CN103226542A - Method for simulating wavelet base frequency domain approximation - Google Patents

Method for simulating wavelet base frequency domain approximation Download PDF

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CN103226542A
CN103226542A CN2013101649170A CN201310164917A CN103226542A CN 103226542 A CN103226542 A CN 103226542A CN 2013101649170 A CN2013101649170 A CN 2013101649170A CN 201310164917 A CN201310164917 A CN 201310164917A CN 103226542 A CN103226542 A CN 103226542A
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wavelet basis
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何怡刚
童耀南
尹柏强
于文新
龙英
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Hefei University of Technology
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Abstract

A method for simulating wavelet base frequency domain approximation comprises the following steps: (1) achieving reversion, time shifting, integration and Fourier transform of a wavelet base function to obtain a wavelet base step response function; (2) solving the magnitude spectrum and the phase spectrum of the wavelet base step response function; (3) solving a wavelet base step response rational approximation function by adopting the plural rational approximation method; and (4) differentiating the wavelet base step response rational approximation function to solve a wavelet base rational approximation function. Compared with the conventional Pade approximation method and the Maclaurin series approximation method, for comprehensive approximation of the magnitude spectrum and the phase spectrum of the wavelet base step response function, the defect that the conventional frequency domain approximation method only considers the magnitude spectrum but not focus on the phase spectrum is avoided, the approximation precisions of the wavelet base at both the time domain and the frequency domain are effectively improved, and the stability of a wavelet filter is guaranteed. The high-precision wavelet base rational approximation function obtained by adopting the method is suitable for being further realized by an analogue circuit.

Description

A kind of simulation wavelet basis frequency domain approach method
Technical field
The present invention relates to a kind of simulation wavelet basis frequency domain approach method.
Background technology
Wavelet transformation is with its good time-frequency local characteristics, is widely used in non-stationary and transient signal and handles, and now become one of efficient mathematical instrument of each engineering field signal Processing.Wavelet transformation can be realized also can realizing with hardware with software.Realize wavelet transformation with hardware, particularly realize,, more and more come into one's own owing to have low in energy consumption, fireballing advantage with mimic channel.Mimic channel realization wavelet transformation can be regarded the linear combination of the yardstick wavelet filter different with displacement as, and therefore, the design of research wavelet filter has most important theories meaning and engineering actual value.The wavelet filter design has two critical step, and the one, the reasonable of wavelet basis function approaches, and the 2nd, the circuit design of wavelet filter.Wavelet basis function reasonable approaches and comes down to seek stable, an available circuit system function that realize, rational form, we wish no matter still be frequency domain, no matter be amplitude spectrum or phase spectrum that in time domain this system function is all similar as far as possible to wavelet basis function.The reasonable approach method of the wavelet basis of having reported can be divided into time domain and approach with frequency domain and approach two classes.The frequency domain approximatioss typically has Pade method and Maclaurin series approximatioss.For example: document " the Switched-Current Circuit design and the realization of wavelet filter " (Chinese journal of scientific instrument, the 27th the 9th phase of volume) and document " based on the filter circuit realization of the wavelet transformation of switched current technique " (Acta Physica Sinica, the 55th the 2nd phase of volume) adopt the Pade method to ask for the frequency domain transfer function of wavelet filter, but the Pade method exists approximation accuracy poor, can not guarantee the stability of approaching automatically.The patent No. is that 201110298934.4 patent of invention discloses " a kind of wavelet filter method for designing ", it adopts the Maclaurin series method to carry out approaching of wavelet basis, the Maclaurin series approximatioss can not guarantee the stability of approaching equally, be that 2 Gauss wavelet base carries out five rank when approaching at the time shift amount for example, the Gauss wavelet base that approaches is unstable.Above-mentioned frequency domain approach method all is to approach at the wavelet basis impulse response in frequency field, only considered approaching of amplitude spectrum, do not consider approaching of phase spectrum, more seriously the above-mentioned frequency domain approach method of event exists approximation accuracy poor, and can not guarantee that the reasonable approximating function of wavelet basis has the defective of stability.
