Basis Expansion Models And Diversity Techniques For Blind … · 2012. 8. 16. · Keywords— Adaptive equalizers, diversity methods, Doppler effect, fading channels, identiﬁcation,

Basis Expansion Models and DiversityTechniques for Blind Identification andEqualization of Time-Varying Channels

GEORGIOS B. GIANNAKIS,FELLOW, IEEE, AND CIHAN TEPEDELENLIOGLU

Invited Paper

The time-varying impulse response of rapidly fading mobilecommunication channels is expanded over a basis of complexexponentials that arise due to Doppler effects encountered withmultipath propagation. Blind methods are reviewed for estimatingthe bases’ parameters and the model orders. Existing second-ordermethods are critiqued and novel algorithms are developed for blindidentification, direct, zero-forcing equalization and minimum meansquare error (MMSE) equalization by combining channel diversitywith temporal (fractional sampling) and/or spatial diversity whichbecomes available with multiple receivers. Illustrative simulationsare also presented.

Keywords—Adaptive equalizers, diversity methods, Dopplereffect, fading channels, identification, least mean square methods,mobile communication, time-varying channels.

I. INTRODUCTION

Blind techniques for identification of linear time-invariant(TI) systems have found widespread applications in timeseries modeling, econometrics, exploration seismology, andequalization of communication channels, just to name afew. With no access to the input, many blind methodshave relied on stationary high-order statistics [13], [18],[34], [51] and cyclostationary or multivariate second-orderstatistics [12], [14], [33], [43], [45], [52] of the outputdata in order to: 1) either estimate TI systems or 2) theirinverses when input recovery is the ultimate goal. Such self-recovering schemes are important, for example, in digitalbroadcasting because transmission is not interrupted to trainnew users entering the cell. Similarly, in wireless environ-ments bandwidth is utilized efficiently when cold start-upis possible and in multipoint data networks throughputincreases and management overhead drops when trainingis obviated [18].

Manuscript received August 14, 1997; revised March 6, 1998. Thiswork was supported by NSF Grant MIP 9424305.

The authors are with the Department of Electrical Engineering, Uni-versity of Virginia, Charlottesville, VA 22903-2442 USA (e-mail: [email protected]).

Publisher Item Identifier S 0018-9219(98)06973-4.

However, many systems violate the time-invariance as-sumption. In cellular telephony, the multipath propaga-tion channel not only exhibits frequency selectivity, whichcauses intersymbol interference (ISI), but also changes asthe mobile communicators move [1], [2], [7], [28], [37],[42]. Temperature and salinity variations cause underwaterchannels to vary [24], [25], [44], and fluctuations in theionosphere give rise to deep fades in the data receivedvia microwave links [20], [35]. For channel variations withcoherence time in the order of hundreds of symbols (slowfading) adaptive variants of algorithms developed for TIsystems offer a valuable alternative, although periodic re-training is recommended to avoid runaway effects [2], [24],[36], [42]. Recursive least-squares (RLS) and least mean-square (LMS) are adaptive algorithms which are knownto diverge when channel variations exceed the algorithms’convergence time. In such cases explicit incorporation ofthe channel’s time-varying (TV) characteristics is called for.

Most explicit models of TV communication channelstreat the TV taps as uncorrelated stationary random pro-cesses which are assumed to be low-pass, Gaussian, withzero mean (Rayleigh fading) or nonzero mean (Ricianfading) depending on whether line-of-sight propagation isabsent or present [6], [23] [25], [35], [50]. Correlations ofthe unknown taps capture average channel characteristicsand are used to track the channel’s time evolution usingKalman filtering estimators [8], [9], [25], [50]. The unob-servable channel statistics are either fixed to experimentallycomputed values [8], [25] or estimated from the data duringthe decision directed mode [6], [9], [50]. Hidden Markovmodels have also been used in modeling the tap variations[3].

Statistical modeling is well motivated when TV pathdelays arise due to a large number of scatterers (e.g., inover-the-horizon communications). But recently determin-istic basis expansion models have gained popularity forcellular radio applications, especially when the multipath

0018–9219/98$10.00 1998 IEEE

PROCEEDINGS OF THE IEEE, VOL. 86, NO. 10, OCTOBER 1998 1969

is caused by a few strong reflectors and path delays exhibitvariations due to the kinematics of the mobiles [1], [17],[28, p. 383], [39], [46], [47]. The TV taps are expressedas a superposition of TV bases (e.g., complex exponentialswhen modeling Doppler effects) with TI coefficients. By as-signing time variations to the bases, rapidly fading channelswith coherence time as small as a few tens of symbols canbe captured. Such finitely parameterized expansions renderTV channel estimation tractable and have been previouslyused in modeling speech and economic time series [21],[29]. They are also encountered in Doppler radar and sonarapplications when scintillating point targets give rise todelays which change (linearly or quadratically) with timeand cause Doppler shifts in the carrier frequency [38]. In[28, p. 383], it is argued that such Doppler-induced varia-tions are equivalent to the random coefficient model sincenarrow-band Gaussian processes are well approximated bysuperimposed sinusoids having constant amplitudes andrandom phases.

Time- and frequency-selective channels are special casesof the basis expansion models considered here. Althoughmost existing blind equalization research has focused onfrequency selective channels, modeling time-selective ef-fects are well motivated due to local oscillator drifts and/orrelative motion encountered in mobile communications.

Finite basis expansions offer well-structured parsimo-nious modeling which allows for blind identification of TVchannels. In [47], this important feature was establishedfirst based on second- and fourth-order output correlations.The high-variance of high-order TV statistics with moderatedata records, prompted recent second-order methods whichrely on complementing the TV channel’s diversity with timediversity (offered when oversampling the continuous-timeoutput) and/or with spatial diversity (appearing when outputdata are collected from multiple antennas) [10], [16], [31],[32], [49].

