View
213
Download
0
Category
Preview:
Citation preview
Quantifying and cancelation memory effectsin high power amplifier for OFDM systems
Haleh Karkhaneh • Ayaz Ghorbani •
Hamidreza Amindavar
Received: 27 March 2011 / Revised: 20 October 2011 / Accepted: 27 October 2011 / Published online: 15 November 2011
� Springer Science+Business Media, LLC 2011
Abstract A modeling approach to power amplifier design
for implementation in OFDM radio units is presented. The
power amplifier model assesses the impact of linear
memory effects within the system using a Wiener repre-
sentation, and employs a linear novel parametric estimation
technique using Hilbert space. In addition, in order to
model the nonlinear memory effects the previous topology
is generalized by inserting the truncated Volterra filter
before the static nonlinearity. Predistortion based on the
Hammerstein model is introduced to deal with the non-
linear response. The new general algorithm is proposed to
evaluate the Hammerstein model parameters for an OFDM
system. A representative test bed was designed and
implemented. The assessment of the new methods for PA
and PD modeling are confirmed by experimental mea-
surements. The measurement results reveal the preference
of the new techniques over the existing approaches.
Keywords Adaptive predistortion � Hammerstein
system � Memory effects � Nonlinear distortion �Power amplifier (PA) � Wiener system
1 Introduction
Orthogonal frequency division multiplexing (OFDM) is
currently under significant investigation due to a high
spectral efficiency and immunity to multipath fading and
impulse noise. Usage of an appropriate guard interval in
OFDM can easily prevent intersymbol interference (ISI)
and interchannel interference (ICI), whereas powerful
equalization techniques are required for single carrier
modulation. However, OFDM-based systems are subject to
be significantly sensitive to nonlinear distortion, usually
caused by a high power amplifier [1]. In system-level
simulation, behavioral models are often employed to model
the PA nonlinearities. These measurement-based empirical
models provide a computationally efficient means to relate
the input complex envelope to the output complex enve-
lope without resorting to a physical level analysis of the
PAs. Behavioral models for PAs can be classified into three
categories depending on the existence of memory effects
[2–4]: memoryless nonlinear systems, quasi-memoryless
nonlinear systems, and nonlinear systems with memory.
For the memoryless nonlinear system, the PA block is
represented by the narrow-band AM/AM transfer function.
For the quasi-memoryless nonlinear system, with memory
time constants on the order of the period of the RF carrier,
the PA block is often represented by AM/AM and AM/PM
functions. Usually, AM/AM and AM/PM are measured by
sweeping the power of a single tone in the center frequency
of the passband of the RF PA. For a nonlinear system with
long-term memory effects, on the order of the period of the
envelope signal, the system response depends not only on
the input envelope amplitude, but also its frequency. An
alternate view is that the AM/AM and AM/PM functions
appear to change as a function of past input levels [5]. Such
effects occur in high power amplifiers (HPA) from thermal
H. Karkhaneh (&) � A. Ghorbani � H. Amindavar
Department of Electrical Engineering,
Amirkabir University of Technology, Tehran, Iran
e-mail: h_karkhane@aut.ac.ir
A. Ghorbani
e-mail: ghorbani@aut.ac.i
H. Amindavar
e-mail: hamidami@aut.ac.ir
123
Analog Integr Circ Sig Process (2012) 72:303–312
DOI 10.1007/s10470-011-9805-4
effects, as well as the long time constants in the DC bias
networks such as OFDM and WCDMA systems. A non-
linear system with memory can be represented by Volterra
series, which are characterized by Volterra kernels [4].
However, the computation of the Volterra kernels for a
nonlinear system is often difficult and time consuming for
strongly nonlinear devices. So, the Wiener model, which is
cascade connection of linear time invariant (LTI) system
and a memoryless nonlinear system, has been used to
model nonlinear PAs with memory with lower complexity
than volterra [2, 6–9]. Meanwhile in order to reduce per-
formance degradation in OFDM systems, compensation of
nonlinear distortion is required. Several prefiltering tech-
niques for a memoryless nonlinear system, preceded by a
linear system, have been reported [10–12]. In this paper,
we propose a more accurate model based on the Wiener
model developed by Schetzen [4]. Using two-tone signals,
AM/AM and AM/PM curves are extracted for each enve-
lope frequency by measuring inter-modulation distor-
tion(IMD) products. The derivation of the AM/AM and
AM/PM complex function from two-tone measurement is
proposed in Sect. 2. In Sect. 3, the linear novel parametric
estimation technique for the adaptive modeling of a PA
with linear memory effects for OFDM signal and gen-
eralized wiener to model PA with memory effects are
presented. An adaptive nonlinear predistorter in an OFDM
system is proposed in Sect. 4. To demonstrate the validity
of this design strategy, we take into account the output
spectrum by measurement. Finally, the new algorithms are
compared analytically with previous approaches with
respect to accuracy and adaptation time.
