# Method and apparatus for approximating signals

FIELD: physics, computer engineering.

SUBSTANCE: invention relates to digital signal processing and information measuring equipment and can be used to linearise the transform function of units and systems, as well as interpolation and extrapolation of measurement results. The apparatus has an analogue-to-digital converter, an integrator, a unit for calculating sequences of quotients and differences and a unit for calculating parameters of the approximating unit, connected in series, as well as a unit for setting integration intervals, the output of which is connected to the control input of the integrator, and a threshold circuit whose input is connected to the output of the unit for calculating sequences of quotients and differences, and the output is connected to the control input of the unit for calculating sequences of quotients and differences.

EFFECT: high accuracy of approximation.

3 cl, 5 dwg

The invention relates to the field of digital signal processing and information-measuring devices and can be used for linearization of the transformation function blocks and systems, interpolation and extrapolation of the measurement results, the compression and recovery of signals, and also for measuring parameters of complex waveforms, namely the parameters of the oscillatory component of the signal.

Well-known and widely used methods and implement their devices for polynomial, spline and Fourier-signal approximation [1, 2, 3]. Their main disadvantage is the low accuracy of approximation of the alternating signals and a significant effect of random noise is inevitably superimposed on the signal and noise quantization (digitization) on the results obtained. There is also known a method of approximation by linear combinations of exponentials, based on the method of least squares [3], and device for its implementation [4]. The disadvantage of this method is the complexity of calculating the parameters of the approximating unit that hinder its implementation on the microcontroller, and the use of mobile systems in real time.

Known method of approximating the closest in technical essence is the interpolation method of exponential sums, based on the algorithm of private and different the TEI (Rutishauser, The algorithm private and differences. - M.: Publishing house of foreign. LEATER. 1960. - 93 S. [5], pp.54-60).

In this method, set the number of n exponentials used for approximation of continuous function discretizing with a constant sampling interval h, equal to the length of the interval of approximation, divided by 2n-1, and calculate based on the algorithm of private and differences obtained by the discrete values of the parameters of the Exhibitor.

In the method prototype for approximation of continuous functions F(t) is the sum of exponents (the final sum Dirichlet)

coinciding with the function F(t) in 2n points t_{ν}=t_{0}+νh, ν=0, 1,..., 2n-1, defined with a constant sampling interval h. The problem is reduced to the representation as a sum of simple fractions rational functions f(z)represents the z-transform of the original function F(t), for which

ν=0, 1,..., 2n-1.

Then [5, p.54]

where,.

In figure 1, 2, 3 and 4 shows the form of a separate component of the approximation function F(t_{ν}) for different types α_{k}. Because the source data for the approximation of values of the function F(t) are real numbers, then*a*_{k}and α_{k}can be real numbers, and to meet complex-conjugate pair and*
a*_{k},and α_{k},. If α_{k}=0, the component is a DC voltage, depicted in figure 1. If α_{k}- a valid number, then the component is an exponent, shown in figure 2. If the real part of the complex conjugate pair α_{k},equal to zero, then the component is a harmonic oscillation, shown in figure 3. If the real part of the complex conjugate pair α_{k},not equal to zero, then the corresponding component of the signal is a damped harmonic oscillation, shown in figure 4. *a*_{k}affects only the phase of the harmonic oscillations.

The known method implements the following steps.

1. On the basis of a priori known character functions and the required accuracy of approximation, set the order of the approximation model, i.e. the number n of exponents used for approximation.

2. A continuous function F(t) discretizing with a constant sampling interval h, that is, represent it by a sequence ofcontaining 2n discrete digital values F(t_{ν})=s_{ν}.

3. Apply the algorithm of private and differences to define the parameters of the exposure the element as follows.

3.1. For values of s_{ν}calculate privateand the differenceand σ=1, 2,..., c using rules of diamond [5, s]

provided that, ν=0, 1,...,2n, and all numberszero.

3.2. Findandand allput equal to zero.

Its polesthe function f(z) found in item 3.2 sequences using progressive forms of private algorithm and differences [5, p.37-41,43-53], considering allzero

3.4. Find the results PP and 3.3 factorsin the decomposition of the function f(z) on simple fractions using modifications of the algorithm private and differences [5, p.28-31, p.57-60].

4. Calculated by the obtained values ofandthese parameters approximating unit α_{k}=h^{-1}ln(λ_{k}and, k=1,..., n.

In the result, an approximate function is described by a linear combination of exponentials, in 2n points coincides with the original continuous function F(t).

The disadvantage of the prototype method in its practical realization I had is the low accuracy of approximation signals, due to the fact that the number of samples approximated signal is fixed a priori. When a small number of samples, the error of approximation in the intermediate points of the original function F(t)describing the signal is large and with a large number of samples occurs complexity of the calculations and shows the instability of the algorithm private and differences. In addition, the actual signals are inevitably accompanied by superimposed noise and digitization of signals occur, the quantization noise. It also leads to less accurate approximation, as an approximate function is the same as in the point approximation signal with superimposed noise, and in some cases to instability of the algorithm and the impossibility of approximation.

