# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 1 Introduction
Theoretical analysis of low-pass filters, both with hand-made and electronically adjustable significant frequencies and quality factors, is the subject of [34; 35; 36]. Similarly, FO high-pass filtering structures is the topic of [37]. However, network structures proposed here can be considered as standard, with direct replacement of all capacitors with FO immittance equivalents. The presented results, i.e., frequency responses and filter parameters, do not exhibit significant benefits. Several different reconfigurable FO filters, i.e., multi-ports with different accessible frequency responses, can be found in [38; 39; 40; 41]. In these publications, reaching different slopes seems to be the focus of authors. Specific classes of the so-called (\(1+\alpha\)) filters, i.e., circuits having a mathematical order between one and two, are thoroughly studied in [42; 43]. The network presented and analyzed in [44] also belongs to this class of filter, but allows for full adjustment of the filter's quality factor. This useful feature can be incorporated into the standard configuration of the Kerwin-Huelsman-Newcomb (KHN) filter, but at the cost of an additional active element connected as an amplifier. Synthesis and analysis of the higher-order FO filters is thoroughly discussed in [45]. The design process of active filters (including those with FO elements) can be significantly sped up by using a field programmable analog array (FPAA) [46; 47]. Besides the necessary frequency limitations of these development kits (defined by intrinsic low-pass filters), there is also a restricted area dedicated to circuit design. Thus, only simple active topologies can be implemented using an FPAA. An interesting question is the correct operation and network topology leading to an FO all-pass filter. The simple interchange of common accumulation elements with an FO equivalent does not lead to a module frequency response that is constant and independent of frequency. This problem has been recognized, and the authors of [48] tried to solve it with only partial success, since the module frequency response is not flat. CPEs are core circuit elements in a class of building blocks where the module frequency response is not as important as the phase frequency response. Asymptotical phase shifters are devoted to the phase transition between two arbitrary values for frequency, moving from DC to infinity. There are two related types with increasing [49] and decreasing [50] phases. The core idea behind both papers is to practically realize predefined complex transfer functions as a composition of a simple (order is equal to or less than unity) sub-parts. This is a task suitable for cascade synthesis. Individual two-port sections can be, for example, implemented by a sequence of inverting amplifiers connected in a common configuration with voltage-feedback operational amplifiers and grounded non-inverting inputs. The only condition is a sufficiently large value of input impedance for each section. This input impedance corresponds to the module of impedance connected to the input terminal. Of course, there are other possibilities for implementing high-input impedance voltage-mode two-port configurable transfer functions. The design of modern feedback regulators has been intentionally affected by the presence of an FO element, either two-port or two-terminal. Differences and similarities of integer-order and FO Proportional-Integration-Derivation (PID) control is discussed in [51]. The reason for this is to avoid abrupt changes in the dynamics of plant processes [52; 53], or due to the necessity of gaining smooth control of the derivation and integration process. FO circuit elements are included in the single-input single-output configurations of PID regulators (see tutorial [54]). Therein, a few interesting examples of the very fast processes that are successfully controlled, and the corresponding measured responses, are provided. The design of PID controllers for fractional-order systems with time delays is presented in [55].