Timedomain neural network characterization for dynamic behavioral models of power amplifiers
ABSTRACT This paper presents a blackbox model that can be applied to characterize the nonlinear dynamic behavior of power amplifiers. We show that timedelay feedforward neural networks can be used to make a largesignal inputoutput timedomain characterization, and to provide an analytical form to predict the amplifier response to multitone excitations. Furthermore, a new technique to immediately extract Volterra series models from the neural network parameters has been described. An experiment based on a power amplifier, characterized with a twotone power swept stimulus to extract the behavioral model, validated with spectra measurements, is demonstrated.

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ABSTRACT: This paper presents a new RF dynamic behavioral model based on a neural network (NN) approach suitable for FET devices in a wide range of working classes, and capable to identify the device response, through the training procedure, for a wide range of input power levels. The presented model has been effectively applied to GaNbased devices at 1 GHz, working in class A and BIntegrated Nonlinear Microwave and MillimeterWave Circuits, 2006 International Workshop on; 03/2006  [Show abstract] [Hide abstract]
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ABSTRACT: This paper presents a new approach to build RF dynamic behavioral models, based on timedelay neural networks (TDNNs), suitable for FET devices, and capable to identify the working class and to characterize both short and longterm device memory, through a timedomain training procedure, for a wide range of input power levels. The presented model has been effectively applied to GaNbased devices, working in class A, AB and BEuropean Microwave Integrated Circuits Conference, 2006. The 1st; 10/2006
Page 1
TimeDomain Neural Network Characterization for
Dynamic Behavioral Models of Power Amplifiers
G.Orengo1, P.Colantonio1, A.Serino1, F.Giannini1, G.Ghione2, M.Pirola2, G.Stegmayer2
1Dpt. Ingegneria Elettronica, Univ. Tor Vergata, via Politecnico 1, 00133 Roma, Italy, orengo@ing.uniroma2.it
2Electronics Dpt, Politecnico, Cso. Duca degli Abruzzi 24, 10129 Torino, Italy, marco.pirola@polito.it
Abstract — This paper presents a blackbox model that
can be applied to characterize the nonlinear dynamic
behavior of power amplifiers. We show that timedelay
feedforward Neural Networks can be used to make a large
signal inputoutput timedomain characterization, and to
provide an analytical form to predict the amplifier response
to multitone excitations. Furthermore, a new technique to
immediately extract Volterra series models from the Neural
Network parameters has been described. An experiment
based on a power amplifier, characterized with a twotone
power swept stimulus to extract the behavioral model,
validated with spectra measurements, is demonstrated.
I. INTRODUCTION
The nonlinear analysis of electronic systems often
requires an analytical model for each nonlinear element (i.e.
an equation representing the inputoutput relationship), that
allows to draw conclusions about the system performance.
This approach aims to extract a nonlinear relationship from
a relatively simple characterization set, in order to build an
inputoutput model able to generalize the nonlinear
dynamic behavior of electronic components for input
waveform not used in the characterization set.
Behavioral models try to accurately express the measured
behavior of an object, linear or nonlinear, formulating a
single closed form equation that represents a measured
parameter, which might be a function of multiple
independent variables. The process of converting measured
data into equations relies on curvefitting techniques [1].
However, many of the most common techniques are useful
where data trace is well behaved over a defined
independent variable range and where behavior of an object
is known to follow a specific mathematical model, but
problems arise when the object’s complex internal
parameters cause the data trace to exhibit sharp inflections.
In that case, data ceases to be well behaved and common
curvefitting techniques become useless. There is a clear
need for a new curve fitting technique that provides
smoothness and continuity through plotted trace having
sharp inflection.
A new technique that could overcome this problem could
be the use of Neural Networks. They can help building a
behavioral model of a nonlinear element or device. In fact,
the Neural Network approach for electronic device
modeling has received increasing attention, especially in
recent years [2], since model tailoring to the element under
study only needs a training procedure based on simulation
data or measurements of the physical circuit. Our proposal
is not only to use a Neural Network to build a behavioral
model for a nonlinear element, but also to obtain an
analytical expression for the model, either as neural
analytical model and Volterra series expansion, calculated
as function of the neural network model parameters (Fig.1).
Fig. 1. Novel Behavioral NeuralNetworkbased approach
As well as the frequency performance of linear devices
has been successfully represented by a linear convolution,
Volterra series represents its natural extension to nonlinear
devices. In such way both a linear and nonlinear dynamic
behavior can be usefully represented in a system chain with
a blackbox model. So far, however, behavioral models
based on the Volterra series hold their validity only for
weak nonlinearities and require heavy characterization
efforts to extract the kernels, especially when multitone
intermodulation is a matter of interest.
