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Generalized spectrum analyzer of the M-element random vector

Estimation of the energy spectrums of reflections in pulse Doppler weather radars. Part 1. Modifications of the spectral estimation algorithms

David I. Lekhovytskiy, Dmytro V. Atamanskiy, Dmytro S. Rachkov, Andrii V. Semeniaka


This is the first paper in a series of papers devoted to the peculiarities of estimation of the continuous energy spectra of random processes of different nature, which are defined by their samples at discrete moments of time. In the paper we consider two kinds of the generalized spectrum analyzers (GSA), whose structure fits the majority of classical (nonparametric) and modern noneigenstructured spectral estimation (SE) methods. It has been demonstrated that a number of known superresolution SE methods may be considered as particular cases of parametric GSA based on whitening or inversing filters of the input process. We focus on the autoregressive models of analyzed processes with continuous energy spectrums, for which the whitening or inversing filters are the transversal filters of various structures with proper parameters. The utilized interpretation allows one to modify the well-known superresolution SE methods for the problem of continuous spectrums reconstruction and, what is more important, to establish their new varieties with practically useful properties, that are going to be explored in the following two papers.

In the article we used the following acronyms, abbreviations and mathematical notations:

  • MO — meteorological objects;
  • DWR — Doppler weather radar;
  • SE — spectral estimation;
  • GSA — generalized spectrum analyzer;
  • CM — correlation matrix;
  • GF — generating filter;
  • AR-p process — autoregressive process of the pth order;
  • AR MO — reflections from MO, that are approximated by the AR process;
  • LF — linear filter;
  • CC — correlation coefficient;
  • PR — pulse response of the linear filter;
  • HPD — Hermitian positive definite matrix;
  • LP, ME, MV, TN, BL, MCA — linear prediction method, Berg method of maximal entropy, Capon method of minimal variance, method of thermal noise, Borgiotti-Lagunas method, modified Capon algorithm [12, 13, 24], respectively;
  • ALF — adaptive lattice filter;
  • PFR — power frequency response, which is equal to the squared magnitude of frequency response (FR) of the linear filter;
  • FIC — the first integral criterion of resolution–reconstruction [30];
  • SIC — the second integral criterion of resolution–reconstruction [51];
  • x(f) — M-dimensional vector of equally spaced time samples of complex harmonic with normalized Doppler frequency  or the scanning (searching) vector;
  • SF — spectral function;
  • MPR — matrix pulse response;
  • ELF — elementary lattice filter;
  • RC — resolution capability;
  • (T), ( ~ ), (*),() — the signs of transposition, complex conjugation, Hermitian conjugation and statistical averaging, respectively;
  • el(L)*  — lth row of the identity LxL matrix IL;
  • JMMxM  permutation matrix with units on the secondary diagonal;
  • p* and pN-dimensional row with elements pi and column with elements pi~;
  • Ф)  — estimation of  matrix Ф;
  • (A–1)T, (A–1)~, (A–1)* — the notations of transposed, complex conjugate and Hermitian conjugate inverse matrices.


continuous energy spectrum of random processes; spectral function; spectral estimation methods; autoregressive model; meteorological object; Doppler weather radar; whitening filter

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