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This is the third paper in a series of papers dedicated to the peculiarities of estimation of the continuous energy spectrums of random processes of different nature, which are determined by their samples at discrete moments of time. In the article we justify the methodology and present the quantitative results of analytical and experimental investigation and comparison of statistical characteristics of classical and “parametric” methods of energy spectrums reconstruction for interperiod fluctuations of different nature reflections (including the ones from meteorological objects) in pulse radars. The methodology is followed by quantitative results which correspond to and obtained for a real-world “adaptive” case. Under the latter, a priori unknown echoes’ covariance matrix is replaced with different-kind estimates formed from finite-size training samples. Based on the results obtained, we substantiate the spectral estimation algorithms reasonable for utilization in different-purpose radars, in particular in pulse Doppler weather ones. Discussion of efficient ways for their practical implementation on a unified basis of adaptive lattice filters concludes the paper.

This paper is the first paper of the sequence devoted to modern methods of protection of coherent pulse radars against combined interferences (additive internal noise mixture masking the active jamming and clutter (passive jamming)). It compares the ultimate capabilities of the known and relatively new varieties of the interference protection (anti-jam and anti-clutter) systems under the hypothetical conditions of exact knowledge of statistical characteristics of signals and interferences. The ultimate capabilities of systems are understood in the sense that their efficiency is calculated for the hypothetical conditions of exact knowledge of statistical characteristics of input actions. The obtained estimates determine the upper bounds of efficiency in the real conditions of a priori uncertainty of parameters of signals and interferences. The losses of efficiency related to the transition to simplified systems of space-time signal processing (STSP) are also analyzed. The second paper deals with peculiarities (high-speed) of the considered anti-jam and anti-clutter systems in real conditions of parametric a priori uncertainty that is overcome by using different kinds of estimates of a priori unknown parameters of interferences. The third paper is devoted to the substantiation of general-purpose STSP system based on adaptive lattice filters.

Adaptive systems protecting pulse radars from non-stationary in time (range) clutter echoes are usually tuned using training vectors composed of complex amplitudes of input signals and comprising a finite-length “sliding window” of data. From any current range gate to a subsequent one, a training sample is partially updated (or modified) by means of excluding the “old” training vectors (correspond to the current range gate) and including the “new” ones (correspond to the next range gate). As a consequence, respective estimates of adaptive system parameters are corrected according to a modified sample correlation matrix (CM), which is typically a sum of an initial CM and a modifying matrix of rank K ≥ 1. In this case it is possible to avoid re-computing these parameters based on a new training sample of full size and, instead of this, we correct them in an “economical” way employing K-rank modification of a matrix inverse to the CM estimate.This paper is devoted to comparative analysis of various (K ≥ 1)-rank modification algorithms that correct the parameters of adaptive lattice filters (ALF). Main attention is paid to synthesis as well as theoretical and experimental study of algorithms of direct (K > 1)-rank modification of the ALF parameters. These algorithms attain the said objective omitting the K-fold application of known rank-one (K = 1) modification algorithms. We also synthesize a combined algorithm (CA) of (K ≥ 1)-rank modification of the ALF parameters that is more computationally simple and more numerically robust compared to known algorithms. The ALF employing the CA can serve as an effective tool for solving various tasks of space-time adaptive signal processing in pulse radars of different purpose.

This is the second paper in a series of papers dedicated to the peculiarities of estimation of the continuous energy spectra of random processes of different nature, which are determined by their samples at discrete moments of time. In this paper we analyze extreme performance of the reconstruction of continuous energy spectra, in particular, the ones of interperiod fluctuations of reflections from meteorological objects in pulse Doppler weather radars under the hypothetical conditions of a priori known covariance matrix of the analyzed processes. The reasons, which cause known disadvantages of classical (nonparametric) spectral estimation (SE) methods for energy spectrums shape reconstruction, are discussed. We have considered known and suggested criteria, using which the extreme performance of classical SE methods and parametric superresolution ones has been quantitatively compared. It has been demonstrated that the extreme performance of SE methods contains important but not comprehensive information. In order to choose a SE method appropriate for operation under real-world conditions, this information should be used together with information on a corresponding method’s adaptive performance under a priori unknown statistical characteristics of input effects.In the article we used the acronyms, abbreviations and mathematical notations, which were presented in the first paper (see link [1]).

The authors analyze the algorithms intended for correction of adaptive lattice filters (ALF) parameters under K-rank (K i 1) modification of estimate correlation matrix within a “sliding” over the time (range) data window. The drawbacks of methods that correct the ALF parameters based on K-fold utilization of known algorithms of rank-one (K = 1) modification are discussed. The combined algorithm (CA) of K-rank (K i 1) modification is synthesized. Under considered conditions, the only one-fold utilization of the CA solves the task of ALF parameters correction. The paper demonstrates, that proposed CA reduces the computational complexity and enhances the numerical stability of procedure of ALF parameters correction as compared with the competing methods based on algorithms of rank-one modification.

