1140 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 8, AUGUST 1999

Transactions Papers

Blind Estimation of Multipath Channel Parameters:A Modal Analysis Approach

Insung Kang,Member, IEEE,Michael P. Fitz,Member, IEEE,and Saul B. Gelfand,Member, IEEE

Abstract—In this work, we propose a novel approach to effi-ciently estimate multipath channel parameters, which is particu-larly useful in sparse multipath channels. Conventional methods[1]–[4] do not fully exploit inherent structure present in thecombined channel response; the excess number of parametersto be estimated by conventional methods makes the identificationdifficult. By utilizing a priori knowledge of the transmission datapulse, the channel identification problem is transformed into themode estimation problem. Then, the parameters directly relatedto the multipath propagations are extracted in the modal analysisframework, and hence, the number of estimation parameters aresignificantly reduced. Finally, the multipath channel parametersare obtained by inverse-transforming the mode parameters. Sim-ulation results show significant improvement in normalized meansquare error over existing approaches.

Index Terms—Frequency domain analysis, HDTV, multipathchannels, parameter estimation.

I. INTRODUCTION

I DENTIFICATION and equalization of a digital commu-nication channel are important issues in high-performance

reliable communications system design. Adaptive equalizersare often used to compensate for intersymbol interferencein communication systems that have a frequency-selectivechannel response. Adaptive equalizers typically use trainingsequences periodically to prevent catastrophic error propaga-tion [5]. As a result, a portion of transmission bandwidth iswasted. Blind identification and equalization algorithms arerequired when dedicated training sequences are not availableto initialize demodulation. Example applications are retrainingafter deep fades in wireless systems and node initialization ina broadcast communication system.

Paper approved by R. A. Kennedy, the Editor for Data CommunicationsModulation and Signal Design of the IEEE Communications Society. Man-uscript received May 27, 1997; revised May 2, 1998 and December 14,1998. This work was supported in part by National Science Foundation underContract NCR-9406073 and in part by Thomson Consumer Electronics. Thispaper was presented in part at the Communication Theory Mini-Conferenceof GLOBECOM’97, November 1997, Phoenix, AZ.

I. Kang is with the Network Solution Sectors, Motorola, Arlington Heights,IL 60004 USA.

M. P. Fitz is with the Department of Electrical Engineering, The OhioState University, Columbus, OH 43210-1272 USA (e-mail: fitz@ee.eng.ohio-state.edu).

S. B. Gelfand is with the School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47907-1285 USA.

Publisher Item Identifier S 0090-6778(99)06307-2.

Fig. 1. A sample sparse channel used by ATTC where sample rate= 10:76

MHz, i.e., two samples per symbol.

Depending on the applications, there are different technicalchallenges to the blind channel identification algorithms. Thefast fading characteristics of mobile wireless channels imposea limitation on the number of data samples, which makesblind identification algorithms less successful. Practicallyspeaking, fixed wireless channels found in applications suchas wireless local loop and wireless digital broadcasts mightbe the ideal applications for the blind channel identificationalgorithms. However, the high-speed data transmission indigital broadcast potentially results in a sparse multipathchannel. Sparsity of a multipath channel can be defined asthe ratio of the time duration (in symbol periods) spannedby the multipaths and the number of multipaths. Thus,for a fixed-channel impulse response, a higher symboltransmission rate yields a more sparse multipath channel.These sparse multipath channels are difficult to identify withtraditional parameter estimation schemes like [6] or channelidentification schemes based on second-order statistics dueto both the large number of discrete time channel coefficientsand the numerical sensitivity of these approaches.

In this work, we propose a novel blind identification al-gorithm for the fixed wireless sparse multipath channels. Asparse multipath channel typical for video data transmissionis shown in Fig. 1 [7]. The proposed approach transformsthe channel identification problem into the mode estimation

0090–6778/99$10.00 1999 IEEE

KANG et al.: BLIND ESTIMATION OF MULTIPATH CHANNEL PARAMETERS 1141

problem by utilizinga priori knowledge of the transmissiondata pulse. The multipath related parameters are extracted inthe modal analysis framework and the number of the estimatedparameters can be significantly reduced compared to a standardformulation. Finally, the multipath channel parameters areobtained from the mode parameters. The proposed approachis unique in the sense that the phase information is not usedin reconstructing the channel response, but in resolving thephase ambiguity inherent in the channel estimation when onlythe magnitude spectrum is used.

