Channel estimation using implicit training

240 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004

Channel Estimation Using Implicit TrainingAldo G. Orozco-Lugo, Member, IEEE, M. Mauricio Lara, Member, IEEE, and Des C. McLernon, Member, IEEE

Abstract—In this paper, a new method to perform channel es-timation is presented. It is shown that accurate estimation can beobtained when a training sequence is actually arithmetically addedto the information data as opposed to being placed in a separateempty time slot: hence, the word “implicit.” A closed-form solutionfor the estimation variance is derived, as well as the Cramér–Raolower bound. Conditions are derived for the training sequencesthat result in a channel estimation performance that is indepen-dent of the channel characteristics. In addition, estimation perfor-mance is shown to be independent of the modulation format. A pro-cedure to synthesize optimal training sequences is presented, andthe problem of synchronization is solved. The performance of thealgorithm is then compared with other methods that use explicittraining under GSM-like environmental conditions, and the newalgorithm is shown to be competitive with these. Finally, compar-isons are also carried out against blind methods over realistic ban-dlimited channels, and these show that the new method exhibitsgood performance.

Index Terms—Channel estimation, cyclostationarity, equaliza-tion, synchronization.

I. INTRODUCTION

B LIND channel estimation and equalization have been in-tensely studied since the original work of Sato in 1975

[1]. Although there appear to be so many different approachesand algorithms to solve the problem, all of them can be clas-sified into four basic types: Bussgang statistics, higher orderstatistics, cyclostationary statistics induced at the receiver viaover-sampling and/or multiple antennas, and cyclostationarystatistics induced at the transmitter. Recently, the concept ofsemi-blind estimation has appeared, where the transmitter sendssome known training symbols in specific positions, and a func-tional is constructed that depends on both known symbols andthe unknown ones. This way, the information carried out by theknown part is traditionally exploited, but the use of the unknownpart also enhances channel estimation performance. Therefore,the length of the training sequence can be shortened, which con-sequently improves bandwidth efficiency [2], [3].

The method to be presented in this paper belongs to the classof methods that induce cyclostationary statistics at the trans-mitter. Although the method could be considered blind in abroad sense, we have decided not to call it blind because thetraining sequence is added with the sole purpose of aiding the

Manuscript received June 5, 2002; revised April 19, 2003. The associate ed-itor coordinating the review of this paper and approving it for publication wasDr. Helmut Bölcskei.

A. G. Orozco-Lugo and M. M. Lara are with CINVESTAV-IPN, Sección deComunicaciones, CP. 07360, México City, México (e-mail: [email protected]; mlara@mail. cinvestav.mx).

D. C. McLernon is with the Institute of Integrated Information Systems,School of Electronic and Electrical Engineering, University of Leeds, LeedsLS2 9JT, U.K. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2003.819993

channel estimation. The notion of “implicit training” is used inthe paper title to distinguish the proposed method (where thetraining sequence is actually arithmetically added to the infor-mation data) from one where the training sequence is allocatedan empty time slot that is separate from the information data(as in GSM). This way, no bandwidth is lost in sending trainingdata, and since the “training sequence” cannot be seen explic-itly in the transmitted signal, channel estimation must be car-ried out using statistical information. The crux of the matter is,however, if the training sequence is periodic, then the receiveddata will exhibit cyclostationary statistics (specifically, a peri-odically time-varying mean) that can be exploited to performaccurate channel estimation.

While writing this paper (motivated by our earlier publishedresults [4]–[6]), we became aware of the work in [7] and the re-cently published works in [8] and [9], which propose a similarapproach. It is worth mentioning that the works in [8] and [9]overlooked the original contribution in [7]. Now, although thereare similarities between this paper and [7]–[9], there are key is-sues that were not treated in these publications but are fully ad-dressed here. These are as follows. First, there is no analyticaltreatment for the channel estimation performance in [7] and [9](moreover, the channel estimation method proposed in [9] is dif-ferent). Although the work in [7] mentions the synchronizationproblem, the proposed solution of rotating the estimated samplesso that their peak is positioned at the middle is incorrect becausethere is no guarantee that the channel peak will be the center ofthe response. This fact will be clear later when we deal with thesynchronization problem. Neither the selection of training se-quences is given nor is the synchronization problem tackled in[9]. Second, the work in [8] is different from that in [7] and [9].It provides analytical results. However, the work in [8] assumesthat the receiver’s reference is synchronized to the transmittedtraining sequence, which is not a realistic assumption. In thiswork, we address the synchronization problem and propose analgorithm to solve it. Another key improvement over the workin [8] consists of the general analysis for the variance of the es-timation error for arbitrary training sequences. While [8] onlyconsiders a very simple training sequence (an impulse everysymbols), here, we obtain results for a general class of trainingsequences (of which that of [8] is a special case) and includethe Cramér–Rao lower bound (CRLB). More importantly, we arenow able to determine the characteristics that a training sequenceshould have in order to obtain an optimal (in a sense defined later)channel estimation performance. Moreover, a procedure to syn-thesize such training sequences with these desired properties isdeveloped. It turns out that these sequences were mentioned in[7] but without any proof of optimality when applied to the cur-rent scenario. Another important improvement over the contri-butions in [7]–[9] relies on the fact that our model includes the

1053-587X/04$20.00 © 2004 IEEE

OROZCO-LUGO et al.: CHANNEL ESTIMATION USING IMPLICIT TRAINING 241

possibility of having an unknown dc offset at the receiver. Thisis important in the present context because channel estimation iscarried out using first-order statistics, and therefore, the dc-offsetcannot be neglected nor simply estimated if the method were towork in a direct conversion receiver.1 In conclusion, this papergreatly develops and extends the initial contributions of [7]–[9]from both an analytical and experimental nature.

The paper is organized as follows. Section II is devoted to thetheoretical derivation of the concept of implicit training and thedevelopment of the channel estimation algorithm. In Section III,a closed-form solution for the variance of the channel estimate isobtained. Using the results of Section III, Section IV gives theconditions for the selection of appropriate training sequences,and a method for their synthesis is presented. Section V high-lights the problem of estimation ambiguity due to both unknowndc-offset and lack of synchronization between the starting phaseof the transmitted training sequence and the receiver’s refer-ence, and a procedure to resolve this ambiguity is given. Ex-tensive simulation results are given in Section VI, and finally,Section VII offers conclusions. Appendix A presents a proof ofa particular claim made in Section V. Finally, the CRLB for theestimation error variance is derived in Appendix B.

