Adaptive Interference Suppression for CDMA Systems using Interpolated FIR Filters with Adaptive Interpolators in Multipath Channels

0.72 MB·English

by  Rodrigo C. de Lamare

Adaptive Interference Suppression for CDMA
Systems using Interpolated FIR Filters with
Adaptive Interpolators in Multipath Channels
Rodrigo C. de Lamare and Raimundo Sampaio-Neto
Abstract—Inthisworkweproposeanadaptivelinearreceiver are required and the sample average approach for estimating
structure based on interpolated finite impulse response (FIR) the covariance matrix R = E[r(i)rH(i)] of the observed
3
1 filters with adaptive interpolators for direct sequence code divi- data r(i) has to be replaced by Rˆ = PPH +σ2I, which is
sion multipleaccess (DS-CDMA)systems in multipathchannels.
0 constructedwith a matrix Pcontainingtheeffectivesignature
Theinterpolatedminimummean-squarederror(MMSE)andthe
2 interpolatedconstrainedminimumvariance(CMV)solutionsare sequence of users and the variance σ2 of the receiver noise
n described for a novel scheme where the interpolator is rendered [10].Inaddition,withsomerecentadvancesinrandommatrix
a time-varying in order to mitigate multiple access interference theory [11] it is also possible to deploy techniques originally
J (MAI) and multiple-path propagation effects. Based upon the developed for systems with short codes in implementations
interpolated MMSE and CMV solutions we present compu-
1 with long codes. Furthermore,the adaptive receiverstructures
tationally efficient stochastic gradient (SG) and exponentially
2 weighted recursive least squares type (RLS) algorithms for both reportedsofar[1]-[10]canbeeasilyextendedtoasynchronous

receiver and interpolator filters in the supervised and blind systems in uplink connections. In presence of large relative
]
T modes of operation. A convergence analysis of the algorithms delaysamongsttheusers,theobservationwindowofeachuser
andadiscussionoftheconvergencepropertiesofthemethodare
I should be expandedin orderto consideran increased number
. carriedoutforbothmodesofoperation.Simulationexperiments
s foradownlinkscenarioshowthattheproposedstructuresachieve of samples derived from the offsets amongst users. Alterna-
c a superior BER convergence and steady-state performance to tivelyforsmallrelativedelaysamongstusers,thedesignercan
[
previously reported reduced-rank receivers at lower complexity. utilizechipoversamplingtocompensatefortherandomtiming
1 Index Terms—DS-CDMA, multiuser detection, reduced-rank re- offsets. These remediesimply in augmented filter lengths and
ceivers, adaptive algorithms.
v consequently increased computational complexity.
3 In this context, a problem arises when the processing
0
gain used in the system and the number of parameters for
0 I. INTRODUCTION
estimation is large. In these scenarios, the receiver has to
5
. ADAPTIVElinearreceivers[1], [19], [3], [4] are a highly cope with difficulties such as significant computational bur-
1 effective structure for combatting interference in DS- den, increased amount of training and poor convergence and
0
CDMA systems, since they usually show good performance trackingperformance.In general,whenan adaptivefilterwith
3
1 andhavesimpleadaptiveimplementation.Thelinearminimum alargenumberoftapsisusedtosuppressinterference,then it
: mean-squared error (MMSE) receiver [3], [4] implemented implies slow response to changing interference and channel
v with an adaptive filter is one of the most prominent design conditions. Reduced-rank interference suppression for DS-
i
X criteria for DS-CDMA systems. Such receiver only requires CDMA[12]-[18]wasoriginallymotivatedbysituationswhere
r the timing of the desired user and a training sequence in the number of elements in the receiver is large and it is
a order to suppress interference. Conversely, when a receiver desirable to work with fewer parameters for complexity and
loses track of the desired user and a training sequence is not convergencereasons.Earlyworksinreduced-rankinterference
available,a blind linearminimumvariance(MV) receiver[5], suppression for DS-CDMA systems [12], [13], [14] were
[6] that trades off the need for a training sequence in favor based on principal components(PC) of the covariance matrix
of the knowledge of the desired user’s spreading code can be R of the observation data. This requires a computationally
used to retrieve the desired signal. expensive eigen-decomposition to extract the signal subspace
The works in [1]-[6] were restricted to systems with short which leads to poor performance in systems with moderate
codes, where the spreading sequences are periodic. How- to heavy loads. An attempt to reduce the complexity of PC
ever, adaptive techniques are also applicable to systems with approaches was reported in [15] with the partial despreading
long codes provided some modifications are carried out. The (PD) method, where the authors report a simple technique
designer can resort to chip equalization [7] followed by a that allows the selection of the performance between the
despreader for downlink scenarios. For an uplink solution, matched filter and the full-rank MMSE receiver. A promising
channelestimationalgorithmsforaperiodicsequences[8],[9] reduced-rank technique for interference suppression, denoted
multistage Wiener filter (MWF), was developed by Goldstein
ThisworkwassupportedbytheBrazilianCouncilforScientificandTech-
et al. in [17] and was later extended to SG and recursive
nologicalDevelopment(CNPq).Dr.R.C.deLamarewaswithCETUC/PUC-
RIO and is now a Lecturer with the Communications Research Group, adaptive versions by Honig and Goldstein in [18]. A problem
Department of Electronics, University of York, York Y010 5DD, United withtheMWFapproachisthat,althoughlesscomplexthanthe
KingdomandProf.R.Sampaio-Neto iswithCETUC/PUC-RIO,22453-900,
full-ranksolution,itstillpresentsaconsiderablecomputational
Rio de Janeiro, Brazil. Phone: [phone] Fax: [phone]. E-
mails:[email protected], [email protected] burden and numerical problems for implementation. In this
2
work, we present an alternative reduced-rank interference tive AIFIR structure jointly optimizes two filters, namely the
suppression scheme based on interpolated FIR filters with interpolator and the reduced-rank, resulting in reduced-rank
adaptive interpolators that gathers simplicity, great flexibility, filters with fewer taps and faster convergence than PD since
low complexity and high performance. the interpolator helps with the compensation of the discarded
samples. In comparison with the MWF the proposed scheme
The interpolated FIR (IFIR) filter is a single rate structure
issimpler,moreflexibleandmoresuitableforimplementation
that is mathematically related to signal decimation followed
because the MWF has numerical problems in fixed point
by filtering with a reduced number of elements [20], [21].
implementations.
The basic idea is to exploit the coefficient redundancy in
A convergence analysis of the algorithms and a discussion
order to remove a number of impulse response samples,
of the global convergence properties of the method, which
whicharerecreatedusinganinterpolationscheme.Thesavings
are not treated in [24]-[26], are undertakenfor both modes of
are obtained by interpolating the impulse response and by
operation.Specifically,westudytheconvergencepropertiesof
decimating the interpolated signal. This technique exhibits
the proposed joint adaptive interpolator and receiver scheme
desirable properties, such as guaranteed stability, absence of
and conclude that it leads to an optimization problem with
limit cycles and low computational complexity. Thus, adap-
multipleglobalminimaandnolocalminima.Inthisregardand
tive IFIR (AIFIR) filters [22],[23] represent an interesting
based on the analyzed convergence properties of the method,
alternative for substituting classical adaptive FIR filters. In
we show that the prediction of the excess mean square error
some applications they show better convergence rate and can
(MSE) of both blind and supervised adaptive algorithms is
reduce the computational burden for filtering and coefficient
rendered possible through the study of the MSE trajectory of
updating, due to the reduced number of adaptive elements.
only one of the jointly optimized parameter vectors, i.e. the
These structures have been extensively applied in the context
interpolator or the reduced-rank filters. Then, using common
ofdigitalfiltering,althoughtheiruse forparameterestimation
assumptions of the adaptive filtering literature, such as the
in communications remains unexplored.
