Blind Adaptive Constrained Constant-Modulus Reduced-Rank Interference Suppression Algorithms Based on Interpolation, Switched Decimation and Filtering

0.37 MB·English

by  Rodrigo C. de Lamare

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Blind Adaptive Constrained Constant-Modulus
Reduced-Rank Interference Suppression Algorithms
Based on Interpolation, Switched Decimation and
Filtering
Rodrigo C. de Lamare, Raimundo Sampaio-Neto and Martin Haardt
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1 Abstract—his work proposes a blind adaptive reduced-rank The proposed set of constraints ensures that the multi-path
0 schemeandconstrainedconstant-modulus(CCM)adaptivealgo- componentsofthechannelarecombinedpriortodimensionality
2 rithms for interference suppression in wireless communications reduction. In order to cost-effectively design the BARC scheme,
systems. The proposed scheme and algorithms are based on a we develop low-complexity decimation techniques, stochastic
n
two-stageprocessingframeworkthatconsistsofatransformation gradient and recursive least squares reduced-rank estimation
a
matrix that performs dimensionality reduction followed by a algorithms. A model-order selection algorithm for adjusting
J
reduced-rank estimator. The complex structure of the transfor- the length of the estimators is devised along with techniques
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mation matrix of existing methods motivates the development for determining the required number of switching branches to

of a blind adaptive reduced-rank constrained (BARC) scheme attain apredefinedperformance. An analysisof theconvergence
]
T along with a low-complexity reduced-rank decomposition. The propertiesandissuesoftheproposedoptimizationandalgorithms
proposedBARCschemeandareduced-rankdecompositionbased is carried out, and the key features of the optimization problem
I
. on the concept of joint interpolation, switched decimation and are discussed. We consider the application of the proposed
s
reduced-rank estimation subject to a set of constraints are algorithmstointerferencesuppressioninDS-CDMAsystems.The
c
then detailed. The proposed set of constraints ensures that the results show that the proposed algorithms outperform the best
[
multi-path components of the channel are combined prior to knownreduced-rankschemes, whilerequiringlower complexity.

1 dimensionality reduction. In order to cost-effectively design the T
v BARCscheme,wedeveloplow-complexitydecimationtechniques,
Index Terms—nterference suppression, blind adaptive estima-
2 stochastic gradient and recursive least squares reduced-rank
tion, reduced-rank techniques, iterative methods, spread spec-
1 estimation algorithms. A model-order selection algorithm for
trumsystems.nterferencesuppression,blindadaptiveestimation,
7 adjusting the length of the estimators is devised along with
reduced-rank techniques, iterative methods, spread spectrum
1 techniques for determining the required number of switching
systems.I
. branches to attain a predefinedperformance. An analysis of the
1
convergence properties and issues of the proposed optimization
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3 and algorithms is carried out, and the key features of the I. INTRODUCTION
optimizationproblemarediscussed.Weconsidertheapplication
1 Interference suppression in wireless communications has
of the proposed algorithms to interference suppression in DS-
v: CDMA systems. The results show that the proposed algorithms attracted a great deal of attention in the last decades [1], [2].
i outperformthebest knownreduced-rankschemes, whilerequir- Motivated by the need to counteract the effects of wireless
X
ing lower complexity. channels,to increase the capacity of multipleaccess schemes,
r his work proposes a blind adaptive reduced-rank scheme
and to enhance the quality of wireless links, a plethora of
a
and constrained constant-modulus (CCM) adaptive algorithms
schemes and algorithms have been proposed for equalization,
forinterferencesuppressioninwirelesscommunicationssystems.
The proposed scheme and algorithms are based on a two-stage multiuser detection and beamforming. These techniques have
processing framework that consists of a transformation matrix beenappliedtoavarietyofstandardsthatincludespreadspec-
that performs dimensionality reduction followed by a reduced- trum[3],orthogonalfrequency-divisionmultiplexing(OFDM)
rank estimator. The complex structure of the transformation
[4] and multi-input multi-output (MIMO) systems [5] and
matrix of existingmethodsmotivates thedevelopment of ablind
continue to play a key role in the design of wireless com-
adaptive reduced-rank constrained (BARC) scheme along with
a low-complexity reduced-rank decomposition. The proposed munications systems.
BARC scheme and a reduced-rank decomposition based on the
conceptof jointinterpolation,switcheddecimation andreduced-
A. Prior Work
rank estimation subject to a set of constraints are then detailed.
In order to design interference mitigation techniques, de-
signers are required to employ estimation algorithms for
Copyright (c) 2010 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be computing the parameters of the filters used at the receiver
obtained fromtheIEEEbysendingarequest [email protected]. or at the transmitter. In the literature of estimation algo-
This work was partially funded by the Ministry of Defence (MoD), UK,
rithms,onecan broadlydividethemintosupervisedandblind
ContractNo.RT/COM/S/021.Dr.R.C.deLamareiswiththeDepartmentof
Electronics, University ofYork,YorkY0105DD,United Kingdom,Prof.R. techniques. Blind methods are appealing because they can
Sampaio-Neto iswith CETUC/PUC-RIO,22453-900, RiodeJaneiro, Brazil alleviate the need for training sequences or pilots, thereby
and Prof. Haardt is with the Communications Research Laboratory, Ilme-
increasingthe throughputandefficiencyofwireless networks.
nau University of Technology, Germany. E-mails: [email protected],
[email protected] [email protected] Inparticular,blindestimationalgorithmsbasedonconstrained
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optimizationtechniquesareimportantinseveralareasofsignal are jointly optimized using the CCM design criterion. In
processing and communications such as beamforming and the BARC system, the number of elements for estimation is
interference suppression [6]. The constrained optimizations substantiallyreducedincomparisonwithexistingfull-rankand
in these applications usually deal with linear constraints that reduced-rankschemes,resultinginconsiderablecomputational
correspond to prior knowledge of certain parameters such savingsandimprovedconvergenceandtrackingperformances.
as direction of arrival (DoA) of users’ signals in antenna- AuniquefeatureoftheBARCandtheproposedalgorithmsis
array processing [7] and the signature sequence of the de- that, unlike existing blind schemes, they do not rely on the
siredsignalinCDMAinterferencesuppression[8].Numerous full-rank covariance matrix R for performing dimensional-
blind estimation algorithms with different trade-offs between ity reduction. The BARC and proposed algorithms skip the
performance and complexity have been reported in the last processing stage with R and directly obtain the subspace
decades [8]-[16]. The designs based on the constrained con- of interest and constraints via a set of simple interpolation,
stantmodulus(CCM)criterion[11],[12],[13],[14],[16]have decimation and reduced-rank estimation operations, which
shown increased robustness against signature mismatch and leads to much faster convergenceand improved performance.