II. PROPOSED ALGORITHM

A. System Model

Consider the single input/single output (SISO) discrete-timebaseband, complex envelope equivalent, communications linkshown in Fig. 1, where is a sequence of elements drawnfrom a finite alphabet representing the information symbols.We make the usual assumption that the information sequence

is zero mean and that . A special trainingsequence (to be defined) is added to to produce thetransmitted signal . The sequence is periodic with pe-riod so that , and the average power in

is . Under these conditionswill be a cyclostationary sequence, possessing a periodicallytime-varying mean . Returning to Fig. 1,travels trough a channel with discrete-time finite impulse re-sponse , and after the output is combined with AWGN andan unknown dc-offset , we get the received signal . Be-cause our method will exploit the knowledge of the periodicallytime-varying mean induced at the transmitter (and so the esti-mation will be based on first order statistics), it is very impor-tant to consider a possible dc-offset at the receiver. When theestimation is based on second or higher order statistics, it isnot necessary to consider the unknown dc-offset because it canbe estimated and substracted in a simple way. The problem ofdc-offsets is of considerable importance in a direct conversionreceiver [10], and if it is not taken into account, the performancecould be seriously degraded [11].

The channel estimation task posed in this paper consists ofidentifying (the unknown dc-offset is then obtained as abyproduct) with knowledge of only the received sequenceand the training sequence . Once the channel is correctlyestimated, the training sequence and dc-offset can then both be

1We would like to thank one of the anonymous reviewers for pointing out thisimportant issue.

)(kb

)(kc

Training Sequence

Transmitter

Channel

)(kh

)(kn

)(kx

Noise

ReceivedSequence

)(ks

Data

+ +

d

DC Offset

Fig. 1. Discrete-time model of a communications link.

removed from , and channel equalization can be carried outto recover the information signal . Note that the advantageof this method is increased information data rate (i.e. no trainingsignal is needed in its own time slot) but at the expense of re-duced SNR (due to some transmitter power being allocated to

). However, as we will later surmise, the advantages out-weigh the disadvantages.

B. Basic Considerations

From Fig. 1

(1)

where is the number of nonzero taps in the channel impulseresponse. Although we consider here that the dc-offset is timeinvariant, a slowly time-varying offset will not affect the formu-lation if we assume an observation interval short enough so thatthe offset could be deemed constant within the interval. Practicalreceivers use long-term averages to estimate this offset [10]when the transmitted signal possesses zero mean. However, if thesystem operates in a frequency-hopping fashion, the dc-offsetshouldbeestimated inaburstbyburstbasis [11], and thedc-offsetcan be considered constant within the burst. Now, define

(2)

(3)

(4)

In (4), the first and third expectations are zero since andare zero mean, and clearly, the second and fourth expecta-

tion operators can be removed (deterministic quantities) to give

(5)


In (5), indicates arithmetic modulo- asuniquely defines the periodic sequence . Equation (5)represents a set of linear equations (one for each value of) for the channel impulse response coefficients

and the complex-valued dc-offset . First, assume that thereis no dc-offset in the system so that . In order to obtaina unique solution to (5), under this condition, must bein principle equal to , and the coefficient matrix shouldbe full rank. In this case, exact knowledge of the channelorder is required, which in practice may not be known. Onthe other hand, if the channel model order is known onlyby its upper bound, will be in general greater than . Inthis case, provided the coefficient matrix is full rank, therewill still be a unique solution where the channel coefficients

will all be zero. With this inmind, define , wherethe operation “circ” produces a circulant matrix [12]. As anexample, take ;then, is the (4 4) matrix

(6)

Now, letting and(where stands for trans-

pose), it is possible to rewrite (5) in matrix notation as

(7)

Then, taking into account the previous remarks, all that isneeded to achieve channel identification based on solving (7)is to do the following. a) Select , and b) design thecoefficient matrix to be full rank.

C. Proposed Algorithm Assuming Zero dc-Offset

The basic problem, though, is to estimate the elements offrom a finite amount of received samples. Letting

(we assume that is a multiple of ), then from (2),given the usual assumptions of stationarity and ergodicity, wemay use the following estimate for :

(8)

which is asymptotically unbiased. From (7) and (8), our channelestimate becomes

(9)

where , and.

D. Impact of a Nonzero dc-Offset

When the dc-offset is not equal to zero, we can still use themethod for channel estimation proposed in the previous section,but now, the channel estimate will be ambiguous within a con-stant term. That is, the channel will still be estimated using (9),but the vector for finite , or when

, where .Therefore, after using(9), we will have, when , .Note that will also be a circulant matrix; therefore,will be equal to with the sum of the elements of anyrow in . Therefore, if , then the unknown offset willnot be a problem because the estimated channel will tend to thetrue channel as grows, irrespective of the value of . How-ever, we will later see that it is not possible to find a trainingsequence having and complying at the same time withother desiderable restrictions. Therefore, we will then later pro-pose one method to estimate the ambiguity factor .

E. Extension to Multiple Channel Systems

Finally, note that while we have assumed a single input–single output (SISO) scenario, the extension of the proposedmethod to the single input–multiple output (SIMO) setup issimple. The straightforward way to achieve this is to estimateeach subchannel separately from the rest, as if it were the onlyone in the system. Then, the overall channel is constructedfrom the estimations obtained for the individual subchannels.This particular approach will be used later in the simulations.

III. PERFORMANCE ANALYSIS

A. Variance of the Estimation Error

We assume in this section that because we havepreviously seen that a nonzero only produces a channelambiguity within a constant added term. Under this condition,it is clear from (8) and (9) that (i.e., unbiased channelestimate); therefore, this section is devoted to the investigationof the variance of the error between the estimated channeland the true channel , where . We might expectthe error to depend on the channel characteristics, trainingsequence, modulation format, number of received symbols,signal-to-noise ratio (SNR), information sequence power, andtraining sequence power. Some further definitions are nowgiven. A filtering matrix is defined as(10), shown at the bottom of the page. The information datavector is

(11)

......