independence theory, we analyze the trajectory of the mean
Interference suppression with IFIR filters and time-varying
tap vectorof the joint optimizationof the interpolatorand the
interpolators with batch methods, which require matrix in- receiverandMSEtrajectory.Wealsoprovidesomemathemati-
versions, were reported in [24]. In this work, we investigate
calconditionswhichexplainwhythenewschemewithSGand
the suppression of MAI and intersymbol interference (ISI) RLS type algorithms is able to converge faster than the full-
with adaptive IFIR filters (that do not need matrix inver- rank scheme. Althoughthe novelstructure and algorithmsare
sions) for both supervised and blind modes of operation in
examined in a synchronous downlink scenario with periodic
synchronous DS-CDMA systems with short codes. A novel signature sequences in this work, it should be remarked that
AIFIR scheme where the interpolator is rendered adaptive is theycanbeextendedtolongcodesandasynchronoussystems
discussed and designed with both MMSE and MV design
provided the designer adopts the modifications explained in
criteria. The new scheme, introduced in [25], [26], yields a the works reported in [7]-[11].
performance superior to conventional AIFIR schemes [22],
This paper is organized as follows. Section II describes
[23] (where the interpolatoris fixed) and a faster convergence
theDS-CDMAsystemmodel.Thelinearinterpolatedreceiver
performance than full-rank and other existing reduced-rank
principle and design criteria, namely the MMSE and con-
receivers. Computationally efficient stochastic gradient (SG)
strained MV (CMV) are described in Section III. Section
andrecursiveleastsquares(RLS)typeadaptivealgorithmsare
IV is dedicated to the derivation of adaptive algorithms and
developed for the new structure based upon the MMSE and
Section V is devoted to the global convergence properties of
MV performancecriteria with appropriateconstraints to miti-
the method and the convergence analysis of the algorithms.
gateMAI,ISIandjointlyestimatethechannel.Themotivation
Section VI presents and discusses the simulation results and
for the novel structure is to exploit the redundancy found in
Section VII gives the concluding remarks.
DS-CDMA signals that operate in multipath, by removing a
numberof samples of the received signal and retrieving them
through interpolation. The gains in convergence performance II. DS-CDMASYSTEM MODEL
over full-rank solutions are obtained through the reduction of Let us consider the downlink of a synchronousDS-CDMA
the number of parameters for estimation, leading to a faster systemwithK users,N chipspersymbolandL propagation
p
acquisition of the required statistics of the method and a paths. The signal broadcastedby the base station intendedfor
smaller misadjustment noise provided by the smaller filter user k has a baseband representation given by:
[1], [19]. Furthermore, the use of an adaptive interpolator
∞
can provide a time-varying and rapid means of compensation
x (t)=A b (i)s (t iT) (1)
k k k k
for the decimation process and the discarded samples. The −
i=−∞
novel scheme has the potential and flexibility to consistently X
yield faster convergence than the full-rank approach since where bk(i) 1 denotes the i-th symbol for user k, the
∈ {± }
the designer can choose the number of adaptive elements real valued spreading waveform and the amplitude associated
duringthetransientprocessanduponconvergenceincreasethe with user k are sk(t) and Ak, respectively. The spreading
numberofelementsuptothefull-rank.UnlikePCtechniques, waveforms are expressed by s (t) = N a (i)φ(t iT ),
k i=1 k − c
our scheme is very simple because it does not require eigen- where a (i) 1/√N , φ(t) is the chip waveform, T
k c
∈ {± } P
decomposition and its performance is not severely degraded is the chip duration and N = T/T is the processing gain.
c
whenthesystemloadisincreased.IncontrasttoPD,theadap- Assumingthatthereceiverissynchronizedwiththemainpath,
3
the coherently demodulated composite received signal is
K Lp−1
r(t)= h (t)x (t τ )+n(t) (2)
l k l
−
k=1 l=0
X X
where h (t) and τ are, respectively, the channel coefficient
l l
and the delay associated with the l-th path. Assuming that
Fig.1. Proposed adaptive reduced-rank receiver structure.
τ =lT ,thechannelisconstantduringeachsymbolinterval
k,l c
and the spreadingcodes are repeated from symbol to symbol,
thereceivedsignalr(t) afterfilteringbyachip-pulsematched
filter and sampled at chip rate yields the M = N +L 1 The projected interpolated observation vector ¯r (i) =
p k
−
dimensional received vector Dr (i) is obtained with the aid of the M/L M projection
k
×
matrix D which is mathematically equivalent to signal deci-
K
mation on the M 1 vector r (i). An interpolated receiver
r(i)=H(i) A S b (i)+n(i) (3) k
k k k ×
with decimation factor L can be designed by choosing D as:
k=1
X
1 0 0 0 0 ... 0 0 0 0 0
where n(i) = [n (i) ... n i)]T is the complex Gaussian
1 M . . . . . . . . . . .
noise vector with E[n(i)nH(i)]=σ2I, where (·)T and (·)H  .. .. .. .. .. .. .. .. .. .. .. 
denotes transpose and Hermitian transpose, respectively, and 0 ... 0 1 0 ... 0 0 0 0 0
E[.] is the expected value, the k-th user symbol vector is  
D= (m−1)L zeros 
Lbk(iis)t=he[IbSkI(is+paLnsa−nd1)th.e..((b2kL(i) .1.). bkN(i)−L(2sL+1)]1T),mwahterirxe  ...| {z... } ... ... ... ... ... ... ... ... ... 
s s− × × s−  
S with non-overlapping shifted versions of the signature of  0 0 0 0 0 ... 0 1 0 ... 0 
k  
user k is  
sk 0 ... 0  (M/L−1)L zeros (L−1) z(e6ro)s 
S = 0 sk ... 0  (4) wstrhaetreegym,th(amt|al=low1s,2u,s.t.o.d,Me{vzi/sLe)sodleuntiootnessftohr}ebomth-thin|treorwpo{.zlTathoer}
k  ... ... ... ...  and receiver, is to express the estimated symbol x (i) =
  k
 0 ... ... sk  wkH(i)¯rk(i) as a function of wk(i) and vk(i) (we will drop
  the subscriptk andsymbolindex(i)forease ofpresentation):
where the signature sequence for the k-th user is s =
k
[ak(1) ... ak(N)]T and the M ((2Ls 1) N) channel xk(i)=w0∗vkHr˙0+w1∗vkHr˙1+...+wM∗ /L−1vkHr˙M/L−1
× − ×
matrix H(i) is
=vH(i) r˙(i) ... r˙(i) w∗(i)=vH(i)ℜ(i)w∗(i)
k 0 | | M/L−1 k k k
h0(i) ... hLp−1(i) ... 0 0 h i (7)
0 h (i) ... h (i) ... 0
H(i)= ... 0... ... Lp.−..1 ... ...  wefhfiecrieenutsk(oif)w=ℜ(i)(ia)nwdk∗t(hie) iNs aneleNmIe×nts1ovfecvto(ri,)thaereMas/sLumceod-
  k I k
 0 0 ... h0(i) ... hLp−1(i) to be complex, the asterisk denotes complex conjugation and
 (5) r˙ (i) is a length N segment of the received vector r(i)
s I
where hl(i)=hl(iTc) for l=0,...,Lp−1. TheMAI arises beginning at rs×L(i) and
from the non-orthogonality between the received signals,
whereasthe ISI span L dependsonthe lengthof the channel r(i) r(i) ... r(i)
s 0 L (M/L−1)L
response, which is related to the length of the chip sequence.  r(i) r(i) ... r(i) 
FNor<LLp =612,NL,Ls ==13(naondISsIo),ofonr. 1<Lp 6N,Ls =2, for ℜ(i)= 1... L...+1 ... (M/L−...1)L+1 
p s  
 rN(i)I−1 rL(i+)NI−1 ... r((Mi)/L−1)L+NI−1 
III. LINEARINTERPOLATEDCDMA RECEIVERS  (8)
The interpolated linear receiver design is equivalent to deter-
The underlying principles of the proposed CDMA receiver
mininganFIRfilterw (i)withM/Lcoefficientsthatprovide
structure are detailed here. Fig. 1 shows the structure of k
an estimate of the desired symbol:
an IFIR receiver, where an interpolator and a reduced-rank
receiver that are time-varying are employed. The M 1 ˆb (i)=sgn Re wH(i)¯r (i) (9)
received vector r(i) = [r(i) ... r(i) ]T is filtered by×the k k k
0 M−1 (cid:16) h i(cid:17)
interpolator filter v (i) = [v(i) ...v(i) ]T, yielding the whereRe()selectstherealpart,sgn()isthesignumfunction
k k,0 k,NI−1 and the rec·eiver parameter vector w ·is optimized according
interpolated received vector r (i). The vector r (i) is then k
k k
to a selected design criterion.
