ieee transactions on signal processing, vol. 53, no. 3

IEEE

Proo

f

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005 1

Timing Phase Offset Recovery Basedon Dispersion Minimization

Wonzoo Chung, William A. Sethares, and C. Richard Johnson, Jr., Fellow, IEEE

Abstract—This paper presents a blind timing phase offset re-covery scheme that attempts to optimize the baud spaced equalizeroutput mean square error (MSE) for a realistic equalizer lengththat is usually shorter than the ideal length. Among the existingblind timing recovery schemes, few are designed for equalizeroutput MSE optimization, and none are designed for the realisticcase when the equalizer is short. The proposed algorithm (thatis based on a cost function that minimizes the dispersion of thereceived signal) attempts to minimize the MSE of a one-tapequalizer output. It also exhibits good performance for relativelyshort equalizers. Conditions for the unimodality of the dispersionminimization cost are investigated, and a geometric relationship tothe minimum MSE (MMSE) timing offset is shown qualitatively.The detailed MSE performance of the algorithm is investigated forthe representing classes of channels by comparing existing blindtiming offset estimation schemes.

Index Terms—Adaptive blind synchronization, MMSE timingoffset, timing offset recovery.

I. INTRODUCTION

I N most digital communication systems, digital sourcesare conveyed via excess bandwidth analog pulses that

are designed in satisfaction of the Nyquist Criterion [1] toprevent aliasing for a proper sampling offset in the absenceof a multipath channel. Due to aliasing, the performance ofthe receiver with a symbol-spaced equalizer is sensitive tothe sampling offset, especially in the presence of multipathinterference, where there is (in general) no sampling offset thatcan perfectly eradicate aliasing. Furthermore, for the scenariowhere multisensor outputs are combined in order to increaseSNR, a proper sampling offset for each sensor can contributeto significant performance improvement [2]. In both cases,estimation of an optimal timing offset is an important task, butthe criterion for “optimality” is not obvious when the timingoffsets must be determined at the synchronization stage. Anatural approach is to minimize the overall MSE performanceat the input to the detection device, i.e., to minimize the MSE

Manuscript received November 14, 2003; revised February 12, 2004. W.Chung and C. R. Johnson Jr. were supported in part by the National ScienceFoundation under Grant ECS-9811297 and NxtWave Communications (nowATI). The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Gregori Vazquez.

W. Chung is with Dotcast Inc., Kent, WA 98032 (e-mail: [email protected]).

W. A. Sethares is with the Department of Electrical and Computer En-gineering, University of Wisconsin, Madison, WI 53706 USA (e-mail:[email protected]).

C. R. Johnson, Jr. is with the School of Electrical and Computer Engineering,Cornell University, Ithaca, NY 14853 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2004.842211

at the output of the equalizer [3]. Unfortunately, direct esti-mation of the timing offset that minimizes the overall MSErequires knowledge of the analog channel, and the procedureis computationally complex. Adaptive approaches based ontraining sequences or decisions from detection devices can beused in place of direct computation, but directly coupling theupdates of the timing offset and the equalizer parameters isconsidered undesirable in actual implementation for a varietyof reasons [4]. Since reliable reference or decision data is oftenunavailable during the acquisition phase, the estimation ofthe synchronization parameters benefits from the use of blind(nondata aided) methods. In order to avoid the sensitivity ofsampling offset in symbol spaced processing, sampling abovethe Nyquist rate of the analog pulses (so-called fractionallysampling) may also be used [5]. However, fractionally spacedsampling and equalization is not always feasible, especiallywhen the additional hardware requirements cannot be met. Forexample, in digital audio/video broadcasting, the signals areoften transmitted over a channel with delay spread that is toolong to be equalized with a fractionally spaced equalizer usinga restricted number of taps [6].

Therefore, a desirable timing offset estimation scheme for asymbol spaced receiver should i) be blind, ii) not rely on theequalizer output, and, at the same time, iii) optimize the MSEof the equalizer output.

