Radio Frequency Power Amplifiers Behavioral Modeling, Parameter-Reduction,

and Digital Predistortion

Magnus Isaksson

Doctoral Thesis in Telecommunications Stockholm, Sweden 2007

II

TRITA–EE 2007:010 KTH School of Electrical Engineering ISSN 1653–5146 SE-100 44 Stockholm ISBN 978–91–7178–589–3 SWEDEN Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i telekommunikation fredagen den 25 maj klockan 13.00 i sal 99:131 på Högskolan i Gävle, Kungsbäcksvägen 47, Gävle. © Magnus Isaksson, mars 2007 Tryckt av Universitetsservice US AB

III

Abstract This work considers behavioral modeling, parameter-reduction, and digital predistortion

of radio frequency power amplifiers. Due to the use of modern digital modulation methods, contemporary power amplifiers are frequently subjected to signals characterized by considerable bandwidths and fast changing envelopes. As a result, traditional quasi-memoryless amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) characteristics are no longer sufficient to describe and model the behavior of power amplifiers; neither can they be successfully used for linearization.

In this thesis, sampled input and output data are used for identification and validation of several block structure models with memory. The time-discrete Volterra model, the Wiener model, the Hammerstein model, and the radial-basis function neural network are all identified and compared with respect to in-band and out-of-band errors. Two different signal types (multitones and noise), with different powers, peak-to-average ratios, and bandwidths have been used as inputs to the amplifier. Furthermore, two different power amplifiers were investigated, one designed for third generation mobile telecommunication systems and one for second generation systems.

A stepped three-tone measurement technique based on digitally modulated baseband signals is also presented. The third-order Volterra kernel parameters were determined from identified intermodulation products. The symmetry properties of the Volterra kernel along various portions of the three dimensional frequency space were analyzed and compared with the symmetry of the Wiener and Hammerstein systems.

The Kautz-Volterra model, a new type of behavioral model for radio frequency power amplifier modeling, is proposed and compared with existing behavioral models with respect both to accuracy and complexity. The proposed model has the same general properties as the Volterra model and has been shown experimentally to be appropriate for use as a either direct or inverse model. Furthermore, the high accuracy of the Kautz-Volterra model allows use of a frequency weighting algorithm to suppress the out-of-band error to low levels. This thesis gives the experimental results of the Kautz-Volterra model used with frequency weighted data as well as reports on the use of this model as a digital predistortion algorithm for radio frequency power amplifiers.

The main results in this thesis include two novel behavioral radio frequency power amplifier models, a method to measure and analyze Volterra kernels along certain paths in the frequency domain, and the conclusion that when designing a PA for use with digital predistortion it is most important to reduce the memory effects in the linear term since these will have the most significant impact on the inverse function.

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Sammanfattning Det här arbetet behandlar beteendemodeller, parameterreduktion och digital predistorsion

av effektförstärkare på radiofrekvens. På grund av användandet av moderna digitala modulationsmetoder är nutida effektförstärkare ofta utsatta för signaler med betydande bandbredder och snabba förändringar i sin amplitud. Detta resulterar i att traditionella kvasi-minneslösa amplitud-till-amplitud (AM/AM) och amplitud-till-fas (AM/PM) distorsionskurvor inte längre är tillräckliga beskrivningar av effektförstärkares beteende. Inte heller kan de med framgång användas för linjärisering.

I den här avhandlingen används samplade in- och utdata för identifiering och validering av flera blockmodeller med minne. Den tidsdiskreta Volterramodellen, Wienermodellen, Hammersteinmodellen och det radial-bas funktions baserade neurala nätet är alla identifierade och jämförda med hänsyn till felen såväl inom som utanför bandet. Två olika typer av signaler (multitonsignaler och brus) med olika effekter, topp-till-medelvärdes förhållanden och bandbredder har användts som insignaler till förstärkaren. Dessutom har två olika typer av effektförstärkare ingått i undersökningen, den ena utvecklad för tredje generationens mobiltelefonisystem och den andra för andra generationens system.

En stegad tretonsmätteknik baserad på digitalt modulerade basbandssignaler presenteras också. Parametrarna för den tredje ordningens Volterrakärna bestämdes med hjälp av identifierade intermodulationsprodukter. Volterrakärnans symmetriegenskaper analyserades i olika delområden av den tredimensionella frekvensrymden och jämfördes med symmetrierna hos Wiener och Hammersteinsystem.

Kautz-Volterramodellen, som är en ny typ av beteendemodell för effektförstärkare på radiofrekvens, föreslås och jämförs med avseende på noggrannhet såväl som komplexitet. Den föreslagna modellen har samma generella egenskaper som Volterramodellen och har experimentellt visat sig användbar både som direkt och invers modell. Vidare, Kautz-Volterramodellens höga noggrannhet tillåter användandet av en frekvensviktningsalgoritm för att undertrycka felen utanför bandet till låga nivåer. Den här avhandlingen ger såväl experimentella resultat av Kautz-Volterramodellen tillsammans med frekvensviktat data som rapporterar användandet av modellen som en digital predistorsionsalgoritm för effektförstärkare på radiofrekvens.

Avhandlingens huvudsakliga resultat inkluderar två stycken nya beteendemodeller för effektförstärkare på radiofrekvens, en metod för att mäta och analysera Volterrakärnor längs bestämda vägar i frekvensdomänen och slutsatsen om att det vid design av en effektförstärkare som skall användas för digital predistorsion är minneseffekterna i den linjära termen som är av störst betydelse eftersom dessa kommer att påverka den inversa funktionen mest.

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Acknowledgments

First, I would like to thank my supervisor and co-author, Dr. Daniel Rönnow, and my supervisor, Prof. Peter Händel. I want to thank Daniel for his invaluable academic and scientific supervision, for his tireless reading of manuscripts, and for his friendship and companionship throughout our travels around the world. Prof. Händel, my supervisor in the later stages of my doctoral studies, managed this responsibility respectfully. I also want to thank my co-author David Wisell for a rewarding cooperative relationship and for being a good friend. The steering group of this project and particularly Prof. Niclas Keskitalo, for his excellent leadership, are all acknowledged. Prof. Edvard Nordlander and Tekn. Lic. Lars Pettersson, for their inspiring attitudes and for talking me in to graduate studies in the beginning of this journey, are both acknowledged as well.

My thanks also to all the people in the department of ITB/Electronics at the University of Gävle for their contributions towards creating a very pleasant working environment, and specifically to my Ph.D. student colleagues Niclas Björsell, Olof Bengtsson, and José Chilo for interesting and fruitful discussions, and to Prof. Claes Beckman, whose dynamic mentality has been a source of inspiration.

This work has been financially supported by the University of Gävle, the Knowledge Foundation (KKS), Ericsson AB, Note AB, Syntronic AB, Racomna AB, and Rohde & Schwarz GmbH. The author especially wants to thank Ericsson AB for providing the power amplifiers and Rohde & Schwarz GmbH for providing the signal generator and the vector signal analyzer to our radio frequency laboratory at the University of Gävle.

I also want to thank my family, and especially my wife Linnéa, for their wonderful support along the whole journey. Without you this thesis would never have been completed.

Magnus Isaksson Gävle, March 2007.

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VII

TABLE OF CONTENTS

APPENDED PAPERS....................................................................................... IX

RELATED PAPERS NOT INCLUDED IN THE THESIS ................................... IX

1 INTRODUCTION............................................................................................. 1

2 BEHAVIORAL MODELING OF RADIO FREQUENCY POWER AMPLIFIERS .................................................................................... 5

2.1 Nonlinear dynamic description of an RF PA ................................................6

3 MEASUREMENTS AND MODEL VALIDATION .......................................... 11

3.1 Measurement set-up ......................................................................................11

3.2 Experimental Power Amplifiers...................................................................12

3.3 Digital Signal Processing Techniques ..........................................................13

3.4 Model Validation ...........................................................................................15

4 MODELING RADIO FREQUENCY POWER AMPLIFIERS USING RBFNNS ........................................................................................... 19

4.1 The Neural Network......................................................................................19

4.2 RBFNN Modeling Results.............................................................................21

5 COMPARISON OF BEHAVIORAL PA MODELS......................................... 25

5.1 A Comparative Analysis ...............................................................................26

6 IDENTIFICATION OF FREQUENCY DOMAIN VOLTERRA KERNELS ...... 29

6.1 The Third-Order Kernel...............................................................................29

6.2 Investigation of Symmetry Properties .........................................................32

7 A PARAMETER-REDUCED VOLTERRA MODEL....................................... 35

8 DIGITAL PREDISTORTION OF RF PAS...................................................... 41

VIII

9 CONCLUDING REMARKS........................................................................... 47

REFERENCES................................................................................................. 49

SUMMARY OF APPENDED PAPERS ............................................................ 57

IX

Appended Papers

PAPER I M. Isaksson, D. Wisell, and D. Rönnow, "Wide-Band Dynamic Modeling of Power Amplifiers Using Radial-Basis Function Neural Networks," IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3422-3428, Nov. 2005.

PAPER II M. Isaksson, D. Wisell, and D. Rönnow, "A Comparative Analysis of

Behavioral Models for RF Power Amplifiers," IEEE Trans. Microwave Theory Tech., vol. 54, no. 1, pp. 348-359, Jan. 2006.

PAPER III D. Rönnow, D. Wisell, and M. Isaksson, "Three-Tone Characterization

of Nonlinear Memory Effects in Radio Frequency Power Amplifiers," IEEE Trans. Instrum. Meas. Accepted.

PAPER IV M. Isaksson and D. Rönnow, "A Parameter-Reduced Volterra Model

for Dynamic RF Power Amplifier Modeling based on Orthonormal Basis Functions," Int. J. RF and Microwave Computer-Aided Eng. In Press.

PAPER V D. Rönnow and M. Isaksson, "Digital predistortion of radio frequency

power amplifiers using a Kautz-Volterra model," Electron. Lett., vol. 42, no. 13, pp. 780-782, June 2006.

Related Papers Not Included in the Thesis VI M. Isaksson and D. Wisell, "Extension of the Hammerstein Model for

Power Amplifier Applications," in 63rd ARFTG Conf. Dig., Fort Worth, TX, 2004, pp. 131-137.

VII M. Isaksson, D. Wisell, and D. Rönnow, "Nonlinear Behavioral Modeling

of Power Amplifiers Using Radial-Basis Function Neural Networks," in IEEE MTT-S Int. Microwave Symp. Dig., Long Beach, CA, 2005, pp. 1967-1970.