projected onto an M/L 1 vector ¯r (i). This procedure
k
×
corresponds to removing L 1 samples of r (i) of each set
k
−
of L consecutive ones. Then the inner product of ¯r (i) with A. MMSE Reduced-Rank Interpolated Receiver Design
k
the M/L-dimensional vector of filter coefficients wk(i) = TheMMSEsolutionsforwk(i)andvk(i)canbecomputed
[wk(i,)0 ... wk(i,)M/L−1]T is computedto yield the outputxk(i). if we consider the optimization problem whose cost function
4
is = w (i)= R¯−1(i)DC λ
⇒ k − k k
JMSE(wk(i),vk(i))=E |bk(i)−vkH(i)ℜ(i)wk∗(i)|2 (10) Using the constraint CHk DHwk(i) = g(i) and substituting
where bk(i) is the desiredhsymbol for user k at timeiindex λwk=(i)=(C−HR¯D−kH1(Ri¯)−D1C(ik)DλCwe)a−r1rgive(ia)t. The resulting expres-
(i). By fixingtheinterpolatorvk(i)andminimizing(10)with sion f−or thek receivekr is k k
respectto w (i) the interpolatedWienerfilter/receiverweight
k
vector is w (i)=α (v )=R¯ (i)−1DC (CHDHR¯ (i)−1DC )−1g(i)
wk(i)=α(vk)=R¯−k1(i)p¯k(i) (11) k o k k k k k k(16)
where R¯ (i) = E[¯r (i)¯rH(i)], p¯ (i) = E[b∗(i)¯r (i)], and the associate minimum output variance is
k k k k k k
¯rk(i)=ℜT(i)vk∗(i)andbyfixingwk(i)andminimizing(10) J (v )=J (α (v ),v )=wH(i)R¯ (i)w (i)
with respect to v (i) the interpolator weight vector is o k MV o k k k k k (17)
k =gH(i)(CHDHR¯ (i)−1DC )−1g(i)
v (i)=β(w )=R−1(i)p (i) (12) k k k
k k uk uk By fixing w , the solution that minimizes (15) is:
where R (i)=E[u (i)uH(i)], p (i)=E[b∗(i)u (i)] and k
uk k k uk k k
uk(i)=ℜ(i)wk∗(i). The associated MSE expressions are vk(i)=βo(wk)=argmvin vHRuk(i)v (18)
J(v )=J (α(v ),v )=σ2 p¯H(i)R−1(i)p¯ (i) (13)
k MSE k k b− k k k subjectto vk(i) =1.Therefore,thesolutionfortheinterpo-
J (w ,β(w ))=σ2 pH (i)R−1(i)p (i) (14) latoristhe||norma|l|izedeigenvectorofR correspondingtoits
MSE k k b − uk uk uk uk
minimumeigenvalue,viasingularvaluedecomposition(SVD).
where σ2 = E[b(i)2]. Note that points of
b | | As occurs with the MMSE approach we iterate (16) and (18)
global minimum of (10) can be obtained by
withaninitialguesstoobtaina CMVsolution[24].Notealso
v = argmin J(v ) and w = α(v )
k,opt vk k k,opt k,opt that (16) assumes the knowledge of the channel parameters.
or w = argmin J (w ,β(w )) and
k,opt wk MSE k k However, in applications where multipath is present these
v = β(w ). At the minimum point (13) equals
k,opt k,opt parameters are not known and thus channel estimation is
(14) and the MMSE for the proposed structure is achieved.
required. To blindly estimate the channel we use the method
We remark that (11) and (12) are not closed-form solutions
of [6], [27]:
for w (i) and v (i) since (11) is a function of v (i) and
k k k
(12) depends on wk(i) and thus it is necessary to iterate gˆ(i)=argmin gHCHk R−m(i)Ckg (19)
(11) and (12) with an initial guess to obtain a solution, as in g
[24]. An iterative MMSE solution can be sought via adaptive
subjectto gˆ =1, where R(i)=E[r(i)rH(i)], m is a finite
algorithms. || ||
powerand whose solution is the eigenvectorcorrespondingto
theminimumeigenvalueoftheL L matrix CTR(i)−mC
B. Constrained Minimum Variance (CMV) Reduced-Rank In- p× p k k
throughSVD. Note that in this work we restrict the values of
terpolated Receiver Design
mto1althoughtheperformanceofthechannelestimatorand
TheinterpolatedCMVreceiverparametervectorwk andthe consequentlyofthereceivercanbeimprovedbyincreasingm.
interpolator parameter vector v are obtained by minimizing Inthenextsection,weproposeiterativesolutionsviaadaptive
k
algorithms.
J (w ,v )=E x (i)2 =E vH(i)ℜ(i)w∗(i)2
MV k k | k | | k k |
=whH(i)R¯ wi (i)=h vH(i)R v (i) i
k k k k uk k IV. ADAPTIVE ALGORITHMS
(15)
We describe SG and RLS algorithms [31] (see Chapters
subject to the proposed constraints CHk DHwk(i) = g(i) 9 and 13) that adjust the parameters of the receiver and the
and vk(i) = 1, where the M Lp constraint matrix Ck interpolatorbased ontheMMSE criterionandtheconstrained
|| || ×
contains one-chip shifted versions of the signature sequence minimization of the MV cost function [25], [26]. The novel
of user k, g(i) is an Lp-dimensional parameter vector to be structure, shown in Fig. 1 and denoted INT, for the receivers
determined. The vector of constraints g(i) can be chosen gathersfastconvergence,low complexityandadditionalflexi-
amongst various criteria although in this work we adopt bilitysincethedesignercanadjustthedecimationfactorLand
g(i) as the channel parameter vector (g = [h0...hLp−1]T) thelengthoftheinterpolatorNI dependingontheneedsofthe
because it provides better performance than other choices application and the hostility of the environment. Based upon
as reported in [6]. The proposed constraint vk(i) = 1 the MMSE and CMV design criteria, the proposed receiver
|| ||
ensures adequate design values for the interpolator filter vk, structurehasthefollowingmodesofoperation:trainingmode,
whereas CHk DHwk(i) = g(i) avoids the suppression of the whereitemploysaknowntrainingsequence;decisiondirected
desired signal. By fixing vk, taking the gradient of the La- mode, which uses past decisions in order to estimate the
grangianfunctionJl (w ,v )=E [vH(i)ℜ(i)w∗(i)2 + receiver parameters; and blind mode, which is based on the
MV k k | k k |
CMVcriterionandtrades-offthetrainingsequenceagainstthe
Re (CHDHw (i) g(i))Hλ , whehre λ is a vector of Lia-
k k − knowledgeofthesignaturesequence.Thecomplexityinterms
grange multipliers, with respect to w and setting it to 0 we
h i k of arithmetic operations of the algorithms associated with the
get:
INT and the existing techniques is included as a function of
E ¯rk(i)¯rHk (i) wk(i)+DCkλ=0 the number of adaptive elements for comparison purposes.
