Mathematical Problems in Engineering

Volume 2015, Article ID 585962, 29 pages

http://dx.doi.org/10.1155/2015/585962

## Conflict Set and Waveform Modelling for Power Amplifier Design

Department of Power, Electronics and Communication Engineering, Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia

Received 19 July 2014; Accepted 7 February 2015

Academic Editor: Ben T. Nohara

Copyright © 2015 Anamarija Juhas and Ladislav A. Novak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Various classes of nonnegative waveforms containing dc component, fundamental and harmonic , which proved to be of interest in waveform modelling for power amplifier (PA) design, are considered in this paper. In optimization of PA efficiency, nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) play an important role. Optimal waveforms have multiple global minima and this fact closely relates the problem of optimization of PA efficiency to the concept of conflict set. There is also keen interest in finding descriptions for various classes of suboptimal waveforms, such as nonnegative waveforms with at least one zero, nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of harmonic, and nonnegative cosine waveforms with at least one zero. Closed form descriptions for all these suboptimal types of waveforms are provided in this paper. Suboptimal waveforms may also have multiple global minima and therefore be related to the concept of conflict set. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA modelling are also provided.

#### 1. Introduction

The origin of the concept of conflict set goes back to J. C. Maxwell (Maxwell 1831–1879), who informally introduced most of features of what today is called conflict set [1]. From this reason Maxwell set or Maxwell stratum is also used as synonyms for conflict set. Roughly speaking, conflict set associated with a smooth function with parameters is the set of -tuples in parameter space for which has multiple global minima. Conflict set is also intimately related to singularity theory and catastrophe theory [1].

Although without explicit reference, many max-min/min-max engineering design problems related to nonsmooth optimizations in parameter spaces (e.g., see [2]), including problems related to the optimization of efficiency of power amplifiers (PAs) (e.g., see [3–12]), are connected to the concept of conflict set. The concept of conflict set has been also used in mathematics (e.g., see [13–16]) and physics (e.g., see [17, 18]), including subjects like black holes [19].

Nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) have multiple global minima and therefore are closely related to the concept of conflict set. The suboptimal waveforms such as(i)nonnegative waveforms with at least one zero,(ii)nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic,(iii)nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic,(iv)nonnegative cosine waveforms with at least one zeromay also have multiple global minima [9, 11, 12] and therefore be related to the concept of conflict set, as well. These suboptimal waveforms are clearly of interest in shaping/modelling drain (collector/plate) waveforms in PA design (e.g., see [3–12, 20, 21]).

Fejér in his seminal paper [22] provided general description of all nonnegative trigonometric polynomials with consecutive harmonics in terms of parameters satisfying one nonlinear constraint. He also derived closed form solution to the problem of finding maximum possible amplitude of the first harmonic of nonnegative cosine polynomials with consecutive harmonics.

Fuzik [3] (see also [10]) considered cosine polynomials with dc, fundamental and harmonic, for arbitrary and provided closed form solution for coefficients of optimal waveform. Rhodes in [7] provided closed form expression for maximum possible amplitude of fundamental harmonic of nonnegative waveforms containing consecutive odd harmonics. A subclass of nonnegative cosine waveforms with dc, fundamental and third harmonic, having factorized form description has been considered in [23].

High efficiency PA with arbitrary output harmonic terminations has been analysed in [9], along with maximal efficiency, fundamental output power, and load impedance.

Factorized form of nonnegative waveforms up to second harmonic with at least one zero has been suggested in [11] in the context of continuous class B/J mode of PA operation.

General description of all nonnegative waveforms up to second harmonic in terms of four independent parameters has been provided in [12]. This includes nonnegative waveforms with at least one zero, as a special case.

End point of conflict set normally corresponds to so-called maximally flat waveform, which also belongs to class of suboptimal waveforms. First comprehensive usage of maximally flat waveforms, in the context of analysis of PA, goes to Raab [20]. General description of maximally flat waveforms with arbitrary number of harmonics has been presented in [21], along with closed form expressions for efficiency of class-F and inverse class-F PA with maximally flat waveforms. Description of maximally flat cosine waveforms with consecutive harmonics has been presented in [8] in the context of finite harmonic class-C PA.

In this paper we provide general descriptions of a number of optimal and suboptimal nonnegative waveforms containing dc component, fundamental and an arbitrary th harmonic, , and show how they are related to the concept of conflict set. According to our best knowledge, this paper provides the very first usage of conflict set in the course of solving problems related to optimization of PA efficiency. Main results are stated in six propositions (Propositions 1, 6, 9, 18, 22, and 26), four corollaries (Corollaries 2–5), twenty remarks, and three algorithms. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA efficiency analysis are considered in detail in Section 7.

