On the equivalence of blind equalizers based on MRE and subspace intersections

ON THE EQUIV ALENCE OF BLIND EQUALIZERS BASED ON MRE AND SUBSPACEINTERSECTIONSAlle-Jan van der Veen1,David Gesbert2,and Arogyaswami Paulraj2 1Delft Univ.of Technology,Dept.Elec.Eng./DIMES,2628CD Delft,The Netherlands 2Information Systems Laboratory,Stanford University,Stanford,CA94305Two classes of algorithms proposed for the blind equaliza-tion of multiple channels driven by a single source are the Mutually Referenced Equalizers(MRE)method by Gesbert et al.,and the Subspace Intersection(SSI)method by Van der Veen et al.Although these methods seem atfirst sight unrelated,we show here that a variant of the SSI method and a particular member in the class of MRE methods provide mathematically identical solutions.1.INTRODUCTIONBlind equalization has been an active research area during the last two decades,but especially during the last few years. Two major factors appear to drive the wide interest in this topic.Firstly,there is an increasing number of interesting and promising applications in the area of digital communi-cations,wireless or not.Secondly,the fact was recognized that channel oversampling,either temporally(fractionally spaced equalizers)or in space(antenna arrays),offers sev-eral new leverages for solving the blind equalization prob-lem,and thus enhances its applicability.By oversampling an analog channel carrying a single discrete source signal,this signal is effectively seen through several parallel linear channels.These may be the polyphase representations of a single physical channel or correspond to different physical links in the case of multiple antennas,or combinations of the two.In any case,we obtain a single-input,multiple-output(SIMO)channel model.From an algebraic perspective,oversampling leads to a low-rank model for the output vector signal.This has been extensively exploited in the so-called second-order statistics and algebraic methods for the SIMO identifica-tion problem[1].At least three classes can be identified. Thefirst addresses the problem of estimating the channels, viz.e.g.[2–4],the second considers the estimation of chan-nel inverses(equalizers)[5–7],and the third attempts to recover the transmitted symbols directly from a(typically small)batch of output samples without resorting to chan-nel/equalizer estimates[8,9].Categories2and3have the advantage of by-passing the channel estimation and inversion step,which in cases re-sults in increased robustness.The direct symbol-estimationmethods[8,9]have sometimes been called row-span meth-ods as they exploit the row-span information of the data ma-trix tofind the vector of unknown symbols.Following aseemingly different strategy,MRE techniques[5]estimate a collection of channel equalizers by forcing them to producethe same(unknown)output sequence,up tofixed equaliza-tion lags.As we show here,these two methods are in fact more or less identical,with small differences depending onthe implementation.Wefirst review the row-span method of[8].The symbol estimates produced by this technique can be regarded as theoutputs of linear equalizers,averaged across all equalization lags.We show that these equalizers optimize a maximal co-herence(MC)criterion.Finally,we demonstrate the equiv-alence between the MC criterion and a particular member in the class of MRE criteria.2.DATA MODEL2.1.Data matricesSuppose that a digital symbol sequence s i is transmit-ted through a medium and received by an array of M sen-sors.The received signals are sampled at a rate which is Ptimes faster than the symbol rate,here normalized to T 1. Hence,during each symbol period,a total of MP scalarmeasurements are available,which can be stacked into MP-dimensional vectors x i asx ix1i...x MPiAssuming an FIR channel,we can model x i as the output of an MP-dimensional vector channel with impulse response h0h1···h L−1,where L denotes the channel length in symbols.In the noise free case,a sample vector x i is then given by the convolutionx iL−1∑k0h k s i−k(1)Consider a finite block of data and define the mMP ×N block-Toeplitz data matrixix ix i1...x iN −1x i −1x i ..................x i −m1.........N is the block length,while m can be interpreted as the mem-ory of an equalizer acting on the rows ofi .Let n L m −1.From (1),i has a factorization as ii ,where is the mMP ×n channel matrix,h 0···h L −1 0h 0···h L −1(2)andis is i 1...s iN −1............s i −n1.........We will assume that is full rank and that m and N are suffi-ciently large,so that this is a low rank factorization (mMP ≥L m −1≤N ).In that case,we can recover any row of i by taking linear combinations of the rows of i .2.2.EqualizersAn equalizer with delay k acting on itries to recon-struct the k 1-st row of i :w ∗kis i −ks i −k1···See figure 1a .Since i has n rows,there is a total of npossible delays,and hence there are n different equalizers w k (k 0···n −1).Note in particular thatw ∗iis 0s 1···so thatw ∗iiw ∗kki k0···n −1Hence we can make the n equalizers all produce the sameoutput sequence by properly delaying the data sequences.2.3.NormalizationIf m is sufficiently large,then iis rank deficient.This is a source of non-uniqueness for the equalizers {w k },since any vector from the left null space of i may be added.Op-timization algorithms avoid this by asking for equalizers ofminimal norm (this also helps in reducing noise amplifica-tion).Alternatively,we can achieve the same effect by defin-ing equalizers to act on a minimal basis for the row span of i ,rather than i itself.Thus introduce the SVDs:iU i Σi Vii0···n −1If i has rank n ,then U i has n orthonormal columns,V i has n orthonormal rows,and Σi is a diagonal matrix contain-ing the n nonzero singular values.The rows of V i form an orthonormal basis for the row span of i .A “normalized”equalizer acting on V i is called t i ,which is related to w i viat iΣi U ∗i w ihence also satisfiest ∗i Vis 0s 1···(3)t ∗i Vit ∗k Vki k0···n −12.4.Super-equalizersDefineX T...n −1V TV...Vn −1(4)“Super-equalizers”are long vectors that collect several equal-izers with different delays,each reconstructing the same se-quence s 0s 1···.They act on the data X T or on the normal-ized data V T ,respectively:w ∗w ∗0···w ∗n −1t ∗t ∗0···t ∗n −1Hence the super-equalizer combines the outputs of the reg-ular equalizers,forming an average over all admissible de-lays.(By itself,it can also be interpreted as an ordinary equalizer of length n m −1at delay n −1.)See figure 1b .There is an issue of how to weight the outputs of each equal-izer and combine them in an optimal fashion.3.BLIND EQUALIZATION3.1.Subspace intersection methodFrom an algebraic perspective,the problem of blind equal-ization is,for given a data matrix ,to find a factoriza-tion where meets the required Toeplitz structure.Since a Toeplitz matrix is generated by a single vector in a linear way,this translates to findingss 0s 1···s N −1such thats ∈row 0s ∈row1...s ∈rown −1ax is i−kbs ix i n−1Fig.1.a Equalizer with delay k;b Super-equalizer,combining the outputs of several equalizers at different delays where‘row·’stands for the row span.Thus,we have tofind a single(hopefully unique)vector s which is in the in-tersection of all n subspaces.Numerically,there are several ways to compute the inter-section.The usual way is to construct the union of the com-plement of all subspaces,and take the complement again.The problem with this is that the complementary spacescan be highly dimensional.It was proven in[8]that,sincethe rows of V i form a minimal and orthonormal basis forrow i,the same result can be obtained by constructingthe matrix V T in(4)and looking for the right singular vec-tor corresponding to the largest singular values of V T.In thenoise-free case it is equal to the vector in the intersection;with noise perturbations,wefind a sequence that“best”fitsall subspaces.The corresponding left singular vector can beinterpreted as the equalizer that returns this sequence.In particular,it is proven in[8]that,if t ssi is the principalleft singular vector of V T and n L m−1,thent∗ssi V Tαs0s1···s N−1whereαis some nonzero scalar that makes the output se-quence have norm1.The reason,essentially,is that be-cause of the normalization,the largest singular value ofV T is bounded bysubject to the constraint that the sum of the powers is kept to a constant.The SSI method tends to maximize the coher-ence of the equalizers outputs.Indeed,in the noise-free case, all equalizers return the same output sequence s0s1···,up to a common scaling.Note that this result holds true only in the case of the constraint specified in(7).3.3.The MRE methodThe idea behind the mutually referenced equalizer(MRE) method for blind equalization[5]is tofind a vector of n equalizers w w∗0···w∗n−1∗that optimizes1min w n−2∑i0w∗i i−w∗i1i12(8)Each w i plays the role of an equalizer with delay i and servesas a training for the next equalizer.The criterion provides a collection of exact channel inverses in the noise free case[5].To avoid trivial solutions,w should be constrained,e.g.by fixing one of its entries or its norm.Another suitable con-straint is one that keeps the sum of output powers to a con-stant,w∗0w 1.The motivation for this particular choice is that it avoids trivial null space solutions w∗i i0∀i, which is necessary at least in the noise free case.The MRE method can be extended to look at all avail-able cross-differences[5],i.e.to solvew mre:arg minw∗0w1n−1∑i0n−1∑k0w∗i i−w∗k k2(9)Elaborating on this expression,wefindJ mre:n−1∑i0n−1∑k0w∗i i−w∗k k2n−1∑i0n−1∑k0w∗i R ii w i w k R kk w k−w∗i R ik w k−w∗k R ki w i2w∗n−1R00−R01···−R0n−1−R10n−1R11·......−R n−10···n−1R n−1n−1wIt thus follows thatJ mre2J ssi2n w∗0w Under the constraint w∗0w1,wefinally obtainmin w∗0w1J mre2n−maxw∗0w1J ssiso thatw mre w ssiHence we conclude that the SSI method and the extended MRE method under output power constraint provide identi-cal solutions.。