After a brief overview of the prior work and the identi-fiability of the sparse multipath channels, Section II definesthe problem and translates the channel estimation probleminto a modal analysis framework. In Section III, the proposedidentification algorithm is presented. Numerical results for atypical sparse multipath channel are followed in Section IV.Finally, the paper concludes in Section V.

A. Prior Work

Approaches using the second-order statistics have beenpopular in the literature since Tonget al. showed that under acertain identifiability condition, blind channel identification isachievable using only the second-order statistics [1], [8]. Thisresult is based on the fact that most communication systemoutput is cyclostationary rather than stationary [9].

Moulines et al. [2] exploited the orthogonality propertybetween the signal and noise subspaces. This is called thesubspace method. The subspace method reduces the numberof estimation parameters to the number of the samples ofthe combined channel response. Baccala and Roy’s work [3]and further refinement by Shellet al. [4] were developedby noting that the subchannel responses should produce thesame signals if they are used to excite each other by thecommutivity of linear systems. This is called the subchannelresponse matching.

Shell et al. showed performance improvement over theprevious work by exploitinga priori knowledge of the trans-mission data pulse to further reduce the number of parametersto the length of the channel impulse response (not the com-bined pulse and channel response). A similar attempt forthe subspace method was published by Ding [10]. It wasnamed the multipath subspace method. For a sparse multipathchannel, this reduction may not be sufficient to achieve thedesired performance improvement. The trend here is to reducethe number of estimation parameters to increase the estimationaccuracy.

The common advantage of the approaches based on thesecond-order statistics is that they show faster convergence.The key drawback is that the methods work poorly forchannels that do not satisfy the identifiability conditionsoverviewed in the next section.

B. Identifiability of the Sparse Multipath Channels

The second-order statistics-based blind identification algo-rithms commonly require identifiability conditions in orderto obtain consistent estimates. Identifiability ensures nonsin-gularity of the linear equations solved in the second-order

TABLE IA SPARSE MULTIPATH CHANNEL USED BY ATTC

Fig. 2. Root locus of the channel given in Fig. 1.

approaches. Although the sparse channels encountered inbroadcast applications are identifiable, the linear equationsare near singular. The resulting numerical sensitivity yieldsunusable estimates for reasonable observation length.

We review the identifiability condition in the frequencydomain since it is illustrative for our purpose. Let

(1)

for (2)

where is the th time sample of the thsubchannel and is the oversampling ratio. Then, a finiteimpulse channel response is identifiable if and only if

do not share any common roots [11].Since the sparse multipath channels are typical in high-speed

digital communications and digital television is a practicalexample of high-speed data communications, we have selectedone of the Advanced Television Test Center (ATTC) testchannels [7] for our demonstration. The channel is a discretemultipath channel with the characteristics seen in Table I.In fact, Fig. 1 is a plot of the combined channel impulseresponse when a raised-cosine pulse with an excess bandwidthof 11.52% and a pulse duration of ten symbol periods are used.The symbol rate is 5.38 MHz (6-MHz bandwidth).

Recall that the identifiability condition for second-orderstatistics-based channel identifications requires that the-transform of the two subchannels do not share commonroots (zeros). Unfortunately, sparse multipath channels pro-duce subchannels with many closely spaced roots. Fig. 2shows the root locus plot for the two subchannels of the

1142 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 8, AUGUST 1999

Fig. 3. Eigenvalues when true covariance matrix for the channel given inFig. 1 is used.

impulse response shown in Fig. 1. While none of the roots areidentical, they are close enough to cause numerical problemsin any subspace method. Fig. 3 shows the eigenvalues of thetrue covariance matrix sorted by the magnitude where the noisevariance was subtracted off (i.e., signal subspace). The noisesubspace eigenvalues for white noise become all zeros whenthe noise variance is subtracted off. Notice that there are aboutseven orders of magnitude difference in the signal subspaceeigenvalues, and thus, the small signal subspace eigenvaluesare virtually indiscernible from the noise subspace eigenvalues.This is the reason why the subspace method fails with anysignificant noise in the estimated correlation functions.