VIII M. Isaksson, D. Wisell, and D. Rönnow, "Behavioural Amplifier

Modelling Using Sampled Complex Envelope Measurement Data," in Proc. Gigahertz 2005, Uppsala, Sweden, 2005, pp. 213-216.

IX D. Wisell, M. Isaksson, and D. Rönnow, "Validation of Behavioural

Power Amplifier Models Using Coherent Averaging," in Proc. Gigahertz 2005, Uppsala, Sweden, 2005, pp. 318-321.

X

X M. Isaksson and D. Rönnow, "A Kautz-Volterra Behavioral Model for RF Power Amplifiers," in IEEE MTT-S Int. Microwave Symp. Dig., San Francisco, CA, 2006, pp. 485-488.

XI D. Wisell, M. Isaksson, N. Keskitalo, and D. Rönnow, "Wideband

Characterization of a Doherty Amplifier Using Behavioral Modeling," in 67th ARFTG Conf. Dig., San Francisco, CA, 2006, pp. 190-199.

XII D. Wisell, M. Isaksson, and N. Keskitalo, "A General Evaluation Criteria

for Behavioral PA Models," accepted for presentation at the 69th ARFTG Conf., Honolulu, HI, June 2007.

XIII M. Isaksson and D. Wisell, A. Eng, and D. Rönnow, "A Study of a

Variable-Capacitance Drain Network's Influence on Dynamic Behavioral Modeling of an RF PA," accepted for presentation at the 69th ARFTG Conf., Honolulu, HI, June 2007.

1

1 Introduction

The radio frequency (RF) power amplifier (PA) is a key component in modern telecommunication systems since its power consumption dominates the other parts in the system. The purpose of the RF PA is to amplify the radio signal to a necessary power level for transmission to the receiver. The transmission is performed through the air. It is important to handle the PA's conflicting behaviors of efficiency and linearity in this process.

RF PAs are divided into different classes, i.e. A, AB, B etc., with respect to their power efficiency. In RF PAs there is a trade-off between efficiency (defined as the ratio of the generated and consumed RF and DC powers, respectively) and linearity. High efficiency and high linearity cannot be achieved at the same time. Class A RF PAs belong to the group of weakly nonlinear systems [1]. The inherent nonlinearity of the PA causes interference with other transmitting channels and is something that one wants to avoid. Modern digital modulation techniques offer high data rates but use high bandwidths and peak-to-average ratios. Together with the nonlinear behavior of the PA, it enlarges the interference problem and leads to spreading of the transmitted spectrum which is often referred to as spectral regrowth. The only solution to the above problem is the use of linear PAs. The design of linear and efficient RF PAs in modern radio telecommunication systems has been described in the literature as one of the most challenging design problems [1, 2].

A number of different techniques have been proposed over the years. Among the many promising techniques are analog and digital predistortion, cross-cancellation, envelope elimination and restoration, and Chireix and Doherty amplifier techniques. All of these techniques have their pros and cons and none of them can be said to be the obvious choice for the future. For an introduction to these techniques see [1, 3, 4].

One of the most promising techniques is the technique of digital predistortion. In order to work, digital predistortion requires knowledge of the nonlinear characteristics of the amplifier since it, in principle, applies the inverse of the raw amplifier to the signal prior to amplification. Accurate nonlinear characterization of the amplifiers is necessary for several of these techniques and for optimizing the amplifier design. Behavioral models, also denoted as black-box models, have attracted interest as a means for characterizing PAs.

In behavioral modeling the input and output relationship is described as an explicit model, i.e. a model in which the system response is expressed as an explicit operation on the system input. One of the first who discussed theories of nonlinear systems as explicit models was Vito Volterra [5]. The book by Norbert Wiener [6] is another classical example of a book presenting theory in the field of nonlinear systems. An overview of the Volterra and Wiener nonlinear theories can be found in Martin Schetzen's book published in 1980 [7].

2

Volterra theory [5, 7] is used to describe the response of a nonlinear system. This theory is valid for weakly nonlinear time invariant systems with fading memory, an example of which includes PAs utilized in telecommunication applications. According to Volterra theory, a nonlinear system H, with input and output signals

( )u t and ( )y t , respectively, is described as a “Taylor series with memory” [8, 9]. In this section ( )u t , ( )y t and ( )h t are real-valued.

[ ] 1 2 3( ) H ( ) ( ) ( ) ( ) ...y t u t y t y t y t= = + + + (1)

where

[ ] ( ) ( )1 1 1( ) H ( )y t u t h u t dτ τ τ= = −∫ (2)

is the linear term with ( )1h t being the first-order time-domain response function (i.e. the impulse response), or Volterra kernel. The higher order terms are given by

[ ]

( ) ( ) ( )1 1 1

( ) H ( )

... ,..., ... ...

n n

n n n n

y t u t

h u t u t d dτ τ τ τ τ τ

=

= − −∫ ∫ (3)

where ( )1,...,n nh t t is the n-order time domain response function. Time domain Volterra models, both full and reduced Volterra models, have been used in behavioral modeling of RF PAs, e.g. [10-14]. Even though the full Volterra series can describe a wide range of nonlinear systems with memory it suffers from slow convergence [15] and is often a poor choice for PA modeling due to the large increase of parameters with respect to nonlinear order and memory length.

An oft-recurring phenomenon in behavioral modeling of RF PAs is the distinction between linear and nonlinear memory effects. It should be pointed out that memory effects are equivalent to frequency dependence, and a couple of methods for quantifying memory effects in RF PAs have recently been reported, e.g. [13, 16]. Volterra theory describe nonlinear memory effects as the inherent dynamic in the n-order Volterra kernels ( )1...n nh t t for 1n > , e.g. (3).

This thesis discusses behavioral modeling, parameter-reduction, and digital predistortion of RF PAs. The five papers referenced above are included as appendices to this document. A mathematical description has been performed in the digital domain and is, in most cases, presented in complex-envelope notation due to the band-pass nature of the studied device (i.e. the RF PA).

The statistical signal processing approach to RF PA descriptions presented in this thesis is not conventionally employed in RF engineering; the proper statistical signal processing technique is generally not common knowledge among engineers working with RF measurements and design. This thesis is interdisciplinary in that it is focuses on signal processing in the field of traditional RF engineering.

The identification and validation of behavioral models, parameter-reduction, and digital predistortion techniques depend on the availability of high-quality data. Analyzed data must provide a highly dynamic description of the PA over a large bandwidth. These requirements place strenuous conditions on the RF measurements

3

and signal processing techniques used to analyze the data. All of the results provided in this thesis are based on measurements of commercial RF PAs in an RF laboratory.

4

5

2 Behavioral Modeling of Radio Frequency Power Amplifiers

The requirements on RF PAs increase with each generation of mobile telecommunication systems, due to the increasing bandwidth and peak-to-average ratio of the signals they must accommodate. Nonlinear properties and memory effects, such as frequency dependence, have attracted much attention in the last few years due to the development of 3rd generation mobile systems [1, 2]. Measures of these effects include dynamic amplitude distortion (AM/AM) and phase distortion (AM/PM) [17] characterization and swept 2-tone measurements [10, 11, 18].

Behavioral models (also denoted black-box models) that account for nonlinearities and memory effects have attracted interest as a means for characterizing PAs as well. Typically, these models relate sampled input and output signals. An advantage of these models is that realistic signals can be used for exciting the PA. Behavioral models are used for modeling the properties of subsystems of telecommunication systems that are too complex for circuit level or transistor level simulations [19]. Physical models are based on a physical description of the components of the PA and how they interact [20], and are typically formulated as equivalent circuit models. When used for analyzing a PA, physical models can give insight into the source of the nonlinear properties and memory effects [21]. Both behavioral and physical models are thus useful in the design process. Physical models and equivalent circuits of active nonlinear RF microwave components are treated in several previously published works [2, 22-24].

Behavioral models are used for digital predistortion of PAs. In digital predistortion, either a model-based or look-up table inverse model of the PA's response function is used. The signal is predistorted before entering the PA in such a way that the output signal of the PA is similar to the desired output signal according to a given linearization criteria [1]. The increasing technical importance of digital predistortion, and hence of PA modeling, is indicated by the number of patents filed per year on this topic, which has increased more than a factor of four in five years [25].

Nonlinear dynamic behavioral PA models can, in most cases, be categorized as either neural networks [26] or Volterra models [9]. A review of behavioral PA models, including a detailed classification, is given in [27]. A Volterra model can be seen as a Taylor series with memory [8]. Volterra models are suitable for systems with moderate nonlinear effects, which is the case for many RF PAs that are driven moderately into compression [28]. However, a Volterra model has a disadvantage in that the number of parameters can become large and, in practice, such models are often not well identified. Consequently, the Volterra series converges slowly both in nonlinear order and memory depth. Behavioral models are usually reductions of a general Volterra model [8], and are often block structures of linear time invariant

6

filters and memoryless nonlinearities. Models with a linear filter, H, followed by a nonlinearity, N, known as Wiener models (H-N), have also been used [17, 29-31], as have Hammerstein models, whose structures are denoted N-H [31-33]. In [34] an extension of the Wiener model was presented, in which parallel Wiener systems of different order, each possessing a linear filter, were added. Similar extensions of the Hammerstein model have been reported [33], [12, 13, 35]. Models of the type H-N-H [36-39], as well as longer chains of blocks [40], have also been described. The memoryless nonlinearity is usually described using a polynomial, but can alternatively be represented with a series of Bessel functions [32]. Linear filters arise naturally in the time domain, but linear filters described by analytic functions in frequency domain are also represented in literature [31, 32, 36, 37].

Recently, methods using sparse delays [13] and pruned Volterra series [41, 42] have been employed for optimizing model performance. These models identify and make use of the most significant parameters, resulting in improved convergence of the Volterra series. An algorithm that simplify the computation of Volterra kernels in the frequency domain has also recently been proposed [43].

Volterra and block models are identified or extracted using various techniques. The choice of identification technique is largely determined by the details of the model structure. In some cases different sets of measured signals are used for identifying different blocks. A linear filter can be identified using a small-signal network analysis, and a static nonlinearity can be determined from amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) curves. The PA can subsequently be excited by modulated signals and polyspectral techniques [8] can be used to identify all the blocks of the PA model [36, 37].

