In this paper general solution to the problem of finding maximally flat waveforms with finite number of harmonics (maximally flat trigonometric polynomials) is provided. Waveform coefficients are expressed in closed form as functions of harmonic orders. Two special cases of maximally flat waveforms (so-called maximally flat even harmonic and maximally flat odd harmonic waveforms), which proved to play an important role in class-Fand inverse class-Fpower amplifier (PA) operations, are also considered. For these two special types of waveforms, coefficients are expressed as functions of two parameters only. Closed form expressions for efficiency and power output capability of class-Fand inverse class-FPA operations with maximally flat waveforms are also provided as explicit functions of number of a harmonics.

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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

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