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Recently I am stucking in frequency estimation of a solvable chaos waveform. Its local analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi),mT\leq t<(m+1)T $$ where $u_m \sim U(-1,1)$ , $s_m=sgn[u_m]$ and $s_m\in[+1,-1]$. $T=2\pi/\omega_0$ and $\beta=T \mathrm{ln}2$. And $m = 0,1,2,...$.

And its waveform is shown in following ($\omega_0 = 200\pi$) enter image description here

Its spectrum shows spikes at base frequency $f_0 =2\pi/\omega_0$ and its harmonics (here we set the base frequency as 100Hz). Besides, the spectrum has regularly repeated shape which implies some kind of determined solvable expression of Fourier Transform.

enter image description here

To begin with, I tried to calculate the FT directly, as $$ Z(\omega)=\int_{-\infty}^\infty z(t)e^{j\omega t}\mathrm{d}t\\ =\sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) \int_{mT}^{(m+1)T} e^{\beta(t-mT)}\cos(\omega_0 t+\varphi) e^{j\omega t}\mathrm{d}t\\ =\frac 1 2 \sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) \int_{mT}^{(m+1)T} e^{\beta(t-mT)} \{e^{j[(\omega_0+\omega) t+\varphi]}+e^{-j[(\omega_0-\omega) t+\varphi]}\}\mathrm{d}t $$

However, the integeration is limted in a window with width of $T$ and it is hard to calculate the FT from it.

Then I tried another way to prove the existence of spikes at base frequency and its harmonics. Here, I calcute the integration of $z(t)\sin(n\omega_0 t)$ as $$ a_0(n\omega_0)=\int_{-\infty}^\infty z(t)\sin (n\omega_0 t)\mathrm{d}t\\ =\sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) \int_{mT}^{(m+1)T} e^{\beta(t-mT)}\cos(\omega_0 t+\varphi) \sin (n\omega_0 t)\mathrm{d}t\\ =\sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) \int_{0}^{T} e^{\beta t}\cos(\omega_0 t+\varphi) \sin (n\omega_0 t)\mathrm{d}t\\ =\int_{0}^{T} e^{\beta t}\cos(\omega_0 t+\varphi) \sin (n\omega_0 t)\mathrm{d}t\sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) $$

where $n=1,2,3...$ . And $$ b_0(n\omega_0)=\int_{-\infty}^\infty z(t)\cos (n\omega_0 t)\mathrm{d}t\\ =\int_{0}^{T} e^{\beta t}\cos(\omega_0 t+\varphi) \cos (n\omega_0 t)\mathrm{d}t\sum_{m=-\infty} ^{\infty}s_m(u_m-s_m) $$

And the amplitude of spectrum at $n\omega_0$ is $$ c(n\omega_0)=\sqrt{a_0(n\omega_0)^2+b_0(n\omega_0)^2} $$

Still, it is difficult to prove the existence of delta funcions in the spectrum from the amplitude at base frequency.

Hence, how to prove the frequency properties shown in Figure 2 ? Is there any possiblity to calculate the FT of $z(t)$?

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  • $\begingroup$ Interesting waveform, 1) What is $t$ and what is $m$? 2) $u$ has the deltas you are looking for. Am I right to assume that $u$ is a sequence of random step functions placed randomly in the time domain? Or is it meant to be impulses? 3) Multiplication in the time domain becomes something else in the frequency domain (?)...but that is not exactly visible here... $\endgroup$ – A_A Oct 18 '18 at 9:00
  • $\begingroup$ 1) $t$ means time, and $m$ is the index of the $m$th sign $S_m$ and $u_m$. 2)$u_m$ is a random number uniformly occuring from -1 to 1. $S_m$ is the sign of $u_m$. And $u_m$ admits a kind of iteration function with $u_{m-1}$. 3) There exists some repeated similar sections in time domain, but I don't know whether it has any relations with the spikes in frequency domain. Thanks for your consideration! $\endgroup$ – Ang Zhou Oct 18 '18 at 9:13
  • $\begingroup$ Thank you for letting me know, I totally misunderstood $u$, do you have a reference for this equation? It looks like a driven damped oscillator that is driven by random impulses (?). What do you mean "...a kind of iteration function with $u_{m-1}$..."? Where does $u_{m-1}$ appear? $\endgroup$ – A_A Oct 18 '18 at 9:24
  • $\begingroup$ The original idea is to conduct frequency estimation of a solvable chaos system proposed in [A matched filter for chaos]( researchgate.net/publication/… ) . $z(t)$is an absolute-valued envelope of equation (15) in this article. And $z(t)=|u(t)|-1$. The iteration function is shown in equation (16) in that reference. $\endgroup$ – Ang Zhou Oct 18 '18 at 12:55
  • $\begingroup$ The signal that you have is one realization of a non-WSS random process and there is no guarantee that the Fourier transform of this particular realization exists in the conventional sense (finite-energy signals), or for that matter, in the generalized sense (power signals) where impulses are allowed in the spectrum. You could try the usual tricks to convert the process into a WSS process and then compute its PSD as the Fourier transform of its autocorrelation function if that would be satisfactory to you. $\endgroup$ – Dilip Sarwate Oct 19 '18 at 4:22

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