Set-Membership Adaptive Algorithms based on Time-Varying Error Bounds for Interference Suppression

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by  Rodrigo C. de Lamare

1
Set-Membership Adaptive Algorithms based on
Time-Varying Error Bounds for Interference
Suppression
Rodrigo C. de Lamare, IEEE Member and Paulo S. R. Diniz, IEEE Fellow
Abstract—This work presents set-membership adaptive is very difficult to obtain in practice due to the lack of
3 algorithms based on time-varying error bounds for CDMA knowledge of the environment and its dynamics. In wireless
1 interference suppression. We introduce a modified family of networkscharacterizedbynon-stationaryenvironments,where
0 set-membership adaptive algorithms for parameter estimation users often enter and exit the system, it is very difficult to
with time-varying error bounds. The algorithms considered
2
include modified versions of the set-membership normalized chooseanerrorboundandtheriskofoverbounding(whenthe

n least mean squares (SM-NLMS), the affine projection (SM-AP) error bound is larger than the actual one) and underbounding
a and the bounding ellipsoidal adaptive constrained (BEACON) (when the error bound is smaller than the actual one) is
J recursive least-squares technique. The important issue of error
significantly increased, leading to performance degradation.
1 bound specification is addressed in a new framework that takes
In addition, when the measured noise in the system is time-
intoaccount parameterestimation dependency,multi-accessand
] inter-symbol interference for DS-CDMA communications. An varying and the multiple access interference (MAI) and the
T algorithm for tracking and estimating the interference power intersymbol interference (ISI) encountered by a receiver in
I is proposed and analyzed. This algorithm is then incorporated a communication system are highly dynamic, the selection
.
s into the proposed time-varying error bound mechanisms. of an error-bound is further complicated. This is especially
c Computer simulations show that the proposed algorithms are
[ capable of outperforming previously reported techniques with a relevant for low-complexity estimation problems encountered

significantly lower number of parameter updates and a reduced in applications such as mobile terminals and wireless sensor
1
risk of overbounding or underbounding. networks[8],wherethesensorshavelimitedsignalprocessing
v
7 capabilities and power consumption is of central importance.
9 Index Terms—Set-membership filtering, adaptive filters, DS- These problems suggest the deployment of mechanisms to
CDMA, interference suppression.
0 automatically adjust the error bound in order to guarantee
0
goodperformanceandasmallupdaterate(UR).Itshouldalso
.
1 be remarked that most of prior work on adaptive algorithms
0 I. INTRODUCTION for interference suppression [18] is restricted to systems with
3
Set-membership filtering (SMF) [1], [2], [3], [4] represents short codes. However, the proposed adaptive techniques are
1
a class of recursiveestimation algorithmsthat, on the basis of also applicable to systems with long codes provided some
:
v a pre-determined error bound, seeks a set of parameters that modifications are carried out. For downlink scenarios, the
i
X yieldboundedfilteroutputerrors.Thesealgorithmshavebeen designer can resort to chip equalization [24] followed by a
used in a varietyof applicationssuch as adaptiveequalization despreader. For an uplink solution, channel estimation algo-
r
a [5] and multi-access interference suppression [6], [7]. The rithms for aperiodic sequences[25], [26] are required and the
SMF algorithms are able to combat conflicting requirements sample averageapproachforestimating the covariancematrix
such as fast convergenceand low misadjustment by introduc- R=E[r(i)rH(i)]oftheobserveddatar(i)hastobereplaced
ingamodificationontheobjectivefunction.Inaddition,these by Rˆ = PPH +σ2I, which is constructed with a matrix P
algorithms exhibit reduced complexity due to data-selective containing the effective signature sequence of users and the
updates, which involve two steps: a) information evaluation variance σ2 of the receiver’s noise [27].
and b) update of parameter estimates. If the filter update Previous works on time-varying error bounds include the
does not occur frequently and the information evaluation tuning of noise bounds in [9], [10], the approach in [13]
does not involve much computational complexity, the overall which assumes that the ”true” error bound is constant, and
complexity can be significantly reduced. the parameter-dependent error bound recently proposed in
The adaptive SMF algorithms usually achieve good con- [11], [12] with frequency-domain estimation algorithms. The
vergence and tracking performance due to an adaptive step techniquessofarreporteddonotintroduceanymechanismfor
size for each update, and reduced complexity resulting from trackingtheMAIandtheISIandincorporatingtheirpoweres-
data selective updating. However, the performance of SMF timatesintheerrorbound.Inaddition,theexistingapproaches
techniques depends on the error-bound specification, which with time-varying bounds have not been considered for more
sophisticated adaptive filtering algorithms such as the affine
R. C. de Lamare is with the Communications Research Group, De- projection (AP) and the least-squares (LS) based techniques.
partment of Electronics, University of York, United Kingdom and Paulo
In this work, we propose and analyze a low-complexity
S. R. Diniz is with LPS/COPPE-UFRJ, Rio de Janeiro, Brazil. E-mails:
[email protected], [email protected]. frameworkfor tracking parameter evolution and MAI and ISI
2
power levels, that relies on simple channel and interference symbolswould interferein total, the currentone, the previous
estimation techniques, and encompasses a family of set- andthesuccessivesymbols.InthecaseofN <L ≤2N then
p
membership algorithms [2], [3], [6], [14] with time-varying 5symbolswouldinterfere,thecurrentone,the2previousand
errorbounds.Specifically,wepresentmodifiedversionsofthe the 2 successive ones. In most practical CDMA systems, we
set-membership normalized least mean squares (SM-NLMS) have that 1 < L ≤ N and then only 3 symbols are usually
p
[2], the affine projection [3] (SM-AP) and the bounding affected.ThereaderisreferredtoUMTSchannelmodels[22],
ellipsoidal adaptive constrained (BEACON) [6], [14] recur- whichrevealthatthechannelusuallyaffectsatmost3symbols
sive least-squares (RLS) algorithm for parameter estimation. (it typically spans a few chips).