The aim of the invention is to improve the accuracy of approximation in the practical implementation of the method for real signals.

This goal is achieved by the fact that in the method of approximation of signals of finite linear combination of exponents, including sampling, digitizing a signal, and a calculation according to the received parameter values, the Exhibitor based on the algorithm of private and differences, pre-produce a splitting of the discrete samples of the signal at M blocks of equal duration, carry out numerical integration of the signal in each block of vechicle the t parameters of the Exhibitor, approximating the integrated signal, calculate the parameters of the Exhibitor, approximating the signal, the number of blocks M is chosen equal to or greater than twice the number of exponents used for the approximation, and the required number of Exhibitor determine a priori and precise in execution algorithm private and differences.

In the present method for approximation of a continuous signal F(t) also used the sum of the Exhibitor (the final sum Dirichlet)

match the original signal F(t) in 2n points t_{ν}=t_{0}+νh, ν=0, 1,..., 2n-1, defined with a constant sampling interval h. For this purpose, the signal F(t) discretizing and then digitizes, presenting its N samples, signal F(t) at time t_{0}, t_{1},..., t_{N-1}, t_{ν}=t_{0}+νh, ν=0, 1,..., N-1 and N>>2n. The samples of the signal is divided into M identical blocks,whose number is not less than twice the number of exponents, and N-1=Mm. Integrating P(t) on the interval [t_{0}+µmh, t_{0}+(µ+1)mh], where µ=0,..., M-1, gives

µ=0,..., M-1.

That is, the original sampling interval h is replaced by a large interval of sampling mh.

These samples the integrated signal find the z-transform of this signal.

where,.

The application of the algorithm private and differences to the sequenceallows you to calculate the polesand oddsdecomposition into simple fractions, which define the parameters of the approximating unit

and, k=1,..., r.

The number r of exponents used for approximation, does not exceed the integer part of M/2 and is in the process of execution of the algorithm private and differences of conditions:less pre-specified margin of error.

The proposed method implements the following steps.

1. Continuous signal F(t) discretizing and digitize with a constant sampling interval h, then there are N samplesat time t_{0}, t_{1},..., t_{N-1}, t_{ν}=t_{0}+νh, ν=0, 1,..., N-1, and the number of samples N lot more than twice the number required for approximation of the Exhibitor.

2. On the basis of a priori known the nature of the signal, the required accuracy of approximation and the number of samples N, choose a number M, which is not less than twice the number of exponentials required for AP is roximate.

3. The samples of the signal is divided into M identical blocks, duration m

,and N-1=Mm.

4. Carry out numerical integration of the sample signal in each of the M blocks and get a new sequence of integrated values of.

5. Apply the algorithm of private and differences to define the parameters of the Exhibitor as follows.

5.1. Valuescalculate privateand the difference, k=1, 2,..., using the rules of a rhombus

provided that, µ=0, 1..., M-1 and all numberszero.

5.2. Findandwhere r does not exceed the integer part of M/2 and is in the process of execution of the algorithm private and differences of conditions:less pre-specified margin of error. The number r limits the number of exponentials needed to approximate the signal with a given error.

5.3. Supposeequal to zero and is found in paragraph 5.2 sequencesanddetermine the poles .

5.4. Find the results PP and 5.3 factorsin the decomposition functionsin simplest fraction.

6. Calculated by the obtained values ofandthese parameters approximating unitandk=1,..., r.

6.1. When measuring the parameters of the oscillatory component of the signal frequency f_{k}[Hz], the attenuation coefficients β_{k}[with^{-1}], the amplitude A_{k}[In] and the initial phase θ_{k}[happy] is calculated by the obtained values of α_{k}and*a*_{k}accordingly:

f_{k}=(2π)^{-1}Im(α_{k}), β_{k}=Re(α_{k}),

,

or directly by the values ofand:

Technical result - increase the accuracy of the approximation of signals is achieved by integrating the discrete samples of the signal within the M blocks, with random noises are averaged, and also due to the choice of the number of exponentials used for approximation, directly in the process of implementation of the method on the basis of a valid POG is enosti.

The technical result is also achieved through the use of a new device for implementing the inventive method approximation signal containing analog-to-digital Converter, an integrator, a computing unit sequences and private differences and the computing unit parameters approximating unit series-connected, and the unit interval of integration, the output of which is connected to the control input of the integrator and threshold circuit, the input of which is connected to the output of the computing unit sequences and private differences, and the output is connected to the control input of the computing unit sequences private and differences.

Figure 5 shows a functional diagram of a device that implements the inventive method of approximation signals.

