Robust behavioral modeling of dynamic nonlinearities using Gegenbauer polynomials with application...

Robust Behavioral Modeling of DynamicNonlinearities Using Gegenbauer Polynomials withApplication to RF Power Amplifiers

Afef Harguem,1 Noureddine Boulejfen,2,3 Fadhel M. Ghannouchi,3 Ali Gharsallah1

1D�epartement de Physique, Facult�e des Sciences de Tunis, Universit�e El-Manar, Tunis, Tunisia2Department of Electrical and Computer Engineering, iRadio Laboratory,Schulich School of Engineering, University of Calgary, Calgary, AB, Canada T2N 1N43Institut Sup�erieure des Sciences Appliqu�ees et de Technologie, Universit�e de Kairouan,Kairouan, Tunisia

Received 30 October 2012; accepted 22 April 2013

ABSTRACT: In this article, we propose a new set of basis functions based on Gegenbauer

polynomials suitable for robust behavioral modeling of nonlinear dynamic systems. These

polynomials can be optimized for maximum model identification stability under different

input signal distributions. The efficiency and robustness of the proposed polynomial models

are demonstrated and compared to the ones of previously published models. The obtained

results revealed an exceptional numerical stability regardless of the input signal statistics,

making the proposed new models suitable for multimode and broadband nonlinear wireless

transmitters. VC 2013 Wiley Periodicals, Inc. Int J RF and Microwave CAE 24:268–279, 2014.

Keywords: distortion power; complex nonlinearity; intermodulation distortion; memory effects;

multitone excitation; power series; spectral regrowth

I. INTRODUCTION

Nonlinearity is the main source of intermodulation distortion

in wireless transmitters, in general, and in radio frequency

(RF) power amplifiers, in particular, causing a spectral

regrowth of the passband signals in digital communication

systems. Without an accurate estimation of the effect of this

nonlinearity, designers are driven to the over specification

of the nonlinear circuits/subsystems to meet the require-

ments of the ever increasing demands of the communication

standards. However, with today’s technological drive toward

higher performance of communication systems, complex

digitally modulated signals, such as wideband code division

multiple access (WCDMA) and multicarrier signals, are

being used more and more frequently. Consequently, blind

over specification of nonlinear systems/subsystems, such as

microwave power amplifiers (PAs), has become inefficient

in predicting their behavior in their field of operation. For

this reason, the accurate modeling of dynamic complex non-

linearities in digital communication systems has become

crucial, in order to estimate their effects and proceed to

their linearization in some cases.

In the last decade, several models have been proposed

to predict the behavior of nonlinear systems. The Volterra

series based model [1,2] is the most comprehensive repre-

sentation of nonlinear systems exhibiting memory effects.

However, the relatively high number of parameters, which

increases exponentially with the nonlinearity order and the

memory depth, as well as the computational complexity

associated with the extraction of the parameters makes the

model practically limited to relatively low nonlinearity

orders (typically third order) and memory depth. This leads

to a poor accuracy since PAs and wireless transmitters usu-

ally exhibit significantly higher nonlinearity orders.

To reduce the complexity of the general Volterra

model, several pruning techniques such as in Ref. 3 have

been proposed. These techniques are based on selectively

discarding the Volterra model coefficients/kernels that do

not have a significant impact on the modeling accuracy.

However, the associated model complexity remained rela-

tively high.

Correspondence to: A. Harguem;

e-mail: [email protected] or [email protected].

DOI: 10.1002/mmce.20758

Published online 12 July 2013 in Wiley Online Library

(wileyonlinelibrary.com).

VC 2013 Wiley Periodicals, Inc.

268

In recent years, memory polynomial models [4,5] have

been widely used. These models are composed of a finite

number of delay taps followed by nonlinear static functions.

They can be seen as a truncation of the general Volterra

series, where only the diagonal terms of the Volterra kernel

are considered. This truncation significantly reduces the num-

ber of parameters in the model. On the other hand, it can be

interpreted as a special case of the parallel Wiener model

[6], where infinite impulse response functions are reduced to

single delay taps. Many attempts have been made to improve

the convergence of this model. As an example, sparse delay

taps have been used by Ku and Kenney [7] to reach conver-

gence with a minimum number of branches.

Polynomial models, however, suffer from a major prob-

lem related to numerical stability. In fact, the parameter

identification associated with these models necessitates

inversion of matrices, which are generally badly condi-

tioned due to the set of basis functions used in these mod-

els. This stability problem is more pronounced with high

polynomial orders and memory depths. To overcome the

stability problem, Raich et al. [8] has proposed the conver-

sion of the conventional basis functions to orthogonal ones.

However, this requires the calculation of converting terms

with exponentially increasing number as the polynomial

order increases. Moreover, the orthogonality of the obtained

set of basis functions is guaranteed for uniformly distrib-

uted and zero reaching input envelope signals only. To

overcome this limitation, associated Laguerre polynomials

have been proposed [9]. The resulting set of basis function

is fully orthogonal for Rayleigh distributed envelope ampli-

tudes, making it suitable for complex Gaussian envelopes.

