# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 3 Approximation of CPE in Frequency Domain and Numerical Results
### Numerical Analysis of Designed Audio and Wideband CPEs  
This subsection demonstrates the numerical verification of designed audio and wideband CPEs using Mathcad. Module and phase frequency responses (lower left plots), together with absolute error (lower right plots), are calculated in the frequency range starting with 10 Hz and ending with 100 kHz, which is suitable for audio CPEs. Within these pictures, the locations of zeroes and poles of the complex admittance function are also provided (upper plot). Individual results are provided in Figures 3 and 4.  
Provided visualizations clearly prove that individual calculated CPE approximations are in very good agreement with ideal CPE, i.e., promised accuracy were reached. Non-symmetrical phase ripples are caused by real-valued RC components inside analyzed CPE.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Rg/Cp** & **R1/C1** & **R2/C2** & **R3/C3** & **R4/C4** & **R5/C5** & **R6/C6** & **R7/C7** \\ \hline
**156 k\(\Omega\)** & **53 k\(\Omega\)** & **13.5 k\(\Omega\)** & **3.44 k\(\Omega\)** & **873 \(\Omega\)** & **222 \(\Omega\)** & **56 \(\Omega\)** & **18 \(\Omega\)** \\
150 k\(\Omega\)\(\angle\) & 1 M\(\Omega\)\(\parallel\) & 12 k\(\Omega\)\(\angle\) & 3.3 k\(\Omega\)\(\angle\) & 860 \(\Omega\)\(\angle\) & 680 \(\Omega\)\(\parallel\) & 56 \(\Omega\) & 18 \(\Omega\) \\
5.6 k\(\Omega\) & 56 k\(\Omega\) & 1.5 k\(\Omega\) & 180 \(\Omega\) & 12 \(\Omega\) & 330 \(\Omega\) & & \\ \hline
**458 nF** & **188 nF** & **161 nF** & **138 nF** & **119 nF** & **102 nF** & **88 nF** & **75 nF** \\
390 nF \(\parallel\) & 180 nF \(\parallel\) & 150 nF \(\parallel\) & 120 nF \(\parallel\) & 560 nF \(\angle\) & 100 nF \(\parallel\) & 330 nF \(\angle\) & 150 nF \(\angle\) \\
68 nF & 8.6 nF & 10 nF & 18 nF & 150 nF & 2.2 nF & 120 nF & 150 nF \\ \hline \hline \end{tabular}
\end{table}
Table 38: CPE with order \(\alpha\) = 9/10; passive ladder RC series-parallel topology of fractal capacitor.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Rs/Cs** & **R1/C1** & **R2/C2** & **R3/C3** & **R4/C4** & **R5/C5** & **R6/C6** & **R7/C7** \\ \hline
**1.3 \(\Omega\)** & **12 k\(\Omega\)** & **3.8 k\(\Omega\)** & **968 \(\Omega\)** & **246 \(\Omega\)** & **62 \(\Omega\)** & **16 \(\Omega\)** & **4 \(\Omega\)** \\
1.2 \(\Omega\) & 12 k\(\Omega\) & 12 k\(\Omega\)\(\parallel\) & 33 k\(\Omega\)\(\parallel\) & 330 \(\Omega\)\(\parallel\) & 220 \(\Omega\)\(\parallel\) & 15 \(\Omega\)\(\angle\) & 15 \(\Omega\)\(\parallel\) \\  & & 5.6 k\(\Omega\) & 1 k\(\Omega\) & 1 kFigure 3: Numerical verification of designed audio CPEs for orders considered in this paper: (**a**) \(\alpha=1/10\), (**b**) \(\alpha=1/9\), (**c**) \(\alpha=1/5\), (**d**) \(\alpha=2/9\), (**e**) \(\alpha=1/4\), (**f**) \(\alpha=3/10\), (**g**) \(\alpha=1/3\), (**h**) \(\alpha=2/5\), (**i**) \(\alpha=4/9\).  
Figure 4: Numerical validation of designed audio CPEs (continuation of previous figure): (a) \(\alpha\) = 1/2, (b) \(\alpha\) = 5/9, (c) \(\alpha\) = 3/5, (d) \(\alpha\) = 2/3, (e) \(\alpha\) = 7/10, (f) \(\alpha\) = 3/4, (g) \(\alpha\) = 7/9, (h) \(\alpha\) = 4/5, (i) \(\alpha\) = 8/9, (j) \(\alpha\) = 9/10.