II. PROPOSEDAPPROACH—MODAL ANALYSIS FRAMEWORK

In our proposed approach, we cast the channel estimationproblem into a modal analysis framework by utilizinga prioriknowledge of the transmission data pulse. The innovationin this work is the transformation of the channel estimationproblem to a mode estimation problem, the implementationof the inverse-mapping of the estimated mode parameters tothe multipath channel parameters, and the resulting decreasein the number of parameters to be estimated. Classical modalestimation typically involves a four-step process [12, Ch. 11]involving:

1) estimating the time series output correlation function;2) estimating the number of sources;3) mode estimation; and4) mode weight estimation.

Additionally to unravel the transformation, the mode andmode weight estimates are used to compute the correspondingdelay and multipath attenuation estimates for use in a channelequalizer. Fig. 4 gives a flow chart of this proposed algorithm,and each of the components is overviewed in the sequel.

A. Signal Model

Consider a digital communication system model shown inFig. 5. It is assumed that the multipath channel parameters areunknown but deterministic, the pulse shape is known, and the

Fig. 4. Proposed channel identification algorithm flow chart where thenumbers in parenthesis represent the corresponding equations in the text.

Fig. 5. A digital communication channel model.

data symbols are random. For this work, we assume a discretemultipath channel model as is typical for wideband terrestrialwireless communication [13]. The received and sampled signalis given by

(3)

where are complex data symbols, is a known trans-mission pulse (bandlimited either by front-end filter and/orby the transmission pulse shape filter), is the number ofthe transmitted symbols, is the oversamplingratio (unless stated explicitly, it is assumed to be an integergreater than one), is the symbol period, is the samplingperiod, is the symbol timing offset, are the multipathdelays ( ), are the complex multipathattenuations, is the number of multipath components, and

KANG et al.: BLIND ESTIMATION OF MULTIPATH CHANNEL PARAMETERS 1143

is an additive complex Gaussian noise. In the sequel, weassume that:

1) the first multipath delay is zero1 ( );2) the transmitted symbols are an independently, identically

distributed (i.i.d.) sequence (uncoded) with the varianceof ;

3) the additive noise is a white Gaussian sequence withthe variance of and independent of the transmittedsymbols.

B. Transformation to a Modal Analysis Paradigm

The channel estimation algorithm herein is based on manip-ulations of the autocorrelation function of . The autocorre-lation functions of the cyclostationary signal and the inputsequence are defined as

(4)

(5)

where denotes the complex conjugate operation. Note thatfor the autocorrelation definition to be consistent, the stationarysymbol rate sequence is expanded to the sampling rate byzero-padding. In practice, must be estimated andthis is done as

(6)

where is the sample size measured in symbol periods. Sincethe autocorrelation function is periodic in with a period ,

is adjusted appropriately in order to confine the time indiceswithin the measurement period. Using the form given in (3)gives

(7)

where is a two-dimensional (2-D) convolution operatorand is the autocorrelation function of the noisesequence .

The th cyclic spectrum of is defined to be theth sliceof the 2-D Fourier transform of the autocorrelation function[14, Ch. 12]. That is

(8)

Then

(9)1The fraction of the first multipath delay is incorporated in the symbol

timing offset, while the integral part of the first multipath delay cannot beestimated blindly.

where is the Fourier transform of the known transmissionpulse , and are the cyclic spectra of theinput and white noise sequences, respectively. For i.i.d. inputsequence and AWGN noise sequence

(10)

Since and are known, we can extract informationabout the multipath parameters from if the noisevariance are available. The noise variance can be eithereasily estimated using the covariance matrix or ignored forhigh signal-to-noise ratios (SNR’s).

We define the th channel cyclic spectrum2 as

(11)

(12)

(13)

where

(14)

(15)

(16)

We call the relative multipath delays, is thenumber of complex modes, are the mode weights whichare nonlinear functions of the complex multipath attenuations(and the relative multipath delays for the nonzeroth-channelcyclic spectrum), and are the modes which are functionsof the relative multipath delays.

