Pruned Volterra Models with Memory Effects for Nonlinear Power Amplifiers

doi:10.4236/cn.2013.53B2102

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Communications and Network, 2013, 5, 570-572

http://dx.doi.org/10.4236/cn.2013.53B2102 Published Online September 2013 (http://www.scirp.org/journal/cn)

Pruned Volterra Models with M emory Effects for

Nonlinear Power Amplifiers

Pengpeng Li1, Qingfang Zhang1, Ping Wang1, Zhongshan Xie2, Bing Liu2

1Shanghai Institute of Microsystem and Information Technology, Shanghai, China

2Nanjing University of Aeronautics and Astronautics, Nanjing, China

Email: [email protected], [email protected]

Received May 2013

ABSTRACT

In this letter, a novel model is proposed for modeling the nonlinearity and memory effects of power amplifiers. The

classical Volterra model is modified through a function of the sum of nonlinearity order with sum of memory length.

The parameters of this model can be extracted in digital domain since the model is analyzed based on the envelope sig-

nals. The model we proposed enables a substantial reduction in the number of coefficients involved, and with excellent

accuracy.

Keywords: Volterra Series; Power Ampliﬁer (PA); Behavioral Model; Memory Effect

1. Introduction

To handle multi-carrier envelope varying signals with

wide bandwidths in modern wireless communication sys-

tem, the signals passing through the transistor and power

amplifiers should be able to predict accurately. In the

behavioral model, the nonlinear component is generally

considered as a “b lack-box ” which is completely charac-

terized by external responses, in terms of input and out-

put signals, through the use of relatively simple mathe-

matical expressions. The Volterra series is a general non-

linear model with memory and has been used by many

researchers to characterize power amplifiers [1,2]. Un-

fortunately, the number of coefficients of the Volterra-

based models becomes unacceptable in practical imple-

mentation as the accuracy level increased, such models

become useless.

To overcome the modeling complexity, various mod-

el-order reduction approaches have been proposed to

simplify the Volterra m odel str ucture. For e xample, Wiener,

Hammerstein, Wiener-Hammerstein [3,4] and Memory

Polynomial [5] models are the most popular approxima-

tions. Physical knowledge has been considered in recently

proposed models by taking into account the real behavior

of the PA [6]. And the device electrical properties were

also considered in the reduction [7].

Although these simplified models have been employed

to characterize PAs with reasonable accuracy under cer-

tain conditions, the number of coefficients to be esti-

mated is still increasing prominently with the degree of

nonlinearity and memory length of the system. While the

interaction be tween the nth order and memory length has

not attracted enough attention.

The model proposed in this work is based on an un-

derlying general Volterra model of the PA. By focusing

on the fact that output items is fading with the non-li-

nearity order and memory length increase, the Volterra

model is modified. As the function of nonlinearity and

memory effects applied, a slow growth of the coefficients

can be obtained while the accuracy can be improved. The

letter is organized as follows: after this introduction, Vol-

terra model is modified based on the nonlinearity and

memory effects analyzed in Section 2. The model’s per-

formance is shown in Se ction 3. At last the conclusion is

presente d in Sect ion 4.

2. Principle of Proposed Model

A Volterra series is a combination of linear convolution

and a nonlinear power series so that it can be used to

describe the input/output relationship of a general nonli-

near, causal and time-invariant system with fading mem-

ory.

(1)

where and represents the input and the

output, P and M are the order of nonlinearities and the

memory length, respectively, and is the

discrete time Volterra kernels of order n.

In the application of wide-band system, memory ef-

∑∏

∑ ∑

= =

−=

jjpp

inxiihny

1 0

)(),...,(...)(

)(

),...,(h 1p

P. P. LI ET AL.

571

fects of the amplifier are very significant and have an

important impact on linear effects. Memory effects [8]

can be classified into electrical and thermal. The electric-

al memory effects arise as a consequence of the variation

of the impedance along the signal bandwidth modulation.

As the output of the power amplifier nonlinearity will

vanish at the infinite order and the memory effects will

fade out with time passing by, there must be an exponen-

tial role in the amplitude function.

