Identification of the impulse response of the communication channel. A mathematical model of a linear communication channel with memory based on characteristic functions and a probabilistic mixture of signal distributions. How the Viterbi equalizer works

Blind signal processing (Blind signal processing) is relative new technology digital signal processing (DSP), which has developed over the past 10-15 years.

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

Uncertainty area Observation area

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, digital television; in radio communication systems; in tasks of digital processing of speech, images.

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

There are two main types of blind signal processing tasks: 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 then be used to estimate the information sequence, i. E. is the first step in blind alignment or correction.

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

4-co y (0 = | n (*, z) x (rYy + y (0, (1) co unknown matrix of impulse responses (IR) with elements hi j (r)); v (t) ~ additive noise (vector random process with values ​​in CT, as a rule with independent components); х (г) - unknown information signal with values ​​in WITH".

The systems described by expression (1) are called multiple-input and multiple-output systems (in the English-language literature, Multiple-Input Multiple-Output, or MIMO).

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

If the components of the matrix H (r) have the form | / yyj (r)), we get the model used in the problems of blind source separation (Blind Source Separation or BSS):

Y (0 = H x (f) + v (f), (3) where: H - m x n unknown, complex (so-called "mixing") matrix with elements (fyjj; x (z) ~ unknown signals ...

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

In this case, the model used in the analysis of independent components is often presented in the form:

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 the vector y onto the 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 is known.

ANC is a 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 model of the system 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) - unknown complex information signal with values ​​in C.

Systems described by models of the form (5) are called systems with one input and multiple output (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

The problems of blind channel identification based on models (5) and (6) will be referred to below as problems of stationary blind identification of a vector and scalar channel, respectively.

The blindly identifiable system is understood as the possibility of restoring the impulse response of the system with an accuracy to a complex multiplier only from the output signals.

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

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

Moreover, in radio channels, ICs are usually nonstationary due to 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 - 30,000 MHz, space communication channels with ionospheric scattering in the frequency range 30 - 300 MHz ...

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

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

Similar problems can arise, for example, in global radio navigation satellite systems. The radio signal from near-horizon spacecraft can arrive at a ground-based mobile object not only in a direct way, but also due to specular reflection from the earth's surface.

At the same time, errors in measuring pseudo-ranges due to multi-beam 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 an increase in the 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 an urgent problem.

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

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

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

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

A0 = \ h (t, T) s0 (z-nT) dT + v (t). (eight)

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

Y (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 on the receiving side.

Uncertainty in the considered context 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 can arise in the tasks of radio intelligence and radio monitoring.

In the case of "complete" (nonparametric) uncertainty about 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 the 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 the end-to-end channel can be considered both deterministic and random function... When the channel is stationary, the output sequence is stationary in discrete time.

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

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 (f) and b (u) are the t-dimensional vectors of the signal in the receiver and the impulse response.

Another case described by the vector channel model (11) arises in the case of spatial diversity of several receive antennas (receive diversity).

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

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

In systems using deterministic chaos, information is introduced into a chaotic signal using amplitude modulation of a noise signal or by changing the parameters of a deterministic chaos source. The use of a special test signal in these systems becomes impractical because the existing problem of synchronization of generators of deterministic chaos 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 the SOS.

In problems of digital television, linear distortions arise as a result of the transmission of a television signal over a radio channel, characterized by re-reflections from relief elements or urban buildings, as well as as a result of bandwidth limitation in analog systems for recording and storing a television signal.

The use of special test signals in this case significantly reduces the speed of information transmission, and postpones the prospect of the appearance of digital television systems using standard radio bands for broadcasting a digital television signal.

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

Key moment in the development of a blind equalizer, it is the development of 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 criterion for the minimum average error probability.

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

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

For the first time, the algorithm for direct blind equalization of the communication channel in digital systems with amplitude modulation was proposed, apparently, 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, these algorithms converge when the output sequence of the equalizer satisfies the Buzgang 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 a non-inertial nonlinear transformation of the output signal, the form of which depends on the used signal-code structure.

