O:SNR = 0 dB (circles),SNR = 12 dB (�-marks), andSNR = 24 dB (stars).

(a) (b)

Fig. 5. MSE of (a)� and (b)feT versus SNR (dB) for different values of the Doppler spread,feT = 0:1; � = 0:5: Estimators (24) and (25):B�T = 0:001(solid line),B�T = 0:01 (dashed line),B�T = 0:05 (dash–dot line). S–O:B�T = 0:001 (stars),B�T = 0:01 (circles),B�T = 0:05 (�-marks).

The improvement with increasing observed in Fig. 3(b)is also expected since the effective data length increases as

increases. Nevertheless, the improvement is negligible forhigher sampling rates and seems to be a good tradeoffbetween performance gain and implementation complexity.

2) Time Epoch Estimation—(25) versus (30):Fig. 4(a)shows the MSE of versus SNR for different values of,and Fig. 4(b) depicts the MSE of versus for differentvalues of SNR. The other parameters of the simulation were

, , and In Fig. 5(a) we plot

the MSE of and in Fig. 5(b) the MSE of , versus SNRfor , , and different Doppler spread values

(very slow fading), (slow fading),and (fast fading). Especially the performance ofthe timing estimator is considerably improved and justifiesthe extra computations involved in the proposed estimatorrelative to S–O.

3) Time Epoch Estimation—(28) versus (30):Fig. 6 showsthe results of comparing the time epoch estimators (28) and(30), assuming an estimation interval of symbols and


(a) (b)

Fig. 6. (a) Bias and (b) MSE of� versus SNR (dB) for different values of the frequency offset. Estimator (28):feT = 0:0 (solid line),feT = 0:1 (dashedline), feT = 0:2 (dash–dot line). S–O:feT = 0:0 (circles), feT = 0:1 (�-marks),feT = 0:2 (stars).

(a) (b)

(c) (d)

Fig. 7. Cyclic spectrumS2x(1; f) estimated from (16) with (a) unknown 4-QAM and (c) known preamble, and from (18) with (b) unknown 4-QAMand (d) known preamble.

averaging over 400 Monte Carlo trials. The other parametersof the simulation were , , , ,and About an order of magnitude improvementis observed in both frequency offset and timing estimationwith the proposed estimator which is FFT-based and thuscomputationally efficient as well.

Fig. 7(a) and (c) shows estimated via (16), whileFig. 7(b) and (d) depicts estimates computed as in (18). ForFig. 7(a) and (b) a 4-QAM modulation was used (nondata-aided case), while Fig. 7(c) and (d) refer to the case wherea particular training sequence is used. It is evident that the

random modulation reduces the resolution in For bothestimates in (16) and (18), a Kaiser window with parameter

[14, Ch. 7] was used. The other parameters of thesimulation were , , ,dB, , , , Note alsothat, as predicted by (27), peaks (approximately) at

To the best of the authors’ knowledge, [18] and [10] are theonly works dealing with nondata-aided open-loop algorithmsfor joint frequency offset and time epoch estimation in flat-fading channels. In [18], the effect of on the estimators


(a) (b)

Fig. 8. (a) Bias (a) and (b) MSE offeT versus SNR (dB) without fading(B�T = 0:0): Estimator (27) with known preamble:feT = 0:0 (solid line),feT = 0:1 (dashed line), andfeT = 0:2 (dash–dot line). M–M with known preamble:feT = 0:0 (circles),feT = 0:1 (�-marks), andfeT = 0:2 (stars).

(a) (b)

Fig. 9. (a) Bias and (b) MSE offeT versus SNR (dB) in the presence of fading(B�T = 0:05): Estimator (27) with known preamble:feT = 0:0 (solid line),feT = 0:1 (dashed line), andfeT = 0:2 (dash–dot line). M–M with known preamble:feT = 0:0 (circles),feT = 0:1 (�-marks), andfeT = 0:2 (stars).

is not evident (in their analytical derivation the authors ignorefading). In [10], the fading distortion was considered constantover the entire burst (i.e., for ),so that , where denotes the Dirac delta(note that in this case (AS5) is satisfied trivially).

Many works exist on data-aided open-loop frequency errorestimation in AWGN (see, e.g., [9] and [11] and referencestherein), but fading effects are not included. In the nextsubsection, we will compare the estimator (27) with theone proposed in [11] [Mengali–Morelli (M–M)], in orderto quantify the performance loss in (27) with respect to analgorithm tailored to a specific data-aided situation. On theother hand, we explore how performance improves by using(27) instead of the M–M estimator when time-selective fadingis present.

