Identification of the impulse response of the communication channel. Application of threshold technique to estimate the impulse response of a communication channel. How the Viterbi equalizer works

Blind signal processing is a relatively new digital signal processing (DSP) technology that has been developed over the past 10-15 years.

In general, the task of blind processing can be formulated as the digital processing of unknown signals that have passed through a linear channel with unknown characteristics against a background of additive noise.

Region of uncertainty Region of observation

X Vector channel GL U

Rice. 1. Blind problem.

“Blind problem” often arises when processing signals in radio engineering systems, including radar systems, radio navigation, radio astronomy, and digital television; in radio communication systems; in problems of digital speech and image processing.

Since SOS problems historically arose in various applications of digital signal and image processing, therefore, quite often the solution to these problems was based on taking into account the specifics of specific applications. With the accumulation of results in recent years, the prerequisites have been created for the construction of a systematic theory for solving the “blind problem”.

There are two main types of blind signal processing problems: blind channel identification (estimation of an unknown impulse response or transfer function), blind channel alignment (or correction) (direct estimation of an information signal). In both cases, only implementations of the observed signal are available for processing.

In the case of blind identification, the impulse response estimate can be further used to estimate the information sequence, i.e. is the first step in blind alignment or correction.

Blind processing tasks require a wide class of models to describe observed signals. In the most general case, a continuous model of a system is described by the following expression:

4-со у(0= |н(*,г)х(гУг + у(0, (1) о where: у(/) - observed vector signal with values ​​in St, Н(?,г) -тх p unknown matrix of impulse responses (IR) with elements hi j (r)); v(t)~ additive noise (vector random process with values ​​in St, usually with independent components); WITH".

Systems described by expression (1) are called systems with multiple input and multiple output (in English literature Multiple-Input Multiple-Output or MIMO).

In the particular case when H(/, r) = H(/-r) we have the case of a stationary system, and (1) has the form: oo y(0= jH(i-r)x(rWr + v(0. ( 2) oo

If the components of the matrix H(r) have the form |/r-yj(r)), we obtain a model used in blind source separation (BSS) problems:

У(0 = Н x(f)+ v(f), (3) where: Н - m x n unknown, complex (so-called “mixing”) matrix with elements (fyjj; x(r)~ unknown signals .

In the particular case when source signals are implementations of stationary random processes that are statistically independent of each other, we have a problem that in recent years is increasingly called independent component analysis (ICA).

In this case, the model used in independent component analysis is often presented as:

Y = H ■ x + v, (4) where: y and v are random vectors, x is a random vector with independent components, H is a deterministic unknown matrix.

The ANC problem is formulated as the problem of finding such a projection of a vector y onto a linear space of vectors x whose components are statistically independent. In this case, only a certain sample of the random vector y is available and the statistics of the noise vector v are known.

PCA is some development of the method of principal components, well known in statistics, where instead of the stronger property of statistical independence, the property of uncorrelatedness is used.

If in (2) u = 1 and m > 1, then the system model can be described by a simpler expression: oo y(i) = Jh(i - r)x(z)dz + v(f), (5)

00 where h(r) is the unknown impulse response of the t-dimensional channel; x(r) is an unknown complex information signal with values ​​in C.

Systems described by models of the form (5) are called single-input multiple-output systems (Single-Input Multiple-Output or SIMO).

If n = 1 and m = 1, then we have a model of a system with one input and output (Single-Input Single-Output or SISO): 00

We will further call the problems of blind channel identification based on models (5) and (6) the problems of stationary blind identification of a vector and scalar channel, respectively.

Blind identifiability of a system means the ability to reconstruct the impulse response of a system accurate to a complex multiplier only from output signals.

At first glance, such a task may seem incorrect, but this is not so if blind channel estimation is based on the use of the channel structure or known properties of its input. Naturally, such properties, in turn, depend on the characteristics of the specific application of blind identification methods.

In the practice of radio engineering information transmission systems designed for high-speed transmission through channels with various types of scattering, the IR radio channel, as a rule, is not known with sufficient accuracy to be able to synthesize optimal modulators and demodulators.

Moreover, in radio channels, IR is usually non-stationary due to the multipath propagation of radio waves along the transmitter-receiver path, the effects of refraction and diffraction of broadband radio signals in the tropospheric and ionospheric layers.

These channels include ionospheric radio communication channels in the frequency range 3-30 MHz, radio communication channels with tropospheric scatter in the frequency range 300 - 3000 MHz and in the frequency band 3000 - 30000 MHz, space communication channels with ionospheric scatter in the frequency range 30 - 300 MHz .

In mobile radio communication systems in the range from 1000 to 2000 MHz, the multipath nature of signal propagation is caused mainly by re-reflections of radio waves from buildings and structures, and terrain features. Similar effects occur in underwater acoustic channels.

In digital trunking communication systems using TBMA, remote radio access systems, and local office radio networks, channels are also characterized by significant time scattering and fading.

Similar problems may arise, for example, in satellite global radio navigation systems. A radio signal from near-horizon spacecraft can reach a ground-based moving object not only directly, but also through specular reflection from the earth's surface.

At the same time, errors in measuring pseudo-ranges due to multipaths can reach 3-9 m in the worst situation, i.e. will account for 10-30% of the total measurement error. In addition to multipath, with increasing measurement accuracy, the problem of compensating for the scattering of broadband signals in the ionosphere may also become relevant in these systems. The use of SOS methods in this case can become a pressing problem.

The development trends of modern communication systems are characterized by increasingly stringent requirements for maximum use of channel volume. In systems for the serial transmission of discrete messages over channels characterized by the occurrence of intersymbol interference, scattering estimation using channel testing with a test pulse is a key technology for implementing various types of equalizers. However, the time (from 20% to 50%) spent on channel testing is an increasingly attractive resource for upgrading TDMA standards, especially in mobile radio systems (for example, in the GSM standard, approximately 18% of the information frame is used to transmit the test pulse).

An alternative to channel testing in these systems is to use blind signal processing techniques.

The model of a discrete message transmission system taking into account scattering in the channel can be presented in the form of the following expression: oo «=+oo y(t)= jh(t,r)- + (7)

Оо «=-оо where: - signal in the receiver; (an) - sequence of information symbols of the alphabet A = ); ¿"¿(r,^) - channel signal corresponding to the A:-th symbol; h(r,t) - impulse response of the communication channel; v(i) - additive noise, T - clock interval. For linear digital modulation (7 ) can be transformed to form (8).