.... . .

. . .. . .

. . .. . .

(10)


and the training sequence vector issimilarly defined. Let be the noise vector

(12)

and let the received data vector be similarlydefined. Using (1) (with ) and (10)–(12), it is possible towrite an expression for the received signal vector as

(13)

where it has been assumed that the sum in (1) now goes up to, and the channel coefficients

are all zero. From (8) and (13)

(14)and then substituting (14) into (9) gives

(15)Thus, the (zero mean) channel estimation error vector is

(16)

where we have used the fact that sinceis periodic in , with defined for (7) and defined in asimilar way to (11). Let the following be a scalar measure of thevariance of the vector channel estimation error

tr tr(17)

where means conjugate transpose, and tr means take thetrace of . So from (16) and (17)

(18)

where . Assuming is i.i.d., is white,and and are uncorrelated, the first expectation termin (18) becomes

otherwise

(19)

where and represent, respectively, the identityand the zero matrices of dimensions .The matrix (dim. ) is given by

, where the operation “semicirc”produces a semicirculant matrix from the argument vector as

defined in [12]. The only element equal to 1 in the argumentappears in position . For example, if ,

is

(20)

because the semicirc operation shifts each time the argumentvector to the right and inserts zeros from the left instead of cir-cling the argument vector. The second and third expectations onthe right-hand side of (18) are

(21)

and

(22)

In (21) and (22), denotes the zero matrix of dimension. The last expectation in (18) is

(23)

with . Now, substituting (19) and (21)–(23)into (18) gives

(24)

where is the channel correlation matrix. Furtherdefining , and after some more sim-plifications, (24) can be written as

(25)

Substituting (25) into (17) gives the final result for

tr

(26)

where is a normalized matrix suchthat the sum of the squares of any row is unity (this normaliza-tion simplifies derivations in Section IV). An important obser-vation is that does not depend on the modulation format of


the information sequence but only on its relative power whencompared with the training sequence power, i.e., . Thisfact makes the proposed method very attractive because it canbe applied with equal success for widely different modulationssuch as BPSK, QAM, etc.

IV. SELECTION AND SYNTHESIS OF THE TRAINING SEQUENCE

This section is devoted to the investigation of the characteris-tics of the training sequence that results in desirableproperties for the channel estimate in (9), and we then pro-pose a method to synthesize these training sequences with thedesired properties.

A. Optimum Training Sequences

Let us first discuss what our objective will be. Due to the un-biased property of the estimate , we seek to minimize . Theunconstrained minimization of (26) is of no use because fromSection II, we know that (and therefore ) must be a fullrank circulant matrix, and the unconstrained minimization willnot necessarily result in a full rank matrix. Imposing this con-straint in (26) results in a minimization problem that is difficultto solve. More importantly, we realize that depends on thechannel characteristics due to the and terms. Therefore, ourobjective will be to look for particular training sequences thatresult in an estimation performance independent of the channelcharacteristics. This objective is appealing because, for pop-ular channel identification methods based on cyclostationarityinduced at the receiver via oversampling, it is known that es-timation performance is affected by the closeness of the zerosbetween distinct subchannels. In fact, it is exactly this propertythat has precluded the application of such techniques for ban-dlimited channels that could have close subchannel zeros [13],[14]. Therefore, as tr tr [15], then (26) becomes

tr

tr tr (27)

Now, from (27), if we choose as unitary (that is), then

tr tr

tr (28)

(29)

In (29), tr follows directly from the construc-tion of matrices and since the main diagonal of the product

is all zero. In addition, is thechannel autocorrelation at zero lag (channel energy), which,without loss of generality, has been normalized to one. Notefrom (28) that as , and so we now have a

consistent estimator for any channel that is also independent ofthe channel characteristics.

For engineering purposes, it would be better to express in(29) as a function of the SNR at the receiver and the loss of trans-mitted data power due to the sending of the training sequence.Therefore, if we denote as the receivedSNR (since ), and the power loss factor as

, then after some manipulation

(30)

Thus, by using periodic training sequences thatproduce unitary circulant matrices , it is possible toachieve estimation performance independent of the channelcharacteristics, and for this estimation error is given in avery simple form (29) or (30). Next, we provide a method forthe synthesis of such desirable training sequences.

B. Synthesis of Optimum Channel Independent TrainingSequences

Define the discrete Fourier transform (DFT) of the pe-riodic training sequence as ,

. Now, if , where is aconstant, then

DFT (31)

where denotes circular convolution. With a little thought, itis clear that this property of circular convolution, along withthe definition of the circulant matrix in (26), impliesthat (and hence ) is unitary. We will call thesespecial types of training sequences optimum channel indepen-dent (OCI) sequences. From the previous discussion, we con-clude that the DFT of an OCI sequence should be of the form

, where the anglestake arbitrary values. For example, if , thenone possible choice could be

(32)

which gives us the real training sequence

(33)

On the other hand, if we choose , then this givesfor the simple training sequence as in [8],

and therefore, our method is more general. Note in passing thatthe requirement for an OCI sequence of having constant DFTprecludes the possibility of obtaining zero mean sequences sincethe dc value must be different from zero. Therefore, thisfact, together with the result that is unitary, implies that

, which is the sum of the elements of any row of ,cannot be zero, as was pointed out in advance in Section II-D.