This paper is organized in the following way. In Section 2 we introduce concepts of minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3). In Sections 3–6 we provide general descriptions of various classes of nonnegative waveforms containing dc component, fundamental and th harmonic with at least one zero. General case of nonnegative waveforms with at least one zero is presented in Section 3.1. The case with exactly two zeros is considered in Section 3.2. An algorithm for calculation of coefficients of fundamental harmonic of nonnegative waveforms with two zeros, for prescribed coefficients of harmonic, is presented in Section 3.3. Description of nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of harmonic is provided in Section 4. Nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of harmonic are considered in Section 5.1. An illustration of results of Section 5.1 for particular case is given in Section 5.2. Section 6.1 is devoted to nonnegative cosine waveforms with at least one zero and arbitrary , whereas Section 6.2 considers cosine waveforms with at least one zero for . In Section 7 four case studies of application of descriptions of nonnegative waveforms with fundamental and harmonic in PA modelling are presented. In the Appendices, list of some finite sums of trigonometric functions, widely used throughout the paper, and brief account of the Chebyshev polynomials are provided.

#### 2. Minimum Function, Gain Function, and Conflict Set

In this section we consider minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3) in the context of nonnegative waveforms with fundamental and harmonic.

We start with provision of a brief account of the facts related to the concepts of minimum function and conflict set. For this purpose let us denote by a family of smooth functions of variables depending on parameters, where is -tuple of variables and is -tuple of parameters. The minimum function , associated with the function , is defined as . Therefore, the domain of the minimum function is parameter space of the function . The minimum function is continuous, but not necessarily smooth function of parameters [13, 24]. It is a smooth function if possesses unique global minimum at nondegenerate critical point [13] (critical point is degenerate if at least first two consecutive derivatives are equal to zero). In this context, the conflict set can be defined as the set of the parameters for which function has global minimum at a degenerate critical point or/and multiple global minima [13].

For a wide class of minimum functions, when the number of parameters is not greater than four, the behaviour of minimum function in a neighbourhood of any point can be described by one of “normal forms” from a finite list as stated in [24]. For example, for smooth function , the minimum function near the origin can be locally reduced to one of the following three normal forms [25]: , or . In this example, the conflict set is the set of all points for which minimum function is not differentiable because function possesses at least two global minima [25].

##### 2.1. Minimum Function and Gain Function

In what follows we consider family of waveforms of typewhere stands for , , , , and . Waveforms of type (1) include all possible shapes which can occur, but not all possible waveforms containing fundamental and th harmonic. However, shifting of waveforms of type (1) along -axis could recover all possible waveforms with fundamental and th harmonic.

The problem of finding nonnegative waveform of type (1) having maximum amplitude of fundamental harmonic plays an important role in optimization of PA efficiency. This extremal problem can be reformulated as problem of finding nonnegative waveform from family (1) having maximum possible value of coefficient . Nonnegative waveform of family (1) with maximum possible value of coefficient is called “optimal” or “extremal” waveform.

Furthermore, let us introduce an auxiliary waveform which is smooth function of one variable and two parameters and . In terms of , the above extremal problem reduces to the problem of finding maximum possible value of coefficient that satisfiesClearly, for any prescribed pair , there is a unique maximal value of coefficient for which inequality (3) holds for all . This maximal value of associated with the pair we denote it by and call it “gain function.”

Letbe the minimum function associated with . According to (3), and satisfy the following relation: . Since is obviously nonzero it follows immediately thatA relation analogue to (5), for (fundamental and second harmonic), has been derived in [4]. According to our best knowledge, it was the first appearance of gain function expressed via associated minimum function. The consideration presented in [4] has been restricted to the particular case when . The same problem for and arbitrary has been investigated in [3] (see also [10]).

According to above consideration, the problem of finding 3-tuple with maximum possible value of for which (3) holds is equivalent to the problem of finding maximum value of gain function and corresponding pair that satisfiesThus the optimal waveform is determined by parameters , , and ; that is, Optimal waveform has two global minima (this claim will be justified in Section 4, Remark 21). Consequently, the pair , which corresponds to maximum of gain function , belongs to conflict set in parameter space.

Figure 1 shows graph of gain function for . Notice that it has sharp ridge and that maximum of gain function (point ) lies on the ridge. This maximum corresponds to the optimal waveform (solution of the considered extremal problem). The beginning of the ridge (point ) corresponds to the waveform which possesses global minimum at degenerate critical point, that is, corresponds to maximally flat waveform (e.g., see [21]). Gain function is not differentiable on the ridge and consequently is not differentiable at the point where it has global maximum. This explains why the approach based on critical points does not work and why conflict set is so important in the considered problem.