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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 2, FEBRUARY 1993
649
The Gamma FilterA
New
Class
of
Adaptive IIR
Filters with Restricted Feedback
Jose C. Principe,
Senior Member, IEEE,
Bert de Vries,
Member, IEEE,
and Pedro
G.
de Oliveira,
Member, IEEE
AbstractIn this paper we introduce the generalized feedfor
ward filter, a new class of adaptive filters that combines at
tractive properties of finite impulse response (FIR) filters with
some of the power of infinite impulse response (IIR) filters. A
particular case, the gamma filter, generalizes Widrow’s adap
tive transversal filter (adaline) to an infinite impulse response
filter. Yet, the stability condition for the gamma filter is trivial,
and least mean square (LMS) adaptation is of the same com
putational complexity as the conventional transversal filter
structure. Preliminary results indicate that the gamma filter is
more efficient than the adaptive transversal filter. We extend
the WienerHopf equation to the gamma filter and develop some
analysis tools.
I. INTRODUCTION
FINITE impulse response (IIR) filters are more effi
T”.
cient than finite impulse response (FIR) filters, but in
adaptive signal processing, FIR systems are used almost
exclusively
[5],
[
121.
This is largely due to the difficulty
of ensuring stability during adaptation of IIR systems.
Moreover, gradient descent adaptive procedures are not
guaranteed to find global optima in the nonconvex error
surfaces of IIR systems
[
101.
Yet IIR systems have an important advantage over FIR
systems. For a Kth order FIR system, both the region of
support of the impulse response and the number of adap
tive parameters equal
K.
For an IIR system, the length of
the impulse response is uncoupled from the order (and
number of parameters) of the system. Since the length of
the impulse response of a filter is closely related to the
depth of memory of the system, IIR systems are preferred
over FIR systems for modeling of systems and signals
characterized by a deep memory and a small number of
free parameters. These features are typical for lowpass
frequency signals, as is the case for most biological and
other realworld signals.
In this paper we introduce the
generalized feedforward
filter,
an IIR filter with restricted feedback architecture.
The
gamma filter,
a particular instance of the generalized
feedforward filter, is analyzed in detail. The gamma filter
borrows desirable features from both IIR and FIR system:
Manuscript received June 17, 1991; revised January 6, 1992. This work
was supported in part by NSF Grants ECS8915218 and DDM8914084.
The work of P. Guedes de Oliveira at the University of Florida was shp
ported in part by JNICT.
J. C. Principe and
B.
de Vries are with the Computational Neuroengi
neering Laboratory, Department of Electrical Engineering, University of
Florida, Gainesville, FL 3261
1.
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This note was uploaded on 06/05/2011 for the course EEL 6502 taught by Professor Principe during the Spring '08 term at University of Florida.
 Spring '08
 PRINCIPE
 Signal Processing

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