Volume 2007, Article ID 38341, 10 pagesdoi:10.1155/2007/38341 Research Article Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application Jian Yang,
Trang 1Volume 2007, Article ID 38341, 10 pages
doi:10.1155/2007/38341
Research Article
Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application
Jian Yang, Feng Yang, Hong-Sheng Xi, Wei Guo, and Yanmin Sheng
Laboratory of Network Communication System and Control, Department of Automation, University of Science
and Technology of China, Hefei, Anhui 230027, China
Received 1 October 2006; Revised 13 February 2007; Accepted 16 April 2007
Recommended by Nicola Mastronardi
We propose a robust adaptive algorithm for generalized eigendecomposition problems that arise in modern signal processing applications To that extent, the generalized eigendecomposition problem is reinterpreted as an unconstrained nonlinear opti-mization problem Starting from the proposed cost function and making use of an approximation of the Hessian matrix, a robust modified Newton algorithm is derived A rigorous analysis of its convergence properties is presented by using stochastic approxi-mation theory We also apply this theory to solve the signal reception problem of multicarrier DS-CDMA to illustrate its practical application The simulation results show that the proposed algorithm has fast convergence and excellent tracking capability, which are important in a practical time-varying communication environment
Copyright © 2007 Jian Yang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Generalized eigendecomposition has extensive applications
in modern signal processing areas, for example, pattern
recognition [1,2], and signal processing for wireless
com-munications [3,4] Many efficient adaptive algorithms have
been proposed for principal component analysis [5 7],
which is a special case of generalized eigendecomposition
However, developing efficient adaptive algorithms for
gen-eralized eigendecomposition has not been addressed so far
This paper aims to propose a novel adaptive online algorithm
for generalized eigendecomposition
Consider the matrix pencil (Ry, Rx), where Ryand Rxare
M × M Hermitian and positive-definite matrices A scalar λ
andM ×1 vector w that satisfy [8,9]
are called the generalized eigenvalue and corresponding
gen-eralized eigenvector of matrix pencil (Ry, Rx), respectively In
this paper, we are interested in finding the generalized
eigen-vector corresponding to the largest eigenvalue
Many numerical methods have been presented for
the generalized eigendecomposition problem [8] However,
these methods are inefficient in a nonstationary signal
envi-ronment, since they are computationally intensive and
be-long to the class of batch processing methods For practi-cal signal processing applications, an adaptive online algo-rithm is preferred, especially in a nonstationary signal envi-ronment Chatterjee et al have presented an online gener-alized eigendecomposition algorithm for linear discriminant analysis (LDA) [10] However, this algorithm as well as those
in [11,12] are based on the gradient method, and their per-formance is largely determined by the step size, which is di ffi-cult to select in a practical application To overcome these dif-ficulties, Rao et al apply a fixed-point algorithm to solve the generalized eigendecomposition problem [13] The result-ing RLS-like algorithm is proven to be more computation-ally feasible and faster than most of the gradient methods Recently, by using the recursive least-square learning rule, Yang et al develop fast adaptive algorithms for the gener-alized eigendecomposition problem [14] Besides RLS tech-niques, the Newton method is also a well-known powerful technique in the area of optimization By constructing a cost function based on the penalty function method, Mathew and Reddy develop a quasi-Newton adaptive algorithm for es-timating the generalized eigenvector corresponding to the smallest generalized eigenvalue [9] However, this method
suffers from the difficulty of selecting an appropriate penalty factor, which requires its priori information of the covariance matrices, which is unavailable in most applications As a re-sult, this will affect the learning performance In addition, for
Trang 2many applications, the generalized eigenvector
correspond-ing to the largest eigenvalue is desired
In this paper, motivated by the work of Mathew and
Reddy [9], we develop an efficient adaptive modified Newton
algorithm to track the adaptive principal generalized
eigen-vector The basic idea is that we reformulate the
general-ized eigendecomposition problem as minimizing an
uncon-strained nonquadratic cost function that has a unique global
minimum and no other local minima, and then apply an
ap-propriate Hessian matrix approximation to derive an
adap-tive modified Newton algorithm The resulting algorithm is
numerically robust no matter whether it is implemented with
infinite or finite precision We also illustrate its application
by using it to solve an adaptive signal reception problem in a
multicarrier DS-CDMA (MC-DS-CDMA) system [15]
The rest of the paper is organized as follows InSection 2,
we formulate the adaptive signal reception problem in an
MC-DS-CDMA system as the principal generalized
eigenvec-tor estimation problem, to show the importance of the
gener-alized eigendecomposition technique InSection 3, the
gen-eralized eigendecomposition problem is reinterpreted as a
nonlinear optimization problem, and a robust adaptive
mod-ified Newton algorithm is developed to estimate the
princi-pal generalized eigenvector The convergence property of the
proposed algorithm is also discussed InSection 4, we present
numerical simulation results to show the performance of the
proposed algorithm Conclusions are drawn inSection 5
APPLICATION
In this section, we show that it is possible to formulate the
signal reception problem in a multicarrier DS-CDMA system
[16] as a generalized eigendecomposition problem
2.1 Signal model of MC-DS-CDMA system
Consider an MC-DS-CDMA system with K simultaneous
users Each one uses the sameM carriers The kth user, for
1≤ k ≤ K, generates a data sequence:
b(k) = , b(0k),b(1k),b(2k), .
