\clearpage
%\section{Название}\label{s:1}
%Некоторая преамбула --- что увидит читатель из нижеследующего текста данного раздела ?

\section{Problem Statement\label{problem}}


Let the noise model be given in the discrete time by the equations
\begin{eqnarray}
x_{\mathrm{n}}(t_{i+1})&=&\Phi_{\mathrm{n}} x_{\mathrm{n}}(t_i)+ G_{\mathrm{n}} w_{\mathrm{n}}(t_i), \quad x_{\mathrm{n}}\in \Rset^n \label{x} \\
z_{\mathrm{n}}(t_i) &=& H_{\mathrm{n}} x_{\mathrm{n}}(t_i) + v_{\mathrm{n}}(t_i), \quad  z_{\mathrm{n}}\in \Rset^1  \label{y}
\end{eqnarray}
with $\{w_{\mathrm{n}}(\cdot)\}$, $\{v_{\mathrm{n}}(\cdot)\}$ being zero mean independent and identically distributed
sequences of covariance matrix $Q_{\mathrm{n}}\ge 0$ and variance $R_{\mathrm{n}}>0$ respectively. So, we assume that the acoustic noise $z_{\mathrm{n}}(t_i)$ is composed of two scalar components: $H_{\mathrm{n}}x_{\mathrm{n}}(t_i)$ is the low frequency component and $v_{\mathrm{n}}(t_i)$ is the high frequency one. 

To represent a quasi-periodic signal $z_{\mathrm{s}}(t_i)$ close  in sounding to speech or music tones, let us start from the Habibi's autoregressive model of 
images \cite{Habibi:1972} given on the finite or infinite plane rectangular grid. 
Using, for convenience, the unit step grid, we write the model in the form of equation
\begin{equation}\label{Habibi}
x_{k,l}=\rho x_{k,l-1} + rx_{k-1,l} - \rho r x_{k-1,l-1} + \beta \xi_{k,l}
\end{equation}
with $\{\xi_{k,l}\}$ being the field of standard  independent  Gaussian values and $k,l$ the two spatial coordinates of the image.
In (\ref{Habibi}), $\rho $ and $r$ are two parameters of the random field correlation function 
\begin{equation}\label{corHab}
V(k,l)\triangleq \E{x_{0,0}x_{k,l}}= \sigma_x^2 r^{|k|} \rho^{|l|}, \quad
\sigma_x^2=\frac{\beta^2}{(1-r^2)(1-\rho^2)}.
\end{equation}
Besides plane images, one can consider the other 2-dimensional images placed, for example, on the cylinder surface  where various grids can be defined.
Instead of standard  cylinder coordinates, we introduce 
a spiral-like grid set on a cylinder surface (Fig.~\ref{spiral}); using in this case the model (\ref{Habibi}), we have $k$ as a coil number and $l$ as a node number within a coil.
\begin{figure}[h]
\input{spiral.txt}
\caption{A spiral-like grid to represent a quasi-periodic signals close to music tones and vowel sounds.}\label{spiral}
\end{figure}