h i
5
A. Least mean squares (LMS) algorithm To avoid the inversion of Rˆ we use the matrix inversion
uk
lemma and again for convenience of computation we define
Given the projected interpolated observation vector ¯r (i)
and the desired symbol bk(i), we consider the following ckost Puk(i)=Rˆ−uk1(i) and the Kalman gain vector Guk(i) as:
function: α−1P (i 1)u (i)
JMSE =|bk(i)−vkH(i)ℜ(i)wk∗(i)|2 (20) Guk(i)= 1+α−1uHku(ki)P−uk(i−k1)uk(i) (27)
and thus we can rewrite (27) as
Takingthegradienttermsof(20)withrespectto w (i),v (i)
k k
and using the gradient descent rules [31] (see Chapter 9, pp. P (i)=α−1P (i 1) α−1G (i)uH(i)P (i 1) (28)
v36k7(i-)37−1)η∇wJvkM∗(iSE+y1i)el=ds:wk(i)−µ∇JwM∗SE and vk(i +1) = B(2y6u)pkrwoeceaerdriivnegaint auksim−ilar−approacuhktothekonetaukken−toobtain
v (i+1)=v (i)+ηe∗(i)u (i) (21)
k k k k v (i)=v (i 1)+G (i)ξ∗(i) (29)
w (i+1)=w (i)+µe∗(i)¯r (i) (22) k k − uk k
k k k k The RLS algorithm for the proposed structure trades off a
where e (i) = b (i) w (i)H¯r (i) is the error for user k, computational complexity of O(M2) against two RLS algo-
k k k k
u =ℜ(i)w (i), µ a−nd η are the step sizes of the algorithm rithmsoperatingin parallel,with complexity O((M/L)2) and
k k
for w (i) and v (i). The LMS algorithm for the proposed O(N2), respectively. Because N is small (N << M, as
k k I I I
structure described in this section has a computational com- will be shown later) the computational advantage of the RLS
plexityO(M/L+N ).Infact,theproposedstructuretradesoff combined with the INT structure is rather significant.
I
one LMS algorithmwith complexityO(M) againsttwo LMS
algorithmswithcomplexityO(M/L)andO(NI),operatingin C. Constrainedminimum variancestochastic gradient(CMV-
parallel. It is worth noting that, for stability and to facilitate SG) algorithm
tuning of parameters, it is useful to employ normalized step
Consider the unconstrained Lagrangian MV cost function:
sizesandconsequentlyNLMStyperecursionswhenoperating
inachangingenvironmentandthuswehaveµ(i)= η0 J =(vH(i)u (i)uH(i)v (i))+λH(CHDHw (i) g(i))
and η(i) = µ0 as the step sizes of the algo¯rriHkth(im)¯rkf(oi)r MV +(kwH(ik)DCk gHk(i))λ k k −
uHk(i)uk(i) k k−
w (i)andv (i),whereµ andη aretheconvergencefactors. (30)
k k 0 0
where λ is a vector of Lagrange multipliers. An SG solution
B. Recursive least squares (RLS) algorithm can be devised by taking the gradient terms of (30) with
Consider the time average estimate of the matrix R¯k, respect to wk(i) and vk(i) as described by wk(i + 1) =
required in (11), given by Rˆ¯k(i) = il=1αi−l¯rk(l)¯rHk (l), wadka(pit)iv−eµlywm(i)in∇imJwizke(si)J,vk(i+wi1th)=resvpke(cit)−toηw(i)∇(iJ)vakn(di)vwh(iic)h.
where α (0 < α 6 1) is the forgetting factor, that can be MV k k
alternativelyexpressedby Rˆ¯ (i)=αRˆ¯P(i 1)+¯r (i)¯rH(i). Substituting the gradient terms the equations become
To avoid the inversion of Rˆ¯kk(i) requirekd in−(11), wke uske the wk(i+1)=wk(i)−µw(i)(x∗k(i)¯rk(i)+DCkλ(i)) (31)
matrix inversion lemma and define Pk(i) = Rˆ¯−k1(i) and the vk(i+1)=vk(i)−η(i)x∗k(i)uk(i) (32)
gain vector G (i) as:
k where x (i) = wH(i)¯r (i) = vH(i)u (i). We use (32) and
k k k k k
Gk(i)= 1+αα−−11¯rPHkk((i)iP−k1(i)¯r−k(1i))¯rk(i) (23) cinatnermpoalkaetorvvkk(i.+It1is)w←orvthk(nio+tin1g)/t|h|avtki(ni+ou1r)s||tutdoieusptdhaetentohre-
malizationon SG algorithmsdoes notlead to differentresults
and thus we can rewrite P (i) as
k from the ones obtained with a non-normalized interpolator
P (i)=α−1P (i 1) α−1G (i)¯rH(i)P (i 1) (24) recursion. In this regard, analyzing the convergence of (32)
k k − − k k k − without normalization is mathematically simpler and gives
By rearranging (23) we have Gk(i) = α−1Pk(i 1)¯rk(i) us the necessary insight into its convergence. By combining
− −
α−1Gk(i)¯rHk (i)Pk(i−1)¯rk(i)=Pk(i)¯rk(i) . By employing the constraint CkDHwk(i) = g(i) and (32) we obtain the
the LS solution (a time average of (11)) and the recursion Lagrange multiplier
pˆ (i)=αpˆ (i 1)+¯r (i)b∗(i) we obtain
k k − k k λ(i)=(CHDHDC )−1 (CHDw (i) µ CHDx∗(i)¯r (i) g(i))
w (i)=Rˆ¯−1(i)pˆ (i)=αP (i)pˆ (i 1)+P (i)¯r (i)b∗(i) k k × k − w k (k33)−
k k k k k − k k (k25) By substituting(33)into(31)wearriveatthe updaterulesfor
Substituting (24) into (25) yields: the estimation of the parameters of the receiver wk:
wk(i)=wk(i−1)+Gk(i)ξk∗(i) (26) wk(i+1)=Πk(wk(i)−µw(i)x∗k(i)¯rk(i))+DCk(CHk DH(D34C)k)−1g(i)
abwkrhe(eid)ree−vtiwhseekHda(biyp−rui1os)ri¯rnikge(is()t1i.m2S)a.imtTioihlnaereerrsertocimurraisstieodnReˆssufcokrirbcteahdnebbineyteoξrbpkto(ailia)nteo=dr tswhtrhaateirnpetrsoΠ.jekct=s wIk−oDntCoka(nCotHkheDrHhyDpCerpkl)a−n1eCtHkoDenHsurise athemactornix-
through Rˆuk(i) = il=1αi−luk(l)uHk (l) and can be alter- Normalized versions of these algorithms can be devised
natively written as Rˆ (i) = αRˆ (i 1)+u (i)uH(i). by substituting (32) and (34) into the MV cost function,
Puk uk − k k
6
differentiating the cost function with respect to µw(i) and Rˆ¯−k1(i)=α−1Rˆ¯−k1(i−1)−α−1G(i)¯rHk (i)Rˆ¯−k1(i−1) (37)
µ (i), setting them to zero and solving the new equations.
