Communications and Network, 2013, 5, 570-572
http://dx.doi.org/10.4236/cn.2013.53B2102 Published Online September 2013 (http://www.scirp.org/journal/cn)
Pruned Volterra Models with M emory Effects for
Nonlinear Power Amplifiers
Pengpeng Li1, Qingfang Zhang1, Ping Wang1, Zhongshan Xie2, Bing Liu2
1Shanghai Institute of Microsystem and Information Technology, Shanghai, China
2Nanjing University of Aeronautics and Astronautics, Nanjing, China
Received May 2013
ABSTRACT
In this letter, a novel model is proposed for modeling the nonlinearity and memory effects of power amplifiers. The
classical Volterra model is modified through a function of the sum of nonlinearity order with sum of memory length.
The parameters of this model can be extracted in digital domain since the model is analyzed based on the envelope sig-
nals. The model we proposed enables a substantial reduction in the number of coefficients involved, and with excellent
accuracy.
Keywords: Volterra Series; Power Amplifier (PA); Behavioral Model; Memory Effect
1. Introduction
To handle multi-carrier envelope varying signals with
wide bandwidths in modern wireless communication sys-
tem, the signals passing through the transistor and power
amplifiers should be able to predict accurately. In the
behavioral model, the nonlinear component is generally
considered as a “b lack-box ” which is completely charac-
terized by external responses, in terms of input and out-
put signals, through the use of relatively simple mathe-
matical expressions. The Volterra series is a general non-
linear model with memory and has been used by many
researchers to characterize power amplifiers [1,2]. Un-
fortunately, the number of coefficients of the Volterra-
based models becomes unacceptable in practical imple-
mentation as the accuracy level increased, such models
become useless.
To overcome the modeling complexity, various mod-
el-order reduction approaches have been proposed to
simplify the Volterra m odel str ucture. For e xample, Wiener,
Hammerstein, Wiener-Hammerstein [3,4] and Memory
Polynomial [5] models are the most popular approxima-
tions. Physical knowledge has been considered in recently
proposed models by taking into account the real behavior
of the PA [6]. And the device electrical properties were
also considered in the reduction [7].
Although these simplified models have been employed
to characterize PAs with reasonable accuracy under cer-
tain conditions, the number of coefficients to be esti-
mated is still increasing prominently with the degree of
nonlinearity and memory length of the system. While the
interaction be tween the nth order and memory length has
not attracted enough attention.
The model proposed in this work is based on an un-
derlying general Volterra model of the PA. By focusing
on the fact that output items is fading with the non-li-
nearity order and memory length increase, the Volterra
model is modified. As the function of nonlinearity and
memory effects applied, a slow growth of the coefficients
can be obtained while the accuracy can be improved. The
letter is organized as follows: after this introduction, Vol-
terra model is modified based on the nonlinearity and
memory effects analyzed in Section 2. The model’s per-
formance is shown in Se ction 3. At last the conclusion is
presente d in Sect ion 4.
2. Principle of Proposed Model
A Volterra series is a combination of linear convolution
and a nonlinear power series so that it can be used to
describe the input/output relationship of a general nonli-
near, causal and time-invariant system with fading mem-
ory.
(1)
where and represents the input and the
output, P and M are the order of nonlinearities and the
memory length, respectively, and is the
discrete time Volterra kernels of order n.
In the application of wide-band system, memory ef-
∑∏ ∑ ∑
== = =
−=
M
i
p
jjpp
p
p
M
i
p
inxiihny
01
1
1 0
)(),...,(...)(
1
)( nx
)( ny
),...,(h 1p
p
ii

P. P. LI ET AL.
571
fects of the amplifier are very significant and have an
important impact on linear effects. Memory effects [8]
can be classified into electrical and thermal. The electric-
al memory effects arise as a consequence of the variation
of the impedance along the signal bandwidth modulation.
As the output of the power amplifier nonlinearity will
vanish at the infinite order and the memory effects will
fade out with time passing by, there must be an exponen-
tial role in the amplitude function.