2 AM/AM and AM/PM curves extraction using two
tone response
A bandpass input signal of a PA can be represented as,
xðtÞ ¼ <fgðtÞejxctg ¼ rðtÞ cosðxct þ hðtÞÞ; ð1Þ
where g(t) is the PA input complex envelope signal, xc is the
carrier center frequency, r(t) and h(t) as its time-varying
amplitude and phase of g(t), respectively. The equivalent
baseband PA output w(t) for a bandpass memoryless nonlin-
earity can be characterized with a polynomial as follows:
wðtÞ ¼ < f ðtÞejxct� �
; ð2Þ
where the output complex envelope f(t) can be acquired as
follows:
f ðtÞ ¼Xn
k¼1
a2k�1jgðtÞj2ðk�1ÞgðtÞ ¼Xn
k¼1
a2k�1rðtÞ2k�1ejh; ð3Þ
where a2k-1 is a complex coefficient. In (3), the odd order
complex power series is defined as:
FðrðtÞÞ ¼Xn
k¼1
a2k�1rðtÞ2k�1: ð4Þ
And
wðtÞ ¼ jFðrðtÞÞj cosðxct þ hðtÞ þ \FðrðtÞÞÞ; ð5Þ
where |F(r(t))| and \FðrðtÞÞ can be represented as AM/AM
and AM/PM characteristic functions. AM/AM and AM/PM
can then be directly related with the two-tone response and
vice versa by using the complex envelope f(t). The two-
tone input, which has magnitude A2
and phase /(t) for each
tone, which tone spacing 2xm , can be described as:
vðtÞ ¼ A
2½cosððxc � xmÞt þ /ðtÞÞ þ cosððxc þ xmÞtþ /ðtÞÞ�
¼ A cosðxmtÞ cosðxct þ /ðtÞÞ: ð6Þ
For this two-tone input, the amplitude of input complex
envelope signal r(t) according to (1) is A cosðxmtÞ. So, the
output complex power series F(t) can then be acquired
from (4) as follows [13]:
FðtÞ ¼Xn
k¼1
a2k�1A2k�1ðcosðxmtÞÞ2k�1
¼Xn
k¼1
d2k�1 cosðð2k � 1ÞxmtÞ; ð7Þ
where
d2k�1 ¼Xn
i¼k
1
4i�1
2i� 1
i� k
� �a2i�1A2i�1: ð8Þ
Equation 7 results that F(t) and consequently AM/AM and
AM/PM depend on tone spacing xm for two tone input. By
sweeping envelope frequency xm, the frequency-dependent
a2k-1(xm) can be derived from two-tone measurements.
Thus, the frequency-dependent complex power series
F(r, xm), considering memory effects, can be described
as
Fðr;xmÞ ¼ a1ðxmÞr þ a3ðxmÞr3 þ � � � þ a2n�1ðxmÞr2n�1
¼Xn
k¼1
a2k�1ðxmÞr2k�1: ð9Þ
Therefore, the output of a PA with memory effect is
wðtÞ ¼ jFðr;xmÞj cosðxct þ hðtÞ þ \Fðr;xmÞÞ: ð10Þ
3 Modeling of a PA that exhibits memory effects
The frequency-dependent complex polynomial in (9) can
be realized by wiener model, a cascade structure of the LTI
system connected in series with a memoryless nonlinear
304 Analog Integr Circ Sig Process (2012) 72:303–312
123
system [14], as shown in Fig. 1 where f ð�Þ is defined from
(3).
On the other hand, the low-pass equivalent OFDM sig-
nal is expressed as:
gðtÞ ¼XN=2�1
k¼�N=2
XkejkDxt; Dx ¼ 2pT
0 \ t \ T; ð11Þ
where {Xk} are QAM data symbols, N is the number of sub-
carriers, and T is the OFDM symbol time. For wireless
applications, the low-pass signal is typically complex-
valued; in which case, the transmitted signal is up-
converted to a carrier frequency fc. In general, the
transmitted signal can be represented as:
xðtÞ ¼ < gðtÞejxct� �
¼XN=2�1
k¼�N=2
jXkjcosððxc þ kDxÞt þ \XkÞ: ð12Þ
Since OFDM input signal is multi-tone, the output PA
signal is frequency-dependant relation in (9) and
consequently, wiener model is suitable for behavioral
modeling of PA in this system.
3.1 Wiener model identification
The input and output relationship of the wiener system is
given by
wðkÞ ¼ FðvðkÞÞ ¼XP
p¼1
c2p�1ðvðkÞÞ2p�1: ð13Þ
vðkÞ ¼XN
n¼0
bnxðk � nÞ; ð14Þ
where N and P respectively, denote the memory length of
the linear filter, and the order of the nonlinear filter. Also,
x(k) and w(k) are the complex kth sampled input and output
of OFDM system at time kTs, respectively and 1/Ts is the
sampling rate and Fð�Þ is the complex envelope describing
the AM/AM and AM/PM responses derived from single-
tone measurements as described in (4). The coefficients cp
and bn are complex scalars that define the model. Using only
the input and output signals of the system, the coefficients of
the system estimator are adjusted to minimize the mean
square error (MSE), J(k) = E{|e(k)|2} between measured
and modeled time-domain outputs [6, 15].
wðkÞ ¼ FXN
n¼0
bnxðk � nÞ !
