IDENTIFICATION OF POWER AMPLIFIER USING MIMO - XLIM-SIC

IDENTIFICATION OF POWER AMPLIFIER
USING MIMO TIME-DOMAIN
REPRESENTATION
Smail Bachir ∗ Mourad Djamai ∗
Claude Duvanaud ∗ Régis Ouvrard ∗
∗
Laboratoire d’Automatique et d’Informatique Industrielle
4 avenue de Varsovie, 16021 Angoulême cedex
Email : [email protected]
URL : http://laii.univ-poitiers.fr
Abstract: In this paper, we present a new technique for modeling and characterization of Power Amplifier (PA) by parameter estimation using continuous-time
representation. Firstly, we study a continuous model which takes into account
nonlinearities and PA filtering. The filter structure includes a non linear polynomial
representing amplitude and phase conversion in addition to a MIMO Laplace plant.
Then, we propose a new approach for PA model characterization based on parameter estimation with several excitations. Using time-domain measurements, this
method deduces recursively an optimal estimation with Non Linear Programming
technique. The experimental results show good agreement and demonstrate the
possibility of this technique to explain power amplifier dynamics.
Keywords: Parameter estimation, Non Linear Programming, initialization
problem, PA modeling, continuous filter, time-domain envelope.
1. INTRODUCTION
Power Amplifiers (PA) are an important component in modern communication systems, providing the transmit signal levels needed to overcome the loss between the transmitter and receiver. However, they also introduce problems.
The amplifier can consume a major fraction of
the power used by the system and also distort the
transmitted signal, introducing additional spectral components within the signal frequency band.
Thus, the key challenge amplifier design for mobile
communication systems is the design of amplifiers, which provide high output power, linearity
and efficiency. For that, the modeling step is indispensable to a good comprehension of amplifier transmittance effects. The standard technique
for PA modeling uses a nonlinear structure with
fixed delay taps and complex coefficients (Clark
et al., 1998)(Launay et al., 2002). A Volterra
series (Kenington, 2000) and Wiener model (Ku
and Kenney, 2003) are largely employed in quantifying Radio frequency (RF) effects. A number of algorithms exist performing the coefficient
adaptations like the Least Mean Squares (LMS)
technique which is the most commonly used algorithm (Young, 1981)(Ljung, 1999)(Johansson,
1994). However, the parameters computation for
a nonlinear system is often difficult and time consuming for strongly nonlinear devices.
The model considered in this paper is described
on continuous-time domain. This structure is similar to PA discrete-time representation which is
including nonlinear transfer function and continuous filter (Clark et al., 1998). The first branch
is set to a memoryless amplitude (AM/AM) and
phase (AM/PM) conversion. Conventionally, the
power series model is used to considered these
transfer functions. To describe a PA dynamics, an
nth MIMO filter is inserted. This element operates on modulating input and represents a lowpass equivalent in envelope signals (Djamai et
al., 2005). The parameters of this model have been
identified by Output-Error technique (Richalet et
al., 1971)(Trigeassou et al., 2003).
In practice, identification technique should be
adapted to the use objectives. In our case,
PA characterization is realized by a parameters
follow-up. Then, it is necessary to work in continuous time representation because achieved parameters have physical significance (gain conversion, cut-off frequency, ...). However, parameters
initialization supposes a preliminary learning to
know approximatively the parameter values. For
this aim, a new methodology of PA identification
based on different excitations has been adapted to
parameters extraction in an effective way.
A special experimental setup dedicated to radio
frequency with baseband transmission has been
performed and used in order to validate this
technique. Experimental investigation exhibit the
good agreement and confirm the possibility to PA
characterization using continuous-time representation.
2.1 Nonlinear Statics functions
A nonlinear amplifier can be modeled by two
parallel nonlinear elements. One is a nonlinear
amplitude element which represent AM/AM conversion and another is a nonlinear phase element
which represent AM/PM conversion. In our study,
the identification is performed to characterize the
PA amplitude and phase nonlinearities with polynomial power series described by scalars parameters {ck } and {dk } (Kenington, 2000)(Launay et
al., 2002), such as:

