The Complex Order Fractional Derivatives and Systems Are Non Hermitian

This paper discusses the concept of fractional derivative with complex order from the application point of view. It is shown that a fractional derivative is hermitian, if and only if the order is real. The hermitian part of complex order derivatives behave like allpass filters with almost logarithmic phase.

Wavelet Filter Banks Using Allpass Filters

Allpass filter is a computationally efficient versatile signal processing building block. The interconnection of allpass filters has found numerous applications in digital filtering and wavelets. In this chapter, we discuss several classes of wavelet filter banks by using allpass filters. Firstly, we describe two classes of orthogonal wavelet filter banks composed of two real allpass filters or a complex allpass filter, and then consider design of orthogonal filter banks without or with symmetry, respectively. Next, we present two classes of filter banks by using allpass filters in lifting scheme. One class is causal stable biorthogonal wavelet filter bank and another class is orthogonal wavelet filter bank, all with approximately linear phase response. We also give several design examples to demonstrate the effectiveness of the proposed method.

High Input Impedances Voltage-mode First-order Filters with Grounded Capacitors using CCIIs

Background: Three high input impedances voltage-mode first-order filters are presented. Methods: The first proposed circuit uses one multi-output second-generation current conveyor, two resistors and one grounded capacitor. The second proposed circuit uses two second-generation current conveyors, three resistors and one grounded capacitor. The third proposed circuit uses one multi-output second-generation current conveyor, one resistor and two grounded capacitors. Results: First-order lowpass and allpass filters can be simultaneously obtained in the first proposed circuit. First-order lowpass, highpass and allpass filters can be simultaneously obtained in the second proposed circuit. The third proposed circuit can realize first-order allpass filter. Conclusion: All the proposed circuits have the advantages of high input impedances and using only grounded capacitors.

Frequency-Dependent Schroeder Allpass Filters

Since the introduction of feedforward–feedback comb allpass filters by Schroeder and Logan, its popularity has not diminished due to its computational efficiency and versatile applicability in artificial reverberation, decorrelation, and dispersive system design. In this work, we present an extension to the Schroeder allpass filter by introducing frequency-dependent feedforward and feedback gains while maintaining the allpass characteristic. By this, we directly improve upon the design of Dahl and Jot which exhibits a frequency-dependent absorption but does not preserve the allpass property. At the same time, we also improve upon Gerzon’s allpass filter as our design is both less restrictive and computationally more efficient. We provide a complete derivation of the filter structure and its properties. Furthermore, we illustrate the usefulness of the structure by designing an allpass decorrelation filter with frequency-dependent decay characteristics.