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Time-Domain 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 black-box model that

can be applied to characterize the nonlinear dynamic

behavior of power amplifiers. We show that time-delay

feed-forward Neural Networks can be used to make a large-

signal input-output time-domain 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 two-tone

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 input-output 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

input-output 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 curve-fitting 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

curve-fitting 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 Neural-Network-based 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 black-box 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 input-output relation, the question is

how to learn the nonlinear behavior response to different

input power levels. The answer is that time-delay Neural

Networks can learn a nonlinear behavior with medium-to-

strong memory effects, along with high-order nonlinearity,

if they are trained with input-output time-delayed 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 three-order kernels. This

fact can be a useful chance for medium-power 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 time-domain device measurements, could train a

Neural Network and generate a black-box 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

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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 black-box 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

feed-forward time-delay Neural Network with three layers,

an input layer composed of the input time-domain 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 input-output 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 input-output 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(t-T)

d

b3

bN

c3

cN

wN1

Vi(t-2T)

Vi(t-MT)

wNM

Fig. 2. Time-delay feed-forward Neural Network model

The analytical forms in (1) and (2) can be used as

input-output time-domain 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 single-input single-output 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 Time-delay Multi-Layer 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 1-2 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

time-domain simulation waveform of the two behavioral

models and the amplifier time-domain 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 large-signal 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 time-domain 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

IST-1-507893-NOE, www.target-net.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.1339-1350, 2003.

[3] T.Liu, S.Boumaiza, and F.M.Ghannouchi, “Dynamic

Behavioral Modeling of 3G Power Amplifiers Using Real-

Valued Time-Delay Neural Networks”, IEEE Trans. on

MTT, vol.52, no.3, pp.1025-1033, March 2004.

[4] A.Ahmed, M.O.Abdalla, E.S.Mengistu, and G.Kompa,

“Power Amplifier Modeling Using Memory Polynomial

with Non-uniform 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.432-437, 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 non-linear 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.7-12, 1992.

[10] V.Z.Marmarelis and X.Zhao, “Volterra models and Three

Layers Perceptron,” IEEE Trans. on Neural Networks,

vol.8, pp.1421-1433, 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 time-domain 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