# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 4 Applications of CPE
### Implementation of Audio CPEs in Bass-Band Corrector  
A proposed bass-band corrector can be considered as a half-circuit known as [PERSON]'s corrector, a well-known building block that can be found in most discrete realizations of audio amplifiers.  
Figure 11: Circuit realization of Case III: phase shifter (**left schematic**), Orcad Pspice time domain circuit simulation and processed harmonic signals with amplitude 1V in plane projection (**right plot**).  
Figure 12: Polar plots associated with phase shifters: Case I (**left**), Case II (**middle**), and Type III (**right**). Lower frequencies (red), active phase shift (blue), and higher frequencies (orange).  
Figure 10: Circuit realization of Case III: phase shifter, arbitrary non-asymptotical phase shifter over entire acoustic range (**left graph**) and magnified area of interest (**right plot**).  
Its fundamental function is to independently strengthen or suppress the module frequency response in low bands (bass correction) and high bands (treble correction). The standard topology of Baxandal's circuit utilizes a single active element--a voltage feedback operational amplifier. Using single CPEs designed in Section 3, the bass-corrector can be separated (see Figure 13). The module frequency response is affected by the actual position of the potentiometer slider; let us mark its location as \(k\). For the left corner, where \(k=0\), and right corner, where \(k=1\), the voltage transfer functions are  
\[\begin{array}{l}K_{left}(s)=-\frac{1}{R_{1}}\frac{s^{\alpha}Y_{0}R_{2}R_{ \varphi}+R_{2}+R_{\varphi}}{s^{\alpha}Y_{0}R_{1}R_{\varphi}+1}\sim-\frac{R_{1} }{R_{2}}\frac{s^{\alpha}+1/(Y_{0}R_{2})}{s^{\alpha}},\\ K_{right}(s)=-R_{2}\frac{s^{\alpha}Y_{0}R_{1}+1}{s^{\alpha}Y_{0}R_{1}R_{ \varphi}+R_{1}+R_{\varphi}}\sim-\frac{R_{1}}{R_{2}}\frac{s^{\alpha}}{s^{ \alpha}+1/(Y_{0}R_{1})},\end{array} \tag{21}\]  
where \(Y_{0}\) is the pseudo-capacitance of a chosen CPE, \(\alpha\) is the mathematical order of this CPE, and the value of potentiometer (\(R_{y}\) in the case of \(K_{left}\) and \(R_{x}\) for \(K_{right}\)) is supposed to be much higher than the impedance of the CPE in the audio band and \(R_{1,2}\). To achieve the same roll-off frequency for bass strengthening and suppression, the condition \(R_{1}=R_{2}\) needs to be satified. For an ideal operational amplifier and general position of the potentiometer's slider \(k\in(0,1)\), the resulting voltage transfer function can be expressed as  
\[K_{general}(s)=-\frac{s^{\alpha}Y_{0}R_{2}R_{pot}+R_{2}+R_{pot}(1-k)}{s^{ \alpha}Y_{0}R_{1}R_{pot}+R_{1}+R_{pot}k}=-K_{0}\frac{s^{\alpha}+\omega_{Z}}{s ^{\alpha}+\omega_{p}}, \tag{22}\]  
where \(R_{pot}\) represents the value of the potentiometer. Obviously, the slider position in the centre of trace \(k=0.5\) means that no correction is applied.  
An Orcad Pspice circuit simulation (AC sweep analysis) of a standard and CPE-based bass-corrector (see Figure 13) is provided in Figure 14. For \(\alpha=1\), the numerical values of circuit components are \(R_{1}\) = \(R_{2}\) = 1 k\(\Omega\) and \(C\) = 100 nF, leading to significant frequency \(f_{0}\) = 1.6 kHz. For \(\alpha=2/3\), the choice of resistances \(R_{1}\) = \(R_{2}\) = 100\(\Omega\) and CPEs from Section 3.13 lead to \(f_{0}\) = 60 Hz. For order \(\alpha=1/2\), the choice of resistances \(R_{1}\) = \(R_{2}\) = 1 k\(\Omega\) and CPEs taken from Section 3.10 lead to \(f_{0}\) = 1.9 Hz. Finally, for \(\alpha=1/3\), the choice of resistances \(R_{1}\) = \(R_{2}\) = 22 k\(\Omega\) and CPEs from Section 3.7 lead to \(f_{0}\) = 0.07 Hz. CPEs allow for smooth continuous corrections over the entire audio frequency band. Of course, the order of CPEs is a part of this \"acoustic effect\".  
Figure 13: Circuit realization of the bass-corrector with CPEs (**left schematic**), decomposition of CPEs and potentiometer introduced to clarify the design formulae (**right picture**).