I. J. Communications, Network and System Sciences. 2008; 1: 1-103
Published Online February 2008 in SciRes (http://www.SRPublishing.org/journal/ijcns/).
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
A Discrete Newton’s Method for Gain Based
Predistorter
Xiaochen LIN1, Minglu JIN2, Aifei LIU
School of Electronic and Information Engineering
Dalian University of Technology, Dalian, P.R.China
Abstract
Gain based predistorter (PD) is a highly effective and simple digital baseband predistorter which compensates for
the nonlinear distortion of PAs. Lookup table (LUT) is the core of the gain based PD. This paper presents a discrete
Newton’s method based adaptive technique to modify LUT. We simplify and convert the hardship of adaptive updating
LUT to the roots finding problem for a system of two element real equations on mathematics. And we deduce discrete
Newton’s method based adaptive iterative formula used for updating LUT. The iterative formula of the proposed
method is in real number field, but secant method previously published is in complex number field. So the proposed
method reduces the number of real multiplications and is implemented with ease by hardware. Furthermore, computer
simulation results verify gain based PD using discrete Newton’s method could rectify nonlinear distortion and improve
system performance. Also, the simulation results reveal the proposed method reaches to the stable statement in fewer
iteration times and less runtime than secant method.
Keywords: Predistortion, Discrete Newton’s Method, Power Amplifiers (PAs), Lookup Table (LUT)
1. Introduction
Radio frequency (RF) power amplifiers (PAs) play an
important role in wireless communication systems, but
are inherently nonlinear. To compensate for nonlinear of
PAs, linearization is an indispensable technique today.
Furthermore, among various linearization techniques,
digital baseband predistortion is more attractive than
others by virtue of its simplicity and ease of
implementation with digital signal processor (DSP)
equivalently in baseband. In this paper, we take into
account the gain based predistortion [1] which employs a
lookup table (LUT) block using random access memory
(RAM).
PAs characteristics drift by reason of aging,
temperature changing, channel switches, and source
voltage variations, so a predistorter (PD) should have the
ability of adaptation. And this paper focuses on adaptive
techniques. The conventional adaptive algorithms
including recursive least squares (RLS) and least mean
squares (LMS) originate from adaptive filter theory.
Based on above, [2] presents a broadcasting adaptive
algorithm more efficient for updating PD. Moreover, [3]
proposes a modified broadcasting adaptive algorithm,
which does not require special form of training sequence.
Meanwhile, in [1], J. K. Cavers generates an idea that the
adaptation issue can be converted to the root finding
problem on mathematics and presents secant method. In
recent years, following J. K. Cavers, [4][5], and [6]
present combining dichotomy with linear method, rapid
secant method, and combing dichotomy with Newton’s
method, respectively. Methods above meet the
requirement of convergence rate and are able to reach to
the stable statement, but the drawback of these methods is
of high computational load. Therefore, in this paper, we
propose discrete Newton’s method to adapt a PD which
has the advantages of fast convergence and low
computational load.
This paper is organized as follows. Section 2 analyses
and deduces new method and formula. Section 3
demonstrates the virtue of new method by computer
simulation. Finally, we conclude the paper in Section 4.
2. Adaptive Gain Based Predistortion
2.1. Gain Based Predistortion
Figure 1 shows the architecture of a communication
system with gain based PD. PD, whose characteristic is
opposite of PA’s, is used in baseband circuit. The symbol
stream v is transferred to a PD via a P/S converter to get
predistorted signal w. Then w is converted to analog
waveforms via a digital-to-analog (D/A) converter.
Finally, the analog signals are quadrature modulated,

