# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 1 Introduction  
Although fractional-order (FO) calculus is an ancient mathematical idea, it has still attracted significant interest from analog design engineers in the last two decades. In the field of signal generation and processing, we can find many applications of FO circuit elements. As mentined in the popularizing paper [1], concepts of circuits with FO elements and with equivalent integer-order parts remain the same, but the system properties and features of the first case can be better. Besided this statement, the author also provides a fundamental background for numerical analysis of FO networks. The application potential of analog circuits with FO elements has been celebrated in a ten year old paper [2], where some possible structures of FO capacitors are described and used in multivibrator circuits. As mentioned, the design of a good FO capacitor is enough ground to start the construction of complex analog FO networks. An FO capacitor can be understood as a two-terminal element that forms a bridge between the resistor and conventional capacitor, which has a module frequency response of admittance linearly increasing with frequency (in a logarithmic frequency scale) slower than the capacitor, and has constant phase shift of current and voltage between 0\({}^{\circ}\) and 90\({}^{\circ}\). In theory, these features should be valid from DC up to very high frequencies. Constant phase properties lead to the denotation of FO capacitors as constant phase elements (CPE). In fact, CPE can be understood as a generalization of FO capacitors since the same abbreviation can be used for FO inductors, FO integrators, and other multi-ports. The specific value of a module's slope and phase shift depends onthe mathematical order of an FO element. In the case of FO capacitors, this order could be any real number between zero and one. Fingerprints that resemble the behavior of true CPEs have been recognized during observations of many physical phenomena in different branches of engineering, including electrical and chemical [3; 4; 5; 6]. Unfortunately, despite huge efforts in the development of suitable materials [7; 8] and fabrication techniques [9], \"good\" FO capacitors that are generally applicable are still not available. Thus, behavior of CPEs need to be modeled by more complex circuitry with appropriately connected simple components with well-defined dynamics. The first official mention of an electronic approximation of CPEs (in this early stage denoted as fractance or a fractional capacitor) can be found in [10], where impedance (1/s)\({}^{1/n}\) has been approximated using an RC network. In particular, the case \(n=2\) is the subject of the paper [11]. Inverse immittance function, that is s\({}^{1/n}\), is briefly studied in [12]. Many years later, passive ladder CPE approximations were re-investigated in [13]. Besides generalizations, it is pointed out that these CPEs have close connections to RC networks with distributed parameters. In fact, sections of transmission lines can be used to model the behavior of constant argument impedances. Mathematical descriptions of these CPEs (for example, half-capacitor) can be based on the primary R, G, L, or C parameters per unit length [14]. For lumped realizations of CPE, the author highly recommends two papers [15; 16] where the calculation routine of immittance-type passive RC approximants CPEs is described in a step-by-step manner. For a deeper investigation of available CPE approximation techniques, including the popular Chareff and Oustaloup methods, readers should consult a comprehensive review [17]. This paper provides advice and formulae how to choose the optimal order of passive RC approximation circuits. Considerable attention was paid to active implementations of CPE. Complex immittance-type CPEs that take advantage of the multiplication of many first-order zero-pole immittances are thoroughly investigated in [18].