Examining (13) for shows that the channel cyclicspectrum has the form of a standard modal analysis problemwhich has a rich history of techniques developed for parameterestimation [15]–[18]. Note also that the mode weightsare separable from the modes in the zeroth-channel cyclicspectrum only. This property can be exploited toreduce the computational complexity in parameter estimationstructures [19]. Additionally since is real, the Her-mitian symmetry can be exploited to reduce the number ofindependent complex modes to .3 For example, when athree-ray multipath channel is considered (i.e., ), onlyeight parameters (four for modes corresponding to the relativemultipath delays and four formode weights ) need to be estimated no matterhow large the channel delay spread.

2We can simply consider thekth channel cyclic spectrum to be the channelrelated component of thekth cyclic spectrum.

3dxe is an integer not smaller thanx.

1144 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 8, AUGUST 1999

The channel cyclic spectrum is obtained from the outputcyclic spectrum for only nonzero4 .

has the longest support, and we can obtain moresamples from the zeroth cyclic spectrum than from otherspectra. Hence, our initial focus will be on estimation basedon . Note that the zeroth cyclic spectrum is differentfrom the magnitude spectrum of the stationary signal (symbolrate measurement) because the magnitude spectrum of thestationary signal is the aliased version of the zeroth cyclicspectrum. Although the zeroth cyclic spectrum is not aliased,the estimates derived from it will have a resulting ambiguitydue to the lack of phase information and is laterused to resolve this ambiguity. Consequently, while a major-ity of our algorithm is concerned with , the phasediscrimination capability of the cyclostationarity of the signalis used.

Suppose that the nonzero support of is, where is the sampling resolution

in frequency. Then by sampling at , where, we have

(17)

where

......

......

......

. . ....

and , and is the number ofmodes. For notational simplicity, we omitted the super-indexand simplified the subindex for the mode weights . Itshould be pointed out that the noise involved in this new signalmodel given by (17) is no longer white nor Gaussian.

C. Estimation of the Number of Sources

The most prevalent techniques for estimating the number ofmodes use the minimum description length (MDL) criterion[20], [21]. For this section, we will define our observations tobe the following snapshot array

......

...(18)

which is parameterized by unique eigenvalues (), eigenvectors ( ), and the noise

4Practically speaking, significantly larger than zero to limit the magnitudeof the distortion due to noise.

variance . Note that refers to the hypothesized signaldimension, and is the effectivenumber of snapshots (i.e., we are now viewing asour time-series snapshots). The necessary condition for theestimation of source cardinality (or the number of sources) isthat the signal dimension should not be smaller than theactual number of sources. The sample correlation matrixis obtained by , where is the hermitiantranspose.

The MDL criterion [22] uses information theoretic argu-ments to show that the best estimate for the source cardinality

is the value that (for large sample size) minimizes over

MDL

(19)

where is a probability density function,, , and are

the ML estimates of the eigenvalues, eigenvectors, andnoise variance, respectively. This formulation is prohibitivelycomplex for finding a useful solution to the posed problem,so we approximate the snapshot vectors as Gaussian randomvectors. Using this Gaussian approximation, the ML estimates[12, Ch. 6] are given as

(20)

(21)

(22)

where and are the eigenvalues andeigenvectors of the sample correlation matrix. Then, the MDLcriterion is given by [20]

MDL

(23)

Finally, the number of modes is determined by

MDL (24)

D. Mode Estimation

There are numerous estimators for the classical modalanalysis signal model, e.g., least squares Prony method, totalleast squares, principal component method, Pisarenko, MUSIC(or root-MUSIC), ESPRIT (or TLS-ESPRIT), or ML-basedtechniques. These methods provide varying degrees of tradeoffbetween performance, generality, and complexity. We havefound ESPRIT [17], [23] (TLS-ESPRIT [24]) most useful andour discussion focuses on this algorithm.

ESPRIT is a mode estimation technique that exploits theconstant lag property of the noiseless signal samples. For

KANG et al.: BLIND ESTIMATION OF MULTIPATH CHANNEL PARAMETERS 1145

the complex exponential signal model given in (17), we canconstruct two data matrices defined by

......

......