(2) with a symbol interval ofT seconds We assume that the data
symbolsb(j k)are independent variables withE[b(j k)]=0 and
E[|b(k)
j |]=1
Thekth user is provided a randomly generated signature
sequence:
a(k) = , a(0k),a(1k), , a(G k) −1, .
whereG is the spreading gain and the elements a(i k)are
mod-elled as independent and identically distributed (i.i.d.)
ran-dom variables such that Pr(a(i k) = −1)=Pr(a(i k) =1)=1/2.
The sequencea(k)is used to spectrally spread the data
sym-bols to form the signal [15]
a k(t) =
∞
=−∞
b( k) i/G a(i k) ψ
t − iT c
wherexdenotes the largest integer less than or equal tox,
the chip intervalT c is given byT c = T/G, G is the number
of chips per symbol interval, andψ(t) is the common chip
waveform for all signals We assume that the chip waveform
ψ(t) is bandlimited, such as the square-root raised-cosine
pulse [17], and normalized so that∞
−∞ ψ(t)2dt = T c Assume a slowly time-varying frequency-selective Ray-leigh fading channel Following the approach [16], by suit-ably choosingM and the bandwidth of ψ(t), we can assume
that each carrier experiences slowly varying flat fading Then, the received signal in complex form is given by [18]
r(t) = K
k =1
M
m =1
2P k α k,m e jω m t
·
∞
i =−∞
b( k) i/G a(i k) ψ
t − iT c − τ k
+n(t),
(5)
whereω mis the frequency of themth carrier, α k,maccounts for the overall effects of phase shifts and fading for the mth
carrier of thekth user, P kandτ k ∈[0,T) represent the power
for each carrier of the transmitted signal and the delay of the
kth user signal, respectively, and n(t) denotes additive white
Gaussian noise
Without loss of generality, throughout the paper we will consider the signal from the first user as the desired signal and the signals from all other users as interfering signals As-sume that synchronization has been achieved with the trans-mitted signal of the desired user Therefore, the delay of the desired signalτ1 can be taken to be zero In order to avoid interchip interference for the desired signal when it is chip-synchronous, the waveform is chosen to satisfy the Nyquist criterion Then the input signal to the first PN correlator (fin-ger) associated with themth carrier is written as
x m[g] = 1
T c
∞
−∞ r(t)ψ ∗
t − gT c
e − jω m t dt
= 2P1α1, b(1) g/G a(1)
g +
K
k =2
i k,m[g] + n m[g],
(6)
whereg is the chip index, n m[g] denotes the component due
to AWGN, and
i k,m[g] =2P k α k,m
∞
i =−∞
b( k) i/G a(i k) R ψ (g − i)T c − τ k
(7)
is the component due to thekth user signal, 2 ≤ k ≤ K The
functionR ψ(·) is the autocorrelation of the chip waveform defined by
R ψ(t) = 1
T c
∞
−∞ ψ(s)ψ ∗(s − t)ds. (8) The input signal vector can be written as
x[g] = x1[g], x2[g], , x m[g]T
=h1b (1)g/G a(1)
g +
K
k =2
hk
∞
i =−∞
b( k) i/G a(i k)
· R ψ (g − i)T c − τ k
+ n[g],
(9)
Trang 3where hk = [
2P k α k,1, ,
2P k α k,m]T, 1 ≤ k ≤ K, and
n[g] = [n1[g], n2[g], , n M[g]] T is a zero-mean Gaussian
random vector with covarianceσ2I.