Then,weincorporatetheproposedmechanismsofinterference The multiuser linear receiver design corresponds to deter-
estimation and tracking into the time-varying error bounds. mining an FIR filter w [i]= w [i] w [i] ... w T with
k 0 1 M−1
In order to evaluate the proposed algorithms, we consider a M coefficientsthatprovidesanestimateofthedesiredsymbol
(cid:2) (cid:3)
DS-CDMA interference suppression application and adaptive as given by
linear multiuser receivers in situations of practical interest.
This work is organized as follows. Section II briefly de- ˆbk[i]=sgn ℜ wkH[i]r[i] =sgn ℜ zk[i] , (3)
scribes the DS-CDMA system model and linear receivers. (cid:16) h i(cid:17) (cid:16) h i(cid:17)
where the quantity ℜ(·) selects the real part and sgn(·) is
Section III reviews the SMF concept with time-varying error
the signum function. The quantity z [i] = wH[i]r[i] is the
boundsandisdevotedtothederivationofadaptivealgorithms. k k
output of the receiver parameter vector w for user k, which
Section IV presents the framework for time-varying error k
is optimized according to a chosen criterion.
bounds and the proposed algorithms for channel, interference
estimationandtracking.SectionVisdedicatedtotheanalysis
of the algorithms for channel, amplitude, interference estima- III. SET-MEMBERSHIP ADAPTIVE ALGORITHMS WITH
tion and their tracking. Section VI shows and discusses the TIME-VARYING ERROR BOUNDS AND PROBLEM
simulations results, while Section VII gives the conclusions. STATEMENT
In this section, we describe a framework that encompasses
modifiedset-membership(SM)adaptivealgorithmswithtime-
II. DS-CDMA SYSTEMMODELAND LINEARRECEIVERS
varying error bounds for communications applications. In an
Letusconsiderthe downlinkofa symbolsynchronousDS-
SM filtering [2] framework, the parameter vector w [i] for
k
CDMA system with K users, N chips per symbol and L
p user k in a multi-access system is designed to achieve a
propagationpaths[18].Weassumethatthedelayisamultiple
specified bound on the magnitude of the estimation error
of the chip rate, the channel is constant during each symbol
e [i] = b [i]−wH[i]r[i]. As a result of this constraint, the
interval and the spreading codes are repeated from symbol k k k
SM adaptive algorithm will only perform filter updates for
to symbol. The received signal r(t) after filtering by a chip-
certain data. Let Θ [i] represent the set containing all w [i]
k k
pulse matched filter and sampled at chip rate yields the M-
thatyieldsanestimationerrorupperboundedinmagnitudeby
dimensional received vector
a time-varying error bound γ [i]. Thus, we can write
k
K
r[i]= Ak[i]bk[i]Ckh[i]+η[i]+n[i], (1) Θk[i]= {wk ∈CM :|ek[i]|≤γk[i]}, (4)
kX=1 (bk[i],\r[i])∈Sk
where M = N +L −1, n[i] = [n [i] ... n [i]]T is the wherer[i]istheobservationvector,S isthesetofallpossible
p 1 M k
complexGaussiannoisevectorwithzeromeanandcovariance data pairs (b [i], r[i]) and the set Θ [i] is referred to as
k k
matrix E[n[i]nH[i]] = σ2I, where (·)T and (·)H denote the feasibility set for user k, and any point in it is a valid
transpose and Hermitian transpose, respectively. The quantity estimate z [i]=wH[i]r[i]. Since it is not practical to predict
k k
E[·] stands for expected value, the user k symbol is b [i] all data pairs, adaptive methods work with the membership
k
and is assumed to be drawn from a general constellation. sets ψ = i H provided by the observations, where
k,i m=1 k,m
The amplitude of user k is A [i] and η[i] is the intersymbol H = {w ∈ CM : |b [m] − z [m]| ≤ γ [m]} is the
k k,m k k k k
T
interference (ISI) for user k. The M×L convolutionmatrix constraint set. It can be seen that the feasibility set Θ [i]
p k
C that contains one-chip shifted versions of the signature is a subset of the exact membership set at any given time
k
sequence for user k expressed by s = [a (1)...a (N)]T instant. The feasibility set Θ [i] is also the limiting set of
k k k k
and the L ×1 vector h[i] with the multipath componentsare the exact membership set, i.e., the two sets will be equal
p
described by: if the training signal traverses all signal pairs belonging to
S . The idea of the SM algorithms is to adaptively find
a (1) 0 k
k an estimate that belongs to the feasibility set Θ [i]. One
 ... ... ak(1)  h0.[i] alternativeistoapplyoneofthemanyOBEalgorithmkssuchas
Ck = .. ,h[i]= .. . theboundingellipsoidaladaptiveconstrained(BEACON)[14],
 a (N) . 
 k  h [i] [6] recursive least-squares (RLS) algorithm, which tries to
 0 ... a (N)   Lp−1  approximatetheexactmembershipsetwithellipsoids.Another
 k 
  (2) way is to computea pointestimate throughprojectionsusing,
Inthismodel,theISIspanandcontributionη [i]arefunctions for example, the information provided by the constraint set
k
of the processing gain N and L . If 1 < L ≤ N then 3 H , as done by the set-membership NLMS (SM-NLMS)
p p k,i
3
and the affine projection [3] (SM-AP) algorithms. In order TheSM-AP adaptivealgorithmwith time-varyingboundscan
to devise an effective SM algorithm, the error bound γ [i] be derived from the optimization problem
k
must be appropriately chosen. Due to time-varying nature of
minimize ||w [i+1]−w [i]||2
many practical environments, this error bound should also k k (12)
be adaptive and adjustable to certain characteristics of the subject to (bk[i]−YH[i]wk[i+1])=gk[i].