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;JM — MxM permutation matrix with units on the secondary diagonal;p* and p — N-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.

The paper explores adaptive performance of some quasi-optimal (QO) systems of inter-period (inter-pulse) processing (IPP) with tapered (windowed) accumulators of coherent batch for the moving target detection (MTD) in pulse radar systems. The main attention is paid to the schemes based on the whitening and quasi-whitening clutter rejection filters. The results for their comparative analysis with the most common DFT-based filter bank under both model and the real-world records of echo-signals are presented. The expediency of their application at the MTD systems with windowed accumulators is substantiated.

In this paper we analyze the shape reproduction errors of continuous spectrum of random Gaussian process (particularly, echo signals from meteorological formations in pulse Doppler weather radar) using various spectral estimation (SE) methods in “adaptive” situation when a priori unknown covariance matrix (CM) is replaced with its estimates of different kind. We also show the significant robustness difference of SE methods in such conditions and evaluate possibility of increasing their robustness applying better estimates and correcting the type of chosen spectral functions (SF)

In this article we investigate the procedures intended for meteorological objects (MO) mean radial velocity unambiguous estimation in pulse Doppler weather radars with constant and staggered pulse repetition times (PRT). It has been demonstrated that the fulfillment of the requirements to the unambiguous velocity measurement range by means of PRT staggering can be accompanied by the undesirable increase of velocity estimation errors. The causes, which yield this drawback in the known algorithms of velocity unambiguous measurement [1–4], have been determined. We demonstrate the possibility to decrease this drawback using modified estimation of the radial velocity. An influence of various factors (namely staggering principle; the width of the spectrum and the order of autoregressive process, which approximates weather echoes; training sample size; and the variant of final velocity estimation) on the accuracy of modified estimation of radial velocity has been investigated. Reasonable parameters of modified procedure and scheme for its practical implementation have been proven.

Under real-world operational conditions, the meteorological objects (MO) echoes received by weather radar could be distorted by ground objects returns or point aerial targets echoes which affects a quality of weather spectral moments estimation. This paper analyzes an impact of such a clutter on an accuracy of weather parameters measurement. We quantitatively evaluate how the spectral moment estimation errors depend on intensity of interfering reflections; distinctions between clutter and weather echoes in terms of radial velocity and width and shape of respective Doppler velocity spectra; and a size of training sample being used for MO parameters measurement. The practical recommendations on construction and parameters of a system intended for weather radar protection from different-nature interfering reflections are derived. The universal lattice filters including adaptive ones are demonstrated to be an effective structural and algorithmic basis for such system implementation.

In this paper, we develop the unified structure of a system for primary interperiod processing (IPP) of pulse Doppler weather radar signals. For this purpose, to solve IPP main tasks (namely to estimate the first three moments and a shape of energy spectrum of returns from meteorological objects, to decrease an influence of interfering reflections from ground objects and point aerial objects on estimation accuracy) it is proposed to use the algorithms, which are chosen based on results of theoretical and scaled-down investigation and contain both specific and common operations. The presence of the latter creates the prerequisites for unification of IPP system in whole. Universal lattice filters including adaptive ones are proposed to be used as a basis for such IPP system. The scheme of respective system of interperiod signal processing is introduced and described. Some results of its utilization in operational Doppler weather radar are shown.

We analyze the algorithms intended for correction of adaptive lattice filters (ALF) parameters under K ≥ 1 - rank modification of estimate correlation matrix within a “sliding” over the time (range) data window. The drawbacks of methods that correct the ALF parameters based on K - fold utilization of known algorithms of rank-one (K = 1) modification are discussed. The combined algorithm (CA) of K ≥ 1 - rank modification is synthesized. Under considered conditions, the only one-fold utilization of the CA solves the task of ALF parameters correction. We demonstrate, through both the theoretical analysis and results of scaled-down experiments, that proposed CA reduces the computational complexity and enhances the numerical stability of procedure of ALF parameters correction as compared with the competing methods based on algorithms of rank-one modification.

The maximum likelihood estimates of mean power of reflections from meteorological objects (MO) are derived for four different cases. The laws of distribution of well-known estimate of mean MO power are derived. Based on them, its statistical characteristics are determined. Their dependences on training sample size, intensity of reflections from MO, and type and level of their interperiod correlation are analyzed.