improved performance over constrained minimum variance We developlow-complexitydecimationtechniques,stochastic
(CMV) approaches [8], [9], [10]. In general, the convergence gradient(SG) and recursive least squares (RLS) reduced-rank
and tracking performances of these algorithms depend on the estimation algorithms.Differentlyfrom [30], these algorithms
eigenvalue spread of the M ×M full-rank covariance matrix are designed with a set of constraints that are alternated in
R of the input data vector r[i] that contains M samples of theoptimizationprocedure.Amodel-orderselectionalgorithm
the signal to be processed, and the number of elements M in for adjusting the length of the filters is devised along with
the estimator [6]. When M is large, blind algorithms require techniques for determining the required number of switching
a largenumberofsamplestoreachtheirsteady-statebehavior branches to attain a predefined performance. The proposed
and may encounter problems in tracking the desired signal. model-order selection differs from [30] as it employs an
Reduced-rank signal processing is a key technique in low- extended filter approach, which is significantly simpler than
samplesupportsituationsandlargeoptimizationproblemsthat theschemein [30] thatusesmultipleschemesinparallel.The
has gained considerable attention in the last few years [17]- algorithms for adjusting the number of branches are based
[31]. The fundamental idea is to devise a transformation in on the constant modulus criterion as opposed to the mean-
such a way that the data vector r[i] can be represented by a squared error (MSE) criterion employed in [30]. An analysis
reducednumberofeffectivefeaturesandyetretainmostofits of the convergence properties and aspects of the proposed
intrinsic informationcontent[17]. The goal is to find the best optimization and algorithms is also presented. We apply the
tradeoff between model bias and variance in a cost-effective proposedBARCandalgorithmstointerferencesuppressionin
way. Prior work on reduced-rank parameter estimation has DS-CDMA systems.
considered eigen-decomposition techniques [18], the multi- This paper is organized as follows. The system model of a
stage Wiener filter (MSWF) [19], [14] that is a Krylov sub- DS-CDMA system and the problem statement are presented
space method, the auxiliary vector filtering (AVF) algorithm in Section II. Section III is dedicatedto the description ofthe
[20],[21],[22],[23],thejointanditerativeoptimization(JIO) BARCschemeandtheCCMreduced-rankestimators.Section
strategy [28], [29], [31] and adaptive interpolated filters [24], IV is devoted to the presentation of the blind adaptive SG
[25], [26]. A major problem with the MSWF, the AVF-based and RLS estimation algorithms, adjustment of model-order
and the JIO schemes is their high complexity. Prior work on selection and the number of switching branches, and their
adaptive interpolated filters [24], [25], [26] has considered complexity.SectionVprovidesananalysisandadiscussionof
MMSE- and CMV-based designs and shown a significant the proposed optimization problem. Section VI presents and
performance degradation for rank reduction with large com- discusses the simulation results and Section VII draws the
pression ratios. This problem has been recently addressed by conclusions.
thejointinterpolation,decimationandfiltering(JIDF)scheme
[27], [30] for supervised training. With the exception of the
CCM-based MSWF of [14] and the JIO of [31], there is no
blindreduced-rankthathaslowcomplexity,goodperformance II. SYSTEMMODELANDPROBLEM STATEMENT
and robustness against signature mismatches.
Let us consider the uplink of a symbol synchronous DS-
CDMA system with K users, N chips per symbol and L
p
B. Contributions of This Work
is the maximum number of propagation paths in chips. A
In this work, we present a low-complexity blind adaptive synchronousmodelis assumedforsimplicitysince itcaptures
reduced-rankconstrainedscheme (BARC) based on the CCM most of the features of asynchronous models with small to
criterion and a reduced-rank decomposition using joint inter- moderate delay spreads. The modulation is assumed to have
polation, switched decimation and reduced-rank estimation. constant modulus. Let us assume that the signal has been
The proposed scheme is simple, flexible, and provides a demodulatedatthebasestation,thechannelisconstantduring
substantial performance advantage over prior art. Unlike the each symbol and the receiver is perfectly synchronized with
JIDF scheme [30], the BARC uses an iterative procedure the main channel path. The received signal after filtering by
in which the interpolation, decimation and estimation tasks a chip-pulse matched filter and sampled at chip rate yields an
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M-dimensional received vector at time i the convergence and tracking performance and reduce the
complexity.Thisisperformedviathereductionofthenumber
K
r[i]= A [i]b [i]C h [i]+η [i]+n[i], (1) ofcoefficientsforcomputationfromM (full-rankschemes)or
k k k k k
kX=1 D+MD (existing blind reduced-rank schemes) to less than
a dozen. The structure of the BARC scheme is shown in Fig.
where M = N + L − 1, n[i] = [n [i] ... n [i]]T
p 1 M 1, where an interpolator, a decimator with several switching
is the complex Gaussian noise vector with zero mean and
decimation branches and a reduced-rank estimator which are
E[n[i]nH[i]] = σ2I whose components are independent and
time-varying are employed.
identically distributed, where (.)T and (.)H denote transpose
and Hermitian transpose, respectively, and E[.] stands for
expected value. The user symbols are denoted by bk[i], the ..
.
amplitude of user k is Ak[i], the first term in (1) represents Select that Output
the usersignals transmittedovermultipathchannelsincluding Interpolator minimizes Reduced-RankEstimator
.
the inter-chip interference (ICI), and ηk[i] is the inter-symbol .. D×1
interference (ISI) for user k from the adjacent symbols. The
M×1
signatureofuserkisrepresentedbys =[a (1)...a (N)]T,
k k k
theM×L constraintmatrixC thatcontainsone-chipshifted DesignofBlind
p k Algorithm
versions of the signature sequence for user k and the L ×1 M×D TransformationMatrix
p
vector h [i] with the multipath components are described by
k
a (1) 0 Fig.1. Proposedblindadaptive reduced-rank estimation structure.
k
 ... ... ak(1)  hk,.0[i] v[iT]h=e[Mv[i×].1..rvec[ie]iv]eTdwveitchtoIrbre[iin]gistfiheltelerendgtbhyotfhethienitnertperoploatloa-r
Ck = ak(N) .. ... ,hk[i]= hk,Lp..−1[i] . tthoeraMnd×y0iMeldcsothnIve−oi1lnutteiropnolmataetdrixveVcto[ir]rwIh[ii]c=h hVasHs[hii]frt[eid],cwohpeieres
 0 . ak(N)  of v[i] as described by
(2)
The multiple access interference (MAI) comes from the v[i] 0 ... 0
0
non-orthogonality between the received signature sequences,  ... v[i] ... ... 
whereasthe ISI span Ls dependsonthe lengthof the channel  0. 
response and how it is related to the length of the chip v[i] .. ... 0 
V[i]= I−1 . (4)
Nseq,Luesnc=e.2F,ofrorLpN=<1L, pL≤s =2N1,L(nso=ISI3),, afonrd s1o <on.LTph≤is  0. vI[i.−]1 .... 0. 
means that at time instant i we will have ISI coming not  . . . . 