Among the existing schemes for timing phase offset re-covery, the output energy maximization (OEM) method comesclosest to satisfying these three conditions. The algorithmfinds a timing offset that maximizes the sampler output energyand consequently maximizes SNR. In [4], it was shown thatmaximizing SNR is approximately equivalent to minimizingthe overall MSE of the receiver when equipped with an infinitelength equalizer. In practice, however, the equalizer length isalways constrained. When the equalizer is short, the timingoffset at that the SNR is maximized may be significantlydifferent from the timing offset at that the MSE is minimized.Thus, OEM may not near optimal for a short equalizer.

This paper examines an alternative blind approach to timingoffset estimation (see also [7]) that attempts to minimize MSEfor a short equalizer, as OEM attempts to optimize MSE for asufficiently long equalizer. The alternative timing offset estima-tion is designed to choose a timing offset minimizing the dis-persion of the sampler output. Simulation results in [2] and [7]suggest that the MSE performance of this dispersion minimiza-tion timing offset estimation method is improved over the OEMmethod for the practical case when the equalizer length is short.This paper presents an analysis of the dispersion minimization

1053-587X/$20.00 © 2005 IEEE

IEEE

Proo

f

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005

timing offset as an approximation of the minimum mean squareerror (MMSE) timing offset for a one-tap equalizer.

Section II defines the optimal timing offset as a timing offsetminimizing MSE of the equalizer output. An existing blindtiming algorithm (OEM-timing) that is optimized for infinitelength equalizers is compared with a training-based timingalgorithm optimized for a short length (one-tap) equalizer.The importance of a timing offset optimized for the one-tapequalizer is emphasized.

Section III proposes a blind adaptive approach for the timingoffset that attempts to minimize the MSE of the one-tap equal-izer output (called DM-timing) and analyzes its cost function.Conditions under which the timing estimation is unimodal arederived, and a geometrical interpretation of the relation betweenDM-timing and OEM-timing is presented.

Section IV investigates how well this DM-timing offsetapproximates the MMSE timing offset for a one-tap equal-izer. Due to the complexity of the closed-form analysis forDM-timing cost function of a general channel, quantitativeanalysis is performed for the representative classes of channels,pure delays channels, clustered channels, and sparse channels.The investigation and simulation results show that DM-timingoffset quite well approximates MMSE timing offset optimizedfor the MMSE one-tap equalizer for mild multipath channel andalways performs better than OEM timing offset for the one-tapMMSE equalizer. The final section concludes the paper.

II. LINEAR EQUALIZER OUTPUT MSE AND TIMING

This section reviews the problem of timing offset in digitalcommunication systems and defines the optimal timing offsetwith respect to the equalizer output MSE. Consider a digitalbaseband communication system in Fig. 1, where the digitalsources drawn from a finite alphabet set are converted to ananalog signal by convolving with the pulse shape function

(1)

where is referred to as the symbol period. The transmittedsignal is subject to distortion due to multipath and noise. Themultipath is characterized by a finite impulse response

(2)

where and denote the delay and the complex path gain ofthe th path, respectively. The noise is modeled as a wide sensestationary (w.s.s.) zero mean additive white Gaussian process

with variance that is uncorrelated with the signal .At the receiver, the received signal is processed by a matchedfilter with output that can be written

(3)

where the analog channel is

(4)

and where denotes the convolution operator. The matchedfilter output is sampled at the symbol rate with a timingoffset .

(5)For practical reasons, the pulse has excess bandwidth (i.e.,the maximum frequency in is larger than ). However,

is designed under the Nyquist criterion [1] so that thesampled signal does not have aliasing for the perfect timingoffset in the absence of a multipath channel.

To examine the effect of the timing offset on the sampledsignal in the presence of a multipath channel, assume that the re-ceived filter is matched to the pulse shape andthat the pulse shape is symmetric (for ex-ample, a square root raised cosine filter [1]). Then, the samplednoise component , which is denoted by

, becomes a white zero mean Gaussian process with vari-ance independent of from the property of the symmetricNyquist filter and the whiteness of . On the otherhand, the signal component is dependent on the timing offset.By defining the sampled channel with respect to the samplingoffset as a vector , i.e.,

(6)

the sampled output can be written in the discrete time domain as

(7)

In the absence of multipath, is given by

(8)

and thus, the correct timing offset perfectly restores thedigital data . In the presence of multipath, cannot bereduced to a single nonzero tap in general. Furthermore, thealiasing due to excess bandwidth of can deepen the spec-tral null of the sampled channel , depending on the timingoffset , and can effect the MSE performance of the equalizer[8]. Thus, in the presence of multipath, it is desirable to deter-mine jointly the timing offset and the equalizer.