Summary of the invention
The technical problem to be solved in the present invention is, overcome the above-mentioned defective that exists in the prior art, a kind of reasonable approximating function of frequency field wavelet basis step response of asking earlier is provided, obtains wavelet basis simulation wavelet basis frequency domain approach method reasonable approximating function, that comprehensively approaching aspect amplitude spectrum and the phase spectrum again by differentiating.
The technical solution adopted for the present invention to solve the technical problems is:
A kind of simulation wavelet basis frequency domain approach method may further comprise the steps: (1) to wavelet basis function overturn, time shift, integration and Fourier transform, obtain the wavelet basis step response functions; (2) ask the amplitude spectrum and the phase spectrum of wavelet basis step response functions; (3) adopt plural reasonable approximatioss, try to achieve the reasonable approximating function of wavelet basis step response; (4) the reasonable approximating function of wavelet basis step response is carried out differential, try to achieve the reasonable approximating function of wavelet basis.
In the step (1), described wavelet basis function is carried out the necessity that time shift handles is because common wavelet function is a non-causal, can not directly comprehensively realize by hardware circuit, makes wavelet function have causality so handle by time shift.
In the step (1), described wavelet basis function is carried out that integration and Fourier transform handle is the wavelet basis step response functions that is converted into frequency domain for the wavelet basis function with time domain.
In the step (2), described amplitude spectrum and the phase spectrum of asking the wavelet basis step response functions is in order to adopt plural reasonable approximation theory, can both obtain to approach preferably effect aspect the amplitude of frequency domain and the phase place.
In the step (3), the plural reasonable approximatioss of described employing is tried to achieve the reasonable approximating function of wavelet basis step response and is meant: the wavelet basis step response functions is being approached aspect amplitude and the phase place by the reasonable approach method of plural number, approaching the result is a rational expression, reasonable the approaching of wavelet basis step response functions is that a transitionality is approached, and final purpose is to approach in order to obtain the reasonable of wavelet basis impulse response.
In the step (4), described the reasonable approximating function of wavelet basis step response is carried out differential, the principle and the particular content of trying to achieve the reasonable approximating function of wavelet basis are: according to the Signals ﹠ Systems theory, the impulse response of system is the differential of its step response, so on the basis of trying to achieve the reasonable approximating function of wavelet basis step response, only need carry out derivation operation just can obtain the reasonable approximating function of wavelet basis impulse response, i.e. the reasonable approximating function of wavelet basis.
Principle of the present invention is: at first, to wavelet basis function overturn, time shift, integration and Fourier transform, obtain the wavelet basis step response functions, try to achieve the amplitude spectrum and the phase spectrum of wavelet basis step response then, then, adopt plural reasonable approximatioss, try to achieve the reasonable approximating function of wavelet basis step response, last, according to the Signals ﹠ Systems theory, the reasonable approximating function of wavelet basis step response is carried out differential, try to achieve the reasonable approximating function of wavelet basis.Carrying out reasonable reason of approaching at wavelet basis step response (existing method be at the small echo impulse response) is that step response is approached to approach than the impulse response and had more robustness, and after the reasonable approximating function that obtains wavelet basis step response, only need simply differentiate and to obtain the reasonable approximating function of wavelet basis impulse response, be i.e. the reasonable approximating function of wavelet basis.
The invention has the beneficial effects as follows: because the present invention has taken all factors into consideration approaching of amplitude spectrum and phase spectrum comprehensively, can effectively improve the approximation accuracy of wavelet basis, guarantee the stability of wavelet filter at time domain, frequency domain.Compare with the Maclaurin series approximatioss with existing P ade approximatioss, the present invention approaches at wavelet basis step response (existing method is at impulse response) in frequency field comprehensively, has taken all factors into consideration amplitude spectrum and phase spectrum; Can effectively improve approximation accuracy, guarantee the stability of the reasonable approximating function of wavelet basis, and improve only consideration amplitude of existing method and approached and do not pay close attention to the defective that phase place is approached.The reasonable approximating function of high precision wavelet basis that the present invention obtains is applicable to further to be realized with mimic channel.