¼XP
p¼1
c2p�1
XN
n¼0
bnxðk � nÞ !2p�1
: ð15Þ
The coefficients of the FIR filter, b, can be acquired by
the adaptive LMS as follows [19]:
bðkþ1Þn ¼ bðkÞn � lrJðkÞ ¼ bðkÞn þ leðkÞ oFðvðkÞÞ
obn
� ��
n ¼ 1; . . .;N ð16Þ
The derivative of Fð�Þ with respect to b� is nonlinear
since the filter coefficients are integrated in the power
series. Therefore, it is evident that the parameter estimation
process presented in [6, 15] is not enough accurate. In order
to achieve a more precise estimation, this paper represents
linear novel approach to resolve the above mentioned. So,
parameter estimation of linear filter and memoryless
nonlinearity of wiener system will be enhanced.
Therefore, it is possible to first estimate the intermediate
variable v(k). To estimate the intermediate variable, it is
necessary to use the following assumptions [16]: (a) the
linear subsystem (the FIR filter) is stable, (b) the nonlinear
function (polynomials) is invertible, and (c) The data is
noiseless. Assuming this, it is possible to calculate the
intermediate variable as described in (17),
vðkÞ ¼ f�1ðwðkÞÞ: ð17Þ
Since f�1ð�Þ is the inverse of the invertible memoryless
nonlinear function of PA shown in Fig. 1, therefore can be
described by a power series such as:
vðkÞ ¼ f�1ðwðkÞÞ ¼XL
‘¼1
n2‘�1ðwðkÞÞ2‘�1: ð18Þ
To perform the best approximation of f�1ð�Þ and f ð�Þ we
use the concept of the best approximation in Hilbert spaces
[17]. To approximate f by a polynomial, let the independent
sequence of vectors be {xj}j=1N , where increasing N
provides a better approximation. The aj coefficients in
f(x) =P
j=1N ajx
jare calculated through a matrix inversion in
(19):
hx; xi � � � hxN ; xihx; x2i � � � hxN ; x2i
..
. ... ..
.
hx; xNi � � � hxN ; xNi
2
6664
3
7775
a1
..
.
aN
2
64
3
75 ¼hf ; xi
..
.
hf ; xNi
2
64
3
75: ð19Þ
where hf ðxÞ; xji ¼R b
a fjf ðfÞdf. In order to find the
polynomial expansion of the f�1 ¼PL
‘¼1 n‘x matrix at the
right side of (19) cannot be evaluated because there is no
Fig. 1 Model for a system with memory using the wiener model
Analog Integr Circ Sig Process (2012) 72:303–312 305
123
analytic form for f-1. So, we proposed this new method
[18] in order to avoid the numerical difficulty associated
with hf�1ðxÞ; xji ¼R b
a fjf�1ðfÞdf. After a change of
variable as f-1(x) = u or f(u) = x, (20) can be written as
hf�1ðxÞ; xji ¼Z f�1ðbÞ
f�1ðaÞ½f ðuÞ�juf 0ðuÞdu; ð20Þ
which finally leads to
hf�1ðxÞ; xji ¼ u
jþ 1½f ðuÞ�jþ1
���f�1ðbÞ
f�1ðaÞ�R f�1ðbÞ
f�1ðaÞ ½f ðuÞ�jþ1du
jþ 1:
ð21Þ
Now, the right side of (19) is obtained without knowing
analytical form for f-1. Then, using (21) in (19) one can
obtain a polynomial representation of degree L of f-1(x) as
df�1ðxÞ ¼XL
l¼1
nlxl: ð22Þ
Now, with knowing f-1, the intermediate variable v(k) can
be obtained from (18). By obtaining v(k), the unknown
Linear Time Invariant (LTI) parameter, bn, (15) can be
acquired based on normalized least mean square (NLMS)
when the quadrative criterion min{P
k=1K (e(k))2} is
minimized [19] as follows:
�bkþ1 ¼ �bk þ lx�ðkÞ
�þ kxðkÞk2eðkÞ; ð23Þ
where eðkÞ ¼ vðkÞ � dvðkÞ in which v(k) is achieved from
(15) and dvðkÞ has been defined in (18). Meanwhile, l is a
normalized step-size constant with 0 \ l\ 2 which con-
trols stability and the convergence speed of the algorithm.
When x(k) is large, the LMS algorithm experiences a
problem with gradient noise amplification. By normalizing
the LMS step size by kxðkÞk2in the NLMS algorithm,
however, this noise amplification problem is diminished.
At the same time, � is some small positive number which is
to use the modification to the NLMS algorithm faced with a
similar problem that occurs when kxðkÞk become too
small. The proposed scheme can be applied to other non-
linear systems as well and in comparison with the method
proposed in [6, 15] is faster and having low complexity
since the major complexity is the number of parameters
used in adaptive identification. Since, the linear filter with
no loss of generality due to the over parameterizations can
be defined as monic, the number of complex-valued
parameters of the conventional wiener model presented in
[6, 15] becomes N?P, while in our proposed approach
using the intermediate variable leads to reduce coefficients
which should be iteratively tracked. In fact, the number of
adaptive coefficients obtained from (23) are N ? 1.
In the last section, to see the performance of proposed
method of dynamic nonlinearity for HPA modeling,
we considered the OFDM system based upon WLAN
IEEE802.11a comparing the results with memoryless model
and presented approaches found in previous literatures.