А0= \h(t,T)s0(z-nT)dT + v(t). (8)

For channels with slow time fading, the following simplification is valid: oo +°o

У(0= Ysan \h(t-T)s0(z-nT)dT + v(t). (9)

In various cases of a priori parametric and structural uncertainty, the channel model contains a number of parameters and/or functions of unknowns at the receiving side.

Uncertainty in the context under consideration can arise not only due to the passage of information signals of transmission systems through an unknown distorting channel, but also in cases of unknown structure and parameters of test signals used in the transmission system. A similar problem may arise in radio reconnaissance and radio monitoring tasks.

In the case of “complete” (nonparametric) uncertainty regarding the impulse response of the channel and the channel signal, we have a discrete-time model of the transmission system in the form (10), corresponding to the model with one input and output (6):

R0 = R")|,=/r = X>("M"-"M/), (10) n=0 where: x(/) is an unknown information sequence described by one or another statistical model, /?( /) is the unknown impulse response of an end-to-end discrete channel of the transmission system, b is the channel memory, y(/) is an unlimited sequence of statistically independent, arbitrarily “colored” noise samples.

The impulse response of an end-to-end channel can be considered both a deterministic and a random function. When the channel is stationary, the output sequence is stationary in discrete time.

For linear, time-constant, deterministic channels, when the sampling rate is higher than the symbol rate (usually a factor of integer m), the sampled signal is cyclostationary, or, equivalently, can be represented as a stationary sequence vector underlying the one-time model. input and multiple output (5), where we stack t - a sequence of input samples, during the reception of the next input symbol.

Then the discrete-time model of the transmission system can be represented as: y(/)=5>(u)x(/!-/)+y(/) (11) n=0

In this expression, y(/) and b(u) are the t-dimensional vectors of the signal at the receiver and the impulse response.

Another case described by the vector channel model (11) occurs in the case of spatial diversity of several receiving antennas (diversity reception).

SOS methods can find effective applications in chaotic communication systems. In recent years, the possibility of using noise signals has attracted great interest among researchers in the field of communications. According to some estimates, such systems can provide transmission speeds in a radio channel of up to 1 Gbit/s (today the experimentally achieved level of transmission speed is tens of Mbit/s).

The main idea here is the use of a noise (chaotic) signal as a carrier oscillation of an information transmission system.

In systems using deterministic chaos, information is introduced into the chaotic signal using amplitude modulation of the noise signal or by changing the parameters of the source of deterministic chaos. The use of a special test signal in these systems becomes impractical, because the existing problem of synchronizing deterministic chaos generators leads to the emergence of a priori uncertainty, including for the test signal.

At the same time, the specificity of the formation, emission and propagation of ultra-wideband signals arising in chaotic communication systems leads to the emergence of significant linear and nonlinear signal distortions, the compensation of which is a problem solved within the framework of SOS.

In digital television problems, linear distortions arise as a result of the transmission of a television signal over a radio channel, characterized by reflections from relief elements or urban development, as well as as a result of bandwidth limitations in analog television signal recording and storage systems.

The use of special test signals in this case significantly reduces the speed of information transmission and delays the prospect of the emergence of digital television systems that use standard radio bands to broadcast a digital television signal.

Today, a fairly large number of approaches to constructing blind equalizers have been developed for communication systems.

The key point in developing a blind equalizer is developing a rule for adjusting the equalizer parameters. In the absence of a test pulse, the receiver does not have access to the channel parameters and cannot use the traditional approach to minimizing the minimum average error probability criterion.

Adapting a blind equalizer requires the use of some special cost function, which of course includes high order statistics of the output signal.

The simplest algorithm in this class minimizes the mean squared error between the equalizer output and the two-way limiter output. The characteristics of the algorithm depend on how well the initial equalizer parameters are selected.

The first algorithm for direct blind equalization of a communication channel in digital systems with amplitude modulation was apparently proposed by Sato in 1975. . Sato's algorithm was subsequently generalized by D. Godard in 1980. for the case of combined amplitude-phase modulation (also known as the “constant modulus algorithm”).

In general, such algorithms converge when the output sequence of the equalizer satisfies the Buzzang property, i.e.:

M(y(/M/ - *)) = M(y(0/M" - *))), (12) where: /( ) is the cost function. Therefore, these algorithms are also called Bazgang algorithms.

In general, algorithms of this type belong to the class of so-called stochastic gradient blind equalization algorithms, which are built on the principle of an adaptive equalizer.

The error signal of the adaptive equalizer in this case is formed by an inertia-free nonlinear transformation of the output signal, the type of which depends on the signal-code design used.

It is essential for algorithms of this type that the input signals in digital communication systems are, as a rule, non-Gaussian, and the influence of dripping, leading to the superposition of a large number of these signals due to the central limit theorem of probability theory, normalizes the observed signal samples at the receiver. Therefore, the error signal in these algorithms is sensitive precisely to these properties of the signals at the equalizer output

The basic limitation of stochastic gradient algorithms is relatively slow convergence, the requirement of reliable initial conditions.

A distinctive advantage of these algorithms is the absence of requirements for stationarity of the IR channel over the evaluation interval. Moreover, we note that the absolute majority of blind identification and correction algorithms, one way or another, require such stationarity.

For communication systems characterized by a finite alphabet of information symbols, the idea of ​​extending the classical maximum likelihood estimation method not only to information symbols, but also to the unknown impulse response of a scalar channel may be justified.

Similar methods are classified in the literature as stochastic maximum likelihood algorithms.

Since the information signal is unknown, we can consider it a random vector with a known distribution. Let us assume, for example, that information symbols take on a finite number of values ​​(x\, x2, -~, xk) with equal probability, and the additive noise is white Gaussian noise with a spectral density N o, then the channel estimation algorithm will have the form:

The application of this algorithm in communication systems was first discussed in. Maximizing the likelihood function (13) is generally a difficult task, since this function is non-convex. However, today it is known ra.

1-X n=0 there is a fairly large number of algorithms that make it possible to obtain high-quality estimates (see bibliography in, as well as). If the regularity conditions are met and the initial approximation is good, these algorithms converge (at least in the root mean square sense) to the true value of the channel impulse response.

The deterministic version of the MP algorithm does not use a statistical model for the information sequence. In other words, the channel vector b and the information vector x are subject to simultaneous evaluation. When the noise vector is Gaussian with zero mathematical expectation and covariance matrix o21 MP, the estimate can be obtained by nonlinear minimum squares optimization.

Joint minimization of the likelihood function with respect to the channel vector and information samples is an even more difficult task than (13). Fortunately, the observed vector is a linear function with respect to the data vector or channel vector, defined by a Toeplitz or Hankel matrix. Therefore, we have a nonlinear minimum squares problem that we can solve sequentially.