environment for the SM estimation technique. The natural
In order to solve the above problem, we employ the method
questionthatarisesis:howtodesignanefficientandeffective
of Lagrange multipliers and consider the unconstrained cost
mechanismto adjustγk[i]? In whatfollows,we will presenta function
modifiedfamilyofSMadaptivealgorithmsthatrelyongeneral
time-varying error bounds. Specifically, we will consider the L=||wk[i+1]−wk[i]||2+2ℜ (bk[i]−YH[i]wk[i+1]−gk[i])Hλ ,
SM-NLMS[2],SM-AP[3]andBEACON[14],[6]algorithms h (13) i
and we will modify them such that they will operate with
where λ is the vector with Lagrange multipliers and g [i]
general time-varying error bounds. k
is a constraint vector. By calculating the gradient terms of
(13) with respect to w [i +1] and λ, setting them to zero
A. SM-NLMS Algorithm with Time-Varying Bounds k
and solving the resulting equationswe arrive at the following
In order to derive an SM-NLMS adaptive algorithm with
algorithm:
time-varying bounds using point estimates, we consider the
following optimization problem tk[i]=(YH[i]Y[i]+δI)−1(ek[i]−g[i]), (14)
minimize ||wk[i+1]−wk[i]||2 (5) wk[i+1]=wk[i]+Y[i]tk[i], (15)
subject to (b [i]−wH[i+1]r[i])=g [i]
k k k where δ is a small constant inserted in addition to the term
In orderto solve the aboveconstrainedoptimization problem, YH[i]Y[i] for improving robustness. If we select e [i] −
k
we resort to the method of Lagrange multipliers [1], [23], g [i] = (e [i] − γsgn(e [i]))u = (1 − γ [i]/|e [i]|)e [i]u,
k k k k k k
which yields the unconstrained cost function where the a posteriori errors e [i−j] are kept constant for
k
j = 1, ...,P − 1 and u = [1 0 ... 0]T, we obtain the
L=||w [i+1]−w [i]||2+2ℜ λ∗(b [i]−wH[i+1]r[i]−g [i]) ,
k k k k k following recursion for the update of t [i]:
k
h (6) i
t [i]=(YH[i]Y[i]+δI)−1(1−γ/|e [i]|)e [i]u. (16)
where∗denotescomplexconjugate,λisaLagrangemultiplier k k k
and g [i] is the time-varying set-membership constraint for Substituting(16)into (15)andusing the boundconstraint,we
k
userk.Takingthegradienttermsof(6)withrespecttow [i+ obtain the following SM-AP algorithm:
k
1] and λ∗, and setting them to zero, leads us to a system of
w [i+1]=w [i]+µ [i]Y[i](YH[i]Y[i]+δI)−1e [i]u, (17)
equations. Solving these equations yields: k k w k
ek[i]=bk[i]−wkH[i]r[i], (7) µw[i]= 0(1−γk[i]/|ek[i]|) oifth|eekrw[ii]s|e>. γk[i], (18)
w [i+1]=w [i]+(rH[i]r[i])−1(e [i]−g [i])∗r[i], (8) (cid:26)
k k k k
TheSM-APalgorithmdescribedherehascomputationalcom-
whereek[i]istheerrorforuserk.Bychoosinggk[i]suchthat plexity of UR×O(PM+2KinvP2), where Kinv is a factor
ek[i] lies on the closest boundary of Θk[i] and considering a requiredtoinvertaP×P matrix[1]andURistheupdaterate.
time-varying error bound γk[i], i.e., gk[i] = γk[i]sgn(ek[i]) Note that the SM-AP is a generalized case of the SM-NLMS
[2], we obtain the following data dependent update rule and where P data vectors are used to increase the convergence
step size speed.
w [i+1]=w [i]+µ [i]e∗[i]r[i], (9)
k k w k
1 (1−γ [i]/|e∗[i]|) if |e∗[i]|>γ [i], C. BEACON Adaptive Algorithm with Time-Varying Bounds
µ [i]= rH[i]r[i] k k k k
w 0 otherwise. Here, we propose a modification for a computationally
(cid:26)
(10) efficientversionof an optimalboundingellipsoidal(OBE) al-
gorithmcalledtheBoundingEllipsoidalAdaptiveConstrained
B. SM-AP Algorithm with Time-Varying Bounds Least-Squares (BEACON) algorithm [6], which is closely
related to a constrained least-squares optimization problem.
In order to describe a modified SM-AP algorithm with
TheproposedtechniqueamountstoderivingtheBEACONal-
time-varyingbounds,let us first define the observationmatrix
gorithm equipped with time-varying bounds. Unlike the other
Y[i] = [r[i] ... r[i − P + 1]], the desired output vector
previouslyreportedlow-complexityalgorithms[2],[3]andthe
b [i] = [b [i] ... b [i−P +1]]T that comprises P outputs
k k k
modified SM-NLMS and SM-AP techniques described in the
and the error vector
previous subsections, the modified BEACON recursion has
b∗[i]−rH[i]w [i]
k k the potential to achieve a very fast convergenceperformance,
.
ek[i]= .. =b∗k[i]−YH[i]wk[i]. which is relatively independent from the eigenvalue spread
b∗[i]−rH[i−P +1]w [i] of the data covariance matrix as compared to the stochastic
 k k 
  (11) gradient algorithms [1].