Description of drawings
Fig. 1 is the invention process FB(flow block);
The Gauss wavelet base step response frequency domain effect figure that Fig. 2 approaches for the present invention:
Fig. 2 (a) is desirable Gauss wavelet base step response amplitude spectrum;
Fig. 2 (b) is desirable Gauss wavelet base step response phase spectrum;
The Gauss wavelet base step response amplitude spectrum that Fig. 2 (c) approaches for the present invention;
The Gauss wavelet base step response phase spectrum that Fig. 2 (d) approaches for the present invention;
The Gauss wavelet base impulse response frequency domain effect figure that Fig. 3 approaches for the present invention:
Fig. 3 (a) is desirable Gauss wavelet base impulse response amplitude spectrum;
Fig. 3 (b) is desirable Gauss wavelet base impulse response phase spectrum;
The Gauss wavelet base impulse response amplitude spectrum that Fig. 3 (c) approaches for the present invention;
The Gauss wavelet base impulse response phase spectrum that Fig. 3 (d) approaches for the present invention;
The Gauss wavelet base step response time domain design sketch that Fig. 4 approaches for the present invention;
Fig. 5 approaches design sketch for the Gauss wavelet base time domain that the present invention obtains;
Fig. 6 approaches the effect contrast figure for the time domain of the present invention and Pade method;
Fig. 7 approaches the effect contrast figure for the time domain of the present invention and Maclaurin series method;
Fig. 8 approaches the effect contrast figure for the frequency domain of the present invention and Pade method and Maclaurin series method;
The pole distribution figure of the reasonable approximating function of Gauss wavelet base that Fig. 9 approaches for the present invention.
Embodiment
The invention will be further described below in conjunction with accompanying drawing and example.
With reference to Fig. 1, the present invention includes following steps: (1) to wavelet basis function overturn, time shift, integration and Fourier transform, obtain the wavelet basis step response functions; (2) ask the amplitude spectrum and the phase spectrum of wavelet basis step response; (3) adopt plural reasonable approximatioss, try to achieve the reasonable approximating function of wavelet basis step response; (4) the reasonable approximating function of wavelet basis step response is carried out differential, try to achieve the reasonable approximating function of wavelet basis.
The inventive method is fit to approaching of any continuous wavelet wave filter.For the convenience that illustrates, approaching with the Gauss wavelet wave filter below is that example describes.
1, to the Gauss wavelet basis function overturn, time shift, integration and Fourier transform, obtain Gauss wavelet base step response functions
The Gauss wavelet basis function is
Figure 2013101649170100002DEST_PATH_IMAGE001
（1），
In the formula,
Figure 160889DEST_PATH_IMAGE002
Be the Gauss wavelet basis function, tBe time variable.
According to the wave filter principle, reach the purpose of wavelet transformation in order to make signal by wavelet filter, need carry out turning operation to formula (1), obtain
Figure 2013101649170100002DEST_PATH_IMAGE003
（2），
In the formula,
Figure 245127DEST_PATH_IMAGE004
Be the Gauss wavelet basis function of upset, tBe time variable.
Wave filter is a causal system, and formula (2) is a non-causal, in order to obtain causal system, formula (2) is carried out time shift operation.The tight supporting domain of time domain of considering the Gauss wavelet base is between-2 to 2, so determine the time shift amount usually t 0=2, obtain Gauss wavelet basis function to be approached
Figure DEST_PATH_IMAGE005
（3），
In the formula,
Figure 235823DEST_PATH_IMAGE006
Be Gauss wavelet function base function to be approached, tBe time variable.
To formula (3) integration, get the time domain expression formula of Gauss wavelet base step response
Figure DEST_PATH_IMAGE007
（4），
In the formula,
Figure 103547DEST_PATH_IMAGE008
Be the time domain expression formula of Gauss wavelet function base step response, tBe time variable.