Model parameters can also be obtained from a set of measured input and output data, often of a modulated signal. The model can be identified by minimizing the mean square error of the modeled and measured output signal using nonlinear least squares techniques [12, 31, 32]. In some cases the model can be linearized and identification methods used for time series in linear systems can be employed [33].

Various neural networks have been used as behavioral models of RF PAs, including multilayer perceptron neural networks [14, 44, 45], and recurrent neural networks [46]. A radial-basis function neural network (RBFNN) was recently used for RF PA behavioral modeling [47]. Paper I reports the use of an RBFNN which shows good performance for large signal bandwidths - up to 20 MHz - and as a digital predistortion algorithm with memory. The proposed model uses the signal’s envelope. The model requires thus less training than a model using both in-phase and quadrature-phase data.

Although there exists a variety of behavioral models for RF PAs, few studies investigate the use of different types of PAs or compare different models for a certain type of PA. In contrast, Paper II reports on such a study.

2.1 Nonlinear dynamic description of an RF PA

RF PAs are baseband in nature, operating in a band-limited region that spans a small fraction of the carrier frequency. A mathematical description of an RF PA can

7

in principle be restricted to that specific frequency region of interest. For mathematical convenience it is therefore practical to use complex-envelope notation when describing an RF PA [48]. Let the input signal to an RF PA be

( ) ( ) ( )[ ]expu t r t j t= Φ (4)

where ( )r t and ( )tΦ represent the envelope and the phase of the input signal, respectively, and 1j = − . The output of the nonlinear PA ( )y t is a distorted version of the input signal ( )u t where the output envelope and phase are influenced by the input envelope ( )r t . The output is

( ) ( )[ ] ( ) ( )[ ]{ }exp .y t g r t j t f r t= Φ + (5)

The functions g and f arise due to the non-ideal behavior of the PA and are often denoted the amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) distortion, respectively. If the AM/AM and AM/PM distortions are fully described with static functions in which the distortion in the PA is only dependent on the present value of the input envelope, the PA is said to be static. However, the intermodulation components generated by the PA vary as a function of many input conditions, such as amplitude and signal bandwidth. These bandwidth-dependent phenomena are often called memory effects [2]. In order to describe the so-called memory effects in the PA, the functions g and f must therefore be dynamic functions that consider not just the present value of the input envelope but also previous ones.

Figure 1 shows schematically the frequency behavior of a nonlinear system. Intermodulation products and harmonics caused by nonlinearities up to the 5th order are shown. Figure 1 illustrates that if the output signal is measured in a bandwidth limited region around a center frequency, only effects of odd-order nonlinearities will be seen. The bandwidth of the input is W Hz, and the bandwidth of the products of the 2nd, 3rd, 4th, and 5th order nonlinearities are 2W, 3W, 4W, and 5W, respectively.

Figure 2 and Figure 3 show examples of AM/AM and AM/PM distortion in a PA. The data are taken from measurements of a solid-state PA from Ericsson AB, which were performed with a 4 MHz-wide noise-like signal at –7.8 dBm input power. The gain of the PA was 52 dB but is normalized to 0 dB in the figure . The input signal had a peak-to-average power ratio of 7.2 dB and the center frequency was 2140 MHz. The AM/AM and AM/PM characteristics were calculated on a sample-by-sample basis. The spread of the data in Figure 2 and Figure 3 is due to memory effects in the PA. If a nonlinear PA with no memory effects would have been measured, the data in Figure 2 and Figure 3 would have rather appeared as thin lines. Hence, an accurate, first spectral zone model of a real RF PA should not only include odd-order nonlinearities (Figure 1) but should also model the dynamic behavior due to the spread in Figure 2 and Figure 3, cf. (5).

Many ways of expanding the functions g and f in (5) have been proposed. Some early attempts at static expansions can be found in e.g. [49, 50]. One of the first attempts to model the dynamic behavior of the PA was proposed by Saleh [38].

8

Power

Power

fc

fc

2fc 3fc

1st

3rd 5th

3rd 2nd

4th 4th 2nd

f

5th f

(b)

(a) W

W

Figure 1. Schematic illustrations of the input (a) and output (b) spectra referring to a nonlinear system. The nonlinear behavior of the system is specified up to the 5th order. The bandwidth of the nonlinear components are 2,3,4, and 5 times wider than the bandwidth of the linear spectrum for the 2nd, 3rd, 4th, and 5th nonlinearity, respectively.

-25 -20 -15 -10 -5 0-1

-0.5

0

0.5

1

Input Power [dBm]

AM

/AM

[dB

]

Figure 2. Measured AM/AM distortion of an RF PA from Ericsson AB with 52 dB gain. The measurements were done with a 4 MHz wide noise-like signal at –7.8 dBm input power. The input signal had 7.2 dB peak-to-average power ratio. The center frequency was 2140 MHz.

9

-25 -20 -15 -10 -5 0-8

-4

0

4

8

Input Power [dBm]

AM

/PM

[de

g]

Figure 3. Same as Figure 2 but the AM/PM distortion is shown.

10

11

3 Measurements and Model Validation

In this thesis, the identified models have typically been extracted and validated with measurement data from a commercial RF PA. The collected data have been down-converted and sampled through a vector signal analyzer, transmitted via a local area network to a computer, and analyzed in the complex-envelope form. The devices under test included PAs designed for second (2G) or third (3G) generation telecommunication systems. Two types of signals with different statistics were used as input stimuli: noise-like signals and multitone signals. The noise-like signal is designed to be equivalent with a signal in a wide-band code division multiple access system (WCDMA) and is hereinafter sometimes also referred to as a WCDMA-like signal.

3.1 Measurement set-up

The measurement set-up consisted of a vector signal generator and a vector signal analyzer as shown in Figure 4. The vector signal generator is capable of producing virtually any signal within its bandwidth limits from complex-envelope data. A R&S SMU200A from Rohde & Schwarz that had two digital-to-analog converters each with clock frequencies up to 100 MHz and capacity for direct up-conversion from baseband to RF was also used. In order to cancel possible distortion from the vector signal generator the signals were sampled both before and after the PA, i.e. the experimental input signal was used in the model identification and validation rather than the input signal to the arbitrary waveform generator. The input and output signals were measured at different time instants and introduces an error due to the difference between the input signal to the PA and the input signal as measured when switched directly to the vector signal analyzer. Repetitive measurements of the input signal have however confirmed that the error was negligible in the present study.

The vector signal analyzer was an R&S FSQ26 from Rohde & Schwarz which samples the signal at a low intermediate frequency using a single analog-to-digital converter at a sampling frequency of up to 81.6 MHz. The experimental signals were generated in the vector signal generator and the input and output signals of the PA were sampled by the vector signal analyzer. Switching was handled manually. The time between two consecutive measurements was typically several minutes and the PA was left running for at least three hours before the measurements were made in order to ensure that all the data were taken with the PA at thermal equilibrium. The measurement system has been used for measurements in other applications as well, e.g. analog-to-digital converter characterization [51]. The key figures of merit of the measurement system are presented in [52] and tabulated in Table 1.

12

Key figure of merit Approximated value Comment

Measurement bandwidth >100 MHz using frequency stitching Dynamic range –75 dB (adjacent channel

leakage ratio as defined in [53])

using coherent averaging

Amplitude ripple <0.1 dB using equalization Phase ripple <1˚

Table 1. The key figures of merit of the measurement system. The terms frequency stitching, coherent averaging, and equalization are put in their proper contexts below.

PC

VSA DUT

DPA VSG AWG

I, Q

I, Q

.

Figure 4. Outline of the measurement set-up. The signals are generated in the arbitrary waveform generator. The output signal from the vector signal generator is fed into the device under test (DUT) via a driver power amplifier (DPA) for the 3G PA. The input and output signals of the device under test are sampled by the vector signal analyzer.

3.2 Experimental Power Amplifiers

Measurements were made on two different base station solid-state PAs from Ericsson AB. The first PA, a 2G PA, was a single carrier GSM-1900 amplifier (GSM is the global system for mobile communications), and the second one, a 3G PA, was a multiple carrier amplifier intended for the 3G system WCDMA. The 2G and 3G PAs had output rated powers at 1 dB gain compression and small signal gains of 48 dBm and 31 dB, and 54 dBm and 52 dB, respectively. The 3G PA was designed for feedforward linearization, but that option was not used during our measurements. Both of the PAs were multiple stage amplifiers. The carrier frequency was 1960 MHz and 2140 MHz for the 2G PA and the 3G PA, respectively.

A single GSM carrier has a constant envelope, whereas a multiple carrier WCDMA signal has a fast changing envelope with a peak-to-average power ratio exceeding 7 dB. Hence, the PAs were designed for different kinds of signals, which possibly could imply that different PA models should be used. The 2G PA is not optimized for higher bandwidths and could be expected to have larger memory effects

13

than the 3G PA, if excited with a wideband signal with high peak-to-average power ratio. The reason for using the 2G PA with signals for which it was not designed was to determine the generally applicability of the models. A good model should be able to model the behavior of a PA excited with different types of signals. Behavioral models of PAs are sometimes validated using two-tone or multi-tone signals, even though the PA may have been designed particularly for wideband modulated signals, see e.g. [13].

For the solid-state PAs of interest here, nonlinear distortion has been attributed to nonlinear memory effects [28]. Models that take into account linear memory effects have been successfully used for TWTs [29]. The results and conclusions presented below are naturally of more general value for solid-state PAs than for TWTs.

3.3 Digital Signal Processing Techniques

To improve the raw performance of the hardware in the measurement system, extensive signal processing was used [52]. The digital signal processing algorithms employed include: coherent averaging for increased dynamic range, ”frequency stitching” for increasing the BW, equalization for compensating for linear distortion in the instruments, and digital predistortion for increasing the dynamic range of the signal generator. The combination of state-of-the art instruments and digital signal processing yields a system with the desired performance for high bandwidth, high dynamic range characterization of PAs.

3.3.1 Coherent Averaging The principle of coherent averaging is straightforward. By using a repetitive

signal as input to the PA and then measure this signal numerous times, time-aligning the measurements and taking the arithmetic average, the noise in the measurement set-up can be reduced by 20log10(N), where N is the number of measurements taken. The technique is further described in [54, 55].