Essential for algorithms of this type is that the input signals in digital communication systems, as a rule, are 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 in the receiver. Therefore, the error signal in these algorithms is sensitive precisely to these properties of the signals at the output of the equalization

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

A distinctive advantage of these algorithms is the absence of requirements for stationarity of the IH channel in the estimation 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 the scalar channel, may be justified.

Such 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. Suppose, for example, that information symbols take a finite number of values ​​(x1, 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:

For the first time, the application of this algorithm in communication systems was considered in. In the general case, maximizing the likelihood function (13) is a difficult problem, since this function non-convex. However, ra is known today.

1-X n = 0 a sufficiently large number of algorithms to obtain estimates High Quality(see bibliography at as well). Under the regularity conditions and with a good initial approximation, these algorithms converge (at least in the rms sense) to the true value of the channel impulse response.

The deterministic version of the MT 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 estimation. When the noise vector is Gaussian with zero mathematical expectation and a covariance matrix of about21 MP, the estimate can be obtained by nonlinear least 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 linear function relative to a data vector or channel vector, given by a Toeplitz or Hankel matrix. Therefore, we have a nonlinear minimum squares problem that we can solve sequentially.

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

Despite the fact that MT estimates usually provide best performance, computational complexity and local maxima are their two main problems.

An important place in communication applications is occupied by the so-called "half-blind" channel identification. These methods of identifying communication channels have received a lot of attention lately because they provide fast and stable channel estimation. In addition, since a large number of serial transmission systems already use test signals, the likelihood of these techniques being introduced into communication practice is higher.

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

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

A stage in the development of methods for blind signal processing 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 the framework of these methods, as a rule, it is possible to find an explicit solution for an unknown channel.

The relatively recently understood possibility of using second-order statistics for blind identification of a vector communication channel (m> 1) has significantly brought the prospect of introducing blind processing technologies into communication systems and has provoked a whole line of work in recent years, within which a whole family of rapidly converging algorithms has been found. identification. At the same time, the presence of at least 2 independent reception channels is essential for channel identification.

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

B Scalar channel to and

Fig. 2. Model of a communication channel unsteady on the input.

The possibility of blind identification in the case of cyclostationarity of the signal at the output was shown in, for forced cyclostationary modulation of the signal at the input (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) a sequence with a passive pause; c) a sequence with an active pause; d) sequence with cyclostationary modulation general view.

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

Vk = ^ k181 + kx1 + k + h> k = (15)

1 = 0 where: / r /, / = 0,., B -1 - impulse response of the communication channel; gi, i = 0,., N + b-2 is a modulating sequence;

X [, 1 = O,., N + b - 2 - information sequence. Depending on the type of the modulating sequence, we can obtain various structures of the 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 blind identification of the communication channel.

At the same time, in systems with an active pause (systems with a test pulse), the maximum time is spent on channel testing. At the same time, in systems with cyclostationary modulation of a general form (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 development effective methods SOS allows 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 more and more broadband electromagnetic pulses for sounding is directly related to an increase in the temporal resolution and, consequently, the information content of these systems.

However, the influence of the path and the propagation medium 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 dimensions.

This problem arises, in particular, during active radar of space objects through the Earth's atmosphere in radar stations for air and space defense, missile attack warning systems. In addition to military applications, such radars are used to control space "debris" big problems for the space activities of mankind.

In this case, a burst of radar sounding signals, passing back and forth through the atmosphere, receives distortions caused by the frequency dependence of the ionospheric refractive index and polarization dispersion arising from the Faraday effect. The magnitude of the effect of this effect is considered in. In accordance with these data, significant dispersion distortions of the radio signal appear already in the S band and rapidly increase with increasing frequency band and wavelength.

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

Ynb) = \ h (t-T-nT)% (r, n) dr + v (t) (16) oo where: yn (t) is a 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 sounding signal. This information can be used to solve problems of recognizing a radar object and obtaining data on its shape.

The geometric structure of the 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-angle 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, object recognition can be carried out by time, polarization, or time-frequency portraits of the radar target (signatures).

In all these tasks, to reconstruct the backscatter coefficient, we must know exactly the shape of the radar probe pulse. At the same time, when the probe pulse propagates, its shape changes when it passes through the atmosphere and the receiving path.