Once again the stability of the resulting model is guaran-

teed only for a Rayleigh distribution, as it can be seen in

Section V. Safari et al. [10] has proposed a cubic spline

based model for modeling PAs and their digital predistor-

ders. As only third-order polynomials are used in this

model, a fair improvement relative to the conventional

polynomial model has been observed. However, the risk of

numerical instability is still high. In Ref. 11, Bouajina

et al. suggested the use of a synthetic signal with a Ray-

leigh like distribution to enhance the robustness of the

identification process. A fair improvement has been

observed over the conventional WCDMA signal. However,

this cannot solve the problem for the adaptive predistortion

linearization where a real time identification process is per-

formed under field conditions. Younes et al. suggested in

Ref. 12 a three-box model based on a parallel connection

of a lookup table a conventional polynomial model and an

envelope polynomial model. With the resulting model the

authors succeeded to reduce the model number of coeffi-

cients, and therefore its complexity, while maintaining the

same output accuracy. However, since the proposed model

is based on the conventional polynomial model, it exhibited

the same instability problem when the nonlinearity order

increases.

In this article, two new models based on Gegenbauer

polynomials are suggested to guarantee a maximum

numerical stability for different signal distributions and

nonlinearity order, with a minimum computational cost.

The organization of the article is such that an introduc-

tion to conventional polynomial models is presented in

Section II. In Section III, two memoryless Gegenbauer

polynomial models (GPMs) are presented. In Section IV,

we extend the proposed models to handle memory effects.

Sections V and VI are devoted to the numerical and

experimental validations of the proposed models respec-

tively; and, finally, Section VII presents the conclusions.

II. MODELING COMPLEX NONLINEARITIES WITHCONVENTIONAL POLYNOMIAL MODELS

A common characterization of complex nonlinearities,

such as those of quasi-memoryless PAs, is based on their

amplitude-amplitude modulation (AM/AM) and

amplitude-phase modulation (AM/PM) conversions. These

characteristics are frequently measured in a static manner

using continuous wave (CW) signals and network analyzer

based setups. However, a more accurate single-tone PA

characterization can be achieved by performing dynamic

AM/AM and AM/PM measurements using modulated sig-

nals and devoted setups [5,6]. To predict the spectral

regrowth of such PAs, the complex envelopes of the RF

input and output signals can be related by:

y~ðtÞ5 x

~ðtÞGðrÞ (1)

where r5j x~ðtÞj, and G(r) is the complex gain of the PA,

with jG(r)j as its gain (AM/AM conversion) and /G(r) as

its output phase shift (AM/PM conversion). Conversely,

G(r) can be represented by a complex power series of a

finite order, K, such that:

y~ðtÞ5

XK

l51

bl j x~ðtÞjl21 x~ðtÞ (2)

To evaluate the coefficients, bl, N samples of the

measured input and output envelopes can be used. Using

the least square estimation approach, a solution such as

bLS5ðUHUÞ21UHy (3)

and

y5UbLS (4)

can be obtained where U5½/1ðxÞ/2ðxÞ… /KðxÞ� is a

NxK matrix with /lðxÞ5½/lðx~1Þ/lðx~2Þ… /lðx~NÞ�T,

x5½x~1 x~2 … x~N �T. y5½y~1 y~2 … y~N �

T, and /lðx~Þ5 x~ðtÞj x~ðtÞ

jl21. Assuming a uniform sampling with period Ts,

x~n5 x~½nTs�, y~n5 y~½nTs�. ðUHUÞ21UH is the Moore-

Penrose pseudo inverse of U and UH is its Hermetian

transpose.

The inversion of the KxK matrix,UHU, represents the

major limitation of the least square (LS) approach.

A higher conditioning number is an indicator of a badly

Gegenbauer Polynomials 269

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

conditioned observation matrix, UHU which makes the

pseudo-inverse calculation very sensitive to slight disturban-

ces. It may also lead to inaccurate results when finite preci-

sion calculation is used. As we are using LS method which

is based on norm-2 error cost function, we use norm-2 con-

dition number, given in the logarithmic scale by:

qðMÞdB510log 10ðkMk2kM21k2Þ510logkmax ðMÞkmin ðMÞ

��

� �(5)

where M is the matrix to be inverted. kmax and kmin are

the highest and smallest Eigen values calculated for the

Vandermonde matrix using single vector decomposition

respectively.

Regardless of the distribution of r the condition num-

ber of the matrix A5UHU, related to the above described

conventional polynomial model is extremely high. This is

due to the inappropriate choice of the set of basis func-

tions, /lðx~Þ, used in the PM. In the next section, a new

set of complex basis functions is proposed, and its per-

formance is compared to other published models.

