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.

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−20 −15 −13 −12 −10 10 12 13 15 20−70

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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