TABLE OF CONTENTS As concluded in the previous section, other methods are sometimes required in order to obtain network functions for unusual filter requirements. A method involving the use of the Fourier series approximation to a periodic function provides a brute-force solution to almost any requirement that can be imagined. This Fourier method, which is developed first, leads to a simple iterative structure. The method also introduces the Hilbert transform relationship between the network magnitude function and its required minimum phase function. The chapter concludes by presenting an algorithm that finds a network function from the associated minimum phase to fit magnitude (attenuation) specifications that are described by a series of straight-line connected points on a magnitude (in decibels) frequency response curve.
Fourier series have the advantage that by simply adding more terms, convergence to a desired (objective) function is obtained. Normally, the requirement for a large number of terms is undesirable, but in the case to be presented here, each pole is obtained from a two-transistor unit-gain amplifier, and so, from an integrated circuit point of view, the iterated circuit thus obtained is sometimes a practical realization. Consequently, this section presents the theory for an inefficient Fourier series approach that leads to a more efficient pole-placement method, which concludes the chapter. Theoretically, however, we gain more understanding about the network function and the properties of its sinusoidal steady-state magnitude and phase function dependencies.
In order to obtain a Fourier series, we need a periodic...
