Microwave Filters for Communication Systems: Fundamentals, Design and Applications

CHAPTER 3

CHARACTERIZATION OF LOSSLESS LOWPASS PROTOTYPE FILTER FUNCTIONS

This chapter describes the synthesis process for the characteristic polynomials to realize the ideal, classical, prototype filters: the maximally flat, Chebyshev, and elliptic function filters. The chapter includes a discussion of filters that are not symmetric with respect to their center frequency. This leads to transfer function polynomials (with certain restrictions) with complex coefficients, a distinct departure from the more familiar characteristic polynomials with rational and real coefficients. This provides a basis for analysis of the most general class of filter functions in the lowpass prototype domain, minimum and nonminimum phase filters, exhibiting a symmetric or an asymmetric frequency response.

3.1 THE IDEAL FILTER

In communication systems, filter networks are required to transmit and attenuate signals in specified frequency bands. Ideally, this must be accomplished with the minimum of distortion and loss of energy of the transmitted signal.

3.1.1 Distortionless Transmission

A signal is characterized by the amplitude and phase of its frequency components, referred to as the waveform of the signal. For distortionless transmission, the waveform must be preserved; that is, the output signal must be an exact replica of the input signal. This can be achieved only if the filter network has a constant amplitude and time delay for all the frequencies in the passband. A constant time delay implies ...