Consider an MMSE equalizer for a given channel withrespect to a given timing . The optimal timing offset isto minimize the MSE of the equalizer output in thepresence of a multipath channel, i.e.,

(9)

where the delay applied to the source (or training) sequenceis called the system delay. This definition of timing offset

IEEE

Proo

f

CHUNG et al.: TIMING PHASE OFFSET RECOVERY BASED ON DISPERSION MINIMIZATION 3

Fig. 1. Digital communication systems model.

optimality can be extended to the problem of combined mul-tiantenna inputs. In this form, determining the optimal timingoffset can be viewed as a nonlinear quadratic optimizationproblem over the channel vectors under a constraint imposedby the parameter . This problem is difficult because a direct up-date for the timing offset based on feedback from the equalizeroutput is undesirable not only from an implementation point ofview [4] but also due to the latency caused by the convergenceof the equalizer. Furthermore, a closed-form estimation ofthe optimal timing offset is infeasible due to the extremelycomplicated nonlinear relation between the parameter andthe resulting sampled channel .

However, relatively simple solutions exist for the extremecases of equalizer length; for a one-tap equalizer, ,and for an infinite length equalizer, . For the MMSEequalizer of length , , the MMSE with respect to thesystem delay and timing is given by

MMSE (10)

MMSE (11)

where is the noise-to-signal ratio defined by

(12)

A detailed derivation of (10) and (11) can be found in [9].Now, consider the timing offset for a one-tap linear

equalizer. Since the delay term appears only in inMMSE , the attainable minimum MMSE withrespect to the is given by MMSE , andhence, the optimal timing offset that minimizes the MMSE forthe one-tap equalizer is given by

(13)

where denotes the norm.The MMSE-timing function (13) provides a geometrical in-

terpretation of the timing offset optimization for a one-tap equal-izer. Consider normalized ,

(14)

lies on an -dimensional sphere as varies from 1/2to 1/2, and is chosen so that (13) is as large as possible,which occurs when is maximized. Thus, the effect of

using the optimal timing offset for a one parameter equalizercan be seen directly by looking at the taps of . Intersymbolinterference (ISI) is minimized by maximizing the magnitude ofthe tap corresponding to the system delay and, at the same time,suppressing all others.

On the other hand, for the timing offset optimized for infinitelength equalizers, in [4], the following approximation of (11) isproposed:

MMSE (15)

In this case, minimizing MMSE is equivalent to maximizingthe channel power and, consequently, improving thesignal (including ISI components) to noise ratio, since an in-finite length equalizer can perfectly restore signals from ISI.

Apparently, increasing total signal power does not necessarilyimprove the equalizer output MSE, especially when the multi-path is severe and the equalizer is not long enough to mitigatethe multipath. The following example confirms that applying thetiming offset optimized for an infinite length equalizer to a finiteequalizer (15) performs worse than the timing offset optimizedfor the one-tap equalizer (13).

Example 1: Consider a multipath channel

(16)

using a raised-cosine pulse-shaping filter with a roll-off factorunder SNR dB. Numerical calculation shows that

the optimal timing offsets for , and the optimaltiming offset for , are given by

(17)

For these two timing offsets, Fig. 2 plots the MSE of a) MMSElinear of equalizers of all lengths from 1 to 25 and b) the MMSEdecision feedback equalizer (DFE) with feedback filter of fixedlength 12 and feedforward equalizers with all lengths from 1to 25. The optimal timing offset for each finite equalizer lengthhas been calculated numerically, and its MMSE equalizer MSEis depicted as a dotted line. In all cases, when the timing is op-timized for a one-tap equalizer, the performance is better thanwhen it is optimized for an infinite-length equalizer.

This example does not represent a peculiar case. In fact, onecan confirm this trend for a set of randomly generated chan-nels, as reported in [10]. The next section presents an adaptivescheme approximating the timing offset for the one-tap equal-izer without help of known training sequence.