III. MEMORYLESS GEGENBAUER POLYNOMIAL MODEL

To overcome the aforementioned limitation and ensure

numerical stability of the matrix inversion, a new set of

complex basis functions, wlðx~Þ, is proposed such that:

y~ðtÞ5

XK

l51

bl wlðx~Þ (6)

where

wlðx~Þ5 x

~Ck

l21ðr0Þ (7)

Ckl ðr0Þ are the Gegenbauer polynomials [13] and best

expressed with their recurrence relations Ck0ðr0Þ51,

Ck1ðr0Þ52kr0 and

Ckl ðr0Þ5

2

lðl1k21Þr0Ck

l21ðr0Þ21

lðl12k22ÞCk

l22ðr0Þ (8)

for l >1. As Ckl ðr0Þ are defined and orthogonal over the

interval [21, 1], its argument, r0, is given by r05 2r2b2ab2a ,

where r is considered to be in the interval [a, b]. Table I

shows the first five basis functions, wlðx~Þ for l 5 1…5.

The Gegenbauer polynomials are orthogonal with respect

to the weighting function ð12r02Þk21=2, [14] such that:

ð121

ð12r02Þk21=2Ck

l ðr0ÞCkpðr0Þdz5

(2122kp

Cðl12kÞðl1kÞC2ðkÞCðl11Þ

for l5p

0 for l 6¼ p

(9)

where C is the Gamma function defined by:

CðzÞ5ð10

tz21e2tdt (10)

Under certain circumstances, odd-order only models are

preferred. In this case, Gegenbauer polynomials can still be

very attractive in constructing a set of basis functions that

guarantee a stable numerical computation. For an odd-order

only Gengenbauer Polynomial based model (odd-only GPM)

and a nonlinearity order of P 5 2K11, y~ðtÞ can be given by:

y~ðtÞ5

XK

l50

b2l11 n2l11ðx~Þ (11)

where n2l11ðx~Þ is proposed to be:

n2l11ðx~Þ5 x

~Ck

l ðr00Þ (12)

In this case, Ckl ðr00Þ must be a polynomial of order 2l

in terms of r. Moreover, r00 must fall in the interval [0,1].

Assuming r 2 ½a; b� we obtain r005 2r22b22a2

b22a2 .

Similar to the conventional polynomial model, the LS

method can be applied to the two proposed odd-even and

odd-only GPMs to identify the bLS coefficients such that

bLS5ðWHWÞ21WHy and bLS5ðNHNÞ21NHy respectively.

In this case W5½w1ðxÞ w2ðxÞ… wKðxÞ� while N5½n1ðxÞn3ðxÞ… n2K11ðxÞ�.

Ideally, an orthogonal set of basis functions, (say wlðx~Þ),results in a matrix A5wHw with a low condition number

leading to a numerically stable matrix inversion. However,

perfectly orthogonal basis functions regardless of the input

signal statistic is not a goal in itself. In fact, a carefully

selected set of basis functions could lead to very satisfactory

results. Moreover, it is very difficult to guarantee the ortho-

gonality of a given set of basis functions for all kinds of

distributions, unless a computation cost is paid. In this con-

text, the proposed basis functions wlðx~Þ and n2l11ðx~Þ offer

a remarkable tradeoff between complexity and efficiency. In

fact, the k parameter in the Gegenbauer polynomials can be

optimized to achieve maximum numerical stability for any

given distribution of r 5 j x~ðtÞj and matrix size K.

A. Odd-Even GPMFor each input signal distribution, a minimization tech-

nique based on the Golden Section search with parabolic

interpolation [15] can be used to find kopt for which the

TABLE I First Five Gegenbauer Based Polynomials forOdd and Even Order Model

w1ðx~Þ5 x~

w2ðx~Þ5 x~2kr0

w3ðx~Þ5 2kð11kÞ x~r02k x~

w4 x~ð Þ5 43k 11kð Þ 21kð Þ x~r0322k 11kð Þ x~r0

w5 x~ð Þ5 23k 11kð Þ 21kð Þ 31kð Þ x~r0422

k 11kð Þ 21kð Þ x~r021 13k 11kð Þ x~

270 Harguem et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 2, March 2014

condition number of the expected value of A, qðE½A�Þ is

minimum. The (m,n)th element of E½A� is given by:

E½wHmðxÞwnðxÞ�5E½

XN

i51

w�mðx~iÞwnðx~iÞ�5NE½w�mðx~Þwnðx

~Þ�

(13)

Let fr(r) be the probability density function (pdf) of r, the

substitution of Eq. (7) in Eq. (13) yields:

E½wHmðxÞwnðxÞ�5N

ð121

r2Ckm21ðr0ÞCk

n21ðr0ÞfrðrÞdr (14)

For r uniformly distributed over an interval [0,1]

E½wHmðxÞwnðxÞ�5a

ð1

21

ðr11Þ2Ckm21ðrÞCk

n21ðrÞdr (15)

with a 5 N/8. Using (15) qðE½A�Þ can be calculated and

used in the minimization procedure to obtain an optimum

kopt 5 0.837 for K 5 11. For verification purpose and as

shown in Figure 1, qðE½A�Þ has been calculated for K 5

11 and swept values of k between 0.5 and 1.5. The figure

shows the uniqueness of kopt and confirms its value

obtained using the minimization procedure. The same

minimization procedure has been repeated for K 5 2…12,

leading to a set of points for kopt(K) as shown in Figure 2.