We remark on the fact that the number of OCI sequences thatcan be obtained by the proposed technique above is infinite,yet all of them will produce the same . Still, different OCIsequences could exhibit different peak-to-average power ratio:a factor that could be relevant in cases where highly efficientlinear r.f. transmitter amplifiers are required. Note, for example,that while the very simple training sequence used in [8],


belongs to the set of OCI sequences, it is undesirablein practice, as the authors in [8] recognized, since it actually pos-sesses the worst peak-to-average power ratio possible for anytraining sequence. Thus, it may be desirable to impose on theOCI sequences the extra constraint of having a peak-to-averagepower ratio as close to one as possible (note that the best pos-sible value is one). We will presently show that for any value of

, there exist complex OCI sequences with a peak-to-averagepower ratio of one. For the preceding condition to hold, both thetime sequence modulus and that of its DFT should be constant,that is, and , or equivalently,

and , whereis the unnormalized circular autocorrelation of . Is it

possible to have sequences with constant modulus in both do-mains for any value of ? If so, is there a general way to param-eterize such sequences? The authors are unaware of a generalprocedure that could be used to find all possible sequences satis-fying the above property; however, it has been possible to find aspecial class of sequences possessing such a property. These se-quences were found as the solution of a difference equation thatconsidered the relationship between the phases of two consecu-tive values of . The complete design procedure is somewhatlengthy, and it is not the main objective of the present paper.Therefore, we will only give the final result and show that it ful-fils the aforementioned requirements. Therefore, a special classof OCI sequences possessing optimum peak-to-average powerratio is given by

odd even

(34)

Note that the magnitude of is constant; thus, the peak-to-average power ratio of such sequences equals one. Now, it iseasy to show from (34) that , and therefore,the unnormalized circular autocorrelation of the sequence canbe computed as

(35)

which shows that is effectively an OCI sequence. Whilethe previous class of sequences show that it is always possibleto find OCI sequences with ideal peak-to-average power ratio,questions about the uniqueness of these sequences and the ex-istence of more general methods of sequence design for a givenpeak-to-average power ratio are still open to further research.Note, however, that for the present application, the above ex-pression for could be sufficient.

Sometime after we developed the previous material on OCIsequences with ideal peak-to-average power ratio, we becameaware of the works in [7] and [16]. From these, we found theworks in [17] and [18], which dealt with the problem of findingcodes with good periodic correlation properties. Those code se-quences are basically very similar to the ones we have designedin (34).

V. SYNCHRONIZATION ISSUES

A. Estimation Ambiguity

As is the case in many papers on channel estimation and/orequalization, issues around synchronization are often conve-niently overlooked with the phrase “assuming perfect synchro-nization .” Thus far, we too have implicitly assumed the ex-istence of synchronization between the starting phase of thetraining sequence at the transmitter and that of the receiver, butthis timing cannot be assumed known in the actual scenario andwas overlooked in [8] and [9], whereas in [7], timing estimationwas not adequately tackled. The important question is then howa potential mistiming affects the channel estimation capabilityof the proposed algorithm. We will now show that the estimationperformance is not compromised, but what happens is that theestimated channel response vector in (9) will simply be a cir-cular shift of the that would have been obtained with perfectsynchronization. That is, ifis obtained via (9) fromand then is given a positive circular shift of symbols as in

(36)

then the new from (9) will be

(37)

which also exhibits a similar circular shift of symbols. Thiscan be verified from (9), due to the fact that a mistiming leadsto a circular shift in and, as is circulant, it produces asimilar shift in .

B. Resolution of the Estimation Ambiguity

We will now show how we can resolve the ambiguity for thechannel estimate in (37), and at the same time, we will providea method to resolve the ambiguity caused by the dc-offset. Thatis, we will try to identify which one of the possible channelestimates in (37) , which can arise fromdue to the lack of synchronization and unknown dc-offset, is“closest” to the true channel in (7) and Fig. 1.

Let us first start by forming a new sequence , which isobtained by subtracting, from in Fig. 1, the estimate of itstime-varying mean [ in (8)]

(38)which follows from (1) and (5). To estimate the dc-offset ambi-guity , we propose a correlation matching approach as fol-lows. First, define

(39)


where is the estimated channel vector possessing thedc-offset ambiguity. Therefore, an estimate of the true channelvector (without the dc-offset) given by can be obtained by

(40)Note that is directly obtained from (9), but because isunknown, cannot be directly obtained from (40). Denote theautocorrelation of (excluding the zero lag term) as

(41)

In addition, observe that the autocorrelation of (excludingthe zero lag term to eliminate the influence of white noise) isconsistently estimated by the empirical estimate (using sampleaveraging) of the second-order correlation function of ,

. Therefore, consider the following development, wherewe denote . Now, from (38), we can say

(42)

and

(43)where

(44)

Note that (43) represents a set of equations for the un-known (and ): one for each value of . Now,because in (42) is only an approximation of the true

(which will always happen when using a finite amountof received data), no value of will fulfil all the equa-tions at the same time. Therefore, to estimate , we propose touse a nonlinear least squares approach as follows. Define

(45)

and a cost function ; then, can be esti-mated as the value that minimizes . Note that is a functionof both and . We next provide a procedure of solution. First,note that from (45), can be written as

(46)

where

The objective function can then be ex-pressed as

(47)

where , ,, and 1, 2, 3. Now, to find the critical points,

we proceed as follows:

(48)

and solving for , we have

(49)

Now, if (48) holds, then the following relation [which is just thecomplex conjugate of (48)] also holds

(50)Finally, substituting (49) into (50) and after some manipula-tions, we obtain a polynomial of fifth degree in given by

(51)

where

Now, since is a scalar quantity, the optimization problemcan be easily solved by polynomial rooting applied to (51).However, while (51) gives five solutions for , we can show


that not all of these five solutions will necessarily correspondto saddle points, minima, etc., of in (47). That is, if is asolution to (51) (always five solutions), it need not always give

in (48), where the number of solutionsvaries from one to five, depending on the coefficients in (48).However, a solution to (48) will always appear in (51). This sit-uation is due to (48) not satisfying the fundamental theorem ofalgebra [by virtue of the complex conjugated terms in (48)].

A solution to the above is to substitute the five roots of (51)into (47) and check for the one giving smallest cost function

. However, it is shown in Appendix A that the cost functiondefined above can have either one or two global minimums.

Therefore, we propose that if the two smallest values of the costare similar, then both of the values obtained for could be

considered valid, and the selection of the true one is deferred tothe synchronization stage, as will be explained next.