III. TIMING OFFSET BASED ON DISPERSION MINIMIZATION

A. DM-Timing Offsets

Instead of directly minimizing the MSE, which requirestraining signal or analog channel estimation, [7] has proposedminimizing the mean dispersion of the sampler output signal,i.e.,

(18)

IEEE

Proo

f


Fig. 2. MMSE equalizer output MSE and Timing offset. (a) Linear equalizer. (b) DFE with feed back filter length 12.

where is a constant corresponding to the source statistics. (; for details, see [11].) The cost function is known

as the constant modulus (CM) cost and is widely used for blindequalization [11]. In order to approximate MMSEwith respect to a one-tap equalizer , we proposed to rewrite thecost function (18) as

(19)

as illustrated in Fig. 3. Using a stochastic gradient algorithm[12] to minimize (19) by recursive selection of and givesthe dispersion minimization (DM) timing algorithm

(20)

(21)

where can be obtained numerically when a closedform is unavailable.

Expansion of the DM-cost function (19) provides detailed in-formation on the DM timing offsets. Setting andassuming , the DM cost function (18) is given by thefollowing (for the derivations, see Appendix A):

(22)

where and are called kurtosis of the source and the noise, respectively, and are defined by

(23)

(24)

where denotes the norm. Note that for a realGaussian and for a complex Gaussian . Since

Fig. 3. Dispersion minimization timing offset.

the cost function (22) is a quadratic function of , there existsa unique optimum for each . Using this fact, (22) can bewritten as a function of only , which leads to the followingresult.

Theorem 1 (DM-Minima): The DM-timing offset for a singlesensor is given by

(25)

Proof: See Appendix B.There is a similarity between MMSE-timing and DM-timing.

Recall the notation . In the absence of noise,the optimal MMSE timing offset for a one-tap equalizer is givenby , whereas the optimal DM-timing offsetis . The only difference between these twotiming offsets is the norm used ( or ). In both cases, with

constrained to , the optimal lies on theunit ( or ) sphere.

B. Unimodality Condition of DM-Timing

This section establishes a concrete set of conditions under thatthe DM cost function bears a unique maximum, which seems to

IEEE

Proo

f


TABLE IDM-TIMING FUNCTION COEFFICIENTS VERSUS ROLL-OFF FACTOR OF RAISED COSINE

be satisfied in most practical situations. The immediate impli-cation of this result is that the algorithm’s asymptotic behavioris insensitive to the initial conditions. Call

(26)

the DM-timing cost. The following analysis focuses on the uni-modality of . Let denote the Fourier transform of ,which is the analog channel, and denote

(27)

where indicates convolution operator. From the Fourier seriesexpansion of and

(28)

(29)

Assuming , since is bandlimited, we have

for for (30)

From (28) and (29)

(31)

(32)

The values of are determined by the multipathchannel and the pulse-shaping filter . In practice, acommonly used pulse-shaping filter is a raised cosine functionwith spectrum

(33)where is the roll-off factor determining the excess bandwidth.A roll-off factor from 0.1 to 0.3 is commonly used in practice(for example, [13] and [14]).

For a multipath , is givenby

(34)

Since for given complex func-tions and , we have

(35)

(For the detailed derivation, see Appendix E.) For a roll-offfactor from 0.1 to 0.3, the tails of are quite small, and con-sequently, the values , , and are vanish-ingly small in comparison with other values such as and

(see Table I). Thus, a good approximation of is

(36)

which simplifies the DM-timing function to

(37)

For the rest of this paper, we use the approximation (36) forfurther investigation in order to avoid extremely difficult com-putational complexity due to the high-frequency term in (32).

Theorem 2 (Unimodality): Denote

(38)

Then

(39)

is unimodal, i.e. has a unique maximum, when the followingcondition holds:

(40)

Proof: See Appendix C.Notice that the condition (40) in Theorem 2 is satisfied when

the tails and decrease rapidly. Table I shows thatfor small roll-off factors , (40) is satisfied, which impliesthat the DM-algorithm has only one minimum. This holds for

IEEE

Proo

f


any multipath channel. In fact, (40), which is given in Theorem2, is a loose bound. A tighter bound can be found under slightlymore complicated conditions, as shown in [10].

