CN104506164B - Method for optimally designing graph filter banks on basis of two-step process - Google Patents
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Abstract
The invention discloses a method for optimally designing graph filter banks on the basis of a two-step process. The method includes converting requirements on reconstruction characteristics and frequency characteristics of the filter banks into functions related to coefficients of prototype filters; converting design problems of the graph filter banks into a constrained optimization problem according to design requirements; ultimately solving the optimal coefficients of the prototype filters by the aid of the two-step process. The method has the advantages that optimization ideas are introduced into designs of the graph filter banks for the first time, and the reconstruction characteristics and the frequency characteristics of the graph filter banks are taken into consideration by the aid of factor regulating modes; the reconstruction characteristics and the frequency characteristics of the filter banks can be taken into consideration by optimization paths as compared with existing methods on the basis of polynomial decomposition; compromising effects can be realized between the reconstruction characteristics and the frequency characteristics by means of regulating parameters epsilon r under the condition of given lengths of graph filters.
Description
Technical field
The invention belongs to field of signal processing, and in particular to a kind of figure wave filter group optimization design side based on two-step method
Method.
Background technology
In recent years, the process of network big data causes extensive concern, and these data are often defined on irregular several
In what structure, such as social networkies, wireless sensor network, banking network etc..The model of traditional rule-based position definition
It is not used in the process of such data.Figure (graph) is considered as a kind of model for effectively describing such data, the node of figure
The position that data are located is represented, the size of data can be represented with node of graph signal, and the line between figure interior joint can be with
Represent the relatedness between different pieces of information node.On this basis, it is processed into for key based on the data signal of figure.In many
Using in, data volume is very huge, and the process to full figure will bring very huge computation complexity, cannot often realize, from
And expedited the emergence of the research work of the multiresolution analysis of figure signal.
For this purpose, there is many scholars to propose the wavelet transformation/wave filter group suitable for figure signal processing.Typical example
Have：Suitable for the wavelet-like transform of transportation network figure, it is adaptable to which two passages of wireless sensor network figure can inverse filter group.
In these work, two passage biorthogonal figure wave filter groups are one of structures the most outstanding, it possess threshold sampling, compact schemes,
The advantages of Perfect Reconstruction.However, only having a kind of method to be used for the design of the class formation, and existing method based on multinomial point at present
Solution method, cannot take into account the frequency characteristic of ptototype filter, including pass-band flatness and stopband attenuation in design.From optimization angle
Document report be yet there are no to design two passage biorthogonal figure wave filter groups.
The content of the invention
The technical problem to be solved is that existing two passages biorthogonal figure wave filter group cannot take into account prototype filtering
A kind of deficiency of the frequency characteristic of device, there is provided figure wave filter group Optimization Design based on two-step method.
To solve the above problems, the present invention is achieved by the following technical solutions：
Based on the figure wave filter group Optimization Design of two-step method, comprise the steps：
Step 1, the analysis ptototype filter of figure wave filter group and comprehensive ptototype filter are expressed as with regard to characteristic root
Polynomial function；
Step 2, by pass-band flatness E of analysis ptototype filterp(h0) be converted into regard to analyzing ptototype filter coefficient
h0, and by pass-band flatness E of comprehensive ptototype filterp(g0) be converted into regard to comprehensive ptototype filter coefficient g0；
Step 3, by the stopband ENERGY E of analysis ptototype filters(h0) be converted into regard to analyzing ptototype filter coefficient h0,
And by the stopband ENERGY E of comprehensive ptototype filters(g0) be converted into regard to comprehensive ptototype filter coefficient g0Function；
Step 4, sets the perfect reconstruction filter bank of figure wave filter group；
The design problem of figure wave filter group is attributed to following optimization problem, i.e., by step 5
①
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Ep(g0) represent the logical of comprehensive ptototype filter
Band flatness, Es(h0) represent the stopband energy for analyzing ptototype filter, Es(g0) represent the stopband energy of comprehensive ptototype filter
Amount, α represent weight, h0Represent analysis ptototype filter coefficient, g0Analysis ptototype filter coefficient is represented,Represent analysis
Ptototype filter frequency vector is in λkThe value at place,Represent comprehensive ptototype filter frequency vector in λkThe value at place,Represent analysis ptototype filter frequency vector in 2- λkThe value at place,Represent comprehensive prototype filtering
Device frequency vector is in 2- λkThe value at place, subscript T represent transposition, subscript LhRepresent analysis ptototype filter coefficient number, subscript Lg
Represent analysis prototype synthesis filter coefficients number, λkFrequency vector is represented, N represents discrete interval number, εrRepresent that control reconfiguration is missed
Poor parameter, k represent discrete point coordinates.
