Stochastic Analysis of the LMS and NLMS Algorithms for Cyclostationary White Gaussian Inputs

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2238 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 9, MAY 1, 2014
Stochastic Analysis of the LMS and NLMS
Algorithms for Cyclostationary White
Gaussian Inputs
Neil J. Bershad, Fellow, IEEE, Eweda Eweda, Fellow, IEEE, and José C. M. Bermudez, Senior Member, IEEE
Abstract—This paper studies the stochastic behavior of the
LMS and NLMS algorithms for a system identification framework
when the input signal is a cyclostationary white Gaussian process.
The input cyclostationary signal is modeled by a white Gaussian
random process with periodically time-varying power. Mathemat-
ical models are derived for the mean and mean-square-deviation
(MSD) behavior of the adaptive weights with the input cyclosta-
tionarity. These models are also applied to the non-stationary
system with a random walk variation of the optimal weights.
Monte Carlo simulations of the two algorithms provide strong
support for the theory. Finally, the performance of the two algo-
rithms is compared for a variety of scenarios.
Index Terms—Adaptive filters, analysis, LMS algorithm, NLMS
algorithm, stochastic algorithms.
I. INTRODUCTION
AN important aspect of adaptive filter performance is the
ability to track time variations of the underlying signal
statistics [1], [2]. The standard analytical model assumes the
input signal is stationary (see e.g., [3]–[9]).
However, a non-stationary signal model can be provided by
a random walk model for the optimum weights. The form of
the mean-square error performance surface remains unaltered
while the surface moves in the weight space over time. This
model provides the conditions for the adaptive algorithm to
track the optimum solution [1]. Alternatively, the input signal
can be modeled as a cyclostationary process in many practical
applications [10]–[12]. In these cases, the form of the perfor-
mance surface is periodic with the same period as the input
autocorrelation matrix [13]. This performance surface defor-
mation affects the adaptive filter convergence and is indepen-
dent of changes in the optimum weights. This transient perfor-
mance surface deformation can be modeled by standard ana-
lytical models. However, it is still desirable to understand the
Manuscript received June 17, 2013; revised December 06, 2013; accepted
February 06, 2014. Date of publication February 20, 2014; date of current ver-
sion April 04, 2014. The associate editor coordinating the review of this manu-
script and approving it for publication was Dr. Suleyman S. Kozat. This work
has been partially supported by CNPq under grant No. 307071/2013-8 and by
Capes.
N. J. Bershad is with the Department of Electrical Engineering and Com-
puter Science, University of California, Irvine, Newport Beach, CA 92660 USA
(e-mail: bershad@ece.uci.edu).
E. Eweda is with the National Knowledge Center, Abu Dhabi, United Arab
Emirates (e-mail: eweda@ieee.org).
J. C. M. Bermudez is with the Department of Electrical Engineering, Fed-
eral University of Santa Catarina, Florianópolis, SC 88040-900, Brazil (e-mail:
j.bermudez@ieee.org).
Digital Object Identifier 10.1109/TSP.2014.2307278
adaptive performance with non-stationary inputs. This type of
analysis is essentially absent from the technical literature [1],
[2]. A first analysis of the LMS behavior for cyclostationary in-
puts [14] studied only its convergence in the mean. The spe-
cial case of a pulsed variation of the input power and a linear
combiner structure has recently been studied for both LMS and
NLMS algorithms [15], [16]. An analysis of the Least Mean
Fourth (LMF) algorithm behavior for nonstationary inputs has
been recently presented in [17]. The analytical model derived
for the LMF behavior in [17] was valid only for a specific form
of the input autocorrelation matrix, and cannot be easily ex-
tended to a general time-varying input statistics. Also, as the
LMF weight update equation is a function of a higher power of
the estimation error, the statistical assumptions used in [17] are
necessarily different from those required for the analysis of the
LMS and NLMS algorithms. Hence, the study of the behaviors
of the LMS and NLMS algorithms under cyclostationary inputs
cannot be inferred from the analysis in [17] and new models
must be derived.
Adaptive solutions involving cyclostationary signals have
been sought for many application areas [18]. In particular,
communication, radar, and sonar systems frequently need
such solutions, as several man-made signals encountered in
these areas have parameters that vary periodically with time
[19]–[21]. [21] studied adaptive beamforming algorithms for
applications where input signals are cyclostationary. [22]
proposed an adaptive minimum variance equalizer that ex-
ploits the cyclostationarity properties of the intersymbol and
adjacent-channel interferences. [20] proposed a gradient-based
adaptive beamforming algorithm that exploits the cyclostation-
arity of the input signal. [23] employed adaptive filtering to
extract cyclostationary interference from speech signals. The
reader is directed to [18] for a survey on the importance of
cyclostationary signals in several areas, including communica-
tions, channel identification and equalization. Thus, a statistical
analysis of adaptive algorithms under cyclostationary inputs
could have a significant impact on a wide variety of problems
involving cyclostationary processes.
The analysis of the adaptive filter behavior for cyclosta-
tionary inputs is not easy because of the difficulty of modeling
the input cyclostationarity in a mathematically treatable way1.
Thus, relatively simple models are needed from which to infer
algorithm behavior for inputs with time-varying statistics.
1A similar difficulty exists for the time-varying weight model. The first order
Markov weight variation model is simplified to a vector random walk model
with independent components [24].
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