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

In radio intelligence systems and electronic warfare and radio countermeasures, the problem of blind separation of radio emission sources, adaptation of the directional patterns of active phased arrays to the interference environment created by the enemy is urgent.

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 engineering device and, accordingly, the lack of information about the mixing matrix coefficients in (2) or (3).

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

From the application 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.

Expansion of SAR applications 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 errors in trajectory measurements, the influence of the propagation medium, and target movement, becomes more and more significant.

The problem of automatic focusing of images of synthetic aperture radars became urgent for the first time in connection with the increase in the spatial resolution of aircraft SAR 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 a spacecraft (SC) could not provide with the required accuracy the measurement of the trajectory of the phase center of the SAR antenna, which is a necessary condition for obtaining a high spatial resolution.

If the parameters of the relative motion of the object and the radar are known, then using the methods of direct or inverse synthesis of the aperture, 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:<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 the methods of inverse aperture synthesis, telescopic SAR, the size of the integration region t> (f, z) is much larger than the size of the object in the z plane, the signal model (14) can be represented as a two-dimensional convolution: y (*> z) = N °) % (0, st) s1ws1su + v (tig) (18) V

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

Fig. 4. Image formation in PCA.

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 and constitutes the essence of the problem of parametric focusing of radio images [19,155,220,223,217,214,232].

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

Two main groups of autofocusing algorithms are widely known, these are: algorithms based on the use of a quality criterion in the form of local statistics of SAR images and algorithms that use 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 the SAR is installed on light aircraft (small aircraft, helicopters, unmanned aircraft), the 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 trajectory signal models and more efficient autofocusing algorithms.

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

1 in general.

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

That. in radar, the solution of a blind problem is in many cases an uncontested technology for achieving high tactical and technical characteristics, it is sometimes the only opportunity for mastering new frequency ranges and levels of resolution, increasing the detecting characteristics and, in general, the information content of radar systems.

One of the characteristic features of the formulation of a blind problem under these conditions is the absence of a priori statistical information about the observed object, which creates additional restrictions for existing methods of blind identification and correction.

The problem of compensating for distortions in imaging systems is one of the most widespread applications of SOS. In contrast to active radar, the correction of linear distortions of images of various origins (radiometric, radio astronomical, optical, acoustic, X-ray, infrared) is the problem of recovering 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% (b, a) are positive, spatially bounded 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 distortion are, for example, defocusing of the lens of the optical imaging system, speed shift (blur) of the image due to the movement of the object during exposure, various diffraction restrictions (i.e., limitation of the spatial spectrum of the image by a 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 that distorts the image, then the 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 deconvolution is a problem arising in the absence of a priori information about their formation channel. The problem of blind correction of linear distortions of images in problems of remote sensing of the Earth, astronomy, and medicine is especially urgent.

Possibilities of blind identification of scalar two-dimensional channels are somewhat wider than that of one-dimensional ones. This circumstance has been noted more than once in the literature and has historically led to a more intensive introduction 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 identification of the channel by the modulus of the transfer function is possible only for a narrow class of systems with a minimum phase.

It is interesting that, generally speaking, this is not the case for discrete random fields. Those. for two-dimensional discrete signals, the possibilities of phase reconstruction by the modulus of the transfer function are much wider. This somewhat unexpected result was obtained by the method of mathematical modeling by Fienap in 1978. (see overview).

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 sufficiently 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 the polynomial into irreducible factors, we can restore the discrete signal from its autocorrelation or, which is equivalent, from its amplitude spectrum.

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

Consider a two-dimensional discrete convolution model:

The same relation can be written in the form of a product of polynomials of the ring C: y (z \, z2) = h (z 1> Z2MZ1> Z2) (21) where: y (21'22) = XX y (!> PU \ r2 ; ") = XX ^" K-r2; i / n

If the polynomials / 2 (21,22) and nr ^^) are irreducible in the ring C ^^], then factorizing ^ (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 by the presence of noise.