Similar to the uniform distribution, the same approach

can be applied to the most popular distributions used to

statistically model digital communication signals. As an

example, the WCDMA signal is considered as a complex

Gaussian signal with envelope amplitude obeying the

truncated Rayleigh distribution. This distribution is char-

acterized by a probability density function, fr(r), such that:

frðrÞ5re2r2=2r2

r2ð12e21=2r2Þ for 0 � r � 1

0 otherwise

8><>: (16)

for r >0, leading to


ð1

21

ðr11Þ3e2ðr11Þ2

8r2 Ckm21ðrÞCk

n21ðrÞdr

(17)

with a5 N16r2ð12e21=r2 Þ.

Likewise, the truncated exponential distribution is

widely used and characterized by:

frðrÞ5be2br

12e2bfor 0 � r � 1

0 otherwise

8<: (18)

where b >0 is the rate of change of r. This leads to


ð1

21

ðr11Þ2e2bðr11Þ2 Ck

m21ðrÞCkn21ðrÞdr

(19)

with a5 bN8ð12e2bÞ.

Finally, we consider the truncated chi-squared distribu-

tion characterized by

Figure 1 qðE½A�Þ for r uniformly distribution over [0,1] and a

nonlinearity order K 5 11.

Figure 2 Optimal k for a minimum qðE½A�Þ under uniform

distribution.

Figure 3 Probability density functions of the truncated Ray-

leigh, exponential and chi-square distributions.



frðrÞ5rd=221e2r=2

2d=2Cðd=2ÞPð1=2; d=2Þfor 0 � r � 1

0 otherwise

8><>: (20)

where P(x,a) is the incomplete Gamma function defined by

Pðx; aÞ5 1

CðaÞ

ðx0

ta21e2tdt

and the degree of freedom d is a positive integer number.

For the distribution E½wHmðxÞwnðxÞ� is given by


ð1

21

ðr11Þd=211e2ðr11Þ

4 Ckm21ðrÞCk

n21ðrÞdr

(21)

with a5 N2d12Cðd=2ÞPð1=2;d=2Þ.

Figure 3 shows the three distributions for r 5 0.3, a5 2 and b 5 3.

Without loss of generality and for simplification pur-

poses, a can be set to 1 for all the three distributions. In

fact for each distribution, a is the same for all the

E½wHmðxÞwnðxÞ� elements and hence it does not affect the

value of qðE½A�Þ.Figure 4 shows the variation of kopt for the truncated

Rayleigh distribution as a function of r and for different

values of the nonlinearity order K ranging from 3 to 12.

Similarly, Figure 5 shows the variation of kopt for the expo-

nential distribution as a function of its rate of change b, for

the same rage of K. Finally Figure 6 shows the variation of

kopt for the Chi-squared distribution as a function of its

degree of freedom d and for the same rang of K. the three

figures reveal that for each distribution the variation of kopt

exhibits comparable asymptotic behavior for the different

nonlinearity orders K. Moreover, it is important to note that

in Figures 4 and 5 kopt converges to approximately 0.84

when the pdf is approaching that of a uniform distribution.

B. Odd-Only GPMA similar analysis has been applied to the odd-only GPM.

This has led to the minimization of qðE½A�Þ where now

A 5 NHN. The mnth element of the Matrix E½A� of size

K11 is given by:

E½nH2m11ðxÞn2n11ðxÞ�5N

ð121

r2Ckmðr00ÞCk

nðr00ÞfrðrÞdr (22)

For a uniform distribution over [0, 1] such that

r005 2r2 21 we obtain:

E½nH2m11ðxÞn2n11ðxÞ�5a

ð1

21

ðr11Þ1=2CkmðrÞCk

nðrÞdr (23)

with a5N=4ffiffiffi2p

.Similarly, for a truncated Rayleigh distri-

bution over [0,1] we obtain


ð1

21

ðr11ÞCkmðrÞCk

nðrÞe2ðr11Þ4r2 dr

(24)

with a5 N8r2ð12e21=r2 Þ.

For an exponential distribution over the same range

[0,1] the elements of E½NHN� are found to be:


ð1

21

ðr11Þ1=2CkmðrÞCk

nðrÞe2bffiffiffiffiffiffiffiðr11Þ

2

pdr

(25)

Figure 4 Optimal k for the odd-even GPM under truncated

Rayleigh distribution.


Exponential distribution.


Chi-squared distribution.

272 Harguem et al.


with a5bN

4ffiffi2pð12e2bÞ. However, for a Chi-squared distribu-

tion we obtain:


ð1

21

ðr11Þd=4CkmðrÞCk

nðrÞe2

ffiffiffiffiffiffiffiðr11Þ

8

pdr

(26)

with a5 N23d=412Cðd=2ÞPð1=2;d=2Þ. Once again a can be set to

one in the calculation of the condition number of the

E½NHN� matrices for the four considered distributions.

Figures 2 and 7–9 show the variation of kopt for the

uniform, truncated Rayleigh, exponential and Chi-square

distributions respectively, using the odd-only GPM. The

curves are obtained by sweeping the parameter of the cor-

responding distribution and for different nonlinearity

orders P 5 2K11. Similarly to the case of the odd-even

GPM, the three figures reveal that for each distribution

the variation of kopt exhibits comparable asymptotic

behavior for the different values of P.