The property of a finite alphabet of an information sequence can also be used within the framework of a deterministic MT approach. Such an algorithm is proposed in and uses the generalized Viterbi algorithm. The convergence of these approaches is not guaranteed in the general case.

Although MP estimates generally provide better performance, computational complexity and local maxima are their two main problems.

An important place in communication applications is occupied by the so-called “semi-blind” channel identification. These communication channel identification methods have recently attracted much attention because they provide fast and robust channel estimation. In addition, since a large number of serial transmission systems already use test signals, the likelihood of introducing these methods into communication practice is higher.

Semi-blind identification uses additional knowledge about the input information sequence, since part of the input data is known.

In this case, both stochastic and deterministic MP estimates are used, naturally taking into account the modification of likelihood functions by introducing a priori input data.

A milestone in the development of blind signal processing methods in communication systems was the use of high-order statistics to identify channels whose input signals are described by a model of stationary non-Gaussian random processes. Within these methods, as a rule, it is possible to find an explicit solution for an unknown channel.

The relatively recently understood possibility of using 2nd order statistics for blind identification of a vector communication channel (t > 1) has significantly brought closer the prospect of introducing blind processing technologies into communication systems and provoked a whole line of work in recent years, within which a whole family of fast-converging algorithms has been found to date identification. In this case, for channel identification, the presence of at least 2 independent reception channels is essential.

The use of 2nd order statistics for blind identification of a scalar channel (m = 1) is possible in general for a non-stationary model of the input signal and in the particular case of a periodically correlated (cyclostationary) signal.

b Scalar channel k and

Fig.2. Model of a non-stationary input communication channel.

The possibility of blind identification in the case of cyclostationary signal at the output was shown in, for forced cyclostationary modulation of the input signal in (Fig. 2), in the general case for a non-stationary input it was independently shown by the author in for radar applications.

Fig.3. Input signals of the transmission system: a) stationary sequence; b) sequence with a passive pause; c) sequence with an active pause; d) a sequence with cyclostationary modulation of a general form.

A discrete-time model of a wide class of discrete message transmission systems can be written as:

Ук = ^к181+кх1+к+Ч>к = (15)

1=0 where: /g/,/ = 0,.,b -1 - impulse response of the communication channel; g¡,i = O,., N + b-2 - modulating sequence;

X[ ,1 = O,., N + b - 2 - information sequence. Depending on the type of modulating sequence, we can obtain different structures of transmitted signals (Fig. 3).

Systems with modulating sequences shown in Fig. 3.b, c, d belong to the class of systems with a non-stationary input. The presence of this type of nonstationarity in the input signals is already a sufficient condition for the identification of a blind communication channel.

At the same time, in systems with an active pause (systems with a test pulse), the maximum time is spent testing the channel. At the same time, in systems with general cyclostationary modulation (Fig. 3.d), as well as in systems with a stationary input, we do not waste time testing an unknown communication channel.

That. in the tasks of developing radio engineering systems for transmitting information over radio channels characterized by significant scattering and fading, the development of effective SOS methods makes it possible to increase the throughput of systems using various types of channel testing methods. In this case, blind channel identification is an alternative technology and the developer should be given the opportunity to optimize the main parameters of the system: transmission speed, reliability, cost.

In modern radar, the use of increasingly broadband electromagnetic pulses for sounding is directly related to an increase in the time resolution and, consequently, the information content of these systems.

However, the influence of the path and propagation environment of radio waves increases in proportion to the frequency band of the signals used, which often leads to a loss of system coherence. This effect is especially significant for ultra-wideband radar.

The problem of blind signal processing in this case can be formulated as the problem of optimal coherent reception of unknown signals reflected from an extended object of finite size.

This problem arises in particular with the active radar of space objects through the Earth’s atmosphere in air and space defense radars and missile attack warning systems. In addition to military applications, such radars are used in monitoring space “junk”, which over the 40 years of the space era, filling near-Earth space, creates increasing problems for mankind’s space activities.

In this case, a packet of radar sounding signals, passing back and forth through the atmosphere, receives distortions caused by the frequency dependence of the refractive index of the ionosphere and polarization dispersion arising due to the Faraday effect. The scale of influence of this effect is discussed in. In accordance with these data, significant dispersion distortions of the radio signal appear already in the S band and quickly increase with increasing frequency band and wavelength.

In most cases, the model of a radar signal reflected from a spatially distributed target can be represented as: oo

Vnb)= \h(t-T-nT)%(r,n)dr+ v(t) (16) oo where: yn(t) - sequence of reflected pulses;<^(т,п) - коэффициент обратного рассеяния лоцируемого объекта; h{t) - искаженный зондирующий импульс РЛС.

The backscatter coefficient depends on the structure and geometry of the object, the orientation of the object and the radar, their relative motion, and the parameters of the probing signal. This information can be used to solve problems of recognizing a radar object and obtaining data about its shape.

The geometric structure of a radar object can be restored with a sufficiently large spatial separation of the radar receivers (radar base). In this case, the possibility of obtaining multi-view projections is realized, and the task is reduced to the use of tomographic methods.

In the case of locating an object from one point in space, recognition of the object can be carried out using time, polarization or time-frequency portraits of the radar target (signatures).

In all these problems, to restore the backscattering coefficient, we must accurately know the shape of the radar probe pulse. At the same time, as the probing pulse propagates, its shape changes as it passes through the atmosphere and the receiving path.

In this case, to restore the backscattering coefficient of the located object, we have the task of blind identification of a scalar or vector radar channel. Moreover, in contrast to blind identification applications in communication systems, where it is almost always possible to use the technique of test pulses to identify an unknown channel, in radar such an approach is practically impossible.

In radio reconnaissance systems and electronic warfare and radio countermeasure systems, the problem of blind separation of radio emission sources and adaptation of the radiation patterns of active phased arrays to the interference environment created by the enemy is relevant.

The emergence of a blind problem here is associated with the lack of a priori information about the coordinates of the sources, their orientation relative to the antenna of the radio device and, accordingly, the lack of information about the coefficients of the mixing matrix in (2) or (3).

Radar of the Earth's surface from aircraft using synthetic aperture radars (SARs) over the past 30 years has gone from isolated scientific experiments to a steadily developing industry of Earth remote sensing (ERS).

From the use of these systems, the scientific community expects in the near future significant progress in solving such global problems as predicting earthquakes and volcanic eruptions, understanding the processes of global climate change and in Earth science in general.

In addition to scientific purposes, these systems today are a unique tool for solving such practical problems as emergency control, environmental monitoring, cartography, agriculture, navigation in ice, etc. It should also be noted that these systems are one of the effective tools for monitoring the implementation of disarmament treaties.

