Regularization of Inverse Signal Recovery Problems

Author Name(s): Evgeny G. Zhilyakov, Sergei P. Belov, Ivan I. Oleinik, Ekaterina I. Prokhorenko
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The compensation for the distortions that occur during their registration is called the recovery of signals. The main problem studied in the literature review of this paper is the instability of the calculated estimations of input actions regarding the effects of feedback registration errors. Therefore, various techniques have been developed for regularizing the original equations on the basis of converting them into another equation, whose solution is calculated stably. The most famous technique is the regularization method developed by A. N. Tikhonov. At the same time, this paper shows that the response may lack some information about the input effect, that is, even in the absence of measurement errors, the resulting solution will be approximate. A method is proposed for estimating the nonrecoverable distortions caused by recording system operator itself, which can be used at the stage of its synthesis. A linear form of representation of an impact component accessible to recovery through an impulse response is obtained so that the recovery task is reduced to the calculation of its coefficients. A method of regularization of systems of linear algebraic equations arising ,in this case, is proposed on the basis of adaptive estimation of registration error levels directly from the registered response.


In the framework of this work, a signal is a function of time, the parameters of which contain some information about real processes or phenomena. Such signals are, in particular, channel signals of information transmission systems, responses to probing effects in radar, input influences in information-measuring systems, etc. Real signals arrive at the inputs of some systems, and manifest themselves in the form of responses at their outputs. In many cases, an integral model of signal conversion in recording systems is the integral relation of the convolution type [1-3].


In the present paper, the problem of sustainable signal recovery in linear systems with constant parameters is considered. It is shown that in the response of the system, some of the information about the input effect may be missing, and a method has been developed for the priori analysis of the component available for recovery. A projection method for the stable restoration of signals based on singular decomposition of matrices is proposed. The basic relations have been obtained, which make it possible to regularize recovery problem and calculate the regularization parameter directly from the measurement data.

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