4
The proposed BEACON algorithm with a time-varying A. Parameter Dependent Bound
boundcanbederivedfromthefollowingoptimizationproblem
Here, we describe a parameter dependent bound (PDB),
[6]
that is similar to the one proposed in [11] and considers the
i−1 evolution of the parameter vector w [i] for the desired user
k
minimize λi−l[l]|bk[l]−wkH[i]r[l]|2 (19) (user k). The proposed PDB recursion computes a bound for
Xl=1 SM adaptive algorithms and is described by:
subject to |b [i]−w [i]Hr[i]|2 =γ2[i].
k k k
γ [i+1]=(1−β)γ [i]+β α||w [i]||2σˆ2[i], (26)
The constrained problem above can be recast as an uncon- k k k v
strained one and be solved via an unconstrainedleast-squares whereβ isaforgettingfactorthatsphouldbeadjustedtoensure
cost function using the method of Lagrange multipliers given
an appropriatetime-averagedestimate ofthe evolutionsofthe
by parametervector w [i], α||w [i]||2σˆ2[i] is the variance of the
k k v
i−1 inner product of wk[i] with n[i] that provides information on
L= λk[l]i−l|bk[l]−wkH[i]r[l]|2+λk[i](|bk[i]−wkH[i]r[i]|2−theγk2e[vio])l,ution of wk[i], α is a tuning parameter and σˆv2[i] is
l=1 an estimate of the noise power. This kind of recursion helps
X
(20)
avoiding too high or low values of the squared norm of w [i]
k
where λ [l] plays the role of Lagrange multiplier and forget- and provides a smoother evolution of its trajectory for use in
k
ting factor at the same time for user k. The solution to the thetime-varyingbound.Thenoisepoweratthereceivershould
aboveoptimizationproblemisobtainedbytakingthegradient be estimated via a time average recursion. In this work, we
terms with respect to w [i] and making them equal to zero. will assume that it is known at the receiver.
k
After some mathematical manipulations we have
w [i]=w [i−1]+λ [i]P [i]r[i](b [i]−wH[i]r[i])∗. (21) B. Parameter and Interference Dependent Bound
k k k k k k
By using the constraint b [i] − wH[i]r[i] = In this part, we develop an interference estimation and
k k
(ξ [i]/|ξ [i]|)γ [i] = ξ [i] − (λ [i]G [i](ξ [i]/|ξ [i]|)γ [i]), trackingproceduretobecombinedwithaparameterdependent
k k k k k k k k k
and the matrix inversion lemma [1], we can arrive at the bound and incorporated into a time-varying error bound for
BEACON algorithm with time-varying bounds described by SM recursions. The MAI and ISI power estimation scheme,
outlined in Fig. 2, employs both the RAKE receiver and the
λ [i]P [i−1]r[i]rH[i]P [i−1]
P [i]=P [i−1]− k k k , (22) linearreceiverdescribedin(3)forsubtractingthedesireduser
k k
1+λ [i]t [i]
k k signal from r[i] and estimating MAI and ISI power levels.
wk[i]=wk[i−1]+λk[i]Pk[i]r[i]ξk∗[i], (23) With the aid of adaptive algorithms, we design the linear
where the prediction error is ξ [i] = b [i]−wH[i−1]r[i], receiver,estimate the channelmodeledas an FIR filter forthe
t [i] = rH[i]P [i − 1]r[i], aknd λ [i]k ≥ 0. kIn order to RAKEreceiverandobtainthedetectedsymbolˆbk[i],whichis
k k k combinedwithanamplitudeestimateAˆ [i]forsubtractingthe
compute the optimal value for λ [i], the algorithm considers k
k
desired signal from the output x [i] of the RAKE. Then, the
the following cost function [6]: k
difference d [i] between the desired signal and x [i] is used
k k
C =λ [i] ξk[i] 1 −1 . (24) to estimate MAI and ISI power.
λk[i] k γ2[i] 1+λ [i]t [i]
" k k k ! #
The minimization of the cost function in (24) leads to the
innovation check of the proposed BEACON algorithm:
1 |ξk[i]| −1 if |ξ∗[i]|>γ [i],
λk[i]= tk[i] γk[i] ! k k (25)
 0 otherwise.
IV. ALGORITHMS FOR TIME-VARYING ERROR BOUNDS,
INTERFERENCEESTIMATIONAND TRACKING
Thissectionisdevotedtotime-varyingerrorboundsthatin-
corporateparameterandinterferencedependency.We propose
a low-complexity framework for time-varying error bounds,
interference estimation and tracking. A simple and effective
algorithm for taking into considerationparameter dependency
is introduced and incorporatedinto the error bound. A proce-
dureforestimatingMAIandISIpowerlevelsisalsopresented
and employedin the adaptive error boundfor SM algorithms. Fig.1. Blockdiagramoftheproposedinterference estimationandtracking
algorithm.
The proposedalgorithmsare based on the use of simple rules
and parameters that behave as forgetting factors, regulate the
pace of time averages and selectively weigh some quantities.
5
Channel Estimation: where fH[i]˜s [i] = ρ [i] and fH[i]˜s [i] = ρ [i] for j 6=
k k k k j 1,j
1. The symbol ρ represents the cross-correlation (or inner
k
Letusfirstpresentasimplechannelestimationalgorithmfor product) between the effective signature and the RAKE with
designing the RAKE receiver. Consider the constraint matrix perfect channel estimates. The symbol ρ [i] represents the
1,j
C that contains one-chip shifted versions of the signature cross-correlationbetweentheRAKEreceiverandtheeffective
k
sequenceforuser k definedin (2)andthe assumptionthatthe signature of user j. The second-order statistics of the output
symbolsb [i]areindependentandidenticallydistributed(i.i.d) of the RAKE in (31) are described by:
k
and statistically independent from the symbols of the other
E[|x [i]|2]=A2[i]ρ2[i]E[|b [i]|2]
users. The proposedchannelestimation algorithmis based on k k p k
the following cost function →1
K K
C(hˆ[i],Aˆk[i])=E ||Aˆk[i]bk[i]Ckhˆ[i]−r[i]||2 (27) + | A2j{[zi]E[}bj[i]b∗l[i]]fjH˜sj˜sHl fj
=E(cid:2)||Aˆk[i]bk[i]ˆfk[i]−r[i]||2 , (cid:3) Xjj6==k1 Xjl=6=1k
where hˆ[i] is an estim(cid:2)ate of the channel h[i](cid:3)and ˆfk[i] = →PKj=1,j6=kfjH˜sj˜sHj [i]fj
Ckhˆ[i] is the RAKE receiver for user k with the estimated +f|HE[η[i]ηH[i]]f {+z fHE[n[i]nH[i]]}f .
