General Description of Nonnegative Waveforms up to Second Harmonic for Power Amplifier Modelling

Abstract

General description of nonnegative waveforms up to second harmonic in terms of independent (unconstrained) parameters is provided. Three important subclasses of the class of nonnegative waveforms are also fully characterised: nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of second harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of second harmonic, and nonnegative waveforms with at least one zero. We prove that members of the first two subclasses have at least one zero; that is, they also belong to the third subclass. Nonnegative cosine waveforms up to second harmonic are also considered and characterised. A number of case studies of practical interest for power amplifier (PA) design, involving nonnegative waveforms up to second harmonic, are also considered.for at least one 0 (Section 5). We prove that this subclass includes all nonnegative waveforms with maximal amplitude of fundamental harmonic, when second harmonic amplitude

Conflict Set and Waveform Modelling for Power Amplifier Design

“…Relations (12) and (13) say that ( ; , ) has minima at and , while relations (11) and (14) imply that these minima are equal and global.…”

“…providing that satisfy (40) and | | ≤ . Three examples of nonnegative waveforms with at least one zero for = 3 are presented in Figure 4 (examples of nonnegative waveforms with at least one zero for = 2 can be found in [12]). For all three waveforms presented in Figure 4, we assume that 0 = /6 and = 3 /4.…”

“…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][4][5][6][7][8][9][10][11][12]), are connected to the concept of conflict set. The concept of conflict set has been also used in mathematics (e.g., see [13][14][15][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 zero may 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]).…”

“…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.…”