**Properties Frequency Spectrum (uncertainty principle).**
**Properties Frequency Spectrum (uncertainty principle).**
The following got me puzzled the last weeks:

When the pulsewidth in the time domain gets smaller the frequency spectrum of the divisors tend to be identified more clear. From Fourier transform properties one would expect the frequency bandwidth to become wider as the time domain pulse gets narrow the: Uncertainty principle.

The spectrum of the wave divisor function seems to behave opposite to the uncertainty principle. Below the z-score of the wave divisor spectra is calculated. The z-score describes the behavior and the uncertainty principle is remained.

Time domain $f(x)$:

$$\large \Re(\sigma_{0})\rightarrow \sum_{\mathbb{X}=2}^{\infty}e^{a x^{2}} \cos (b x) $$

The pulsewidth in the time domain is determined by: $L$ pulseheight at position $\Delta x$. In the equations described later we will vary the pulsewidth in the time domain. Onward we set $L=0.5$ as an constant and the time domain pulsewidth is varied by reducing $\Delta x \rightarrow 0$.

$$\large a=\frac{\log(L) \space}{\Delta x^{2}}=constant$$

$$\large b(\mathbb{X}) = \frac{N}{\mathbb{X}}\pi \approx - \frac{2 \space \log(L)}{\pi \space \Delta x^{2}} \mathbb{X} = constant \cdot \mathbb{X}$$

Frequency domain $\hat{f} (\xi)$:

$$\large \hat{\sigma}_{0}(\xi)= \frac{\sqrt{\pi}}{2 \sqrt{-a}} \left( e^{(b-2 \pi \xi)^{2} /4a} + e^{(b+2 \pi \xi)^{2} /4a} \right) $$

The frequency pulses can be seen as normal distributions. The standard deviation of a pulse in the frequency domain is proportional to:

$$\large Stdev(\hat{\sigma}_{0}(\xi)) \propto \sqrt{-a}$$

The minimal frequency distance between two neighbor pulses is:

$$\large \Delta \xi = b(\mathbb{X}+1)-b(\mathbb{X})=b(1)$$

The z-score between two neighbor frequency pulses then is:

$$\large Z \propto \frac{b(1)}{\sqrt{-a}} \propto \frac{1}{\Delta x}$$

When the time domain pulse gets narrow $\Delta x \rightarrow 0$ the $z-score$ in the frequency domain gets bigger. Thus the individual pulses in the frequency domain become better identified. One can say that the pulsewidth in frequency domain $\sqrt{-a}$ grows more slowly then the frequency difference between two neighbor divisors $b$.

More information and simulation:

**https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20Audio.ipynb**
New Question:

In the first post 3 equations where given for the wave divisor function (in time domain as to say).

$$1) \space \sigma_{0}(x)=\sum_{\mathbb{X}=2}^{\infty} 2^{(-N)} \sum_{k=0}^{N} \binom{N}{k} e^{-i\left( \frac{\pi}{\mathbb{X}}kx \right)} $$

$$2) \space \Re(\sigma_{0}(x))=\sum_{\mathbb{X}=2}^{\infty} \cos^{N} \left( \frac{\pi}{\mathbb{X}}x \right) \cos \left( \frac{N\pi}{\mathbb{X}}x \right) $$

$$3) \space \Im(\sigma_{0}(x))=-i \sum_{\mathbb{X}=2}^{\infty}\cos^{N} \left( \frac{\pi}{\mathbb{X}}x \right) \sin \left( \frac{N\pi}{\mathbb{X}}x \right) $$

After more thinking I still think :spin: $1)$ is a sort of frequency spectrum of $2)$ and $3)$ is this correct? Has someone seen these 3 equations in maybe another form or application?

Best regards,

Vince