# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 1 Introduction
Electronic tunability (via external DC signals) of the trans-conductances associated with voltage-controlled current-sources can be used to set the phase shift of CPE approximation, as pointed out in [19]. However, since trans-conductances vary with temperature and age, sensitivity issues need to be addressed. From a practical point of view, the cascading of simple bilinear sections based on AD844 integrated circuits could be a better active solution for CPEs [20]. Because a pair of bilinear filters can be interchanged by a general biquadratic section, two-port CPEs can be approximated using second-order filters [21]. Comparative studies of active realizations of both impedance-type and admittance-type two-terminal CPE approximants can be found in [22]. A similar analysis of different active implementations of two-port CPEs is the topic of [23]. A change of the trans-conductance of operational trans-conductance amplifiers can be used to change the FO nature of a two-terminal element. This idea was first introduced briefly in [24], and, considering one active alternative, developed into complete circuitry described in detail in [25]. CPEs have been advantageously incorporated into many different building blocks of analog signal processing. First, let us discuss a few applications of CPE in the signal generation process. The most straightforward approach to how FO oscillators can be obtained is by direct replacement of capacitors or inductors with an FO equivalent. This works well for the common structures of feedback oscillators [26], oscillators based on integrator block schematics [27], as well as for oscillators based on state variable circuit synthesis [28]. However, detailed study of these and similar publications reveals that such a substitution does not bring significant benefits over conventional integer-order harmonic oscillators. Moreover, in the case of the negative resistance concept of an oscillator, the negative slope of the nonlinear component needs to be adjusted; a linear sub-circuit is \"more stable\" since the roots of the describing characteristic polynomial have different locations in complex planes (see [29] for clarification). Probably the most promising advantage of employing FO elements inside a generator of harmonic waveforms can be noticed in the multi-phase oscillators [30] or electronically adjustable oscillators with a constant phase shift between two circuit nodes over the whole tunability range. Note that the second case can be implemented by using two integer-order all-pass filters in a cascade; only the control of frequency change becomes much more complicated (additional circuitry needs to be included). The author highly recommends [30] because it can be considered as a cookbook for designers of FO oscillators. Circuit elements with FO dynamics can be utilized also to enhance the behavior of relaxation oscillators. Easy replacement of a common capacitor with an FO equivalent can generate time domain waveforms [31] that make use of a memory property (oil-like effect). The second area, where many sound applications, discoveries, and interesting publications can be found, is signal processing. Although fractionality can be introduced into continuous-time as well as discrete-time dynamic systems, let us focus on the linear continuous-time systems, starting with frequency filters with CPEs. The simplest ideas, including transfer functions and associated frequency responses of the first-order filters generalized to FO domains, are presented in [32]. Since this interesting work describes the behavior of a near zero-order filter under idealized conditions, it also serves as a springboard for many upcoming studies, research papers, and applications. Much more versatile transfer functions in the Laplace transform, leading to a large variety of frequency responses, are provided in [33]. Therein, analog frequency filters with a total order between one and two are addressed.