Using residue sums to estimate high‐order Fourier harmonics of piecewise‐continuous transcendental functions: Application to Class A‐, B‐ and F‐type amplifier circuits

Abstract

AbstractA discrete fast‐Fourier transform (DFFT) is the preferred method of choice for the rapid evaluation of a set of harmonics of a piecewise‐continuous and periodic transcendental form. For large sets of Fourier components specified to some stringent error criterion, the approach becomes increasingly unattractive owing to the presence of round‐off errors that result from the switching of one transcendental form to another. As an alternative, it might be wondered whether the high‐frequency components can be more efficiently estimated by employing a combination of residue sums and boundary integrals in the complex plane z = Reiωt, where ω is the fundamental frequency and R = ∣z∣.The starting point is the construction of suitable contours that divide the complex plane into a number of sectors in accordance with the number of intervals of smooth behaviour of a periodic piecewise‐continuous real function along ∣z∣ = 1. Each sector encompasses the analytic extension of a real transcendental function on ∣z∣ = 1 to yield p(z)T(f(z)), where T(ζ) is meromorphic and p(z), f(z) are Laurent series. Fourier coefficients are subsequently expressed in terms of residue series and constant‐phase boundary integrals from each of the various sectors associated with a given p(z)T(f(z)). This approach is applied to the model for the drain current of a field effect transistor (FET), where in this case T(ζ) = tanh(ζ), which is subject to the modes of operation: ‘Class A’, ‘Class B’ and approximate ‘Class F’. In contrast to Classes A and B, the Fourier coefficients in the ‘Class F’ drain current decay slowly with frequency, suggesting that this mode might be more suitably analysed using a combined DFFT/residue procedure. Copyright © 2007 John Wiley & Sons, Ltd.