For both proposed GPM models, the above-described

optimization approach has led to a unique solution, kopt,

for each combination of nonlinearity order and amplitude

distribution of the input envelope signal.

IV. MEMORY EFFECTS

As polynomial models, the static (memoryless) GPMs,

proposed in the previous section, can be extended to han-

dle memory effects. To do so, the sampled output com-

plex envelope for the odd-even GPM is expressed by:

y~n5XK

l51

XQ

q50

bl;q wlðx~n2qÞ (27)

where Q is the number of memory branches (memory

depth) to be added to the first branch (q 5 0) that repre-

sents the static model. As K is the model nonlinearity

order, K(Q11) coefficients bl,q need to be identified.

Using the least square approach as in Eq. (3), the follow-

ing solution can be obtained such that:

bLS5ðWHWÞ21WHy (28)

where

W5

w1ðx~1Þ… wKðx~1Þ 0 : : : 0

w1ðx~2Þ… wKðx~2Þw1ðx~221Þ… wKðx~221Þ 0 :::::0

:

:

:

w1ðx~NÞ… wKðx~NÞ :::::::::… w1ðx~N2QÞ… wKðx~N2QÞ

2666666666664

3777777777775

(29)

and

bLS5½b1;0 b1;1 … b1;Q b2;0 b2;1 … b2;Q … bK;0 bK;1 … bK;Q�T

(30)

From Eq. (28), one can note that the matrix, w, is

formed by the set of basis functions repeatedly applied to

shifted versions of the input envelope, x~ðtÞ. This definitely

increases the condition number of the matrix, (wHw,

regardless of the set of basis functions used. For this rea-

son and for simplification purposes, the validation of the

proposed models will be limited to the static (memory-

less) case.

V. NUMERICAL VALIDATION

To estimate the performance of the proposed GPMs, it is

essential to evaluate the numerical stability of their identifi-

cation process, for different combinations of input envelop

signal amplitude distribution and model nonlinearity order.

To do so, 10 random realizations have been taken from

each of the uniform distribution, and the truncated Ray-

leigh, exponential and Chi-squared distributions with r 5

0.23, b 5 7 and d 5 20 respectively, over r 2 ½0; 1�. The

generation of a random variable obeying a given distribu-

tion can be performed using the inverse transformation

method described in Ref. 16 such that r 5 F-1(u) where uis a random variable uniformly distributed over [0,1]. F(x)

is the corresponding cumulative distribution function such

thatFðxÞ5Ðx

21frðrÞdr. For the truncated Rayleigh distribu-

tion we obtain r52ð1=bÞln ½12uð12e2bÞ� while for the

truncated exponential distribution r is given by

r5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22r2ln ½12uð12e21=2r2Þ�

p. The calculation of F21(x)

is tedious for the Chi-squared distribution as F(x) is defined

by integral. For this reason the acceptance-rejection method

described in Ref. 16 has been adopted to generate the sam-

ples of r instead of the inverse transformation.

Each of these realizations represents 200,000 samples

of the input envelope signal amplitude, r 5 j x~ðtÞj. Each

set of 10 realizations corresponding to one distribution

has been used to compute an average qðE½A�Þ for K in the

range of [3…11]. For comparison purposes, the same pro-

cedure was repeated for the conventional (PM), orthogo-

nal (OPM), Associated Laguerre (ALPM) and cubic spline

(CSPM) polynomial models briefly described in the intro-

duction section.

Figure 10 shows the condition number of A, q(A) for

the proposed odd-even GPM and the published PM, OMP,

ALPM and CSPM, with respect to a uniformly distributed

r over the interval [0, 1]. The figure shows that the PM

exhibits the worst condition number with the highest risk

of numerical instability. The CSPM fared better, but it still

suffers from a relatively high condition number. The

ALPM exhibits a reasonable condition number for low non-

linearity order K. However its performances deteriorate as

K increases. In contrast, the OPM exhibited the lowest con-

dition number for the whole range of K. This was expected

as this model is optimized for the uniform distribution.

Theoretically, its resulting condition number should be 1 (0

dB). However, since the 10 considered realizations do not



necessarily obey to a perfect uniform distribution, the

resulting condition number is slightly higher than 1.

Finally, the figure shows the performance of the pro-

posed odd-even GPM. This model exhibited a higher con-

dition number compared to the OPM. However, it still

retains an acceptable performance with a condition number

not exceeding 35 dB for K 5 11. It has to be mentioned

here that the condition number resulting from the OPM

increases dramatically if the lower limit a of the interval

[a, b], such that r 5 ½a; b�, is higher than zero [8]. This

limitation seriously impacts the stability and accuracy of

the OPM when used for predistortion based linearization

purposes with a peak-to-average ratio reduced signal lead-

ing to a narrower input envelope signal swing interval with

its lower limit above zero. These techniques are often des-

ignated as “hole punching” techniques into the vector dia-

gram of digital signals Ref. 17. The proposed odd-even and

odd-only GPMs do not suffer from this limitation, as no

constraints are imposed on the limits a and b.