Define , , and ,, , as the em-

pirical estimates (using sample averaging) for the second-ordercorrelation function and fourth-order cumulant of , respec-tively. Now, consider the analytical expressions for two similarfunctions, as above, but where now, the “ ” sign in (38) is re-placed by an equality “ ,” and the true channel isreplaced by any one of the possible (with the dc-offset re-moved using the previous estimation procedure) channel esti-mates from (37) to give

(52)

(53)

where is the th element of the vector in (37), andis the kurtosis of . Note that by not using in (52),

we avoid the contribution of the AWGN in Fig. 1. Finally,we may reasonably assume that and

if our choice ofis “close” to the true channel . Therefore, the proposed algo-rithm will select the minimum , where, for

(54)

When in the solution of (51) there exists an ambiguity be-tween two values of giving global minimums for (47), theabove algorithm is applied to each of these values, and the solu-tion is found as the best match among the now channel esti-mates ; (note that the ambiguity willbe resolved because fourth-order cumulants are unique). Oncethe best estimate of the channel has been obtained, the estimateof the noise variance can be acquired by subtracting the channelenergy from the second-order correlation function of eval-uated at zero lag

(55)

as long as this quantity is positive. Otherwise, is set to asmall quantity (let us say ) or zero. This synchroniza-tion process for a single-input single-output (SISO) system inFig. 1 can easily be extended to the single-input multiple-output(SIMO) case, but where the variables are now obtained bysumming the contributions coming from all subchannels.

Now, before continuing to the next section, an important ar-tifact of this synchronization algorithm should be noted. Letthe true channel be ,and let the channel estimate with proper synchronization be

. Now, may have nu-merically small values at the beginning or end (or both) for thefollowing reasons: either because we have overestimated thechannel order , thus giving zeros at one end of the es-timated channel impulse response, and/or because the channelis time limited and approximately band limited, and thus, theimpulse response “tails” will contain little energy. Either way,this means that some circular shifts [as defined in (37)] will onlylook like delays/advances in the estimated channel impulse re-sponse, and because the statistics in (52) and (53) are invariantto advances/delays in , then a number of different in(37) could give very similar values in (54). That is, circu-larly shifting [0,0,1,4,6,2,0,0] by two gives [ 0,0,0,0,1,4,6,2] (adelay that will not be identified by (54)), but a shift of four gives[6,2,0,0,0,0,1,4] [not a delay, and so, it will be identified by(54)]. However, a delay ambiguity does not cause any problemswith equalizer design as the equalizer output will simply also bedelayed.

Note that the solution for the synchronization problem of ro-tating the estimated channel until the peak is positioned at themiddle, as proposed in [7], is not correct. This is so becausean actual channel will not necessarily have the maximum at thecenter of the response as the position of the maximum stronglydepends on the magnitude, phase, and delay positions of the in-dividual multipath components.

VI. SIMULATION RESULTS

This section (broken down into five parts) is devoted to thepresentation and analysis of extensive simulation results con-cerning channel estimation and equalization. The objective hereis to try to ascertain the quality of performance of the proposedimplicit training (IT) algorithm when used not only as a channelestimator but also to construct an FIR channel equalizer. Thiswill be compared with a choice of both trained and blind channel


0 2 4 6 8 10 1210 -2

10 -1

100

ER

RO

R V

AR

IAN

CE

SNR (dB)

IT [loss=2(dB)]IT [loss=3(dB)]IT [loss=4(dB)]CRLB [loss=4{dB}]LSCC

Fig. 2. Error variance of LS, CC, and the proposed IT method using differentloss factors in Section VI (Section VI-A). The CRLB relates only to the ITmethod and is calculated as shown in Appendix B.

estimators and equalizers, which are each representative of theirclass and are extensively quoted in the literature. We shouldadmit that because of the differences between blind, semi-blind,and trained methods, and because of the many different parame-ters that must be selected for each algorithm, there is a difficultywhen trying to compare “like with like.” However, we have en-deavored throughout to be as fair and consistent as possible andbelieve that we have achieved this.

A. Comparison of the IT Method With Explicit TrainedMethods for Channel Estimation

We wish to compare analytic results for the IT channel esti-mation error [ in (30)] with similar expressions for the trainedchannel estimation methods of least-squares (LS) and cross-cor-relation (CC) used in GSM receivers; for a detailed referenceon these methods, see [19] and [20]. The training sequencesfor both the CC and LS methods use 26 symbols (as defined inthe GSM standard for a single frame), whereas the IT methoduses received symbols to perform the estimation [theOCI training sequence in (33) is used]. We choseto approximate the number of bits in a GSM frame while ful-filling integer. The channel dispersion lengthis set equal to six symbol intervals (the worst-case GSM sce-nario), and so, we chose training period , and thus,

[see (8)]. The actual channel used was. Fig. 2 shows the perfor-

mance for each of the distinct methods, with plotted againstreceiver SNR (defined as previously). For the IT method, the“loss” parameter (loss (in decibels)) is a measureof the information data power loss due to the training sequence,with defined for (30). Included in Fig. 2 is the Cramèr–Raolower bound (CRLB) for the IT method calculated for a loss pa-rameter of 4 dB. The derivation of the CRLB for in (17) isgiven in Appendix B. Note that the channel estimates obtainedin Fig. 2 using the IT method are competitive with those ob-tained using explicit training for the received SNR of interest inthe GSM environment (6–12 dB). The deterministic CC and LS

methods improve for high SNR giving (as expected) zero errorfor infinite SNR, whereas for low SNR, the IT method is supe-rior due to the use of 144 symbols. In fact, the performance ofthe IT method is not so far from the CRLB.

Note that while the IT method employs some power forthe transmission of the training sequence , it also savesbandwidth because information data is continuously trans-mitted. While a more precise evaluation is beyond the scopeof this present paper, future work will present bit-error-rate(BER) performance of both approaches (trained and implicitlytrained), assuming the same conditions, i.e. equal transmittedpower and information data rate. Preliminary results suggestthat by means of channel coding (using the 26 saved GSMtraining symbols) and Viterbi equalization based on the ITchannel estimates, it is possible to show an advantage (in termsof BER) by using the IT method.