If the time domain approach is chosen in order to
characterize the memory effects adding enough time
delayed inputs to the inputoutput relation, the question is
how to learn the nonlinear behavior response to different
input power levels. The answer is that timedelay Neural
Networks can learn a nonlinear behavior with mediumto
strong memory effects, along with highorder nonlinearity,
if they are trained with inputoutput timedelayed data
samples at different power levels, simultaneously [3][4].
This fact turns out of outstanding importance to build
behavioral models of power amplifiers which are able to
simulate the nonlinear performance with different input
spectra and power levels.
A further advantage of this approach is that a new
algorithm to extract the Volterra kernels directly from the
neural network parameters has been found [5], and the
resulting model represents a very good approximation of
the nonlinear behavior, with only threeorder kernels. This
fact can be a useful chance for mediumpower analysis,
because the neural analytical models can be more complex
to implement into simulation CAD tools then compact
models based on Volterra series.
In other words, the objective of our work is to develop a
new kind of behavioral model for nonlinear RF elements,
independent of the physical circuit modeled, which fitting
only timedomain device measurements, could train a
Neural Network and generate a blackbox model on the one
hand, and could provide an analytical model for the
nonlinear behavior, also in Volterra series form, on the
other hand.
13th GAAS Symposium  Paris, 2005 189
Page 2
In this paper we show, as a case of study, the results
obtained from a power amplifier input/output time
domain characterization to build both neural and Volterra
series based blackbox models. The organization of the
paper is the following: in the next Section, the Neural
Network model proposed is described; in Section III the
building of a Volterra model from the neural network
parameters is explained; in Section IV the power
amplifier characterization and the modeling results are
presented. Finally, the conclusions appear in Section V.
II. NEURAL NETWORK MODEL
The neural network frame used in this application is a
feedforward timedelay Neural Network with three layers,
an input layer composed of the input timedomain voltage
samples and their delayed replies, an hidden layer with
nonlinear activation functions, and a linear output layer.
The architecture is shown in Fig.2, whereas (1) and (2) are
the corresponding inputoutput analytical expression, for
hyperbolic tangent and polynomial activation functions,
respectively
( )
t
()
∑
=
n
∑
=
k
−++=
N
1
M
0
inknn0o
kTtVwbtanhcdV
(1)
( )
t
()
∑
=
n
∑
=
k
−++=
N
1
P
M
0
inknkn0o
kTtVwbcdV
(2)
where M is the input memory, N is the number of hidden
neurons, and P is the polynomial degree. The particular
form of polynomial development in (2) has been chosen
because it can be directly implemented in neural network
training tools.
The input and output waveform are expressed in terms of
their samples in the time domain. The input memory (M)
should be chosen in order to adequately represent the
memory effects of the behavioral model, in the same
manner as done with linear filters, where the number of
input taps represents the accuracy in bandwidth shaping.
The number of hidden neurons (N) is chosen to perform the
best fitting to inputoutput data without overfitting
problems. The Neural Network is trained with a
backpropagation algorithm, based on the Levemberg
Marquardt algorithm for network parameters optimization.
w11
w21
w31
c1
c2
Vo(t)
b1
b2
Vi(t)
Vi(tT)
d
b3
bN
c3
cN
wN1
Vi(t2T)
Vi(tMT)
wNM
Fig. 2. Timedelay feedforward Neural Network model
The analytical forms in (1) and (2) can be used as
inputoutput timedomain characterizations for the
nonlinear element to model. Furthemore, a Volterra series
expansion, calculated in function of the neural network
parameters can be extracted from the polynomial output
network in (2) as well. This is explained in the next
section.
III. VOLTERRA MODEL
A nonlinear dynamic system can be represented exactly
by a converging infinite series, that reports the dynamic
expansion of a singleinput singleoutput system. This
equation is known as the Volterra series expansion,
which for third degree expansion can be expressed in the
time domain as follows
( )
t
( ) (
Vk
)
( ) (
V
) (
V
)
( ) (
V
) (
V
) (
V
)
∑ ∑ ∑
==
0kk
12
∑ ∑
=
0k
1
∑
=
k
=
=
−−−+
+−−+
+−+=
MM
0
M
0k
3i2i1i3213
MM
0k
2i1i212
M
0
i10o
3
2
TktTktTkt k , k ,kh
TktTkt k ,kh
kTthhV
(3)
The functions h0 h1 h2 h3 are known as the Volterra
kernels of the system.