P
X


k

V
=
ck Vin

 e
k=1
Q

X

k


ϕ
=
ϕ
+
dk Vin
e
in

(1)
k=1
where Ve and ϕe are respectively the amplitude
and phase of the distorted signal.
Vin and ϕin resp. the input envelope amplitude
and phase of the Radio Frequency (RF) signal
obtained by the trigonometric relations :

q

2 + Q2
 Vin = Iin
in Qin

 ϕin = arctan
Iin
where Iin and Qin are the baseband inputs.
2. CONTINUOUS POWER AMPLIFIER
MODEL
2.2 Continuous filter
The nonlinear amplifier model used in this paper is an extension of the discrete time-model at
continuous one (Clark et al., 1998). The major
disadvantage of the discrete representation systems is that the parameters used have no physical significance, contrary to continuous representation where parameters keep their real aspect
(Unbehauen and Rao, 1997). This is very important when advanced PA applications are considered such as linearization or real time control.
AM/AM
Ι
Q
in
V
V
in
Gain and
phase
conversion
ϕ
ϕ
in
e
I
I
e
in
out
e
Baseband
conversion
Q
 H
e


0
( s )
0
H
As shows in fig. 1, the inputs to outputs relationship of this nth order filter may be represented
with a differential equation:

n−1
m
X
X

dn
dk
dk


I
+
a
I
=
b
Ie

out
k
out
k
 dtn
dtk
dtk
k=0
k=0
(2)
n−1
m
X
X

dn
dk
dk



ak k Qout =
bk k Q e
 dtn Qout +
dt
dt
k=0
k=0
Ie (t), Qe (t) and Iout (t), Qout (t) are respectively
the filter input vector and output one where:


