# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 3 Approximation of CPE in Frequency Domain and Numerical Results
Audio CPE for Mathematical Order \(\alpha=\) 1/9, Prescribed Phase Shift \(\varphi=\pm 10^{\circ}\)  
This CPE uses start-up frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 4\cdot 10^{4}\cdot 10^{-6})=4\) Hz and uses an impedance constant of \(10^{3}\). Numerical values can be found in Tables 4 and 5, and pseudo-capacitance is 1.85 \(\mu\)F/s\({}^{8/9}\) (type I) and 2.886 \(\mu\)F/s\({}^{8/9}\) (type II). Audio CPE for Mathematical Order \(\alpha=\) 1/5, Prescribed Phase Shift \(\varphi=\pm 18^{\circ}\)  
This CPE has start-up frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 3.4\cdot 10^{4}\cdot 10^{-6})=4.7\) Hz and uses an impedance constant of \(10^{4}\). Numerically, this CPE is defined by means of Tables 6 and 7, and pseudo-capacitance is about 119 \(\mu\)F/s\({}^{4/5}\) (type I) and 4.8 \(\mu\)F/s\({}^{4/5}\) (type II). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**R/Cp** & **R/C\({}_{1}\)** & **R/C\({}_{2}\)** & **R/C\({}_{3}\)** & **R/C\({}_{4}\)** & **R/C\({}_{5}\)** & **R/C\({}_{6}\)** & **R/C\({}_{7}\)** \\ \hline
**430 \(\Omega\)** & **2300 \(\Omega\)** & **1940 \(\Omega\)** & **1640 \(\Omega\)** & **1380 \(\Omega\)** & **1170 \(\Omega\)** & **987 \(\Omega\)** & **833 \(\Omega\)** \\
330 \(\Omega\)\(\angle\) & 2.2 k\(\Omega\)\(\angle\) & 1.8 k\(\Omega\)\(\angle\) & 1.5 k\(\Omega\)\(\angle\) & 1.2 k\(\Omega\)\(\angle\) & 1 k\(\Omega\)\(\angle\) & 1 k\(\Omega\) & 820 \(\Omega\) \\
100 \(\Omega\) & 100 \(\Omega\) & 150 \(\Omega\) & 150 \(\Omega\) & 180 \(\Omega\) & 180 \(\Omega\) & & \\ \hline
**1.8 nF** & **17.4**\(\mu\)F & **4.5**\(\mu\)F & **1.2**\(\mu\)F & **300 nF** & **77.6**\(\mu\)F** & **20 nF** & **5.2 nF** \\
1.8 nF & 15 \(\mu\)F \(\parallel\) & 3.9 \(\mu\)F \(\parallel\) & 1.2 \(\mu\)F & 150 nF \(\parallel\) & 56 nF \(\parallel\) & 10 nF \(\parallel\) & 3.3 nF \(\parallel\) \\  & 2.2 \(\mu\)F & 560 nF & & 150 nF & 22 nF & 10 nF & 1.8 nF \\ \hline \end{tabular}
\end{table} 
Table 4: CPE with math order \(\alpha=\) 1/9; fully passive RC series-parallel topology of fractal capacitor. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**R/Cp** & **R/C\({}_{1}\)** & **R/C\({}_{2}\)** & **R/C\({}_{3}\)** & **R/C\({}_{4}\)** & **R/C\({}_{5}\)** & **R/C\({}_{6}\)** & **R/C\({}_{7}\)** \\ \hline
**409 \(\Omega\)** & **2.5 k\(\Omega\)** & **2.1 k\(\Omega\)** & **1840 \(\Omega\)** &  
Audio CPE for Mathematical Order \(\alpha=\) 2/9, Prescribed Phase Shift \(\varphi=\pm 20^{\circ}\)  
This CPE has starting frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{4}\cdot 3\cdot 10^{-6})=5.3\) Hz and the impedance constant is \(10^{4}\). CPE is defined by values provided in Tables 8 and 9, and pseudo-capacitance is \(114.3\) uF/s\({}^{7/9}\) (type I) and \(14.6\) uF/s\({}^{7/9}\) (type II). Audio CPE for Mathematical Order \(\alpha=\) 1/4, Prescribed Phase Shift \(\varphi=\pm 22.5^{\circ}\)  
This approximation of CPE begins with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 3.4\cdot 10^{4}\cdot 10^{-6})=4.7\) Hz and uses an impedance constant of \(10^{4}\). Calculated numerical values are provided in Tables 10 and 11, and pseudo-capacitance is \(104.9\) uF/s\({}^{3/4}\) (type I) and \(4.78\) uF/s\({}^{3/4}\) (type II). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Rp/Cp** & **R1/C1** & **R2/C2** & **R3/C3** & **R  
Audio CPE for Mathematical Order \(\alpha=3/10\), Prescribed Phase Shift \(\varphi=\pm 27^{\circ}\)  
This CPE begins with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{4}\cdot 2\cdot 10^{-6})=8\) Hz and uses an impedance constant of \(10^{4}\). Values for construction of this CPE are given in Tables 12 and 13, and pseudo-capacitance is about 97.3 \(\mu\)F/s\({}^{7/10}\) (type I) and 12.1 \(\mu\)F/s\({}^{7/10}\) (type II). Audio CPE for Mathematical Order \(\alpha=1/3\), Prescribed Phase Shift \(\varphi=\pm 30^{\circ}\)  
This approximation of CPE has a start-up frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 3\cdot 10^{4}\cdot 10^{-6})=5.3\) Hz and utilizes an impedance constant of \(10^{4}\).