B. Comparison of the IT Method With Other Blind Methodsfor Channel Estimation

Unlike Section VI-A, because we are now considering blindtechniques, we do not have closed-form expressions for , andtherefore, we must now carry out computer simulations. Forthis subsection, we have assumed perfect synchronization andzero dc-offset. These assumptions will be relaxed in the simu-lations to come. Here, we consider a single channel scenario,with symbols drawn from a quaternary pulse am-plitude modulation constella-tion that were transmitted through the three-ray multipath con-tinuous-time channel with impulse response

(56)

where is a truncated trans-mitted pulse shape, with a rectangular windowstarting at and ending at , while isa raised cosine pulse (comprising the overall response of thetransmit and receive filters) with roll-off factor equal to

. The carrier frequency is 900 MHz, and the symbolperiod is . The multipath amplitudes and delays

are randomly varying at each realization. The amplitudes areGaussian distributed with an average power per path equal to

, , and . The delays are indepen-dent and uniformly distributed between 2 to 4 . The singlechannel baseband discrete-time impulse response for this simu-lation was obtained by sampling the continuous-time channel ateach realization at time instants , .

Two blind channel estimation techniques were simulatedfor comparison purposes—the MTCSA method in [21] thatis based on modulation inducing cyclostationary and the EVImethod of [22] based on higher order statistics (HOS). TheMTCSA method was implemented using the modulatingsequence for the first 126 symbols and

for the following 126. The EVI method [22]was implemented using the MATLAB program available onthe Web page of the authors [22], and the default values wereused. The IT method used an OCI modulating sequence withperiod . The MTCSA and EVI methods were given


0 2 4 6 8 10 12 14 16 18 2010 -2

10 -1

100

SNR (dB)

ER

RO

R V

AR

IAN

CE

EVIMTCSAIT

Fig. 3. Error variance for channel estimation in Section VI (Section VI-B).

the true channel model order. In addition, due to the fact thatMTCSA and EVI give channel estimates that are ambiguous upto a complex scale factor, the estimated channel was correctedby a complex factor (to allow reasonable comparisonsbetween different methods). This was calculated according to

and , where isthe true channel vector (which was normalized in such a waythat ), are the initial (ambiguous up toa complex scale) channels estimates, and .Therefore, the scaled channel estimates obtained by MTCSAand EVI were calculated asand . The channel estimate by the ITmethod was also scaled but only to enforce .Synchronization was assumed known for both the modulatingsequence of the MTCSA method and for the training sequenceof the IT method. We estimated , with“X” standing for MTCSA, EVI, or IT, by using 300independent Monte Carlo realizations.

The estimation results are shown in Fig. 3 with plottedagainst received SNR. The IT method introduced a loss of 3 dBin data power (the training sequence in this and the followingsimulation possesses half the transmitted power), but Fig. 3 wasdrawn for total received power. So, it is necessary to discount3 dB for the IT method when it comes to recovering the datavia equalization. However, MTCSA and EVI both introduce arotational ambiguity and need differential encoding, which in-troduces an SNR loss on the order of 2–3 dB (this loss, of course,depends on the type of modulation and SNR working level).Therefore, the comparison between the methods may be deemedfair. Therefore, from Fig. 3, it is clear that IT clearly outperforms[as far as is concerned] the other two blind methods at allSNR levels.

C. Equalization Based on the Channel Estimates inSection VI-B

Now, consider what happens when an equalizer is constructedfrom the estimated channel using the different methods in Sec-tion VI-B. Note that the signal to be processed by the equal-izer is in (38), where the training sequence has now beenremoved from the received signal . The performance mea-sure that will be used is the average signal to interference plusnoise ratio (SINR) at the equalizer output. This is defined as in(57), shown at bottom of the page, where stands for realiza-tion number, (in this case 300) is the number of MonteCarlo realizations, is the combined channel plus equal-izer impulse response at symbol rate for the th realization,

is the maximum value in the combined impulseresponse, is the variance of the transmitted signal (informa-tion data plus training), is the noise variance at the equalizerinput, is the th subchannel equalization filter impulse re-sponse for the th realization (in this case, because of baud-ratesampling, there is only one subchannel), represents the over-sampling ratio (number of samples per symbol period, in thiscase, ), and is the equalizer length in symbol periods(here, ).

The coefficients of the equalizer were calculated accordingto the minimum mean square error (MMSE) criterion [20]using the estimated channel and the true noise variance forthe MTCSA and IT methods. Instead of using an equalizercalculated from the EVI channel estimate, the EVA method[23] (the equalization counterpart of EVI) was used. EVA wasimplemented using four iterations with an initial referenceequalizer equal to a centered spike. Finally, as a benchmark, theperformance of the MMSE equalizer using the exact channeland noise variance was also considered.

The results are shown in Fig. 4. Note that again, the proposedIT method outperforms the other two methods by a significantmargin. The MTCSA method shows good behavior at low SNR,whereas the opposite happens for EVA. The IT method perfor-mance is very close to the optimum MMSE solution at low SNR,and for high SNR, the IT method’s performance starts to flatten,as expected. This is due to the stochastic nature of the IT methodthat does not give perfect estimation (even when noise is absent)when using only a finite amount of data. Note that the other twomethods also exhibit similar behavior.

D. Equalization Based on Fractional Sampling at the Receiver

In this experiment, a QPSK constellation was used to sendinformation data over the multipath channel considered in [24].This time, the sampling rate at the receiver was four times thesymbol rate, thereby producing an equivalent multiple channelsystem with four subchannels. In this experiment, ,

SINR (57)


0 2 4 6 8 10 12 14 16 18 20 –10

–5

0

5

10

15

20

SIN

R (

dB)

SNR (dB)

MMSEMTCSAITEVA

Fig. 4. Baudrate equalizer performance for Section VI (Section VI-C).

0 5 10 15 20 25 30 35 40–5

0

5

10

15

20

25

30

35

40

45

SNR (dB)

SIN

R (

dB)

MMSEITA-ITMDCMCRTXKJLSS

Fig. 5. Nonbaud rate equalizer performance for Section VI (Section VI-D).

and the number of transmitted symbols was . A frac-tional spaced equalizer (FSE) with symbol rate temporal length

was used at the receiver. The equalizer coefficients werecalculated using the estimates resulting from seven differentmethods. The proposed method (IT) has been implementedusing an OCI training sequence with period . Twoversions of the IT method were simulated. The first one (calledsimply “IT” in Fig. 5) assumes that perfect synchronization isknown in advance, whereas the second version (called “A-IT”)is asynchronous (hence, the “A”) insofar as the algorithm hasto derive timing information using the method proposed inSection V (dc-offset has been assumed zero). For comparisonpurposes, the methods in [25] (abbreviated to MDCM), [24](abbreviated to TXK), [26] (abbreviated to CR), and [27](abbreviated to JLSS) are all implemented. As a benchmark,the performance obtainable using the optimum (constructedfrom knowledge of the true channel and noise variance) MMSEfractionally spaced equalizer (FSE) is also considered.