It is the objective of this paper to review, unify, andextend these second-order diversity combining approachesfor blind identification and equalization of finite impulseresponse (FIR) TV communication channels where the vari-ation of the channel is modeled by a basis expansion. To putTV approaches in context, the random model is reviewedbriefly in Section II, followed by the basis expansion modelintroduced in Section III. With the rapidly fading mobilechannel as a paradigm, subsequent presentation focuses oncyclostationary methods used to estimate the frequencies ofthe Fourier bases. Section IV describes blind TV channelestimation methods which utilize the whiteness of the inputand rely upon output samples collected at one or twosensors. A deterministic approach is also reviewed alongwith order selection techniques, which are developed todetermine not only the channel memory, but also the num-ber of bases necessary in the expansion (this corresponds tothe number of dominant reflectors in a multipath terrain).Mean-square error (MSE, Wiener) equalizers are presentedin Section V along with direct blind equalizers derivedin a deterministic framework. The latter lend themselvesnaturally to adaptive schemes and allow almost perfect

equalization when the signal-to-noise ratio is high, whileimposing minimal assumptions on the input. Representativesimulations are given in Section VI, while conclusions,topics not covered, and thoughts for future research aredelineated in Section VII (more technical proofs can befound in the Appendixes).

Bold upper (lower) case will denote matrices (columnvectors). Prime will stand for Hermittian transpose,forconjugate, for transpose, for pseudo-inverse, forKronecker product, for range, and for null space.

II. FADING CHANNELS: RANDOM MODELS

In some communication schemes, unpredictable changesin the medium warrant modeling the TV impulse response(TVIR) as a stochastic process in the time variable. Using central limit theorem arguments, the TVIR is

usually approximated as a complex Gaussian process. Itis common practice to assume the channel to be wide sensestationary for a fixed lag and uncorrelated for differ-ent lags (i.e., wide sense stationary uncorrelated channel(WSSUC) assumption) [35]. The channel spectral densityfor a fixed is called the scattering function ,and it fully characterizes the second-order statistics of theWSSUC. There are several other functions that rely on thesecond-order statistics of the random channel. The integral

is called the Doppler spectrum,and its extent, the Doppler spread, is a measure of thechannel’s time variation. The so-called multipath inten-sity profile describes how the outputpower varies as a function of the delay, and the length ofits support is called the multipath spread and is a measureof the average extent of the multipath.

The characterization of the random channel mainly hasbeen used to analyze and simulate existing methods ratherthan to undo the TV distortion the fading channel has onthe input signal. However, recent work in [6], [25], and[50] has addressed the TV channel identification problemby casting it in discrete time and using a Kalman filter totrack the channel parameters.

Consider the fading communication system model ofFig. 1 before the sampler with the input/output (I/O) re-lationship

(1)

where subscript denotes continuous time, is theconvolution of the spectral pulse , the TV impulseresponse , the receive-filter , is thesequence of input symbols, and is the noise process.If the output is sampled at the symbol rate ,Fig. 1 can be simplified into Fig. 2 with an I/O relation indiscrete-time

(2)

1970 PROCEEDINGS OF THE IEEE, VOL. 86, NO. 10, OCTOBER 1998

Fig. 1. Continuous-time TV communication system.

Fig. 2. TV channel model.

where is truncated to an order ,which is common practice in communications applications.Note that if , then (2) yields a time in-variant frequency selective channel and the I/O relationshipbecomes . On the otherhand, if , (2) yields a time selective channel with anI/O relationship, . One approach incharacterizing the variation of the impulse responseis to consider it as a stochastic process in the time index.In communications, tracking the variations of the channeltaps is of importance [35]. To this end, fitting parametricmodels to the variation of the channel coefficients hasbeen proposed in [25] and [50]. In [50], the challengingtask of estimating random channel parameters from I/Odata has been tackled. The channel wasassumed to obey an vector AR( ) model

(3)

where , and is an in-dependently, identically distributed (i.i.d) circular complexGaussian vector process whose components are uncorre-lated with each other. The coefficient matriceswere estimated using the multichannel Yule–Walker equa-tions

(4)

where is the channel cor-relation matrix whose entries were estimated consistentlyfrom output statistics conditioned on the input [50]. With

available, we can solve for using (4). Oncethe AR parameter matrices are estimated, a Kalman filteris employed to track the channel coefficients after castingthe AR model in (3) in a state-space form

(5)

where is the channelstate vector, is a constant matrix consisting of the ARparameter matrices in its first block row,

identity matrices in its first sub-block diagonal and zeroelsewhere, and . Finally, a decision feedbackequalizer using the channel estimates is utilized to obtainan estimate of the input (see [50] and references therein).

III. FADING CHANNELS: BASIS EXPANSION MODELS

Consider the random variation in one tap,, of amultipath mobile radio channel [27], [28]

(6)

where is the amplitude of the th path, is a uni-formly distributed random variable in , is thewavelength corresponding to the carrier frequency, andis the speed of the mobile [28, p. 382]. For sufficientlylarge , the amplitude of (6) approximates a Rayleighprobability density function (pdf), and the power spectrumof (6) provides a discrete approximation to experimentallymeasured fading spectra which are of the form

, where is a constantdetermining the power of tap , and is the carrierfrequency [27], [28].

As an alternative to the random channel assumption ofthe previous section, where is a realization of astochastic process, the variation in the impulse response canbe captured deterministically by means of a basis expansion

(7)

where the TI parameters , together with the

bases characterize the system. It is clear that(7) with and with

have the same functional formas the model in (6). In Fig. 3, we depict how time selec-tivity, frequency selectivity, and time-frequency selectivitymanifest themselves in plots generated by (6), which issubsumed by the basis expansion model discussed in thissection. As the number of paths increases (chosen tobe ten in Fig. 3), the basis expansion model approximatesthe well-known random coefficient fading models used tosimulate mobile communication channels [27], [28].

In summary, random coefficient models are used eitherfor identification of the model parameters, which determinethe evolution of the channel coefficients, from the stationarymoments of the output, or they are used for simulatingfading channels with certain spectral properties. Interest-ingly, random coefficient models used to simulate mobilechannels can be obtained from the basis expansion modelwith random parameters. In this paper we will focus onterrains entailing only a few reflectors so that the Dopplerand multipath parameters can be considered deterministic.We will rely on the basis expansion channel model toperform blind identification and equalization.

GIANNAKIS AND TEPEDELENLIOGLU: BLIND IDENTIFICATION OF TIME-VARYING CHANNELS 1971

Fig. 3. Fading channels generated by the basis expansion model.