IV. PERFORMANCE OF DM-TIMING

The MSE performance of the dispersion minimization ap-proach has been intensively studied for the equalization appli-cations under the assumption that perfect equalization is pos-sible [15], [16]. It has been shown that the dispersion minimiza-tion approach can achieve near-MMSE performance. However,this result cannot be applied to the timing offset problem here,since the channel vectors parameterized by timing offsets areconstrained. In fact, the geometrical arguments used to showthe near-optimal MSE performance of dispersion minimization[17], [18] fail for the constrained cases [10]. On the other hand,due to the nonlinearity of the cost functions for DM-timingand MMSE-timing, it is a virtually impossible task to estab-lish a rigorous MSE bound for a general channel. Therefore, theMSE performance of DM-timing is investigated by comparingthe location of the three different timing offsets, DM-timing,MMSE-timing for a one-tap equalizer, and MMSE-timing offsetfor an infinite length equalizer (the existing blind timing offsetmethod). This comparison demonstrates the relative MSE per-formance of the three timing methods when used with shortequalizers and provides intuition about the different approaches.

Consider a multipath channel of rays with a raised-cosine pulse shape and its Fourier transformation, as given in(33) and (34)

Recall that and that ,, , and . Let

(41)

The MMSE timing offset for infinite length MMSE equalizer,which is denoted by , is given from (15)

(42)

On the other hand, the MMSE timing offset for the one-tapMMSE equalizer, which is denoted by , is given by

(43)

(44)

These formulas imply that is shifted fromtoward , and is shifted from to-ward . For example, satisfies (45), shown atthe bottom of the page. Notice that when ,

, and consequently, . Depending on thechannel, , , and agree or disagree each other. Ingeneral, for mild multipath channels, is a reasonableapproximation of , and thus, is close to ,whereas usually deviates from . As the severity of thechannel increases, tends to locate away from and closeto , and consequently, its MSE performance is degraded.In the next subsections, we will further investigate these MSEproperties of DM-timing for the cases where can beapproximated by a manageable function. To proceed further, weneed the following Lemma.

Lemma 1: Let and denote delays in the time domain.Then, we have

(46)

(47)

(48)

Proof: See Appendix D.

A. Case 1: Pure Delay

First, as a simple example, consider a pure delay channel, i.e.,. Since is a raised cosine delayed by

and sampled at , the one-tap MMSE equalizer as well as theinfinite length equalizer has the minimum MSE. On the otherhand

(49)

Since is real valued, . Thus, from (45),we have . In this pure delay case, , ,and agree with each other.

B. Case 2: Clustered Channels

Assume that the multipath channel given by

(50)

is clustered in the sense that .Since the delays are clustered, the peaks for each delay merge

(45)

IEEE

Proo

f


Fig. 4. DM-timing and MSE-timing for two-tap channels. (a) Timing offset locations. (b) MSE performance of timing offsets.

Fig. 5. DM-timing and MSE-timing for clustered channels. a) MSE deviation mean; b) MSE deviation variance.

into a single peak. This can be shown by the following approx-imation in the region :

sinc sinc (51)

where we have used

sinc

(52)

Therefore

for

(53)

for a magnitude and phase . From, and are given by

(54)

Thus, we have

(55)

for all .On the other hand

(56)

IEEE

Proo

f


Fig. 6. DM-timing and MSE-timing for mild multipath channels. (a) MSE deviation mean. (b) MSE deviation variance.

(57)

(58)

where, in (57), we have used property ii) of Lemma 1, and theassumption that the multipath channel is clustered. Similarly,since is again real and symmetric, Lemma 1 shows that

(59)

and consequently

(60)

(61)

Therefore, we have

(62)

This implies that and are shifted from the same lo-cation toward , and thus, they are located close eachother (close MSE performance), whereas may not be lo-cated close to (different MSE performance), as the simula-tion results in Section IV-D confirms.