The MDCM method was implemented using the full noisesubspace with a temporal window of width equal to anda quadratic constraint. The correct model order was assumed.The CR method was implemented using a quadratic constraint.The TXK method was implemented assuming knowledge ofthe true noise variance and employing an observation windowof length equal to . Finally, the JLSS method was givenan upper bound on the channel model order equal to the truemodel order.

Using the (incorrect) channel estimates obtained via IT,MDCM, CR, TXK, and JLSS, a “nonoptimum” MMSE FSEwas implemented (the true input SNR was assumed known),and the performance obtained from the different algorithmswas evaluated in terms of SINR as specified in (57) but, now,for fractional sampling. From the results in Fig. 5, we canconclude that the proposed method outperforms all the othersat low SNR. In addition, the synchronization algorithm doesan excellent job as the curve for the asynchronous implicittraining method (i.e., synchronization has to be estimated asin Section V—labeled “A-IT”), virtually coincides with thecurve obtained using the synchronous IT method (i.e., fullsynchronization is known—labeled “IT”). In the high SNRregion, the performance of the proposed IT method flattens,as expected, due to the finite number of data samples usedin the channel estimation. This region is where the highlydata-efficient deterministic methods like MDCM, CR, andJLSS show their full potential. Note, again, that the IT methodintroduced a power loss of 3 dB (so it is necessary to discount3 dB when it comes to recover the information data ), butthe other methods need differential encoding to compensate forthe rotation ambiguity.

E. Performance of the Method Under dc-Offset and Lack ofSynchronization

This subsection is dedicated to the presentation of the com-bined performance of the dc-offset estimation and channel syn-chronization algorithms. The transmitted signal was drawn froma QPSK constellation, and the channel impulse response wastaken to be the first subchannel belonging to the channel in theprevious subsection. Therefore, the equalizer now operates atbaud-rate and uses coefficients. There weresymbols used for the estimation. The noise variance was not as-sumed known but was evaluated using (55). The signal power todc-offset power was taken to be 10 dB. The training sequenceand the loss parameter have been kept the same as in the pre-vious simulation. In addition, the period of the implicit trainingsequence is . Results are presented in Fig. 6. The BERcurve obtained when using the optimum MMSE equalizer is in-cluded in the figure as a benchmark. This curve assumes thatthe transmitted signal does not possess a superimposed trainingsequence, and therefore, the full transmission power is allocatedto the data. Note that the IT curve is not so far from the MMSEbound, which shows that the performance of the dc-offset andsynchronization algorithms is quite good. This is so becausemost of the performance degradation is due to the fact that 3 dBin transmitted power were assigned to the training sequence;therefore, the data is 3 dB weaker.


0 2 4 6 8 10 1210 -3

10 -2

10 -1

100

BE

R

SNR (dB)

MMSEIT (S/DC=10dB)

Fig. 6. Baud-rate bit error rate equalizer performance for Section VI(Section VI-E).

VII. CONCLUSIONS

We have presented a method for blind channel identificationbased on implicit training. From the theoretical analysis andsimulation results, it can be concluded that this technique offersinteresting estimation properties, like those of having perfor-mance independentof thechannelcharacteristicsandmodulationformat. A closed-form solution for the variance of the estimatorwas derived and, from the result, guidelines emerged on how tobest select the implicit training sequence. A procedure to synthe-size optimal channel independent training sequences was alsogiven. The problems of synchronization or timing uncertaintyand that of having an unknown dc-offset were addressed, andan algorithm was proposed for their solution. The performanceof the proposed method was compared with several blind andtrainedmethods, and the results showed that it is averysimpleandcompetitive technique. In addition, unlike totally blind methods,no complex scaling ambiguity exists for the channel estimate.The way to extend the algorithm for use in multichannel systemswas presented. Issues such as selecting the optimum distributionof power between the training sequence and the informationdata deserve special attention in practical applications and willdepend on the specific system conditions such as operating SNR,receiver structure, and coding format. These are left as particularquestions for each specific system. Inaddition, among the infinitenumber of OCI sequences, there could be some preferred onesdue to factors like peak to average transmitter power, which couldpotentially havean impact on theoverall performance of the radiofrequency stages. Specific training sequences with no penaltyregarding the peak-to-average power ratio, while at the same timebeing optimum for channel estimation, were given. Finally, theextension of the IT method for efficient channel equalizationand multiuser interference cancellation problems are currentlyunder investigation by the authors.

APPENDIX APROOF THAT THE COST FUNCTION IN (47) CAN ONLY HAVE

EITHER ONE OR TWO GLOBAL MINIMA

First, we assume that we have perfectly estimated the second-order correlation function of . Second, we also assume that

the estimated channel is equal to the true channel plus a constantdc-offset term, as given in (39), but with equal to the true .Let the deterministic ACF of the true channel be

Then, we have

(A.1)

We claim that there exists zero or one other thathas the following two properties:

and

(A.2)

where is a complex constant, is the unit step function, and; .

In other words, we want to prove that for a given autocorrela-tion function, there can only be at most two channels possessingthis autocorrelation function and at the same time related by

, where is the numberof channel coefficients. Therefore, we start from

(A.3)

where .Now

(A.4)

and because should hold,we have

(A.5)First, from (A.5), is given by

(A.6)

Now, if (A.5) holds, then (A.7) below also holds

(A.7)Substituting (A.6) into (A.7) and after some manipulations, wearrive at

(A.8)

which is a polynomial equation of the second degree in , whichhas two solutions given by

(A.9)


and

(A.10)

Note that while (A.9) will always be a solution, (A.10) couldor could not be a valid solution. This is because is restrictedto be a complex scalar, and a quotient of polynomials in doesnot necessarily reduce to a constant. Therefore, we have nowshown that the problem at hand will have either one solution ortwo solutions. Note that this is the case even though in (47), weremoved the zero lag of the correlation to avoid the influenceof white noise. This can be seen from the similarity that (A.4)possesses with the individual errors terms [given by (45)] thatmake up .