( s )
Q
out
AM/PM
Fig. 1. High frequency power amplifier model
As shown in fig. 1, the MIMO model structure is a
nonlinear transfer function added to an nth order
Laplace plant. A similar filtering H(s) operates
on the two baseband inputs I/Q. The nonlinear functions defined an AM/AM and AM/PM
characteristics and is placed before the filter to
complete the PA model.
Ie = Ve cos(ϕe )
Qe = Ve sin(ϕe )
The coefficients {ak } and {bk } are real scalars that
define the model. Note that the filter structure is
the same on the two axes I and Q, which gives
a decoupled MIMO plant. Thus, the input-output
relation can be expressed in Laplace-domain with
the transfer-function H(s), as so:
H(s) =
Pm
k
k=0 bk · s
P
n−1
sn + k=0 ak sk
(3)
2.3 Identification algorithm
Oscillator
Parameter estimation is the procedure that allows
the determination of the mathematical representation of a real system from experimental data.
Two classes of identification techniques can be
used to estimate the parameters of continuous
time systems: Equation Error and Output Error
(Richalet et al., 1971)(Ljung, 1999)(Trigeassou et
al., 2003)
I
Modulation
Demodulation
I
0°
0°
PA
90°
90°
Q
Q
PA Model
• Equation Error techniques are based on the
minimization of quadratic criterion by ordinary least-squares (Ljung, 1999). The advantage of these techniques is that they are
simple and require few computations. However, there are severe drawbacks, especially
for the identification of physical parameters,
not acceptable in PA characterization, such
as the bias caused by the output noise and
the modeling errors.
• Output Error (OE) techniques are based on
iterative minimization of an output error
quadratic criterion by a Non Linear Programming (NLP) algorithm (Young, 1981).
These techniques require much more computation and do not converge to an unique optimum. But, OE methods present very attractive features, because the simulation of the
output model is based only on the knowledge
of the input, so the parameter estimation are
unbiased (Trigeassou et al., 2003). Moreover,
OE methods can be used to identify non
linear systems. For these advantages, the OE
methods are more appropriate for PA characterization.
Parameter identification is based on the definition
of a model. For power amplifier, we consider the
previous mathematical model (Eqs. 1-3) and we
define the parameter vector:
θ = [ a0 · · · an−1 b0 · · · bm c1 · · · cP d1 · · · dQ ]
T
AM/AM
0
0 _
Q̂
ˆ
I
+
+
_
AM/PM
Identification
Algorithm
Fig. 2. Continuous-time PA identification schemes
As a general rule, parameter estimation with OE
technique is based on minimization of a quadratic
multivariable criterion defined as :
J=
K
X
(εI 2k + εQ 2k ) = εTI εI + εTQ εQ
(5)
k=1
We obtain the optimal values of θ by Non Linear
Programming techniques. Practically, we use Marquardt’s algorithm (Marquardt, 1963)(Trigeassou
et al., 2003) for off-line estimation:
00
θ̂ i+1 = θ̂i − {[Jθθ
+ λ · I]−1 .Jθ0 }θ̂=θ
i
(6)
00
with Jθ0 and Jθθ
are respectively gradient and
hessian such as:
PK Jθ0 = −2 k=1 εTIk · σ Ik,θ + εTQk · σ Qk,θ
PK 00
Jθθ
≈ 2 k=1 σ Ik,θ · σ TIk,θ + σ Qk,θ · σ TQk,θ
λ : monitoring parameter
T
where [.] denotes transposition operation.
σ Ik,θ =
Assume that we have measured K values of
input vector (Iin (t), Qin (t)) and output vector
∗
(Iout
(t), Q∗out (t)) with t = k · Te and 1/Te is the
sampling rate, the identification problem is then
to estimate the values of the parameters θ. Thus,
we define the output prediction errors:
∗
εIk = Iout
− Iˆoutk (θ̂, Iin , Qin )
k
∗
εQk = Qoutk − Q̂outk (θ̂, Iin , Qin )
(4)
where predicted outputs Iˆoutk and Q̂outk are obtained by numerical simulation of the PA model
and θ̂ is an estimation of true parameter vector θ.
σ Qk,θ =
∂ Iˆout
∂θ
∂ Q̂
out
∂θ
: output sensitivity on I axis
: output sensitivity on Q axis
where the sensitivity functions σ are obtained, for
each parameter, by numerical integration of their
differential system (Bachir, 2002).
All discrete-time models are deduced from the
continuous one by second order serie expansion
of the transition matrix (Bachir, 2002).
2.4 Initialization problems
Usually, for engineering process, one has good
knowledge on physical parameters, necessary to
initializing the recursive algorithm (Eq. 6). In
our case, PA users have not sufficient information
on parameter vector θ, especially on AM/AM