Device for signal approximation contains analog-to-digital Converter 1, the integrator 2, block 3 calculation sequences and private differences, block 4 calculation of the parameters α_{k}and*a*_{k}approximating unit, block 5 intervals of integration and threshold circuit 6. Analog-to-digital Converter 1, the integrator 2, block 3 calculation sequences and private differences and unit 4 calculation of the parameters α_{k}and*a*_{k}approximating unit are connected in series. Output b is the eye 5 of the interval of integration is connected with the control input of the integrator 2.
The input of threshold circuit 6 is connected to the output unit 3 calculation sequences and private differences, and its output to the control unit 3 compute sequences of private and differences.

All the elements included in the device, can be implemented as separate functional units, such as blocks 2-6 on the programmable logic integrated circuits (FPGA) or programmatically using a microcontroller equipped with an analog-to-digital Converter.

A device for signal approximation as follows. Analog-to-digital Converter 1 performs sampling and digitizing approximated continuous signal F(t), converting it into a sequence of N digital samplesat time t_{0}, t_{1},..., t_{N-1}, t_{ν}=t_{0}+νh, ν=0, 1,..., N-1. The sampling frequency is high and the capability of the analog-to-digital Converter 1. In the integrator 2 is performed numerical integration of the sample signal at time intervals M, defined by block 5 of the interval of integration. The number of intervals M is set to no less than double the number of exponentials required to approximate the signal with a given accuracy, and depends on the expected nature of the signal F(t.
The sequence of the integrated values of signalwhere µ=0,..., M-1, enters the block 3 calculation sequences and private differences, where using the algorithm of the private and differences calculate privateand the difference, k=1, 2,..., which are sequencesand. In the process of computing these sequences each regular value ofcompares the threshold circuit 6 with a prescribed accuracy ε. If the next value isbecomes less than ε, the calculation is terminated. If this condition is not achieved, then the computation stops when the number of members of the sequence becomes equal to the integer part of M/2. In block 4 parameter calculation approximating unit values,determine the polesfind the coefficientsin the decomposition functionsfor simple fractions and calculate the desired parameters of the approximating unit by the following formulas

and, k=1,..., r.

Modeling is known and before aguinaga methods of approximation in the environment "MAPLE" revealed when using the exact values of the signal standard deviation of the approximating function from the signal for both ways does not exceed a few hundredths of a percent. When using the quantized signal values to four decimal places as the source of data to be fit in the known method shows the instability of the algorithm and in some cases the approximation is not possible, while in the proposed method, the standard deviation of the approximating function from signal does not exceed a tenth of a percent. When applying on the approximated signal is normal noise with signal-to-noise ratio up to 500 and use of digitized signal values to four decimal places as the source data for the approximation of the known method is practically stop working, while the proposed method retains the same value of the standard deviation of the approximating function from the signal. There is a method of approximation does not work if the number of samples exceeds the number of points in the approximation.

The present invention can be applied to solve various tasks of information-measuring devices. The parameters of the approximating unit α_{k}and*a*_{k}can be applied for calculations of correction factors when Lina is the polarization conversion functions.
In compression-recovery of signals by telemetry channels appropriate to transfer or sequence of private and differences,or the parameters of the approximating unit α_{k}and*a*_{k}. The amount of information transmitted is not changed. The restoration of the signal can be performed for the parameters of the approximating unit α_{k}and*a*_{k}calculated at the receiving side, and the parameters (frequency, amplitude, initial phase and attenuation coefficients) of the oscillatory component of the signal.

Sources used

1. Popov, B.A., Tesler G.S. Calculation functions on the computer. The Handbook. - Kiev: Naukova Dumka, 1984. - 600 C.

2. Patent RU 2010324, G06F 15/31. Device for approximation of functions. Publ. 30.03.1994,

3. Marple.-Jr. S.L. Digital spectral analysis and its applications: TRANS. from English. - M.: Mir, 1990. - 584 S.

4. Ions SV, Myasnikova MG, Tsypin BV, Shapoval ACTING Modular system for measuring parameters and spectral analysis of weak signals // Measurement techniques. - 2011. No. 4.

5. Rutishauser, the Algorithm private and differences. - M.: Publishing house of foreign. letters., 1960. - 93 S.

1. The method of approximation of signals of finite linear combination of exponents, including sampling, digitizing the signal and computation on the resulting values of the parameters of the Exhibitor on the basis of the algorithm of private and differences, wherein the pre-produce a splitting of the discrete samples of the signal at M blocks of equal duration, carry out numerical integration of the signal in each block, compute the parameters of the Exhibitor, approximating the integrated signal, and they calculate the parameters of the Exhibitor, approximating the signal.

2. The method according to claim 1, characterized in that the number of blocks M is chosen equal to or greater required for approximating twice the number of exponents, and the required number of exponents determined during execution of the algorithm private and differences.

3. Device for approximation signal containing analog-to-digital Converter, an integrator, a computing unit sequences and private differences and the computing unit parameters approximating unit series-connected, and the unit interval of integration, the output of which is connected to the control input of the integrator and threshold circuit, the input of which is connected to the output of the computing unit sequences and private differences, and the output is connected to the control input of the computing unit sequences and private differences.

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