An algorithm of some practical importance and based on the property of irreducibility of polynomials (21) is known as the "zero sheet" algorithm was proposed in. The algorithm uses the properties of surfaces, the points of which are the roots of the channel polynomials and the true image. A conceptually similar algorithm was proposed in.

Some additional limitation of the field of application of this approach is the use of the assumption of the spatial limitation of signals.

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

One of the central problems in the practice of neural network applications, statistics, DSP tasks, 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 such problems, the method of finding a linear transformation that ensures the independence of the components - ANC - has attracted wide attention. The ANC problem is formulated as the problem of finding such a projection of a vector onto a linear vector space, the components of which would be statistically independent. In this case, only a certain statistical sample of the values ​​of a random vector is available for analysis. In this sense, the tasks and methods of ANC are related to the tasks and methods of SOS.

One of the promising directions in the development of modern ERS systems is synchronous imaging of the earth's surface in various ranges of the electromagnetic spectrum. Joint processing of multispectral optical images, multifrequency and multipolarized radar images, radiometric images, a promising area of ​​research and practical applications of recent times.

The development of technologies for the 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 signs of automatic classification of objects, to provide their visual representation (visualization), to reduce the amount of stored information. ANC methods can be a powerful tool for collaborative image analysis.

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

That. in tasks of digital image processing effective solution blind problem is in many cases a necessary, non-alternative stage of preliminary, primary processing, providing the possibility of subsequent analysis. In problems of joint analysis of images of different nature, methods of analysis of independent components can become an effective tool.

Biomedical computer technology is a classic application of ANC and methods of blind source separation.

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

A feature of the application of these methods is the need to separate the signals of the organs under study from noises of various origins and interfering signals (for example, separation of the cardiograms of the mother and the child).

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

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

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

In this case, the parameters of the sensor, as well as the parameters of the medium, vary enormously. Handsets vary in degree of distortion, spectral content, and signal strength. Microphones are made in a variety of ways and are positioned in different positions on the handset, with holes of different sizes, located at different points within the sound field around the mouth. A recognition device that works well for one specific sensor in one specific environment might work 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 reconstruct the original speech signal.

Reverberation is necessary when the original speech signal is distorted by the acoustics of the environment. the acoustics of the environment depend on the geometry and materials of the room and the location of the microphone.

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

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

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

In the tasks of geology, seismological studies, technologies are used to register signals from sources of mechanical vibrations, both artificial (laying dynamite in the pit) and natural (earthquake). These signals are used to estimate the reflection coefficients of various layers of the earth's crust.

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

That. The considered problems arising in various fields of radio engineering and communication, as well as in numerous other applications of signal processing, confirm the thesis about the relevance of the problem of developing new SOS methods, 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. Soifer, V.F. Kravchenko, D.D. Klovsky, V.I. Tikhonova., Yu.G. Sosulin, V.G. Repin, 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 Karjakin 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. 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 the SOS were prepared by numerous results in the field of adaptive methods for recovering space-time signals, including parametric methods for evaluating their 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. Potekhin, A.P. Reutova, Yu.A. Feoktistova, A.A. Kosty-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. and etc.

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

A. Well-varinen, A. Cichocki, S. Amari, J.-F. Cardoso, P. Comon, M. Rosenblatt, C. Ya. Shatskikh, S. A. Ayvazyan, L. D. Meshalkin, etc.

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

In addition, to provide the possibility of widespread implementation of SOS methods in radio engineering, it is necessary to create new SOS technologies, characterized by a high convergence rate, providing the possibility of blind identification in the absence of a priori information about the statistics of the information signal, providing the possibility of identifying a non-stationary channel and non-stationary information signals.

A new class of SOS methods potentially providing 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 to polynomial rings in many variables with random coefficients and use the methods of commutative algebra, algebraic geometry, and computer algebra that 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 corresponding 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-kom, O. Zariskiy and others.

Goals and objectives of the study. The aim of the thesis is to develop theoretical foundations, methods and algorithms for blind signal processing and their application in some problems of radio engineering, communication, 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 radar images of SAR, 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 blind signal processing methods formulated in this work require the creation of a new mathematical apparatus based on the compilation of methods of probability theory, commutative algebra and algebraic geometry. In addition, the use of classical methods of probability theory, statistical radio engineering, numerical methods, methods of computer simulation and computer algebra.