C. Case 3: Sparse Channels

Now assume that the multipath channelis sparse in the sense that

for all (63)

Without loss of generality, is assumed to be the maximummultipath magnitude, and . Since the multipath is sparse,near the highest peak , the contribution from the tails ofother peaks can be ignored, i.e., locally at

(64)

Thus

(65)

On the other hand

(66)

(67)

where we have used property iii) of Lemma 1 and the assump-tion of channel sparsity. Similarly

(68)

IEEE

Proo

f


Fig. 7. DM-timing and MSE-timing for severe multipath channels. (a) MSE deviation mean. (b) MSE deviation variance.

and consequently

(69)

(70)

Notice that and are the nonlinear average of ’s weightedby the power of the magnitude of multipath. when

for (pure delay) and for all (the worstmultipath). For the other cases, is closer to thansince is smaller than . For example, in a two-tap channelcase, assuming , and are given by

(71)

and for mod , this is simply( , 4). Fig. 4(a) plots three different timing offset for

as increases 0 to 1. follows the trajectory offrom 0 to 1/8 , and is

shifted toward (or, equivalently, ) from. is located close to until and rapidly

approaches as the multipath becomes more severe. Thecorresponding MSE performance is shown in Fig. 4(b). Noticethat the MSE exhibits the same pattern for timing offset loca-tions in Fig. 4(a).

This two-tap example suggests that under mild channelsthe DM-timing offset yields near MMSE performance for theone-tap MMSE equalizer, and under severe multipath distor-tion, the DM-timing offset achieves better MSE performancethan the conventional OEM algorithm. The simulation resultsin the following subsection experimentally confirms this MSEperformance of DM-timing for general channels.

D. DM-Timing Offset MSE Performance Simulation

This subsection presents simulation results comparing MSEperformance of different timing offset , , and

for a one-tap equalizer under 30-dB noise. For the numberof multipath rays from 2 to 10, 10 000 different randomlygenerated multipath channels are used to compute mean andvariance of MSE difference, i.e., MSE MSE ,

MSE MSE , Var MSE MSE ,and Var MSE MSE . For the clustered scenario,the delays are confined to . For the mild channelscenario, the magnitude of the cursor tap is set to 1, and the restare confined to below 0.5 (less than 3 dB multipath), and forsevere multipath, the rest are set to above 0.5. The simulationresults shows the MSE performance of the DM-timing offsetpredicted in the previous subsections.

For clustered channels, mean and variance of the MSE differ-ence between and are significantly lower than that be-tween and . This indicates that is a good approx-imation for , whereas may not be for certain channels.

For the nonclustered channels, outperforms . Formild channels, the mean and variance are small, which indicatesthat is still a good approximation for . Since the mag-nitude of the multipath echos rarely exceeds 3 dB in terrestrialdigital broadcasting, can be used as a reliable blind replace-ment for .

V. CONCLUSION

The timing phase-offset recovery problem has been statedas a constrained channel optimization for a finite-lengthsymbol-spaced equalizer with respect to the MSE of the equal-izer output. A blind adaptive approach based on dispersionminimization can be used to achieve this optimality for a shortequalizer. A geometrical argument is presented that allowscomparison with the MMSE approach and issues of uni-modality of the DM-solution have been addressed. The MSEperformance of DM-timing has been investigated by comparingdispersion minimization timing offset with standard timingoffsets. DM-timing offset achieves close MMSE performancefor clustered multipath channels, regardless of the severity ofthe multipath. For sparse channels, the MSE performance ofDM-timing degrades as the severity of the multipath increases

IEEE

Proo

f


but still achieves better performance than the conventionaloutput energy maximization timing offset (see Figs. 5–7).

APPENDIX

A. Proof of (22): Expansion of DM-Cost Function

For a given parameter set , denote . Then, thecost function (22) is written as

(72)

First, from the fact is i.i.d.

(73)

where is the second moment, and is the fourth moment.Since we assume is circular when it is complex

complex casereal case

(74)Therefore

(75)

(76)

(77)

On the other hand

(78)

(79)

Similarly

complex casereal case.