3.3.2 Frequency Stitching The frequency stitching technique is a special case of the filter bank method

proposed in [56] and described in [54]. It is used to extend the digital bandwidth of a vector signal analyzer by taking multiple measurements at different center frequencies. The basic principle relies on use of a repetitive signal as input to the PA. Due to the nonlinearities in the PA, the output signal will have a significantly larger bandwidth than the input signal, but will be repeated with the same repetition frequency as the input signal. The output signal is then passed through a filter, normally the resolution bandwidth filter in the vector signal analyzer, at different center frequencies. If sufficient energy exists in the frequency overlap when measurements are made at different center frequencies, the measurements are “stitched” together as described in [54] to obtain a wideband signal.

14

3.3.3 Digital Predistortion of the Signal Generator and Driver Amplifier

In order to generate an input signal to the PA with low distortion, digital predistortion was applied to the signal generator and driver amplifier. A parallel Hammerstein structure was used for the digital predistortion. Having a low distortion input signal to the PA may not, initially, seem necessary since both the input and output signals of the PA are measured. While this holds true for linear distortion and for identification of the nonlinear model, it does not hold true for model validation as described in [55], and for which an input signal with a level of distortion significantly below the level of the model error is a necessity.

3.3.4 Equalization To accurately characterize the PA for memory effects, it is necessary that the

vector signal analyzer does not introduce linear or nonlinear distortion to the measurements. Most advanced vector signal analyzers, like the one used in this paper, are highly linear and do not introduce notable nonlinear distortion if used correctly. However their linear response, normally caused by the resolution bandwidth, is not negligible. Thus compensation in the form of equalization is sometimes needed, especially if the frequency stitching technique described above is used. Several techniques for compensation can be employed. A vector signal generator with excellent performance could be used as a calibration standard in order to calibrate a narrowband, low performance vector signal analyzer. Amplitude, but not phase, can be calibrated using a stepped sine wave which produces good results even for large bandwidths (if necessary the power of the stepped sine wave should be calibrated using a power meter). The phase calibration for a large bandwidth signal, as is necessary here, is trickier. One plausible method is to calibrate a vector signal generator with, for the application, large enough bandwidth using a digital sampling oscilloscope that has superior performance over the bandwidths used here, and then to use the calibrated vector signal generator to calibrate the vector signal analyzer.

3.3.5 One Receiver – Cancellation Effects Several differing measurement set-ups for behavioral modeling of PAs have

been proposed [10, 57-60]. The adopted set-up for this work used the same receiver to measure both the input and output signals of the PA, as seen in Figure 4. This set-up has the advantage that the linear distortion in the receiver (i.e. the vector signal analyzer) is cancelled to some extent between the two measurements. A drawback is that the measurements are taken at different times, but the effects are negligible if the device under test is time-invariant over a few seconds or minutes, as is the case here.

15

3.4 Model Validation

The "true system" is an esoteric entity that cannot be attained through practical modeling. Hence, model validation is a difficult task to master and is usually one of the weakest points in system identification studies. The essence of model validation can be described as follows: having chosen the "best" model within some structure, one must decide if that model is "good enough." Most model validation techniques focus on the question of how well the model agrees with the observed data. There have been a number of suggested model validation techniques, such as Akaike's final prediction-error criterion [61], statistical hypothesis tests [62], and whiteness tests [63]. A modern survey on model validation techniques can be found in [64].

An attractive way of comparing two models is to evaluate their performance on validation data. A validation data set is one that has not been used to help construct any of the models that we would like to evaluate. This procedure is often called cross-validation [65, 66]. The cross-validation technique is attractive because it makes sense without any probabilistic arguments and assumptions about the true system [64]; it is therefore the technique used throughout this thesis.

The identified models were validated with a new set of measured input and output data. The model error e(n) is the difference between the measured, ( )measy n , and simulated, ( )modely n , output signals

meas model( ) ( ) ( ) .e n y n y n= − (6)

The sum square error ε can be calculated

( ) ( ) 2meas model

1

L

n

y n y nε=

= −∑ (7)

and the normalized mean square error (NMSE) is one of the quality numbers used for comparison between the models. Throughout this thesis the NMSE is defined as

( )

102

meas1

NMSE 10 log .L

n

y n

ε

=

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑ (8)

Another important quality number used is the adjacent channel error power ratio (ACEPR), defined as

( )

( )

( )

2

21,2meas

ACEPR max .iadj

i

ch

E f df

Y f df=

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

∫∫

(9)

As seen in (9), the ACEPR is defined as the larger of the values evaluated for both the upper and the lower adjacent channels. Here ( )E f and ( )measY f are the time-

16

discrete Fourier transforms of the model error ( )e n in (6) and measured output ( )measy n , respectively. The channel and the adjacent channels are denoted ( )ch and

( )iadj , respectively. Loosely speaking one can say that the NMSE is a measure of the model accuracy in-band and that the ACEPR measures out-of-band model accuracy. Often, in these kind of applications, it is the out-of-band measurement that is of most interest, e.g. [33].

3.4.1 RF PA Model Validation A common technique used in this thesis for comparing RF PA models is to plot

the absolute value of ( )E f relative to the measured absolute value of the output signal ( )Y f . Figure 5 shows a typical plot. Here, a realistic WCDMA signal has been used to validate two different models. The channel bandwidth and the bandwidths of the adjacent channels are 3.84 MHz. The ellipses in the figure denote the frequency regions of the lower and upper adjacent channels, respectively. The adjacent channels are centered ±5 MHz relative to the center frequency. The NMSE, which corresponds to the total model error, is strongly dependent on the error within the channel (cf. Figure 5). Due to spectrum masks [53], the ACEPR is often of greater importance than the NMSE value. However, a low NMSE can, under certain circumstances, be used to further decrease the ACEPR by the use of frequency weighting (e.g. Paper IV). Notice the benefit of showing the model error in a plot; in instance of low model errors, the model and the measured outputs are hard to distinguish, and a model comparison is difficult to do.

17

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-80

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-40

-20

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dBc]

Measured OutputModel Output Error Signal 1

Input Signal

Error Signal 2

Figure 5. Typical plot used for model comparisons. The ellipses highlight the lower and upper adjacent channels. The measured PA was a lateral-diffused metal-oxide semiconductor (LDMOS) PA intended for 3G. The output power was 44 dBm and the center frequency was 2140 MHz.

18

19

4 Modeling Radio Frequency Power Amplifiers Using RBFNNs

Paper I proposed a method for modeling RF PAs using RBFNNs. The model uses the envelope of the signal, in agreement with the procedure outlined in (5). A neural network is a natural choice for modeling nonlinear dynamic systems, since it can approximate any continuous function arbitrarily well [26]. Hence, neural networks have been used for modeling communication systems [67] like satellite channels [44], or subsystems like PAs [19, 44], and microwave circuits [68]. Neural networks have also been used recently for digital predistortion [69, 70], and for modeling radio-frequency and microwave components and circuits - both active and passive - other than PAs [68]. The most frequently used type of neural networks for behavioral RF PA modeling are multilayer perceptrons [14, 19, 44, 45], but recurrent neural networks have also been used [46]. In [71], an RBFNN was trained using simulated input and output data for a PA. In Paper I experimental data were used.

Behavioral RF PA models typically use sampled input and output signals represented as in-phase and quadrature-phase (I and Q) signals. A common approach in neural networks is to use two real-valued neural networks, one for I (or the amplitude) and one for Q (or the phase), or to use complex-valued neural networks [67]. The RF PA modeled in [72] utilized a single real-valued neural network, but included separate I and Q input signals.

In Paper I we used a radial-basis function neural network (RBFNN) for modeling the dynamic amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) distortion of an RF PA. The envelope of the sampled input and output signals was used, rather than I and Q. In this RBFNN, the phase of the input signal was not passed through the RBFNN but was rather applied at the output. The model result for the AM/AM and AM/PM distortion was a simple network that required less training than earlier proposed models that used sampled IQ-data [72]. The RBFNN was used for modeling a PA designed for a third generation mobile communication system (3G). Signals with bandwidths of 4 and 20 MHz were used, as they correspond to bandwidths used in 3G systems. The performance of the proposed RBFNN was compared to a parallel Hammerstein model, a block model which has recently been shown to be an excellent choice for PA-modeling [13, 33].

4.1 The Neural Network

Figure 6 shows the RBFNN used in Paper I. An RBFNN consists of three layers: an input layer, a hidden layer, and an output layer. The transformation from the input layer space to the hidden layer space is nonlinear while the transformation from the hidden layer space to the output layer space is linear. The network's nonlinear behavior was implemented using multivariate Gaussian functions (so-called

20

Green's functions). It can be seen in Figure 6 how the phase of the input signal was led beside and not through the RBFNN. The phase was applied at the output. The dynamic AM/AM and AM/PM distortions (i.e. the g and f functions in (5)), were modeled from the input layer to 1y and 2y , respectively. The model used sampled data, which explains why (5) can be simplified to

( ) ( ) ( )[ ]

( ) ( ) ( ) ( )[ ][ ]{ }

( ) , 1 ,...,

exp , 1 ,...,

y n g r n r n r n M

j n f r n r n r n M

= − − ×

Φ + − − (10)

where ( )r n i− is the i sample delayed replica of the input envelope ( )r n . M is the network's memory length.

Excluding the phase of the input signal in the RBFNN significantly simplifies the model and requires considerably less training than when the I and Q signals are used as inputs. The proposed RBFNN has ( 3) 2K M + + real-valued parameters. Hence, if both the I and Q signals were used as input, the number of parameters would increase by a factor of KM .

rn

rn-M

rn-1

ϕ1

y1 ϕ2

ϕK

ϕ = 1

w0,1=b1

y2

w1,1

w2,1

wM,1

wM,2

w2,2

w1,2

w0,2=b2

Input Layer

Hidden Layer

Output Layer

∑

∑

u(n) y(n)

Φ(n)

. . . .. .

Figure 6. A radial basis function neural network with 1M + input nodes, K nodes in the hidden layer, and two output nodes. The two output nodes correspond to the AM/AM and AM/PM distortion, respectively.

Initially the hidden layer has no neurons. The following steps are repeated until the network's mean squared error falls below the goal, or until the number of neurons reaches the predefined number K . The goal here is set to zero.

Step 1) Simulate the network. Step 2) Find the input vector with the greatest error. Step 3) Add a neuron with weights equal to that vector. Step 4) Redesign the output layer weights to minimize error.