(25)

where is the overfitting parameter. Then, the roots of thepencil of matrices produce the estimates for the modes

. That is, the modes are obtained byfinding the generalized eigenvalues satisfying

(26)

where is the corresponding eigenvector.An intuitive form for exploiting the pure sinusoidal prop-

erty of the zeroth-channel cyclic spectrum is the forward-and-backward matrix pencil [25]. In that case, arereplaced by with

(27)

where is a reverse permutation matrix. If a mode isthe root of the forward matrix pencil, its inverse should be theroot of the backward matrix pencil. Thus intuitively, a moderesides on the unit circle if it satisfies the forward–backwardmatrix pencil. In practice, it is very close to the unit circlewhen the overfitting parameteris sufficiently larger than thenumber of modes . Once we compute number of modes,only the modes5 closest to the unit circle are retained.

The mode weights are obtained by the least squares estimategiven the mode estimates. That is

(28)

III. PROPOSEDAPPROACH—INVERSE TRANSFORMATION

At this point in the algorithm, an estimate of the number ofmodes, the mode values (equivalently the relative multipathdelays), and the mode weights have been obtained and needto be transformed into estimates of the number of multipaths,the individual multipath delays, and the complex multipathattenuations. This inverse-transformation is nonlinear and theoptimal solution is not obvious. In the following subsections,we propose a suboptimal technique to accomplish this trans-formation.

5In order to compensate the consistently underestimated number of modesat low SNR (short observation window),� is adjusted to the nearest possible� = L(L� 1) + 1 � � for the smallestL.

Fig. 6. The difference table.

A. Number of Multipaths

The first problem is the estimation of the number of mul-tipaths . Recall that if all the relative multipath delays areunique, then is obtained from . If thereis a multiplicity in a value of the relative multipath delays,then the mode weights are modified slightly and the numberof modes decreases, i.e.,

(29)

where is the cardinality of the relative multipath delaymultiplicity. Thus . In estimating thenumber of multipaths, an accurate estimate for the numberof modes and the multiplicity of the relative multipath delaysplay an essential role. Since the multiplicity of the relativemultipath delays is generally not available, only the potentialrange for is known. If the source cardinality has beenestimated with reasonable accuracy, the potential range foris

(30)

This gives the potential range over which to search for anoptimal estimate of . This range appears large, but it will beseen later that the proposed algorithm further reduces the rangewithout knowing the multiplicity of the relative multipathdelays.

B. Search over Delays

Before discussing the search procedures, it is important tooverview how the actual multipath delaysproduce the relative multipath delays in the cyclic spectrum

. The represent all thepossible unique time-delay differences between the multipathdelays. An intuitive construct for understanding the relationbetween the and the isthe difference table. We have only shown the upper-right halfbecause the nonnegative are the independent parametersof interest. The difference table is an table that hasa top row consisting of the multipath delays ordered in size(recall ). The th row is this top row minus . Thisconstruction is seen in Fig. 6. The number of unique elementsin this difference table is and . We definetwo difference tables to be equivalent if all of the values andmultiplicity of the elements comprising the two differencetables are identical. The following example would illustratethis concept.

1146 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 8, AUGUST 1999

(a) (b)

(c)

Fig. 7. Equivalence among difference tables: (a) and (b) are equivalent,while (c) is not equivalent to either (a) or (b).

Example 1: In Fig. 7, table (a) is equivalent to table (b),while they are not equivalent to table (c).

The fact that the actual delays are partof the estimated relative multipath delays,

enables a simplified search mechanism. Since we haveassumed that , the modes associated with the positivedelay differences produced by examining the zeroth-channelcyclic spectrum contain the actual values of the absolutemultipath delay. Consequently, a brute force exhaustive searchthat chooses of the estimated as candidate solutions isfeasible. The complexity of the exhaustive search amounts tothe number of ways to pick items out ofitems.6

Unfortunately, using the zeroth-channel cyclic spectrumproduces an estimation ambiguity. This ambiguity arises be-cause a difference table is equivalent to a difference tableformed using the last column of the original difference tableas the first row (appropriately ordered). The delays in thiscolumn are denoted asthe reflected set.Note this is exactlyhow the two equivalent difference tables were constructed inExample 1. Thus, it seems impossible to resolve which valuesof multipath delays produced the zeroth-order channel cyclicspectrum. This ambiguity can be resolved by using the first-order cyclic spectrum , and this will be discussed inthe sequel.