To make it easy to understand the proposed model,
first we consider the output of the memoryless system is
a function of nonlinearity order p, which can be ex-
pressed as follows:
(2)
When the polynomial contains memory, the change of
the amplitude must be the function of memory which can
be written as
(3)
According to Equation (3), the function of memory
can be written as
(4)
When we consider the error of the polynomial, if
, the polynomial will be replaced by
ε (ε can be calculated by the value of IMD and gain of
the PA). Obviously the output can get the maximum val-
ue when the input takes the maximum value, so we can
get the function of the memory with the nonlinearity or-
der p as follo w ing (the input has been norm a lized):
(5)
Considering the memory effects of PAs, the function
of memor y g(m) would fade gradually, as g(0) = 1, g(∞)
= 0 , we can assume and that corr esponds
with the characteristic of capacitance or inductive com-
ponents which can release the energy, then Equation (5)
can be written as
(6)
And the multiply items such as
can be written as
(7)
where is a function which depends on the nonli-
nearity of the PAs and decreased as the nonlinearity or-
der increased. To simplify the analyzing, we can assume
(has been normalized as same as input),
then last-written equation can be written as
(8)
To attain the value of β and
ε
, the input p ower, IMD
and gain of the PA must be known. For example, assum-
ing the parameters of a definite PA are , IMD5
= I5 and gain = G. From the definition of IMD5, I5 can be
written as
in
P
P
I
5
5log10=
(9)
as and , so
ε
can be
determined by
(1 0)
Take (10) into (8), since , then the β can be
determined by
(11)
Note that α is the only unknown variable in (8) and α
depends on the memory effects of PA. However, the
memory effects of PA cannot be tested, so we attain a
relative accurate α through comparing model perfor-
mance of different α.
Evidently the function of and p is a decreasing
function so that the coefficients can be decreased rapidly
when the system’s order increased.
Maintaining the Integrity of the Specifications
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3. Model Performance
In order to validate the proposed behavioral model in a
real system, a Doherty PA was tested. This PA was oper-
ated at 460 MHz and excited by an OFDM signal with 20
MHz bandwidth.
The parameters of tested PA were Pin = 10 W, IMD5
= −40 dBc and gain = 20 dB. So the ε and β can be fixed
by (10) and (11). Thus (8) can be written as
2.48.0 ≤+
∑∑
pmα
(12 )
∑∑
==
==
p
p
p
p
pp
nxpfnyny
11
)()()()(
)()()()( mnxmgpfmny
p
p
−=−
)( mg
)()(
)(
)(
mnxpf
mny
mg
p
p
−
−
=
ε <− ))(( mnyMax
p
)
)(
()(
1
pf
gpm ε
−
≤
n
engα
−
=)(
)
)(
ln(
1
)(
pf
pm ε
α
−≤
)()( 21
21
mnxmnx
pp
−−
)
)(
ln(
1
∏
∑
−≤
pf
mε
α
)( pf
p
epf
β −
=)(
εβα ln−≤+
∑∑
pm
0
PP
in
=
502
12
5
5
y
P =
Gy ×= ε
max
5
G
P
I 20/
0
5
10
−
×
=
ε
5=
∑
p
G
P
I 20/
0
5
10
ln
5
1
−
×
−= β
∑
m

P. P. LI ET AL.
572
In this test, the nonlinearity of the model was truncated
to order 5. For comparison, the value of α was set from
0.2 to 2. To evaluate the model’s fidelity in the time do-
main, the NMSEs and number of coefficients for each
partial model were calculated. These results are shown in
Figure 1. Due to the fact that memory effects of different
PAs were not the same，so the value of α was different
for different PAs. In this paper, the value of α was set to
1.6 while the performance of proposed model for the
tested PA was better and with lesser number of coeffi-
cients.
To show the model accuracy in the frequency domain,
the spectra of modeled errors are plotted in Figure 2. It
can be seen that the error signal spectrum of our pro-
posed model is very small, while significant errors are
generated in the output predicted by the memoryless mod-
el. For reference, the spectrum of the simulated output is
also plot t e d in Figure 2.
4. Conclusion
An efficient and effective Volterra model pruning me-
Figure 1. Model performance in the time domain.
Figure 2. Sample frequency domain output and modeled
error spectra.
thod for RF PAs has been presented in this letter, which
based on a function of the sum of nonlinearity order with
the sum of memory length. The advantage of this model
reduction approach is that it allows ef ficient reduction of
the model complexity, while keeping the essential prop-
erties caused by memory effects of a real PA. With a
Doherty PA tested, the proposed model can be employed
to characterize a nonlinear PA with memory effects in
high accuracy.
5. Acknowledgements
The research work is supported by Chinese Major Na-
tional Science and Technology Projects
(No.2010ZX03007-003).
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