In order to find the optimum input nodes (or optimum previous samples) of the RBFNN sparse delays were used [13]. For nonlinear models with memory, the number of model parameters becomes large even for moderate memory depths, which

21

may result in overmodeling. When using sparse delays, only the most significant nodes from the set r(n), r(n-1),…, r(n-M) are used. The maximum memory length, M, can thus be longer for the same number of model parameters. The most significant delays were found by an iterative procedure. In the first step the model performance was optimized by stepping through nodes r(n) to r(n-M) and choosing the one that exhibited the best model performance. In the second step the previously identified significant node was retained and the model performance was further optimized by stepping the second node through nodes r(n) to r(n-M). The procedure was continued until no further model improvement was found. This technique has previously been used in block models of RF PA [13], but to our knowledge has not been applied to neural network models.

4.2 RBFNN Modeling Results

Low model errors (i.e. NMSEs) can be achieved by using a RBFNN. For a RBFNN with a memory length of three samples, NMSE values below –46 dB have been reported [47]. In Figure 7, Figure 8, and Figure 9 experimental and modeling results of an RF PA are shown. Measurements were taken on a base station PA intended for the 3G system WCDMA. The PA was designed for feed-forward linearization, although this feature was not activated during measurement acquisition. A noise-like signal with a peak-to-average power ratio of about 7 dB and a bandwidth of 4 MHz (the 3-dB bandwidth was 3.84 MHz) was used as the input. The center frequency was 2.14 GHz. The signals were designed to be realistic in terms of bandwidth and peak-to-average power ratio; the PA was designed to handles these types of signals.

The validation data were measured 150 times, added coherently in order to reduce the effects of noise (cf. Section 3.3.1). After noise reduction the input signal had an adjacent channel power ratio below –75 dBc (see Figure 7).

In Figure 7 the normalized spectra for the input signal, ( )u n , the measured output signal, ( )measy n , the RBFNN output, ( )RBFNNy n , and the error signal,

( )RBFNNe n , are shown. The spectrum of ( )measy n has a clear out-of-band asymmetry indicating significant memory effects. The spectrum of the output signal of the RBFNN exhibits a similar asymmetry; the RBFNN seems to reproduce these memory effects well. However, it is seen in Figure 7 that the model still leaves room for improvement, since the spectrum of the error signal is about 20 dB higher than that of the input signal in the adjacent channels.

Finally, we compared the performance of the RBFNN with that of a parallel Hammerstein model [13, 33] Paper I and Paper II. The spectrum of the error signal,

( )PHe n , is shown in Figure 7. For a single element delay ( 1M = ), the best parallel Hammerstein model (9th order) gave an NMSE of about –40.7 dB and an ACEPR of –54 dB. This is poorer than the performance of the best RBFNN reported in [47] and Paper I, although the difference is not large. A PH model with three delay terms (or more) and nonlinear terms up to order 13 achieved an NMSE of about –41.6 dB and an ACEPR of –62 dB. The RBFNN with 1M = reached a NMSE below –46 dB and an ACEPR below –56 dB. Hence, the RBFNN does a significantly better job of

22

modeling the total error, but seems to have more difficulty modeling the higher order nonlinear behavior of the PA, compared to the parallel Hammerstein model. Figure 8 and Figure 9 show excellent agreement between the measured and model output signals for the dynamic AM/AM and AM/PM distortions, respectively.

-10 -6 -2 0 2 6 10-100

-80

-60

-40

-20

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dBc]

Measured OutputRBFNN Model Error Signal PH

Input Signal

Error Signal RBFNN

Figure 7. Normalized spectra for the input signal, ( )u n , the measured output signal, ymeas(n), the RBFNN output, yRBFNN(n), the error signal of the RBFNN, eRBFNN(n), and the 9th order Parallel Hammerstein error signal, ePH(n). All model curves have a memory length M=1.

-25 -20 -15 -10 -5 0-1

-0.5

0

0.5

1

Input Power [dBm]

AM

/AM

[dB

]

MeasuredRBFNN Model

Figure 8. Experimental and modeled instantaneous AM/AM distortion of an RF PA. An RBFNN with memory length M=1 and K=10 was used.

23

-25 -20 -15 -10 -5 0-8

-4

0

4

8

Input Power [dBm]

AM

/PM

[deg

]

MeasuredRBFNN Model

Figure 9. Experimental and modeled instantaneous AM/PM distortion of an RF PA. An RBFNN with memory length M=1 and K=10 was used.

Paper I reports the use of the RBFNN for wide-band RF PA modeling. Signals with bandwidths up to 20 MHz was used for modeling and model validation. The distorted output bandwidth from the PA exceeds 100 MHz at high input power levels, and direct use of a vector signal analyzer will not support the required bandwidth and dynamic range, considering the aforementioned orders of nonlinearities (see Figure 1).

The usable digitized bandwidth of the vector signal analyzer used in this the measurement set-up (Figure 4) was only 20 MHz. In order to obtain a wideband signal, "frequency stitching" (cf. Section 3.3.2) was used [54]. A virtual sampling frequency of 288 MHz was obtained.

In Figure 10 we compare the performance of the RBFNN with that of a parallel Hammerstein model of order 13 and sample memory length M = 5 (i.e. a PH(13, 5) model) for an input signal bandwidth of 20 MHz. The RBFNN model had a memory length 1M = and 10 neurons in the hidden layer. It can be seen that the relative difference in ACEPR between the models is now leveled out. Hence, the RBFNN with a one unit sample delay models the nonlinearities of the PA almost as well as the best parallel Hammerstein model.

In summary; the RBFNN has a lower total error but a higher out-of-band error compared to the well-established parallel Hammerstein model. This type of behavior is in qualitative agreement for both the 4 and 20-MHz bandwidth signals tested.

24

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-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dBc] Output Signal

RBFNN Model

Input Signal

Error SignalPH

Error SignalRBFNN

Figure 10. Normalized spectra for the input signal, the measured output signal, the RBFNN output signal, yRBFNN(n), and the error signals for the RBFNN and the PH(13,5) models, eRBFNN(n) and ePH(n). For model identification a 20-MHz noise-like signal with mean power of –7.8 dBm was used, M=1 and K=10. The ACEPR values were calculated to be –57.3 dB and –58.0 dB, and the NMSE values were determined to be –38.4 dB and –38.6 dB for the RBFNN and the PH(13,5), respectively.

25

5 Comparison of Behavioral PA Models

Due to the development of the third-generation of mobile communications (3G) there is substantial industrial and academic interest world-wide in characterizing RF PAs which are known to be one of the key components in the communication link (see Chapter 2 ). Many research groups have focused on behavioral modeling of PAs and report on the results such models acquired by simulation and occasionally through actual measurement of the responses of real RF PAs (see Chapter 2 ). Through these efforts, several types of behavioral models of RF PAs have been reported. However, comparing different model structures from different scientific papers is a difficult task, since validation results depend not only on the model itself but also on other factors such as the measurement set-up, the chosen stimuli signals, the output power of the PA, the PA type, etc. Hence, the appropriateness of the proposed model compared to other models is hard to evaluate. The difficulty of conducting such a comparative analysis might explain why such papers are seldom found. In contrast, Paper II reports on such a study.

Few studies compare different models for a certain type of PA or investigate different types of PAs. For example, the review paper [27] does not include any experimental studies of different models. In [14] an experimental comparative study of behavioral models was presented, and included neural network, Volterra and N-H models. The work presented in Paper II extends the work done in [14] in that it examines two different types of PAs (one for 2G and one for 3G), uses different signal types (noise-like signals and multitones of different bandwidths), and cross-validates the models by validating them with a different signal type than the one used to identify the model. We also used realistic signals with high peak-to-average power ratios and bandwidths up to 20 MHz, and measured over a dynamic amplitude range greater than that of relevant standards [53]. Such rigorous parameters have not, to our knowledge, hitherto been used for validating behavioral models of solid-state PAs for mobile telecommunication applications, though the technical importance of such choices for characterization is obvious. Behavioral models of traveling wave tube (TWT) amplifiers have, on the other hand, been evaluated using time domain signals of several GHz [29]. The investigation of PA modeling at different bandwidths is motivated in part by the variable bandwidth of emerging WiMAX technology [73]. The 2G PA was designed for GSM signals which have a constant envelope, but for our purposes was excited with different types of signals.

For practical reasons, only a limited number of all models reported in the literature can be included in a comparative study. Furthermore, there are many ways to perform model identification; a full investigation of the different identification methods merits separate study. Neither could a comparative study realistically include all reported types of PAs and signals. In Paper II we have tried to strike a balance between the number of models, signal types, and PAs investigated. Our investigation

26

was therefore limited to models that can be identified by sampled input and output data of wide-band modulated signals.

5.1 A Comparative Analysis

Paper II presents a comparative analysis of behavioral models. The paper included an analysis based on more than 7000 model identifications and validations. Table 2 gives a summary of what kind of behavioral models, PAs, and signals that were used in the study.

Behavioral Models Polynomial (memoryless), RBFNN (Section 4.1),

Wiener, Hammerstein, parallel Hammerstein, and "full Volterra"

PA types PAs intended for 2G and 3G Bandwidths 0.8, 4, and 20 MHz Signal types Noise-like (WCDMA-like) and eight-tone signals Power levels Two different power levels

Table 2. A summary of what kind of behavioral models, PAs, and signals that were used in the study reported in Paper II.

All validations were analyzed in terms of in-band (NMSE) and out-of-band (ACEPR) errors as functions of the model nonlinear orders and memory lengths. For the ACEPR calculations the adjacent channels were centered at ±1, ±5, and ±25 MHz for the 0.8, 4, and 20 MHz bandwidth data sets, respectively (cf. (9)). In the case of validation with eight-tone signals, the out-of band error analysis included a frequency-by-frequency inspection of the model’s ability to predict spurious frequencies. Figure 11 depicts a study involving validation with an eight-tone signal, and shows the overlapping spectra of the measured and model output signals of the PH(7,5) model. The model was identified with an eight-tone signal with a mean power of –10.8 dBm and 9 dB peak-to-average power ratio, and validated with another eight-tone signal with the same mean power and peak-to-average power ratio. Measurements were made on the 3G PA.

27

-6 -2 0 2 6

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [d

Bx]

Output DataModel OutputPH(7,5)

Figure 11. The spectra of the measured and model output signals of the PH(7,5) model. The model was identified with an eight-tone signal with a mean power of –10.8 dBm and 9 dB peak-to-average power ratio, and validated with another eight-tone signal with the same mean power and peak-to-average power ratio. Measurements were made on the 3G PA.

