# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 4 Applications of CPE
### Application of Audio CPEs in Non-Asymptotical Phase Shifters  
Acoustic effects are a class of signal processing tasks which operates with continuous waveforms and frequency bands, beginning with 20 Hz and ending with 20 kHz. There are three sources of audio signals: human performance, instrumentation music, and computer-aided sound synthesis. After generation, all sources subjected to further processing that can be marked as basic, such as amplification, attenuation, mixing, or equalization. Advanced audio operations, such as the surround effects and  
Figure 5: Experimental printed circuit board (PCB) of the fully passive ladder CPE with seven series-parallel RC sections and the so-called three-quarter capacitor (**upper photos**). Modular realization of active CPEs in the sense of Figure 1i, zeroes and poles tunable using variable resistors, and a fragment with four cells (**lower photo**). 3D illusion design, require either black-boxed digital signal processing or a sequence of two-ports with predefined phase frequency responses. Fully analog realization of the arbitrary phase shifters proposed in this example can be easily adopted by readers and re-used in their specific situations, i.e., for different shapes of rooms or the location of audio sources, speakers, or the listener. Basically, there are two types of integer-order analogue phase shifters. The first class perform asymptotical phase change, i.e., the phase approaches an initial and final value that is \(\pm 90^{\circ}\)\(k\), where \(k\) is the natural number, for DC and infinite frequency, respectively. Obviously, all types of frequency filters, including all-pass sections, are members of this extensive class of analog electronic systems. Standard filters can also be utilized for a non-asymptotical phase shifting. However, we cannot arbitrarily choose and connect two points on the phase frequency response by using a curve that is monotonically decreasing. On the other hand, the appropriate combination of integer-order and FO two-ports can treat all the problems mentioned. Missing operation, i.e., fully arbitrary phase shifting in a selected frequency range, becomes the topic of this first example. However, remember that phase shifting using an FO circuit causes the deformation of group delay and linear distortion. A common strategy for the utilization of CPEs in a continuous-time filtering application is to replace the standard linear capacitor with CPEs. However, such a substitution changes the definition of a significant frequency and, in specific circumstances, the type of filter itself. For example, a bad identification of the filter type can be found in [68], where an FO all-pass filter is incorrectly reported. For a phase shifting purpose, we can directly adopt the transfer function of the first-order low-pass filter generalized into an FO domain. For the given starting point \(\{f_{1},\varphi_{1}\}\) and final point \(\{f_{2},\varphi_{2}\}\), both located on a phase frequency response, the significant frequency \(\omega_{0}\) and total circuit order \(\alpha\) is the solution of the two nonlinear equations  
\[\varphi_{k}(\omega)=-atan\left[\frac{\left(2\cdot\pi\cdot f_{k}\right)^{\alpha }\cdot sin\left(\frac{\pi\alpha}{2}\right)}{\omega_{0}+\left(2\cdot\pi\cdot f _{k}\right)^{\alpha}\cdot cos\left(\frac{\pi\cdot\alpha}{2}\right)}\right], \tag{16}\]  
where \(k=1\), \(2\) is the index of a point on the phase frequency response. If a larger phase shift between the two selected frequency points is requested and final realization in the form of single two-port is required, than the transfer function of an integer-order all-pass filter can be generalized into an FO domain. In this case, simple replacement of the standard capacitor with CPEs does not produce a flat module frequency response, and can no longer be referred to as an all-pass filter.