 . . . . 
only from the previous Ls − 1 time instants but also from  0 0 ... v[i]
the nextLs−1 symbols. The linearmodel in (1) can be used  0 
to representotherwireless communicationssystemsincluding Let us now express the M × 1 vector r [i] in a way that
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MIMO and OFDM systems. For example, the user signatures is suitable for algebraic manipulation as a function of the
ofaDS-CDMAsystemareequivalenttothespatialsignatures interpolator v[i]:
of MIMO system.
r [i]=VH[i]r[i]=ℜ [i]v∗[i], (5)
A reduced-rank interference suppression scheme processes I o
thereceivedvectorr[i]intwostages.Thefirststageperforms wheretheM×I Hankelmatrix[32]withthereceivedsamples
a dimensionality reduction via a decomposition of r[i] into a of r[i] performs the convolution and is described by
lower dimensional subspace. The second stage is carried out
by a reduced-rank estimator. The output of a reduced-rank r[i] r[i] ... r[i]
0 1 I−1
scheme corresponding to the ith time instant is  r[i] r[i] ... r[i] 
1 2 I
z[i]=w¯H[i]SHD[i]r[i]=w¯H[i]r¯[i], (3) ℜo[i]= ... ... ... ... . (6)
where SD[i] is an M × D decomposition matrix  rM[i]−2 rM[i]−1 ... 0 
which performs dimensionality reduction and  r[i] 0 ... 0 
 M−1 
w¯[i] = [w¯[i] w¯[i] ...w¯[i]]T is the D × 1 parameter
1 2 D The M × 1 vector r [i] is transformed by a decimation
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vector of the reduced-rank estimator. The basic problem is
unitthatcontainsB switching decimationpatternsin parallel,
how to cost-effectively and blindly design the M ×D matrix
leading to B different D × 1 vectors r¯ [i], b = 1,...,B,
b
S [i] that transforms the M × 1 vector r[i] into a D × 1
D where L is the decimation factor and D = M/L is the rank
reduced-rank vector r¯[i] using the CM criterion .
of the BARC system. This is inspired by diversity techniques
found in wireless communications [35], whose principle is
III. PROPOSED BARC SCHEME to collect different copies of signals and combine them to
In this section we introduce the proposed BARC scheme increasethesignal-to-noiseratio,andswitchedcontrolsystems
and detail its key features. The motivation is to improve [36] that exploit switching rules to stabilize and design a
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system. The decimation procedure corresponds to discarding r¯[i] = D [i]VH[i]r[i] that will be used in the following
b
M−D samplesofr [i]with differentpatterns,resultingin B procedure for the design of v[i] and w¯[i].
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differentD×1 decimatedvectorsr¯ [i]. The D×1 decimated By using the method of Lagrange multipliers, fixing w¯[i]
b
vector for branch b is given by and minimizing the Lagrangian with respect to v[i], the
expression for the interpolator becomes
r¯ [i]=D [i]r [i], b=1,...,B (7)
b b I
v[i+1]=R¯−1[i] d¯ [i]−(p¯H[i]R¯−1[i]p¯ [i])−1
whereeachrowof Db[i] containsasingle1andM−1zeros. u u w u w (12)
TheD×M decimationmatrixDb[i]isequivalenttoremoving ·p¯w[i] (cid:2)p¯Hw[i]R¯−u1[i]d¯u[i]−ν ,
M−D samplesofrI[i].ThematricesDb[i]aredesignedoff- where R¯ [i] = E[|(cid:0)z[i]|2u[i]uH[i]], d¯ [i](cid:1)(cid:3)= E[z∗[i]u[i]],
u u
line, stored at the receiver and the best Db[i] is selected to u[i]=ℜT[i]w¯∗[i] and p¯ [i]=PT[i]w¯[i]. The D×I matrix
minimize a desired objective function. The output zb[i] of the P [i]=Db [i]ℜ [i] ariseswfrom theoconstraint and the equiva-
o p
BARCschemecorrespondstofilteringr¯b[i]withw¯[i]andthen lence w¯H[i]SH[i]p[i]= w¯H[i]PT[i]v∗[i] = vH[i]p¯ [i] = ν,
selecting the branch that minimizes the desired criterion. The where ℜk [i] iDs a D × MkHankoel matrix witkh elewments of
p
outputz [i] is a functionof w¯[i], D [i] and v[i] expressedby
b b the effective signature p[i] shifted in a similar way to (6).
z [i]=w¯H[i]SH [i]r[i]=w¯H[i] D [i]VH[i]r[i] By fixing the interpolatorv[i] andminimizingthe Lagrangian
b D,b b
=w¯H[i] D [i]ℜ [i] v∗[i]=(cid:0)w¯H[i]ℜ [i]v∗[i](cid:1) (8) with respect to w¯[i], we obtain
b o b
=vH[i](cid:0)ℜTb[i]w¯∗[i](cid:1)=vH[i]u[i], w¯[i+1]=R¯−z1[i] d¯z[i]−(p¯H[i]R¯−z1[i]p¯[i])−1 (13)
whereu[i]=ℜT(cid:0)[i]w¯∗[i]isa(cid:1)nI×1vector,the D coefficients ·p¯[i] p¯(cid:2)H[i]R¯−z1[i]d¯z[i]−ν ,
b
of w¯[i] and the I elements of v[i] are assumed complex and where R¯ [i] = E(cid:0)[|z[i]|2r¯[i]r¯H[i]] = (cid:1)S(cid:3)H[i]R [i]S [i]
z D z D
theD×I matrixℜb[i]isℜb[i]=Db[i]ℜo[i].Inwhatfollows, , Rz[i] = E[|z[i]|2r[i]rH[i]] , d¯z[i] = E[z∗[i]r¯[i]] =
we will develop constrained constant modulus (CCM)-based SH[i]E[z∗[i]r[i], p¯[i]=SH[i]p[i] and S [i]=D [i]VH[i].
D D D b
estimatorsanddescribehowtheswitchingruleisincorporated We remark that (11), (12) and (13) depend on each other and
into the proposed blind design. theirpreviousvalues.Therefore,itisnecessaryto iterate(11),
(12)and(13)inanalternatedform(onefollowedbytheother)
A. Joint Iterative CCM Design of Estimators and Discrete with an initial value to obtain a solution. The expectations
Optimization can be estimated either via time averages or by instantaneous
The design of the BARC scheme is equivalent to solving a estimates as will be described by the adaptive algorithms.
joint optimization problem with v[i], D [i] and w¯[i] using
b
a strategy based on fixing two parameters and optimizing B. Design of Decimation Schemes
one,andalternatingthe procedureamongthe parametersuntil We are interested in developing decimation schemes that
convergence.A key feature of this problem is that it involves are cost-effective and easy to employ with the proposed
a combination of continuous and discrete optimization proce- BARC scheme. This can be done by imposing constraints
dures. Specifically, the design corresponds to the constrained on the structure of D [i]. Since the operator D [i] performs
b b
continuous minimization of the estimators v[i] and w¯[i] and decimation, the structure of D [i] is constrained to contain
b
the discrete minimization of D[i] according to the CCM only zerosand D ones. Thus, the decimationoperationof the
design criterion. BARC scheme amounts to discarding samples in conjunction
Let us describe the CCM estimators design of the BARC with filtering by v[i] and w¯[i]. The decimation matrix D [i]
b
structure. The CCM design for v[i], D [i] and w¯[i] can be is selected so to minimize the square of the instantaneous
b
computed through the optimization problem constantmoduluserrorobtainedforthe B branchesemployed
as follows
v ,D ,w¯ =arg min J (v[i],D [i],w¯[i]),
opt opt opt CM b
(cid:8) (cid:9) v[i],Db[i],w¯[i] D [i]=D [i] when b =arg min (e [i])2, (14)
subject to w¯H[i]SH[i]p[i]=ν, b bs s 1≤b≤B b
k D
(9) where e [i] = |w¯H[i]SH [i]r[i]|2 − 1. The design of the
b D,b
where the parameter ν is a constant employed to enforce decimation matrix Db[i] considers a general framework that
convexity and can be used for any decimation scheme and is illustrated by
JCM(v[i],Db[i],w¯[i])=E |w¯H[i]ℜ[i]v∗[i]|2−1 2 . (10) dT1,b[i]
.
h(cid:0) (cid:1) i  .. 