This idea of the symmetry in the difference tablecan be exploited to formulate a reduced complexitysearch algorithm. A delay in the solution set

has a delay difference in thereflected set , whichsummed together adds up to . We call this a constituentpair. To illustrate this idea consider the next example.

Example 2: The set is the reflection of theset . Then (1, 6) and (3, 4) are the constituentpairs which add up to 7.

Thus, a reduced complexity search would only select valuesfor for which an element in

exists that adds up to . If we have( ) constituent pairs, then we can find the solutionwith the order of complexity. A selection for

will produce a set of relative multipath delays

6There are at mostd(�� 1)=2e positive relative multipath delays amongthe�� 0

isince~�x(0; �) is real valued.

, which constitute a difference table and this must match thedifference table comprised of the relative multipath delaysestimated by the mode estimation algorithm. We use the leastsquares error criterion to select the best match. By regardingthe difference tables as matrices, this is accomplished byminimizing the Frobenius norm [26, Ch. 2] of

where denotes a difference table.We summarize the suboptimal search algorithm as follows.Algorithm 1—A Suboptimal Search Algorithm:

1) Find the minimum and the maximum multipath delays

•

• .

2) Find all the constituent pairs of numbers which add upto . There are at least such pairs. We choose

pairs, the sum of which is the closest to .3) Construct a candidate for the solution by selecting one

number from each pair. By minimizing the Frobeniusnorm of , we construct a mapping fromthe multipath attenuations to the mode weights.

Through step two of the suboptimal search algorithm, wecan further confine the potential range for the number ofmultipaths . Suppose we have constituent pairs, thenapparently . Thus, combined with (30)

for

for (31)

The Frobenius norm can be used to further limit the searchspace by selecting the best choices out ofpossibilities. In fact, our numerical results to be presented inSection IV were obtained using the algorithm with . Thissuboptimal algorithm works well when the estimate foris quite reliable. The reliability of is easily ensured bythresholding the mode weight corresponding to the maximumrelative multipath delay. This technique has worked for pathsthat are over 20 dB below the largest path in simulations. Oftentimes, the minimum power of the multipath gains is known bythe desired operating SNR. Our thresholding is designed toaccommodate the mode weight corresponding tothe maximum relative multipath delay, which is 46 dB belowthe largest path.

C. Multistart Steepest Descent for Optimizationover Complex Multipath Amplitudes

This section discusses an approach to finding the truemultipath channel values given the modes and mode weightsof the channel cyclic spectrum. Considering (15) and denoting

to be the number of unique elements in the differencetable produced by the delay estimate, we now haveindependent equations and unknowns. Since we have atleast as many equations as unknowns and usually more equa-tions than unknowns, a least squares solution is an obviousapproach.

For a moment, we assume that all the entries in thedifference table are distinct. Then for each selection of the

KANG et al.: BLIND ESTIMATION OF MULTIPATH CHANNEL PARAMETERS 1147

multipath delay estimates, we have from (14) and (15)

(32)

(33)

A different selection of multipath delays results in a differentmapping .

Then, an optimization index in the least squares sense wouldbe

(34)

where

A well-known and stable approach to solve this set of non-linear equations is the steepest descent algorithm. Since theoptimization index is multimodal, multiple initializationsare necessary to find the global minimum. Since we have only

multipath components, a systematic initialization schemecan be found easily. For example, we can assumewithout loss of generality. Then, initializations will beenough by initializing for only one

and setting zeros for the rest of multipath attenuationcoefficients.

The steepest descent algorithm [27, Ch. 4] is given by

(35)

where is a positive constant and is the gradientat th iteration.