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

The 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 nonzero correlation are defined;

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

A number of algorithms for blind identification of a scalar channel with a non-stationary input according to second-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 nonstationarity 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, as a problem of solving a system of polynomial equations in many variables, are determined;

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

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

Algorithms for blind identification have been developed, based on the properties of symmetric polynomial cumulants, observed signals;

The problem of identifying a vector channel in a polynomial interpretation is considered, the main theorems of identifiability are proved, a polynomial interpretation of the method of mutual relations (BO) is proposed - the zero subspace algorithm (ANP), expressions of the relative error of identification are obtained, a comparison is made with other methods;

The possibilities of using the developed methods of blind identification in radio engineering systems for transmitting information are considered; 2nd order statisticians;

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

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

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 the spaceborne 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 bands;

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

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

Fast algorithms for the formation of SAR images, based on the use of the rotation vector technique;

Algorithm for nonlinear analysis of independent components based on nonlinear transformation of independence and kernel estimates of integral functions of multivariate 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 of 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 FSUE Research Institute "Vector" (St. Petersburg) in 2002-2003

The results of the research and development carried out are part of a number of research and development work carried out at FSUE GNP RKTs "TsSKB-PROGRESS" (Samara) to create radar space and aircraft ERS systems in 1988-2000. (ROC for the creation of space systems "Sapphire-S", "Resource-Spectrum", "Resurs-DK", research work "Elnik-UN", "Mirror").

The research results were used at FSUE TsNIIMASH (Moscow) to substantiate 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 the formation of requirements for the promising dual-purpose radar surveillance space system "Arkon-2".

The developed algorithms and programs for blind identification of the radar channel were used at FSUE NII TP (Moscow) in the preparation of aircraft tests and the processing of radar data of the IK-VR aviation radar complex in 1994-1995, as well as in the analysis of the influence of the atmosphere and forecast accuracy for the resolution of space SAR 14V201 for the 17F117 spacecraft, Luch-M for the Resurs-DK-R1 spacecraft.

The results of the work have found application in the educational process at GOUVPO PGATI, in particular in the courses of lectures "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 the relevant implementation documents.

1. BASIC BLIND IDENTIFICATION THEOREM

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 complex enough; there should be enough output counts available.

2. The conditions for the statistically identifiable deterministic vector channel can be discussed in a broader context. For example, if the number of available samples at the channel output 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 a stationary random process (including a Gaussian) at the input can be identified, the system can be identified if the statistics of the second order of the output are known exactly and the joint zeros of the channel polynomials are inside the unit circle (the phase minimum condition).

3. Both in the case of deterministic and statistical identification of a vector channel, for the channel identification it is necessary or sufficient the absence of common roots for the polynomials (r). This means that channel cross-links are used explicitly or implicitly to identify a vector channel.

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

5. The rigid 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 these methods.

6. For the statistical identifiability of the 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 using algorithms for solving a system of homogeneous equations to solve the variational problem of the method of minimal squares.

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

9. The values ​​of the formal variables.

10. The choice of the values ​​of the formal variables = exp (-y "2m / M), 1 = \,., M, and r ^ = exp (-j2m / r"), / = 0,.,? - 1 under the condition ? = r "= r provide the minimum value of the relative error of the channel estimation, when? = r" Ф г 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 the parameters of additive noise in different subchannels, correlation of noise samples, the choice of cross sections should be carried out by minimizing the right side of (2.24).

11. The relative error of 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. With an increase in the channel length, the error grows linearly; however, with an increase in the number of channels for large ratios, the signal-to-noise ratio of the channel length practically does not affect the magnitude of the error.