v
Hence, the CMV-SG algorithm proposed here for the INT where0<α61istheforgettingfactor.Thealgorithmcanbe
receiver structure adopts the normalized step sizes µw(i) = initialized with R¯−k1(0) = δI and R−uk1(0) = δI, where δ is
µ0 and η(i) = η0 where µ and η are the a large positive number. For the computation of the reduced-
c¯roHkn(vi)eΠrgk¯rekn(cie) factors for wuHkaunk(di) v , respec0tively. 0 rankreceiverparametervectorwk weusethematrixinversion
k k lemma[31]toestimate(CHDHR¯−1(i)DC )−1 asgivenby:
The channel estimate gˆ(i) is obtained through the power k k k
method and the SG technique described in [27]. The method
1 Γ−1(i 1)γ (i)γH(i)Γ−1(i 1)
is an SG adaptive version of the blind channel estimation Γ−1(i)= Γ−1(i 1) k − k k k −
algorithmdescribedin(19)andintroducedin[28]thatrequires k 1−α" k − − 1−αα +γHk (i)Γ−k1(i)γk(i) #
only O(L2) arithmetic operations to estimate the channel, (38)
against O(pL3) of its SVD version. In terms of computational where Γ (i) is an estimate of (CHDHRˆ¯−1(i)DC ) and
p k k k k
complexity, for the rejection of MAI and ISI, the proposed γ (i)=CHDHr (i)andthenwe constructthereduced-rank
k k k
blind interpolated receiver trades off one blind algorithm receiver as :
with complexity O(M) against two blind algorithms with
complexity O(M/L) and O(NI), operating in parallel. wk(i)=Rˆ¯k(i)−1DCkΓ−k1(i)gˆ(i) (39)
The channel estimate gˆ(i) is obtained through the power
D. Constrained minimum variance recursive least squares method and the RLS technique described in [28]. Following
(CMV-RLS) algorithm thisapproach,theSVDontheL L matrixCHR−1(i)C ,
p× p k k
Based upon the expressions for the receiver wk and inter- as stated in (19) and that requires O(L3p), is avoided and
polator v in (16) and (18) of the interpolatedCMV receiver, replaced by a single matrix-vectormultiplication, resulting in
k
we developa computationallyefficient RLS algorithm for the the reduction of the corresponding computational complexity
INT structure that estimates the parameters of w and v . on one orderof magnitude and no performanceloss. In terms
k k
Theiterativepowermethod[29](seepp.405 408),[30](see of computational complexity, the CMV-RLS algorithm with
pp. 314 333) is used in numerical analysis t−o compute the the interpolated receiver trades off one blind algorithm with
eigenvec−tor corresponding to the largest singular value of a complexity O(M2)againsttwowithcomplexity O(M2/L2)
matrix.Inordertoobtainanestimateofvk andavoidtheSVD and O(NI2) operating in parallel. Since NI is small as
ontheestimateofthematrixR (i)weresorttoavariationof compared to M, it turns out that the new algorithms offer
theiterativepowermethodtoobutkaintheeigenvectorofR (i) a significant computational advantage over conventional RLS
which corresponds to the minimum eigenvalue. uk algorithms.
Specifically, we apply the power method to the difference
betweenR (i)andtheidentitymatrixI,ratherthanapplying
uk E. Computational Complexity
it to the inverse of R (i). This approach, known as shift
uk
iterations[30](seepp.319),leadstocomputationalsavingson In this section we illustrate the computational complexity
one order of magnitude since direct SVD requires O(N3), of the proposed INT structure and algorithms. In Table 1
I
while our approach needs O(N2). The simulations carried we consider supervised algorithms, whereas the complexity
I
out reveal that this method exhibits no performance loss. of blind algorithms is depicted in Table 2. Specifically, we
Hence, we estimate R (i) via the recursion Rˆ (i) = compare the full-rank, the proposed INT structure, the PD,
uk uk
i αi−nu (n)uH(n) and then obtain the interpolator vˆ thePCandtheMWFstructureswithSG andRLSalgorithms.
n=0 k k k
with a one step iteration given by: In general, the INT structure introduces the term M/L,
P which can reduce the complexity by choosing the decimation
vˆk(i)=(I−νk(i)Rˆuk(i))vˆk(i−1) (35) factor L > 2. This is relevant for algorithms which have
where ν (i) = 1/tr[Rˆ (i)]. After that, we make vˆ (i) quadraticcomputationalcostwithM,i.e.theblindandtrained
k uk k ← RLS and the blind SG, because the decimation factor L in
vˆ (i)/ vˆ (i) to normalize the interpolator. This procedure
k || k || the denominator favors the proposed scheme which requires
is based on the following result.
complexity O((M/L)2). This complexity advantage is not
Lemma: Let R be a positive semi-definite Hermitian sym-
verified with linear complexity recursions. For instance, with
metric matrix and q the eigenvector associated to the
min NLMSalgorithmstheproposedINThasacomplexityslightly
smallest eigenvalue. If q is unique and of unit norm, then
min superior to the full-rank. Among the other methods, the PD
with ν = 1/tr[R] the sequence of vectors v(i) = vˆ(i)
||vˆ(i)|| is slightly more complex than the INT. A drawback of PC
with vˆ(i)=(I ν(i)R)vˆ(i 1)convergesto q , provided
− − min methods is that they require an SVD with associated cost
that vˆ(0) is not orthogonal to q . A proof is shown in the
min O(M3) in order to compute the desired subspace. Although
Appendix.
the subspace of interest can be retrieved via computationally
To recursively estimate the matrix R¯ (i) and avoid its
k efficient tracking algorithms [13], [14], these algorithms are
inversionweusethematrixinversionlemmaandKalmanRLS
still complex (O(M)2) and lead to performance degradation
recursions [31]:
as compared to the actual SVD. The MWF technique has a
α−1Rˆ¯−1(i 1)¯r (i) complexity O(DM¯2), where the variable dimension of the
G(i)= k − k (36) vectors M¯ = M d varies according to the orthogonal
1+α−1¯rH(i)Rˆ¯−1(i 1)¯r (i) decomposition and t−he rank d=1,...,D.
k k − k
7
TABLEII
108 (a) Supervised Recursive Algorithms 108 (b) Blind Recursive Algorithms COMPUTATIONALCOAMLPGLOERXIITTHYMOSF.BLINDADAPTATION
107 107 Numberofoperations persymbol
Algorithm Additions Multiplications
106 106 CMV-SG- M2+MLp+2M +1 M2+MLp+3M
Number of operations110045 FINuTll−(Lr=an2k) Number of operations110045 CMMFuVWl-lS-FrGa-S-nIGkNT D+4+(((NMM2L2(IMM)LM−¯2++−1+)(N21(ML)I+ML2)++)MLN2p22I) D((+2MLM7M¯L)M22N+++2IN+(+3IMMLM¯4)MLL+p5)
1110001230 50 100 1M5IIPPPPM0NNCCDDWTT((((((DDDDFLL(======2DKKMM340=))0))//4−−24)SS −ttaaVorppenD2sscg5))u 0&rs Rivoey300 1110001230 50 100FIIIPPMNNNuCCWTTTll((−(((DDF1LLLMr5(=====aD0KKn234=k)))))4−−)W S−2raoe0nnc0ggu r&&s iPRv2eoo5oy0r−SV3D00 PCCMC-FWMVu-alVRln--rLgRa&SLn-PkSINo-oTr +4((+MMLL3+LM+(N2p−−L33+pI(11+LM−))2pN22L1+−+(Ip)M2+31+(+L)ML−(2MpN2)−1−2IMLL)1+2−p1LL1p)2p2 O+(M++7M+L(3LMMLLLp)2pp+++N)+2+LI22+2pM+MLL+2pNL2NML+p+II24M
MWF-Recursive D(3(M¯ −1)2+2M¯) D(3M¯2+2M¯ +3)
Fig.2. Complexity interms of arithmeticoperations versus number
of received samples (M) for (a) supervised and (b) blind recursive
adaptation algorithms.