To incorporate the multiplicity information in the modeweights, (33) should be changed slightly as follows: supposethat has a multiplicity of , that is,

for different pairs.Then (33) becomes

(36)

D. Joint Estimation of Symbol Timing Offset andDetermination of the Best Channel Estimate

As we have pointed out previously, the solution obtainedusing the zeroth-channel cyclic spectrum is not unique. Oncethe identifiability conditions are satisfied, the cyclic spectracontain the sufficient information for the blind identificationof the linear time-invariant finite impulse response channel[28]. The zeroth cyclic spectrum contains only the magnitudeinformation. Thus, the phase information should be providedby the nonzeroth cyclic spectrum. Note that among the

nonzeroth cyclic spectrum, only first and th (orequivalently first) cyclic spectra are nonzero for a spectrallyefficient transmission data pulse. Thus, we use the first and

first cyclic spectra to determine the best channel estimate.Also, we need to estimate the symbol timing offset simulta-neously from the nonzeroth cyclic spectrum. From (12), theleast squares estimate for the symbol timing offset is given by

(37)

where denotes the angle of the quantity in radiansbetween and , the summation index is over the nonzerosupport of , is the length of thenonzero support, and can be either 1, or both. The bestchannel estimate is chosen to produce the least squares phaseerror given

(38)

where is obtained in (37).

E. Complexity Consideration

For the computational complexity of the proposed method,we will thoroughly examine the algorithm flowchart shown inFig. 4. The proposed algorithm can be divided into three steps.

The first step includes the computations of the autocorre-lation function, the cyclic spectrum by Fourier Transform,and the channel cyclic spectrum. Without considering thecomputation of the autocorrelation function, since its compu-tation is evenly distributed over the observation window, thecomplexity of this step is in the order of denotedby , where is the length of the combinedchannel response in symbol periods.

The second step is to estimate the mode parameters thatinclude the number of sources, the modes , andthe mode weights . The number of sources is obtained byapplying MDL principle to a snapshot correlation matrixwhere is the hypothesized signal dimension. Since MDLuses an eigenvalue decomposition, the complexity is .The modes are estimated by ESPRIT. ESPRIT’s complexityis where is the overfitting parameter. The modeweights are obtained by the least squares estimate in which an

matrix inverse is involved, hence .The third step is to inverse-transform the mode parameters

into the channel parameters. It consists of two outer loopsgenerating a hypothesis for the multipath delay. For eachhypothesis, the multipath attenuation estimation using the

1148 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 8, AUGUST 1999

(a) (b)

Fig. 8. One hundred channel estimates and the 100 sample mean produced by the proposed method. The data record length is 0.2 s,Es=N0 = 25 dB, = 80, sample rate= 10:76 MHz: (a) channel estimates and (b) estimates mean.

steepest descent algorithm, the symbol timing estimation, andthe phase error given by (38) are computed. The first outerloop is to search over the number of multipaths, which hasa potential range given by (31). Let denote the numberof runs for this loop. The second loop is to search over themultipath delay , which has a range between 2 and .A large complexity reduction is achieved by controlling thisloop. We use to denote the number of runs for this loop.The steepest descent algorithm has a variable complexitydepending on the convergence threshold. Its complexity is

where is the number of runs to converge. Forcomplexity reduction, the convergence threshold is set large tokeep small. The symbol timing estimation and the phaseerror computation is linear in , where is the length ofthe nonzero support of the nonzeroth-channel cyclic spectrum.For the example considered here (digital television broadcast),

will be relatively small since the excess bandwidth is only12%.

Thus, the total complexity of the proposed algorithm is

Complexity

(39)

In contrast, the subspace method [2] has a complexity of.

IV. NUMERICAL RESULTS

To test the resulting performance of the proposed algorithm,we use the same sparse multipath channel as documented inSection I-B. We use a symbol rate of 5.38 MHz (6-MHzbandwidth), uncoded 64-QAM modulation with two timesoversampling ( ). The transmission data pulse is araised-cosine pulse with an excess bandwidth of 11.52% anda pulse duration of ten symbol periods. The autocorrelationfunction is assumed to have a length of 239 symbol periods.It is long enough to include the combined channel response,

which has a length of 117 symbol periods ( ). Thezeroth-channel cyclic spectrum has a length of 240samples. Throughout the simulation, the parameter settingsare as follows:

(40)

The suboptimal search algorithm comes up with two besthypothesis ( ) minimizing the Frobenius norm of

, where denotes a difference table. Thecomplexity of the steepest descent algorithm depends on whenit is terminated. The steepest descent algorithm terminateseither when or when

, where is a threshold. In order to reduce thecomplexity of the proposed algorithm, the steepest descentalgorithm was implemented with resulting infairly coarse estimation of the multipath gains. The parameter

controlling the speed of convergence is optimally selectedand, thus, time-varying (see [29, Ch. 10] for details).