and AM/PM parameters. It is then essential to
define a global strategy which makes it possible
to obtain an approximative parameters values. So
we propose an optimal search method based on
Equation Error techniques to obtain initial values
of PA parameters with two steps.
2.4.1. Initialization of AM/AM and AM/PM parameters
The first step consists in searching the nonlinear
parameters {ck } and {dk } using the envelope magnitude and phase (Eq. 1). Thus, the AM/AM and
AM/PM characteristic is employed to optimize a
polynomial function by Least Squares technique
(Ljung, 1999). A solution for the coefficients is
obtained by minimizing the mean-squared error
∗
between the measured (Iout
, Q∗out ) and the modeled output (Iout , Qout ) such as:
(
θ̂ c = (φTc φc )−1 φTc V ∗e
θ̂ d = (φTd φd )−1 φTd (ϕ∗e − ϕin )
(7)
the linear filter effects. Thus, we define the filter
parameter vector:
θf = [ a0 a1 · · · an−1 b0 b1 · · · bm ]T
(8)
Parameter estimation is performed by iterative Instrumental Variable based on Reinitilised Partial
Moments (see also (Trigeassou, 1987)(Garnier et
al., 2003)).
3. TESTS RESULTS
In this section, we illustrate through experimentations, performance of the PA time-continuous
characterization method based on parameter estimation.
In our experimental study, the Cartesian structure
(I and Q axes) has been used in order to characterize the power amplifier. This transmission
structure overcomes the problems associated with
the wide bandwidth of the signal phase by applying modulation in I and Q components. Since the
I and Q components are the natural outputs of a
modern DSP, the Cartesian loop is widely used in
mobile-radio systems.
θ̂c = [ c1 c2 · · · cp ]T : AM/AM parameters
T
θ̂d = [ d1 d2 · · · dQ ] : AM/PM parameters
∗
e
ϕ∗e
V and
respectively magnitude and phase of
output signal
h
i
φc = ϕ1 ϕ2 · · · ϕK : AM/AM regression
c
c
c
matrix
T
2
P
ϕk = Vink Vin
· · · Vin
: regression vector
k
k
c
i
h
φd = ϕ1 ϕ2 · · · ϕK
: AM/PM regression
d
d
d
matrix
h
iT
Q
2
ϕk = Vink Vin
·
·
·
V
: regression vector
in
k
k
3.1 Experimental setup
For the experimental investigation, a commercial 700-MHz/4,2-GHz MINI-CIRCUITS is used
(ZHL-42 Model). Inputs and outputs data are
obtained from YOKOGAWA Digital Oscilloscope with a sampling period equal to 10 ns.
DATA Acquisition
Qin
Qout
Iin
Iout
d
Vink is a k th sampled input.
Noted that for these estimations, the regression
vector is not correlated with the output.
In practice, the PA characteristics is performed by
a sinusoid input at fixed low frequency and higher
input level. In these conditions, the PA filtering
characteristics are assume neglected according to
non linear dynamics. The input-output curves are
often obtained by measuring the output gain and
phase as a function of input power.
PA
0°
0°
90°
90°
Arbitrary Waveform
Generator
φ
Modulation
De-modulation
Phase
shifter
Local Oscillator
Fig. 3. PA Experimental setup
2.4.2. Filter initialization
The initial values of the filter coefficients are
obtained for small input (linear case). Indeed,
in this case, the signal distorsion is negligible,
which makes it possible to take into account only
Identification algorithm needs persistent excitation to provide appropriate estimation. Indeed,
modulated signals are required to excite both
steady-state (low frequency) and process dynamics (medium to high frequency). This excitation
is performed with a Pseudo Random Binary Sequence (P.R.B.S) baseband pulse as the input
modulation to the transmitter (fig. 5.a).
AM/AM characteristic
0.25
Measured Data
0.2
Output amplitude (V)
All modulation signals I and Q are delivered by
a TTi 40 MHz Arbitrary Waveform Generator
connected to PC control. The local oscillator frequency is 900 MHz obtained from 300-KHz/2,75GHz Digital Modulation Signal Generator (Anritsu MG 3660A).
Estimation
0.15
0.1
0.05
0
0
3.2 Experimental results
0.05
0.1
0.15
0.2
Input amplitude (V)
3.2.1. AM/AM and AM/PM characteristics identification
Polynomial parameters are extracted from the input/output transfer function. The measured characteristics is obtain by sweeping the power of an
input signal in the center-band frequency equal
to 900 MHz of the RF PA band width. In our
case, we used the 3th and 2nd order polynomial
respectively to amplitude and phase conversion
(Djamai et al., 2005):

