# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 5 Discussion  
The orientation of this paper is two-sided. Firstly, it provides deep investigation of research papers focused on the design and applications of CPEs in analog electrical engineering. This section reveals the current progress in this emerging scientific field such that an interested reader can find their own path to future research topics and answer the as-yet unsolved problems. CPEs have significant potential in audio applications, for example, to generate specific sound effects, create audio filters or user-defined audio correctors, equalizers, etc. Because of these reasons, passive ladder-type realizations of CPEs dedicated to the frequency range beginning with 20 Hz and ending at 20 kHz for 19 different mathematical orders have been proposed. Each approximation utilizes resistors and capacitors taken from the fabrication series E12 and arranged in basic series and/or parallel combinations to concretize real experimental implementations of CPE two-terminal devices. A short program in Matlab for easy design of passive CPE approximants has been created; check Figure 17 for a visualization of a certain example. The calculation engine is a direct implementation of algorithms proposed by Professor [PERSON] and is summarized in [15,16]. Software is able to analyze designed passive CPEs from the viewpoint of fabrication tolerances. If a curious reader is interested in  
Figure 16: Orcad Pspice circuit simulation of two-port CPE with phase shift between output and input voltage 20\({}^{\circ}\) in frequency domain: (**a**) phase response, (**b**) module response, and in time domain: (**c**) signal frequency 100 \(\upmu\)Hz, (**d**) signal frequency 100 mHz, input (blue), and output (red) voltage. Figure 15: Two-port CPEs implemented by general bilinear filtering sections with controlled sources: voltage-controlled voltage-source E and current-controlled current-source F.  
either problems connected with CPE synthesis or the mentioned piece of software, please do not hesitate to contact the author (preferably via email). A free copy of this Matlab software can be delivered. The software works well with Matlab version 2013 and higher, together with a monitor/display resolution of \(1920\times 1080\) and higher. Some authors run stochastic optimization methods to reach the best numerical values of the components of passive or active CPEs. For example, a genetic algorithm was used in [69]. However, in this case, an optimal design is given by the exact formulas and the application of additional mathematical tools, especially techniques dedicated to global optimization, is questionable. On the other hand, a slight adjustment of the first and last zero-pole location can create phase overshoot and improve or enlarge the frequency band for the application of CPEs. Section 4 suggests a new two-port circuit element that can perform arbitrary phase shifting, i.e., a non-asymptotical phase change that can connect any two points on a phase frequency response. This can be understood as generalization of asymptotical phase shifting. For a phase change from DC to infinity that begins with a value other than \(90\cdot n^{\circ}\) (\(n\) being integer number), only an FO integrator or FO differentiator should be placed before an FO phase shifter [49; 50] (if the phase starts in the first of fourth segment of the polar plane of a complex transfer function) together with an inverting amplifier (to begin the phase shifting process in the second or third segment of the polar plane of a complex transfer function). Remember that CPEs suggested in Section 3 of this paper are designed with pre-defined values of impedance norm \(\xi\). This factor can be adjusted (change of the pseudo-capacitance), and this process is equivalent to a shift in module frequency response vertically upward (\(\xi>1\)) or downward (\(\xi<1\)) while the phase frequency response remains unchanged. To make renormalization, each capacitor needs to be divided by \(\xi\) while each resistor is multiplied by the same amount \(\xi\).