A. Exponential Basis Expansion Model

To appreciate the usefulness of complex exponentialbases, consider a communication signal

Re

transmitted through a TV multipath channel

where are the input symbols, is the number of paths,and , denote each path’s TV attenuation anddelay, respectively. With reference to Fig. 1, we convolve

with and remove the carrier to arrive atthe received signal-plus-noise model in baseband form:

. To suppressthe additive white noise Gaussian (AWGN) , we filter

through the receive-filter and obtain

(8)

Let denote the time in-variant (TI) transmit-receive filters in cascade, and assumethe following:

a1) constant attenuation and delay over a symbol, i.e.,const. , for ,

const. , for ;

a2) linearly varying delays across symbols (valid forapproximately constant path velocity), i.e.,

, where is proportional to the path velocityand . This is a first-order approximation of thedelay variation. Existence of higher order termswould yield polynomial phase signals, which bringsup the tradeoff between accuracy and complexity;this is outside the scope of this paper.

Under a1), we can pull and outside theintegral in (8), using the definition of and after achange of variables we have

, where

(9)After sampling the output at the symbol rate

(fractional sampling will be considered in Section IV-B),and using a2), we obtain

. Ifwe further assume that the

is approximately constant withrespect to since it is changing slowly compared to theexponential, we obtain

(10)

with and . Notice thatwith we arrive at the basis ex-pansion model in (7); so the exponentials’ dependence onlag can be included in the parameters, yielding an I/Orelationship (see also Fig. 4)

(11)


Fig. 4. Multichannel discrete-time equivalent of a TV basis ex-pansion model.

The complex exponentials in (11) can be viewed as eachpath’s Doppler arising due to motion—an effect also en-countered in radar and sonar where moving targets in-duce TV delays which for narrow-band signals manifestthemselves as TV phases [38]. Since the same input ismodulated by different complex exponentials in Fig. 4,some redundancy is introduced at the output which we callchannel (or Doppler) diversity, a term also used in [39] ina time-frequency context.

Our TV channel parametrization is not unique. An alter-native one is proposed in [40] as

(12)

where is the delay (expressed in multiples of the symbolinterval ), and denotes the Doppler frequency shift(normalized by the symbol rate ) of the th path relativeto the zeroth path. The I/O relation in (12) is simpler sinceit only involves a single sum, but it can be shown that thebasis expansion model can capture a more general class oftime variations, and hence (11) subsumes (12). In view ofthis fact, we focus on the basis expansion model in (11).

In this paper, given , we wouldlike to do the following : 1) estimate ; 2) determinethe channel length and the number of bases; and 3)estimate , or the equalizers which, whenconvolved with the data , yield input estimates ,and estimate the equalizer length. First, we will addressfrequency estimation.

B. Estimating the Exponential Bases

In all the methods that follow, to estimate the TI pa-rameters , we will assume the knowledge of thebases. So for complex exponential bases, the question ofestimating the frequencies from the outputin (11) needs to be addressed.

The idea is to exploit the cyclostationarity of anduse its TV moments that only depend on the time indexthrough the complex exponentials [46]. The frequencies ofthese exponentials are calculated from the so-called cyclicmoments, the Fourier series of the TV moments (e.g.,see [12] for detailed definitions). The input will beassumed to be independent of the noise , white, withmean and variance . Let

(13)

where ( ) is the number of conjugated (unconjugated)terms, so that, e.g., , and

. In this notation, the dependence onwill be dropped when the process is stationary. Clearly,if the input has nonzero mean, the frequencies can befound by computing the Fourier Series of

. A simple calcula-tion on (11) will reveal that if any moment of the input iszero, then the corresponding moment of the output will alsobe zero, thereby preventing us from estimating . Ifthe input is coming from a real, zero-mean constellationsuch as binary PAM, then and

(14)

enables us to find the frequencies, since the only dependence of

(14) on is through the exponentials. The zero lag ischosen for convenience, but if the term in the bracketsin (14) is small for some pair , then different lags

could be utilized. From it is possible toobtain as follows: let . Then,

from which we can find . The nextsmallest frequency is , from which could befound. Knowing , we can discard from , sincewe do not know whether , and find from

. This procedure enables the computationfrom the knowledge of .

Unfortunately, for a class of important constellations (4-quadrature amplitude modulation (QAM), 16-QAM), due totheir symmetry, the unconjugated correlation of the input

, therefore when the symbolsare equiprobable. Thus, we are prompted to use

which enables the estimation of. But it is not possible to obtain

from , so higher order moments of theoutput must be used. Due to their symmetry, all oddordered moments of many constellations are zero, but theirfourth-order moments are nonzero. It is possible to obtain

from . The method discussed earlierto obtain from can also be employed,with slight modifications, to calculate the frequencies from

. Since second-order statistics generally havelower variance than higher order statistics, knowledge ofthe difference frequencies , whose estimates rely onsecond order statistics, can be incorporated in the aboveprocedure [46].

The frequencies can be obtained using sample estimatesof the cyclic moments which are defined as the Fourier


Series of (13) with respect to. The estimators for thecyclic moments are [12]

(15)

which can be computed efficiently by taking the fast Fouriertransform (FFT) of the output product. For example, in thecase of a pulse amplitude modulation (PAM) constellation,as explained earlier in this section, the smallest frequencycan be obtained by .Cyclic moment estimators are known to be asymptoticallynormal and mean square consistent when the input has finitemoments and the subchannels are of finite length, so that theoutput satisfies the necessary mixing conditions [5], [12].

IV. BLIND TV CHANNEL IDENTIFICATION

Throughout the rest of the paper, the channel coefficientswill be assumed to be deterministic, but some ap-

proaches (which we term “statistical”) will require the inputto be random and white. The bases are

assumed to be known.