The decimation matrix D [i] is selected to minimize the
square of the instantaneous cbonstant modulus error obtained Db[i]= dTj,b[i] , (15)
for all the B branches according to  ... 
 
Db[i]=Dbs[i] when bs =arg min (eb[i])2, (11)  dTD,b[i] 
1≤b≤B
where each row of the matrix D [i] is structured as
wheretheconstantmoduluserrorsignaloftheBARC scheme b
is eb[i] = |w¯H[i]SHD,b[i]r[i]|2 − 1. With the selected deci- dj,b[i]=[0 ... 0 1 0 ... 0 ]T, (16)
mation matrix Db[i], we can form the reduced-rank vector γj zeros (M−γj−1) zeros
| {z } | {z }
5
and the index j (j = 1,2,...,D) denotes the j-th row of determining the minimum number of branches required to
the matrix, the rank of the matrix D [i] is D = M/L, the achieve a predetermined performance. The model-order and
b
decimation factor is L and B corresponds to the number of number of branches selection algorithms are decoupled in
parallel branches. The quantity γ is the number of zeros order to reduce the search space and the computational cost.
j
chosen according to a given design criterion. We have tested a joint search over I, D and B and this has
Given the constrained structure of D [i], it is possible to notresultedinperformancegainsovertheseparatesearchover
b
deviseanoptimalprocedurefordesigningD [i]viaanexhaus- B and over I and D. Unlike prior work [30] with the MSE
b
tive search of all possible design patterns with the adjustment criterion, the proposed algorithms employ the CM approach
of the variable γ , where an exhaustive procedure that selects and rely on a set of linear constraints. The complexity of the
j
D samples out of M possible candidates is performed. The proposed SG, RLS and model-order selection algorithms is
total number of patterns B is equal to compared with existing methods in terms of additions and
ex
multiplications.
M
B =M ·(M −1)...(M −D+1)= .
ex (cid:18) D (cid:19)
A. SG Algorithms for The BARC Scheme
D terms
We can view| this exhaust{izve procedure }as a combinatorial To design the estimators v[i] and w¯[i] and the decimation
matrix D[i], we consider the Lagrangian
problemthathasM samplesaspossiblecandidatesforthefirst
row of Db[i] and considersM−j+1 positionsas candidates L(v[i],D[i],w¯[i])=E |w¯H[i]ℜb[i]v∗[i]|2−1
for the following D −1 rows of the matrix Db[i], where j h(cid:0) (cid:1)i (17)
is the index used to denote jth row of the matrix D [i]. The +2ℜ w¯H[i]SH[i]p[i]−ν λ ,
b D
exhaustive scheme described above is, however, too complex h(cid:0) (cid:1) i
where λ is a Lagrange multiplier and ℜ[·] denotes the real
for practical use because it requires D permutations of M
partoftheargument.Theinputvectorr[i]isprocessedbythe
samplesforeachsymbolintervalandM−1candidatesforthe
interpolatorv[i],yieldingr [i]=VH[i]r[i].Wethencompute
positions,andcarriesoutanextensivesearchoverallpossible I
the decimated interpolated vectors r [i] for the B branches
patterns. b
withthedecimationmatrixD [i],where1≤b≤B.Oncethe
It is highly desirable to employ decimation schemes that b
B candidate vectors r¯ [i] are computed, we select the vector
arecost-effectiveandgatherimportantpropertiessuchas low- b
r¯ [i] which minimizes the square of
requirementsofstorageandcomputationalcomplexityandcan b
work with a small number of branches B. By adjusting the e [i]=|w¯H[i]S [i]r[i]|2−1. (18)
b D,b
variable γj in the framework depicted in (15), we can obtain where S [i]=V[i]DH[i]. Based onthe selection ofD [i],
the following sub-optimal schemes: D,b b b
we choose the corresponding reduced-rank vector r¯[i] and
A. Uniform(U)DecimationwithB =1.Wemakeγj = select the errorof the proposedSG algorithm e[i] as the error
(j−1)L and this correspondsto the use of a single e [i] with the smallest squared magnitude of the B branches
b
branch(B =1)onthedecimationunit(noswitching according to
and optimization of branches), and is equivalent to
S [i]=S , r¯[i]=r¯ [i] and e[i]=e [i]
the scheme in [26]. D D,bs bs bs
B. Pre-Stored (PS) Decimation. We select γ = (j − when (19)
j
1)L+(b−1) which corresponds to the utilization b =arg min (e [i])2.
s b
of uniform decimation for each branch b out of B 1≤b≤B
branches and the different patterns are obtained by In order to derive an SG algorithm for v[i], we need to
picking out adjacent samples with respect to the transform the proposed constraint in (9) and obtain a suit-
previous and succeeding decimation patterns. able and equivalent form for use with v[i]. We can write
C. Random(R)Decimation.We chooseγj asadiscrete w¯H[i]SHD[i]p[i] = w¯H[i]PTo[i]v∗[i] = vH[i]p¯w[i] = ν,
uniform random variable, which is independent for where p¯w[i] = PTo[i]w¯[i] and the D ×I matrix Po[i] is a
eachrowjoutofBbranchesandwhosevaluesrange functionofDb[i] andp[i]andisgivenbyPo[i]=D[i]ℜp[i],
between 0 and M −1. A constraint is included to where ℜp[i] is a D×M Hankel matrix with elements of the
avoid rows with repetitive patterns. effective signature p[i] shifted in a similar way to (6). We
need to construct p¯ [i] foreach symbolfrom P [i] and w¯[i].
w o
IV. BLIND ADAPTIVEESTIMATIONALGORITHMS Minimizing (17) and using the proposedequivalentconstraint
vH[i]p¯ [i]=ν, we obtain
In this section, we develop SG and RLS estimation algo- w
rithms[6] forestimating the parametersof the BARC scheme v[i+1]=v[i]−µ e[i]z∗[i] I−(p¯H[i]p¯ [i])−1p¯ [i]p¯H[i] u[i],
(v[i],D[i] and w¯[i]). The SG algorithmsrequirethe setting of v (cid:18) w w w w (cid:19)
stepsizesandareindicatedforsituationswheretheeigenvalue (20)
spread of R¯−1[i] is small. The RLS algorithms need the where µv is the step size. Minimizing (17) and using the
setting of fozrgetting factors and are suitable for scenarios constraint w¯H[i]SHD[i]p[i]=ν, we obtain
in which R¯−1[i] has a large eigenvalue spread. We also
z w¯[i+1]=w¯[i]−µ e[i]z∗[i] I −(p¯H[i]p¯[i])−1p¯[i]p¯H[i] r¯[i],
present blind model-order selection algorithms for adjusting w (cid:18) (cid:19)
the lengths D and I of the estimators and algorithms for (21)
6
whereµw isthestepsize. TheSGalgorithmfortheBARChas k¯ [i]= α−1Rˆ¯−z1[i−1]z[i]r¯[i] (30)
acomputationalcomplexityO(D+NI)andemploysequations z 1+α−1r¯H[i]z[i]Rˆ¯−1[i−1]z∗[i]r¯[i]
(19)-(21). In fact, the BARC scheme trades off one SG z
algorithm with complexity O(M) against two SG algorithms
withcomplexityO(D)andO(I),operatingsimultaneouslyand Rˆ¯−z1[i]=α−1Rˆ¯−z1[i−1]−α−1k¯z[i]z∗[i]r¯H[i]Rˆ¯−z1[i−1]
exchanging information. (31)
and the initial values of the recursions are Rˆ−1[i] = δ I
z w
and dˆ [0]=ρ , where δ and ρ are small positive scalars.
