基于物理知识的Volterra级数功放行为模型简化方法
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 5, MAY 2007 813
Pruning the Volterra Series for Behavioral Modeling
of Power Amplifiers Using Physical Knowledge
Anding Zhu, Member, IEEE, José Carlos Pedro, Fellow, IEEE, and Telmo Reis Cunha, Member, IEEE
Abstract—This paper presents an efficient and effective ap-
proach to pruning the Volterra series for behavioral modeling of
RF and microwave power amplifiers. Rather than adopting a pure
“black-box” approach, this model pruning technique is derived
from a physically meaningful block model, which has a clear
linkage to the underlying physical behavior of the device. This
allows all essential physical properties of the PA to be retained, but
significantly reduces model complexity by removing unnecessary
coefficients from the general Volterra series. A reduced-order
model of this kind can be easily extracted from standard time/fre-
quency-domain measurements or simulations, and may be simply
implemented in system-level simulators. A complete physical
analysis and a systematic derivation are presented, together with
both computer simulations and experimental validations.
Index Terms—Behavioral model, power amplifiers (PAs),
Volterra series.
I. INTRODUCTION
B
EHAVIORAL modeling for RF and microwave power
amplifiers (PAs) has received much attention from many
researchers in recent years. It provides a convenient and ef-
ficient way to predict system-level performance without the
computational complexity of full simulation or the physical
analysis of nonlinear circuits, thereby significantly speeding up
system design and verification process. As wireless communi-
cation is evolving towards broadband services, we increasingly
encounter frequency-dependent behavior, i.e., memory effects,
in RF PAs. To accurately model a PA, we have to take into
account both nonlinearities and memory effects.
The Volterra series is a multidimensional combination of a
linear convolution and a nonlinear power series [1]. It provides
a general way to model a nonlinear dynamic system so that it
can be employed to characterize a nonlinear PA with memory
effects. However, since all nonlinearities and memory effects
Manuscript received August 9, 2006; revised December 22, 2006. This work
was supported by the Science Foundation Ireland under the Principal Investi-
gator Award. This work was supported in part by the Network of Excellence
TARGET under the Sixth Framework Program funded by the European Com-
mission, and in part by the Portuguese Science Foundation under the ModEx
Project.
A. Zhu is with the School of Electrical, Electronic and Mechanical Engi-
neering, University College Dublin, Dublin 4, Ireland (e-mail: anding.zhu@ucd.
ie).
J. C. Pedro and T. R. Cunha are with the Institute of Telecommunications,
University of Aveiro, 3810-193 Aveiro, Portugal (e-mail: jcpedro@det.ua.pt;
trcunha@det.ua.pt).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMTT.2007.895155
are treated equally in the classical Volterra model, the number
of coefficients to be estimated increases exponentially with
the degree of nonlinearity and memory length of the system.
Therefore, it has been very difficult to find a practically con-
venient procedure for extracting full Volterra kernels of order
greater than five, which restricts the practical use of the general
Volterra model to the characterization of relatively weakly
nonlinear PAs.
To overcome the modeling complexity, various model-order
reduction approaches have been proposed to simplify the
Volterra model structure. For example, in the Wiener- or
Hammerstein-like models [2]–[4], memory effects are rep-
resented by linear filters, while nonlinearity is characterized
by static/memoryless polynomials in a cascade arrangement.
However, in a Wiener system, the
th-order Volterra kernel
must be proportional to the
-folded product of their linear
elements; while a Hammerstein model requires that the Volterra
kernels are only nonzero along their diagonals and each kernel
diagonal is proportional to the impulse response of the linear
subsystem. All off-diagonal coefficients are set to zero in a
memory polynomial model [5], while near-diagonality reduc-
tion-based models [6] only keep the coefficients on and near the
main diagonal lines. Polyspectral models [7] are again based on
filter/static-nonlinearity cascades, where the multidimensional
nonlinear filters are approximated by 1-D versions. In the mod-
ified/dynamic Volterra series [8]–[11], high-order dynamics are
normally omitted since they are considered to have little effect
on the output of a PA. Orthonormal basis functions, like the
Laguerre [12] and Kautz [13] functions, were employed as the
basis for the Volterra expansion to efficiently model long-term
memory effects. However, it was found difficult to locate the
pre-decided poles.
Although these simplified models have been employed
to characterize PAs with reasonable accuracy under certain
conditions, there is no systematic way to verify if the model
structure chosen is truly appropriate to the PA under study.
Indeed, because behavioral models developed to date have been
mainly based on a pure “black-box” approach, or were mostly
constructed from “blind” nonlinear system identification algo-
rithms (where the amplifier was considered to be a complete,
or very general nonlinear system), we cannot guarantee that the
relevant conditions are satisfied when doing a specific model
truncation. In particular, little or no PA physical knowledge
was taken into account during the model development or
model-order truncation.
In this paper, we seek to construct a behavioral model for RF
PAs from a physical, rather than a pure “black-box” perspective,
so that we may have a clear idea on how to select a proper model
0018-9480/$25.00 © 2007 IEEE
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