mathematical model of an existential filtering
The circuit design of an existential filtering of a signal in SOASZI is resulted in drawing 4.13.
The problem consists in definition of values Wij (t).vyhod SOASZI is directly connected with sampling of a criteriaon of performance of functioning of system.
Primary goal OEU MPI, entering in SOASZI, measurement of any parametre of the signal coming on an entry of system, and a signal reproduction (on one or its several parametres) with its maximum accuracy [217] is.Criteriaon of performance of such system the average quadratic lapse of measurement (estimation) of the reproduced parametre of a signal, for example, its time or attitude is considered, amplitudes and so on. Systems which provide the minimum average quadratic lapse, are called as optimum detecting filters.
Drawing 4.13 - the Circuit design of an existential filtering of a signal in SOASZI: S (t) - a legitimate signal vector on elements OEU MPI (such, that S (t) = (S ∏ (t)... S ∏ (t), S21 (t)... S2 ι (t)..., Sml (t)... Sm ι (t))), ∙ V (t) - a vector of external handicapes on elements OEU (such, that V τ (t) = (Vn (t)... Vn (t).V21 (t)... V2 ι (t)., Vrn1 (t). Vmi (t)));
W (t) - a vector vnutripikselnyh coefficients of amplification (such, that W τ (t) = (W11 (t). W1l (t). W21 (t). W2 ι (t) Wm1 (t). Wmi (t))); y (t) - exit OEU;
d (t) - a reference signal
At use of criterion of an average quadratic deviation (SKO) exit SOASZI is set by a reference signal d (t) (drawing 4.14).
Drawing 4.14 - the Circuit design of definition SKO of a reproduction system
The difference between a reference signal and a real exit y (t) is called as an error signal and is equal its square is equal
And the mathematical expectation is expressed as
Reference signal sampling d (t) is the independent problem which solution in each separate case considerably influences quality of functioning OEU, a part SOASZI.
At synthesis of blocks of formation of images OEU which is carrying out tune of weight numbers for each element SOASZI, use of various criteria of efficiency, thus optimum solutions is is possible will be connected among themselves through so-called optimum vinerovskoe the solution [218,219]. Therefore sampling of concrete criterion of efficiency usually is not of great importance. More important is sampling of the control algorithm intended for realisation of an existential filtering as it directly influences both speed of transients, and on complexity of technical implementation SOASZI.
Further we will observe process of a sizing up and a filtering in sampled-data systems. Systems with discrete time are presented by the difference equations of an aspect where k - discrete readout of a time; W (k) - state vector from nкомпонент; τ - a digitization step according to Euler's difference circuit design; f (W (k), k) - some function.
For the linear determined installations expression (4.80) we will write down:
Where Φ (k+1, k) - a transitive matrix of a condition of a size n? n.
If on system the casual signal co (k) in sampled-data systems its statistical characteristics look like acts
And then stochastic systems of discrete type will register in an aspect:
Processes of an aspect (4.84) are discrete in a time markovskimi sequences.
If there is still a quantization and on level, that is W (k) can accept values only from discrete assemblage of possible conditions, at hit of value W (k) in intervalemu on the defined
To rule one of values in the interval (for example, its bottom boundary line) is put in conformity.
The average quadratic deviation of sampled-data systems looks like
At a filtering of handicapes in such systems methods of the theory of automatic steering of criterion of efficiency Jслужит an average square of an error between unknown optimum weight vector Wopt (k) and adjusted vector W ():
Thus the reference signal participates in process of construction of an estimation for formation of observations of dynamic model of optimum weight numbers.
Let
Where ξ () - measurement noise, that is internal noise of system (OEU MPI and the block of formation of images) which is approximated by sequence gaussovskih nekorrelirovannyh with a zero average and the set dispersion
Expression (4. 88) it is fair owing to that adjusted on optimum reception OEU MPI should provide the "desirable" exit set in the form of a reference signal on an exit, and in the presence of the noise of measurement originating owing to act of internal noise OEU, it is possible to write down as where u (k) - some random process with an unknown dispersion.
Taking into account (4.89) expression (4.88) will register as follows:
Thus, at implementation of the digital actuation device working with discrete samples of continuous process, the installation registers in the form of discrete in a time of stochastic linear process G (k), Q (k) - the dispersive matrixes which elements characterise extent of indeterminate form of conditions of reception; δks - a Kronecher's simbol.
Accordingly the system exit is presented as
The problem consists in search of a vector of estimations W () for vector Wopt (k), such which would minimise funktsional (4.85) or (4.86). Thus vector Wopt (k) contains a set of optimum values of elements of vector W (k), matching to any concrete conditions of reception OEU MPI of a mix of a legitimate signal and external handicapes, and change of conditions of signal detection calls change necessary for qualitative reception of a legitimate signal of values of elements of a matrix of optimum coefficients of amplification.
Let's muster a condition of observability of such linear non-stationary system with discrete time.
Let's write down a matrix of criterion of observability [220]:
Such matrix can have a rank rank (H) 11 [221]:
4.7.2.
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