# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 4 Applications of CPE
### Application of Audio CPEs in Non-Asymptotical Phase Shifters
The FO pole angular frequency, \(\omega_{0}\), and the mathematical order, \(\alpha\), is the solution to the following coupled equations  
\[\varphi_{k}(\omega)=atan\left[\frac{\left(2\cdot\pi\cdot f_{k}\right)^{\alpha }\cdot sin\left(\frac{\pi\cdot\alpha}{2}\right)}{\left(2\cdot\pi\cdot f_{k} \right)^{\alpha}\cdot cos\left(\frac{\pi\cdot\alpha}{2}\right)-\omega_{0}} \right]-atan\left[\frac{\left(2\cdot\pi\cdot f_{k}\right)^{\alpha}\cdot sin \left(\frac{\pi\cdot\alpha}{2}\right)}{\left(2\cdot\pi\cdot f_{k}\right)^{ \alpha}\cdot cos\left(\frac{\pi\cdot\alpha}{2}\right)\ +\ \omega_{0}}\right], \tag{17}\]  
where, again, \(k=1\), \(2\). Example I: Suppose that a bass-band phase shift transformation \(\{30\) Hz, \(-10^{\circ}\}\)\(\rightarrow\)(\(280\) Hz, \(-30^{\circ}\)) with an uncorrected group delay is requested. By solving System (16), we obtain the solution for significant frequency \(f_{0}=23\) Hz and order \(\alpha=2/3\). Ideal and real voltage transfer functions in the Laplace transform (to be implemented as electronic circuit) can be expressed as  
\[K_{ital}(s)=\frac{2\pi f_{0}}{s^{2/3}+2\pi f_{0}}\ K_{real}(s)=\frac{1/R}{Y(s)+ 1/R}, \tag{18}\]  
where \(Y(s)\) is the admittance function of CPE approximation (1), and the time constant of the phase shifter is formed by the pseudo-capacitance taken from Section 3.13 together with resistance \(R=70\)\(\Omega\). A graphical illustration of the obtained results (Orcad Pspice circuit simulation data imported to Mathcad, AC sweep analysis) is provided in Figure 6. Here, green and blue crosses represent the start and end points in the phase frequency response, respectively. One simple network realization with off-the-shelf active elements of this kind of phase shifter is demonstrated in Figure 7. Note that the major parasitic properties of AD844--namely input resistance of a low-impedance input inverting terminal, as well as output impedance of the current output C terminal--can be neglected, or, speaking more precisely, included into values of working resistances \(R\). The infinite input impedance makes this two-port a good candidate for a cascade synthesis and can cooperate with other voltage-mode filters. Example II: Assume a treble-band phase shift transformation represented by connection of two points {6 kHz, \(-55^{\circ}\)}\(\rightarrow\){12150 Hz, \(-80^{\circ}\)}. The solution of System (14) yields significant frequency \(f_{0}\) = 27 kHz and order \(\alpha\) = 1.151. Simple replacement of a capacitor with standard CPEs (implemented using passive RC approximation) cannot be done, and some sort of immittance multiplier should be employed. However, by solving Formula (17) with the given input parameters yields an FO pole frequency \(f_{0}\) = 49 Hz and order \(\alpha\) = 0.5. Thus, CPEs suitable for this application can be taken directly from Section 3.10. Corresponding simulation outputs are shown in Figure 8, while possible circuit realization is demonstrated in Figure 9, namely in the upper right schematic. The proposed two-port is derived from the common structure of operational-amplifier-based first-order voltage-mode all-pass filters. The ideal and real voltage transfer functions required by this application are  
\[K_{ideal}(s)=-\frac{s^{1/2}-2\pi f_{0}}{s^{1/2}+2\pi f_{0}}\quad K_{real}(s)=- \frac{Y(s)-1/R}{Y(s)+1/R}, \tag{19}\]  
where \(Y(s)\) is admittance function of CPE approximation (1). If pseudo-capacitance 85.25 \(\mu\)F/s\({}^{1/2}\) is adopted, numerical values of resistors should be chosen as \(R_{X}\) = 10 k\(\Omega\), \(R_{Y}\) = 36 \(\Omega\), and \(R_{Z}\) = 1 k\(\Omega\). Figure 6: Phase frequency responses and group delays of Example I: arbitrary non-asymptotical phase shifter over entire acoustic range (**left graph**) and magnified area as specified by task (**right plot**). Figure 7: Circuit realization of Case I phase shifter (**left picture**), Orcad Pspice time domain circuit simulation, and processed harmonic signals with amplitude 1V in plane projection (**right plot**).