3.2 Generalized wiener model
Memory effects as explained earlier can generally be cat-
egorized as linear and nonlinear memory effects. The first
group arise from time delays or phase shift in the PA
matching network while the latter group is due to the
trapping effects and bias network [4]. One possible accu-
rate algorithm to model the linear memory effect has been
explained in previous section. However an LTI filter does
not take into account the nonlinear memory effects and
cross-terms due to the interaction between the previous
samples. In order to model the nonlinear memory effects
the previous topology is improved by inserting the trun-
cated Volterra filter before the static nonlinearity as shown
in Fig. 2. As it is reported in [20] the extracted transfer
functions of the dynamic memory effect model show a
weak nonlinear behavior. Thus, a truncated Volterra filter
is sufficient to represent this weak nonlinearity. Including
the cross-terms in the model results in a more accurate
representation.
In discrete-time the second order Volterra filter with
finite memory systems can be given as follows [4]:
vðkÞ ¼XN1
i¼0
h1ðiÞxðk� iÞ þXN1
i¼0
XN2
j¼0
h2ði; jÞxðk� iÞjxðk� jÞj;
ð24Þ
where N1 and N2 are the memory durations of the first and
second order terms. x(k) and v(k) are the complex input and
output, respectively. h1(i) and h2(i,j) are the complex
Volterra kernels of the nonlinear order 1 and 2. Since we
need the Volterra filter for a mild nonlinearity, choosing
the second order of nonlinearity as well as cross-terms
would be sufficient which itself reduces the complexity of
the algorithm. The series can be written in vector form a
follows:
vðkÞ ¼ HT X; ð25Þ
where N1 = N2 = N and,
Fig. 2 Generalized Wiener model for PA
306 Analog Integr Circ Sig Process (2012) 72:303–312
123
H ¼ ½h1ð0Þ; . . .; h1ðNÞ; h2ð0; 0Þ; . . .; h2ð0;NÞ; h2ð1; 0Þ; . . .;
h2ð1;NÞ; . . .; h2ðN;NÞ�;X ¼ ½xðkÞ; . . .; xðk � NÞ; xðkÞjxðkÞj; . . .; xðkÞjxðk � NÞj;
xðk � 1ÞjxðkÞj; . . .; xðk � 1Þjxðk � NÞj; . . .;
xðk � NÞjxðk � NÞj�
It can be observed the Volterra filter is linear in the
coefficients. The number of unknown adaptive parameters
in the second order generalized wiener model are
(N ? 1) 9 (N ? 2). Applying the NLMS algorithm, the
filter coefficients are acquired by:
Hðkþ1Þ ¼ HðkÞ þ le�ðkÞXk
�þ kXkk2: ð26Þ
eðkÞ ¼ vðkÞ � vðkÞ in which v(k) and vðkÞ are given in (25)
and (18), respectively. The results of evaluating the models
by measured data will be presented in the last section.
4 An adaptive nonlinear predistorter in an OFDM
system
4.1 Generalized Hammerstein predistorter
identification
Considering the nonlinearity compensator section, an
adaptive predistorter, which is ideally the inverse of the
Wiener system, can be designed. The usual inverse struc-
ture of the Wiener system is the Hammerstein model [21].
As it can be seen the predistorter in Fig. 3 is constructed by
a memoryless nonlinear inverse filter cascaded by a linear
inverse filter. By using a polynomial form of finite order as
the memoryless nonlinear inverse filter, the predistorter can
be expressed as
uðkÞ ¼XM
m¼0
km
XQ
q¼1
aqxqðk � mÞ !
; ð27Þ
where M denotes the memory length of the linear inverse
filter and Q denotes the order of the nonlinearity.
vðkÞ ¼XN
n¼0
bn
XM
m¼0
km
XQ
q¼1
aqxqðk � m� nÞ ! !
: ð28Þ
And
wðkÞ ¼ f ðvðkÞÞ ¼XP
p¼1
c2p�1 vðkÞð Þ2p�1: ð29Þ
The Nonlinear subsystem in PA is constructed by PA
transfer function, AM/AM and AM/PM, obtained from
single tone measured. As it was shown previously, the filter
coefficients can not be evaluated precisely, since the output
equation is nonlinear versus unknown parameters. Thus,
the intermediate variable approach is introduced again to
estimate unknown parameters of the predistorter.
Also, we can derive (28) as follows:
vðkÞ ¼XN
n¼0
XM
m¼0
XQ
q¼1
hq;n;mxqðk � m� nÞ: ð30Þ
So, this is an extension Hammerstein model. Rewriting
(30) in matrix notation, we obtain (31):
vðkÞ ¼ CkXk; ð31Þ
where
Ck ¼ ½h0;0;1; . . .;h0;0;Q;h0;1;1; . . .;h0;M;Q;h1;0;1; . . .;hN;M;Q�;Xk
¼ ½xðkÞ; . . .;xQðkÞ;xðk� 1Þ; . . .;xQðk�MÞ;xðk� 1Þ; . . .;xQðk�M�NÞÞ�T
Ck; Xk are (N ? 1) 9 (M ? 1) 9 Q dimensional vectors
and also, ð�ÞT denotes ordinary transposition.