A DISCRETE NEWTON’S METHOD FOR GAIN BASED PREDISTORTER 17
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
upconverted, amplified, and transmitted to the air in
sequence.
||⋅
≈
≈
≈
≈
Figure 1. Block diagram for communication system with
gain based PD
In Figure 2, we find the block diagram of a gain based
PD [8]. The complex gains of PD are saved in a LUT
whose entries are indexed. Supposed indexing function
() Q⋅maps each input signal amplitude to LUT entry
index. Also, efficient
()
Q⋅ can improve PD performance.
LUT entries are either uniform of amplitude or power, or
nonuniform diversely due to
() Q⋅. With mapping input
signal to a certain LUT entry, adaptive algorithm begins
to update the gain value memorized in LUT to get the
optimum gain.
()
Q ⋅ ⋅
⊗
i
v
i
r
y
()
F
y
d
v
o
v
f
v
Figure 2. Block diagram of gain based PD
2.2. Discrete Newton’s Method
As illustrated from Figure 2 d vand o vrepresent input
complex signals and output complex signals of PA,
respectively. Then PA is characterized by
(
)
2
od d
vvGv= (1)
where () G⋅denotes the complex characteristic function of
PA, and2
⋅is defined as squared amplitude of signals.
Describing input complex signals of PD asi v , we can
express PD characteristic function as
()
2
di i
vvFv= (2)
where ()
F
⋅
denotes the complex characteristic function
of PD.
By substituting (2) into (1), we can obtain the whole
linear characteristic function of PD and PA in series
(
)()
(
)
2
22
ii iii
vFvG vFvkv= (3)
where the whole linear gain k is a positive constant, and
usually less than PA’s midrange gain. Simplifying (3), we
have
(
)()
(
)
2
222 0
iii
FvGvFvk−= (4)
Hence, how to get the optimum gain F of LUT is
transferred to the root finding problem for a nonlinear
complex equation.
As follows, we decompose the complex equation (4)
into two real equations separately representing amplitude
and phase characteristic. |()|
⋅
and ()∠⋅ label amplitude
and phase of a complex number, respectively.
Then, amplitude and phase characteristic functions of
PA are given as follows
()
2
adad
vvGv= (5)
and
(
)
2
apd d
vGvv
∠
=+∠ (6)
where
(
)
a
G
⋅
and
(
)
p
G
⋅
denote amplitude-modulated
amplitude-distortion(AM/AM) and amplitude-modulated
phase-distortion (AM/PM) of PA, respectively.
Similarly
(
)
a
F
⋅
and
(
)
p
F⋅represent AM/AM and
AM/PM of PD, respectively. We obtain amplitude and
phase characteristic functions of PD
()
2
diai
vvFv= (7)
and
(
)
2
dpi i
vFv v
∠
=+∠ (8)
Inserting (7) into (5), we get
(
)
()
()
2
22
aiaiai ai
vvFv GvFv= (9)
Similarly, combing (8) to (6), hence,
(
)
(
)
()
2
22 2
apiaipi i
vGvFvFv v
∠
=++∠ (10)

18 X.C. LIN ET AL.
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
The whole gain of PD and PA in series is k, in other
words, the whole amplitude gain is k, and the whole
phase doesn’t change at all. Therefore, we can obtain
()
()
()
22 2
aiaiai aii
vvFv GvFvkv== (11)
()
()
()
2
22 2
apiaipii i
vGvFvFvv v∠=+ +∠=∠
(12)
Simplifying and composing (11) and (12), we write
equations as follows
()
(
)
()
()
()
()
2
222
2
22 2
0
0
ai aiai
piai pi
Fv GvFvk
GvFvFv
−=
+=
⎧
⎪
⎨
⎪
⎩
(13)
In the above system of equations,
(
)
a
F⋅and
(
)
p
F
⋅
are
separately amplitude and phase gain memorized in LUT,
as a result, the issue of updating LUT is converted to the
roots finding problem for system of two element
nonlinear equations. (13) is simply marked by H
()
(
)
()
1
2
,0
,
,0
ap
ap
ap
hFF
HFF
hFF
⎧=
⎪
=⎨=
⎪
⎩
(14)
We can apply Newton’s method for system of
equations (9) to (14)
1
1
kk
aa
pb
FFJH
FF
+
−
⎛⎞ ⎛⎞
=−
⎜⎟ ⎜⎟
⎝⎠
⎝⎠ (15)
where J is Jocabian matrix, defined as
11
22
ap
ap
hh
FF
Jhh
FF
∂∂
⎛⎞
⎜⎟
∂∂
⎜⎟
=⎜⎟
∂∂
⎜⎟
⎜⎟
∂∂
⎝⎠
(16)
However, in fact, since the derivatives of
()
1
h⋅and
(
)
2
h⋅ are not available, we can not apply
Newton method. Instead, we make use of discrete
Newton’s method which substitutes differential entropy
for derivative.
(
)
(
)
(
)
(
)
()( )()( )
11
11 11
11
11
11 11
11
,,,,
,,,,
kk kkkk kk
ap apap ap
kk kk
aa aa
kk kkkk kk
ap apap ap
kk kk
pp pp
hF FhFFhF FhF F
FF FF
J
hF FhFFhF FhF F
FF FF
−−
−−
−−
−−
⎛⎞
−−
⎜⎟
−−
⎜⎟
′=⎜⎟
−−
⎜⎟
⎜⎟
−−
⎝⎠
(17)
Only if J is a nonsingular matrix, iterative formula of
discrete Newton’s method is
()
1
1
kk
aa
pb
FFJH
FF
+
−
⎛⎞ ⎛⎞ ′
=−
⎜⎟ ⎜⎟
⎝⎠
⎝⎠ (18)
Moreover, we know of a simple way to get the inverse
matrix of 2-D square matrix, it is that to say, supposed a
2-D square matrix11 12
21 22
aa
A
aa
=⎛⎞
⎜⎟
⎝⎠
, hence, we obtain
22 12 1
21 11
11 221221
1aa
Aaa
aa aa
−−
=−
−
⎛⎞
⎜⎟
⎝⎠
(19)
Compared discrete Newton’s method for system of
equations with secant method for an equation, they have
the same order of convergence, super-linear convergence.
However, the computational load of discrete Newton’s
method is lower than that of secant method, due to secant
method finding a root of a complex equation, concretely,
each iteration requiring 4 complex multiplications, 2
complex additions, and 2 complex by real divisions, in
total, 20 real multiplications and 14 real additions. On the
contrary, system of equations using discrete Newton’s
method are always in real number field, and each iteration
only needs 6 real multiplications, 12 real additions and 4
real divisions. Obviously, real multiplications of the latter
are fewer than those of former, so the computational load
of discrete Newton’s method is much lower than that of
secant under large numbers of symbols.
From Figure 3, we can see the block diagram of a gain
based PD with discrete Newton’s method. The drawback
of the method is one more R/P.
(
)
Q
⋅
⋅
⊗
i
v
i
r
y
()
Fy
o
v
f
v
⊕
d
v
Figure 3. Block diagram of gain based PD with discrete
Newton’s method
3. Simulation Results
The baseband simulation model block diagram of
adaptive PD is shown in Figure 4. In our simulation, we
used 16QAM signal and square root raised cosine filter
(SRRCF). The channel is assumed to be an additive white
Gaussian noise (AWGN) channel. Input backoff (IBO) is
4dB. In this paper, the simulation is restricted to the
memoryless distortion. Because PA widely used for

