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\title{Antis}
\author{Fassio Blatter}
\maketitle

\chapter{Introduction}
\section{Specific}
Notation: Below I refer to representation as training data for machine-learning, not the representations from ~\cite[Spline\_Axioms.tex]{Axioms}.\\\\
Sound source separation is a $preliminary$ for $rare sound detection$. It can be argued, that rare sound detection can be performed without separation. but in most cases it also depends on it.\\\\
I have made a few Mathematica scripts (tbd.) based on the theory that the full $time domain$, or if assumptions are made on the waveform, even the $sequence$ of events can be $discarded$ in the machine-learning representations for accoustic scene classification.\\
For short frames or time bins, a $statistical representation$ should be enough to correctly identify the scene.\\
I have also adjusted the representations, so that $energy$ scalograms are applied $pulse$-length or $waveform$-length wise. This can enhance the representation for machine learning.\\
Furthermore I have used derivatives of the time signal and applied similar signal processing methods to them. For visualisation I have added them to the original scalogram. The use of $derivatives$ is very much $distorted$ if the sound is not separated.
\section{Context}
Geometric overlay comparison as in Stopeight Comparator works for unseparated signals, but it is only of limited use, because it is inaccurate. The goal of separation is to clean up the signal, so that impulses or notes as they may be referred to in a musical context can be found. It is important to note that the individual pulses have a causal relationship that connects them. Most oscillations are subject to amplitude and/or frequency-modulation, but they are usually repetitive, so they can be removed by statistical means. Truly random/stochastic signals on the other hand can not be filtered.\\\\
In signal analysis, there is the notion of source separation vs. spectral separation of pulses. The method of testing a signal for spectral separation referred to in this text is meant to work $blind$ in a scenario where no prior knowledge of the likelyhood of occurence of waveforms is made.

\subsection{Pulse}
A symmetric pulse has two half pulses with the same wavelength/amplitude correlation.

\subsection{Impulses}
Impulses are consecutive half-pulses of variable wavelengths which originate from the same source and are separated from other overlaping impulses.

\chapter{Source Separation}
Stopeight Comparator may provide a means of either finding exact frames (of length pulse or impulse?) in the original signal $s(t)$ (See Grapher), or in the signal as in parametric curve differences.\\\\
Also Vector Graph manipulations ~\cite{Analyzer} such as finding/counting intersections and setting oriented $ori$ section type occurence limitations (ex. Spiral $\Rightarrow$ scale) could yield interesting results.

\subsection{Signs}
The difference of the signs over an interval.
\begin{align}
dsi(a,b) = card(\mu_{+}\subseteq [a,b])-card(\mu_{-} \subseteq [a,b])\\
ori(a,b) = \frac{dsi(a,b)}{\lvert dsi(a,b) \rvert}
\end{align}
A sequence of sections between $[a,b]$ is alternating $\mu_{-}\cup \mu_{zero}\cup\mu_{+}=\mu$.
\begin{enumerate}
\item $dsi(a,b)= 0; \exists(\mu_{n}\subseteq \mu_{+}) \land \exists(\mu_{n+1}\subseteq \mu_{zero}) \land \exists(\mu_{n+2}\subseteq \mu_{-}) \Leftrightarrow alt(a,b)=n$
\item $dsi(a,b)= 0; \exists(\mu_{n}\subseteq \mu_{-}) \land \exists(\mu_{n+1}\subseteq \mu_{zero}) \land \exists(\mu_{n+2}\subseteq \mu_{+}) \Leftrightarrow alt(a,b)=-n$
\item $dsi(a,b)= 1; \exists(\mu_{n}\subseteq \mu_{+}) \land \exists(\mu_{n+1}\subseteq \mu_{zero}) \land \nexists(\mu_{n+2}\subseteq \mu_{-}) \Leftrightarrow alt(a,b)=\frac{1}{n}$
\item $dsi(a,b)= -1; \exists(\mu_{n}\subseteq \mu_{-}) \land \exists(\mu_{n+1}\subseteq \mu_{zero}) \land \nexists(\mu_{n+2}\subseteq \mu_{+}) \Leftrightarrow alt(a,b)=-\frac{1}{n}$
\item $dsi(a,b)> \pm 1\Leftrightarrow alt(a,b)=0$;
\end{enumerate}
When the difference of the signs $dsi(a,b)$ is larger than one, orientation $ori$ serves as a weak expression of the imbalance.