B. RLS Algorithms for the BARC Scheme z w w w
The RLS algorithm for the BARC has a computational cost
In order to design the estimators v[i], w¯[i] and the matrix of O(D2)+O(I2) and consists of equations (23)-(31).
D[i] with RLS algorithms, we consider the Lagrangian
i
L (v[i],D[i],w¯[i])= αi−l |w¯H[i]ℜ[l]v∗[i]|2−1 C. Model-Order Selection Algorithms
LS
Xl=1 (cid:0) (cid:1) This part develops model-order selection algorithms for
+2ℜ w¯H[i]SH[i]p[i]−ν λ , automatically adjusting the lengths of the estimators used in
D
h(cid:0) (cid:1) i(22) the BARC scheme. Prior work in this area has focused on
methodsformodel-orderselectionwhichutilizeMSWF-based
where λ is a Lagrange multiplier and α is a forgetting factor. algorithms [19] or AVF-based recursions [20], [21], [22]. In
We perform the signal processing according to the block the proposedapproachwe constrainthe search within a range
diagram of Fig. 1. Based on the choice of D [i], we select ofappropriatevaluesandrelyonaCCM-basedLScriterionto
b
the corresponding reduced-rank vector r¯[i] and the error e[i] determine the lengths of v[i] and w¯[i] that can be adjusted in
as the error e [i]=|w¯H[i]S [i]r[i]|2−1 with the smallest aflexiblestructure.Theproposedschemewithextendedfilters
b D,b
squared magnitude of the B branches as follows issignificantlyless complexthanthe multiplefilters approach
reported in [30]. The model-order selection algorithm for the
S [i]=S , r¯[i]=r¯ [i] and e[i]=e [i]
D D,bs bs bs BARC is called Auto-Rank and minimizes
when (23)
i
bs =arg min (eb[i])2. C(v[i],D[i],w¯[i])= αi−l |w¯H[i]D[i]ℜ [l]v∗[i]|2−1 ,
1≤b≤B o
Xl=1 (cid:16) (cid:17)
Minimizing (22) with respect to v[i], using the constraint (32)
vH[i]p [i]=ν and the matrix inversion lemma [6], we get The order of v[i], D[i], w¯[i], and the associated matrices
w
Rˆ¯ [i], and Rˆ¯ [i] defined in (27) and (31), respectively, that
u z
v[i+1]=Rˆ−1[i] dˆ [i]+(pH[i]Rˆ−1[i]p [i])−1 are necessary for the computation of v[i] and w¯[i] require
u (cid:18) u w u w
(24) adjustment.Tothisend,wepredefinev[i]andw¯[i]asfollows:
·p [i](dH[i]Rˆ−1[i]p [i]−ν) ,
w w u w (cid:19) v[i]= v [i] v [i] ... v [i] ... v [i] T
1 2 Imin Imax
where w¯[i]=(cid:2) w [i] w [i] ... w [i] ... w (cid:3)[i] T
1 2 Dmin Dmax
(cid:2) ((cid:3)33)
dˆ [i]=αdˆ [i−1]+(1−α)z∗[i]u[i] (25)
u u
For each data symbol we select the best order for the
α−1Rˆ−1[i−1]z[i]u[i] model. The proposed Auto-Rank algorithm that chooses the
k [i]= u (26)
u 1+α−1uH[i]z[i]Rˆ−1[i−1]z∗[i]u[i] best lengths Dopt[i] and Iopt[i] for the filters v[i] and w¯[i],
u respectively, is given by
Rˆ−1[i]=α−1Rˆ−1[i−1]−α−1k [i]z∗[i]uH[i]Rˆ−1[i−1]
u u u u {D [i],I [i]}=arg min C(v[i],D[i],w¯[i]) (34)
(27) opt opt Imin≤n≤Imax
and the initial values of the recursions are Rˆ−1[i] = δ I Dmin≤d≤Dmax
u v
and dˆu[0] = ρv, where δv and ρv are small positive scalars. where d and n are integers, Dmin and Dmax, and Imin
Minimizing (22) with respect to w¯[i], using the constraint and I are the minimum and maximum ranks allowed
max
w¯H[i]SH[i]p[i]= ν and the the matrix inversion lemma [6], for the reduced-rank filter and the interpolator, respectively.
k D
we obtain The additional complexity of the Auto-Rank algorithm is
that it requires the update of all involved quantities with the
w¯[i+1]=Rˆ¯z−1[i](cid:18)dˆ¯z[i]+(p¯H[i]Rˆ¯−z1[i]p¯[i])−1 maximum allowed rank Dmax and Imax and the computation
(28) of the cost function in (32). This procedure can significantly
·p¯[i](p¯H[i]Rˆ¯−1[i]dˆ¯ [i]−ν) , improve the convergence performance and can be relaxed
z z (cid:19)
(the rank can be made fixed) once the algorithm reaches
where steady state. An inadequate rank for adaptation may lead to
a performance degradation, which gradually increases as the
dˆ¯ [i]=αdˆ¯ [i−1]+(1−α)z∗[i]r¯[i] (29) adaptation rank deviates from the optimal rank.
z z
7
D. Automatic Selection of the Number of Branches the complexity are the length D of w¯[i] or the number of
auxiliaryvectors(AVs)fortheAVFalgorithm[20],[21],[22],
In this subsection we propose algorithms for automatically
selecting the number of branches necessary to achieve a the numberof samples M of r[i], the numberof branches B,
predeterminedperformance. This performancemeasure is de- the length I of v[i] and the number Lp of assumed multipath
components.
terminedoff-lineasaquantityrelatedtotheconstantmodulus
cost function. The first algorithm, termed selection of the
TABLEI
numberof branches(SNB), relies ona simple search overthe COMPUTATIONALCOMPLEXITY OFSG ALGORITHMS.
parallel branches of the BARC scheme and tests whether the
predeterminedperformancehasbeenattainedviaacomparison Numberofoperations persymbol
Algorithm Additions Multiplications
with a threshold ρ. The second algorithm builds on the SNB
Full-rank-trained [6] 2M 2M +1
algorithm and incorporates prior statistical knowledge about (eq. (9.5)-(9.7) of [6]) 2M 2M +1
the use ofthe branchesvia sortingand isdenotedSNB-S. Let MSWF-trained [19] 2(D−1)2+2D(M −1) D2+3D+2DM
us first define for each time interval i the branch cost as (eq. (53)-(62) of [19]) +(D−1)(M−1)+M +M+1
Full-rank-CCM [12] 8M +MLp 7M +MLp
C (v[i], D [i],w¯[i])=(e [i])2 (35) (eq. (10),(11),(13) of [12]) +2 +2
branch b b MSWF-CCM [14] DM2+3(D−1)2+2D DM2+2D2+7D
where (table II of [14]) +3DM +4M +3 +2DM +MLp+2
eb[i]=|w¯H[i]Db[i]ℜo[i]v∗[i]|2−1 (eq.JI(1O4-)C-(C15M) o[f?][?]) 4+D2MD+−M2 4+D7MD++M6
Proposed BARC-CCM 4D+BD+4I 4D+B(D+1)
is the error signal for each branch. The proposed algorithms (eq. (18)-(21) +(I−1)M−2 +5I+IM+4
for automatically selecting the number of branches perform
the following optimization
TABLEII
Bs[i]=argmin min Cbranch(v[i], Db[i],w¯[i]) COMPUTATIONALCOMPLEXITY OFRLSAND AVF-BASED
Bmax 1≤b≤Bmax ALGORITHMS.