The expansion of SAR application areas stimulates the constant growth of requirements for their spatial resolution, as well as the development of new frequency ranges.

At the same time, the effect of degradation of the spatial resolution of radar images (defocusing), which occurs in these systems due to the error in trajectory measurements, the influence of the propagation medium, and target movement, is becoming increasingly significant.

The problem of automatic focusing of synthetic aperture radar images first became relevant in connection with the increase in the spatial resolution of aviation SARs to the level of several meters in the late 80s and the first half of the 90s. The problem was caused by the fact that the navigation systems of an aircraft or spacecraft could not accurately measure the trajectory of the phase center of the SAR antenna, which is a necessary condition for obtaining high spatial resolution.

If the parameters of the relative motion of the object and the radar are known, then using direct or inverse aperture synthesis methods it is possible to construct a radar image of the object. In this case, the model of the reflected signal can be represented in the form: y(r,t)= ¡¡/1((,t,&,cg)£(&,cg)L6M<т + у(г,г) (17) вМ где: I- комплексный коэффициент отражения подстилающей поверхности; к({,т,в,сг) - пространственно-временной сигнал РЛС с синтезированной апертурой, отраженный точечной целью (импульсная характеристика радиолокационного канала); в,<7 - временные координаты элемента подстилающей поверхности (азимут, дальность); - временные координаты двумерного отраженного сигнала.

In systems using inverse aperture synthesis methods, telescopic SARs, the size of the integration region £>(/,r) is significantly larger than the size of the object in the z plane, the signal model (14) can be represented as a two-dimensional convolution: y(*>r)= No °) %(0,st)s1vs1su + v(tig) (18) V

Qualitatively, the process of forming radar images in SAR is shown in Fig. 4.

Fig.4. Image formation in SAR.

In general, the problem of forming radar images belongs to the class of inverse problems. Uncertainty about one or more parameters of a pseudoinverse or regularizing operator

H"1 is the essence of the problem of parametric focusing of radio images [19,155,220,223,217,214,232].

In this formulation, the problem in most cases was successfully solved by the development of algorithms for digital autofocusing of SAR images.

Two main groups of autofocus algorithms are widely known: algorithms based on the use of quality criteria in the form of local statistics of SAR images and algorithms using the correlation properties of defocused images.

In most cases, these algorithms ensure the achievement of a given level of resolution, however, in the case when SAR is installed on light aircraft (small aircraft, helicopters, unmanned aircraft), variations in focusing parameters become comparable to the aperture synthesis interval. In this case, obtaining a given level of resolution requires the use of more adequate models of the trajectory signal and more efficient autofocus algorithms.

In contrast to the problem of parametric focusing, when one or more parameters of the trajectory signal are unknown; in the problem of nonparametric focusing it is necessary to reconstruct the unknown operator H

1 overall.

The problem of nonparametric focusing (blind identification) arises mainly due to the effects of SAR signal propagation in the atmosphere and is more typical for space-based SAR and airborne SAR, the level of spatial resolution of which reaches several centimeters and requires the use of ultra-wideband signals.

That. in radar, solving a blind problem is in many cases a non-alternative technology for achieving high tactical and technical characteristics, and is sometimes the only opportunity for mastering new frequency ranges and resolution levels, increasing the detection characteristics and, in general, the information content of radar systems.

One of the characteristic features of posing a blind problem in these conditions is the lack of a priori statistical information about the observed object, which creates additional limitations for existing methods of blind identification and correction.

The task of compensating for distortions in imaging systems is one of the most widespread applications of SOS. In contrast to active radar, correction of linear distortions of images of various origins (radiometric, radioastronomical, optical, acoustic, x-ray, infrared) is the task of restoring a two-dimensional, spatially limited, non-negative signal distorted by a linear operator.

The model of such a signal can also be described by expressions (17) or (18), taking into account the fact that y^,m) and χ(b,a) are positive, spatially limited functions. In cases where the image is formed as the field intensity of some coherent source, the model of such an image can be represented as:

Sources of linear distortions include, for example, defocusing the lens of an optical imaging system, speed shift (blur) of the image due to the movement of the object during exposure, various types of diffraction limitations (i.e., limitation of the spatial spectrum of the image by the recording device), the influence of the propagation medium (for example, atmospheric turbulence).

Often the researcher knows the shape of the impulse response of the channel distorting the image, then image correction can be carried out with a linear optimal or suboptimal filter, according to

19) And built in accordance with one or another regularization strategy.

Blind image correction (blind image deconvolution) is a problem that arises in the absence of a priori information about their formation channel. The task of blind correction of linear image distortions is especially relevant in tasks of remote sensing of the Earth, astronomy, and medicine.

The possibilities for blind identification of scalar two-dimensional channels are somewhat wider than those of one-dimensional ones. This circumstance has been noted more than once in the literature and historically led to a more intensive implementation of blind processing methods in this case.

It is well known, for example, that the covariance functions of a stationary process at the output of a linear system do not contain information about the phase of its transfer function, and blind channel identification modulo the transfer function is possible only for a narrow class of systems with a minimum phase.

Interestingly, for discrete random fields this is generally not the case. Those. for two-dimensional discrete signals, the possibilities of reconstructing the phase modulo the transfer function are much wider. This somewhat unexpected result was obtained by mathematical modeling by Fienap in 1978. (see review).

The explanation for this fact is that in the ring of polynomials in two or more variables over the field of complex numbers there is a fairly powerful set of irreducible polynomials, in contrast to the ring of polynomials in one variable where, as is known, there are no irreducible polynomials whose degree is greater than 1.

Therefore, if a two-dimensional discrete signal has a z-transform that is indecomposable into simpler factors, then obviously using the uniqueness of the factorization of a polynomial into irreducible factors, we can restore the discrete signal from its autocorrelation or, equivalently, from its amplitude spectrum.

Naturally, this property of two-dimensional signals can also be used to solve the problem of deterministic blind identification of an image formation channel.

Consider a two-dimensional discrete convolution model:

The same relationship can be written as a product of polynomials of the ring C: y(z\,z2)=h(z 1>Z2MZ1>Z2) (21) where: y(21" 22) = X X y(!> PU\r2 ; ") = X X ^ "K-g2;

If the polynomials /2(21,22) and ^^^) are irreducible in the ring C^^], then by factoring ^(21,22) we solve the problem of blind identification.

Of course, the practical application of this approach is significantly limited by the complexity of the procedure for factoring polynomials in many variables and the presence of noise.