The study in Paper II answered several questions regarding behavioral PA modeling. In summary:

i. The RBFNN gave the lowest NMSE and the parallel Hammerstein the lowest

ACEPR. ii. RBFNN is the best choice if the total error is of utmost importance.

iii. The Wiener and Hammerstein models do not have lower model error compared to the polynomial (memoryless) model.

iv. The model errors NMSE and ACEPR increase with increasing bandwidth for both the 2G and the 3G PA.

v. The 2G PA exhibited larger memory effects than the 3G PA when both PAs are excited with the same type of signal.

vi. The decrease in the model errors NMSE and ACEPR between a memoryless model and the best models with memory was larger for the 2G PA than for the 3G PA.

vii. Parallel Hammerstein demonstrated the best model performance for cross-validation.

viii. The 2G PA showed better "cross-validation" performance than the 3G PA, probably due to larger memory effects.

Conclusions i and ii, which assert that the RBFNN is a good model if the total

model error is important and for a constant input signal type, are confirmed by the reported results in Paper I. If the type of input signal is subject to change, the RBFNN performance will decrease more than the comparative parallel Hammerstein

28

model (cf. Conclusion vii). The fact that the Wiener and Hammerstein models with memory did not outperform a memoryless model (Conclusion iii) is initially surprising but can be explained by the model’s inability to deal separately with linear and nonlinear memory effects. That conclusion is further investigated in Papers III, IV, and V. Conclusions iv, v, and vi are somewhat expected, given the large disadvantage of all the models in this test because of their sensitivity to changes in signal statistics. In Figure 12 we see a type of "cross-validation" where the model is identified with a 4-MHz noise-signal and validated with an eight-tone signal (cf. Figure 11). In comparing the model's ability to predict the spurious frequencies in Figure 11 and Figure 12, we see a substantial increase in out-of-band model error in Figure 12. The parallel Hammerstein model is a sub-class of the full Volterra model but does not share the general properties of the Volterra model, as discussed in Paper IV. The full Volterra model requires identification of too many parameters for practical purposes when a models of high nonlinear order and significant memory length are needed. Hence, a future challenge in the field of PA modeling is to develop general, nonlinear, low-complexity behavioral models. A proposal that addresses this is reported in Paper IV.

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-10

0

Frequency [MHz]

Nor

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ized

Pow

er [d

Bx]

Output DataModel OutputPH(7,5)

Figure 12. Similar to Figure 11 but using a model identified with a WCDMA-like signal with a mean power of –6.7 dBm and 7.7 dB peak-to-average power ratio.

29

6 Identification of Frequency Domain Volterra Kernels

Volterra theory is valid for weakly nonlinear time invariant systems with fading memory, an example of which is PAs in telecommunication applications. In Paper III we report a method to measure and characterize the third-order Volterra kernel. In the frequency domain, the output signal ( )Y ω can be calculated by

[ ] 1 2 3( ) H ( ) ( ) ( ) ( ) ...Y U Y Y Yω ω ω ω ω= = + + + (11)

where ( )U ω is the input and H represents the frequency response function. The n-order output is

( )1 2 1 11

1 1 1 2 1

1( ) ... H ( , ,..., )

2( ).... ( ) ... ,

n n nn

n n

Y

U U d d d

ω ω ω ω ω ωπ

ω ω ω ω ω ω

−−

− −

= − − ×

−

∫ ∫ (12)

where

( ) ( ) ( )1 1 ...1 1 1H ,..., ... ,..., ....n nj

n n n n nh e d dω τ ω τω ω τ τ τ τ− + += ∫ ∫ (13)

is the nth order frequency domain response function or Volterra kernel. Even though (13) represents a n-dimensional frequency function, when considering real-valued signals, it nevertheless has the same types of symmetry properties as the linear frequency function 1H (see Paper III). Taking into account the symmetry properties for real-valued signals, it is clear that only a part of the third-order space is being excited. Figure 13 shows those parts of the third-order frequency space that are being excited for some maximum frequency (fmax = 50). The frequencies range from 0 (DC) to fmax (dotted), and occupy a band-limited region (line). The lower (upper) part of the figure gives rise to an output near fc (3fc). Hence, representation of a nonlinear system in the first spectral zone (i.e. around cf ), requires excitement of an irreducible volume in the n-order frequency domain excited. That irreducible volume is, in the case of the third-order nonlinear response function in Figure 13, represented by the lower prism (cf. Figure 1).

6.1 The Third-Order Kernel

In Paper III we used a stepped three-tone test for characterization of the third-order nonlinear function 3H . Measurements were made on an LDMOS PA intended for 3G communication at a center frequency of 2140 MHz. A three-tone signal can be expressed in the time domain as

30

( ) ( ) ( )( ) cos cos cosu t A t A t A tα ω α ω α ω′ ′ ′ ′′ ′′ ′′ ′′′ ′′′ ′′′= + Δ + + Δ + + Δ (14)

where A is the amplitude (α′ , α′′ , and α′′′ are dimensionless constants), ω ′ , ω ′′ , and ω ′′′ the angular frequencies, and ′Δ , ′′Δ , and ′′′Δ are the phases of the three tones, respectively. In order to excite the irreducible volume, the input signal for a stepped three-tone measurement was generated as a signal with sub sequences for each three-tone combination given in Paper III. Six different scans were made at each of the power levels used. In Figure 14 the irreducible volume (cf. the lower prism in Figure 13), is shown for our specific experiment. The six types of scans described in Paper III could excite all parts of the volume. By considering all possible outputs of a third-order nonlinear system excited by three-tone signals, the third-order nonlinear frequency response was identified. For a more detailed description of the identification procedure see Paper III. It should be noted that the red dotted line in Figure 14 corresponds to the part of 3f that was excited by a traditional two-tone test. Thus, a two-tone test is seen to be insufficient for obtaining full information about the third-order nonlinearity. Illustrating the third-order frequency response 3H is a pedagogical challenge; the excited region is three-dimensional and an illustration of the frequency function should, without restrictions, be a four-dimensional plot. Hence, we have plotted 3H as a function of planes and paths in 3f . In Figure 15 the magnitude of H3 along paths in the grey plane in Figure 14 are depicted. The red circles are the parts that would be excited by a frequency-scanned two-tone measurement. For a memoryless system, the surface in Figure 15 would be flat and

3H would remain at a constant level. Apart from the red circles, the frequency dependence seen in Figure 15 would not be detected by a scanned two-tone measurement. The dominant structure in Figure 15 is a minimum that follows the diagonal that of a scanned two-tone (the red diagonal in Figure 14).

31

0 20 40 60020

4060-60

-40

-20

0

20

40

60

f1

f2

f 3

Figure 13. Parts of the frequency (f1,f2,f3)-space of H3 that are being excited. The frequencies range from 0 (DC) to fmax (dotted), and occupy a band-limited region (line). The frequency maximum fmax is here chosen to be 50. The lower (upper) part of the figure gives rise to an output near fc (3fc).

2135 2140 21452135

2140

2145-2145

-2140

-2135

f1 [MHz]

f2 [MHz]

f 3 [M

Hz]

Figure 14. The irreducible volume (thin lines) in the 3f space that can be excited by a signal in the frequency range 2135.6 – 2144.4 MHz and that causes intermodulation products in the fundamental zone. |H3| paths in the grey plane are given in Figure 15. The red dotted diagonal line is the path in the irreducible zone that is excited in a scanned two-tone test and the open planes with dashed borders are those parts that give output signals at f ′ , f ′′ and f ′′′ .

32

33

35

37

39

|H3| [

dB]

(2145,2145,-2135)

(2145,2145,-2145)

(2135,2135,-2145)

Figure 15. The magnitude of H3 along paths in the grey plane of Figure 14. The red circles are the parts that would be excited by a frequency scanned two-tone measurement. The 1 2 3( , , )f f f coordinates of three corners are given.

6.2 Investigation of Symmetry Properties

The symmetry properties of the dominating block structure are more appropriately obtained by analyzing 3H along paths rather than in planes perpendicular to a single frequency axis [74] (see Paper III). The analysis of 3H sought to determine with which system – a Wiener (H-N), Hammerstein (N-H), or N-H-N system – its symmetry properties best corresponded. In the present investigation the determined 3H , rather than statistical measures of random in- and output signals, was analyzed; the latter have previously been used for both frequency [75] and time domain [76] analysis of block structures. Such analysis is advantageous for RF PAs, since they need only be excited by signals of small relative bandwidths. Moreover, this method for analyzing the symmetry properties of 3H is not limited to RF PAs, but should be of general interest for analyzing nonlinear systems. Each IM product yields a certain path for each three-tone scan (see Paper III), and the same path may be obtained for several tonal combinations. The 3H analysis was not performed along all obtained paths, rather the focus was put on those that give insight into the type of nonlinear system that best describes the properties of the PA. Figure 16 gives examples of some of the paths in 3f that were analyzed in Paper III. In Figure 17,

3H and 3H∠ are shown along paths (a) – (d) from Figure 16. The dominant structure in all the 3H curves is a minimum that drops to 35 dB from 38 dB. The positions of the minima are –3, –1, 1, and 3 MHz relative to fc at 2140 MHz. At these minima there is a corresponding step in 3H∠ from 170º to 140º. Due to the fact that Wiener and Hammerstein systems are sub-classes of a full Volterra system, some of the frequency properties of 3H must hold, see Paper III. The frequency response of

3H shown in Figure 17 could be neither the response of a Wiener nor a Hammerstein

33

system. However, the response of 3H in Figure 17 corresponds to the properties of an N-H-N block structure. The curves in Figure 17 thus indicate that the frequency dependence of H3 has the properties of a N-H-N model, and that the frequency dependence occurs at baseband. The frequency dependence of any cubic term appears to be negligible compared to that of the quadratic effects (see Paper III). The investigated paths in 3f that are reported in Paper III all show the same type of symmetry behavior.

21352140

21452135

2140

2145-2145

-2140

-2135

(e)

(f)

(g)

(h)

(h)

(d)

(d)

(g)

(c)

(c)

(f)

f1 [MHz]

(b)

(b)

(e)

(a)

(a)

f2 [MHz]

f 3 [M

Hz]

Figure 16. Paths in the irreducible 3f space for which two frequencies were constant.

