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In our simulations, the array response matrix ${\widehat P}(\omega)$[see definition in (Pt)] in the frequency domain is symmetric but not Hermitian. In general (as for array elements with nonisotropic radiation patterns), it is neither Hermitian nor symmetric, but with slight modifications our methods apply to this case as well. The eigenvectors of $\widehat{P}(\omega)\widehat{P}^H(\omega)$ having unit norm are denoted by $\widehat{\bf U}_r(\omega)$, for $r = 1,\ldots, N$.The eigenvalues of $\widehat{P}(\omega)\widehat{P}^H(\omega)$ are $\sigma_r^2(\omega)$, with $\sigma_r(\omega)$ being the singular values of ${\widehat P}(\omega)$.The significant singular vectors $\widehat{\bf U}_r(\omega)$ [i.e., those in the range of ${\widehat P}(\omega)$] have singular values $\sigma_r(\omega) \gt 0$ for $1 \le r \le M$, where M is either the number of targets, or the size of the array (N) -- whichever is smaller. We assume that the number of targets is smaller than the array size N, so that M is in fact the number of distinguishable targets; this assumption is required by the imaging methods we employ (such as MUSIC) as will become clear while presenting the method.

The notation used here is the same as in Borcea et al. (2002). We denote by $\widehat{{\bf g}}_0({\bf y},\omega)$ the deterministic source vector observed at the array for a source located at ${\bf y}^s$. Then, $\widehat{{\bf g}}_0({\bf y},\omega)$ is given by  
\widehat{{\bf g}}_0({\bf y}^s,\omega) = \left( \begin{array}...
\widehat{G}_0({\bf y}^s,{\bf x}_N,\omega)\end{array} \right),\end{displaymath} (2)
where $\widehat{G}_0({\bf y}^s,{\bf x}_j,\omega)$ is the deterministic two-point Green's function, and ${\bf x}_j$ is the location of the j-th array element.

We also define the projection ${\cal P}_N\widehat{{\bf g}}_0({\bf y},\omega)$ of $\widehat{{\bf g}}_0({\bf y}^s,\omega)$ onto the null-space of $\widehat{P}\widehat{P}^H(\omega)$ by

{\cal P}_N \widehat{\bf g}_0({\bf y}^s,\om...
 ...f y}^s,\omega) 
\widehat{{\bf U}}_r(\omega),\end{array}\end{displaymath} (3)
for each frequency in the support of the probing pulse $\widehat{f}(\omega)$.

The method we describe here is a time domain variant of MUSIC (Schmidt, 1979; 1986; Cheney, 2001; Devaney, 2002) which we label DOA, because it gives very stable estimates of the direction of arrival. Frequency domain MUSIC takes a replica (or trial) vector, which is the impulse response or Green's function for a point source at some point in the space, and dots this vector into an observed singular vector at the array. With appropriate normalization, this dot product acts like a direction cosine of the angle between the replica vector and the data vector. If the sum of the squares of these direction cosines is very close to unity, then it is correct to presume that the source point of that replica vector is in fact a target location since it lies wholely in the range of the array response matrix. Crudely speaking, imaging is accomplished by plotting $1/[1-\cos^2(\cdot)]$, which will have a strong peak when the replica source point is close to the target location.

We form the sum  
{\cal G}^{(j)}({\bf y}^s) = {\displaystyle \sum_{p=1}^N}
 ... F}^{(j)}_p\left({\bf y}^s,t_p({\bf y}^s)\right)
\right\vert^2,\end{displaymath} (4)
\hbox{\boldmath{${\cal F}$}}^{(j)}({\bf y... \right]
\widehat{{\bf U}}_r(\omega) d \omega,\end{array}\end{displaymath} (5)
and display the objective functional  
{\cal R}_{\mbox{DOA}}({\bf y}^s) = {\displaystyle \sum_{j=1}...
 ...s} }\, {\cal G}^{(j)}({\bf y}^s)}
{ {\cal G}^{(j)}({\bf y}^s)},\end{displaymath} (6)
for points ${\bf y}^s$ in the target domain.

The arrival time $t_p({\bf y}^s)$ is the deterministic travel time from the p-th transducer to the search point,  
t_p({\bf y}^s)=\frac{\mid {\bf x}_p -{\bf y}^s \mid}{c_0} .\end{displaymath} (7)

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Next: EXAMPLES AND RANGE ESTIMATION Up: Berryman et al.: Time-reversal Previous: IMAGING PROBLEM
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