k k k k
channel. By taking the gradient terms of (27), making them
equaltozero,wecandeviseastochasticgradient(SG)channel →σ2fkHfk
(32)
estimation algorithm as follows: | {z }
hˆ[i+1]=hˆ[i]−µ Aˆ [i]CHb∗[i] b [i]C hˆ[i]−r[i] . (28) From the analysis above, we conclude that through the
h k k k k k second-order statistics one can identify the sum of the power
Amplitude Estimation: (cid:0) (cid:1) levelsofMAI,ISIandthenoiseterms.Therefore,ourstrategy
is to obtain instantaneous estimates of the MAI, the ISI and
The amplitude has also to be estimated at the receiver in the noise from the output of a RAKE receiver, subtract the
order to provide this information for different tasks such as detected symbol in (3) from this output (using the more reli-
interferencecancellation and power control. The proposedin- able multiuser receiver (wk[i])) and to track the interference
terferenceestimationandtrackingalgorithmneedssomeform (MAI + ISI + noise) power as shown in Fig. 2. Let us define
ofamplitudeestimationinordertoproceedwiththeestimation the difference between the output of the RAKE receiver and
of the interference power. To estimate the amplitudes of the the detected symbol for user 1:
associatedusersignals,wedescribethefollowingoptimization
K
problem with the cost function in (27) d [i]=x [i]−Aˆ [i]ˆb [i]≈ A [i]b [i]fH[i]˜s [i]+fH[i]η[i]
k k k k k k k k k
Aˆk[i]=argminC(hˆ[i],Aˆk[i]) kX=2 ISI
Ak[i] (29) MAI
=argminE[||Ak[i]ˆbk[i]Ckhˆ[i]−r[i]||2]. +f|kH[i]n[i]. {z } | {z }
Ak[i]
noise
In order to solve the problem above efficiently we describe a (33)
simple SG algorithm to estimate the amplitude of user k, as | {z }
given by By taking expectationson |dk[i]|2 and takinginto accountthe
assumptionthatMAI,ISIandnoiseareuncorrelatedwehave:
Aˆ [i+1]=Aˆ [i]−µ b∗[i]hˆH[i]CH Aˆ [i]b [i]C hˆ[i]−r[i] .
k k A k k k k k K
(cid:16) (30) (cid:17) E[|d [i]|2]≈ fH[i]˜s [i]˜sH[i]f [i]
k k k k k (34)
Interference Estimation and Tracking: Xk=2
+fH[i]E[η[i]ηH[i]]f [i]+σ2fH[i]f [i],
k k k k
Let us consider the RAKE receiver with perfect channel where the above equation represents the interference power.
knowledge,whose parametervector fk[i]=Ckh[i] for user k Based on time averages of the instantaneous values of the
(desired one) estimates the effective signature sequenceat the interference power, we introduce the following algorithm to
receiver,i.e.Ckh[i]=˜sk[i].TheoutputoftheRAKEreceiver estimate and track E[|dk[i]|2]:
is given by:
vˆ[i+1]=(1−β)vˆ[i]+β|d [i]|2, (35)
k
K
xk[i]=fkH[i]r[i]=Ak[i]bk[i]fkH[i]˜sk[i]+ Aj[i]bj[i]fjH[i]˜sj[i]where β is a forgetting factor. To incorporate parameter
j=2 dependency and interference power for computing a more
desired signal Xj6=k
effective bound, we propose the parameter and interference
| {z }
MAI dependent bound (PIDB):
+fH[i]η[i]+fH[i]n[i],
k k | {z }
γ [i+1]=(1−β)γ [i]+β τ vˆ2[i]+ α||w ||2σˆ2[i] ,
ISI noise k k k v
(31) (cid:16)p p (3(cid:17)6)
| {z } | {z }
6
wherevˆ[i]istheestimatedinterferencepowerinthemultiuser algorithm, the magnitude of this geometric ratio must be less
system and τ is a weighting parameter that must be set. than 1 for all k (|1−µ σ2 σ2CHC |<1). This means that
h Ak b k k
Similarly to (26), the equations in (35) and (36) are time- 0 < µ < 2 . The reviewer is referred to [1], [23]
averaged recursions that are aimed at tracking the quantities for furhther dσeb2tσaiA2lks.λmax
|d [i]|2 and ( τvˆ2[i]+ α||w ||2σˆ2[i]), respectively. The
k k v
equations in p(35) and (3q6) also avoid undesirable too high B. Amplitude Estimator
or low instantaneous values which may lead to inappropriate
InordertoanalyzetheSGamplitudeestimatordescribedin
time-varying bound γ [i].
k (30),letusfirstdefineanerrorsignalǫ [i]=Aˆ [i]−A .