Then, the output signal of the first PN correlator to
ex-tract the signal at themth carrier can be written as
y m[n] = √1
G
G−1
l =0
a(1)l+Gn x m[Gn + l] (10) and the output signal vector can be expressed as
y[n] = y1[n], y2[n], , y M[n]T
=h1
Gb(1)
n +
K
k =2
hk √1
G
G−1
l =0
a(1)l+Gn
∞
i =−∞
b( k) i/G
· a(i k) R ψ (l + Gn − i)T c − τ k
+ n1[n],
(11)
where
n1(n) = √1
G
G−1
l =0
a(1)l+Gnn[l + Gn] (12)
is the noise component withE{n1[n]n H
1[n]} = σ2I The re-ceived signal vectors x[g] and y[n] are referred to as
un-despreaded and un-despreaded received signal vectors of the
de-sired user
2.2 MSINR signal reception problem
From (11), the despreaded signal vector can be rewritten as
y[n] =s[n] + u[n], (13)
where s[n] =h1√
Gb[(1)n]denotes the desired signal vector, and
u[n] is the undesired signal vector.
The optimal weight vector under the MSINR
perfor-mance criterion can be found as [15]
wMSINR=arg max
w
wHRsw
where Rs = E{s[n]s H[n]}and Ru = E{u[n]u H[n]}are the
covariance matrices of the desired and undesired signals,
re-spectively It is obvious that the optimal weight vector wMSINR
is the generalized eigenvector corresponding to the
maxi-mum generalized eigenvalue of the matrix pencil (Rs, Ru),
that is,
RswMSINR= λmaxRuwMSINR, (15)
whereλmaxis the maximum generalized eigenvalue
Unfortu-nately, because s[n] and u[n] cannot be separately obtained
from the received signal y[n], it seems difficult to obtain
wMSINRfrom (14) In the following, we will propose an
im-proved criterion equivalent to MSINR to overcome the above
difficulty
According to (9) and (11), after some calculations, the
autocorrelation matrices Rx = E{x[g]x H[g]} and Ry = E{y[n]y H[n]}are given by, respectively,
Rx =h1hH1 +
K
k =2
hkhH k
∞
i =−∞
R ψ
iT c − τ k2
+σ2I,
Ry = Gh1hH1 +
K
k =2
hkhH k
∞
i =−∞
R ψ
iT c − τ k2
+σ2I.
(16) Hence, we have
Rx = 1
GRs+ Ru,
Ry =Rs+ Ru
(17) Let us consider the following function:
f (w) =wHRyw
wHRxw = G − G −1
γ/G + 1, (18)
where
γ = wHRsw
for any w except for wHRuw=0 If Ruis full rank, this
func-tion is valid for any w 0 According to (18), we can see that
ifG > 1, the weight vector w that maximizes f (w)
eventu-ally maximizesγ Therefore, the optimal weight vector can
be found as
w
wHRyw
Hereby, estimating the MSINR weight vector from (20) in-stead of (14), we do not need to know or estimate the
co-variance matrices of s[n] and u[n], which are basically not
available at the receiving end Obviously, this is the problem
of estimating the principal generalized eigenvector from two
observed sample sequences y[n] and x[g].
ALGORITHM FOR GENERALIZED EIGENDECOMPOSITION
To solve a class of signal processing problems similar to that
inSection 2, we construct a novel unconstrained cost func-tion Then, starting from this cost function, a robust mod-ified Newton algorithm is derived Its convergence is rigor-ously analyzed by using stochastic approximation theory
3.1 Generalized eigendecomposition problem reinterpretation
Letλ iand ui(1≤ i ≤ M) be the generalized eigenvalue and
the corresponding Rx-orthonormalized generalized
eigen-vector of the matrix pencil (Ry, Rx), that is, [9]
Ryui = λ iRxui,
uH i Rxuj = δ i j, (21) whereδ i jis the Kronecker delta function
Trang 4Consider the following nonlinear scalar cost function:
J(w) =wHRxw−ln
wHRyw
As will be shown next, this is a novel criterion for the
gen-eralized eigendecomposition problem In the following
the-orem, we assume that the maximum generalized eigenvalue
of (Ry, Rx) has multiplicity 1 The case when the multiplicity
of the maximum generalized eigenvalue is larger than 1 will
be discussed later
Theorem 1 Let λ1 > λ2 ≥ · · · ≥ λ M > 0 be the generalized
eigenvalues of the matrix pencil (R y, Rx ) Then w =u1is the
unique global minimal point of J(w) and the others are saddle
points of J(w).