1. step 6, solve formula using two-step method, is derived from the optimum analysis ptototype filter coefficient h of figure wave filter0
With comprehensive ptototype filter coefficient g0。
In the step 6, ptototype filter coefficient h is analyzed0For：
②
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Es(h0) represent the resistance for analyzing ptototype filter
Band energy, α represent weight,Analysis value of the ptototype filter frequency vector at λ=1, h0Represent analysis prototype filtering
Device coefficient.
In the step 6, comprehensive ptototype filter coefficient g0For：
③
In formula, Ep(g0) represent the pass-band flatness of comprehensive ptototype filter, Es(g0) represent the resistance of comprehensive ptototype filter
Band energy, α represent weight, aT(λi), i=0 ..., N represent the constrained vector of comprehensive ptototype filter, and b represents length for N+1's
Element is 2 vector entirely, εrRepresent the parameter of control reconfiguration error, g0Represent analysis ptototype filter coefficient.
In such scheme, N, εrIt is with α and is manually set value, the wherein span of N is set to 100～200, εrValue
Scope is set to 10-5～0-12, the span of α is set to 0.1～0.001.
The present invention is by requiring to be converted into regard to ptototype filter coefficient the reconstruction property of wave filter group and frequency characteristic
Function, and then the design problem of figure wave filter group is converted into according to design requirement the optimization problem of a belt restraining, finally
The optimal coefficient of ptototype filter is solved using two-step method.The method introduces excellent for the first time in the design of figure wave filter group
Change thinking, the reconstruction property and frequency characteristic of figure wave filter group have been taken into account by the way of regulatory factor.Compared to existing base
In the method for Factoring Polynomials, approach of the inventive method using optimization, the reconstruction property and frequency for considering wave filter group can be taken into account
Rate characteristic.In the case of the length of figure wave filter is given, by regulation parameter εrTo seek between reconstruction property and frequency characteristic
Compromise.
Description of the drawings
Fig. 1 is the structure chart of figure wave filter group.
Fig. 2 is the Performance comparision of the inventive method and existing method.
Fig. 3 is the optimum results comparison diagram of existing method and this method in ptototype filter.
Fig. 4 is the ptototype filter coefficient figure of the inventive method design.
Specific embodiment
A kind of figure wave filter group Optimization Design based on two-step method, is characterized in that comprising the steps：
The first step：The structure of figure wave filter group is as shown in Figure 1.Each ptototype filter of figure wave filter group is expressed as closing
In the polynomial function of characteristic root：
Second step：The pass-band flatness of ptototype filter is converted into regard to ptototype filter coefficient h0、g0Function：
3rd step：The stopband energy of ptototype filter is converted into regard to ptototype filter coefficient h0、g0Function：
4th step：The perfect reconstruction filter bank of figure wave filter group is expressed as：
5th step：The design problem of figure wave filter group is attributed to into following optimization problem：
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Ep(g0) represent the logical of comprehensive ptototype filter
Band flatness, Es(h0) represent the stopband energy for analyzing ptototype filter, Es(g0) represent the stopband energy of comprehensive ptototype filter
Amount, α represent weight, h0Represent analysis ptototype filter coefficient, g0Analysis ptototype filter coefficient is represented,Represent analysis
Ptototype filter frequency vector is in λkThe value at place,Represent comprehensive ptototype filter frequency vector in λkThe value at place,Represent analysis ptototype filter frequency vector in 2- λkThe value at place,Represent comprehensive prototype filtering
Device frequency vector is in 2- λkThe value at place, subscript T represent transposition, subscript LhRepresent analysis ptototype filter coefficient number, subscript Lg
Represent analysis prototype synthesis filter coefficients number, λkFrequency vector is represented, N represents the discrete interval number of interval [0,2], εrTable
Show the parameter of control reconfiguration error, k represents discrete point coordinates.
6th step：The optimal coefficient of figure wave filter is solved using two-step method：
(1) an analysis ptototype filter coefficient h with good frequency characteristic is designed using following optimization problem0
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Es(h0) represent the resistance for analyzing ptototype filter
Band energy, α represent weight,Analysis value of the ptototype filter frequency vector at λ=1, h0Represent analysis prototype filtering
Device coefficient.