The model so constructed is essentially causal and can be equivalently
expressed as a random process $\{y_i\}$ being the evolvent along the spiral, that is
\begin{equation}\label{evolvent}
y_{i}=\rho y_{i-1} + ry_{i-T} - \rho r y_{i-T-1} + \beta \xi_{i}, \quad
i=kT+l,\quad \E{\xi_{i}}=0, \quad \E{\xi_{i}^2}=1
\end{equation}
where $\{\xi_{i}\}$ is a standard independent Gaussian sequence.
Going to the commonly accepted vector-matrix notations
\begin{equation}\label{vector-matrix}
\Phi_{\mathrm{s}} =
\left[\begin{array}{r@{\enspace}c@{\enspace}c@{\enspace}c@{\enspace}c@{\enspace}c}
0 & 1 & 0 & \cdots & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots  & \vdots  & \ddots & \vdots  & \vdots  \\
0 & 0 & 0 & \cdots & 0 & 1 \\
-r\rho & r & 0 & \cdots & 0 & \rho 
\end{array}
\right],\quad
%
x_{\mathrm{s}}(t_i)=\left[\begin{array}{c}
y_{i-T}\\
y_{i-T+1}  \\
\vdots  \\
y_{i-1} \\
y_i 
\end{array}
\right],\quad
%
G_{\mathrm{s}} =\left[\begin{array}{c}
0 \\
0 \\
\vdots  \\
0 \\
\beta 
\end{array}
\right],\quad
%
w_{\mathrm{s}}(t_i) =\xi_{i}
\end{equation}
yields
\begin{equation}\label{signal}
z_{\mathrm{s}}(t_i)=H_{\mathrm{s}}x_{\mathrm{s}}(t_i),\quad
H_{\mathrm{s}}= \left[\begin{array}{r@{\enspace}c@{\enspace}c@{\enspace}c@{\enspace}c@{\enspace}c}
0 & 0 & 0 & \cdots & 0 & 1
\end{array}
\right].
\end{equation}
This makes feasible a generalized notation
\begin{eqnarray}
x_{\mathrm{s}}(t_{i+1})&=&\Phi_{\mathrm{s}} x_{\mathrm{s}}(t_i)+ G_{\mathrm{s}} w_{\mathrm{s}}(t_i), \quad x_{\mathrm{s}}\in \Rset^m \label{xs} \\
z_{\mathrm{s}}(t_i) &=& H_{\mathrm{s}} x_{\mathrm{s}}(t_i), \quad  z_{\mathrm{s}}\in \Rset^1  \label{ys}
\end{eqnarray}
to be used together with (\ref{x}) and (\ref{y}) for stating the problem as follows (Fig.~\ref{speech}).

\begin{figure}[h]
\input{speech.txt}
\caption{Mathematical models of a signal and two-component noise.}\label{speech}
\medskip
\end{figure}

Given are the state equation
\begin{equation}\label{state}
\left[\begin{array}{c}
x_{\mathrm{s}}(t_{i+1})   \\
x_{\mathrm{n}}(t_{i+1}) 
\end{array}
\right]=
\left[\begin{array}{c@{\enspace}c}
\Phi_{\mathrm{s}}  &  0 \\
0  & \Phi_{\mathrm{n}}
\end{array}
\right]
\left[\begin{array}{c}
x_{\mathrm{s}}(t_{i})   \\
x_{\mathrm{n}}(t_{i}) 
\end{array}
\right]+
\left[\begin{array}{c@{\enspace}c}
G_{\mathrm{s}}  &  0 \\
0  & G_{\mathrm{n}}
\end{array}
\right]
\left[\begin{array}{c}
w_{\mathrm{s}}(t_{i})   \\
w_{\mathrm{n}}(t_{i}) 
\end{array}
\right]
\end{equation}
and the measurement equation, the latter holds for one of two hypotheses:
\begin{equation}\label{measurement}
\left\{
\begin{aligned}
\Hyp_{0}:\quad
z(t_i)&=\left[\begin{array}{c@{\enspace\enspace\,}c}
0  &  H_{\mathrm{n}} 
\end{array}
\right]
\left[\begin{array}{c}
x_{\mathrm{s}}(t_{i})   \\
x_{\mathrm{n}}(t_{i}) 
\end{array}
\right]+v(t_{i})
  \\
\Hyp_{1}:\quad
z(t_i)&=
\left[\begin{array}{c@{\enspace}c}
H_{\mathrm{s}}  &  H_{\mathrm{n}} 
\end{array}
\right]
\left[\begin{array}{c}
x_{\mathrm{s}}(t_{i})   \\
x_{\mathrm{n}}(t_{i}) 
\end{array}
\right]+v(t_{i})
\end{aligned}
\right.
\end{equation}
Hypotheses $\Hyp_{0}$ and $\Hyp_{1}$ are ``switching'' alternatively as
shown in Fig.~\ref{switches} at unknown time instants $t_k^{\mathrm{on}}$ and $t_k^{\mathrm{off}}$. 
It is necessary to make decisions concerning
the values of $t_k^{\mathrm{on}}$ and $t_k^{\mathrm{off}}$ for all possible $k$. 


\begin{figure}[h]
\input{switches.txt}
\caption{Switch points to be detected.}\label{switches}
\end{figure}








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