Proof SeeAppendix A
Theorem 1 shows that if the maximum generalized
ei-genvalue has multiplicity 1,J(w) has a global minimum and
no other local minima, and global convergence is guaranteed
when one seeks the Rx-orthonormalized generalized
vector corresponding to the maximum generalized
eigen-value of (Ry, Rx) by iterative methods When the
multiplic-ity of the maximum generalized eigenvalue is more than 1,
there are some local minima Hence, the iterative algorithm
will converge to one of these local minima Nevertheless, it
is not a hindrance for one to seek the principal generalized
eigenvector, because these local minima themselves are the
Rx-orthonormalized generalized eigenvectors corresponding
to the maximum generalized eigenvalue Therefore, the
prin-cipal generalized eigenvector estimation problem can be
re-formulated as the following unconstrained nonlinear
opti-mization problem:
min
3.2 Adaptive modified Newton algorithm derivation
The Hessian matrix ofJ(w) with respect to w is derived in
Appendix Aas
H=Rx −Ry
wHRyw−1
+
wHRyw−2
RywwHRy (24)
In order to simplify the Hessian matrix, we drop the second
term on the right-hand side of (24) Therefore, an
approxi-mation to the Hessian matrix can be written as:
H=Rx+
wHRyw−2
RywwHRy (25) The inverse Hessian matrix is given by
H−1=R−1− R−
wHRyw2
+ wHRyR−1Ryw. (26) Then the modified Newton algorithm for updating the
weight vector w[n + 1] can be written as
w[n + 1] =w[n] − H−1∇J(w)
w=w[n]
= 2R−
wHRyw2
+ wHRyR−1Ryw
w=w[n] (27)
Remark 2 In the derivation of the updating rule (27), we
approximate the Hessian matrix H by dropping a term so
as to make the Hessian matrixH positive definite, and con- sequently make the resultant algorithm more robust, since for stabilizing the Newton-type algorithms it is necessary to guarantee that the Hessian matrix is positive definite Al-though the approximation causes the resultant Hessian ma-trix to deviate from the true Hessian mama-trix, as shown in
Section 4, the derived algorithm (27) can asymptotically con-verge to the principal generalized eigenvector of the matrix
pencil (Ry, Rx) In addition, the numerical simulation results show that the approximation has little influence on conver-gence speed and estimation accuracy Therefore, the approx-imation is a reasonable step in developing the adaptive mod-ified Newton algorithm
We apply the following equations to recursively estimate
Rxand Ry:
Rx[n + 1] = μR x[n] + x[n + 1]x H[n + 1], (28)
Ry[n + 1] = βR y[n] + (1 − β)y[n + 1]y H[n + 1], (29) where 0< μ, β < 1 are the forgetting factors.
Let P[n + 1] =R−1[n + 1] Then we get
P[n + 1] =1
μP[n]
I− x[n + 1]x H[n + 1]P[n]
μ + x H[n + 1]P[n]x[n + 1]
.
(30) Postmultiplying both sides of (29) with w[n], we have
Ry[n + 1]w[n] = βR y[n]w[n]
+ (1− β)y[n + 1]y H[n + 1]w[n]. (31)
Applying the projection approximation [5] yields
r[n + 1] =Ry[n + 1]w[n + 1] ≈Ry[n + 1]w[n]. (32) Then (31) can be rewritten as
r[n + 1] = βr[n] + (1 − β)y[n + 1]c ∗[n + 1], (33) wherec[n + 1] =wH[n]y[n + 1] In addition, we define d[n +
1]=wH[n]R y[n + 1]w[n] Then according to (29) we obtain
d[n + 1] = βd[n] + (1 − β)c ∗[n + 1]c[n + 1]. (34) Let
w[n + 1] = Ry[n + 1]w[n]
wH[n]R y[n + 1]w[n] (35)
so that the update rule of w[n + 1] can be rewritten as
w[n + 1] = 2P[n + 1]w[ n + 1]
1 +wH[n + 1]P[n + 1]w[ n + 1] . (36)
Trang 5Thus, the adaptive modified Newton algorithm can be
sum-marized as
P[n + 1] =1
μP[n]
I− x[n + 1]x H[n + 1]P[n]
μ + x H[n + 1]P[n]x[n + 1]
,
c[n + 1] =wH[n]y[n + 1],
r[n + 1] = βr[n] + (1 − β)y[n + 1]c ∗[n + 1],
d[n + 1] = βd[n] + (1 − β)c[n + 1]c ∗[n + 1],
w[n + 1] = r[n + 1]
d[n + 1],
w[n + 1] = 2P[n + 1]w[ n + 1]
1 +wH[n + 1]P[n + 1]w[ n + 1] .