An algorithm that has some practical significance and is based on the irreducibility property of polynomials (21) is known as the “zero sheet” algorithm was proposed in. The algorithm uses the properties of surfaces whose points are the roots of the polynomials of the channel and the true image. A conceptually similar algorithm was proposed in .

An additional limitation to the scope of this approach is the use of the assumption that signals are spatially limited.

In addition to the properties of 2-transformations from signals of finite length, the non-negativity of the true image and various parametric models are also used for blind identification (see review).

One of the central problems in the practice of applications of neural networks, statistics, and DSP problems is the task of finding the most compact representation of data. This is important for subsequent analysis, which can be pattern recognition, classification and decision making, data compression, noise filtering, visualization.

Relatively recently, to solve similar problems, the method of finding a linear transformation that ensures the independence of components - ANC - has attracted widespread attention. The ANC problem is formulated as the problem of finding such a projection of a vector onto a linear space of vectors, components of which would be statistically independent. In this case, only a certain statistical sample of random vector values ​​is available for analysis. In this sense, the tasks and methods of ANC relate to the tasks and methods of SOS.

One of the promising areas for the development of modern remote sensing systems is synchronous photography of the earth's surface in various ranges of the electromagnetic spectrum. Joint processing of multispectral optical images, multi-frequency and multi-polarization radar images, radiometric images is a promising area of ​​research and practical applications of recent times.

The development of technologies for joint analysis of images of various natures includes the development of methods for visualization, classification, segmentation, and data compression. At the same time, as a rule, they strive to reduce the number of features for automatic classification of objects, provide their visual representation (visualization), and reduce the volume of stored information. ANC methods can become a powerful tool for joint image analysis.

Since the statistics of images generated by radio engineering systems (side-scan radars, SARs, radiometers) have significantly non-Gaussian statistics, the use of nonlinear ANC methods can significantly expand the capabilities of these applications.

That. in problems of digital image processing, an effective solution to a blind problem is in many cases a necessary, non-alternative stage of preliminary, primary processing, which provides the possibility of subsequent analysis. In problems of joint analysis of images of various natures, methods of independent component analysis can become an effective tool.

A classic application of ANC and blind source separation methods is in biomedical computing.

The capabilities of digital processing of electrocardiograms, encephalograms, electromyograms, and magnetoencephalograms have significantly expanded the capabilities of diagnosing a wide class of diseases.

A peculiarity of the use of these methods is the need to separate the signals of the organs being studied from noise of various origins and interfering signals (for example, separation of cardiograms of mother and child).

These technologies directly apply methods of blind source separation and independent component analysis. The observed signal models used in these applications are described by expressions (2) and (3).

The problem of speech recognition is a key problem in many areas of robotics and cybernetics. Speech recognition technologies can be used to control the operation of various types of machines and mechanisms, enter and search for data in a computer, etc.

In an audio information recording system, the signal available for recognition is a convolution of the original speech signal and the impulse response of the sensor and the environment.

In this case, the sensor parameters, as well as the environmental parameters, change extremely. Handsets vary in degree of distortion, spectral composition and signal strength. Microphones are manufactured in a variety of ways and are positioned in various positions on the handset, with openings of various sizes, located at various points within the sound field around the mouth. A sensing device that works well for one specific sensor in one specific environment might perform very poorly in other conditions. Therefore, it is desirable that these parameters do not affect the operation of the recognition algorithm. Blind identification is used in this task to restore the original speech signal.

The fight against reverberation is necessary in cases where the original speech signal is distorted by the acoustics of the environment, because environmental acoustics depend on the geometry and materials of the room and the location of the microphone.

Since the initial speech signal is indistinguishable and the environmental acoustics are unknown, blind identification can be used in adaptive reverberation control.

One of the illustrative tasks illustrating the problems of blind separation of independent sources is the so-called. the problem of separating the desired conversation from the background of other talking people, music, extraneous noise (cocktail party problem). We can notice that our brain easily copes with this, but at the same time, for a computer it is a very difficult task.

This problem has practical significance, for example, for the development of adaptive listening systems when recording audio information on several microphones installed in a room.

In tasks of geology and seismological research, technologies for recording signals from sources of mechanical vibrations, both artificial (dynamite in a pit) and natural (earthquake), are used. These signals are used to estimate the reflectance of various layers of the earth's crust.

The blind problem arises here due to the unpredictability and, accordingly, uncertainty of the shape of the exciting impulse.

That. The considered problems arising in various fields of radio engineering and communications, as well as numerous other signal processing applications, confirm the thesis about the relevance of the task of developing new SOS methods and expanding the areas of its applications.

The solution to the “blind” problem in communication problems was prepared by numerous scientific results in the field of statistical communication theory concerning adaptive methods for transmitting discrete messages over channels with various types of scattering and fading, the creation of new methods and devices for signal processing obtained in the works of C.V. Helstrom, T. Kailath, H.L. Van Trees, J.G. Proakis, G.D. Forni, M.E. Austin, B.A. Kotelnikov, B.R. Levina, B.A. Soifera, V.F. Kravchenko, D.D. Klovsky, V.I. Tikhonova, Yu.G. Sosulina, V.G. Repina, G.P. Tartakovsky, P.JI. Stratonovich, A.P. Trifonova, Yu.S. Shinakova, J1.M. Finka, S.M. Shirokova, V.Ya. Kontorovich, B.I. Nikolaeva, V.G. Kartashevsky, B.JL Karyakin and others.

In the development of SOS in communication systems and a number of other areas, the research of such scientists as: G. Xu, H. Liu, L. Tong, T. Kailath, P. Comon, Y. Sato, D.N. played a major role. Godard, E. Serpedin, G.B. Giannakis, E. Moulines, P. Duhamel, J.-F. Cardoso, S. Mayrargue, A. Chevreuil, P. Loubaton, W.A. Gardner, G.K. Kaleh, R. Valler, N. Seshadri, C.L. Nikias, V.R. Raghuveer, D.R. Brillinger, R. A. Wiggins, D. Donoho and many others.

In radar in general and in survey PJ1C in particular, the capabilities of SOS were prepared by numerous results in the field of adaptive methods for reconstructing spatio-temporal signals, including parametric methods for estimating IR radar channels obtained in the works of S.E. Falkovich, V.I. Ponomareva, V.F. Kravchenko, Yu.V. Shkvarko, P.A. Bakuta, I.A. Bolshakova, A.K. Zhuravleva, H.A. Armanda, G.S. Kondratenkova, V.A. Potekhina, A.P. Reutova, Yu.A. Feoktistova, A.A. Kosta-leva, V.I. Kosheleva, Ya.D. Shirman, A. Ishimary, A. Moreiro, R. Klem, S. Madsen, R.G. White, D. Blackneil, A. Freeman, J.W. Wood, C.J. Oliver, C. Mrazek, S. McCandless, A. Monti-Guarnieri, C. Prati, E. Damonti. etc.