V. GLOBAL CONVERGENCE PROPERTIES OF THEMETHOD
AND CONVERGENCE ANALYSIS OFALGORITHMS
In this section we discuss the global convergence of the
In order to illustrate the complexity trend in a compre- method and its properties, the trajectory of the mean tap
hensive way, we depict in Fig. 2 curves which describe the vectors, of the excess mean square error and the convergence
computationalcomplexityintermsofthearithmeticoperations speed. Specifically, we study the convergence properties of
(additionsand multiplications)as a functionof the numberof the proposed joint adaptive interpolator and receiver scheme
parameters M for recursive algorithms. For these curves, we and conclude that it leads to an optimization problem with
consider L = 6 and assume that D is equal to M/2 for the multipleglobalminimaandnolocalminima.Inthisregardand
p
eigen-decompositionapproaches.We also include the compu- based on the analyzed convergence properties of the method,
tational cost of the algorithm of Song and Roy [14], which is it suffices to examine the MSE trajectory of only one of the
capable of significantly reducing the cost required by SVD. jointly optimized parameter vectors (wk or vk) in order to
In comparison with the existing reduced-rank techniques, the predictthe excessMSE of bothblind and supervisedadaptive
proposed INT scheme is significantly less complex than the algorithms. We also provide a discussion of the speed of
PCandthe MWFandslightlylesscomplexthanthe PD. This convergenceof the INT as compared to the full-rank.
is because the analyzed algorithms have quadratic cost (PC
with SVD has cubic cost), whereas the INT has complexity A. Global Convergence of the Method and its Properties
O((M/L)2), as shown in Tables I and II.
1) Interpolated MMSE Design: Let us first consider the
trained receiver case and recall the associated MSE ex-
TABLEI pressions in (13) and (14), namely JMSE(vk,α(vk)) =
COMPUTATIONALCOMPLEXITYOFSUPERVISEDADAPTATION J(v )=σ2 p¯H(i)R¯−1(i)p¯ (i)andJ (β(w ),w )=
ALGORITHMS. σ2 k p¯Hb(ki)−R¯−k1(i)p k(i) wkhere σ2 M=SEE[b (ik)2]. kNote
bk − uk uk uk bk | k |
that points of global minimum of J (w (i),v (i)) =
Numberofoperations persymbol MSE k k
Algorithm Additions Multiplications E b (i) vH(i)ℜ(i)w∗(i)2 can be obtained by v =
LMS-Full-rank 2M 2M+1 | k − k k | opt
LLMMSS--IPNCT +NIOM2(MML+3+()ML+2N)2NIDI +2 M3+3ML(+ML+2)D2NN+II1 maarrihggnimmmiiunnmwvkkpJoMJin(StvEJkM()βS(Eawn(dkv)k,,wwαok(p)vtika)n)=deqvuαoapl(tsvJo=pMt)SβE(o(wrβo(pwwt)ko.p)t,Awtk=a)
LMS-PD (D−1)2+2D+1 D2+2D+2 and the MMSE for the proposed structure is achieved. We
MWF-SG D(2(M¯ −1)2+M¯ +3) D(2M¯2+5M¯ +7)further note that since J(v ) = J(tv ), for every t = 0,
k k
RLS-Full-rank 3(M−1)2+M2+2M 6M2+2M +2 then if v⋆ is a pointof globalminimum of J(v ) then tv6 ⋆ is
3(ML −1)2+3(NI −1)2 6(ML)2+6NI2 also a pokint of global minimum. Therefore, pokints of glokbal
RRLLSS--IPNCT +M+N+N3I2(+I+MLM32−(+MDL1()−+MNL12I))N22I O(M++NML33)INM+L+I62D2 pmvok⋆iinn=itmoaufrmgglmo(boinapl|t|imvmkiu|n|=mim1Juinm(tverokpf)o.JlSa(itvnokcr)efiftlohterer|es|xv)iksc|ta|enn=cbe1eoifsoabgttualaeinraaesndttoebneyde
+D2+2D +2D+2 bythetheoremofWeierstrass[32](seeChapter2,Section2.1,
RLS-PD 4(D−1)2+D2+2D 7D2+2D+2 Appendix B), then the existence of (infinite) points of global
MWF-Recursive D(4(M¯ −1)2+2M¯) D(4M¯2+2M¯ +3)minimum is also guaranteed for the cost function in (10).
In the context of global convergence, a sufficient but not
necessary condition is the convexity, which is verified if its
8
Hessian matrix is positive semi-definite, that is aHHa > 0, interpolator the curves shown in Fig. 3 (a) and (b),
for any vector a. Firstly, let us consider the minimization of indicate that the function is no longer convex but it also
J (w (i),v (i)) = E b (i) vH(i)ℜ(i)w∗(i)2 with does not exhibit local minima.
fixMeSdEintekrpolatokrs. Such ohp|tikmiza−tionkleads to thke fo|lliowing • By taking into account that for small interpolator filter
Hessian H = ∂ (JMSE(.)) = E[r (i)rH(i)] = R (i), length NI (NI 6 3), vk can be expressed in spherical
which is positive∂wsekHmi-d∂ewfiknite and enskureskthe convexitky of coordinatesandasurfacecanbeconstructed.Specifically,
we expressed the parameter vector v as follows: v =
the cost function for the case of fixed interpolators. Let us k k
r[cos(θ)cos(φ) cos(θ)sin(φ) sin(θ)]T,wherer isthe
nowconsiderthejointoptimizationofthe interpolator v and
k radius,θandφwerevariedfrom π/2toπ/2and πto
receiver wk through an equivalent cost function to (10): π,respectively,and(13)wasplott−edforvariousscen−arios
J˜ (z)=E[b zHBz 2] (40) andconditions(SNR,differentchannels,etc).Theplotof
MSE | − k k|
the error-performancesurface of J(v ), depicted in Fig.
0 0 k
where B = is an (N + N/L) (N + N/L) 3 (c), reveals that J(v ) has a global minimum value
ℜ 0 I × I k
(cid:20) (cid:21) (as it should) but do not exhibit local minima, which
matrix, the Hessian (H) with respect to z = [wT vT]T is
k k k implies that (40) has no local minima either. It should
H= ∂z∂Hk ∂(J˜M∂zSkE(.)) =E[(zHk Bzk−bk)BH]+E[(zHk BHzk− be noted that if the cost function in (40) had a point of
b∗)B]+E[Bz zHBH]+E[BHz zHB].ByexaminingHwe local minimum then J(v ) in (13) should also exhibit
k k k k k k
notethatthethirdandfourthtermsyieldpositivesemi-definite a point of local minimum even though the reciprocal is
matrices (aHE[Bz zHBH]a > 0 and aHE[BHz zHB]a > not necessarily true: a point of local minimum of J(v )
k k k k k
0, z = 0) whereas the first and second terms are indefinite may correspond to a saddle point of J (v ,w ), if
k MSE k k
6
matrices. Thus, the cost function cannot be classified as it exists. Note also that the latitude X longitude plot in
convex. However, for a gradient search algorithm, a desirable Fig.3(c)depictsitstwosymmetricglobalminimainthe
property of the cost function is that it shows no points of unit sphere.
local minimum, i.e. every point of minimum is a point of • An important feature that advocates the non-existence
global minimum (convexity is a sufficient, but not necessary, of local minima is that the algorithm always converge
condition for this property to hold) and it is conjectured that to the same minimum value, for a given experiment,
the problem in (40) has this property. independentlyofanyinterpolatorinitialization(exceptfor
v(0)=[0 ... 0]T that eliminates the signal) for a wide
(a) Surface of J(w,v) = (1−w*r*v)2 (b) Contour of J(w,v) = (1−w*r*v)2 range of SNR values and channels.