\chapter{Spectral Separation}
\section{Indicator}
Every zero crossing $t_{m}$ of the 2nd derivative
\begin{equation}
\{t_{m}\}_{m \in \mathbb{Z} , f''(t_{m})=0}
\end{equation}
Note: I have mentioned the derivatives of the time signal as well in the introduction, effectively giving different bounds f''' for the signal f' and the bounds of the bounds f'''' for f''.\\\\
For all the Intervals $t_{m}-t_{m-1}=T$ of a signal $f: \mathbb{R} \rightarrow \mathbb{R}$
\begin{equation}
\eta(T)=\{t \in \mathbb{R}; t_{m} - t_{m-1}=T \vert f''(t_{m})=0, f''(t_{m-1})=0, t \le t_{m}, t \ge t_{m-1} \}
\end{equation}
Note: $\eta$ is a Lebesgue.\\\\
For all the positive and negative function values $f(t)$ respectively
\begin{align}
\eta_{+}(T)=\{{\eta \vert f(t) \ge 0}\}\\
\eta_{-}(T)=\{{\eta \vert f(t) \le 0}\}
\end{align}
In Parseval's theorem
\begin{equation}
\underbrace{\int\limits_{-\infty}^{\infty} \vert f(t) \vert ^2 \mathrm{d} t}_{\text{Signal Energy}} = 1/2\pi * \int\limits_{-\infty}^{\infty} \vert \hat{f}(1/T) \vert ^2  \mathrm{d} T
\end{equation}
The signal energy can be split into positive and negative parts
\begin{align}
\sqrt{\underbrace{\int \int f(t)*\mathrm{e}^{\pi/2} \mathrm{d} t \mathrm{d} \eta_{+}}_{z_{1}}*\underbrace{\int \int f(t) *\mathrm{e}^{-\pi/2}\mathrm{d} t \mathrm{d} \eta_{-} }_{z_{2}} } ^2 = - \int\limits_{-\infty}^{\infty} \vert f(t) \vert ^2 \mathrm{d} t 
\end{align}
Note: Because the complex angle is the same within each integral, the complex radius is preserved. The Euclidean norm $(\mathbb{C},\norm{\cdot}_2)$ equals the taxicab norm $(\mathbb{C},\norm{\cdot}_1)$ in this special case.\\\\
Because periods $T$ in a signal are not continuous, Dirac notation has to be used
\begin{equation}
\int\limits_{-\infty}^{\infty} (\underbrace{\int\limits_{-\infty}^{\infty} \delta (U - T) \mathrm{d} U}_{1} * \sqrt{ \underbrace{z_{1}*z_{2}}_{\text{$\eta$ depends on T}} }^2 )\mathrm{d} T = -1/2\pi * \int\limits_{-\infty}^{\infty} \vert\hat{f}(1/T)\vert ^2 \mathrm{d} T
\end{equation}
Note: Major $U$ is a unique period, not a point in time. $U$ is not continuous, $\delta$ is the discrete Dirac Delta.\\\\
The convolution of the signal with a periodicity T is equal the distributed energy of two semi-pulse functions of the same periodicity. Thus giving
\begin{align*}
\int\limits_{-\infty}^{\infty}(\int\limits_{-\infty}^{\infty} \delta (U - T) \mathrm{d} U * \int \int f(t) \mathrm{d} t \mathrm{d} \eta_{+}*\int \int f(t) \mathrm{d} t \mathrm{d} \eta_{-} ) \mathrm{d} T \\= 1/2\pi * \int\limits_{-\infty}^{\infty} \vert \int\limits_{-\infty}^{\infty}  f(t) * (cos(-t/T)+\mathrm{i} sin(-t/T)) \mathrm{d} t \vert^2 \mathrm{d} T
\end{align*}
This equation holds for every signal with frequency components separated in the time domain and symmetric in $mod(n/2)=0$ equal length parts. If this is not the case, we have either:\\
1. Longer wavelength pulses overlapping the frequency component, which results in a amplitude modulation of the combined signal.\\
2. Shorter wavelength pulses intersecting the segments of the frequency component, which results in frequency modulation of the combined signal.\\
3. Non-symmetric waveforms and/or different length half-pulses, which can not be removed with this procedure. See Source Separation for further discussion of such signals.\\\\
(A longer wavelength, undetected harmonic can be excluded in this scenario since source separated harmonics are symmetric pulses of integer multiples of the wavelength and we imply that the shortest $symmetric$ pulse is found beforehand in the removal procedure below, i.e. the higher harmonics are $not symmetric$ because of criteria 1)\\\\
Conclusion: This method provides an indication for the spectral separation of the signal. For blind removal of individual pulses, the $waveform$ of the pulse needs to be known. Longer wavelength pulses have the highest likelyhood to contain pulses which violate the above criteria, so their $amplitude$ can not be determined. We have to start removal with the shortest isolated pulses which can be found in the sample.\\\\
The main drawback of Fourier Transformation is that frames (integration bounds) and frequency window (measure) have to be adjusted. There are methods for finding bases in order to remove functions from the combined signal. Fourier Transformation does fulfill the task of spectral separation, but arbitrary precision in the time and frequency domain can not be found.

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