C. Comparison with M–M

With the notation adopted so far, the M–M estimator canbe expressed as [11]

(31)

where is theunbiased sample autocorrelation sequence of the datacollected by sampling at the symbol rate, andis a window of length The estimator (31) assumes apreamble of known data and accurate timing information. Tosatisfy these assumptions, we chose as zero-mean preamble

, as requiredby (AS1). This preamble could come from a 4-phase-shiftkeying (PSK) constellation (as in [11]) or from a 4-QAMconstellation.

Figs. 8 and 9 compare (27) with (31) under the assump-tion of perfect timing (even if (27) does not require timingrecovery), known preamble, no fading effects, and AWGN

For the plots in Fig. 8 we averaged over 200 MonteCarlo trials with

As expected, the performance of the M–Malgorithm is slightly better, although with training symbolsthe performance of (27) is close to that of M–M. Furthermore,there is no need to change the structure of the estimator in(27) so long as the preamble is zero mean. M–M performsbetter because it corresponds to an approximate ML approachunder the above assumptions. In [11] it is also shown thatthe M–M algorithm is efficient, so the corresponding MSEshown in Fig. 8(b) coincides with the CRLB for AWGNchannel. However, simulations show that the performance of


(a) (b)

Fig. 10. (a) Bias and (b) MSE offeT versus SNR (dB) in the presence of fading(B�T = 0:05): Estimator (24):feT = 0:0 (solid line), feT = 0:1(dashed line), andfeT = 0:2 (dash–dot line). Estimator (27):feT = 0:0 (circles),feT = 0:1 (�-marks), andfeT = 0:2 (stars).

(31) degrades more rapidly than that of (27) or (24). Thisis illustrated in Fig. 9, under perfect timing recovery, knownpreamble and AWGN, but in the presence of fading distortionwith Doppler spread The M–M algorithm doesnot guarantee consistent estimates in this case, while ouralgorithm does and, from this point of view, (27) and (24) mayhave more general use. Moreover, they do not employ trainingdata, but require more samples relative to those required bydata-aided algorithms (typically in the order of 100 symbols[11]).

It is interesting now to explore the relationship betweenestimators (31) and (24), which at first glance is not evident.Recall that the sequence is stationary and is related tothe oversampled CS sequence in (2) via3 ;hence,

Let us now derive the expression of the zero-cyclecyclic moment that corresponds to the time-invariant (station-ary) component of From (15), we have

, where is the number ofsymbols. Thus, we can write

where we defined; can be thought of as

the correlation of the th stationary componentof the CS It is thus evident that

3In [11] multiplication byw�(n) is also included (so thatw�(n)w(n) =�2

w= 2) in order to remove the linear modulation. We can incorporate these

terms w.l.o.g. inx(n):

and hence

(32)

Note that we could use the estimator (32) withinstead of in order to reduce additive and patternnoise. The price to be paid is in terms of complexity, since

requires oversampling by a factor The linkbetween (24) and (31) is now evident if we recall (5), wherefrequency offset is present in the form of a complex harmonicw.r.t. the lag From (32), we see that the M–M algorithmexploits the fact that frequency offset generates a phase shift

among adjacent values of the correlation sequencecomputed at the symbol rate. It uses only the zeroth cyclesince the data are stationary. The summation w.r.t. the lagin(32) and the window serve the purpose of reducingestimation variance. Our estimator in (24) also exploits thephase shift due to , but invokes cycles different from zeroto obtain an estimator tolerant to stationary additive noise(whatever is the noise color or/and the distribution). Further-more, we have shown that once the frequency offset has beenestimated, timing recovery is possible with minimal overhead.This estimate can represent a reasonable initialization for moresophisticated nonlinear timing estimators, such as the oneproposed in [5], that require good initial estimates to avoidconvergence to local minima.

Finally, in Fig. 10 we compare (27) and (24). The plotswere generated with 4-QAM i.i.d. symbols and estimates wereaveraged over 400 Monte Carlo runs. The other parameterswere , , , ,

, and In this case the cyclic-correlation-based method exhibited better performance, and additionalnumerical results (not reported here) have illustrated that thecyclic spectrum outperforms cyclic-correlation-based methodsin the data-aided case. With regard to oversampling and noisespectrum shaping, the following remark is of interest.

Remark 3: In all of our simulations we assumed rolloff fac-tor , so that had bandwidthAs a consequence, the noise samples , obtained by sam-pling with a rate at the output of , were cor-


related. Here, we reiterate that the proposed algorithms arem.s.s. consistent independent of the noise color; thus, we donot need to use a receiving filter with bandwidth as in[17], or as in [2], in order to obtain uncorrelated noisesamples. Note that increasing the bandwidth of allowsmore noise power to pass, thereby reducing the SNR at thesampler.