(37)
The simplest way to choose the initial values is to set P[0]=
η i(i =1, 2, 3) are appropriate positive values During
deriv-ing the algorithm (37), we have adopted the projection
ap-proximation approach [5] The rationality of using
projec-tion approximaprojec-tion has been concretely explained in [5] In
this paper, the numerical results show that using the
projec-tion approximaprojec-tion has little impact on the performance of
the proposed algorithm
Note that the update step for P[n] involves subtraction.
Hence, the numerical error may cause P[n] to lose the
Her-mitian positive definiteness, while P[n] is theoretically
Her-mitian positive definite An efficient and robust way is to
ap-ply the QR-update method to calculate the square root
matri-ces P1/2[n] [19] Because P[n] =P1/2[n]P H/2[n], the
Hermi-tian positive definiteness remains regardless of any numerical
error
3.3 Convergence analysis
In this section, we apply the stochastic approximation
method, which is developed by Ljung [20], and Kushner and
Clark [21], to analyze the convergence property of the
pro-posed algorithm based on updating rule (27) According to
the stochastic approximation theory, a deterministic
ordi-nary differential equation (ODE) can be associated with the
recursive stochastic approximation algorithm, and the
con-vergence of the algorithm can be studied in terms of this
dif-ferential equation
The ordinary differential equation corresponding to the
proposed algorithm based on updating rule (27) can be
writ-ten as
dw(t)
wH(t)R yw(t)2
+ wH(t)R yR−1Ryw(t) −w(t).
(38)
We have the following theorem to demonstrate the
conver-gence of w(t).
Theorem 3 Given the matrix pencil (R y, Rx ), whose largest
generalized eigenvalue λ has multiplicity 1, and assuming that
uH1Rxw(0) 0, then the ODE (38) has a global
asymptoti-cally stable equilibrium state at (λ1,γu1), where γ is a constant complex number with norm γ = 1.
Proof SeeAppendix B Note that ifγ =1,γu1is also the Rx-orthornormalized generalized eigenvector corresponding to the maximum
gen-eralized eigenvalue of (Ry, Rx).Theorem 3also shows that al-though we approximate the Hessian matrix when deriving the updating rule (27), the resultant algorithm can asymp-totically converge to the principal generalized eigenvector
In this section, we apply the proposed algorithm to the signal reception problem in multicarrier DS-CDMA, and perform numerical simulation to investigate its performance For each run, the proposed algorithm in this paper, the direct eigen-decomposition method, the TTJ algorithm [15], and sam-ple matrix/iterative (SMIT) [12] are implemented simultane-ously in the simulations The data in each plot is the average over 100 independent runs
We consider aK-user asynchronous MC-DS-CDMA
sys-tem ofM =12 carriers with processing gainG =32 The sys-tem uses a square-root raised-cosine chip pulse with roll-off factor of 0.8 [17] It is customary to truncateψ(t) such that it
spans only several chips [18], and we assume that the dura-tion of the pulse is 4T c Throughout this section, the signal-to-noise ratio (SNR) of the desired user is fixed at 20 dB
To evaluate the convergence speed and the estimate ac-curacy, the direction cosine and the normalized projection error (NPE) [22] are defined, respectively, as
direction cosine= wH(k)wMSINR
w(k)wMSINR,
w(k)wMSINR2,
(39)
where wMSINRis the theoretically optimal combining weight vector and can be computed by [23]
We use the MSINR performance to assess the MAI sup-pression capability of the proposed algorithm The expres-sion for calculating the SINR at thenth iteration is given by
SINR(n) =10 logw
H[n]R sw[n]
wH[n]R uw[n] . (41)
The proposed algorithm starts with initial values r[0]=
w[0]=[1 0 · · · 0]T,d[0] =1, P[0]=0.01I, μ =0.995,
andβ =0.8 For the direct eigendecomposition method, we
use the same method as (28) and (29) to estimate the Rx
and Ryat thenth iteration The initial values R x[0]=0.1I,
Ry[0] =0.1I, and a forgetting factor of 0.9 are set We also
start the TTJ algorithm with w[0]=[1 0 · · · 0]T But its step size should be regulated according to different simula-tion environments
Trang 60 100 200 300 400 500
Number of symbol intervals 2
4
6
8
10
12
14
16
18
20
Maximum Eigen method Proposed algorithm
TTJ algorithm SMIT (a)