On the other hand, by knowing f-1 from the 3rd section,
v(k) can be determined from the following equation
vðkÞ ¼ f�1ðdðkÞÞ; ð32Þ
where the desired signal d(k) is the delayed version of the
input signal x(k) by r samples to account for causality of
the predistorter. Now, by knowing ^vðkÞ; xðkÞCk is obtained
by minimizing the mean square error, E{|e(k)|2}, of the
system. An adaptive algorithm for updating the coefficients
of the predistorter is given by applying the Normalized
Least Square Error method as follows:
Cðkþ1Þ ¼ CðkÞ þ le�ðkÞXk
�þ kXkk2: ð33Þ
In the (33), eðkÞ ¼ vðkÞ � vðkÞ in which v(k) and vðkÞ are
substituted from (31) and (32), respectively. Therefore, all
unknown parameters are calculated. So, the total system
behavior are estimated. Also, the mean square error,
E{|eT(k)|2}, of overall system can be evaluated by assuming
eT ¼ dðkÞ � dðkÞ where d(k) = f(v(k)) in which v(k) is the
outcome of estimating all unknown parameters. This pro-
cedure is illustrated in Fig. 4.Fig. 3 Wiener PA and Hammerstein PD for HPA with memory effect
Analog Integr Circ Sig Process (2012) 72:303–312 307
123
5 Experimental setup
For the experimental validation, two-tone output was
measured versus tone-spacing (10-20000 KHz) and input
power sweeping of the MRF7S38010HR3 from Freescale
Semiconductor. The HPA is a 4 watt N-Channel
Enhancement-Mode Lateral MOSFET class-AB PA with
22-dB gain that has the operating frequency at 3.5 GHz.
The third order intermodulation distortion (IMD3) of the
PA is plotted in Fig. 5. The PA versus frequency and input
power level is quite variable, which result memory effect.
Since the contribution of the memory effect branches is
large for the HPA, using the proposed predistortion con-
sidering memory is essential. Consequently, it can be easily
shown that the improvement achieved by using memory-
less predistortion is nearly negligible. The measured data
and corresponding seventh-term power series representa-
tions of AM/AM and AM/PM characteristics are described
in Fig. 6.
The experimental setup shown in Fig. 7 is used to
demonstrate the validity of the proposed PA model and the
predistortion technique for compensation of a class-AB
PA. The test environment used to measure and model the
transmitter integrates an Agilent Vector Signal Generator
(ESG) (E4433B, Agilent Technologies), an Agilent Vector
Signal Analyzer (VSA) (E4406, Agilent Technologies).
Therefore, the transmitter prototype including the RF
vector modulator, digital-to-analog converter is physically
implemented with the ESG and PA. The host digital signal
processor (DSP) is implemented with a personal computer
(PC), where the in-phase/quadrature (I/Q) signal is initially
synthesized using Agilent Signal Studio-IEEE 802.11a
(OFDM). In this paper, a block of 1024 64-QAM symbols
at a data rate of 54 Mbps with the coding rate of 3/4 using
IFFT to modulate them to the 52 subcarriers is generated.
Also, the idle interval period between frames is set
100 lsec to avoid spectrum overlapping [22]. The sam-
pling rate used in the experimental is equal to 20 MHz. As
shown in Fig. 7, both the VSA is synchronized by a
10-MHz reference signal from the ESG.
The baseband I/Q signal is firstly preprocessed by the
predistortion algorithm using Matlab software and then
downloaded to the I/Q arbitrary waveform generator of the
ESG via the general-purpose interface bus (GPIB) interface
with the help of the dynamic link existing between the
ADS and ESG. After that, the predistorted baseband signal
is modulated to an RF carrier in the ESG and fed to the PA.
In this way, the ADS in the host DSP, the ESG, and the PA
work together to form a baseband linearized transmitter
prototype. The baseband data at the output of the trans-
mitter is captured by the VSA as an RF receiver, consisting
of RF/IF down-converter, a high-speed analog-to-digital
converter, a digital down-converter, and the host DSP. In
this study, the receiver prototype is physically constructed
by the VSA and a PC. The received signal is down-con-
verted and digitized and converted to digital baseband I and
Q signals. Next, the baseband I and Q data is captured by
ADS in the PC via the LAN interface. The time delay
between input and output baseband I and Q sequences
should be estimated by the covariance-based algorithm
Fig. 4 An adaptive predistorter
for the HPA preceded by a
linear filter
−30−25
−20−15
−10−5
00.5
11.5
2
x 107
−60
−50
−40
−30
−20
−10
Input Power (dBm)Tone Spacing (Hz)
IMD
3(dB
)
Fig. 5 Measured IMD3 versus tone spacing and power input for
MRF183
308 Analog Integr Circ Sig Process (2012) 72:303–312
123
[23]. The captured baseband data at the input and output of
the transmitter are processed in MATLAB in order to
estimate the parameters of the predistorter. Finally, the
obtained predistorter parameters are sent to the ADS to
update the corresponding predistorter parameters. The
evaluation of performance of the different predistorters
will be confirmed by comparing the output spectra of
the transmitter obtained with the various predistorter
approaches.