Remark 4: Comparing Fig. 8(b) with Fig. 9(b), it is clearthat the multiplicative effect due to fading (or pattern) noiseinfluences MSE performance more so than the AWGN (notethat data (or self) noise is present in both cases). Fadingcauses a floor effect in the MSE curves [c.f. Figs. 9(b) and10(b)] reminiscent to that encountered with laser phase noisein optical communications. An algorithm capable of trackingand removing fading effects could remove such an error floorand constitutes an interesting future direction.

D. A Unifying CS Viewpoint

Although cyclostationarity induced by oversampling hasbeen implicitly used in the past to estimate synchronizationparameters, the role of cyclic statistics in synchronization hasnot been studied systematically. Formulating the estimationproblem in a more general framework allows not only thor-ough understanding of existing solutions but also suggests newdirections to improve performance when, e.g., colored possiblynon-Gaussian additive noise or fading effects are present.

To illustrate this point further, consider the well-knownOerder–Meyr synchronizer [13] (used also in [11]). With ournotation, it can be expressed as

(33)

The estimator in (33) uses only the zeroth lag (w.r.t.of thecycle-frequency In contrast, (25) exploits thefull information available in the data.

So far we have considered only second-order statistics of, but it is clear that our analysis can be extended to

higher (than second)-order cyclic statistics. Joint nondata-aided and estimators based on the fourth-order cyclicmoment, were proposed in [10] for minimum-shift keying(MSK) modulation. In particular the timing estimator readsas follows:

(34)

with , and

(35)

The presence of in (34) removes If, on the other hand,or it has been estimated and removed from the data,

can be dropped from (34). In this case, inserting (35) into

(34), we obtain

(36)

Making use of the identity in footnote 1, we can express theestimator (34) as

(37)

where, by definition,

is the fourth-order cyclic moment ofthe process ; is periodic with period and isestimated via

(38)

Under (AS4), in (38) is asymptotically unbiased andm.s.s. consistent [4], and the same property carries over to

in (37). Note that this is true whatever the color and/orthe distribution of the additive noise , provided that itis stationary and mixing. This is an interesting property notacknowledged in [10].

Once more, viewing synchronization parameter estimateswithin the context of CS statistics gives access to a wealth oftools useful to understand statistical properties of estimationalgorithms and suggests means of improving them, e.g., weexpect that the estimator (37) can be improved by looking atthe “global” relationship between the time epochand thefourth-order cyclic moment By exploitinginformation present in lags different thanand , or, by looking at cycles other than , weexpect noticeable performance gain at the expense of moderateincrease in computations.

VII. CONCLUSIONS AND DISCUSSION

In this paper we proposed methods for estimating fre-quency offset and symbol timing of a linearly modulatedwaveform transmitted through a frequency-flat fading channel.The methods relied on the full exploitation of the receivedsignal’s cyclostationarity, which is introduced in the datawhen oversampling the matched filter output. The resultingestimators have desirable features, as they do not requiretraining data and, under mild conditions, they are m.s.s.consistent independent of the color and distribution of thefading distortion or the additive noise. We also illustrated thelinks with existing estimators relying on cyclostationarity. Thisunifying interpretation allowed us to establish consistency andsuggested means of improving them.


Thorough performance analysis including (even asymptotic)variance expression of and is an interesting researchdirection along with possible extensions to frequency-selectivefading environments. Additional topics include nonlinear least-squares criteria weighted with the inverse covariances ofto yield estimators with smallest asymptotic variance amongall estimators employing sample cyclic correlations.

ACKNOWLEDGMENT

The first author would like to thank Prof. U. Mengali forintroducing him to the synchronization problem, for bringinghis attention to [18], and for kindly providing a draft versionof [11].

REFERENCES

[1] W. R. Bennett, “Statistics of regenerative digital transmission,”BellSystem Tech. J., vol. 37, pp. 1501–1542, 1958.

[2] W. C. Dam and D. P. Taylor, “An adaptive maximum-likelihood receiverfor correlated Rayleigh-fading channel,”IEEE Trans. Commun., vol. 42,pp. 2684–2692, Sept. 1994.

[3] A. V. Dandawate and G. B. Giannakis, “Nonparametric polyspectral es-timators forkth-order (almost) cyclostationary processes,”IEEE Trans.Inform. Theory, vol. 40, pp. 67–84, Jan. 1994.

[4] , “Asymptotic theory of mixed time averages andkth-ordercyclic-moment and cumulant statistics,”IEEE Trans. Inform. Theory,vol. 41, pp. 216–232, Jan. 1995.

[5] A. N. D’Andrea, U. Mengali, and M. Morelli, “Symbol timing esti-mation with CPM modulation,”IEEE Trans. Commun., vol. 44, pp.1362–1371, Oct. 1996.

[6] M. P. Fitz, “Further results in the fast frequency estimation of a singlefrequency,”IEEE Trans. Commun., vol. 42, pp. 862–864, Feb./Mar./Apr.1994.