To make it easy to understand the proposed model,

first we consider the output of the memoryless system is

a function of nonlinearity order p, which can be ex-

pressed as follows:

(2)

When the polynomial contains memory, the change of

the amplitude must be the function of memory which can

be written as

(3)

According to Equation (3), the function of memory

can be written as

(4)

When we consider the error of the polynomial, if

, the polynomial will be replaced by

ε (ε can be calculated by the value of IMD and gain of

the PA). Obviously the output can get the maximum val-

ue when the input takes the maximum value, so we can

get the function of the memory with the nonlinearity or-

der p as follo w ing (the input has been norm a lized):

(5)

Considering the memory effects of PAs, the function

of memor y g(m) would fade gradually, as g(0) = 1, g(∞)

= 0 , we can assume and that corr esponds

with the characteristic of capacitance or inductive com-

ponents which can release the energy, then Equation (5)

can be written as

(6)

And the multiply items such as

can be written as

(7)

where is a function which depends on the nonli-

nearity of the PAs and decreased as the nonlinearity or-

der increased. To simplify the analyzing, we can assume

(has been normalized as same as input),

then last-written equation can be written as

(8)

To attain the value of β and

, the input p ower, IMD

and gain of the PA must be known. For example, assum-

ing the parameters of a definite PA are , IMD5

= I5 and gain = G. From the definition of IMD5, I5 can be

written as

5log10=

(9)

as and , so

can be

determined by

(1 0)

Take (10) into (8), since , then the β can be

determined by

(11)

Note that α is the only unknown variable in (8) and α

depends on the memory effects of PA. However, the

memory effects of PA cannot be tested, so we attain a

relative accurate α through comparing model perfor-

mance of different α.

Evidently the function of and p is a decreasing

function so that the coefficients can be decreased rapidly

when the system’s order increased.

Maintaining the Integrity of the Specifications

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Please do not revis e an y of the current designations.

3. Model Performance

In order to validate the proposed behavioral model in a

real system, a Doherty PA was tested. This PA was oper-

ated at 460 MHz and excited by an OFDM signal with 20

MHz bandwidth.

The parameters of tested PA were Pin = 10 W, IMD5

= −40 dBc and gain = 20 dB. So the ε and β can be fixed

by (10) and (11). Thus (8) can be written as

2.48.0 ≤+

∑∑

pmα

(12 )

∑∑

nxpfnyny

)()()()(

mnxmgpfmny

−=−

)(

)()(

)(

mnxpf

mny

−

<− ))((

mnyMax

)

)(

()(

gpm ε

−

≤

engα

−

=)(

)

)(

ln(

)(

pm ε

−≤

)()( 21

mnxmnx

−−

)

)(

ln(

∏

∑

−≤

mε

)(

epf

−

=)(

εβα

ln−≤+

∑∑

502

×=

max

20/

−

∑

20/

−

−=

∑

P. P. LI ET AL.

572

In this test, the nonlinearity of the model was truncated

to order 5. For comparison, the value of α was set from

0.2 to 2. To evaluate the model’s fidelity in the time do-

main, the NMSEs and number of coefficients for each

partial model were calculated. These results are shown in

Figure 1. Due to the fact that memory effects of different

PAs were not the same，so the value of α was different

for different PAs. In this paper, the value of α was set to

1.6 while the performance of proposed model for the

tested PA was better and with lesser number of coeffi-

cients.

To show the model accuracy in the frequency domain,

the spectra of modeled errors are plotted in Figure 2. It

can be seen that the error signal spectrum of our pro-

posed model is very small, while significant errors are

generated in the output predicted by the memoryless mod-

el. For reference, the spectrum of the simulated output is

also plot t e d in Figure 2.

4. Conclusion

An efficient and effective Volterra model pruning me-

Figure 1. Model performance in the time domain.

Figure 2. Sample frequency domain output and modeled

error spectra.

thod for RF PAs has been presented in this letter, which

based on a function of the sum of nonlinearity order with

the sum of memory length. The advantage of this model

reduction approach is that it allows ef ficient reduction of

the model complexity, while keeping the essential prop-

erties caused by memory effects of a real PA. With a

Doherty PA tested, the proposed model can be employed

to characterize a nonlinear PA with memory effects in

high accuracy.

5. Acknowledgements

The research work is supported by Chinese Major Na-

tional Science and Technology Projects

(No.2010ZX03007-003).

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