A. Statistical Approach 1: One Sensor

Here we will not necessarily assume the bases are com-plex exponentials, for reasons that will soon be given. Fromthe correlations of in (11), we can obtain

(16)

Given , if the product sequencesare linearly

independent, by solving the linear equations in (16) wecan obtain the (deterministic) correlations of all possibleTI coefficient pairs of channels

(17)

The problem of obtaining from (17) can besolved with conventional subspace approaches ifhas been estimated from (16). Subspace approaches havebeen used to estimate a set of coprime TI channels excitedby a common white input (see Fig. 5) [33], [45].Notice that the output auto and cross correlations of thesingle-input multiple-output (SIMO) system in Fig. 5 pro-vides (up to a scale) all possible deterministic correlationsin (17) of a set of FIR channels to be estimated.Hence, both the problem of estimating from

, as encountered in the TI SIMO blind identification

Fig. 5. TI SIMO model.

problem, and the TV single-input single-output (SISO)problem of this section can be solved by the same subspacemethod.

Theorem 1 [49]: Sufficient conditions for identifiabil-ity of from are: 1) for everyfixed the product sequences

are linearly independent; 2) the poly-nomials , do nothave common roots; and 3) the product sequences

are bounded , and

is invertible,where .

The method developed in [49] entails two steps: firstis obtained from , where we need 1),

since a matrix whose columns are formed byneeds to be inverted. Second, needs to be

recovered from , where 2) becomes necessary. Thenontrivial task of estimating the TV statistics ishandled by using an instantaneous estimate ,and consistency of is established under 3), whichrequires some additional boundedness conditions on thebases, and 1) to hold in the limit.

The problem with the method derived from Theorem 1is that the linear independence assumption a2) on the basesis often not satisfied in practice. Nevertheless, the singlesensor approach illustrates nicely that TV channels offerdiversity not available with TI channels, and from this pointof view blind identification based on second-order statisticsis easier for TV channels of structured variation thanTI channels. However, the linear independence conditionnecessary with a single sensor does not hold for complexexponentials, since for ,

. This brings about looking for alternative ways ofobtaining complementary diversity.

B. Statistical Approach 2: Two Sensors

Just like the TI case [15], [52], sampling faster thanthe symbol rate creates diversity that enables the problemto be cast into a SIMO framework. Suppose in(1) is sampled at a rate , whereis given in (9). We obtain the discrete time model

, whereand .

Oversampling offers diversity manifested in thesubprocesses defined as .In the filterbank literature, are termed thepolyphase components of and can be expressedin terms of the subchannels and the


corresponding noise as

(18)

where

.Because the variation of and with respect to isoften negligible relative to that of the exponential, it isreasonable to assume the following:

a3.1) ;a3.2)

.

Based on a1)–a3) we have

(19)

where. Combining (18) and (19) and stacking

the -channel data andwe obtain

(20)

where, as in (11), the exponentials’ dependence onisabsorbed in the TI vector impulse response.

Multichannel diversity can also be achieved by usingmultiple antennas at the receiver [31], the number ofwhich will be denoted also by . With the availabilityof oversampling (time diversity) or multiple sensors (spacediversity) the following question arises: what is , theminimum in order to guarantee identifiability withoutrestrictions on the frequencies of the exponential bases?As we will see in Section IV-C, zero-forcing FIR solutionsrequire to be on the order of . If the input canbe assumed to be white and random, on the other hand,

, which does not depend on and motivates thetwo-sensor approach of this section.

Consider the I/O relation in (20). Given, a white input sequence ,

and a distinct set of cycles , thegoal is to identify .Since the input is white, the output correlations of the twochannels are

(21)

The Fourier Series coefficients of the (almost) periodicsequence of in (21) are

(22)

Taking the -transform of (22) with respect to andassuming that , we arrive at the so-called cross-cyclicspectrum

(23)

Identification of the TI subchannels isachieved by choosing the appropriate cyclesin thecyclic spectra of (23) so that only a few unknownterms out of the summation survive. In the set ,defined in Section III-B, at least one difference (namely,

) lets a single term survive out of (23), whichis the product . InAppendix I it is shown that this product enables estimationof the subchannels corresponding to the minimumand the maximum frequencies:

. After estimating all subchannels correspondingto frequencies in , that force all but one term in (23)to be zero, it can be shown that there is a way to use (23)by choosing from in decreasing order so thatthe sum will only contain two products that have unknownsubchannels in it. This is all summarized in Theorem 2(see Appendix I for a proof).

Theorem 2: For , ,and any set of frequencies , the followingis possible.

1) To identify the subchannels for , such thatthere exists a with

[in other words, subchannels for suchthat there exists a that enables only one product tosurvive out of (23)] with the choice .The identifiability condition for these subchannelsis that are coprime for all

.2) After estimating all subchannels characterized in 1)

(among which are ),to identify the remaining subchannels using

. Identifiability is guaranteed if

and are coprimefor , whenever there exists a with

.


Multichannel diversity removes the severe conditions onthe basis functions from which [49] suffers. In additionto allowing the minimum diversity ( ) for bases ofarbitrary frequencies, the two-cycle method also identifiesthe channel coefficients by use of the cyclic correlations thatavoid the zero cycle ( ). This makes additivestationary noise tolerable down to low SNR’s [12], a featurealso illustrated in the simulations.

C. Indirect Deterministic Approach

In this section we will show how, with sufficient diver-sity, it is possible to estimate the subchannels and obtainperfect estimates of the input in the absence of noise.Similar to the approaches in [15] and [52] for TI systems,these (so called “deterministic”) methods do not requirethe input to be white or random, thereby allowing the useof coded inputs. Unlike the statistical approaches, reliableidentification will be possible with short data records if theSNR is high enough.

First, we will discuss a subspace approach that weterm “indirect approach,” introduced in [31] and [32]. InSection V-B, direct blind equalizers will be derived underalmost identical assumptions.

In order to cast (20) in matrix form, we let

and define for each theblock Toeplitz matrix

......

...

.... . .

...

(24)

Consider (20) in the noise-free case and form theblock Hankel data matrix

......

... (25)

where the modulated input matrixand the channel matrix

are given by

......

...

... (26)

In (26), are Hankel matricesconstructed from .