A,k k k,opt
By subtracting A from the equation in (30) we obtain
k,opt
V. ANALYSIS OF THEALGORITHMS
ǫ [i+1]=ǫ [i]−µ b∗[i]hˆH[i]CH Aˆ [i]b [i]C hˆ[i]−r[i]
In this section we analyze the channel estimation and Ak Ak A k k k k k
interference estimation algorithms described in the previous =ǫAk[i]−µAb∗k[i]hˆH[i]CHk (cid:0)(ǫA,k[i] (cid:1)
section. We focus on the convergence properties of the al- +Ak,opt)bk[i]Ckhˆ[i]−r[i(cid:0)]
gorithms in terms of the step size parameters µ , µ and β
usedforthechannel,theamplitudeandtheinterfehrencAepower =ǫAk[i]−µA|bk[i]|2hˆ[i]CHk C(cid:1)khˆ[i]ǫAk[i]
estimators, respectively. −µ b∗[i]hˆH[i]CH A b [i]C hˆ[i]−r[i]
A k k k,opt k k
= 1−µA|bk[i]|2hˆ[i]C(cid:0) Hk Ckhˆ[i] ǫAk[i] (cid:1)
A. Channel Estimator −(cid:0) µAb∗k[i]hˆH[i]CHk eA,opt[i]. (cid:1)
In order to analyze the SG channelestimator given in (28), (40)
let us first define an error vector ǫ [i] = hˆ[i] − h . By
h opt By considering that the error ǫ [i], e [i] =
subtracting h from the SG recursion in (28), we get A A,opt
opt A b [i]C hˆ[i]−r[i], the signal components K x [i]
k,opt k k k=1 k
ǫ [i+1]=ǫ [i]−µ Aˆ [i]CHb∗[i] Aˆ [i]b [i]C hˆ[i] fromthedatavectorr[i]= K x [i]+n[i]givenby(1)and
h h h k k k k k k k=1 k P
−r[i] n[i]arestatisticallyindependent,andbycomputingthemean-
(cid:0) squared error, i.e., MSE(AP[i])[i] = K [i] = E[|ǫ[i]|2], we
=ǫh[i]−(cid:1) µhAˆk[i]b∗k[i]CHk Aˆk[i]bk[i]Ck(ǫh[i]+hopt)get k Ak
−r[i]
(cid:0) MSE(A [i])[i+1]= 1−µ σ2E[||hH[i]CH||2] 2K [i]
= I−µ(cid:1)hAˆk[i]|bk[i]|2CHk Ck ǫh[i] k +µ2σ2AE[b||hH[ik]CH||k2]MSE Ak ,
−(cid:2) µhAˆk[i]b∗k[i]CHk Aˆk[i]bk(cid:3)[i]Ckhopt−r[i] (cid:0) A b k k (cid:1)A,min(41)
= I−µhAˆk[i]|bk[i]|(cid:0)2CHk Ck ǫh[i] (cid:1) where MSE =E[||e [i]||2]. The cross multiplication
−(cid:2) µhAˆk[i]b∗k[i]CHk ek,opt[i].(cid:3) between theA,tmerimns will vAan,oipsth as a result of the statistical
(37)
independence between them. The general algorithm in (30)
By considering that the error vector ǫ [i], e [i], the sig- will converge asymptotically and in an unbiased way to the
h k,opt
nal components K x [i] from the data vector r[i] = Ak,opt provided the step size µA is chosen such that
k=1 k
K x [i] + n[i] given by (1) and n[i] are statistically 2
k=1 k P 0<µ < . (42)
independentand computingthe covariancematrixof the error A σ2E[||hH[i]CH||2]
Pvector, i.e. K[i]=E ǫ [i]ǫH[i] , we obtain b k k
h h The above range of values has to be tuned in order to ensure
K[i+1]= I−µ σ(cid:2)2 σ2CHC(cid:3) K[i] I−µ σ2 σ2CHC good convergenceand tracking of the amplitude.
h Ak b k k h Ak b k k
+µ2σ2 σ2||CHC ||2MSE ,
(cid:2) h Ak b k k (cid:3) (cid:2)min (cid:3)
(38) C. Interference Estimation and Tracking Algorithm
Let us describe in a general form the mechanisms for
where σ = E[|b [i]|2], σ2 = E[|A [i]|2] and MSE =
b k Ak k min interference estimation and tracking given in (26) and (36).
E[||e [i]||2] stands for the minimum MSE achieved by the
k,opt
We can write without loss of generality
estimator. The recursive rule/algorithm in (28) is asymptot-
ically unbiased and will converge to the optimum channel γ [i+1]=(1−β)γ [i]+β Po[i], (43)
k k
estimator h if the step size µ satisfies the following
opt h
condition where Po[i] can account either for a parameter dependent
2 bound(PDB),asdescribedinSectionIV.A,orforaparameter
0<µ < , (39)
h σb2σA2kλmax and interferencedependentbound(PIDB), as the one detailed
in Section IV.B.
where λ is the largest eigenvalue of the matrix CHC .
max k k Our goal is to establish the convergence of a general
The condition above with concern to the step size µ arises
h stochastic recursion, as the one given in (43), in terms of the
from difference equations. The quantities generated in (38)
mean-squared error (MSE) at iteration [i], as described by
represent a geometric series with a geometric ratio equal to
(1 − µ σ2 σ2CHC ). For stability or convergence of this MSE(γ[i])[i]=E |γ [i]−γ |2 , (44)
h Ak b k k k k,opt
(cid:2) (cid:3)
7
where γ is the optimal parameter estimate for γ [i]. scenario and the desired receiver processes the transmissions
k,opt k
We will show that under certain conditions on β, the intendedtootherusers(MAI)overthesamechannelasitsown
sequence converges to the optimal γ in the mean-square signal.Thechannelcoefficientsareh [i]=p α[i],whereα[i]
k,opt l l l l
sense. Let us define the error (l =0,1,...,L −1) are obtained with Clarke’s model [21].