The Volterra series analysis is well suited to the
simulation of nonlinear microwave devices and circuits,
in particular in the weakly and mildly nonlinear regime
where a few number of kernels are able to capture the
device behavior (e.g. for PA distortion analysis) [6]. The
Volterra kernels allow the inference of device
characteristics of great concern for the microwave
designer. However, the number of terms in the kernels of
the series increases exponentially with the order of the
kernel and this is the most difficult problem with this
approach.
In the Biology field, Wray & Green [8] have outlined a
method for extracting the Volterra kernels from the weights
and bias values of a Timedelay MultiLayer Perceptron
Neural Network. Based on this idea, there have been
several proposals for kernels calculation with different,
often non standard, neural networks topologies [8][9]. Our
proposed model, instead, it is more general because it could
potentially represent not only a dependence on one input
variable, but also a function depending on two or more
input variables.
Concerning the use of laboratory measurements for
feeding a neural model, different models and networks
topologies are used and compared, which need complex
measurements to train the networks and are based, actually,
on numerical estimations. Our proposed approach (applied
here to the particular case of power amplifiers), instead,
needs only a time domain characterization for the training
of the proposed neural network model.
Developing the network output expressed in (2), yields
19013th GAAS Symposium  Paris, 2005
Page 3
( )
t
()
[
V
]
()
( ) (
V
)
( ) (
V
) (
V
)
∑
=
n
∑ ∑ ∑
==
0kk
12
∑
=
n
∑ ∑
=
0k
1
∑
=
N
∑
=
M
∑ ∑
=
1n
∑
=
M
=
=
=
−−−+
+−−+
−+
+−++=
N
1
MM
0
M
0k
3i2i1ik , n k , nk , nn
1
M
0k
2i1ik , nk , nnn
N
1n0k
i k , n
2
nn
NM
0k
3
i
3
k , nn
N
1n
3
nno
3
321
2
21
TktTktTktVwwwc3
TktTktVwwbc3
kTtVwbc3
kTtwcbcdV
(4)
Comparing terms in (3) and (4), the Volterra kernels of a
Volterra series expansion can be easily calculated according
to (5)
N
3
0 n n
n 1
( )
()
()
()
()
=
=
=
N
=
=
N
=
=+
=
=
=
=
===
∑
∑
n 1
∑
n 1
∑
n 1
∑
n 1
∑
n 1
12
12
123
N
2
1 n nn,k
N
n nn,kn,k
212
n nn,kn,k12
N
nn,k n,kn,k
3123
3
n,kn123
hdc b
h k 3c b w
6c b ww
h k ,k
3c b wwkk
3c www
h k ,k ,k
c wkkkk
(5)
The Volterra model extracted in this way is perfectly
equivalent and performs the same degree of accuracy as the
Neural Network itself. This is of great importance because
is very easy and fast to train a polynomial Neural Network
and to extract the correspondent Volterra model.
IV. MODEL TRAINING AND VALIDATION
For training purpose, a Cernex 2266 power amplifier,
with a 12 GHz bandwidth, a 29 dB gain, and 1 dB
compression at 30 dBm, has been stimulated with two tones
with central frequency at 2 GHz and frequency spacing 100
MHz, from two synthesized sweepers, each one ranging the
power from –20 to +1 dB, that is 2 dB over the 1 dB
compression point of the amplifier. The amplifier output
has been connected to a Tek11801B Digital Sampling
Oscilloscope, and 5120 samples has been collected on a 20
ns window. The oscilloscope has been triggered with the
common RF reference at 10 MHz from the generators; two
commensurate frequencies multiple of the trigger frequency
have been used for this purpose. The data samples have
been read with a Labview program from a PC, connected to
the oscilloscope via an GPIB interface. The characterization
setup is shown in Fig.3.
Input data vectors from different input levels have been first
joined together, to train the Neural Network with all power
levels, simultaneously; the resulting vector has been copied
and delayed as many times, to represent the network input,
as necessary to take into account memory effects. The tap
delay must be a multiple or equal to the data sampling time
TS, and is
(
===
SS
TnTn F2* BW
calculated from
)()
−−
11
to avoid spectral
aliasing, where FS is the data sampling frequency and BW
the desired characterization bandwidth.
Two type of networks have been trained: the former, with
sigmoidal activation function, in the entire power range,
with 3 dB power step, to perform a very large signal
nonlinear model, the second, with third degree polynomial
activation function, in a smaller power range, below 1 dB
compression point. Both have been trained with 8 input
delays and 9 hidden neurons. Training results are shown in
Figg.4 and 5.
On the other hand, for validation purpose, frequency
domain amplifier response, obtained from FFT transform of
timedomain simulation waveform of the two behavioral
models and the amplifier timedomain measurements, has
been compared, to demonstrate the validity of the modeling
approach also in the frequency domain. Results are shown
in Figg. 6 and 7. As it can be seen spectra are very close in
the amplifier bandwidth, both for low and high distortion.