Number of symbol intervals 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Eigen method
Proposed algorithm
TTJ algorithm SMIT (b)
Number of symbol intervals
0.4
0.5
0.6
0.7
0.8
0.9
1
Eigen method
Proposed algorithm
TTJ algorithm SMIT (c)
Figure 1: (a) SINR performance in the case of two interferers (b)
Normalized projection error in the case of two interferers (c)
Di-rection cosine performance in the case of two interferers
In the first simulation experiment, we consider the case
when there are two interferers whose received powers are
10 dB stronger than the desired user.Figure 1shows the
sim-ulation results It can be observed that the eigenmethod and
the proposed algorithm outperform the TTJ algorithm The
reason is that the TTJ algorithm belongs to the stochastic
Number of symbol intervals
−25
−20
−15
−10
−5 0 5 10 15 20
Maximum Eigen method Proposed algorithm
TTJ algorithm SMIT (a)
0 100 200 300 400 500 Number of symbol intervals 0
0.2
0.4
0.6
0.8
1
Eigen method Proposed algorithm
TTJ algorithm SMIT (b)
0 100 200 300 400 500 Number of symbol intervals 0
0.2
0.4
0.6
0.8
1
Eigen method Proposed algorithm
TTJ algorithm SMIT (c)
Figure 2: (a) SINR performance in the case of five interferers (b) Normalized projection error in the case of five interferers (c) Di-rection cosine performance in the case of five interferers
dient algorithm class and its fixed step size is chosen based on some tradeoff between tracking capability and accuracy; too small a value will bring on slow convergence and too large
a value will lead to overshoot and instability [19] The eigen method and SMIT have the best performance However, their computational complexity is very high Compared to these
Trang 70 200 400 600 800 Number of symbol intervals
−30
−20
−10
0
10
20
Maximum Eigen method Proposed algorithm
TTJ algorithm SMIT (a)
Number of symbol intervals 0
0.2
0.4
0.6
0.8
1
Eigen method Proposed algorithm
TTJ algorithm SMIT (b)
Number of symbol intervals 0
0.2
0.4
0.6
0.8
1
Eigen method Proposed algorithm
TTJ algorithm SMIT (c)
Figure 3: (a) SINR performance in the dynamical signal
environ-ment (b) Normalized projection error in the dynamical signal
envi-ronment (c) Direction cosine performance in the dynamical signal
environment
methods, the complexity of the proposed algorithm has been
greatly reduced, while its performance degrades only slightly
The simulation results also show that the approximation of
the Hessian matrix and the projection approximation have
little influence on the performance of the proposed
algo-rithm, since its performance approaches that of the eigen method, which uses neither of these approximation tech-niques
In the next simulation experiment, we investigate the per-formance of the proposed algorithm in a signal environment with strong interference We assume that there are two 10 dB, two 20 dB, and one 30 dB interferers The simulation results
inFigure 2show that the performance of the eigen method and the proposed algorithm hardly changes, whereas the per-formance of the TTJ algorithm degrades rapidly This is not surprising because at each step the TTJ algorithm uses a sin-gle instantaneous sample to update the weight vector, and
as a result, the estimated weight vector oscillates around the MSINR combining weight vector As the number and pow-ers of the interferpow-ers increase, the oscillation becomes more dramatic and the amplitude increases Consequently, the av-eraged performance degrades greatly in this scenario In con-trast, the proposed algorithm uses all of the data samples available up to the time instantn + 1 to estimate the
opti-mal weight vector, and as a result, it performs well in a sig-nal environment with strong interference This experiment also shows that in the case with strong interferers, using the Hessian matrix approximation and the projection approxi-mation has only a slight impact on the performance of the proposed algorithm
In the final experiment, we study the tracking capabil-ity of the proposed algorithm in a dynamic environment At the beginning, there are two 10 dB interferers, and at symbol interval 400, three 20 dB, one 30 dB, and one 40 dB interfer-ers are added.Figure 3shows the simulation results Because there are few interferers and their powers are not very strong
in the first phase, the TTJ algorithm performs very well But