In problems of image processing of various natures, numerous SOS methods were proposed in the works of V.P. Bakalova, N.P. Russkikh, P.A. Bakuta, V.A. Soifera, V.V. Sergeeva, D. Kundur, D. Hatzinakos, R.L. Lagendijk, R. G. Lane, R.H.T. Bates and many others.

A. Nu-varinen, A. Cichocki, S. Amari, J.-F. made significant contributions to the development of the fundamentals and methods of AIC. Cardoso, P. Comon, M. Rosenblatt, S.Y. Shat-skikh, S. A. Ayvazyan, L.D. Meshalkin et al.

With the accumulation of results in recent years, the prerequisites have been created for the construction of a systematic theory for solving the “blind problem”.

In addition, to ensure the possibility of widespread implementation of SOS methods in radio engineering, they require the creation of new SOS technologies characterized by a high convergence rate, providing blind identification capabilities in the absence of a priori information about the statistics of the information signal, providing the ability to identify a non-stationary channel and non-stationary information signals.

A new class of SOS methods that potentially provides an effective solution to the problem of statistical identification in the absence of a priori information about the statistics of information signals can be obtained by using polynomial representations of signals.

In this case, we can transfer the problem to be solved from commonly used complex vector spaces into polynomial rings of many variables with random coefficients and use the methods of commutative algebra, algebraic geometry, and computer algebra, which have been intensively developing in recent years.

In the particular case of choosing the values ​​of the formal variable of polynomials on the unit circle of the complex plane, we obtain SOS methods based on polyspectra.

The possibilities of this path are prepared by fundamental results in the relevant branches of mathematics obtained by D. Hilbert, B. Buchberger, H.J. Stetter, W. Auzinger, W. Trinks, K. Farahmand, H.M. Moller, M. Kas, I.M. Gelfand, I.R. Shafarevich, I.A. Ibragimov, Yu.V. Linni-com, O. Zariski and others.

Goals and objectives of the study. The purpose of the dissertation work is to develop the theoretical foundations, methods and algorithms for blind signal processing and their application in some problems of radio engineering, communications, and joint processing of images obtained in various ranges of the electromagnetic spectrum.

Achieving this goal requires solving the following tasks:

Development of a systematic theory for solving SOS problems based on polynomial representations of discrete signals;

Development of new effective methods and algorithms for SOS in the absence of a priori information about the statistics of the information signal;

Development of SOS methods for a non-stationary model of input signals;

Development of algorithms for correcting diffraction distortions of radar sounding signals when reflected from spatially distributed targets;

Development of methods for blind reconstruction of SAR radar images, including space SAR, operating in the R, UNR ranges;

Development of robust nonlinear ANC methods in the problem of joint processing of radar, radiometric and optical images.

Research methods. The tasks of constructing methods for blind signal processing, formulated in this work, require the creation of a new mathematical apparatus, which is based on a compilation of methods from probability theory, commutative algebra and algebraic geometry. In addition, the use of classical methods of probability theory, statistical radio engineering, numerical methods, computer simulation methods and computer algebra.

The scientific novelty of the work is manifested in the fact that for the first time

A description of random vectors based on polynomial moments and cumulants is used, the properties of such a description are determined, concepts are introduced and the properties of affine manifolds of non-zero correlation are defined;

A theorem on sufficient conditions for the identifiability of a scalar stationary channel with a nonstationary input is proved;

A number of algorithms for blind identification of a scalar channel with a non-stationary input using 2nd order statistics have been proposed, including a two-diagonal algorithm for blind channel identification, which does not require a priori knowledge of the type of non-stationarity of information signals;

The problem is formulated, the main algorithms for solving the problem of identifying a channel with a stationary and non-stationary input are determined, as a problem of solving a system of polynomial equations in many variables;

Blind identification algorithms have been developed based on factorization of affine manifolds of zero correlation, which do not require a priori information about the statistics of information signals;

Blind identification algorithms have been developed based on the proposed transformations of non-zero pair correlation;

Blind identification algorithms have been developed based on the properties of symmetric polynomial cumulants of observed signals;

The problem of identifying a vector channel in a polynomial interpretation is considered, the main theorems of identifiability are proven, a polynomial interpretation of the method of mutual relations (MR) is proposed - the zero subspace algorithm (NSA), expressions for the relative error of identification are obtained, and a comparison with other methods is carried out;

The possibilities of using the developed methods of blind identification in radio engineering information transmission systems are considered, the reliability of communication systems is compared by modeling, when using the developed methods of blind identification in comparison with the technique of using test signals, the issues of choosing non-stationary modulation in digital communication systems, providing the possibility of blind identification by 2nd order statisticians;

When solving the problem of blind formation of SAR images: a model of the space-time channel of space SAR was developed, taking into account the influence of atmospheric effects; two-dimensional characteristics of phase fluctuations of the SAR signal in the P, UHF, VHF ranges were obtained; algorithms have been developed for correcting diffraction distortions of PJIC sounding signals when reflected from spatially distributed targets (“blind” matched filter), including an algorithm for blind identification of a radar channel by signed correlations; within the framework of the contrast function method, algorithms for blind formation of SAR images have been developed, including those based on the minimum entropy method;

An algorithm for nonlinear analysis of independent components is proposed based on independence transformations and kernel estimates of integral functions of multidimensional distributions.

The following main provisions and results of the dissertation are submitted for defense:

Methods for blind identification of scalar channels based on polynomial statistics;

Methods for blind identification of scalar channels with non-stationary input;

Zero subspace algorithm for vector channel identification;

Algorithm for identifying the type of digital modulation of a radio communication system, based on the Kullback-Leibler distance;

Model of the space-time channel of space SAR taking into account the influence of atmospheric effects, as well as two-dimensional characteristics of phase fluctuations of the SAR signal in the P, UHF, VHF ranges;

Algorithms for correcting diffraction distortions of PJ1C sounding signals when reflected from spatially distributed targets (“blind” matched filter), including an algorithm for blind identification of a radar channel by signed correlations;

Algorithms for blind generation of SAR images, including those based on the minimum entropy method;

Fast algorithms for generating SAR images, based on the use of rotation vector techniques;

Algorithm for nonlinear analysis of independent components based on nonlinear independence transformation and kernel estimates of integral functions of multidimensional distributions.

Practical value and implementation of work results.