The third order Volterra model is well behaved near 1 dB
compression (Pin = 5 dB), whereas the sigmoidal model
hold its validity up to Pin = +1 dB, 2 dB over the 1 dB
compression.
V. CONCLUSIONS
A new largesignal behavioral model, based on Time
Delay Neural Networks, for the nonlinear dynamic
modeling of power amplifiers, has been developed.
Moreover, an easy procedure to extract Volterra kernels
from the polynomial network parameters provide a very
compact and accurate model to be used below 1dB
compression point. Future developments, which rely on a
more accurate timedomain characterization with a Large
Signal VNA, could enhance the accuracy of the modeling
approach.
ACKNOWLEDGEMENT
Research reported here was performed in the context of
the network TARGET– “Top Amplifier Research Groups
in a European Team” and supported by the Information
Society Technologies Programme of the EU under contract
IST1507893NOE, www.targetnet.org.
REFERENCES
[1] T. R. Turlington, Behavioral Modeling of Nonlinear RF
and Microwave devices, Artech House, 2000.
[2] Q.J.Zhang, K.C.Gupta and V.K.Devabhaktuni, “Artificial
Neural Networks for RF and Microwave Design – From
Theory to practice”, IEEE Trans. on MTT, vol.51,
pp.13391350, 2003.
[3] T.Liu, S.Boumaiza, and F.M.Ghannouchi, “Dynamic
Behavioral Modeling of 3G Power Amplifiers Using Real
Valued TimeDelay Neural Networks”, IEEE Trans. on
MTT, vol.52, no.3, pp.10251033, March 2004.
[4] A.Ahmed, M.O.Abdalla, E.S.Mengistu, and G.Kompa,
“Power Amplifier Modeling Using Memory Polynomial
with Nonuniform Delay Taps”, IEEE 34th European
Microwave Conf. Proc., Amsterdam, Oct. 2004, pp. 1457
1460.
[5] G.Stegmayer, M.Pirola,
“Towards a Volterra series representation from a Neural
Network model”, WSEAS Trans. on Systems, vol.3,
pp.432437, 2004.
G.Orengo and O.Chiotti,
13th GAAS Symposium  Paris, 2005191
Page 4
[6] M.Schetzen, The Volterra and Wiener Theories of
Nonlinear Systems, John Wiley & Sons, 1980.
[7] S.A.Maas, Nonlinear Microwave Circuits, Artech House,
1988.
[8] J.Wray and G.G.R.Green, “Calculation of the Volterra
kernels of nonlinear dynamic systems using an artificial
neural network,” Biological Cybernetics, vol.71, pp.187
195, 1994.
[9] D.I.Soloway and J.T.Bialasiewicz, “Neural Networks
modeling of nonlinear systems based on Volterra series
extension of a linear model,” IEEE Inter. Symp. on
Intelligent Control, pp.712, 1992.
[10] V.Z.Marmarelis and X.Zhao, “Volterra models and Three
Layers Perceptron,” IEEE Trans. on Neural Networks,
vol.8, pp.14211433, 1997.
CW generator
10 MHz Ref RF Out
CW generator
10 MHz Ref RF Out
a
10dB
a
10dB
GPIB link
DSO
TRIGGER CH
a
16dB
DUT
PC
Fig.3. Measurement setup for PA timedomain characterization.
05 1015 20
8
6
4
2
0
2
4
6
8
time [ns]
Vout [V]
Neural model  Meas
Fig.4. Sigmoidal neural model simulation and amplifier
measurement comparison for three input power levels (20, 12, +1
dBm).
05 1015 20
6
4
2
0
2
4
6
time [ns]
Vout [V]
Volterra model  Meas
Fig.5. Volterra model simulation and amplifier measurement
comparison for three input power levels (20, 12, 7 dBm).
1 1.522.53
60
40
20
0
20
Pout [dBm]
Pin=12 dBm
11.522.53
40
20
0
20
40
freq [GHz]
Pout [dBm]
Pin=+1 dBm
Fig.6. Sigmoidal neural model (o) and amplifier (x) output spectra
comparison for two input power levels (12, +1 dBm).
11.522.53
60
40
20
0
20
Pout [dBm]
Pin=12 dBm
1 1.522.53
60
40
20
0
20
40
freq [GHz]
Pout [dBm]
Pin=5 dBm
Fig.7. Volterra model (o) and amplifier (x) output spectra
comparison for two input power levels (12, 5 dBm).
192 13th GAAS Symposium  Paris, 2005
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