Figure 11 shows q(A) with respect to the Rayleigh dis-

tribution for the same models. Similar to Figure 10, it can

be seen from Figure 11 that the PM and CSPM represent

the highest condition number. Theoretically, the ALPM is

fully orthonormal with respect to the Rayleigh distribution.

However the figure shows that a nonideal Rayleigh distri-

bution has led to a condition number exceeding 30 dB and

converging to that of the OPM for P > 7. This can be

assigned to the exponential term in the weighting function

of the Associated Laguerre Polynomials that makes the

ALPM sensitive to any deviation from the Rayleigh distri-

bution. In the other hand, the figure reveals the superiority

and the stability of the proposed odd-even GPM with the

lowest condition number for the whole range of K.

Next, q(A) is shown in Figures 12 and 13 for the same

models and with respect to the exponential and chi-square

distributions, respectively. Once again, the results reveal

that the PM and CSPM represent the highest risk of numer-

ical instability. Moreover, the sensitivity of the ALPM to

the distribution of r was confirmed by the divergence of its

resulting condition number as K increased in the Chi-

squared distribution. Similarly, since this distribution is

very different from the uniform one, the performance of

OPM has been drastically deteriorated with a condition

number reaching 100 dB for K 5 11. However, the pro-

posed even-odd GPM prevailed all the other models with

the lowest condition number and hence the higher numeri-

cal stability for almost all the range of K.

It has to be noted here that for the CSPM model P rep-

resents the size of the square matrix A5UHU with P 5

K13 while K is the number of intervals into which the

input amplitude is subdivided [10]. In Figures 10–13 the

number of intervals starts from 3 leading to P 5 6.

Figure 7 Optimal k for the odd-only GPM under truncated

Rayleigh distribution.


Exponential distribution.


Chi-squared distribution.

Figure 10 q(A) for r uniformly distributed over [0, 1].

274 Harguem et al.


The performance of the proposed odd-only GPM has

been evaluated in Figure 14. Its resulting condition num-

ber has been compared to that of the OPM only. In fact,

it has been shown from Figures 11–13 that the OPM is

the main competitor to the proposed GPMs. The figure

shows an impressive superiority of the odd-only GPM

over the OPM for all the four considered distributions.

It has to be noted here that the additional numerical

stability gained by the proposed GPMs compared to the

OPM and the ALPM does not increase the complexity or

the CPU time of the model identification process. In fact,

a lookup table can be used to select the appropriate kopt

for the corresponding model nonlinearity order and input

signal distribution.

So far, the above described numerical simulations have

demonstrated the remarkable performances of the pro-

posed GPMs and their very low sensitivity to the input

signal distribution. However, it is always important to ver-

ify their accuracy through an experimental validation.

VI. EXPERIMENTAL VALIDATION

To illustrate the efficiency of the proposed GPMs, a high-

power laterally diffused metal oxide semiconductor

(LDMOS)-based Doherty amplifier has been used. The

device under test (DUT) was designed for 3G applications

in the range of 2110–2170 MHz with a typical small sig-

nal gain of 63 dB, an input 1-dB gain compression power,

P1dB, of 27.3 dBm and a peak output power of 300 Watt.

A. DUT CharacterizationTo characterize the DUT, an input baseband signal, gener-

ated using the package ADS from Agilent Inc, is down-

loaded into a vector signal generator that drives the DUT

with the corresponding RF signal. The DUT output signal

is attenuated with a 61.3 dB attenuator and then down con-

verted, digitized and demodulated within the vector signal

analyzer using a sampling frequency, fs, of 92.16 MHz.

For a dynamic characterization of the DUT, it has

been excited with a four-carrier WCDMA signal of carrier

configuration ON-OFF-OFF-ON (1001). The total band-

width of the signal is 20 MHz centered at 2140 GHz with

an average power of 216 dBm which corresponds to an

input power back-off of 8.7 dB. The input and output

baseband waveforms were sampled within a time window

of 2-ms long, leading to 184,239 samples. As shown in

Figure 15, the complementary cumulative distribution

Figure 13 q(A) for the chi-squared distribution with d 5 20.

Figure 14 q(A) for the uniform, Rayleigh, exponential and

Chi-squared distributions with r 5 0.38, b 5 7 and d 5 20,

respectively.

Figure 11 q(A) for the Rayleigh distribution with r 5 0.23.

Figure 12 q(A) for the exponential distribution with b 5 7.



function (CCDF) of the input signal exhibits a peak-to-

average power ratio (PAPR) of 10.4 dB. The figure shows

also the compression of the output signal due to the nonli-

nearity of the DUT.

The input and output baseband signals have been used

to calculate the conversion characteristics of the DUT.

Figures 16 and 17 show the AM/AM and AM/PM charac-

teristics, confirming the values of linear gain and the 1

dB compression input power of the DUT. In addition,

both characteristics highlighted the DUT electrical mem-

ory effect triggered by the relatively large bandwidth of

the four-carrier WCDMA signal.