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

13. If at the input there is a random process non-stationary in terms of the mean value, and = where x "(/) is a stationary process with zero mathematical expectation, then the channel is identified by statistics of the 1st order;

14. If at the input there is a non-stationary in terms of variance random process = where is a stationary process with zero m.o. and then the channel is identified by statistics of the 2nd order;

15. If at the input х (?) Is a random process with a non-stationary frequency structure in time, i.e. = - where х "(() is a stationary process with zero mathematical expectation and //" (?)> О, then the channel is identified by statistics of the 2nd order;

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

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

18. For a non-stationary input signal in terms of 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 the two-diagonal blind identification algorithm), and to obtain the estimate, no prior knowledge of the statistical characteristics of the information signal is required;

20. The error in estimating the transfer function for 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 used estimation algorithm and the type of non-stationarity;

21. Polynomial representation of discrete random signals of finite length allows you 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, affine manifolds generated by polynomial cumulants (called manifolds of nonzero correlation) have a number of unique properties, namely, the dimension that 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, from unknown channel coefficients. Choosing a set of polynomial cumulants corresponding to the specifics of the problem, we can synthesize the corresponding identification algorithm. At the same time, the proposed approach to the synthesis of blind identification algorithms based on polynomial statistics allows one to synthesize various blind identification algorithms for scalar channels with stationary and non-stationary inputs, various distributions of input symbols. In contrast to the approach based on polyspectra, in this case, the uncertainty in the choice of a set of cumulant functions can be reduced, at least with respect to the procedure for synthesizing the algorithm.

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 build 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 longer than the channel length. At the same time, the quality of the assessment approaches the assessment by the test signal.

25. The blind identification algorithm based on the properties of manifolds of zero correlation, using the non-stationary channel model, allows separating 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 about two orders of magnitude fewer implementations, but has a lower noise immunity. In addition, the algorithm error increases significantly with increasing channel length.

26. The blind channel identification algorithm based on the use of nonzero correlation manifolds, in contrast to the blind identification algorithm based on the factorization of affine manifolds, has a sufficiently high convergence rate, providing high quality estimates even at a signal-to-noise ratio of 15-20D6. However, when constructing a nonzero pair correlation transformation, we need to know 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 rather promising technology for channel equalization in serial communication systems in scattered channels. The analysis shows that if we consider the blind assessment as an alternative to the test pulse, then the latter almost always wins in terms of convergence rate and noise immunity, but the blind assessment always wins in transmission speed.

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

30. The answer to the question: "to use blind channel estimation in each specific case or not?" Requires a compromise solution from the developer of the communication system.

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

32. The influence of trajectory and especially atmospheric errors leads to a significant limitation of the spatial resolution of spaceborne SARs, while the degree of degradation increases 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 problem of obtaining a radar image under conditions of a strong influence of trajectory and atmospheric errors as the main problem limiting the development of spaceborne 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 distortions of radar images.

33. The influence of the atmosphere on the resolution of the SAR begins to affect already, starting from 10 cm, and significantly increases from 23 cm. In the long-wavelength range (> 70 cm), the degradation of radar images in spatial resolution with a disturbed ionosphere can reach two orders of magnitude. Moreover, in this range, the resolving power is practically independent of the resolving power without taking into account the destructive influence of the atmosphere and is determined mainly by the effective coherence interval, which in turn is determined exclusively by the parameters of the atmosphere. The degree of degradation increases with an increase in flight altitude, and especially with an increase in 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 the resolution of the SAR in range can be carried out using a two-diagonal blind identification algorithm using sign correlation.

35. Compensation for the effects of degradation of the resolution of SAR 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 readouts of the SAR signal in the basis of the rotation vectors.

36. The proposed ANC method, using an independence transformation based on a kernel estimate of a 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 a single algorithm.

37. The possibility of constructing an independence transformation of an n-dimensional random vector using paired independence transformations for non-Gaussian random vectors significantly expands the scope of this approach. The ANC algorithm described in this section can be used in the tasks 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 the mechanism for converting the information signal into the observable. signal is unknown.

CONCLUSION

The result of the thesis is the development of theoretical foundations, methods and algorithms for blind signal processing and their application in some problems of radio engineering, communication, 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 of SOS problem solving based on polynomial representations of discrete signals has been developed;

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

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

Possibilities have been investigated and algorithms have been developed for blind correction of diffraction distortions of radar probing signals when reflected from spatially distributed targets;

Methods and algorithms for blind reconstruction of SAR radar images in 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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