6 Experimental results
In this section, first of all we investigate the effectiveness
of modeling of PA by our new approach. Figure 8 illus-
trates the spectra of measured output signal of transmitter,
the memoryless modeling of PA and two wiener models of
PA with proposed approaches. It clarifies that the Wiener
model produced an output spectrum closed to the measured
one,while the memoryless one do not contribute precisely
to the modeling performance. However, as illustrated in
Fig. 9 the spectrum differences between the actual trans-
mitter and Wiener model is considerable. Indeed, this
0 0.1 0.2 0.3 0.4 0.5−88
−86
−84
−82
−80
−78
−76
−74
−72
−70
−68
Input Amplitude (v)
Out
put P
hase
(de
g.)
Simulated DataPower Series Fit
(a)
0 0.1 0.2 0.3 0.4 0.50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Input Amplitude (v)
Out
put A
mpl
itude
(v)
Simulated DataPower Series Fit
(b)
Fig. 6 a SSPA AM-PM characteristic. b SSPA AM-AM characteristic
Fig. 7 Experimental setup for modeling PA and PD
−1.5 −1 −0.5 0 0.5 1 1.5
x 107
−50
−45
−40
−35
−30
−25
−20
−15
−10
Normalized frequency(Hz)
PS
D (
dBm
)
Transmitter measurementMemoryless modelWiener modelGeneralized Wiener model
Fig. 8 Output PSD response of measured signal and proposed
models for PA modeling and memoryless modeling
−1.5 −1 −0.5 0 0.5 1 1.5
x 107
−3
−2
−1
0
1
2
3
Normalized frequency (Hz)
PS
D d
iffer
ence
(dB
)
(a)
(b)
Fig. 9 Spectrum difference between a Wiener model and b general-
ized Wiener model with transmitter measured signal
Analog Integr Circ Sig Process (2012) 72:303–312 309
123
algorithm modeled the nonlinearity with memory effect but
it is not sufficient to represent the contribution of the
memory effect in out-of-band emission. On the other word,
the linear filter in the wiener model considers the linear
distortion and not the nonlinear even-order distortion
sources. It implies weak nonlinearity still exists after
removing the static nonlinearity. For this reason the
improved inclusive memory effect model has been pro-
posed to model the nonlinearity of transmitter with mem-
ory effect accurately. It can be seen that there is a very
close correspondence between the Generalized Wiener
model and the measured output in presence of the wide-
band modulated signal. At the same time, comparing
accuracy and convergence speed of the proposed technique
with similar literatures, Table 1 depicts the normalized
mean square error and the number of symbols required to
converge the algorithms. Also, the the number of param-
eters in different approaches are compared in Table 1
because as previously mentioned, the number of adaptive
parameters is a measure of the model complexity. It can be
seen that if the nonlinearity of PA has weak memory
effects, the wiener model with our novel approach to
estimate coefficients is sufficient from the point of view of
the complexity, the accuracy and number of symbols
required to converge. Nevertheless, the Generalized wiener
model is well accurate with reasonable complexity and
convergence speed to model nonlinearity with stronger
memory effects in comparison with similar literatures. In
addition, as described in Table 1, the number of adaptive
parameters in wiener model [6] and our proposed wiener
are equal, while the gradient function given in (16) in [6] is
nonlinear versus unknown parameters. So, it requires larger
number of symbols to reach the similar accuracy with our
presented method in which the gradient function is linear
versus unknown parameters. This issue also exists between
volterra series in [24] and generalized wiener model.
Indeed, the volterra series is time consuming for strongly
memory effects and it might be impractical in order to
reach better accuracy.
Figure 10 displays the normalized spectra of transmitter
for OBO = 4 dB in different situations: measured output
signal without PD, with memoryless PD, Generalized
Hammerstein PD and desired linearized output. It shows
the presented method of PD provides output back-off,
Table 1 Summary of the results of normalized mse for PA models
Model Order NMSE
[dB]
No. of
Param.
No. of
symbolsLin. Non
Lin.
Mem. less PA – 7 -20.53 7
Wiener PA [6] 5 3 -45 8 60000
Proposed wiener
PA
7 7 -47 8 4400
Generalized
wiener PA
2 7 -50 12 12000
Volterra PA [24] 1 3 -39 12
Memory
polynomial
[10]
4 5 -38.5 20 24000
RFBNN [7] Memory
depth
1
-38.3 42
Table 2 The normalized mse values for different pd models
Model OBO
[dB]
Order NMSE
[dB]
No. of
Symb.Lin. Non
Lin.
Mem. less PD 4 – 7 -30 non-Adapt.
Generalized
Hamm. PD
4 3 7 -51 8200
PA [6] 4 3 7 -45 11500
PA [21] 10 3 -37.7 5000
3.485 3.49 3.495 3.5 3.505 3.51 3.515x 10
9
−50
−45
−40
−35
−30
−25
−20
−15
Frequency (Hz)
Nor
mal
ized
PS
D (
dB)
Mask
Memoryless PD
Generalized Hamm. PD
Desired Signal
PA without PD
(a)
3.481 3.483 3.485 3.487 3.489 3.491
x 109
−50
−45
−40
−35
−30
−25
Frequency (Hz)
Nor
mal
ized
PS
D (
dB)
Memoryless PD
PA without PD
Generalized Hamm. PD
Desired Signal
(b)
Fig. 10 Spectrum comparison of the transmitter with different
Hammerstein PDs and memoryless PD. a Full spectrum comparison.