3
X


k

V
=
ck Vin

e

Fig. 4. AM/AM response and estimation
After running a recursive algorithm, we obtained:

â0 = 3, 38.1013


 â = 3, 12.106
G = 1, 325
1
⇒
fc = 0, 925 MHz
b̂0 = 4, 48.1013



b̂1 = 1, 84.106
where the PA gain G and the resonant frequency
fc are calculated using θ̂ values. Noted that the
gain in the two steps are in the same proportion.
0.2
Fig. a
Magnitude (V)
0.1
k=1
2
X


k


ϕ
=
ϕ
+
dk Vin
in
 e
0
−0.1
k=1
where objective is to obtain the initialization
values of ck and dk using Least Squares algorithm.
Table 1 shows the estimation results. It is shown
that the PA gain is equal to G = ĉ1 = 1, 52.
−0.2
0
5
10
15
20
25
30
35
40
45
50
45
50
Fig. b
Magnitude (V)
Measured data
Estimation
0.3
0.2
0.1
Table 1. Estimation results of AM/AM
and AM/PM functions
θ
θ̂
c1
1, 52
c2
1.7.10−4
c3
−9, 61
d1
−1, 1.10−4
d2
3, 1.10−5
For illustration, it may be observed in fig. 4 a
good agreement between a measured AM/AM
characteristic and estimation. This confirms that
a Taylor series is a realistic way to approximate a
PA nonlinearities.
3.2.2. Filter identification
In this step, we search an initialization of a following resonant 2nd order system which is a dynamical relationship between baseband inputs (Iin ,
Iin ) and outputs one (Iout , Qout ) in Laplace domain:
Qout (s)
b1 s + b 0
Iout (s)
H(s) =
=
= 2
Iin (s)
Qin (s)
s + a1 s + a 0
Thus, we define the estimated parameter vector:
θ f = [ a 0 a 1 b0 b1 ]
T
0
−0.1
−0.2
−0.3
0
5
10
15
20
25
30
35
40
Time (µs)
Fig. 5. (a) Input signal. (b) Comparison of timedomain measurement and estimation
For small power, figure (5.b) proves that the PA
dynamic is equivalent to a 2nd order resonant
system.
3.2.3. Global PA Model identification
The unknown system in this case is a global PA
model composed by:
• non-linear AM-AM and AM/PM functions,
• second order system.
The input signals is performed by adding some
P.R.B.Sequence at different levels (fig. 6.a). The
aim is to sweep all AM/AM characteristic in order
to take into account the linear and nonlinear area.
Thus, the PA is driven near small power and
saturation to exploit maximum power efficiency.
Figure (6.a) shows the input signal applied to
perform global PA identification.
After 20 iterations, we obtained:
â0 = 3, 42.1013 ĉ = 1, 55
−4
ˆ
â1 = 3, 15.106 1
−4 d1 = −1, 5.10
ĉ
=
1,
9.10
13 2
−5
ˆ
b̂0 = 4, 53.10 d2 = 3, 6.10
ĉ3 = −9, 48
b̂1 = 1, 86.106
Fig. a
Magnitude (V)
0.2
0
−0.2
0
0.4
5
10
15
20
25
30
35
40
45
50
45
50
Fig. b
Magnitude (V)
Measured data
Estimation
0.2
0
−0.2
−0.4
0
5
10
15
20
25
30
35
40
Time (µs)
Fig. 6. (a) Input signal. (b) Comparison of timedomain measurement and estimation
For a different experiment, model simulation with
the obtained parameters exhibit good approximation of measured data on I axe (fig. 6.b).
4. CONCLUSION
A model is described which offers a simple way to
modeling PA dynamics, based on continuous-time
representation. This model is able of accounting
the magnitude and phase amplifier nonlinearities
such as the saturation effects.
Test results illustrate the efficiency of this technique for use in off-line identification. The continuous approach was found to be accurate in
predicting the PA response in dynamical mode.
Estimation results show that the described amplifier acts like a resonant system coupled with a
polynomial series.
The proposed technique considers the transfer
function approximately constant, which allows to
describe the amplifier as memoryless. Our next
objective will be to characterize PA with memory
effects contributions. So, it will be necessary to
develop a transfer function able to explain these
additional dynamics.
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