Formula (4) is carried out Fourier transform, obtain Gauss wavelet base step response functions, i.e. the frequency-domain expression of Gauss wavelet function base step response
Figure DEST_PATH_IMAGE009
（5），
In the formula,
Figure 1840DEST_PATH_IMAGE010
Be Gauss wavelet base step response functions, jBe imaginary number, ω is a frequency.
2, ask the amplitude spectrum and the phase spectrum of Gauss wavelet base step response functions
According to Gauss wavelet base step response functions, i.e. formula (5), the amplitude spectrum that can try to achieve wavelet basis step response is
（6），
In the formula, The amplitude spectrum of expression Gauss wavelet base step response,
Figure DEST_PATH_IMAGE013
Expression is asked
Figure 164498DEST_PATH_IMAGE010
Mould, ω represents frequency.According to the amplitude spectrum of Gauss wavelet base step response, i.e. formula (6), the Gauss wavelet function base step response amplitude spectrum of drafting is shown in Fig. 2 (a).
According to Gauss wavelet base step response functions, i.e. formula (5), the phase spectrum that can try to achieve wavelet basis step response is
（7），
In the formula,
Figure DEST_PATH_IMAGE015
The phase spectrum of expression Gauss wavelet base step response, ω represents frequency, k represents coefficient, makes
Figure 346135DEST_PATH_IMAGE015
Be in-π and π between.According to the phase spectrum of Gauss wavelet base step response, i.e. formula (7), the phase spectrum of the Gauss wavelet base step response of drafting is shown in Fig. 2 (b).
3, adopt plural reasonable approximatioss, ask the reasonable approximating function of Gauss wavelet base step response functions
According to the signal decomposition principle, Gauss wavelet base step response functions can be decomposed into
Figure 657162DEST_PATH_IMAGE016
（8），
In the formula,
Figure 262366DEST_PATH_IMAGE010
Be Gauss wavelet base step response functions, jBe imaginary number, ω is a frequency,
Figure 26054DEST_PATH_IMAGE012
With
Figure 721609DEST_PATH_IMAGE015
Amplitude spectrum and the phase spectrum of representing Gauss wavelet base step response respectively.
Adopt plural reasonable approximatioss to approach, at Frequency point ω nOn, the jump error of response and desirable step response of order of approximation is designated as
（9），
In the formula, n is a subscript, expression sample frequency point sequence number, The expression order of approximation jumps response and desirable step response at Frequency point ω nOn error, jBe imaginary number, ω nBe frequency,
Figure DEST_PATH_IMAGE019
With Represent that respectively the amplitude spectrum of Gauss wavelet base step response and phase spectrum are at Frequency point ω nOn numerical value,
Figure DEST_PATH_IMAGE021
With
Figure 451646DEST_PATH_IMAGE022
Represent that respectively the molecule of the reasonable approximating function of Gauss wavelet base step response and denominator are at Frequency point ω nOn numerical value.