-2 135-2 140-2 145120

140

160

180

Pha

se(H

3) [D

eg]

-(fc+s) [MHz]

34

36

38

|H3|

[dB

]

(a) (b) (c) (d)

Figure 17. The magnitude (upper) and phase (lower) of H3 along the paths (a), (b), (c), and (d) given in Figure 16.

34

35

7 A Parameter-Reduced Volterra Model

The Volterra model can be used to describe any weakly nonlinear stable system with fading memory with an arbitrary small error. However, due to the model's dramatic increase in the number of parameters with respect to nonlinear order and memory length, in practice it is impossible to use it for systems of high nonlinear orders and memory lengths. The complex-envelope output from a truncated O-order Volterra model is

1 2 1 3

1 2 1 ( 1)/2 ( 1)/2 ( 3)/2 1

,10

*,3 1 2 3 1 2 3

0 0

, 1 20 0

1 2 ( 1)/2

( ) ( ) ( )

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

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

( ) ( )... ( )

O O O O O

M

l l lm

M M M

l l l lm m m m

M M M M M

l O Om m m m m m m m

l l l O

y n h m u n m

h m m m u n m u n m u n m

h m m m

u n m u n m u n m

+ − + −

=

= = =

= = = = =

+

= −

+ − − − +

+ ×

− − −

∑

∑ ∑ ∑

∑ ∑ ∑ ∑ ∑* *

( 3)/2( )... ( ).l O l Ou n m u n m+− −

(15)

In (15) lu and ly denote the input and output complex-envelope signals, respectively, M represents the model's memory length, ,l ih the complex Volterra kernel of order i , and (* ) the complex conjugate. The redundant items due to kernel symmetry have been removed. The number of complex-valued parameters in (15) is given by the following expression (see Paper II):

( 1)/2 2

2 20

(( )!) ( 1).

( !) ( !) ( 1)

O

p

M p M pM p p

−

=

+ + ++∑ (16)

When modeling an RF PA, Paper I and II show a reduction in the model error for a system of approximately 9th nonlinear order and memory length of 4 to 5 samples in the sub-classes of Volterra models (Wiener and Hammerstein structures) reported. The standard Volterra model would require consideration of 11875 (40116) complex-valued parameters for a 9th order system of 4 (or 5) memory length samples. In contrast, the Wiener and Hammerstein models would only require 9 (10) complex-valued parameters for the same order and memory, and achieve the same model error as a static polynomial model (see Paper II and Paper III).

In Paper IV we report the results of an odd-order behavioral PA model that uses a Kautz function expansion; we denote it a Kautz-Volterra (KV) model. The model was briefly described in [77]. Previously, a PA model using Laguerre functions with real poles was been presented [78], but differs from our model in that we use complex poles that are different for different nonlinear orders. This is motivated by the need to incorporate nonlinear memory effects (see above). The KV

36

model has been proven to work successfully as a digital predistortion algorithm for PAs, see Paper V.

Our work complements previous work on Volterra models with orthonormal basis functions since we use complex-valued rather than real-valued signals. Moreover, we use experimental data, which extends previous theoretical treatments of models using simulated data.

Previous work [79] interprets (15) as a filter bank of M+1 linear filters (filter i is defined as ( 1)( ) i

iG z z− −= ) followed by an M+1 input, static, nonlinear Volterra system. The slow convergence of (15) when modeling RF PAs motivates an alternative realization of the filters ( )iG z .

Use of Laguerre and Kautz functions, the Laplace transforms of a class of orthonormal exponentials, for approximating stable linear dynamic systems has been suggested in several papers. One example is [80]. To avoid working with the large number of parameters in (15) caused by the high model order O and memory length M normally required for accurate modeling of an RF PA, (see Paper II), we choose

1( )i

iG z

z ξ=

− (17)

where the poles { }iξ are chosen to a priori knowledge of the system properties. The KV model output is defined as

1

3 3 3

1 2 3 1 2 3

1 2 1 3

1

1 2 1 ( 1)/2 ( 1)/2 ( 3)/2 1 1

1 2 ( 1)/2 ( 3)/2

1,1

*3, 3, 3,

1 1

1

* *, , , , ,

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

O O O O O

O

O O O O O

O O O

N

KV l ll

N N N

l l l l l ll l l l

N N N N N

l ll l l l l l l l

O l O l O l O l O l

y n b x n

b x n x n x n

b

x n x n x n x n x

+ − + = −

+ +

=

= = =

= = = =

=

+ +

+ ×

∑

∑∑ ∑

∑∑ ∑ ∑ ∑

( )n

(18)

where O denotes the nonlinear model order, Ni the number of basis functions for model order i, and (*) the complex conjugate. Furthermore, xi,l(n) is the ith nonlinear order output of the filter Gi,l(z)

1 *2

,1

(1 )( ) 1 ,

( ) ( )

1,2,..,

lq

i l ll qq

zzG z

z z

l N

ξξ

ξ ξ

−

=

−= −

− −

=

∏ . (19)

By judicious choice of the filters Gi,l(z), a model with good accuracy can be achieved with a small number of basis functions, Ni. A KV model can be identified from measured input and output data using standard techniques (see Paper II).

In Paper IV the KV model is used for modeling a real RF PA intended for a 3G communication system. The frequency dependence of the KV model error for two different input bandwidths is investigated along with the calculated NMSE and ACEPR values. The possibility of using frequency weighting for suppressing the

37

out-of-band error (i.e. the ACEPR value) is also investigated. In addition, we investigated the use of a KV model as an inverse PA model and present the relationship between the optimal pole placement for direct and inverse models.

Figure 18 shows an example of how the NMSE varies for different pole placements of the third order pole. The pole can thereafter be held fixed to the position with the lowest NMSE when finding the poles for higher nonlinear orders. It can be seen in the figure that the optimum position is clearly identified. More data on optimal poles are given in [77]. The KV model presented in Paper IV was found to have low model error compared with other behavioral models (cf. Paper II). In Figure 19 the model error spectrum of a KV model is shown. The spectra of the model error for a static polynomial model and the measured output of the PA are shown for comparison.

Optimal pole placements for different identification situations were studied in Paper IV. In Figure 20 we show optimal pole placements for three conditions. The blue circles and the red squares are optimal pole placements for 3rd, 5th, 7th, and 9th orders for two different output powers of the PA, respectively, the latter being twice as high as the former. The black triangles in Figure 20 mark the optimal pole positions when frequency weighted data is used.

The poles are concentrated within a limited sector (about ±30 degrees ) in the z-plane. The poles are real, or close to the real axis, except for those who were found with frequency weighting. Because frequency weighting introduces a strong frequency dependence, which is equivalent to a memory effect, it affects the location of the optimal poles. The optimal pole placements are tabulated in Table II of Paper IV.

The ability to fix pole placements for different modeling situations is a distinct advantage for the KV model over other block models. The number of complex-valued parameters remains small in conjunction with a low value for the modeling error. With the use of frequency weighting, the low overall modeling error in the KV model can be "concentrated" in the adjacent channels and therefore suppress the out-of-band error (ACEPR) (see Paper IV).

38

-1-0.5

00.5

1

-1-0.5

00.5

1-35

-37

-39

RealImaginary

NM

SE

[dB

]

Figure 18. NMSE vs. third order pole placement.

-10 -5 0 5 10-70

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dB]

Meas. Output

Polynom3.84

KV 1

Figure 19. Model error spectra for a KV and static model. The measured output spectrum is shown for reference. The models were validated with a WCDMA-like signal with an output power of 44 dBm and a 7.2 dB peak-to-average ratio.

39

0.2

330

0

Real

Imag

inar

y

0.4

0.6

Figure 20. Optimal pole placements for three different identification situations. Blue circles, red squares, and black triangles correspond to 44 dBm, 47 dBm output power, and frequency-weighted data, respectively.

40

41

8 Digital Predistortion of RF PAs

One of the most promising applications of dynamic behavioral modeling of RF PAs is for digital predistortion. In digital predistortion the inverse of the PAs response function, H , is first identified and then used for predistorting the signal before it is fed to the PA in such a way that the output signal of the PA is similar to the desired output, according to given linearization criteria [1] (see Figure 21). The output of the digital predistortion algorithm

( ) ( )1H [ ]dx t y t−= (20)

is fed to the PA so that the overall output can calculated as

( ) ( )

( ){ }1

H[ ]

H H [ ] .d

y t x t

y t−

= = (21)

If the inverse function, -1H , is close to the inverse to the real function of the PA, the output of the PA, ( )y t , will be close to the desired signal, ( )dy t . The digital predistortion can be model-based or implemented through a look-up table. It is not guaranteed that a behavioral model with low model error will be able to function as a digital predistortion algorithm with the same accuracy. Different models have different prerequisites for success. Not all behavioral models can be analytically inverted, and most often in practice the inverse function of the PA is identified directly from measured input and output data.

Figure 21. Block diagram describing digital predistortion (DPD). A desired signal is predistorted in the digital predistortion algorithm and then fed to the PA.

Nonlinear dynamic behavioral PA models can usually be categorized as either neural networks [26] or Volterra models [9] (see Chapter 2 ). An example of digital predistortion using the Volterra-type parallel Hammerstein model is given in Figure 22 (cf. [12, 52]), which shows the predistortion results from a commercial Doherty amplifier. In [1] the Doherty PA technique is explained. The improvement in adjacent channel power ratio is in excess of 20 dB compared to a case where digital predistortion is not used.

DPD PA yd(t) x(t) y(t)

42

The most common application of neural networks is for adaptive digital predistortion [70, 81]. Multilayer perceptrons [70] and recurrent neural networks [82] have previously been used for digital predistortion. In [83] a complex-valued RBFNN was used for equalization of simulated communication channels that included nonlinear distortion, which is an application similar to digital predistortion.

In Paper I RBFNNs were also used for digital predistortion of a 4-MHz signal, and the adjacent channel power ratio of the output signal was compared to that of an undistorted signal. In particular, RBFNNs with memory were investigated. In Figure 23 we show the normalized power spectra of the measured output signal for the case of no predistortion and when RBFNNs with M = 0 and with M = 1 were used as predistortion algorithms. The RBFNN with M = 0 clearly suppresses the intermodulation products as compared to the undistorted signal, and the RBFNN with M = 1 suppresses the intermodulation products even further. The polynomial predistorter resulted in an output spectrum (not shown in the figure) very similar to that of the RBFNN with M = 0 (see Paper I)

-15 -10 -5 0 5 10 15 -80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dB]

No DPDDPD

Figure 22. The output signal of the Doherty amplifier without digital predistortion (red) and with digital predistortion (blue).