The paper considers known and new quasioptimal (QO) systems for interperiod processing (IPP) of batch coherent signals against the background of Gaussian autoregressive interfering reflections in pulse radar with batch-to-batch staggered or constant pulse repetition times. The main attention is paid to a robustness of IPP features under conditions when “windowing”, which improves frequency resolution of targets in pulse volume, is applied to impulse responses (IR) of coherent accumulators of pulses of useful signal batch. It is demonstrated that the classical IPP, being an optimal one, when IR is matched, and nonoptimal, when IR is “windowed”, is very sensitive to the windowing level. Under the latter conditions, the effectiveness of classical IPP could be significantly inferior to the effectiveness of simpler, but more robust QO IPP systems. We ascertain the physical mechanism of distinctions in robustness between different IPP systems. The results of quantitative evaluation and comparison of potential performance of various QO IPP systems with windowed IRs of useful signal accumulators are presented. The recommendations on a choice of QO IPP systems for application under real-world operational conditions of a priori uncertainty of input signals' statistical characteristics are proved.

The paper presents results of comparative study of different methods for estimation of continuous energy spectra of random returns in coherent pulse radars. We analyze these methods' “extreme” capabilities on reconstruction of continuous spectra under hypothetic conditions of known statistical characteristics of input signals. We investigate also spectral estimation methods' “real” capabilities under real-world operational conditions with parametric a priori uncertainty and small-size samples being used to estimate a priori unknown parameters of reflections. The causes of known drawbacks of classical (nonparametric) spectral estimation (SE) methods are discussed. Then we consider parametric SE methods, sources of their drawbacks and ways to overcome them.

For a number of known "superresolving" algorithms intended for direction-finding of point sources of noise radiation, we compare the requirements to the size of training sample in the estimates of input signals' covariance matrix. We discuss the reasons of the distinctions between the analyzed algorithms, and propose new kinds of methods with improved robustness to the finite sample size.

Common elements of different problems on space-time adaptive processing of Gaussian signal and a place of adaptive lattice filters (ALF) in their solution are discussed. Discussed are the ALF advantages in comparison with other analogous purpose filters; results of their modeling, scaled-down and scaled testing in adaptive systems of clutter protection and meteorological formation spectrum estimation in operating radars; peculiarities of practical realization on the basis of modern digital elemental base.

We define and compare statistical characteristics of estimate of mean power of signals reflected from meteorological formations (MF). Their dependences on training sample size, power of MF, type and level of interperiod correlation of reflections from MF are analyzed. Rational ways of practical implementation of MF mean power estimate are demonstrated.

We define and compare the statistical characteristics of the estimates of Doppler velocity spectrum (DVS) width of meteorological formations (MF) based on the various modifications of “pulse pair” method. For the conditions, when the form of unimodal DVS differs from the Gaussian one, the errors of proposed modification of “pulse pair” method are evaluated, and possible ways to reduce them are proposed. The recurrent organization of estimation procedure that minimizes the computational costs is proved.

We define and compare the statistical characteristics of different estimates of mean radial velocity of meteorological formations (MF) in pulsed Doppler weather radars (PDWR) with constant and staggered intervals of sounding. The modification of “pulse pair” method for PDWR with staggering of sounding intervals is proposed. This technique allows to expand appreciably the range of unambiguous velocity estimation. The reasonable staggering order and principles of sounding pulses arrangement on time axis are proved. The recurrent organization of proposed algorithm for MF's radial velocity estimation with minimized computational complexity is demonstrated.

The well-known and new estimates of the meteorological formations (MF) parameters for pulsed Doppler weather radars with arbitrary staggering of pulse repetition intervals are considered. Their statistical characteristics are determined and compared. The dependences of estimates accuracy on the training sample size, the power of MF, the type and level of interperiod correlation of signals reflected from MF are analyzed. There are proved the most effective on a number of criteria estimates of MF’s parameters and rational ways of their practical implementation.

The convergence rate of a number of adaptive processing algorithms for coherent signal extraction from Gaussian interference is compared. Our main attention is paid to a relatively new hybrid method of regularization of the maximum-likelihood estimate (MLE) of the covariance matrix (CM) of Gaussian vectors with discrete and continuous power spectra. Relatively high convergence rate and efficient lattice-filter-based implementation for the introduced technique are demonstrated.

This paper presents the analytical derivation of joint probability density functions (pdfs) of the maximum likelihood (ML) estimates of a real and complex persymmetric correlation matrices (PCM) of multivariate Gaussian processes. It is oriented at the modifications of the classical Wishart’s–Goodman’s pdfs adapted to the ML estimates of the data CMs in a wide class of signal processing (SP) problems in systems with centrally symmetric (CS) receive channels. The importance of the derived modified pdfs for such CS systems could be as great as that of the classical Wishart’s–Goodman’s pdfs for systems with arbitrary receive channels. Some properties of the new obtained joint pdfs are featured.