Define an error criterion
Figure DEST_PATH_IMAGE023
（10），
In the formula, JThe expression error criterion, n is a subscript, expression sample frequency point sequence number,
Figure 557749DEST_PATH_IMAGE018
The expression order of approximation jumps response and desirable step response at Frequency point ω nOn error, jBe imaginary number, ω nBe frequency,
Figure 843368DEST_PATH_IMAGE024
The denominator of the reasonable approximating function of expression Gauss wavelet base step response, min represents minimum value.This is a linear minimization problem, adopts five rank rational functions to approach, and adopts criterion of least squares to find the solution, and tries to achieve
Figure 2013101649170100002DEST_PATH_IMAGE025
With As follows respectively:
Figure 2013101649170100002DEST_PATH_IMAGE027
（11），
Figure 253019DEST_PATH_IMAGE028
（12），
In the formula, jBe imaginary number, ω is a frequency, With Represent the molecule and the denominator of the reasonable approximating function in Gauss wavelet base step response five rank respectively, make s= jω,
Figure 2013101649170100002DEST_PATH_IMAGE029
（13），
Figure 134760DEST_PATH_IMAGE030
（14），
In the formula, sBe complex frequency,
Figure 2013101649170100002DEST_PATH_IMAGE031
With
Figure 232772DEST_PATH_IMAGE032
Represent the molecule of the reasonable approximating function in Gauss wavelet base step response five rank and denominator expression formula respectively at complex frequency domain.Therefore, having tried to achieve the reasonable approximating function in Gauss wavelet base step response five rank is
Figure DEST_PATH_IMAGE033
（15），
In the formula, sBe complex frequency,
Figure 886870DEST_PATH_IMAGE034
The reasonable approximating function in expression Gauss wavelet base step response five rank,
Figure 776826DEST_PATH_IMAGE031
With
Figure 779549DEST_PATH_IMAGE032
Molecule and the denominator of representing the reasonable approximating function in Gauss wavelet base step response five rank respectively.
4, the reasonable approximating function of Gauss wavelet base step response is carried out differential, ask the reasonable approximating function of Gauss wavelet base
According to the Signals ﹠ Systems principle, the reasonable approximating function in Gauss wavelet base step response five rank is differentiated, be about to
Figure 554738DEST_PATH_IMAGE034
Multiply by complex frequency s, can try to achieve the reasonable approximating function of Gauss wavelet base
Figure DEST_PATH_IMAGE035
（16），
In the formula, sBe complex frequency,
Figure 806334DEST_PATH_IMAGE036
The reasonable approximating function of expression Gauss wavelet base, this is one five a rank rational function.
The five rank Gauss wavelet base step response frequency domain effect figure that Fig. 2 approaches for the present invention, wherein Fig. 2 (a) is desirable Gauss wavelet base step response amplitude spectrum, Fig. 2 (b) is desirable Gauss wavelet base step response phase spectrum, the five rank Gauss wavelet base step response amplitude spectrums that Fig. 2 (c) approaches for the present invention, the five rank Gauss wavelet base step response phase spectrums that Fig. 2 (d) approaches for the present invention.Analysis chart 2 as can be known, the five rank Gauss wavelet base step response frequency domains that the present invention obtains approach effective, the approximation accuracy height, in 0 to 4rad/s frequency range, the square error of approaching of step response amplitude spectrum and phase spectrum reaches 3.17 * 10 respectively -5With 0.196.
The five rank Gauss wavelet base impulse response frequency domain effect figure that Fig. 3 approaches for the present invention, wherein Fig. 3 (a) is desirable Gauss wavelet base impulse response amplitude spectrum, Fig. 3 (b) is desirable Gauss wavelet base impulse response phase spectrum, the five rank Gauss wavelet base impulse response amplitude spectrums that Fig. 3 (c) approaches for the present invention, the five rank Gauss wavelet base impulse response phase spectrums that Fig. 3 (d) approaches for the present invention.Analysis chart 3 as can be known, the five rank Gauss wavelet base impulse response frequency domains that the present invention obtains approach effective, the approximation accuracy height, in 0 to 4rad/s frequency range, the square error of approaching of impulse response amplitude spectrum and phase spectrum reaches 2.03 * 10 respectively -4With 0.75.
The five rank Gauss wavelet base step response time domain design sketchs that Fig. 4 approaches for the present invention, solid line is the five rank Gauss wavelet base step responses of approaching among the figure, dotted line is desirable Gauss wavelet base step response.Analysis chart 4 as can be known, the five rank Gauss wavelet base step responses that the present invention approaches, are approached square error and are reached 2.16 * 10 in 0 to 8s time range very near ideal situation -5
Fig. 5 approaches design sketch for the five rank Gauss wavelet base time domains that the present invention obtains, and solid line is the five rank Gauss wavelet bases that approach among the figure, and dotted line is desirable Gauss wavelet base.Analysis chart 5 as can be known, the five rank Gauss wavelet bases that the present invention approaches, approach square error and reach 6.05 * 10 in 0 to 8s time range very near ideal situation -4
Fig. 6 approaches the effect contrast figure for the time domain of the present invention and Pade method, and solid line is the five rank Gauss wavelet bases that approach among the figure, the five rank Gauss wavelet bases that dotted line approaches for the Pade method.Analysis chart 6 as can be known, in 0 to 8s time range, the present invention approaches square error and reaches 6.05 * 10 -4, and the Pade method has only 5.95 * 10 -3So the present invention approaches effect in time domain and obviously is better than the Pade method.