The results of the dissertation are part of the research work (code "Water Capacity") on the creation of adaptive universal demodulators for digital communication systems, in the development of methods for optimal signal processing in communication systems under conditions of structural and parametric uncertainty, carried out by the Federal State Unitary Enterprise Research Institute "Vector" (St. Petersburg) in 2002-2003

The results of the research and development carried out are part of a series of research and development work carried out at the Federal State Unitary Enterprise GNP RKTs TsSKB-PROGRESS (Samara) to create radar space and aircraft remote sensing systems in 1988-2000. (R&D work on the creation of space systems “Sapphire-S”, “Resurs-Spectrum”, “Resurs-DK”, research work “Elnik-UN”, “Zerkalo”).

The research results were used in the Federal State Unitary Enterprise TsNIIMASH (Moscow) in substantiating a comprehensive scientific program of experiments on the Russian segment of the International Space Station (experiment “Radar sensing of the Earth in the L- and P-bands”, code “Radar”), as well as in developing requirements for promising dual-use space radar surveillance system "Arkon-2".

The developed algorithms and programs for blind identification of the radar channel were used at the Federal State Unitary Enterprise Research Institute of Technology (Moscow) in preparing aircraft tests and processing radar data from the IK-VR aviation radar complex in 1994-1995, as well as in analyzing the influence of the atmosphere and forecast accuracy on the resolution of space RSA 14V201 for the spacecraft 17F117, “Luch-M” for the Resurs-DK-R1 spacecraft.

The results of the work have found application in the educational process at the State Educational Institution of Higher Professional Education PGATI, in particular in the lecture courses “Statistical Theory of Radio Engineering Systems”, “Radio Engineering Systems”, “Fundamentals of Information Processing and Digital Signal Processing”, in laboratory work, as well as in diploma design.

The use of the work results is confirmed by relevant implementation documents.

1. BASIC THEOREMS OF BLIND IDENTIFICATION

The main results and conclusions of the work are as follows:

1. The conditions for deterministic identifiability of a vector channel essentially guarantee the following requirements: all channels in the system must be different from each other, for example they cannot be identical; the input sequence must be quite complex; There must be enough output samples available.

2. Conditions for the statistical identifiability of a deterministic vector channel can be discussed in a broader context. For example, if the number of available samples at the output of a channel is infinite and the input is a non-Gaussian stationary random process, then the system can be identified exactly by higher-order statistics even when the channel polynomials have common zeros. Or, for example, if the input is a stationary random process (including a Gaussian one), the system can be identified if the second-order statistics of the output are known exactly and the joint zeros of the channel polynomials are inside the unit circle (phase minimum condition).

3. Both in the case of deterministic and statistical identification of a vector channel, for the channel to be identifiable it is necessary or sufficient that the polynomials (d) have no common roots. This means that channel cross-links are used explicitly or implicitly to identify a vector channel.

4. For the identifiability of a deterministic scalar channel, it is necessary that the linear complexity of the information sequence be greater (2b - 2).

5. Severe restrictions on the possibilities of blind identification of a scalar channel in the deterministic case, formulated in Theorem T.6, significantly limit the scope of application of these methods.

6. For the statistical identifiability of a scalar channel, it is sufficient that the samples of the information sequence are described by a model of a strictly non-stationary or non-Gaussian process.

7. The polynomial interpretation of the method of mutual relations allows the use of algorithms for solving a system of homogeneous equations to solve the variational problem of the method of minimal squares.

8. The blind vector channel identification algorithm obtained within this approach, called the zero subspace algorithm (ZSA), is equivalent to the estimate obtained using the least squares method and allows analytical and iterative forms of solution representation.

9. The values ​​of the formal variables.,gg>m must be chosen so as to ensure the maximum value of the signal-to-noise ratio d2^,.^/,^,.,^) and at the same time minimize the value of the condition numbers

10. Choice of values ​​of formal variables = exp(-y"2m/M), 1 = \,.,M, and r^ = exp(-j2m/r"), / = 0,.,?-1 when the condition is met ? = g" = g provide the minimum value of the relative error of channel estimation, with? = g" F g this choice provides a solution close to optimal with the same dispersion of white Gaussian noise in the subchannels. In general, in the presence of concentrated interference, differences in additive noise parameters in different subchannels, correlation of noise samples, the selection of cross sections should be carried out by minimizing the right side of (2.24).

11. The relative error of the ANP significantly depends on the level of additive noise. An acceptable level of error is achieved when the signal-to-noise ratio is more than ZODb. As the channel length increases, the error grows linearly, however, as the number of channels increases for large signal-to-noise ratios, the channel length has virtually no effect on the error value.

12. ANP at large signal-to-noise values ​​practically coincides with the MP algorithms and the classical VO algorithm, however, unlike the ANP, the MP algorithm and the VO algorithm have a sharper increase in error at small signal-to-noise ratios.

13. If the input is a random process that is non-stationary with respect to the average value, and = where x"(/) is a stationary process with zero mathematical expectation, then the channel is identified by 1st order statistics;

14. If the input is a random process that is nonstationary in variance = where is a stationary process with zero m.o. and then we identify the channel using 2nd order statistics;

15. If the input x(?) is a random process with a time-nonstationary frequency structure, i.e. = - where x"(() is a stationary process with zero mathematical expectation and //"(?)> O, then the channel is identified by 2nd order statistics;

16. If the input x(()) is a stationary random process with zero mathematical expectation, then the channel is identified by statistics of the 3rd or more order;

17. If the input is a random periodically correlated random process with zero mathematical expectation, then the channel is identified by 2nd order statistics, under additional conditions: 1) the zeros of the channel are not a multiple of 1/T; 2) for channels with an impulse response limited by a time interval (0, rmax), T > rmax;

18. For an input signal that is nonstationary in dispersion, an estimate of the channel transfer function can be obtained from the covariance matrix of the observed signal in the spectral or time domains;

19. To obtain an estimate of the channel transfer function, it is sufficient to have only 2 diagonals of the covariance matrix in the spectral region (the corresponding algorithm is called a two-diagonal blind identification algorithm), and to obtain an estimate, no a priori knowledge of the statistical characteristics of the information signal is required;

20. The error in estimating the transfer function from spectral moments of the 2nd order depends on the signal-to-noise ratio, the number of processed signal realizations, the degree of non-stationarity of the input signals, the estimation algorithm used and the type of non-stationarity;

21. The polynomial representation of discrete random signals of finite length allows us to describe the statistical characteristics of these signals using polynomial moments and cumulants, which are elements of rings of polynomials in many variables over the field of complex numbers.