subject to C (v[i], D [i],w¯[i])≤ρ
branch b
(36) Numberofoperations persymbol
Algorithm Additions Multiplications
where b is an integer and Bmax is the maximum number of Full-rank-trained [6] 3(M−1)2 3M2
branchesallowedfortheBARCscheme,respectively,Bsisthe (table 13.1 of [6]) +M2+2M +2M +2
numberofbranchesrequiredtoattainthedesiredperformance MSWF-trained [19] D2+2(D−1)2 4D2+3D
and ρ is the prespecified performance. The SNB algorithm (eq. (69)-(71) of [19]) +2D(M −1)+M +2DM +MLp
+(D−1)(M−1) +4
determines the minimum number of branches necessary to
Full-rank-CCM [13] 5(M−1)2 4M2+5M
achieve a predetermined performance ǫ according to the cost (eq. (6)-(10) of [13]) +M2+5M −1 +L2p+MLp
function defined in (35). It iteratively increases the number +3(Lp−1)2 +Lp+4
of branches by one until the predetermined performance ρ is MSWF-CCM [14] DM2+5D2 DM2+6D2
attained. The parameter ρ can be chosen as a function of the (table III of [14]) +2DM +(M−1)Lp +2DM +(D−1)M
MMSE with a penaltyallowed bythe designer.An alternative +(D−1)(M+1) +7D+MLp+4
JIO-CCM [?] 5M2+DM 6M2+(2D+6)M
to the SNB algorithm is to exploit prior statistical knowledge (eq. (10)-(11) of [?]) +5D2+3D−1 +5D2+9D+3
about the most frequently used branches and sort the deci- AVF-trained [20] D(3M2−2M) D(4M2+3M)
mation matrices D [i] in descending order of probability of (eq. (3),(11)-(13) of [20] +2M−1 +4M +2
b
occurrence. The SNB algorithm with sorting will be termed Proposed BARC-CCM 6D2+(B+1)D+6I2 7D2+(B+8)D+7I2
(eq. (23)-(31)) +I+(I−1)M +8 +7I+IM+3
SNB-S and consists of ordering the matrices D [i] which are
b
most likely to be used. This can be done at the beginning of
InFig. 2 we illustrate the maincomplexitytrendsbyshow-
thetransmissionandupdatedwheneverrequired.Animportant
ing the computational complexity in terms of the arithmetic
measure that arises from the SNB and SNB-S algorithms is
the average number of branches B = 1/Q Q B [i] operationsasa functionofthenumberofsamples M.We use
avg i=1 s the same colors for the corresponding SG techniques in Fig.
with Q being the data record, which illustrates tPhe savings
2(a)andtheRLScounterpartsinFig.2 (b).Forthese curves,
in computations of the branches.
we considerL =9,D =5,I =3andB =8 forthe BARC,
p
assume D = 4 for the MSWF-SG based approaches, while
E. Computational Complexity
we use D = 5 for the MSWF-RLS techniques and D = 8
In this section we detail the computational complexity of for the AVF technique with non-orthogonal auxiliary vectors
theproposedandexistingSG,RLS andmodel-orderselection (AVs)[20],[21],[21].Thereasonwhyweusedifferentvalues
algorithms, as shown in Tables I, II and III. This complexity for D is because we must find the most appropriate trade-
refers to an adaptive linear receiver that only requires the off between the model bias and variance [17] by adjusting
timing and the spreading code of the user of interest. The D (AVs for the AVF) and this depends on the scheme. We
computational requirements are described in terms of addi- always use the best values for each scheme. The curves in
tions and multiplications and have been derived by counting Fig. 2 (a) show that the reduced-rank BARC SG algorithms
the necessary operations to compute each of the recursions have a complexity slightly higher than the full-rank trained
required by the analyzed algorithms. The key parameters of SGalgorithmsandsubstantiallylowerthantheotheranalyzed
8
TABLEIII
COMPUTATIONALCOMPLEXITYOFMODEL-ORDERSELECTION Imin) additions, as depicted in the first row of Table III, in
ALGORITHMS. additiontotheoperationsrequiredbytheproposedalgorithms,
whose complexity is shown in the last rows of Tables I and
Algorithm Additions Multiplications
Auto-Rank 2(Dmax−Dmin)+1 − II. For the operation of the MSWF and the AVF algorithms
(Extended Filters) 2(Imax−Imin)+1 with model-order selection algorithms, a designer must add
the complexities in Tables I and II to the complexity of the
Projection with 2(2M −1)× ((M)2+M+m1)o×del-orderselectionalgorithmofinterest,asshownin Table
Stopping Rule [19] (Dmax−Dmin)+1 (Dmax−DminI+II.1T)he model-order selection algorithm with multiple filters
CV [20] (2M−1)× (Dmax−Dmin+ha1s)×a number of arithmetic operations that is substantially
(2(Dmax−Dmin)+1) M+1 higher than the other compared methods and requires the
computationof (D −D +1)+(I −I +1) pairs
2(Dmax−Dmin)+1 +7Dm2ax+9Domfafixlters with costsmfax(D,Im)inand f (Dm,Ia)xformadinditions and
Multiple Filters [30] fa(Dmax,Imax)+... fm(Dmax,Imax)+... a m
(JIDF or BARC) +fa(Dmin,Imin) +fm(Dmin,Imminu)ltiplications,respectively,foreachpairoffilterswithDand
2(Dmax−Dmin)+1 2(Imax−Imin)I+. S1pecifically, these costs are shown as a function of D and
I at the bottom of Table III and we have for the SG version
f (D,I) = 4D +BD +4I +(I −1)M −2 additions and
a
reduced-rankalgorithms. For the RLS algorithms, depicted in f (D,I) =4D+B(D+1)+5I +IM +4 multiplications
m
Fig. 2 (b), we verify that the BARC reduced-rank scheme (seethelastrowsofTableI),whereasfortheRLSversionwe
is much simpler than any full-rank or reduced-rank RLS have f (D,I)=6D2+(B+1)D+6I2+I+(I−1)M+8
a
algorithm. This is because there is a quadratic cost on M additionsandf (D,I)=7D2+(B+8)D+7I2+7I+IM+3
m
rather than D for the full-rank schemes operating with the multiplications(see the last rows of Table II). It . Despite the
RLS algorithm and a high computationalcost associated with cost, its performanceis comparablewith the proposedmodel-
the designof the transformationmatrix SD[i] forall reduced- order selection algorithm with extended filters.
rankmethodsexceptfortheBARC scheme.TheAVFscheme
[20], [21], [22] usually requires extra complexity as it has
V. ANALYSIS OF THEPROPOSED ALGORITHMS
more operations per auxiliary vector (AV) and also requires
a higher number of AVs to ensure a good performance. The In this section, we develop a stability analysis of the
trained AVF employs a cross-correlation vector estimated by proposed method and SG algorithms and study the conver-
pˆ[i]=αpˆ[i−1]+(1−α)b∗[i]r[i]. gence issues of the optimization problem. Specifically, we
k
studytheexistenceofmultiplesolutionsanddiscussstrategies
for dealing with it. We consider particular instances of the
104 (a) Complexity of SG Algorithms 107 (b) Complexity of RLS Algorithms proposed algorithms for which a global minimum may be
encountered by the proposed SG and RLS algorithms. We
also examine cases for which there is no guarantee that the
106 algorithms will converge to the global minimum and may
end up in local minima. It should be mentioned, however,
mber of operations103 Full−rank−trained mber of operations110045 cttIhetosawnttevadetshrgfevoeerprtirofiaoepnadopusipmnerdbothexSreimGsoeafateeanxlpdpypeltRrihicmLeaStesinoaatmnslsgetoharfianitltdthtehmnreusvamawllgeueoreorrseiutshiermxrsectsesepnaneslawicrvtiaeiovylsyes.