Fig. 8 shows 100 channel estimates and the mean channelestimate obtained by the proposed method using a 0.2-s datarecord with an dB. To obtain a performancemeasure for a fair comparison, we use the sample mean squareerror (MSE) defined by

MSE (41)

where is the combined channel and pulse response, isthe channel estimate from theth trial, and is the number oftotal trials. Fig. 9 is a plot of the sample MSE (averaged over100 trials) versus the data record length with andB for the proposed method, the subspace method of Moulineset al., and the multipath subspace method of Ding. For bothsubspace methods, the channel length was assumed to be

KANG et al.: BLIND ESTIMATION OF MULTIPATH CHANNEL PARAMETERS 1149

Fig. 9. Sample mean square channel estimation error versus the data recordlength withEs=N0 = 25 dB: solid line (proposed method with = 80),dashed line (subspace method), and dotted line (multipath subspace method).

Fig. 10. Sample mean square channel estimation error by the proposedmethod versusEs=N0 with data record length of 0.5 s and = 80.

known. In addition to that, the symbol timing is assumed tobe known for the multipath subspace method. A significantperformance improvement is evident over the whole range ofdata record lengths.

The anomalous increase in the MSE at short data recordsshould be interpreted as a failure rate and not simply a highervariance estimate. It is due to the increased probability ofselecting the reflected set explained in Section III-B. Thisfailure is primarily due to the underestimation of the numberof modes at short data lengths and the insufficient phaseinformation.

The applicability of the proposed method is not limited tothe high SNR circumstances. We tested the proposed methodfor various SNR’s ranging from 0 to 30 dB with data recordlength of 0.5 s. In Fig. 10, the result is shown. Except for 0dB , the proposed algorithm showed good results.

To compare the complexities of the proposed method andthe subspace method, we used the same parameters givenin (40). The other parameters such as are data-

TABLE IICOMPLEXITY COMPARISON IN EXECUTION TIME

dependent. Programs written in MATLAB were run 100 timesin a Silicon Graphics R10000 machine, and the elapsed timewas measured for each step. is set to 25 dB and thedata record length is 0.2 s. Table II summarizes the result.

V. CONCLUSION

In this work, we have proposed a new blind channelidentification algorithm that exploitsa priori knowledge of thetransmission data pulse in frequency domain. The algorithmworks well for sparse multipath channels, which are typical inthe high-speed fixed wireless digital communications. The sim-ulation results for a typical HDTV channel show substantialimprovement in MSE over existing algorithms.

ACKNOWLEDGMENT

The authors would like to thank L. Potter, R. Moses, andA. Sabharwal for valuable discussions and comments that ledto a greatly improved manuscript.

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Insung Kang (S’93–A’97–M’99) was born in Pu-san, Korea, in 1960. He received the B.S. degreein control and instrument engineering from SeoulNational University, Seoul, Korea, in 1982, the M.S.degree in electrical engineering from the Universityof Dayton, Dayton, OH, in 1993, and the Ph.D.degree in electrical and computer engineering fromPurdue University, West Lafayette, IN, in 1997,respectively.

From 1982 to 1992, he was a Research Engineerand a Research Manager at a subsidiary of LG

Electronics Inc. (formerly GoldStar Telecommunications Inc.) working onmany projects including magnetic resonance imaging and ISDN video-phone.From May to December in 1993, he was a Research Assistant at theUniversity of Dayton. From August 1994 to May 1997, he was a ResearchAssistant at Purdue University. In June 1997, he joined the Network SolutionsSector of Motorola Inc., Arlington Heights, IL, where he currently designsthe base station receivers for the third-generation DS-CDMA system. Hiscurrent research interests are in signal detection, diversity combining, channelestimation, and synchronization.

Michael P. Fitz (S’82–M’89), for photograph and biography, see p. 537 ofthe April 1999 issue of this TRANSACTIONS.

Saul B. Gelfand (M’88), for photograph and biography, see p. 937 of theJune 1999 issue of this TRANSACTIONS.