Under the following assumptions it will become possibleto estimate the channel matrix up to a matrixambiguity , which agrees with the fact that for the TIcase ( ) the ambiguity is a scalar.

a4) , which is easily satisfied bycollecting sufficient data;

a5) is at least fat, i.e., the quadrupletobeys

(27)

To satisfy (27), a minimum chan-nels are required with a minimum equalizer order

(in the TI case, and[45], [52]).

a6) is full rank, i.e., rank whichrequires that transfer functions

are coprime for every fixed. This is be-cause, if the family of polynomials

have common factors for some, then,will lose rank (see e.g., [45]), and hence willhave linearly dependent rows.

a7) Bases are sufficiently varying andis persistently exciting (p.e.) of sufficient order toassure that rank . We stressthat can be either random or deterministic.

To determine within the matrix ambiguity , let usconsider (25) and the eigendecomposition

(28)

Under a7), . Since the signal subspaceis orthogonal to the noise subspace, if , then

, and using (26), , for .Since is a convolution operator, this can also be writtenas , implying that

, where is a block Toeplitz matrix as definedin (24). Hence, if are basis vectors for ,it follows that there exists a full-rank matrix suchthat

(29)

Let denote the null eigenvectors of ,and and be constructed exactlylike and in (24) and (26). If we deconvolvethe data with , we obtain:

. The latter impliesthat is such that

(30)

where is the th entry of . Keeping in mind thatis a matrix formed by the modulated input sequence

, we can write (30) as

(31)


where is an entry of and. Equation (31) can be rearranged

to obtain

(32)

where is the th column of . Given , (31),and therefore (32), has a unique solution for and(see Appendix II for a proof). Hence, (32) can be cast in amatrix form to obtain and , which after using (29)yields .

In summary, to estimate and the input, we need toperform a singular value decomposition (SVD) on tofind the vector corresponding to the minimum eigenvalueof . Then another SVD is performed on , andthe vectors corresponding to its minimum singularvalues yield estimates of the ambiguous channels .Upon constructing as in (24), we deconvolve the data bycomputing to obtain . Based on , the uniquesolution of (32) can be found by casting it in a matrix formto obtain both and the input estimates.

This method requires at least one vector in the noisesubspace, so is necessary.An alternative method described in Section V-B allows

to be square and calculates the columns of its (right)inverse (vectors of equalizer coefficients) directly, usingthe structure of the input matrix.

D. Order Determination

Up to this point we have assumed that the channel orderand the number of basesand were known. To assert

that these blind methods are applicable, one needs to showthat it is possible to obtain these quantities from outputdata. Using the rank properties of the output data matrixin (22), it is possible to obtain the channel orderand thenumber of bases [10], [31].

Under a4)–a7), matrix in (25) has rank .With denoting known upper bounds on ,corresponding matrices and will have rank

, . It is thus possible to select theorders and using

rank rank

rank

With , available, is chosen to satisfy (29) for agiven .

At low SNR’s, noise will make it difficult to discernsmall significant singular values of and from largeinsignificant ones. More elaborate tests involving informa-tion theoretic criteria, such as the AIC, seem possible butare beyond the scope of this paper.

V. BLIND EQUALIZATION OF TV CHANNELS

In this section we will discuss methods for estimating theinput. Having the channel estimates available, maximum-likelihood decoding can be used for that purpose. Thehigh computational complexity of Viterbi’s algorithm iseven more pronounced for the TV model than the TIcase since the number of bases, as well as the channellength , affects the computational complexity. A decisionfeedback scheme has been proposed in [46] in connectionwith the exponential basis expansion model. Here wewill consider linear options: zero-forcing FIR equalizersrequiring enough diversity (at least ) in Section V-A and minimum mean square error (MMSE) solutions inSection V-B. Optimally weighted equalizers and adaptivealgorithms which pertain to the direct blind equalizationmethod are presented in Sections V-C and V-D.

A. Direct Blind Equalization

This method estimates FIR zero-forcing equalizers thatyield perfect estimates in the absence of noise withouthaving to estimate the channel first. Similar to the indirectmethod of Section IV-C no statistical assumptions on theinput are made. Estimation of the direct blind equalizers isless computationally demanding than the indirect method.Also, the linear form of the solution in this section enablesupdating the equalizer estimates adaptively (see Section V-D).

We seek FIR zero-forcing equalizersthat satisfy (see also Fig. 6)

(33)

where denotes a delay which is inherentlynonidentifiable in blind approaches.

To establish existence and uniqueness of such equalizers,we need in (26) to be fat or square so that a thatsatisfies exists. The thcolumn of is . For the directblind equalization method, beyond assumptions a4), a6),and a7), required also for indirect channel estimation, weallow to be square so that a5)[ in (26) is square] is permissible for the method to work.

In order to find the equalizers we first setin (33) and collect equations to obtain

(34)

where

diag


Fig. 6. Vector TV model and FIR vector equalizers.

We use MATLAB’s notation to denote a sub-matrix of formed by the through rows and allcolumns of . So we define

(35)

Matrix is without its first rows, is withoutits last rows (likewise for the diagonal matrices, and themodulated input matrices are , to be used in AppendixIII).

From (34) and (35) it follows that

(36)

We note that ,which allows us to eliminate the input dependence from theequations in (36) and obtain the cross relation

(37)

The pair of equalizers will be uniquely iden-tifiable (up to a scale) as the eigenvector corresponding tothe minimum eigenvalue of in

(38)provided that the nullity . The result issummarized in Theorem 3 (see Appendix III for the proof).

Theorem 3: Under a5), a6), and a7), consider ,, , and. It then holds that , and hence

(38) has a unique solution. If instead of a5),holds ( is fat), then

, and all vectors in the nullspace of yield equalizers which, when convolvedwith the output data, yield perfect input estimates up to amultiple of a known complex exponential sequence in theabsence of noise.

The periodicity requirement on assumedin Theorem 3 can always be satisfied if .This is possible by using the techniques in Section III-B,with which we can easily infer the lowest frequency.Multiplying both sides of (20) with we can“shift” all frequencies by , so that the first basis functionof will be , which isperiodic with any period.

Requiring , , and only enablesus to find , but this is not a real concern since,

using , equalizers corresponding to other delaysand bases (other columns of) can be found using (37).

Strict inequality in a5) causes every equalizer vectorto lie in an affine space [the set of all vectors

satisfying , where is a unitvector with a 1 in its th position]. This gives us morefreedom in choosing the appropriate equalizer with goodnoise suppression characteristics. As mentioned in [14] and[15], if the noise is white, the equalizer with the minimumnorm will have minimum noise variance at its output.