Nu102 Full−rank−CCM Nu of the initialization. This suggests that the problem may have
MSWF−trained multiple global minima or that every point of minimum is
MSWF−CCM 103
a point of global minimum or that the switching of branches
JOINT−CCM
BARC−CCM allowsthealgorithmstofindtheglobalminimum.Specifically,
AVF−trained
101 102 we are interested in examining three cases of adaptation and
0 100 200 300 0 100 200 300
M M parameter estimation, namely:
• Case i) - SD[i] is fixed, i.e. the interpolator v[i] and the
Fig.2. Complexityintermsofarithmeticoperationsof(a)SGand(b)RLS decimation matrix D[i] are fixed.
algorithms andAVF-basedrecursions. • Caseii)-SD[i]istime-variantwithD[i]beingfixedand
v[i] being time-variant.
Thecomputationalcomplexityoftheproposedmodel-order • Case iii) - SD[i] is time-variant,where D[i] and v[i] are
selection algorithm (Auto-Rank) and the existing rank selec- both time-variant.
tion algorithms is shown in Table III. We can notice that the • Case iv) - SD[i] is time-variant, where D[i] is time-
proposedmodel-orderselectionalgorithmwithextendedfilters variant and v[i] is time-invariant.
is significantly less complex than the existing methods based For the sake of analysis and the convexity issues of the
on projection with stopping rule [19] and the CV approach problem, we have opted for studying the method for the
[20]. Specifically, the proposed rank selection algorithm with four cases previously outlined. This allows us to gain further
extended filters only requires 2(D −D )+2(I − insightanddrawconclusionsonthepropertiesofthedifferent
max min max
9
configurations of the method. A key feature of the proposed where
methodwhichmakesitsconvergencestudyextremely difficult
is the combined use of discrete and continuous optimization A= I−µve[i]Bw[i]u[i]DH[i]rH[i] µve[i]Bw[i]u[i]DH[i]rH[i]SD
techniques.Eventhoughthenecessaryconditionsfortheopti- (cid:20) I(cid:8)−µwe[i]Π[i]SHD[i]r[i]rH[i]SD[(cid:9)i] −µwe[i]Π[i]SHD[i]r[i]rH[i]
mization algorithms are met [33], [34] and the cost functions (cid:8) (cid:9)
udsifefderfeonrtidaebrliev,inthgethdeisScGretaendnaRtuLrSeaolgfotrhiethmdescaimreactioonntinanudoutshlye C ="−µwe[i]+Πµ[vi]eS[iHD]B[i]wr[[ii]]ur[Hi][Di](He[Si]Dr[Hi][(iw¯](oSpDt−[i](II)−+wS¯Dop,to)pt−w¯SopDt),o0ptD)×(M−1)
patternsused make its theoreticalanalysis highlychallenging.
This proof is beyond the scope of this paper and remains a The previous equations imply that the stability of the algo-
very interesting open problem. rithms depends on the spectral radius of A. The parameters
of w¯[i] and S [i] will remain bounded and will converge
D
A. Stability Analysis
asymptotically to the optimal values if the step sizes are
In this part, we examine the stability of the proposed SG chosensuchtheeigenvaluesofAHAarelessthanone.Unlike
algorithms. In order to establish these conditions, we define the stability analysis of most adaptive algorithms [6], in the
the error matrices at time i as proposed approach the terms are more involved and depend
E [i]=S [i]−S and oneachotherasevidencedbytheequationsforAandC.Let
SD D D,opt (37)
us now examine the three cases outlined at the beginning of
e [i]=w¯[i]−w¯ ,
w¯ opt
this section.
where w¯ and S are the optimalparameterestimators.
opt D,opt For case i), the transformation S is fixed and we can
D
Since we are dealing with a joint optimization procedure,
consider only the recursion for the error vector e [i], which
both filters have to be considered jointly. At this point, we w¯
yields
needtointroduceamathematicalmanipulationthatallowsthe
expression of SD[i+1] = V[i+1]DH[i+1] as a function e [i+1]=(I−µ e[i]Π[i]S r[i]rH[i]S )e [i]
w¯ w D D w¯
of the recursion in (20). We can rewrite S [i+1] as (42)
D −µ e[i]Π[i]SHr[i]rH[i]S w¯ .
w D D opt
M
S [i+1]=V[i+1]DH[i+1]= B v[i+1]DH[i+1]
D l Taking expectations on both sides, using the fact that
Xl=1 E e [i] =0 and computing R =E e [i]eH[i] we get
w¯ w¯ w¯ w¯
M M
(cid:2) (cid:3) (cid:2) (cid:3)
= B v[i]DH[i]−µ e[i]z∗[i] B (I −(pH[i]p [i])−1p [i]pH[i])u[i]DH[i]
l v l w¯ wR¯ =(Iw¯−µw¯E[e[i]Π[i]]SHRSH)R (I−µ E[e[i]Π[i]]SHRrH[i]S
Xl=1 Xl=1 w¯ w D D w¯ w D D
=SD[i]−µve[i]z∗[i]Bw[i]u[i]DH[i], µ2wE[|e[i]|2Π[i]]SDRSDw¯HoptSHDRSDE[ΠH[i]],
(43)
(38)
where the M × I matrix B [i] = M B I − where R=E[r[i]rH[i]] is the M ×M covariance matrix of
w l=1 l(cid:18) the input r[i]. Using well-known results from the theory in
P
(pH[i]p [i])−1p [i]pH[i] , and the M ×I matrix B has [6], we have the following stability condition
w¯ w¯ w¯ w¯ (cid:19) l
an I-dimensional identity matrix starting at the l-th row, is 2
0<µ < (44)
shifteddownbyonepositionforeachl andtheotherelements w
tr E e[i]Π[i] SHRS
are zeros. D D
h (cid:2) (cid:3)i
BysubstitutingtheexpressionsofE [i]ande [i]in(38)
SD w¯ Forcaseii)weassumethatD[i]isfixedandv[i]andw¯[i]are
and (21), respectively, and rearranging the terms we obtain
time-variant, which means the trajectories of S [i] and w¯[i]
D
ESD[i+1]= I −µve[i]Bw[i]u[i]DH[i]rH[i] ESD[i] must be considered jointly. Therefore, the equation in (41)
−(cid:8) µ e[i]B [i]u[i]DH[i]rH[i]S [(cid:9)i]e [i] should be used in the analysis. For stability, the step sizes
v w D w¯
+µ e[i]B [i]u[i]DH[i]rH[i](S [i](I −w¯ )−shSouldb)e,adjustedsuchthatthe eigenvaluesofAHAareless
v w D opt D,opt
thanone.Despitethisconditionofstabilitythealgorithmsmay
(39)
convergetolocalminima.Inwhatfollows,wewillstudythis.