PROPERTIES OF A CHANNEL THAT HAS A “TWIN” VIA

dc OFFSET

Now, following from the previous result, it is interesting tofind the characteristics that a channel should possess in order tohave a “twin” channel with the same autocorrelation and at thesame time be related to the original channel via (A.3). In otherwords, what characteristics should the channel have in order forthe quotient in (A.10) to reduce to a complex scalar?

To answer this question, we will proceed as follows. Take(A.5), and express in polar form, that is, . Now, aftersome manipulations, (A.5) will read

(A.11)or

(A.12)

However, , and therefore

(A.13)

or in the time domain

(A.14)

As an example, take a channel with an even number of co-efficients. We can determine the last channel coefficientsthat will result in two possible solutions for (A.8), given the first

channel coefficients and the dc-offset term by means of

(A.15)A numerical example now follows. Take , ,and with

, . Therefore, from (A.15),

, . The “twin” channel is,from (A.3), equal to

(A.16)

It is easy to verify that both channels have the same autocorrela-tion given by (only the values for positive lag are given, and thevalues for negative lags are just the complex conjugates) (A.17),shown at the bottom of the page.

APPENDIX B

This section is devoted to the derivation of the CRLB forvar , assuming that the information symbols pos-sess a Gaussian p.d.f. Note that although the Gaussian assump-tion could not be valid when using PAM or QAM modulationformats, it is fully justified when source shaping is used; see, forexample, [28]. The joint probability density of the receivedsymbols is multivariate complex normal, and it can be expressedas [29]

(B.1)

where (dim. ) is formed from (1) with the samplesof the received signal as ,and (dim. ) represents the vector mean of , whichis the convolution between the training sequence and thechannel impulse response . Therefore, if we define

, then we have .Finally, (dim. ) is simply the covariance matrixof the received data, that is, .Define the following complex parameter vector as

.The CRLB for the estimation variance of is given bythe th element of the inverse of the matrix

, where [30]

(B.2)

and is defined in (B.1). Note that only the first diagonalelements of are needed (which correspond to the CRLBof the parameters , respectively) be-cause the CRLB for is equal to the CRLB for . Theevaluation of (B.2) is cumbersome and cannot be reproduceddue to lack of space. However, we began with the approach out-lined [31, App. A], which is for real parameters and real data,and then carried out significant development to accommodatecomplex parameters and complex data to arrive at

tr

(B.3)

(A.17)


where stands for the th element of vector , and similarlyfor . In addition, represents the th element ofthe matrix . In order to develop expressions for the partialderivatives in (B.3), we will first introduce some definitions. Avector is defined by

. To simplify nomenclature, let usdefine a matrix (dim. ) as being constructed from

accordingly to , where the operation isobvious from (B.4), shown at the bottom of the page. Now, thecovariance matrix can be expressed as ,where and withelements

(B.5)

The partial derivatives of the covariance matrix can also becalculated using the “ ” operation applied to the partial deriva-tives of vector , whose elements follow from (B.5); thus

and (B.6)

Then, and. To evaluate (B.3), we also need the ex-

pressions for the partial derivatives of the vector with re-spect to each of the channel parameters. These follow from

and are

(B.7)

with the expressions and in (B.3) beingcalculated directly from (B.7). Finally, we have now all the for-mulae that allow us to calculate the CRLB for the variance ofany estimate of a channel coefficient ,and var . Clearly, we now havethe final result for the CRLB of in (17), which is given by

.

ACKNOWLEDGMENT

The authors would like to acknowledge the reviewers for theirhelpful suggestions, which they believe improved the quality ofthe paper.

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. . ....

. . .. . .

. . .. . .

. . .. . .

......

. . .. . .

. . .. . .

. . .. . .

. . .. . .

......

. . .. . .

. . .. . .

. . .. . .

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Aldo G. Orozco-Lugo (S’96–M’00) was born in Guadalajara City, México, in1970. He received the B.Sc. degree in electronics and communications fromUniversity of Guadalajara, Guadalajara, Jalisco, México, in 1993, the M.Sc.degree in electrical engineering, specializing in communications, from CIN-VESTAV-IPN, México City, México, in 1997 and the Ph.D. degree in electricalengineering specialized in digital signal processing for communications fromthe University of Leeds, Leeds, U.K., in 2000.

He is currently an associate member of research staff at CINVESTAV-IPN.His research interests include space-time signal processing, wideband channelmodeling, analog and digital communication systems, antenna array technology,and radar.

M. Mauricio Lara (M’90) received the B.Sc. degree from the National Au-tonomous University of México, México City, in 1986, the M.Sc. degree fromthe Center for Research and Advanced Studies, IPN, México City, in 1986, andthe Ph.D. degree form the University of Leeds, Leeds, U.K., in 1990, all in elec-trical engineering.

From 1990 to 1993, he was a Research Fellow with the University of Leeds.Since 1993, he has been with the Center for Research and Advanced Studies,IPN, México City, as a principal member of research staff. His areas of interestinclude wireless systems and signal processing for communications.

Des C. McLernon (M’83) was born in Downpatrick, Northern Ireland. He re-ceived the B.Sc. degree in electronic and electrical engineering in 1975 andthe M.Sc. degree in electronics in 1976, both from the Queen’s University ofBelfast, Belfast, Northern Ireland. After working on radar systems with Fer-ranti Ltd. Edinburgh, U.K., he joined Imperial College, University of London,London, U.K., where he received the Ph.D. degree in multirate two-dimensionalsignal processing in 1982.

After lecturing at South Bank University, London, he joined the School ofElectronic and Electrical Engineering, University of Leeds, Leeds, U.K. Hisresearch interests include blind equalization, adaptive signal processing, LPTVsystems, 2-D filters, and radio resource allocation mechanisms.

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Channel estimation using implicit training

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