p
In particular, we employ standard SG adaptive algorithms for
ǫ [i]=γ [i]−γ ,
γ k k,opt channel estimation and RAKE design in order to concentrate
and substitute the above equation into (44), which yields on the comparison between the analyzed algorithms for re-
ceiver parameter estimation. We show the results in terms of
ǫ [i+1]=(1−β)ǫ [i]+β(Po[i]−γ )
γ γ k,opt (45) the normalized Doppler frequency fdT (cycles/symbol) and
=(1−β)ǫγ[i]+βek,opt[i], use three-path channels with relative powers given by 0, −3
and−6dB,whereineachrunthespacingbetweenpathsisob-
The MSE at time instant [i+1] is given by
tainedfromadiscreteuniformrandomvariablebetween1and
MSE(γ[i+1])[i+1]=ǫ∗γ[i+1]ǫγ[i+1]=|ǫγ[i+1]|2 2 chips. The channel coefficients hl[i] (l = 0,1,...,Lp−1)
=(1−β)2ǫ∗[i]ǫ [i] areconstantduringeachsymbolintervalandchangeaccording
γ γ
to Clarke’smodelovertime. Since thechannelismodelledas
+β2 e [i]e∗ [i]
k,opt k,opt an FIR filter, we employ a channel estimation filter with 6
+(1−β)β ǫ [i]e∗ [i]
γ k,opt taps as an upper bound for the experiments. Note that the
+(1−β)β ǫ∗[i]e [i]. delays of the channel taps are multiples of the chip rate and
γ k,opt
(46) are random. Their range coincides with the maximum length
of the estimation filter which is 6 taps.
By takingthe expectedvalueon both sides and assuming that
The parameters of the algorithms are optimized and shown
ǫ [i] and e [i] are statistically independent we have
γ k,opt for each example, the system has a power distribution for
E MSE[i+1] =E |ǫ [i+1]|2 the interferers that follows a log-normal distribution with
γ
associated standard deviation of 3 dB and experiments are
=K [i+1]
(cid:2) (cid:3) (cid:2)γ (cid:3)
averagedover100independentruns.Thereceiversaretrained
=(1−β)K [i](1−β)+β2|e [i]|2,
γ k,opt with 200 symbols and then switch to decision-directed mode
(47)
for each data packet. We address the dynamic channel by
where |e [i]|2 is the minimum MSE achieved by the esti- adjusting the receiver weights with the training sequences
k,opt
mator. (with length equal to 200 symbols) and then we exploit the
Wecannoticethatthecross-multiplicationtermswillvanish decision-directed mode to track the channel variations. If the
as a result of the statistical independence between the terms. channeldoes not vary too fast then the adaptive receivers can
Thegeneralrecursiverulein(43)willconvergeasymptotically track it with this scheme, as will be shown in what follows.
provided the step size β is chosen such that
A. Interference Estimation and Tracking
0<β <2. (48)
In order to evaluate the effectiveness of the proposed
Theaboverangeofvalueshastobeadjustedinordertoensure interference estimation and tracking algorithms, that are in-
good convergence and tracking of the parameter dependency corporatedinto the time-varyingerror boundsfortrackingthe
and/or the interference modeled here as the quantity Po[i]. MAI and ISI powers, we carried out an experiment, depicted
in Fig. 2. In this scenario, the proposed algorithm estimates
VI. SIMULATIONS of the MAI and ISI powers are compared to the actual
interferencepowerlevelsthataregeneratedbythesimulations.
In this section we assess the performance of the proposed
The results obtained show that the proposed algorithms are
and existing adaptive algorithms in several scenarios of prac-
very effective for estimating and tracking the interference
tical interest:
power in dynamic environments. Specifically, the estimation
• The NLMS, the SM-NLMS [2], the SM-NLMS of Guo
error does not exceed in average 5% of the estimated power
and Huang [11] and the proposed SM-NLMS with the
level,asdepictedinFig.2.Indeed,the algorithmsarecapable
parameter-dependent (PDB) and parameter and interfer-
of accurately estimating the interference power levels and
ence dependent (PIDB) time-varying bounds.
trackingthem,whichcanbeverifiedbytheproximitybetween
• The AP [1], the SM-AP [3] and the proposed SM-AP
the curves obtained with the estimation algorithms and the
with the PDB and PIDB time-varying bounds.
actual values.
• TheBEACON[6]andtheproposedBEACONalgorithms
with the PDB and PIDB time-varying bounds.
B. SINR Performance
We consider for the sake of simplicity binary phase-shift-
In this part, the performance of the proposed algorithms is
keying (BPSK) modulation, a DS-CDMA system with Gold
assessed in terms of output signal-to-interference-plus-noise
sequences of length N = 31 and typical fading channels
ratio (SINR), which is defined as
foundinmobilecommunicationssystemsthatcanbemodeled
according to Clarke’s model [21]. The channels experienced wH[i]R [i]w[i]
s
SINR[i]= , (49)
by different users are identical since we focus on a downlink wH[i]R [i]w[i]
I
8
N=31, K=8 users, fT=0.0001 N=31, K=10 users, E/N=15 dB, fT=0.00005
d b 0 d
0.14 14
12
0.12
10
er 0.1
w
Po B) 8
nce 0.08 R (d
erfere SIN 6
nt
I0.06
4 NLMS
SM−NLMS−UR=27.3%
0.04 SM−NLMS−PDB−UR=22.3%
2
Interference Power SM−NLMS−PIDB−UR=18.1%
Interference Power Estimate SM−NLMS−Huang−UR=22.2%
0.02 0
0 500 1000 1500 2000 0 500 1000 1500
Number of received symbols Number of received symbols
Fig. 2. Performance of the interference power estimation and tracking at Fig. 3. SINR Performance of NLMSalgorithms at Eb/N0 =15 dB with
Eb/N0=12dBandwithβ=0.05. β=0.05,α=8andτ =2.