b Zoom-in spectrum comparison
310 Analog Integr Circ Sig Process (2012) 72:303–312
123
4 dB, without suffering to exceed the the IEEE802.11a
standard spectrum limitation [22], indicating a successful
test result, while the test fails for OBO = 4 dB without any
PD . Considering Fig. 10, we can see the results of pro-
posed model tends toward the ideal linear case. However,
the memoryless PD is not sufficient to compensate the
nonlinearity due to memory effect. To prove the validity of
presented method comparing with previous literatures, we
have made similar considerations on MSE of the method. It
should be noted that the normalized mean square error
between the input of the predistorter and the output of the
PA is calculated. Figure 11 shows the ACPR at the output
of the transmitter block for the various predistorters,
which are assessed for several frequency offsets (20, -15,
-13, -12, -10, 10, 12, 13, 15, and 20 MHz) from the
central frequency, within an 18 MHz bandwidth. It can be
seen that the Generalized Hammerstein predistorter
decreases the ACPR dramatically, closely approaching the
ideal case, however the memory-less predistortion is not
effective in reducing the ACPR.
7 Conclusion
In this paper, we have proposed a linear novel parametric
estimation technique using Hilbert space to model the
nonlinearity of RF PAs with linear memory effects based
on wiener system. At the same time, we present the Gen-
eralized wiener model in order to model the nonlinear
memory effects by inserting the second order Volterra filter
before the static nonlinearity. The models were also com-
pared to memoryless model derived from single-tone
measurements. It was seen that the inclusion of memory
effects afforded by the proposed methods improved the
accuracy of output response dramatically. In addition, the
efficient adaptive predistortion technique is proposed
which can compensate for the distortion caused by an HPA
with a linear filter, in an OFDM system. From measure-
ment results, it is confirmed that the proposed adaptive
predistorter is very effective in reducing the nonlinear
distortion of an HPA with memory effect in OFDM
systems.
References
1. Jeon, W. G., Chang, K. H., & Cho, Y. S. (1997). An adaptive data
predistorter for compensation of nonlinear distortion in OFDM
systems. IEEE Transactions on Communications, 45(10),
1167–1171.
2. Ku, H., & Kenney, J. (2003). Behavioral modeling of nonlinear
RF power amplifiers considering memory effects. IEEE Trans-actions on Microwave Theory and Techniques, 51(12),
2495–2504.
3. Bosch, W., & Gatti, G. (1989). Measurement and simulation of
memory effects in predistortion linearizers. IEEE Transactionson Microwave Theory and Techniques, 37(12), 1885–1890.
4. Zhu, A., Pedro, J. C., & Brazil, T. J. (2006). Dynamic deviation
reductionbased Volterra behavioral modeling of RF power
amplifiers. IEEE Transactions on Microwave Theory and Tech-niques, 54(12), 4323–4332.
5. Kenney, J. S., Woo, W., Ding, L., Raich, R., Ku, H., & Zhou, G.
T. (2001). The impact of memory effects on prediction lineari-
zation of RF power amplifiers. In Proceedings of the 8th inter-national symposium on microwave and optical technology (pp.
189–193). Montreal, QC, Canada, June 19–23, 2001.
6. Kang, H. W., Soo, Y. C., & Youn, D. H. (1999). On compen-
sating nonlinear distortions of an OFDM system using an efficient
adaptive predistorter. IEEE Transactions on Communications,47(4), 522–526.
7. Isaksson, M., Wisell, D., & Ronnow, D. (2006). A comparative
analysis of behavioral models for RF power amplifiers. IEEETransactions on Microwave Theory and Techniques, 56(1),
348–359.
8. Ku, H., McKinley, M. D., Kenney, & Kenney J. S. (2002).
Quantifying memory effects in RF power amplifiers. IEEETransactions on Microwave Theory and Techniques, 50(12).
9. Saleh, A. A. M. (1981). Frequency-independent and frequency-
dependent nonlinear models of TWT amplifiers. IEEE Transac-tions on Communications COM-29(11), 1715–1720.
10. Morgan, D., Ma, Z., Zierdt, M. G., & Pastalan, J. (2006). A
generalized memory polynomial model for digital predistortion
of RF power amplifiers. IEEE Transactions on Signal Processing,54(10), 3852–3860.
11. Raich, R., Qian, H., & Zhou, G. T. (2004). Orthogonal polyno-
mials for power amplifier modeling and predistorter design. IEEETransactions on Vehicular Technology, 53(5), 1468–1479.
12. Liu, T., Boumaiza, S., & Ghannouchi, F. M. (Apr. 2006). Aug-
mented hammerstein predistorter for linearization of broad-band
wireless transmitters. IEEE Transactions on Microwave Theoryand Techniques, 54(4).
13. Gard, K. G., Gutierrez, H. M., & Steer, M. B. (1999). Charac-
terization of spectral regrowth in microwave amplifiers based on
the nonlinear transformation of a complex Gaussian processes.
IEEE Transactions on Microwave Theory and Techniques, 47,
1059–1069.