B. Cylic MMSE Equalizers

Consider the I/O relation in (20). We wish to find, foreach , a vector sothat the following MSE is minimized:

(39)

The orthogonality principle yields

(40)

We need to write the set of linear equations in (40) in matrixform and in terms of the estimated channel parameters

and frequencies . To thisend, we define the following:

(41)


diag (42)

......

...

(43)

Using the definitions (41) and (43), it is easy to verify from(20) that the following equation holds:

(44)

where is the corresponding noise matrix. Equation(40) can be rewritten as

vec vec vec

(45)

where vec denotes the vector formed fromconcatenating the columns of . We need to find anexpression for vec using (44), and substituteit into (45) in order to write in terms of

and . Taking the transpose of (44) we have

vec vec

vec vec . Using theidentity vec vec , we can concludethe above string of equalities by

vec vec vec (46)

where . Assuming the noiseis white, and using (45) and (46) we have

vec vec

vec (47)

Next, we define vec vecand using (43) conclude that is an

symmetric block Toeplitz matrixwith a first block-row , wheredenotes an matrix with ones on theth subdiagonal, and zero elsewhere. If we also define

vec , we can express (47) as

(48)

One can also obtain a closed form expression for the MMSEafter substituting (48) into (39)

(49)

In (47), an inversion takes place for each value of,unless the frequencies are commensurate, in which casethe matrix to be inverted is periodic.

C. Weighted Equalizers

Even when is square, the presence of multiple equaliz-ers corresponding to different delays and bases can be usedto improve the multiple input estimates in the presence ofnoise. By aligning, demodulating, and performing weightedcombinations of the estimated columns of, one may getbetter input estimates than using a single equalizer. Let

(50)

We wish to minimize the cost function¸

with respect to the vector of weights, which

after using (33) is

(51)

If we constrain the sum of the elements of to be one,then that minimizes (51) is given by (see Appendix IV)

(52)

where is a vector of all ones, is a block diagonal matrixcontaining the matrices in its diagonal, theelement of which is

(53)

and .Notice that the optimum weights require the knowledge

of perfect equalizer values which cannot be obtained withnoisy data. But with sufficiently high SNR, which enablesaccurate equalizer estimates, the use of the weights oftenimproves the input estimates, as verified in our simulationexamples.

D. Adaptive Equalization

One advantage of the direct method of Section V-Bover the indirect method of Section IV-D is the fact thatequalizer estimates can be linearly related with the outputdata, and can be cast into an adaptive framework. Theadaptive method proposed to estimate the frequencies in[46] can be combined with what follows to construct analgorithm where both the basis frequencies and the channelparameters can be estimated online.

Equation (38) can be recast in a least squares frameworkby setting the first coefficient of to one and can


be rewritten as , where iswithout its first column, is the vector containing theelements of that column, and is without itsfirst element. It is well known that RLS is a recursive wayof computing , which also solves theleast squares problem [22]. We use this algorithm to updatethe vector of equalizer coefficients.

One could also be interested in using the computation-ally less intensive LMS algorithm at the expense of lessaccuracy and slower convergence. In the absence of atraining sequence (desired input), we consider the elementsof as our desired sequence that we would like toestimate. Here, are the rows of and is the estimateof the vector of equalizer coefficients at time. At eachiteration, the vector of equalizer coefficients is updated bythe relations

(54)

where is the step size parameter and denotes thscalar entry of .

It should be noted that rapid variations of the channel aretaken care of by the bases, whereas slower changes in theparameters are tracked by the adaptive algorithm. Since thevariation is built in the model, the algorithms can operateon longer data records with less worry about violating thestationarity assumption.

VI. SIMULATIONS

In this section we illustrate some of the methods and al-gorithms that are discussed and compare them. For this pur-pose we will need the following definitions: the output SNRis defined as SNR ,where is the noise-free output data, and the normalizedroot mean square error (RMSE) between a vectorand itsestimate is computed as follows:

RMSE (55)

where stands for realization and is the number ofrealizations.

Unless otherwise indicated, sensors were usedwith a channel order . The bases werechosen as and . Allplots except the eye diagrams are an average of 100 MonteCarlo runs unless otherwise indicated.

In Fig. 7, we illustrate how the blind algorithm that isdeveloped in [15] for TI channels compares with the oneproposed in Section V-A, when the data comes from arapidly fading TV channel. We see that the TI algorithmis not capable of equalizing the symbols comingfrom a 16-QAM constellation even with a high SNR of 45dB. This motivating example demonstrates the inadequacyof TI equalization algorithms when applied to TV channels.

Fig. 8 illustrates the frequency estimation of Section III-B using data at an SNR = 10 dB. Since there wasonly one nonzero cycle , provided enough

Fig. 7. TI and TV algorithms on TV data.

Fig. 8. Estimation of basis frequencies.

information to estimate it. The plot on the left shows that theFourier Series of has two peaks: one at(due to the stationary noise) and the other . Theplot on the right illustrates the estimation of the same cyclewith fourth-order cyclic moment at lag computed asthe FFT of . We observe peaks at multiples of .The reduced variance of second order statistics relative tofourth-order statistics is also apparent.

In Fig. 9, the five least significant singular values of thematrix are plotted for (left) and (right)and for SNR’s of 50 and 25 dB. Only sampleswere used. The number of least significant singular values(zero singular values in the case of no noise) determines therank of the noise free output data matrix, which, as shownin Section IV-D, enables the estimation of, , and .With an SNR dB, the insignificant singular valuesare still discernible. As the SNR’s get lower (to 25 dB),it becomes more difficult to tell how many zero singularvalues there are since the noise not only increases them butalso perturbs their relative values. The standard deviationof the singular values is also plotted around the mean whichwas estimated from 100 realizations.

Fig. 10 illustrates the two sensor approach of Section IV-B, where the channel coefficients are estimated from thecyclic correlations of the output. The RMSE between thetrue channel coefficients corresponding to

is plotted versus SNR (200 Monte Carlo runs,) and the number of data (500 Monte Carlo runs,

SNR dB). We see that (unlike the deterministicmethods) the channel estimates are consistent and improvesignificantly with the number of data. In addition, the effectof noise is minimal due to the use of nonzero cycles in thecyclic correlations.