N=31, K=10 users, E/N=15 dB, fT=0.00005
whereR [i]istheautocorrelationmatrixofthedesiredsignal b 0 d
s 14
and R [i] is the cross-correlation matrix of the interference
I
and noise in the environment. The goal here is to evaluate 12
the convergenceperformanceofthe proposedalgorithmswith
time-varying error bounds for the modified SM-NLMS, SM- 10
APandBEACONtechniques.Specifically,weconsiderexam-
ples where the adaptive receivers converge to about the same B) 8
d
level of SINR, which illustrates in a fair way the speed of NR (
SI 6
convergenceoftheproposedalgorithmsandtheexistingones.
We also measure the update rate (UR) of all the SM-based
4
algorithms as an important complexity issue.
AP(P=3)
The SINR convergence performance of NLMS, AP and 2 SM−AP(P=3)−UR=24.2%
BEACON algorithms is illustrated via computer experiments SM−AP(P=3)−PDB−UR=20.5%
SM−AP(P=3)−PIDB−UR=15.7%
in Figs. 3, 4 and 5, respectively. The curves in Fig. 3 0
0 500 1000 1500
show that the proposedSM-NLMS algorithms with the PIDB Number of received symbols
time-varying error bounds achieve the fastest convergence,
followed by the proposed SM-NLMS-PDB, the SM-NLMS- Fig. 4. SINR Performance of AP algorithms at Eb/N0 = 15 dB with
Huang [11], the conventional SM-NLMS [2] and the NLMS β=0.05,α=8andτ =2.
[1] recursions. Even though the proposed SM-NLMS-PIDB
andSM-NLMS-PDBalgorithmsenjoythefastestconvergence
rates, they exhibit remarkably lower UR properties, saving
in which they continue to adapt and track the channel varia-
significant computational resources and being substantially
tions.
moreeconomicalthanthe conventionalSM-NLMSalgorithm.
Firstly,weconsiderastudyoftheBERperformanceandthe
By observing the results for the AP and the BEACON
impact on the UR of the fading rate of the channel (f T) in
algorithms, shown in Figs. 4 and 5, one can notice that the d
theexperimentshowninFig.6.Weobservefromthecurvesin
results corroborate those found for the NLMS algorithms.
Fig.6(a)thatthenewBEACONalgorithmsobtainsubstantial
It should be noted that despite their higher complexity than
gainsinBERperformanceovertheoriginalBEACON[6]and
NLMS algorithms, the AP and BEACON techniques have
the RLS algorithm [1] for a wide range of fading rates. In
faster convergence,better SINR steady-state performanceand
addition,asthechannelbecomesmorehostiletheperformance
lower URs.
of the analyzedalgorithmsapproachesone another,indicating
that the adaptive techniques are encountering difficulties in
C. BER Performance
dealingwith the changingenvironmentand interference.With
In this subsection, we focus on the bit error rate (BER) regard to the UR, the curves in Fig. 6 (b) illustrate the
performance of the proposed algorithms. We consider a sim- impact of the fading rate on the UR of the algorithms.
ulation setup where the data packets transmitted have 1500 Indeed, it is again verified that the proposed BEACON-PDB
symbolsand the adaptivereceiversand algorithmsare trained andBEACON-PIDBalgorithmscanobtainsignificantsavings
with 200 symbols and then switch to decision-directed mode, in terms of UR, allowing the mobile receiver to share its
9
14 N=31, K=10 users, Eb/N0=15 dB, fdT=0.00005 10−1 (a) N=31, K=8 users, fdT=0.0001 10−1 (b) N=31, Eb/N0=15 dB, fdT=0.0001
12
10
10−2 10−2
B) 8
NR (d BER BER
SI 6
10−3 10−3
4
RLS AP (P=3)
2 BEACON−UR=9.8% SM−AP (P=3)
BEACON−PDB−UR=7.3% SM−AP (P=3)−PDB
BEACON−PIDB−UR=5.8% SM−AP (P=3)−PIDB
0 10−4 10−4
0 500 1000 1500 4 8 12 16 20 2 4 6 8 10 12 14 16 18 20
Number of received symbols Eb/N0 (dB) Number of users (K)
Fig.5. SINRPerformanceofRLSandBEACONalgorithmsatEb/N0=15 Fig. 7. BER performance versus (a) Eb/N0 dB (b) K for AP algorithms
dBwithβ=0.05,α=5andτ =1.5. withP =3.
(a) N=31, K=16 users, E/N=15 dB (b) N=31, K=16 users, E/N=15 dB
b 0 b 0
10−1 20 more users as compared to the PDB technique for the same
RLS BEACON
BER performance.
BEACON BEACON−PDB
18
BEACON−PDB BEACON−PIDB
BEACON−PIDB
16 VII. CONCLUSIONS
%
R) 14 WeproposedSMadaptivealgorithmsbasedontime-varying
U
ER10−2 ate (12 error bounds. Adaptive algorithms for tracking MAI and ISI
B e R power and taking into account parameter dependency were
at
d10 incorporated into the new time-varying error bounds. Simu-
p
U
lations show that the new algorithms outperform previously
8
reportedtechniquesand exhibita reducednumber of updates.
6 The proposed algorithms can have a significant impact on
the design of low-complexity receivers for spread spectrum
10−130−5 10−4 10−3 10−2 140−5 10−4 10−3 10−2 systems,aswellasforfutureMIMOsystemsemployingeither
fT (cycles/symbol) fT (cycles/symbol) CDMA or OFDM as the multiple access technology. The
d d
proposed algorithms are especially relevant to future wireless
Fig.6. (a)BERperformanceversusfdT and(b)UpdateRate(UR)versus cellular, ad hoc and sensor networks, where their potential
fdT forthe BEACON algorithms fora forgetting factor λ=0.997 forthe to save computational resources may play a significant role
conventional RLS algorithm. The proposed BEACON-PDB abd BEACON-
PIDBalgorithms useβ=0.05,α=5andτ =1.5. giventhe limited batteryresourcesandprocessingcapabilities
of mobile units and sensors.
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