B. Validation of Memoryless GPMsStarting with the memoryless case (Q 5 0) the DUT has

been excited with a one channel WCDMA signal with the

same power characteristics and with a bandwidth of 5

MHz. The input and output signals have been sampled

with fs 5 92.16 MHz and used to evaluate the performan-

ces of the proposed GPMs. First, 20% of the input signal

data has been used to calculate the condition number q(A)

for a nonlinearity order ranging from P 5 3–11. Remem-

ber that P 5 K for odd-even models like the PM, OMPM,

CSPM and the odd-even GPM, while it is equal to 2K11

for the odd-only models such as the ALPM and the odd-

only GPM. As mentioned in section II and as shown in

Figure 18, the amplitude of the WCDMA envelop signal

obeys the Rayleigh distribution with r 5 0.215. Using

Figures 4 and 7 one can obtain the corresponding kopt for

each nonlinearity order for the odd-even and odd-only

GPMs respectively.

Figure 19 shows q(A) for the OPM, ALPM and the

two proposed GPMs. The PM and the CSPM has been

excluded from this comparison. In fact, the numerical val-

idation has demonstrated the very poor performances of

these models as a matter of condition number. In accord-

ance with the numerical validation, the experimental data

confirms the superiority of the proposed GPMs over the

published models by exhibiting the lowest condition num-

bers for almost all the range of P.

The importance of a low condition number resides in

the fact that the PA model is generally used in the PA lin-

earization operation such as the predistortion technique. In

such case the linearization algorithm is implemented on a

DSP or an FPGA board in which a fixed-point processor

with limited number of bits is preferred. In fact, fixed-

point processors are always more effective in terms of

Figure 16 AM/AM characteristic of the DUT.

Figure 17 AM/PM characteristic of the DUT.

Figure 18 Histogram of the amplitude of the envelop WCDMA

signal with comparison to the truncated Rayleigh distribution

with r 5 0.215.

Figure 15 CCDFs of the DUT input and output baseband

signals.

276 Harguem et al.


cost, power consumption and computation time. However,

during the design process the model is generally used for

simulation with a floating-point processor. In such condi-

tions the condition number of the matrix to be inverted

during the model identification is fairly influencing the

accuracy of the model output.

Next, to demonstrate the effect of the numerical stabil-

ity on the accuracy of the model output, the normalized

mean square error (NMSE) has been adopted. The NMSE

is given in the logarithmic scale by

NMSEdB510log 10

PNn51

jyðnTsÞ2 y�ðnTsÞj2

PNn51

jyðnTsÞj2

26664

37775 (31)

y(nTs) and y�ðnTsÞ are the complex envelopes of the meas-

ured and modelled output signals, respectively, when all

the 184,239 input signal samples are used. y�ðnTsÞ has been

calculated using a 64 bit fixed-point processor. Figure 20

shows the performances of the odd-even models for two

different values of the word length. The figure revealed

that for a word length of 24 bits, all the models exhibit a

comparable NMSE converging to approximately 235.5dB

for P 5 11. However, for a fraction length of 16 bits, the

NMSE of the conventional polynomial model deteriorates

drastically, while the OPM and the odd-even GPM retain

the same accuracy. Similarly, Figure 21 compares the per-

formances of the ALPM and the odd-only GPM as a func-

tion of the processor fraction length. The figure shows that

the two model exhibits a very comparable NMSE for a

fraction length of 24 bits. However, the odd-only GPM

retains the same accuracy for a fraction length of 16 bits,

while the NMSE of the ALPM is dangerously deteriorated

to as higher as 222 dB. Therefore, the odd-only GPM pre-

vails the ALPM for relatively low fraction lengths.

Finally, Figure 22 shows the dispersion of the model

coefficients for all the studied models such as

DispersiondB520log 10

bmax

bmin

� �(32)

where bmax and bmin are the highest and lowest model

coefficients, respectively. This figure of merit represents

the ratio between the highest and the smallest coefficients

of the model. The lower the dispersion the smaller the

number of bits required for the coefficients’ storage and

manipulation. The figure shows that the PM exhibits the

Figure 21 NMSE of the output signals for the studied odd-only

polynomial models.

Figure 22 Dispersion of the models’ coefficients.

Figure 19 q(A) for the WCDMA signal.

Figure 20 NMSE of the output signals for the studied odd-

even polynomial models.



highest dispersion exceeding 65 dB for P 5 11. However,

the odd-only GPM prevailed all the models with disper-

sion lower than 35 dB for the same nonlinearity order P.

C. Validation of GPMs with Memory EffectsTo evaluate the performance of the proposed GPMs in the

presence of memory effects, the input and output

WCDMA1001 signals used to characterize the DUT has

been considered. 20% of the output data set was used for

model identification. Figure 23 shows the histogram of

the WCDMA1001 signal that obeys the Rayleigh distribu-

tion with r 5 0.205. The model coefficients for the even-

odd and odd-only GPMs have been calculated for differ-

ent memory lengths Q and for a nonlinearity order P 5

11. For each value of Q the optimization procedure

described in section III has been used to find kopt. Figure

24 shows the condition number q(A) as a function of Q.

The figure revealed an increase of q(A) for all the models.