Fig. 9. Order determination.

Fig. 10. Two sensor (statistical) approach.

Fig. 11. Before and after equalization.

Fig. 11 illustrates MMSE equalization for the estimatedchannel where the channel estimation was done with thetwo-sensor approach. We show the eye diagram for thesymbol estimates at an SNR dB. The unequalizedchannel output is shown on the left plot; the right plotis obtained by using the channel estimates obtained with

data points and then using the cyclic MMSEequalizer of length 15 [ , in (39)].

In Fig. 12 the deterministic methods are compared. Thedirect method is implemented in two different ways. Thefirst one, referred to as “direct,” uses what is suggestedright before (38). The “min-norm” approach substitutes the“direct” estimate in (37) and solves (36) with respectto constraining it to have minimum norm. MSE of

Fig. 12. Direct, indirect, and minimum norm methods versusSNR.

Fig. 13. Performance versus number of data.

Fig. 14. RLS algorithm, approximate initialization.

the input estimates are plotted for two different equalizerlengths . The number of data used for theseestimates was only . Equalizers with minimumnorm consistently outperform those obtained using (38).The direct and minimum norm methods perform better thanthe method in [31] (l-gg) for low ( 25 dB) SNR’s.

To see how much the deterministic methods improve withincreased data length, Fig. 13 compares the minimum norm,direct, and indirect methods. It is seen that the minimumnorm method benefits from the increase of the data lengthmore consistently than the other two methods. An SNR of25 dB was used.

Figs. 14–16 illustrate the performance of the adaptivealgorithms proposed in Section V-D. Here , and

and were chosen. Fig. 14 shows theeye diagrams for the output of an equalizer obtained withthe RLS algorithm. Fig. 15 illustrates the performance ofthe RLS algorithm by plotting the error of the equalizerestimates and also the error in the input estimates. Fig. 16


Fig. 15. RLS with the number of iterations.

Fig. 16. Performance of the LMS algorithm.

Fig. 17. Zero-delay versus average equalizers.

is the same as Fig. 15 except for the LMS algorithm. TheRLS was initialized by , whereas the LMS was initializedwith the batch estimate obtained with the minimum numberof symbols required by a4).

In Fig. 17 effects of weighting of different equalizers onthe input estimates are demonstrated. Here the equalizersweighted by the inverse of their norms (right) yielded betterestimates than , the zero-delay equalizer (left).

These preliminary simulations illustrate the differencebetween the “statistical” and the “deterministic” approachesfor the TV model we have justified and adopted. Whilethe former relies on cyclic correlations and is effectedminimally by the presence of noise, it needs relatively longdata records for accurate estimates. The zero-forcing FIRsolutions, on the other hand, yield good estimates at highSNR’s with short data records, but their noise tolerance israther small.

VII. CONCLUSIONS AND FUTURE DIRECTIONS

Finitely parameterized basis expansions turn SISO TVsystems into multivariate TI systems with inputs formedby modulating a single input with the bases. Fourier basesare well motivated for modeling rapidly fading mobile com-munication channels when multipath propagation caused bya few dominant reflectors gives rise to (Doppler-induced)

linearly varying path delays. Doppler frequencies can beestimated blindly using cyclic statistics, and channel orderscan be determined from rank properties of a received datamatrix.

Structured variations described by bases offer TV channeldiversity, which renders blind identification of TV modelseasier than that of TI models. When channel (or Doppler)diversity is complemented by temporal or spatial diver-sity (available with oversampling or multiple antennas),blind estimators of TV channels along with direct equal-izers become available even with minimal (persistence-of-excitation) assumptions about the input and the bases. Theequalizers are TI, multivariate, zero-forcing (in the absenceof noise), and lend themselves to optimally weighted andadaptive algorithms. The latter provide fine tuning forpossible model mismatch of the bases, which capture thenominal part of the rapidly fading channel. Exploitation ofthe input’s whiteness reduces the amount of spatio/temporaldiversity (only two sensors) needed to identify blindly TVchannels and mitigate their effects using MMSE equalizers.

The blind channel estimation and identification methodspresented in this paper relied on second-order output infor-mation only. In [47], blind higher order statistical methodshave been developed which rely on the independence ofthe input but are capable of identifying TV channels usingsingle sensor data only. Following the start-up mode, blindmethods switch on to a decision-directed mode. Decisionfeedback equalizers for the TV basis expansion model havebeen reported in [46] along with adaptive methods foron-line estimation of the basis frequencies.

A number of interesting directions open up for futureresearch: 1) performance analysis of the channel estimators,especially when model perturbations due to synchroniza-tion effects and Doppler frequency drifts are present; 2)theoretical evaluation in terms of error probability for thezero-forcing equalizers and experimental comparisons withthe MSE equalizers; 3) extensions of blind methods to TVpole-zero channel models; 4) exploitation of input redun-dancy in the form of short training sequences (semi-blindextensions), modulation, codes, or filterbanks in order toidentify TV-basis expansion models without oversamplingor deployment of multiple antennas. Such input-diversitytechniques have gained popularity recently for blind iden-tification of TI channels (see [4], [11], [19], [30], [41],and [48], and references therein); 5) diversity techniquesfor blind identification of random coefficient models andperformance comparisons with the basis expansion modelsusing real data.

APPENDIX IPROOF OF THEOREM 1

We will use the notation to denote a convolutionmatrix with Toeplitz structure associated with the vector

, whose first column is , and first row is, where is the first element of

. The dimensions depend on the size of the vector thatis multiplying and will be clear from the context.


a) Let and be a pair such thatin (23).

