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Increasing the Angular Resolution and Range of Measuring Systems Using Ultra-Wideband Signals

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Abstract

The problem of obtaining three-dimensional radio images of objects with increased resolution based on the use of ultra-wide-band pulse signals and new methods of their digital processing is considered. The inverse problem of reconstructing the image of a signal source with a resolution exceeding the Rayleigh criterion has been solved numerically. Mathematically, the problem is reduced to solving the Fredholm integral equation of the first kind by numerical methods based on the representation of the solution in the form of decomposition into systems of orthogonal functions. The method of selecting the systems of functions used, which increases the stability of solutions, is substantiated. Variational problems of optimizing the shape and duration of ultra-wide-band pulses have been solved, ensuring the maximum possible signal-to-noise ratio during location studies of objects with fully or partially known signal reflection characteristics. The proposed procedures make it possible to increase the range of measuring systems, and also make it possible to increase the stability of solutions to inverse problems. It is shown that the use of the developed methods for achieving super-resolution to process ultra-wideband signals dramatically improves the quality of 3D images of objects in the radio range.

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Funding

The work was partially supported by the Russian Scientific Foundation, project no. 23-29-00448.

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Authors and Affiliations

  1. Russian Technological University (MIREA), Moscow, Russia

    B. A. Lagovsky

  2. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    E. Ya. Rubinovich

Authors
  1. B. A. Lagovsky
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  2. E. Ya. Rubinovich
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Corresponding authors

Correspondence to B. A. Lagovsky or E. Ya. Rubinovich.

Additional information

This paper was recommended for publication by A.A. Bobtsov, a member of the Editorial Board

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APPENDIX

APPENDIX

Proof of Theorem 1. The system of functions Gm(x) is generally non-orthogonal to LG. Let’s make a Gram matrix based on it, i.e., a matrix P of scalar products with elements Pmn:

$${{P}_{{mn}}} = ({{G}_{m}},{{G}_{n}}) = \int\limits_\Phi {{{G}_{m}}(\phi ){{G}_{n}}(\phi )d\phi .} $$
(A.1)

Since the matrix P is symmetric and positively defined, there is a transformation T that leads it to a diagonal form

$${\mathbf{\tilde {P}}} = {{{\mathbf{T}}}^{ \star }}{\mathbf{PT}}.$$
(A.2)

Using the found matrix T, we introduce a new system of functions \({{\tilde {G}}_{m}}(x)\) in the form (9). The resulting system turns out to be orthogonal on the segment LG, which is easily verified by directly calculating scalar products:

$$({{\tilde {G}}_{m}},{{\tilde {G}}_{n}}) = \sum\limits_{j,i = 1}^N {{{T}_{{jm}}}{{T}_{{in}}}\int\limits_\Phi {{{G}_{j}}(\phi ){{G}_{i}}(\phi )d\phi } } = \sum\limits_{j,i = 1}^N {{{T}_{{jm}}}{{T}_{{in}}}{{P}_{{ji}}}} = {{\tilde {P}}_{{mn}}},$$

where \({{\tilde {P}}_{{mn}}}\) are the elements of the diagonal matrix (A.2).

Now let’s find the system of functions \({{\tilde {g}}_{m}}\)(x), which generates the resulting orthogonal in the domain LG the system \({{\tilde {G}}_{m}}\)(x), i.e.

$${{\tilde {G}}_{m}} = {\mathbf{A}}{{\tilde {g}}_{m}}.$$
(A.3)

The required representation (9) follows

$${{\tilde {G}}_{m}} = \sum\limits_{j = 1}^N {{{T}_{{mj}}}{\mathbf{A}}{{g}_{j}}} = {\mathbf{A}}\left( {\sum\limits_{j = 1}^N {{{T}_{{mj}}}{{g}_{j}}} } \right).$$
(A.4)

Comparing (A.3) and (A.4), we obtain

$${{\tilde {g}}_{m}}(x) = \sum\limits_{j = 1}^N {{{T}_{{mj}}}{{g}_{j}}(x).} $$
(A.5)

The found system (A.5) turns out to be orthogonal on the segment Lg. Indeed, due to the orthogonality of the functions gm(x) and the orthogonality of the eigenvectors of the matrix P forming the matrix T, we have

$$({{\tilde {g}}_{m}}(x),{{\tilde {g}}_{n}}(x)) = \sum\limits_{j = 1}^N {{{T}_{{mj}}}{{T}_{{nj}}}({{g}_{j}},{{g}_{i}})} = \left\{ {\begin{array}{*{20}{l}} {0,}&{m \ne n} \\ {{{\lambda }_{m}},\;\;}&{m = n,} \end{array}} \right.\quad {{\lambda }_{m}} = \sum\limits_{j = 1}^N {T_{{mj}}^{2}.} $$

Note that the found system of orthogonal functions \({{\tilde {g}}_{m}}(x)\) is determined by the same linear transformation T as the system of functions \({{\tilde {G}}_{m}}(x)\).

As a result, based on a given system of N orthogonal functions gm(x) on the segment Lg, a new orthogonal system of functions on the same segment is constructed, generating an orthogonal system of functions \({{\tilde {g}}_{m}}(x)\) in the domain Lg. The theorem is proved.

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Lagovsky, B.A., Rubinovich, E.Y. Increasing the Angular Resolution and Range of Measuring Systems Using Ultra-Wideband Signals. Autom Remote Control 84, 1065–1078 (2023). https://doi.org/10.1134/S0005117923100089

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