in the second phase, too much interference and unregulated fixed step size cause the performance to degrade greatly It can be observed that the eigen method, SMIT, and the pro-posed algorithm can rapidly adapt to the suddenly changed signal environment This is because of using the forgetting factor in the recursive covariance matrix estimator The sim-ulation results also show that in time-varying environment the influence of the Hessian matrix approximation and the projection approximation is small
Therefore, from the above simulation results in various signal environments, we conclude that the proposed algo-rithm has rapid convergence, sufficient estimation accuracy, and good tracking capability These properties make it very useful in a practical signal environment, especially when the interfering power increases due to many practical reasons, such as too many interferers, incorrect power control, time-varying channel
In this paper, we have studied the principal generalized eigenvector estimation problem We proposed a new uncon-strained cost function for the generalized eigendecomposi-tion problem Then, based on the proposed cost funceigendecomposi-tion,
we have derived a robust adaptive modified Newton algo-rithm The convergence of the proposed algorithm has been
Trang 8rigorously analyzed In addition, we applied the proposed
al-gorithm to the adaptive signal reception problem in
multi-carrier DS-CDMA systems, and the numerical simulation
re-sults show that the proposed algorithm has fast convergence
and excellent tracking capability, which are very useful for a
practical communication environment
APPENDICES
A PROOF OF THEOREM 1
Proof Let ∇Rand∇Ibe the gradient operators with respect
to the real and imaginary parts of w According to [19], the
complex gradient operator is defined as ∇ = (1/2)[∇ R +
j∇ I] After some calculation, we can derive the gradient of
J(w) as
∇J(w) =Rxw−Ryw
wHRyw−1
When w=ui, it is easy to show that∇J(u i)=0 This implies
that any Rx-orthonormalized generalized eigenvector, ui, of
(Ry, Rx) is the stationary point ofJ(w).
Conversely,∇J(w) =0 means
Ryw=wHRyw
Hence, w is the generalized eigenvector of (Ry, Rx), and the
corresponding generalized eigenvalue is (wHRyw)
Premulti-plying the both sides of (A.2) with wHwe have
wHRyw=wHRyw
wHRxw
Since Ryis positive definite, wHRyw > 0 for w 0
There-fore, we get wHRxw=1 This shows that stationary point, w,
ofJ(w) is the R x-orthonormalized generalized eigenvector of
(Ry, Rx)
From above analysis, we conclude that w is a stationary
point ofJ(w) if and only if w is the R x-orhtonormalized
gen-eralized eigenvector of (Ry, Rx)
Let H= ∇∇ H J(w) be the M × M Hessian matrix [7] of
J(w) with respect to the vector w After some calculations,
the Hessian matrix H is given as
H=Rx −Ry
wHRyw−1
+
wHRyw−2
RywwHRy
(A.4)
Since Rx is positive definite, we have Rx = VVH, where
V is an invertibleM × M matrix Let e i = VHui and C =
V−1Ry(V−1)H According to (21) we obtain
Cei = λ iei,
Obviously,λ iand eiare the eigenvalue and the corresponding
eigenvector of C.
Let e=VHw Then we get
H=V
eHCe+
CeeHC
eHCe2
VH =VF(e)VH, (A.6)
where
F(e)=I− C
eHCe+
CeeHC
eHCe2. (A.7)
From the fact that eH
1Ce1 = λ1and Ce1eH
have
F
±e1
=I− C
λ1
+ e1eH
F
±e1
e1=e1,
F
±e1
ei =
1− λ i
λ1
ei,
(A.8)
wherei =2, , M Since (1 − λ i /λ1)> 0, all the eigenvalues
of F(e1) are positive We can conclude that F(e) is positive definite at the point e= ±e1 Similarly, we can derive
F
ei
e1=
1− λ1
λ i
e1,
F
ei
ei =ei,
(A.9)
wherei =2, , M Because (1 − λ1/λ i)< 0, F(e i) is neither positive definite nor negative definite According to (A.6), we have
H|w =±ui =VF
ei
It is clear that H is positive definite at the stationary point
w =u1 At any other stationary point ui (i =2, , M), H
is neither positive definite nor negative definite This means
that w=u1is the unique global minimal point ofJ(w), and
the other stationary points ui(i =2, , M) are saddle points
ofJ(w).
B PROOF OF THEOREM 3
Proof The vector w(t) can be expressed as a linear
combi-nation ofM generalized eigenvectors u iof (Ry, Rx), which is given by
w(t) = M
i =1
whereα i(t) are complex coefficients
Substituting (B.1) into (38) and premultipying by uH l Rx
yield
dα l(t)
dt =
M
i =1
λ iα i(t)2
2 +
M
i =1
λ2iα i(t)2
−1
·
2λ l α l(t)
M
i =1
λ iα i(t)2
− α l(t).