ew¯[i+1]= I −µwe[i]Π[i]SHD[i]r[i]rH[i]SD[i] ew¯[i] For cases iii) and iv), we consider that D[i], v[i] and
−(cid:8) µ e[i]Π[i]SH[i]r[i]rH[i]E [i] (cid:9) w¯[i] are time-variant and D[i] and w¯[i] are time-variant,
w D SD
−µ e[i]Π[i]SH[i]r[i]rH[i](E [i](w¯ −I)+Srespecw¯tively)., The condition of stability is different from the
w D SD opt pDre,ovpitousopctases since D[i] is a discretely optimized parameter
(40)
and v[i] and w¯[i] are parameter vectors that are continuously
where Π[i] = I − (p¯H[i]p¯[i])−1p¯[i]p¯H[i]. Taking expec- optimized. The equation in (41) still holds but the discrete
tations and considering the two error matrices together, we nature of D[i] makes a precise stability analysis impractical
obtain since D[i] is switched every time instant. In addition, the
E E [i+1] E E [i] problem becomes very difficult to treat since local minima
SD =A SD +C,
(cid:20) E ew¯[i(cid:2)+1] | 0D×(cid:3)(M−1) (cid:21) (cid:20) E ew¯[i(cid:2)] | 0D×(cid:3)(M−1) (cid:21)may arise due to the joint adaptation of D[i], v[i] and w¯[i]
(cid:2) (cid:3) (cid:2) (cid:3) (41) (case iii))and thejointadaptationofD[i] andw¯[i] (caseiv)).
10
B. Analysis of the Optimization Problem the desired signal can be expressed as
Let us now consider an analysis of the joint optimization JCM(q)=E[(qHbbHq)2]−2E[(qHbbHq)]+1
method from the point of view of the cost function and K K K
the constraints. Our strategy is to examine the four cases =8(F + q q∗)2−4F2−4 (qq∗)2−4F −4 (q q∗)+
l l l l l l
previously outlined and draw conclusions on what happens Xl=2 Xl=2 Xl=2
to the nature of the optimization problem. Let us drop the (49)
time index [i] for simplicity and define the cost function where F = q q∗ = A2|tHUHt|2 = ν2A2|hˆHh |2 and b =
k k k p k k k
J (v,D,w¯)=E |w¯HS r|2−1 2 [b1...bK]T is a K×1 vector with the transmitted symbols.
CM D In orderto study the propertiesof the optimization of (49),
h(cid:0) (cid:1) i
=E |w¯HDℜ v∗|2−1 2 we proceed as follows. We take advantage of the constraint
o
h(cid:0) (cid:1) i (45) w¯HSDpk =ν and rewrite (49) as
=E |tHUt|2−1 2
K
h(cid:0) (cid:1) i J˜ (q¯)=8(F+q¯Hq¯)2−4(F2+ (qq∗)2)−4(F+q¯Hq¯)+1,
=E |z|4−2|z|2+1], CM l l
h Xl=2
(50)
where the (D+I)×1 parameter vector t=[w¯T vT]T con- where q¯ = [q2,...,qK]T = TSDw¯, T = A′HP′H, P′ =
siderstogetherthereduced-rankestimatorandtheinterpolator [p ...p ] and A′ =diag(A ...A ).
2 K 2 K
0 0 The previous development allows us to examine the four
and the (D +I)×(D +I) matrix U =
(cid:20) (Dℜo)T 0 (cid:21) cases outlined at the beginning of the section via the compu-
containsthesamplesofthereceivedvectorandthedecimation tation of the Hessian matrix (Θ) using Θ = ∂ ∂(J˜CM(q¯)).
matrix. ∂q¯H ∂q¯
Specifically, Θ is positive definite if mHΘm > 0 for all
The received vector in (1) can be rewritten as r = x + nonzero m ∈ CK−1×K−1 [32]. The computation of Θ is
η+n, where x= K A b p and p =C h . Since the
k=1 k k k k k k given by
symbolsbk,k =1,P...,K arei.i.d.complexrandomvariables
with mean zero and unit variance, bk and n are statistically Θ=16 (F −1/4)I+q¯Hq¯I+q¯q¯H−diag(|q2|2...|qK|2) ,
independent,and we have R=Rx+Rη+σ2I, where Rx = h (51i)
E[xxH] and Rη =E[ηηH]. where the first term depends on F and the selection of some
Let us consider a desired user and its corresponding trans- key parameters, the second term is positive definite, and
formation matrix SD and reduced-rank estimator w¯. We can the third and fourth terms of (51) are positive semi-definite
express the interference free desired signal as matrices. We will now consider the four cases of interest for
our analysis.
qk =AkpHSDw¯ (46) Forcasei),weassumeSD fixedandF yieldsthecondition
and the composite signal as ν2A2k|hˆHk hk|2 ≥1/4, (52)
that ensures the convexity of the optimization problem in the
q =A p , p , ..., p HS w¯ =APHS w¯, (47)
1 2 K D D noiseless case. Since q¯ =TSDw¯ is a linear mapping of SD
(cid:2) (cid:3) and w¯, then J˜ (q¯) is a convex function of q¯ and implies
CM
whereA=diag(A1...AK)isaK×K diagonalmatrixwith that JCM(SD,w) = J˜CM(TSDw¯) is a convex function of
the amplitudes, P =[p1...pK] is a M ×K matrix with the SDw¯.
effective signatures.
For case ii), we suppose that S is time-variantdue to the
D
Now let us make use of the constraint w¯HSDpk = interpolator v and we shall consider v and w¯ jointly via the
w¯HSDCkhk = ν and the relation between SD, w¯, the parameter vector t. In this case, F yields the condition
channel and the signature CHS w¯ = νhˆ [9], [12], [14].
We then have for the desiredkuserDthe equivaklent expressions A2k|tHUHp t|2 ≥1/4, (53)
Althoughtheoptimizationproblemdependsontheparameters
q =A pHS w¯ =A hHCHw¯ =νA hHhˆ
k k k D k 1 k k k k v and w¯ which suggests a nonconvex problem, there is the
=A pHVDHw¯ =A vHℜHDHw¯ =A tHUHt, possibilityofmodifyingtheproblemwiththeconditionabove.
k k k p k p
(48) As the extrema of the cost function can be considered for
small σ2 a slight perturbation of the noise-free case [11], the
0 0 cost function is also convex for small σ2 provided the above
where the (D+I)×(D+I) matrix U =
p (cid:20) (Dℜp)T 0 (cid:21) conditions hold.
and the M×I Hankelmatrix ℜp containsshifted versionsof For case iii), we assume that D, v and w¯ are time-variant.
the effective signature p of the desired user. The discrete natureof D and the switchingbetween branches
k
At this point, we can exploit the previous expressions and are clearly associated with a nonconvex problem for which
substitute them into the cost function in (45). Assuming for there is no easy or known strategy to enforce convexity.
simplicity the absence of noise and ISI, the cost function of Interestingly, the switching does not affect the final values