Fig. 7 approaches the effect contrast figure for the time domain of the present invention and Maclaurin series method.Owing to is 2 o'clock in the time shift amount, the Maclaurin series method can not obtain stable reasonable approximant, for the ease of relatively, adopts the 2.04 close time shift amounts as the Maclaurin series method here.Solid line is the five rank Gauss wavelet bases that approach among the figure, the five rank Gauss wavelet bases that dotted line approaches for the Maclaurin series method.Analysis chart 7 as can be known, in 0 to 8s time range, the present invention approaches square error and reaches 6.05 * 10 -4, and the Maclaurin series method has only 1.57 * 10 -2So the present invention approaches effect in time domain and obviously is better than the Maclaurin series method.
Fig. 8 approaches the effect contrast figure for the frequency domain of the present invention and Pade method and Maclaurin series method, analysis chart 8 as can be known, in 0 to 4rad/s frequency range, the present invention approaches square error and reaches 2.03 * 10 -4, and Pade method and Maclaurin series method have only 1.20 * 10 respectively -2With 4.43 * 10 -2So the present invention approaches effect in frequency field and obviously is better than Pade method and Maclaurin series method.
The pole distribution figure of the reasonable approximating function of five rank Gauss wavelet bases that Fig. 9 approaches for the present invention, analysis chart 9 as can be known, five limits are respectively-0.94+2.70i,-0.94-2.70i,-1.10+1.29i,-1.10-1.29i and-1.14, limit all is positioned at the left half-plane of complex coordinates, so the five rank reasonable approximating functions of Gauss wavelet base that the present invention obtains are stable.
Comprehensive above-mentioned simulation result and the analysis showed that adopts the present invention can effectively improve the approximation accuracy of simulation wavelet basis and the stability that can guarantee to approach wavelet basis.The present invention is suitable for further adopting the Analogical Circuit Technique design and realizes wavelet transformation.

Claims (3)

1. a simulation wavelet basis frequency domain approach method is characterized in that, may further comprise the steps: (1) to wavelet basis function overturn, time shift, integration and Fourier transform, obtain the wavelet basis step response functions; (2) ask the amplitude spectrum and the phase spectrum of wavelet basis step response functions; (3) adopt plural reasonable approximatioss, try to achieve the reasonable approximating function of wavelet basis step response; (4) the reasonable approximating function of wavelet basis step response is carried out differential, try to achieve the reasonable approximating function of wavelet basis.
2. simulation wavelet basis frequency domain approach method according to claim 1, it is characterized in that, in the step (3), the plural reasonable approximatioss of described employing is tried to achieve the reasonable approximating function of wavelet basis step response and is meant: the wavelet basis step response functions is being approached aspect amplitude and the phase place by the reasonable approach method of plural number, approaching the result is a rational expression.
3. simulation wavelet basis frequency domain approach method according to claim 1 and 2, it is characterized in that, in the step (4), described the reasonable approximating function of wavelet basis step response is carried out differential, the principle and the particular content of trying to achieve the reasonable approximating function of wavelet basis are: according to the Signals ﹠ Systems theory, the impulse response of system is the differential of its step response, so on the basis of trying to achieve the reasonable approximating function of wavelet basis step response, only need carry out derivation operation just can obtain the reasonable approximating function of wavelet basis impulse response, i.e. the reasonable approximating function of wavelet basis.
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