22. The properties of polynomial moments and cumulants are in many ways similar to the properties of ordinary moments and cumulants, however, the affine manifolds generated by polynomial cumulants (called non-zero correlation manifolds) have a number of unique properties, namely dimension, which is different for deterministic and random signals. This property can be used for blind identification of channels in the absence of a priori information about the statistics of information signals.

23. The use of polynomial cumulants allows us to formulate the general problem of blind identification as the problem of solving a system of polynomial equations for unknown channel coefficients. By choosing a set of polynomial cumulants that correspond to the specifics of the problem, we can synthesize the appropriate identification algorithm. At the same time, the proposed approach to the synthesis of blind identification algorithms based on polynomial statistics allows us to synthesize various blind identification algorithms for scalar channels with stationary and non-stationary input, various distributions of input symbols. Unlike the polyspectral approach, in this case the uncertainty in choosing a set of cumulant functions can be reduced, at least with respect to the algorithm synthesis procedure.

24. In a scalar channel, blind identification algorithms based on solutions of polynomial equations require some statistical sampling of information blocks at the channel output to construct an estimate. Qualitatively, to obtain a blind estimate in a scalar channel, an information sequence is required, the length of which is usually 2 orders of magnitude greater than the length of the channel. In this case, the quality of the assessment approaches that of the test signal.

25. A blind identification algorithm based on the properties of zero-correlation manifolds, using a non-stationary channel model, allows you to separate manifolds generated by an unknown deterministic channel from manifolds generated by a random information signal. The simulation of this algorithm showed that in comparison with the algorithms of the previous section, as well as algorithms based on the use of high-order spectra, this algorithm requires approximately two orders of magnitude less number of implementations, but has lower noise immunity. In addition, the error of the algorithm increases significantly with increasing channel length.

26. The blind channel identification algorithm, based on the use of non-zero correlation manifolds, in contrast to the blind identification algorithm based on factorization of affine manifolds, has a fairly high convergence rate, providing high-quality estimates already with a signal-to-noise ratio of 15-20D6. However, when constructing a nonzero pairwise correlation transformation, we need knowledge of the covariance matrix of the information sequence.

27. Channel identification, based on the use of the properties of symmetric polynomial cumulants, makes it possible to identify a non-stationary communication channel in the absence of data on the statistics of the information sequence, if 2L > N.

28. Blind signal processing is a fairly promising technology for channel equalization in serial communication systems in scattered channels. The analysis shows that if we consider blind assessment as an alternative to assessment using a test pulse, then the latter almost always wins in terms of convergence speed and noise immunity, but blind assessment always wins in terms of transmission speed.

29. For algorithms using a vector channel model, non-zero correlation transformations, as well as non-stationary modulation, in some cases the gain in reliability of the test pulse estimate can be leveled out or eliminated completely.

30. The answer to the question: “should I use blind channel estimation or not in each specific case?” requires a compromise decision from the developer of the communication system.

31. The algorithm for classifying the type of modulation by signal constellations for large samples comes down to finding the probability distribution that is closest to the point histogram in terms of the Kulbak-Leibler distance. This algorithm turns out to be equivalent to the maximum likelihood algorithm for large samples. The potential characteristics of a two-alternative classification leading to an additive upper bound on the error probability significantly depend on the geometry of the constellation, the level of additive noise and the order of enumeration of the constellations and are completely determined by the Kullback-Leibler distance.

32. The influence of trajectory and especially atmospheric errors leads to a significant limitation in the spatial resolution of space-based SARs, with the degree of degradation increasing sharply with increasing wavelength and potential spatial resolution. In addition, these effects lead to significant geometric and polarization distortions. This allows us to consider the task of obtaining a radar image under conditions of the strong influence of trajectory and atmospheric errors as the main problem limiting the development of space SAR technology in the development of new frequency ranges and resolution levels. One of the most preferable ways to overcome the consequences of these effects is the use of SOS technologies to compensate for distortion of radar images.

33. The influence of the atmosphere on the resolution of SAR begins to affect itself already, starting from 10 cm, and increases significantly from 23 cm. In the long-wave range (>70cm), the degradation of radar images in spatial resolution in a disturbed ionosphere can reach 2 orders of magnitude. Moreover, in this range, the resolution is practically independent of the resolution without taking into account the destructive influence of the atmosphere and is determined primarily by the effective coherence interval, which in turn is determined exclusively by atmospheric parameters. The degree of degradation increases with increasing flight altitude, and especially with increasing ionospheric turbulence. For azimuth resolution in shortwave ranges (<3см), атмосфера влияния практически не оказывает. Влияние атмосферы на РСА, работающих в (Р, UHF, VHF) приводит к существенному снижению их разрешающей способности.

34. Compensation for the effects of degradation of SAR resolution over range can be carried out using a bidiagonal blind identification algorithm using signed correlation.

35. Compensation for the effects of degradation of SAR resolution in azimuth can be carried out using gradient blind correction algorithms based on contrast functions of maximum likelihood or minimum entropy. The computational complexity of the radar image reconstruction algorithm can be significantly reduced by using the representation of complex samples of the SAR signal in the basis of rotation vectors.

36. The proposed ANC method, using the independence transformation, built on the kernel estimate of the multidimensional probability distribution function, can be used in the problem of joint processing of radar, radiometric and optical images. The advantage of this algorithm is the ability to solve linear and nonlinear ANC problems within the framework of one algorithm.

37. The ability to construct an independence transformation of an n-dimensional random vector using paired independence transformations for non-Gaussian random vectors significantly expands the scope of application of this approach. The ANC algorithm described in this section can be used in problems of statistical blind identification and correction, blind separation of radiation sources, in cases where not only about the statistics of the information signal there are only general assumptions (independence), but also a mechanism for converting the information signal into an observed one signal is unknown.

CONCLUSION

The result of the dissertation work is the development of theoretical foundations, methods and algorithms for blind signal processing and their application in some problems of radio engineering, communications, and joint processing of images obtained in various ranges of the electromagnetic spectrum.

In the process of achieving the main goal, the following tasks were solved:

A systematic theory for solving SOS problems based on polynomial representations of discrete signals has been developed;

A class of new effective SOS methods and algorithms has been developed that do not require a priori information about the statistics of the information signal;

New SOS methods and algorithms have been developed for the non-stationary model of input signals;

The possibilities were studied and algorithms were developed for blind correction of diffraction distortions of radar sounding signals when reflected from spatially distributed targets;

Methods and algorithms for blind reconstruction of SAR radar images in the R, UNB ranges have been developed;

A new nonlinear ANC algorithm has been developed, and the possibilities of using this method in the problem of joint processing of radar, radiometric and optical images are considered.

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