(80)

Now, the cost function can be written as

Since , and , and . Furthermore,from the definition of kurtosis

(81)

Thus

(82)

(83)

B. Proof of Theorem 1

From (83) in Appendix A, the DM-cost function jointly witha scalar equalizer can be written as

(84)

Since the cost function is a quadratic of

(85)

Now, can be written with respect to

(86)

(87)

Since is negative, we have

(88)

C. Proof of Theorem 2

We use the following substitution:

(89)

so that we have

(90)

where . Furthermore, we denote

(91)

(92)

Since , the unimodality of is equivalent to that of. Notice that is bounded and periodic, and

. Therefore, it is sufficient to show that the derivative of

IEEE

Proo

f


has only two roots for : One of those isthe maximum, and the other one is the minimum. Consider thederivative

(93)

Because , it is enough to show that the numerator in(93) has only two roots.

(94)

We want to apply the following trigonometry identity to theabove equation:

(95)

In order to do that, we need to check that the coefficient termsof and in (94) are nonzero for some .

First, observe that

(96)

where, in the last step, we have used (40).Second, when , then (94) collapses into the fol-

lowing sinusoid:

(97)

and has only two roots, since the coefficient is nonzero from(96). Therefore, we can assume , which guaranteesthat for all , either

or (98)

Now, we have (99), shown at the bottom of the page, where

(100)

Since the coefficient of the term in (99) is nonzero, it issufficient to show that the term

(101)

has only two roots for in order to prove the theorem.Notice that is a continuous function of and

; thus

(102)

When increases strictly monotonically for, i.e.,

(103)

then

(104)

and has only two roots since

(105)Therefore, we need to show that

or (106)

We will show that . Using the fact

(107)

we have (108)–(112), shown at the bottom of the next page.From (111) to (112), we used the following inequality from (40):

(113)

(114)

From (108) to (109), we utilized the following inequality:

(115)

(99)

IEEE

Proo

f


D. Proof of Lemma 1

Consider

(116)

(117)

From (116) and (117), we have used the change of variableand exploited the fact that

is an anti-symmetric function sinceis symmetric. Property i) follows from the fact that the

integration portion in (117) is real valued. Again, from (117),we have

(118)

(119)

which proves ii). On the other hand, we have

(120)

(121)

(122)

sinc (123)

(124)

where in (122), we used the fact that is non-negative. Thisproves iii).

E. Proof of Inequality (35)

Since can be written as a sum of -convolutions drawnfrom combinations of

(125)

we have

(126)

In order to show that

(127)

consider a two-term-convolution case first:

(128)

(108)

(109)

(110)

(111)

(112)

IEEE

Proo

f


(129)

(130)

In (128), we used the Schwartz Inequality, and in (129), we usedthe fact that and , are non-negative real.

Applying this inequality inductively for each -convolutionterm, inequality (35) can be obtained.

REFERENCES

[1] J. Proakis, Digital Communications. New York: McGraw-Hill, 1995.[2] F. Guglielmi, C. Luschi, and A. Spalvieri, “Joint clock recovery and

baseband combining for the diversity radio channel,” IEEE Trans.Commun., vol. 44, no. 1, pp. 114–117, Jan. 1996.

[3] H. Kobayashi, “Simultaneous adaptive estimation and decision algo-rithm for carrier-modulated data transmission systems,” IEEE Trans.Commun., vol. COM-19, pp. 268–280, Jun. 1971.

[4] D. N. Godard, “Passband timing recovery in all-digital modem receiver,”IEEE Trans. Commun., vol. COM-26, pp. 517–523, May 1978.

[5] R. D. Giltin and S. B. Weinstein, “Fractionally spaced equalization: animproved digital transversal equalizer,” Bell Syst. Tech. J., pp. 301–321,1981.

[6] W. F. Schreiher, “Advanced television systems for terrestrial broad-casting: some problems and proposed solutions,” Proc. IEEE, vol. 83,no. 6, pp. 958–981, Jun. 1995.

[7] W. Chung, J. Balakrishnan, W. Sethares, and C. R. Johnson Jr., “Timingrecovery based on dispersion minimization,” in Proc. Conf. Inform. Sci.Syst., Mar. 2001.

[8] D. L. Lyon, “Timing recovery in synchronous equalized data communi-cation,” IEEE Trans. Commun., vol. 23, no. 2, pp. 269–274, Feb. 1975.

[9] J. E. Mazo, “Optimum timing phase for an infinite equalizer,” Bell Syst.Tech. J., pp. 189–201, Jan. 1975.