[7] H. L. Hurd, “Nonparametric time series analysis of periodically cor-related processes,”IEEE Trans. Inform. Theory, vol. 35, pp. 350–359,Mar. 1989.

[8] R. S. Kennedy,Fading Dispersive Communication Channels. NewYork: Wiley, 1960.

[9] M. Luise and R. Reggiannini, “Carrier recovery in all-digital modemsfor burst-mode transmissions,”IEEE Trans. Commun., vol. 43, pp.1169–1178, Feb./Mar./Apr. 1995.

[10] R. Mehlan, Y. Chen, and H. Meyr, “A fully digital feedforward MSKdemodulator with joint frequency offset and symbol timing estimationfor burst mode mobile radio,”IEEE Trans. Veh. Technol., vol. 42, pp.434–443, Nov. 1993.

[11] U. Mengali and M. Morelli, “Data-aided frequency estimation for burstdigital transmission,”IEEE Trans. Commun., vol. 45, pp. 23–25, Jan.1997.

[12] R. van Nobelen and D. P. Taylor, “Analysis of the pairwise errorprobability of interleaved codes on the Rayleigh-fading channel,”IEEETrans. Commun., vol. 44, pp. 456–463, 1996.

[13] M. Oerder and H. Meyr, “Digital filter and square timing recovery,”IEEE Trans. Commun., vol. 36, pp. 605–611, May 1988.

[14] A. V. Oppenheim and R. W. Schafer,Discrete-Time Signal Processing.Englewood Cliffs, NJ: Prentice-Hall, 1989.

[15] J. G. Proakis,Digital Communications, 3rd ed. New York: McGrawHill, 1995.

[16] T. S. Rappaport,Wireless Communications: Principles and Practice.Englewood Cliffs, NJ: Prentice-Hall, 1996.

[17] J. Riba and G. Vazquez, “Bayesian recursive estimation of frequencyand timing exploiting the cyclostationarity property,”Signal Process.,vol. 40, pp. 21–37, Jan. 1994.

[18] K. E. Scott and E. B. Olasz, “Simultaneous clock phase and frequencyoffset estimation,”IEEE Trans. Commun., vol. 43, pp. 2263–2270, July1995.

[19] M. Tsatsanis, G. B. Giannakis, and G. T. Zhou, “Estimation and equal-ization of fading channels with random coefficients,”Signal Processing,vol. 53, pp. 211–229, Sept. 1996.

[20] G. M. Vitetta, D. P. Taylor, and U. Mengali, “Double-filtering re-ceivers for PSK signals transmitted over Rayleigh frequency-flat fadingchannel,”IEEE Trans. Commun., vol. 44, pp. 686–695, June 1996.

Fulvio Gini , (M’93) for photograph and biography, see p. 60 of the January1998 issue of this TRANSACTIONS.

Georgios B. Giannakis(S’84–M’86–SM’91–F’97)received the Diploma degree in electrical engi-neering from the National Technical University ofAthens, Athens, Greece, in 1981, and the M.Sc.degree in electrical engineering, the M.Sc. degreein mathematics, and the Ph.D. degree in electricalengineering from the University of Southern Cali-fornia, Los Angeles, CA, in 1983, 1986, and 1986,respectively.

After lecturing for a year at USC, he joined theDepartment of Electrical Engineering, University

of Virginia, Charlottesville, where he is currently a Professor. His generalinterests are in the areas of signal processing, communications, estimationand detection theory, and system identification. His current specific researchinterests are focused on diversity techniques for channel estimation andmultiuser communications, nonstationary and CS signal analysis, waveletsin statistical signal processing, and non-Gaussian signal processing withapplications to SAR, array, and image processing.

Dr. Giannakis recieved the IEEE Signal Processing Society’s 1992 PaperAward in the area of Statistical Signal and Array Processing (SSAP). Heco-organized the 1993 IEEE Signal Processing Workshop on Higher OrderStatistics, the 1996 IEEE Workshop on Statistical Signal and Array Processing,and the first IEEE Signal Processing Workshop on Wireless Communicationsin 1997. He was the co-Guest Editor of two special issues on high-orderstatistics (International Journal of Adaptive Control and Signal Processingand the EURASIP journalSignal Processing) and the January 1997 IEEETRANSACTIONS ON SIGNAL PROCESSINGSpecial Issue on Signal Processing forAdvanced Communications. He has served as an Associate Editor for theIEEE TRANSACTIONS ONSIGNAL PROCESSINGand the IEEE SIGNAL PROCESSING

LETTERS, as Secretary of the Signal Processing Conference Board, and as aMember of the Signal Processing Publications Board and the SSAP TechnicalCommittee. He is a member of the IMS and the European Association forSignal Processing.

Date post:	03-Feb-2022
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Frequency Offset And Symbol Timing Recovery In Flat-fading

Documents