2
25 1.5 2) Interpolated CMV Design: For the blind case, let us
20 1 first consider the minimization of JMV(wk(i),vk(i)) =
0.5
15 E vH(i)ℜ(i)w∗(i)2 with fixed interpolators subject to
J(v)10 v−0−.015mglionbimalu m points ChHk|DkHwk(i) =kg(i|)iand ||vk(i)|| = 1. It should be noted
5 that global convergence of the CMV method has been estab-
−1.5
(c−)02 Error P−e1rformanc0e Suvrface1 of Interp2olat−e2d MMS0wE Desig2n: (d) −V−2a2riance Per−fo1rmance Surw0face of Inter1polated CMV2 Design: liinstheerpdoilnato[6rs]aarnedusheedre.Swuechtroepattimaisziamtiiolanrlepardosblteomthewfhoelnlofiwxinedg
N=31, K = 6 users N=31, K=6 Hessian H= ∂ (JMV(.)) =E[r (i)rH(i)]=R (i), which
00..45 00..45 is positive sem∂iw-dkHefini∂twekand ensureks thekconvexitykof the cost
J( v)00..23 J( v)o00..23 funCcotinosnidfeorrtthheejcoaisnet oofptfiimxeizdaitniotenrpooflathtoersi.nterpolator v and
0.1 k
0.1 receiver w via an equivalent cost function to (10):
0 k
0 2
2 1 θ0 −1 −2 −4 −2 φ 0 2 4 1θ0−1−2 −4 −2 φ 0 2 4 J˜MV(z)=E[|zHk Bzk|2] (41)
0 0
subject to CHDHw (i) = g(i), where B =
Fig.3. (a)ErrorPerformanceSurfaceofthefunction f(v,w)=(1− k k ℜ 0
w∗r∗v)2,(b)Contourplotsshowingthatthefunctiondoesnotexhibit is an (N + N/L) (N + N/L) matrix,(cid:20)the Hes(cid:21)-
I I
localminimaandhasmultipleglobalminima,(c)Errorperformance sian (H) with respec×t to z = [wT vT]T is H =
saunrdf(adc)eVoafriinatnecrepopleartefodrmMaMncSeEsruercfeaicveerosfaJtMEVb/(Nv0)=for1C5dMBVforercLeiv=er3s ∂z∂Hk ∂(J˜M∂zSkE(.)) = E[zHk BkzkBH] +k E[zkHk BHzkB] +
−at6Eabn/dN−01=0 1dB5d,BspafcoerdLby=T2c.and channel with paths given by 0, Etha[Bt,zakszoHkcBcuHrs]f+orEth[eBMHMzkSzEHk Bca]s.e,BtyheexthaimrdinainndgfHourwthetenromtes
yield positive semi-definite matrices (aHE[Bz zHBH]a>0
k k
To supportthis claim, we carried out the followingstudies: and aHE[BHz zHB]a > 0, z = 0) whereas the first and
k k k 6
• Let us consider the scalar case of the function in (40), second terms are indefinite matrices. Hence, the cost function
which is defined as f(w,v) = (b w r v)2 = b2 cannot be classified as convex, although we conjecture that it
2bw r v+(w v)2, where r isa co−nstant.By choosin−g does not exhibit local minima. Thus, we proceed similarly to
v (the ”scalar”ℜinterpolator) fixed, it is evident that the theMMSEcasetostudythesurfacesprovidedbytheproblem
resulting function f(w,v) = (b w c)2, where c is a in (41). Then, we carried out the following studies:
−
constant is a convex one, whereas for a time-varying • Wehavealsoplottedthevarianceperformancesurfaceof
9
J (v )in(17),depictedinFig.3(d).Thissurfacereveals By combining (45), (46), (47) and (48) the trajectory of the
o k
thatJ (v )hasaglobalminimum(asitshould)butdoes error vectors is given by:
o k
notexhibitlocal minima,which implies that (41)subject
E[e (i+1)] E[e (i)]
to CHk DHwk(i)=g(i) has no local minima either. E[ew(i+1)] =A E[ew(i)] +B (49)
• Anotherimportantfeaturethatsuggeststhenon-existence (cid:20) v (cid:21) (cid:20) v (cid:21)
of local minima for the blind algorithms is that they where
alwaysconvergeto thesame minimumvalue,fora given (I−µR¯)−µE[¯r(i)vT(i)ℜH(i)] −µE[¯r(i)wT ℜ∗(i)]
experiment, independently of any interpolator initializa- A=(cid:20) −ηE[u(i)voTptℜH(i)] (I−ηR¯u)−ηE[uo(pit)wT(i)ℜ∗(i)](cid:21)
and
tion (except for v(0) = [0 ... 0]T that eliminates the µp¯(i) µE[¯r(i)wT ℜ∗v ]
signal) for a wide range of parameters. B = ηp¯ (i)− ηE[u(i)woTpt ℜ∗vopt ] . Equation (49)
(cid:20) u − opt opt (cid:21)
implies that the stability of the algorithms in the proposed
B. Trajectory of the Mean Tap Vectors structure depends on the matrix A. For stability, the
This part is devoted to the analysis of the trajectory of the convergence factors should be chosen so that the eigenvalues
mean tap vectors of the proposed structure when operating in of AHA are less than one.
blind and supervised modes. In our analysis, we employ the 2) Blind algorithm: The mean vector analysis of the blind
so called Independence Theory [1], [31] (see Chapter 9, pp. algorithm is slightly different from [6] because our approach
390-404) that consists of four points, namely: uses a decoupled SG channel estimation technique [28], that
1.Thereceivedvectorsr(1),...,r(i)andtheirinterpolated yields better channel estimates. Hence, we consider the joint
counterparts¯rk(1),...,¯rk(i) constitute a sequence of statisti- estimation of wk and vk, while g is a decoupled estimation
cally independent vectors. process. To proceed, let us drop the user k index for ease of
2. At time i, r(i) and ¯r (i) are statistically independentof presentation and substitute the expressions of (42) into (32)
k
b (1),...,b (i 1). and (34) that gives:
k k
−
3. At time i, b (i) depends on r(i) and r (i), but is
k k e (i+1)=[I µ¯r(i)¯rH(i)]e (i)+DC(CHDHDC)−1g(i)
independent of previous bk(n), for n=1,...,i 1. w − w
4.Thevectorsr(i)and¯rk(i)andthesamplebk−aremutually −µΠ¯r(i)voTptℜ∗(i)wopt−µΠ¯r(i)woTptℜH(i)ev(i)
Gaussian-distributed random variables. (50)
In the present context, it is worth noting that the indepen-
dence assumption holds for synchronous DS-CDMA systems ev(i+1)=[I−ηu(i)uH(i)]ev(i)−ηu(i)voTptℜ∗(i)ew(i)
[1], which is the present case, but not for asynchronous ηu(i)wT ℜ∗(i)v
− opt opt
models, even though it provides substantial insight. (51)
1) Trained algorithm: To proceed, let us drop the user k
index for ease of presentationand define the tap error vectors where Π = I DC(CHDHDC)−1CHDH and we used
−
e (i) and e (i) at time index i the fact that the scalars have alternative expressions as
w v
(eT(i)ℜH(i)v )T = (eT(i)ℜH(i)v ) = vT ℜ∗(i)e (i)
w opt w opt opt w
ew(i)=w(i) wopt, ev(i)=v(i) vopt (42) and (eT(i)ℜ∗(i)w )T = (eT(i)ℜ∗(i)w ) =
− − v opt v opt
wherewopt andvopt aretheoptimumtapvectorsthatachieve woTptℜH(i)ev(i). By taking expectations on both sides
theMMSEfortheproposedstructure.Substitutingtheexpres- and eliminating the term µΠ¯r(i)voptℜ∗(i)wopt we get
sions in (42) into (21) and (22) we get
E[e (i+1)]=[I µR¯(i)]E[e (i)]+DC(CHDHDC)−1E[g(i)]
w w
ew(i+1)=[I µ¯r(i)¯rH(i)]ew(i)+µ¯r(i)e∗(i) (43) µΠ−E[¯r(i)wT ℜH(i)]E[e (i)]
− − opt v
e (i+1)=[I ηu(i)uH(i)]e (i)+ηu(i)e∗(i) (44) (52)
v v
−
By taking expectations on both sides we have E[e (i+1)]=[I ηR (i)]E[e (i)] ηE[u(i)vT ℜ∗(i)]E[e (i)]
v − u v − opt w
E[ew(i+1)]=[I−µR¯(i)]E[ew(i)]+µE[¯r(i)e∗(i)] (45) −ηE[u(i)woTptℜ∗(i)]vopt
(53)
E[e (i+1)]=[I ηR (i)]E[e (i)]+ηE[u(i)e∗(i)] (46)
v u v
−
By combining(52)and(53)the trajectoryof the errorvectors
Atthispoint,itshouldbenotedthatthetwoerrorvectorshave
for the minimum variance case is given by:
to be considered togetherbecause of the joint optimization of
the interpolator filter and the reduced-rank filter. Rewriting E[e (i+1)] E[e (i)]
w =A w +B (54)
the terms E[¯r(i)e∗(i)] and E[u(i)e∗(i)], using (42) and the E[e (i+1)] MV E[e (i)] MV
v v
(cid:20) (cid:21) (cid:20) (cid:21)
independencetheory[31](seeChap.9,pp.390-404)weobtain
[I µR¯(i)] µΠE[¯r(i)wT ℜH(i)]
E[¯r(i)e∗(i)]=p¯(i) E[¯r(i)vT(i)ℜH(i)]E[ew(i)] whereAMV = ηE[u−(i)vT ℜ∗(i)] − [I ηRo(pit)]]
E[¯r(i)w−T ℜ∗]E[e (i)] E[¯r(i)wT ℜ−∗v ] (cid:20)DC−(CHDHDopCt)−1E[g(i)] − u (cid:21)
opt v − opt opt andB = . Equation(54)
(47) MV ηE[u(i)wT ℜ∗(i)]v
(cid:20) − opt opt (cid:21)
suggests that the stability of the algorithms in the proposed
E[u(i)e∗(i)]=p¯ (i) E[u(i)wT(i)ℜ∗(i)]E[e (i)]
u v structure depends on the matrix A . For stability, the
− MV
E[u(i)vT ℜH]E[e (i)] E[u(i)wT ℜ∗v ] convergence factors should be chosen so that the eigenvalues
− opt w − opt (48op)t of AH A are less than one.
