My Math Forum Audio Analysis Divisor Function
 User Name Remember Me? Password

 Number Theory Number Theory Math Forum

 October 9th, 2019, 10:44 AM #1 Member   Joined: Dec 2014 From: Netherlands Posts: 34 Thanks: 5 Math Focus: hobby Audio Analysis Divisor Function Hello, The divisor function can be written as a summation of repeating pulses with a frequency. It can be represented with the functions below: $$1) \space \sigma_{0}(x)=\sum_{\mathbb{X}=2}^{\infty} 2^{(-N)} \sum_{k=0}^{N} \binom{N}{k} e^{-i\left( \frac{\pi}{\mathbb{X}}kx \right)}$$ $$2) \space \Re(\sigma_{0}(x))=\sum_{\mathbb{X}=2}^{\infty} \cos^{N} \left( \frac{\pi}{\mathbb{X}}x \right) \cos \left( \frac{N\pi}{\mathbb{X}}x \right)$$ $$3) \space \Im(\sigma_{0}(x))=-i \sum_{\mathbb{X}=2}^{\infty}\cos^{N} \left( \frac{\pi}{\mathbb{X}}x \right) \sin \left( \frac{N\pi}{\mathbb{X}}x \right)$$ $N (\mathbb{X})$ is chosen in a way that all pulses have the same pulse width. Note that $N$ should be an even integer. The solution has been calculated up until x=1000 and the corresponding audio signal has been created (more info in description youtube): https://www.youtube.com/watch?v=5oWcu_Qjdn0 I like to study more on the subject. Questions:How to analyse the frequency domain? Can the functions be transformed in frequency domain? Is function $1)$ already a sort of frequency domain of $2)$ and $3)$? Hoped for some input. Hopefully I don't get banned for another question. Last attempt to get some input. Best regards, Vincent (note the described method are excluding 1 as an divisor, solution should actually be +1) Thanks from idontknow
 October 10th, 2019, 04:00 AM #2 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 I dont see it reasonable te get banned for posting advanced math. Thanks from OOOVincentOOO
 October 13th, 2019, 04:01 AM #3 Member   Joined: Dec 2014 From: Netherlands Posts: 34 Thanks: 5 Math Focus: hobby Fourier Transform Wave Divisor Function Fourier Transform Wave Divisor Function. The wave divisor function consists of a pulse outline modulated with a high frequency component. The real solution of the wave divisor function is: $$\large \Re(\sigma_{0})=\sum_{\mathbb{X}=2}^{\infty}\cos^{ N} \left( \frac{\pi}{\mathbb{X}}x \right) \cos \left( \frac{N\pi}{\mathbb{X}}x \right)$$ N is determined by the pulse width of $cos^{N}$ and calculated with ($L$ pulseheight at position $\Delta x$). N should be an positive even integer to obtain positive pulses only: $$\large N(\mathbb{X}) \approx \lim_{\mathbb{X} \rightarrow \infty} \frac{\log(L)}{\log \left( \cos \left( \frac {\pi}{\mathbb{X} } \Delta x \right) \right)} = - \frac{2 \mathbb{X}^2 \log(L)}{\pi^2 \Delta x^2}$$ The first term $cos^N$ can also be simplified, this is the pulse outline. The pulse outline forms a bell shaped distribution arround the origin for $\mathbb{X} \rightarrow \infty$: $$\large O(x)=\lim_{\mathbb{X} \rightarrow \infty}\cos^{N} \left( \frac{\pi}{\mathbb{X}}x \right)= e^{a x^{2}}$$ $$\large a=\frac{\log(L) \space}{\Delta x^{2}}=constant$$ The high frequency component $HF(\mathbb{X})$ scales linear with $\mathbb{X}$ (see link for more information) for: $\mathbb{X} \rightarrow \infty$. $$\large HF(\mathbb{X})= \cos \left( \frac{N\pi}{\mathbb{X}} x \right) \approx \cos (b x)$$ $$\large b(\mathbb{X}) = \frac{N}{\mathbb{X}}\pi \approx \alpha \mathbb{X} = constant \cdot \mathbb{X}$$ So for $\mathbb{X} \rightarrow \infty$ the wave divisor function becomes: $$\large \Re(\sigma_{0})\rightarrow \sum_{\mathbb{X}=2}^{\infty}e^{a x^{2}} \cos (b x)$$ The wave divisor at infinity can be Fourier transformed in the frequency domain. The following Fourier transform definitation was used: $$\large \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x) \space e^{-2 \pi ix \xi} \space dx$$ With help of Wolfram Alpha the Fourier transform is determined (see link below). The frequency spectra of an individual divisor wave will consist of a bell shape mirrored in the y-axis. $$\large \hat{\sigma}_{0}(\xi)= \frac{\sqrt{\pi}}{2 \sqrt{-a}} \left( e^{(b-2 \pi \xi)^{2} /4a} + e^{(b+2 \pi \xi)^{2} /4a} \right)$$ Every number will have at least one divisor wave. Because of the linearity properties of the Fourier transform we can sum the spectra to obtain the complete spectra of a number. The simulation below shows the time domain wave and the frequency spectra. Also the wave has been transposed to an audible signal. https://mybinder.org/v2/gh/oooVincen...%20Audio.ipynb I think I have answered my own question from the first post. My assumption in original post is false: trigonometric and n choose k notation are not each others Fourier complement. Best regards, Vince Last edited by OOOVincentOOO; October 13th, 2019 at 04:05 AM.
 October 14th, 2019, 10:11 AM #4 Member   Joined: Dec 2014 From: Netherlands Posts: 34 Thanks: 5 Math Focus: hobby Hello, Someone notified me on a mistake. Mistake.jpg This would be an better notation: $$\large N(\mathbb{X}) = \frac{\log(L)}{\log \left( \cos \left( \frac {\pi}{\mathbb{X} } \Delta x \right) \right)} \approx - \frac{2 \mathbb{X}^2 \log(L)}{\pi^2 \Delta x^2} \space (\mathbb{X} \rightarrow \infty)$$ I am non math pro, I hope the mad(th) demons leave me alone. Question: Previous post, are the last three formulas of the Fourier transform correct? I only did validation in the Jupyter notebook. Thank you, Vince

 Tags analysis, audio, divisor, function, wave

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post OOOVincentOOO Number Theory 23 September 29th, 2019 10:56 PM Alexrodi Math 1 December 12th, 2016 12:13 AM peppeniello Number Theory 0 June 10th, 2016 12:50 PM MrBibbles Number Theory 0 July 29th, 2013 06:22 AM math4tots Math Books 5 April 14th, 2011 08:26 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top