Then, since and are coprime, the followingholds:

(56)

Define, the matrix

, and. Cross-multiplying in (56), taking

the inverse -transform of both sides, and casting theresulting convolutions in matrix form we obtain

(57)

The solution in (57) is unique up to a scale, since ifchannels also satisfy (56), then

and the numerators and denominators of both sides mustbe equal. This is easily shown by factoring the numeratorand the denominator polynomials into theirfactors. Weshowed how to estimate . Subchan-nels corresponding to can be estimated with a similarprocedure.

b) We showed in a) how to estimate

using , since only thepair and can give rise to the difference .Consider now . If , is theonly pair of frequencies that has the difference

, then is the product of twopolynomials, one of which is known. This will enableestimating . If, on the other hand,

(the only other pair that couldpossibly give rise to this difference), then we have thesum of two products of polynomials where two of the fourpolynomials are known

(58)

After taking the inverse-transform of both sides, (58) canbe cast in matrix form,

(59)

where

and vector contains the inverse -transformof . If and

are coprime, thematrix in (59) has full row rank, which will

enable us to determine up to a scale ambiguity.

After estimating , we can

repeat the same procedure with

. This time, might contain a sumof three products, but if this is the case one of the productshas to involve which hasbeen estimated. Proceeding in this fashion, all subchan-nels can be estimated provided that

and are coprime for ,whenever there exists a with .

APPENDIX IIUNIQUENESS PROOF OF (32)

As mentioned in Section V-B, without loss of generalitywe will assume . Suppose now, inaddition to and , and also satisfy (31).Relating them, we have

(60)

provided that is nonzero. Equating the first element ofboth sides in (60), and likewise the last elements, we obtain

(61)

where denotes the element of matrix defined in(60). Using the last equality in (61) we can relate the firstand last columns of and write it for toobtain the matrix equation (62) as shown at the bottom ofthe next page. Equation (62) has a unique solution (up to ascale) with anddue to the Vandermonde structure ofin (62) and the factthat its first and last columns are identical. This means thefirst and the last elements of (60) are equal toand areindependent of . Thus, (60) implies that ,and ; hence, .


APPENDIX IIIPROOF OF THEOREM 2

Assume first that is square. The nullity of

(63)

is one if and only if the nullity of is one becauseis full rank and square. Because matrix in (26) is fullrank, might lose rank only if columns of one portionare linearly dependent with columns of the other portion.Note that matrices andshare columns. Careful examinationof in (63) will reveal that they involve the products

and , which havecolumns in common. So is a necessarycondition for . But other products such

as and could alsohave common columns. In fact, if is periodicwith period , the first column of

is identical with the last column ofwhenever . Choosing and

will ensure , , and along with, will guarantee that , therefore

.If ( is fat), then

relying on a6) and using the Sylvester inequality, weget rank , therefore,

rank, where the inequality

follows from .

APPENDIX IVDERIVATION OF OPTIMUM WEIGHTS

We wish to minimize (51) subject to the constraint. We will use the method of

Lagrange multipliers to solve this constrained optimizationproblem (e.g., [22, pp. 557–560]). The constanthas noaffect on the relative weights , since the scaling of isabsorbed into the Lagrange multiplier. By writing (51) asa term times its conjugate and moving the expected value

inside we can express it as

(64)

where we usedand . The objective

function to be minimized now becomes

(65)

where is the Lagrange multiplier that will determine thescaling of to fit the constraint, is a block diagonalmatrix containing the matrices in its diagonal, andthe is given by the sum inside the brackets in (64).Since is not Hermitian-symmetric ( ), we write

where is Hermitian symmetric and is Hermittian anti-symmetric. Because (51), and therefore (64), is real, anddue to the Hermitian antisymmetry of, is purelyimaginary; thus it follows that . Retaining theHermitian part of amounts to an extra term in (53) toguarantee . Now we use the standard result

for a Hermitian matrix [22] to takethe gradient of the objective function in (65) and equate itto zero

(66)

Equation (66) enables us to solve for which, afterimposing the constraint, leads to (52) and (53).

ACKNOWLEDGMENT

The authors wish to thank Prof. H. Liu, former graduatestudents, and especially Prof. M. K. Tsatsanis of StevensInstitute of Technology for his collaboration and discussionswhich initiated and formulated a number of topics in thispaper.

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Georgios B. Giannakis(Fellow, IEEE) receivedthe Diploma in electrical engineering fromthe National Technical University of Athens,Greece, in 1981. He received the M.Sc. degreesin electrical engineering and mathematics in1983 and 1986, respectively, and the Ph.D.degree in electrical engineering in 1986 fromthe the University of Southern California (USC),Los Angeles, CA.

After lecturing for one year at USC, he joinedthe University of Virginia, Charlottesville, in

September 1987, where he is now a Professor with the Department ofElectrical Engineering. His general interests are in the areas of signalprocessing, communications, estimation and detection theory, and systemidentification.

Dr. Giannakis received the IEEE Signal Processing Society’s 1992Paper Award in the Statistical Signal and Array Processing (SSAP) area.He co-organized the 1993 IEEE Signal Processing Workshop on Higher-Order Statistics, the 1996 IEEE Workshop on Statistical Signal and ArrayProcessing, and the first IEEE Signal Processing Workshop on WirelessCommunications in 1997. He was a Guest Editor for two special issueson high-order statistics for theInternational Journal of Adaptive Controland Signal Processing,for the EURASIP journalSignal Processing,and for the IEEE TRANSACTIONS ON SIGNAL PROCESSING January 1997Special Issue on Signal Processing for Advanced Communications. Hehas served as an Associate Editor for IEEE TRANSACTIONS ON SIGNAL

PROCESSINGand IEEE SIGNAL PROCESSINGLETTERS, as well as Secretaryof the Signal Processing Conference Board. He has been a member ofthe SP Publications board, the IMS, the European Association for SignalProcessing, Vice-Chair the SSAP Technical Committee, and Chair of theSP Technical Committee for Communications.

Cihan Tepedelenlioglu was born in Ankara,Turkey, in 1973. He received the B.S. degreewith highest honors from Florida Institute ofTechnology, Melbourne, in 1995 and the M.S.degree from the University of Virginia, Char-lottesville, in 1998, both in electrical engineer-ing.

Since 1995 he has been working as a Teachingand Research Assistant at University of Virginiaand is currently a Ph.D. student in the CCSPLabaratory. His research interests include statis-

tical signal processing, system identification, time-varying systems, andequalization of fading channels in digital communications.


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Basis Expansion Models And Diversity Techniques For Blind … · 2012. 8. 16. · Keywords— Adaptive equalizers, diversity methods, Doppler effect, fading channels, identiﬁcation,