MV MV
10
C. Trajectory of Excess MSE where we used the property of trace and QHQ=I. Because
Λ is a diagonal matrix of dimension M/L we have
HerewedescribethetrajectoryoftheexcessMSEatsteady-
state of the trained and the blind SG algorithms. M/L
1) TrainedAlgorithm: Theanalysisforthe LMSalgorithm ξexc(i+1)= λnxn(i+1) (59)
using the proposedinterpolatedstructureand the computation n=1
X
of its steady-state excess MSE resembles the one in [31] (see
where x , n=1,2, ... M/Lare the elementsof the diago-
Chapter 9, pp. 390-404). Here, an interpolated structure with n
nal of X(i). Here, we may use (45), invokethe independence
jointoptimizationofinterpolatorv andreduced-rankreceiver
k theory [31] (see Chapter 9, pp. 390-404) in order to describe
w istakenintoaccount.Despitethejointoptimization,forthe
k the correlation matrix of the weight error vector:
computation of the excess MSE one has to consider only the
ruepdouncecdo-rnavnekrgpeanrcaemebtyer(v1e3c)toarnwdk(1b4e)casuhsoeutlhdebMeStEheattsaaimneed. K(i+1)=E[ew(i+1)eHw(i+1)] (60)
=(I µR¯(i))K(i)(I µR¯(i))+µ2ǫ
Here, we will drop the user k index for ease of presentation. − − min
Consider the MSE at time i+1 as: Next, using the transformations QHR¯Q = Λ, QHKQ = X
andsimilarlyto[31](see Chapter9, pp.390-404),a recursive
ǫ(i+1)=E[b(i+1) wH(i+1)¯r(i+1)2] (55)
equation in terms of X(i) and Λ can be written:
| − |
Byusingw(i+1)=wopt+ew(i+1),wopt,vopt andthefact X(i+1)=(I µΛ)X(i)(I µΛ)+µ2ǫminΛ (61)
− −
that the expressionsin (13)and (14)are equalforthe optimal
parameter vectors, the MSE becomes Because of the structure of the above equation, one can
decouple the elements x (i) from the off-diagonal ones, and
n
ǫ(i+1)=σb2−p¯H(i+1)R¯−1(i+1)p¯(i+1) rthecuusrξseioxnc(:i+1) depends on xn(i) according to the following
p¯H(i+1)e (i+1) eH(i+1)p¯(i+1)
− w − w
wH p¯(i+1)+wH R¯(i+1)w x (i+1)=(1 µλ )2x (i)+µ2ǫ λ (62)
− opt opt opt n − n n min n
+wH R¯(i+1)e (i+1)+eH(i+1)R¯(i+1)w
opt w w optAt this point, it can be noted that such recursive relation
+E[ew(i+1)¯r(i+1)¯rH(i+1)eHw(i+1)] convergesprovidedthat all the roots lie inside the unit circle,
=σb2−p¯H(i+1)R¯−1(i+1)p¯(i+1)+ i. e., (1−µλn)2 <1 for all n, and thus we have for stability
E[ew(i+1)¯r(i+1)¯rH(i+1)eHw(i+1)] 2
0<µ< (63)
=J (w ,v )+ξ (i+1) λ
MMSE opt opt exc max
(56)
where λ is the largest eigenvalue of the matrix R¯. In
max
where p¯(i + 1) = E[b∗(i + 1)¯r(i + 1)], ǫ = practice, tr[R¯] is used as a conservative estimate of λmax.
min
J (w ,v )=σ2 p¯H(i+1)R¯−1(i+1)p¯(i+1) is By taking limi→∞ on both sides of (62), we get xn( ) =
MMSE opt opt b − µ ǫ . Then, taking limits on both sides of (59∞) and
the MMSE achieved by the proposed structure when we have 2+µλn min
w and v and ξ (i+1)=E[eH(i+1)¯r(i+1)¯rH(i+ using xn( ) we obtain the expressionfor the excess MSE at
opt opt exc w ∞
1)e (i+1)] is the excess MSE at time i+1. To compute steady-state
w
the excess MSE one must evaluate the term ξ (i+1). By
invoking the independence assumption and theexpcroperties of ξ ( )=M/Lλ x ( )=M/L µλn ǫ = µ2 tr[R¯] ǫ
trace [31] (see Chap. 9, pp. 390-404) we may reduce it as exc ∞ n n ∞ 2+µλ min 1 µ tr[R¯] min
n=1 n=1 n − 2
follows: X X (64)
Theexpressionin(64)canbeusedtopredictsemi-analytically
E[eH(i+1)¯r(i+1)¯rH(i+1)e (i+1)]=tr R¯(i+1)K(i+1) the excess MSE, where R¯ must be estimated with the aid of
w w
h (57i) computersimulations since it is a function of the interpolator
In the following steps, we assume that i is sufficiently large v(i). Alternatively, one can conduct the analysis for the
such that the matrix R¯(i) = R¯( ) = R¯. To proceed let interpolator v(i), which results in the expression ξexc( ) =
us define some new quantities th∞at will perform a rotation η2 tr[Ru] ǫ , where η is the step size of the interpo∞lator,
of coordinates to facilitate our analysis as advocated in [31]. t1h−eη2mtra[tRriux] Rmin = R ( ) and R (i) = E[u(i)uH(i)], as
DefineQHR¯Q=Λ,whereΛisadiagonalmatrixconsisting defined in conunectionuw∞ith (12). Aumore complete analytical
of the eigenvalues of R¯ and Q is the unitary matrix with
result, expressed as a function of both step sizes, µ and η,
the eigenvectors associated with these eigenvalues. Letting and statistics of the non-interpolated observation vector r(i)
QHKQ=X we get requiresfurtherinvestigationin orderto determine tr[R¯( )],
∞
that depends on η or tr[R ( )], that depends on µ. Never-
u
ξ (i+1)=tr R¯K(i+1) =tr QΛQHQX¯(i+1)QH theless, such investigation is∞beyond the scope of this paper
exc
and it should be remarked that the results would not differ
=trhQΛX¯(i+1i)QH h=tr ΛX¯(i+1) i from the semi-analytical results derived here (that implicitly
h i h i (58) take into account the parameters of v(i)).