43

-6 -4 -2 0 2 4 6

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dB]

No DPD

RBFNN M=1

RBFNN M=0

Figure 23. Normalized spectra of the measured output signal for an undistorted input signal, an input signal predistorted with a memoryless RBFNN, and an input signal predistorted with a RBFNN with one-sample memory.

The predistortion properties of the KV model ((18), (19), and [77]) were investigated in Paper IV and Paper V. The optimal pole placements for direct and inverse models of an RF PA were studied. Figure 24 shows optimal pole placements in the complex plane for 9th order KV models. The blue circles represent the direct model and the red squares the inverse model. It is noticeable that the optimal poles of the inverse model have the same positions as those of the direct model, which indicates that the memory effects of the direct and inverse functions are the same. This result is somewhat unexpected since inverse functions are generally more complex and have longer memory than direct ones [84]. However, an analytical study of the inverse Volterra model of nonlinear systems with small linear memory effects (see Paper IV) easily explains the result presented in Figure 24. There, an investigation up to 5th nonlinear order is presented, and analytical expressions given in Paper IV predict that poles in direct and inverse models are the same if the memory effects in the linear term is negligible.

Paper V reports on digital predistortion of a real RF PA using the KV model. The KV model used was of the 7th nonlinear order and required 34 complex parameters and 4 complex poles. Figure 25 shows the result of the digital predistortion study. A 7th order polynomial predistorter as described in Paper III was used for comparison. Figure 25(a) gives the output spectra for an output power Pout=47 dBm, and Figure 25(b) gives the same for an output power Pout=44 dBm. Results for the case of no digital predistortion are also given in this figure. In Figure 25(a) it is seen that the KV model suppresses the adjacent channel power ratio more than the polynomial model and gives a more symmetric spectrum; the KV model

44

suppresses adjacent channel power ratio 5 to 7 dB more than the polynomial predistorter. Table I of Paper V gives the adjacent channel power ratios.

Poles that were optimal for Pout=47 dBm were used also for the case of Pout=44 dBm. The output spectra, not shown in Figure 25(b), were similar to those found to be optimal at 44dBm. The difference in adjacent channel power ratio between the KV models was ~0.5 dB. The KV model is thus insensitive to the position of the poles with respect to output power (cf. Figure 20). The power efficiency was ~15 % for Pout =47 dBm and ~9 % for Pout=44 dBm.

Because the presented digital predistortion algorithm has the same general properties as a Volterra model, it could likely be used for applications other than PA linearization. Based on IIR filters, it is computationally efficient and needs relatively few parameters for modeling long memory effects. Since wideband signals are more affected by memory effects than narrowband signals, and since inverse models often have longer memory effects than direct models, we believe that the proposed algorithm could be of use in future wideband applications.

0.2

-30

0

Real

Imag

inar

y

0.4

0.6

Figure 24. Optimal pole placements for the direct (blue circle) and the inverse (red square) models.

45

-6 -2 0 2 6-70

-60

-50

-40

-30

-20

-10

0

Frequency [MHz]

Nor

mal

ized

Pow

er [

dBc]

-6 -2 0 2 6Frequency [MHz]

No DPDPolynom.KV

(a) (b)

Figure 25. Normalized output spectra for (a) Pout=47 dBm and (b) Pout =44 dBm. The KV models and the polynomials were of order 7. Two different sets of poles were used, one for 44 dBm and one for 47 dBm, as described in Paper V.

46

47

9 Concluding Remarks

This thesis principally deals with behavioral modeling, parameter-reduction, and digital predistortion of RF PAs. The description is mainly in the digital domain, but it is the result of measurements made in an RF laboratory. Therefore, this thesis provides relevant aspects on RF PA measurement techniques with respect to several important parameters. The two most important parameters are bandwidth and dynamic range. Due to the nonlinear behavior of the device, the bandwidth of the measured output signal must be the nonlinear order times the bandwidth of the input signal, as seen above. The trend of increasing signal bandwidths in the modern communication systems makes the measurement situation even more complicated. Due to spectrum masks, as specified in e.g. [50], the overall linearity of the transmitter, with an implemented linearization technique, must be sufficiently high. The performance of the proposed behavioral PA models that are candidates for an inverse model in a digital predistortion algorithm must have sufficiently low errors out-of-band. Hence, the validation of these models relies on data with a dynamic range in excess of the corresponding model errors.

This thesis presents two new types of behavioral models for RF PA modeling. Both types are compared with other relevant behavioral models with respect to both in-band and out-of-band errors. Both models have been proven to work as inverse models in a digital predistortion algorithm and their capabilities to suppress the power relative to the carrier out-of-band are reported. Moreover, a method to measure and analyze Volterra kernels is presented. This method was applied on the third-order nonlinear behavior properties of an RF PA. This method, which is used to analyze and determine Volterra kernels along certain paths in the frequency domain, was used to investigate the symmetry properties of the PA. The investigation showed that the nonlinear properties of the PA had the properties of an N-H-N system and that the main frequency dependence occurs because of baseband effects. This thesis also concludes that when designing a PA for use with digital predistortion, it is most important to reduce the memory effects in the linear term since these will most significantly affect the inverse function.

Even though the behavioral models, including those used in this thesis, are normally used for modeling the properties of subsystems of telecommunication systems that are too complex for circuit-level or transistor-level simulations, physical models can also provide insight into the source of nonlinear properties and memory effects. Both behavioral and physical models are thus useful in the design process. Physical models are based on a physical description of the components of the PA and on how they interact. An interesting continuation of this work and a possible future research topic could be to explain behavioral models in physical terms to a larger extent. One way to proceed in that direction could be to further analyze the complex poles of the Kautz-Volterra model with respect to e.g. changes in the decoupling networks or biases of the PA as well as different symmetry properties of a nonlinear system.

48

49

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57

Summary of Appended Papers

The author has significantly contributed in all aspects in all the listed papers. In Paper III, the first author has done a majority of the work although the thesis author has taken a significant part in the instrumentation, measurements and writing process. Paper V is a joint work.

Paper I Wide-Band Dynamic Modeling of Power Amplifiers Using Radial- Basis Function Neural Networks

A radial-basis function neural network (RBFNN) was used to model the dynamic nonlinear behavior of an RF power amplifier designed for use with third generation (3G) communication systems. In the model the signal’s envelope is used. The model requires less training than a model using IQ data. Sampled input and output signals derived from noise-like signals with bandwidths of 4 and 20 MHz were used for identification and validation. Subsequently the performance of the RBFNN was compared to that of a parallel Hammerstein (PH) model. The two model types demonstrated similar performance for memoryless systems and similar in- and out-of-band performance for a 20-MHz signal; the RBFNN has better in-band and the PH system better out-of- band performance when memory was used for a 4-MHz signal. When used as a digital predistortion algorithm, the best RBFNN with memory suppressed the lower (upper) adjacent channel power by 7 dB (4 dB) compared to a memoryless nonlinear predistorter, and by 11 dB (13 dB) compared to the case of no predistortion at the same output power for a 4-MHz-wide signal.

Paper II A Comparative Analysis of Behavioral Models for RF Power Amplifiers

A comparative study of nonlinear behavioral models with memory for radio-frequency power amplifier (PAs) is presented. The models included were: a static polynomial, parallel Hammerstein (PH), Volterra, and radial-basis function neural network (RBFNN). In addition, two PAs were investigated: one designed for third generation (3G) mobile telecommunication systems and the other for second generation (2G). The RBFNN reduced the total model error slightly more than the PH, but the out-of-band error was significantly lower for the PH. The Volterra model was found to yield a lower model error than did a PH of the same nonlinear order and memory depth. The PH could achieve a lower model error than the best Volterra, since the former could be identified with a higher nonlinear order and memory depth. The qualitative conclusions are the same for the 2G and 3G PAs, but the model errors are smaller for the latter. For the 3G PA, a static

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polynomial gave as low a model error as the best PH and lower than the RBFNN for the hardest cross-validation test condition. The models with memory, PH, and RBFNN, showed better cross-validation performance, in terms of lower model errors, than a static polynomial for the hardest cross-validation condition in the 2G PA.

Paper III Three-Tone Characterization of Nonlinear Memory Effects in Radio Frequency Power Amplifiers

A stepped three-tone measurement technique based on digitally modulated baseband signals is used for characterizing radio frequency power amplifiers. The bandwidth of the stepped measurement was 8.8 MHz for the input and 26.4 MHz for the output signal. A power amplifier designed for third-generation mobile telecommunication system was analyzed. The amplitude and phase of the third-order Volterra kernel were determined from identified intermodulation products. Properties of the Volterra kernel along several paths in the three-dimensional frequency space were analyzed and compared to various box-models for nonlinear systems. The main symmetry of the third-order Volterra kernel of this power amplifier was found to be of the type given by cascaded quadratic nonlinearities with a linear filter in between (i.e. a Hammerstein-Wiener system) and the frequency dependence (i.e. memory effects), were found to be due to effects at baseband.

Paper IV A Parameter-Reduced Volterra Model for Dynamic RF Power Amplifier Modeling based on Orthonormal Basis Functions

A nonlinear dynamic behavioral model for radio frequency power amplifiers that uses Kautz orthonormal basis functions and different complex poles for each nonlinear order is presented. This model has the same general properties as Volterra models, but the number of parameters is significantly smaller. Frequency weighting was used to reduce the out-of-band model error, and experimental data showed that the optimal poles were the same for different input powers and different nonlinear orders. The optimal poles were also the same for direct and inverse models, which could be theoretically explained as a general property of nonlinear systems with negligible linear memory effects. The model can be used as either a direct or inverse model that yields the same model error for power amplifiers with negligible linear memory effects.

Paper V Digital Predistortion of Radio Frequency Power Amplifiers Using a Kautz-Volterra Model

A digital predistortion algorithm for radio frequency power amplifiers based on orthonormal basis functions is presented. This model has the same general properties as a Volterra model, but fewer parameters.

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Experimentally, using a 3.84 MHz wide signal at 2.14 GHz, the adjacent channel power was suppressed by 12 to 15 dB, which was 5 to 7 dB more suppression than was generated by a polynomial predistorter