[10] W. Chung, “Blind Parameter Estimation for Data Acquisition in DigitalCommunication Systems,” Ph.D. dissertation, Cornell Univ., Ithaca, NY,Aug. 2002.

[11] C. R. Johnson Jr., P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, andR. A. Casas, “Blind equalization using the constant modulus criterion:a review,” Proc. IEEE, vol. 86, pp. 1927–1950, Oct. 1998.

[12] J. R. Treichler, C. R. Johnson Jr., and M. G. Larimore, Theory and De-sign of Adaptive Filters. Englewood Cliffs, NJ: Prentice-Hall, 2001.

[13] ATSC Digital Television Standard (Tech. Rep. Doc. A/53), AdvancedTelevision Systems Committee. (2000, Mar.). www.tsc.org [Online]

[14] H. Holma and A. Toskala, WCDMA for UMTS. New York: Wiley,2001.

[15] H. Zeng, L. Tong, and C. R. Johnson Jr., “Relationships between theconstant modulus and Wiener receivers,” IEEE Trans. Inf. Theory, vol.44, no. 4, pp. 1523–1538, Jul. 1998.

[16] P. Schniter and C. R. Johnson Jr., “Bounds for the MSE performance ofconstant modulus estimators,” IEEE Trans. Inf. Theory, vol. 47, no. 7,pp. 2544–2560, Nov. 2000.

[17] M. Gu and L. Tong, “Geometrical characterization of constant mod-ulus receivers,” IEEE Trans. Signal Processing, vol. 47, no. 10, pp.2745–2756, Oct. 1999.

[18] S. Y. Kung, “Independent component analysis in hybrid mixture:KuicNet learning algorithms and numerical analysis,” in Proc. Int.Symp. Multimedia Inform. Process., Dec. 1997, pp. 368–381.

Wonzoo Chung received the B.A. degree in mathe-matics from Korea University, Seoul, Korea, and theM.S. and Ph.D. degrees in electrical engineering fromCornell University, Ithaca, NY.

He is currently with Dotcast, Inc., Kent, WA, as aSystems Engineer, working on research and develop-ment of signal processing technologies for the dig-ital data broadcasting system. His research interestsinclude blind and adaptive aspects of digital signalprocessing for telecommunication systems.

William A. Sethares received the B.A. degree inmathematics from Brandeis University, Waltham,MA and the M.S. and Ph.D. degrees in electricalengineering from Cornell University, Ithaca, NY.

He has worked at the Raytheon Company,Waltham, as a Systems Engineer and is currentlya Professor with the Department of Electrical andComputer Engineering, University of Wisconsin,Madison. His research interests include adaptationand learning in signal processing, communications,and acoustics. He is is author of Tuning, Timbre,

Spectrum, Scale (New York: Springer 1998) and of Telecommunication Break-down (Englewood Cliffs, NJ: Prentice-Hall 2003).

C. Richard Johnson, Jr. (F’89) was born inMacon, GA, in 1950. He received the Ph.D. degreein electrical engineering with minors in engi-neering-economic systems and art history fromStanford University, Stanford, CA, in 1977.

He is currently a Professor of electrical andcomputer engineering and a member of the GraduateField of Applied Mathematics at Cornell Univer-sity, Ithaca, NY. Since 1990, he has held visitingappointments at Stanford University; the Universityof California–Berkeley; Chalmers University of

Technology, Gothenburg, Sweden; the Technical University of Vienna, Vienna,Austria; National Polytechnic Institute of Grenoble, Grenoble, France; and theAustralian National University, Canberra, Australia. During that period, hisprimary research interest has been blind adaptive fractionally spaced linearand decision feedback equalization for intersymbol and structured multiuserinterference removal from single and multicarrier communication systems.The current broadband adaptive receiver design project of his research groupat Cornell is described at http://bard.ece.Cornell.edu/. The group’s activityis currently supported by the National Science Foundation, Applied SignalTechnology, and Texas Instruments.

Dr. Johnson was elected a Fellow of the IEEE “for contributions to adaptiveparameter estimation theory with applications in digital control and signal pro-cessing” in 1989.

ieee transactions on signal processing, vol. 53, no. 3

Documents