This is expected as the matrix size of A increases propor-

tionally to Q. Even though, the proposed GPMs preserved

their superiority in terms of numerical stability over the

ALPM [9] and orthogonal [8] models.

Figure 25 shows the NMSE obtained by the proposed

GPMs as a function of the memory length Q. As

expected, the NMSE of the two models is improved by

increasing Q to reach a pretty constant value below 236.5

dB for Q 5 5. Similar behavior has been observed for the

orthogonal and ALPM models when calculated in a

floating-point environment, in which numerical stability is

not very sensitive to the condition number q(A).

VII. CONCLUSION

In this article, we proposed two Gegenbauer polynomial

based models that are suitable for behavior modeling of

dynamic nonlinearities with robustness against signal statis-

tics. The proposed models have been validated with high

power RF amplifiers. The simulation results demonstrated

the robustness and numerical stability of the two proposed

models. Particularly, it was proven that the k parameter

related to the Gegenbauer polynomials can be optimized to

guarantee a numerically stable model identification process

for different polynomial orders and input envelope distribu-

tions. In addition, the comparison of the performances of

the proposed models with those of recently published ones

revealed the superiority of the proposed models with basi-

cally no additional complexity or computation cost. This

has been verified for the uniform, Rayleigh, exponential

and Chi-squared distributions while it can be extended to

any distribution by determining its corresponding kopt. The

numerical validation has been confirmed by the experimen-

tal results using WCDMA signals. In addition, these results

highlighted the positive effects of the numerical stability

guaranteed by the proposed GPMs on the model accuracy,

in a fixed-point processing environment.

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Figure 23 Histogram of the amplitude of the envelop

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distribution with r 5 0.205.

Figure 24 q(A) for the WCDMA1001 signal as a function of

the memory length Q and for a nonlinearity order P 5 11.

Figure 25 NMSE of the GPMs under the WCDMA1001 input

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ity order P 5 11.

278 Harguem et al.


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BIOGRAPHIES

Afef Harguem received her B.S. degree in

physics from the University of Montreal,

Montreal, QC, Canada, in 1998, and her

M.S. degree in microwave engineering

from �Ecole Polytechnique de Montr�eal,

Montr�eal, QC, Canada, in 2002. She is cur-

rently pursuing her studies toward her Ph.D.

degree at the Department of Physics at the

Facult�e des Sciences de Tunis, Tunis, Tuni-

sia. She is also an Associate Researcher with the Intelligent RF

Radio Laboratory at the University of Calgary Her research interest

is the modeling, characterization and linearization of RF power

amplifiers and transmitters.

Noureddine Boulejfen received his B.S.

degree in electrical engineering from the�Ecole Nationale des Ing�enieurs de Monas-

tir, Monastir, Tunisia, in 1993, and his M.S.

and Ph.D. degrees from �Ecole Polytechni-

que de Montr�eal, in 1996 and 2000, respec-

tively, both in microwave engineering. He

then joined the Microelectronics Group,

Fiber Optic Department, Nortel Networks

Inc. Canada, where he was an engineer with the On-Wafer Test and

Characterization Laboratory. From 2001 to 2011 he was with the

Electrical Engineering Department of University of Hail, Hail,

KSA as an Assistant Professor. Currently he is an Associate Profes-

sor at The ISSAT, University of Kairouan, Kairouan, Tunisia. He is

also an Associate Researcher with the Intelligent RF Radio Labora-

tory at the University of Calgary. His research interests are in the

fields of nonlinear behavioral modeling applied to the optimization

of wireless transmitters and the design and calibration of micro-

wave multiport measurement systems.

Fadhel M. Ghannouchi is currently a

iCORE professor and Senior Canada

Research Chair in the Department of Elec-

trical and Computer Engineering of the

Schulich School of Engineering at the Uni-

versity of Calgary and is the director of

Intelligent RF Radio Laboratory (www.ira-

dio.ucalgary.ca). He has held numerous

invited positions at several academic and

research institutions in Europe, North America and Japan. He has

provided consulting services to a number of microwave and wire-

less communications companies. His research interests are in the

areas of microwave instrumentation and measurements, nonlinear

modeling of microwave devices and communications systems,

design of power and spectrum efficient microwave amplification

systems and design of intelligent RF transceivers for wireless and

satellite communications. His research activities have led to more

than 400 publications and 10 US patents (3 pending). Professor

Ghannouchi is a Fellow, IEEE, Fellow, IET, and a Distinguish

Microwave Lecturer for IEEE-MTT Society.

Ali Gharsallah received his bachelor

degree in radio frequency engineering from

the higher School of Communications of

Tunis in 1986 and the PhD degree in 1994

from the School of Engineer of Tunis. Since

1991, he was with the Department of

Physics at the Faculty of Sciences of Tunis.

He has been a full Professor and Director of

Engineering studies at the higher minister

of Education of Tunisia. He has authored/co-authored over 80 pub-

lications and one book on Analog Electronics Circuits. His current

research interests include antennas, array signal processing, multi-

layered structures and microwave integrated circuits.



www.iradio.ucalgary.ca

www.iradio.ucalgary.ca

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