A DISCRETE NEWTON’S METHOD FOR GAIN BASED PREDISTORTER 19
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
satellite transmission is Traveling Wave Tube Amplifier
(TWTA), we apply TWTA to our model as well and
simulate the Saleh’s model [10]. In our simulation, input
signal amplitude is normalized to vary from 0 to 1. We
take for uniform amplitude distribution function as
() Q
⋅
.
≈
≈
(
)
T
Hf
(
)
R
Hf
Figure 4. Block diagram of baseband simulation
It is apparent from Figure 5 that the iterative number
of discrete Newton’s method for one input signal is less
by about 10 times than that of secant method. In addition,
when signal source generates 14
2 bits, runtime of secant
is 74.8600s, while that of discrete Newton is 49.6410s;
when there are16
2 bits, runtime of secant is 1536.40s,
while that of discrete Newton is 858.7190s, about a half
of secant’s. And all the above time is on average. Thus
we can conclude the proposed method could reduce
runtime of adaptation. And with input data increasing, the
superiority of discrete Newton’s method is more
outstanding.
Figure 5. Compare for MSE curve
Figure 6 shows constellation for 16QAM modulation
signals. Figure 7 shows constellation for signals distorted
by PA. Further more, Figure 8 and Figure 9 illustrates
constellation of receiver signals predistorted by the gain
based PD with secant and discrete Newton’s method,
respectively. These simulation results prove the gain
based PD with both the two adaptive method compensate
nonlinear distortion, and their distorting effects are
equally.
Figure 6. 16QAM modulation signals constellation
Figure 7. 16QAM signals constellation with PA
Figure 8. 16QAM receiver signals constellation with secant

20 X.C. LIN ET AL.
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
Figure 9. 16QAM receiver signals constellation with discrete
Newton’s method
Figure 10. Compare for BER curve
Figure 11. Comparison for PSD
According to Figure 10, at low signal noise rate (SNR),
bit error rate (BER) of discrete Newton’s method is
similar to that of secant method, and they are much less
than PA’s and close to the theory value. However, with
SNR increasing, BER of discrete Newton’s method is a
bit lower than that of secant method.
Figure 11 shows that the spctrums of 16QAM signals
through PA have spectrum re-growth distortion and the
gain based PD with secant and discrete Newton’s method
depress the spectrum re-growth.
TD performance in Figure 12 versus total degradation
performance of distorted and compensated signals. There
is about 1dB TD decrease with discrete newton’s method
compared with secant.
Figure 12. Comparison for TD curves
4. Conclusion
In this paper, we propose an improved adaptation
technique using discrete Newton’s method efficiently
used in PD. We simplify and transfer adaptation to the
roots finding problem for system of equations from
nonlinear numerical analysis theory. The improved
method can produce better convergence performance. In
addition, due to computation processing completely being
in real number field, the method requires fewer
multiplications, lowers computational load. However, the
shortcoming of proposed method is one more a R/P.
5. References
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Digital Predistorter with Fast Adaptation and Low
Memory Requirements”, IEEE Trans. Veh. Technol.,
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[2] Won Gi Jeon, Kyung Hi Chang, and Yong Soo Cho,
“An Adaptive Data Predistorter of Compensation of
Nonlinear Distortion in OFDM System”, IEEE
Trans. Commun., Vol. 45, No. 10, Oct. 1997, pp.
1167-1171.
[3] Minglu Jin, Sooyoung Kim, Doseob Ahn, Deock-Gil

A DISCRETE NEWTON’S METHOD FOR GAIN BASED PREDISTORTER 21
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
Oh, and Jae Moung Kim, “A Fast LUT Predistorter
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[8] Chih-Hung Lin, Hsin-Hung Chen, Yung-Yi Wang,
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