(B.2)
Under the assumption uH1Rxw(0) 0 we can defineθ l =
α l(t)/α1(t), l =2, , M Then we have
dθ l
dt =
α1(t) dα l(t)
dt − α l(t) dα1(t)
dt
α −2(t). (B.3)
Trang 9Substituting (B.2) into (B.3) yields
dθ l
dt = −λ1− λ l
κ(t)θ l(t), (B.4) where
κ(t) =
2
M
i =1
λ iα i(t)2
×
M
i =1
λ iα i(t)2
2 +
M
i =1
λ2
iα i(t)2
−1
.
(B.5)
Since κ(t) > 0 for all t > 0, lim t →∞ θ l = 0,l = 2, , M.
It follows that limt →∞ α l(t) = 0,l = 2, , M, and w(t) =
α1(t)u1is an asymptotically stable solution of (38)
Therefore, whent is large enough and l =1, (B.2) can be
simplified as
dα1(t)
dt = α1(t) 1−α1(t)2
1 +α1(t)2 . (B.6)
In order to show that limt →∞ α1(t) = 1 we definez(t) =
α1(t)2andV [z(t)] = [z(t) −1]2 Their time derivatives
are
˙z(t) = α ∗1(t) ˙α1(t) + ˙α ∗1(t)α1(t)
=2α1(t)21−α1(t)2
1 +α1(t)2,
˙
V z(t)
=2 z(t) −1
˙z(t)
= −4 1−α1(t)22α1(t)2
1 +α1(t)2 .
(B.7)
According to the theory of Lyapunov stability,V (z) is a
Lya-punov function, andz = 1 is asymptotically stable
More-over, from (B.6) and limt →∞ α1(t) = 1, we can conclude
limt →∞ α1(t) = γ, where γ = 1 Hence, w(t) in (38) will
asymptotically converge to the stable solutionγu1
ACKNOWLEDGMENT
The authors would like to express their sincerest appreciation
to the anonymous reviewers for their comments and
sugges-tions that significantly improve the quality of this work
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Trang 10Jian Yang received the B.S., M.S., and Ph.D.
degrees from the University of Science and
Technology of China (USTC), Hefei, China,
in 2001, 2003, and 2005, respectively He
is currently with the Laboratory of
Net-work Communication System and Control
in USTC His research area is
multime-dia communication and signal processing,
including adaptive streaming media
sys-tem design and performance optimization,
adaptive load balance, adaptive filtering, antenna array signal
pro-cessing, and frequency estimation
Feng Yang received the B.S degree in
elec-trical engineering from Tongji University,
Shanghai, China, in 2001, and the M.S
de-gree from USTC, Hefei, China, in 2003 He
is currently pursuing the Ph.D degree His
current research interests include adaptive
filtering theory, MC-CDMA systems, and
MIMO systems
Hong-Sheng Xi received the B.S and M.S.
degrees in applied mathematics from the
University of Science and Technology of
China (USTC), Hefei, China, in 1980 and
1985, respectively He is currently the Dean
of the Department of Automation at USTC
He also directs the Laboratory of Network
Communication System and Control His
research interests include stochastic
con-trol systems, discrete-event dynamic
sys-tems, network performance analysis and optimization, and wireless
communications
Wei Guo received his B.S degree and Ph.D.
degree in China University of Science and
Technology and Chinese Academy of
Sci-ences in 1983 and 1992, respectively He
worked in Communication Research
Lab-oratory, Japan, and Hong Kong
Univer-sity of Science and Technology, in
1994-1995 and 1998, respectively Professor Wei is
the Member of the Communication Expert
Group, State High Technology Project (863
Project), and the core Member of the Technical Group, China 3G
Mobile Communication System Project His current research
inter-ests are the concept and key technology for the 4G Mobile
Commu-nication system
Yanmin Sheng received the B.S degree in
automation from University of Science and
Technology of China, Hefei, China, in 2002,
the Ph.D degree in control science and
en-gineering from University of Science and
Technology of China, Hefei, China, in 2007
He has worked in areas of wireless
commu-nication, adaptive theory, and application,
and statistical theory His current research
interests include particle filter application in
communication, OFDM, and MIMO
... robust adaptive modified Newton algo-rithm The convergence of the proposed algorithm has been Trang 8rigorously... (36)
Trang 5Thus, the adaptive modified Newton algorithm can be
sum-marized as... DS-CDMA, and perform numerical simulation to investigate its performance For each run, the proposed algorithm in this paper, the direct eigen-decomposition method, the TTJ algorithm [15], and sam-ple