−20 −15 −13 −12 −10 10 12 13 15 20−70
−67
−64
−61
−58
−55
−52
−49
−46
−43
−40
−37
Frequency Offset (MHz)
AC
PR
(dB
c)PA without PDMemoryless PDGeneralized Hamm. PDDesired Signal
Fig. 11 ACPR comparison of the transmitter with different Ham-
merstein predistorters, without predistorter and with memoryless
predistorter
Analog Integr Circ Sig Process (2012) 72:303–312 311
123
14. Pawlak, M., Hasiewicz, Z., & Wachel, P. (2007). On nonpara-
metric identification of wiener systems. IEEE Transactions onSignal Processing, 55(2), 482–492.
15. Eun, C. S., & Powers, E. J. (1995). A predistorter design for a
memoryless nonlinearity preceded by a dynamic linear system. In
Proceedings of GLOBECOM (pp. 152–156).
16. karkhaneh, H., Sadeghpour, T., Ghorbani, A., & Alhameed, R. A.
(2007). Modeling and linearization method for nonlinear power
amplifier with memory effect for wideband application. In
International conference on electromagnetics in advancedapplications, ICEAA’07, Italy.
17. Dudley, D. G. (1994). Mathematical foundations for electro-magnetic theory. Reading, MA: IEEE
18. Aghasi, A., Karkhaneh, H., & Ghorbani, A. (2007). A modified
model and linearization method for solid state power amplifier.
Analog Integrated Circuits and Signal Processing, 51(2), 81–88.
19. Hayes, M. H. (1996). Statistical digital signal processing andmodeling. USA: John Wiley & Sons Inc.
20. Liu, T., Boumaiza, S., & Ghannouchi, F. M. (2005). De-
embedding static nonlinearities and accurately modeling and
identifying memory effects in wideband RF transmitters. IEEETransactions on Microwave Theory and Techniques, 53,
3578–3587.
21. Chiu, Mao., Zeng, C., & Liu, M. (2001). Predistorter based on
frequency domain estimation for compensation of nonlinear
distortion in OFDM systems. IEEE Transactions on VehicularTechnology, 57(2), 938–946.
22. IEEE Std 802.11a-1999. (1999). Part 11: wireless lan mediumaccess control (MAC) and physical layer (PHY) specifications:High-speed physical layer in the 5 GHz band.
23. Liu, T., Ye, Y., Zeng, X., & Ghannouchi, F.M. (2008). Accurate
time-delay estimation and alignment for RF power amplifier/
transmitter characterization. In 4th IEEE interernational confer-ence on circuits and systems for communications (pp. 70–74).
Shanghai, China: ICCSC.
24. Crespo-Cadenas, C., Madero-Ayora, J., & Munoz-Cruzado, M. J.
(2010). A new approach to pruning Volterra models for power
amplifiers. IEEE Transactions on Signal Processing, 58(4),
2113–2120.
Haleh Karkhaneh was born in
Tehran, Iran, on July 6, 1984.
She received the B.Sc. degree
from Shahed University, Teh-
ran, Iran in 2005 and the M.S.
degree from Amirkabir Univer-
sity of Technology (Tehran
Polytechnic), in 2008, both in
electrical engineering. She is
currently working toward the
Ph.D degree at Amirkabir Uni-
versity of Technology, Tehran,
Iran since 2008. Her working
experiences are mobile network
and her research interests
include RF circuit characterization and modeling, and Linearization
of microwave nonlinear components, optical communications. She is
currently working on nonlinear impairments in Optical Coherent
OFDM systems
Ayaz Ghorbani received Post-
graduate Diploma, M.Phil., and
Ph.D. degrees in electrical and
communication engineering as
well as postdoctoral degree
from the University of Bradford,
UK, in 1984, 1985, 1987, and
2004, respectively. Since 1987
up to now he has been teaching
various courses in the Depart-
ment of Electrical and Electrical
Engineering, AmirKabir Uni-
versity of Technology (Tehran
Polytechnic), Tehran, Iran. Also
from 2004 to 2005, he was with
Bradford University for sabbatical leave. He has authored or coau-
thored more than 120 papers in various national and international
conferences as well as refereed journals. In 1987, Dr. Ghorbani
received John Robertshaw Travel Award to visit USA. In 1990, he
received the URSI Young Scientists Award at the General Assembly
of URSI, Prague, Czech Republic. He also received the Seventh and
Tenth Kharazmi International Festival Prize in 1993 and 1995 for
designing and implementation of anti-echo chamber and microwave
subsystems, respectively. His research areas include Radio wave
propagation, antennas bandwidth, nonlinear modeling of HPA, anti-
echo chambers modeling and design, electromagnetic shielding as
well as EMI/EMC analysis and modeling. He has authored one book
in Microwave circuit and devices.
Hamidreza Amindavar recei-
ved B.Sc., M.Sc., M.Sc.
AMATH, and Ph.D. degrees
from the University of Wash-
ington in Seattle, in 1985, 1987,
and 1991, respectively, all in
electrical engineering. He is
currently a Professor in the
Department of Electrical Engi-
neering, Amirkabir University
of Technology, Tehran, Iran.
His research interests include
signal and image processing,
array processing, and multiuser
detection. Prof. Amindavar is a
member of Tau Beta Pi and Eta Kappa Nu.
312 Analog Integr Circ Sig Process (2012) 72:303–312
123
Recommended