(2) based on h0, design the comprehensive ptototype filter coefficient g for meeting Perfect Reconstruction characteristic0
In formula, Ep(g0) represent the pass-band flatness of comprehensive ptototype filter, Es(g0) represent the resistance of comprehensive ptototype filter
Band energy, α represent weight, aT(λi), i=0 ..., N represent the constrained vector of comprehensive ptototype filter, and b represents length for N+1's
Element is 2 vector entirely, εrRepresent the parameter of control reconfiguration error, g0Represent analysis ptototype filter coefficient.
As formula (6) and (7) are all convex optimization problems, therefore can be solved using software kits such as CVX or Sedumi.
To verify the effectiveness of this method, emulation experiment has been carried out.Simulation parameter is：The passband and stopband of ptototype filter
Cut-off frequency is set to：λp=0.8, λs=1.2.Optimization weights are set to α=0.1.Distortion parameter εrWith ptototype filter length
Three kinds of situations of parameter setting, respectively：εr=1 × 10-12,Lh=5, Lg=4, εr=1 × 10-10,Lh=9, Lg=8 and εr=
1×10-7,Lh=15, Lg=14.We are respectively adopted existing method and the inventive method designs above-mentioned wave filter group.Fig. 2 is given
The Performance comparision of two methods.Ptototype filter frequency response (the ε as shown in Figure 3 that gained is designed by two methodsr=1 ×
10-10,Lh=9, Lg=8 length situations).Fig. 4 gives the ptototype filter coefficient of the inventive method design gained.
Can draw from simulation result, the ptototype filter of the inventive method design gained possesses more preferable frequency characteristic,
For example passband is more flat.Although signal to noise ratio is lost, orthogonality has obtained good maintenance.This also illustrates, this
Bright method can take into account reconstruction property and frequency characteristic.
Claims (4)
1. the figure wave filter group Optimization Design based on two-step method, is characterized in that, comprise the steps：
Step 1, the analysis ptototype filter of figure wave filter group and comprehensive ptototype filter are expressed as with regard to the multinomial of characteristic root
Formula function；
Step 2, by pass-band flatness E of analysis ptototype filterp(h0) be converted into regard to analyzing ptototype filter coefficient h0, and
By pass-band flatness E of comprehensive ptototype filterp(g0) be converted into regard to comprehensive ptototype filter coefficient g0；
Step 3, by the stopband ENERGY E of analysis ptototype filters(h0) be converted into regard to analyzing ptototype filter coefficient h0, and will
The stopband ENERGY E of comprehensive ptototype filters(g0) be converted into regard to comprehensive ptototype filter coefficient g0Function；
Step 4, sets the perfect reconstruction filter bank of figure wave filter group；
The design problem of figure wave filter group is attributed to following optimization problem, i.e., by step 5
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Ep(g0) represent that the passband of comprehensive ptototype filter is put down
Smooth property, Es(h0) represent the stopband energy for analyzing ptototype filter, Es(g0) represent the stopband energy of comprehensive ptototype filter, α tables
Show weight, h0Represent analysis ptototype filter coefficient, g0Comprehensive ptototype filter coefficient is represented,Represent analysis prototype filter
Ripple device frequency vector is in λkThe value at place,Represent comprehensive ptototype filter frequency vector in λkThe value at place,
Represent analysis ptototype filter frequency vector in 2- λkThe value at place,Represent comprehensive ptototype filter frequency vector
In 2- λkThe value at place, subscript T represent transposition, subscript LhRepresent analysis ptototype filter coefficient number, subscript LgRepresent that analysis is former
Structural synthesis filter coefficient number, λkFrequency vector is represented, N represents discrete interval number, εrRepresent the parameter of control reconfiguration error, k
Represent discrete point coordinates；
1. step 6, solve formula using two-step method, is derived from the optimum analysis ptototype filter coefficient h of figure wave filter0With it is comprehensive
Close ptototype filter coefficient g0。
2. figure wave filter group Optimization Design according to claim 1 based on two-step method, is characterized in that, the step 6
In, analyze ptototype filter coefficient h0For：
In formula, Ep(h0) represent the pass-band flatness for analyzing ptototype filter, Es(h0) represent the stopband energy for analyzing ptototype filter
Amount, α represent weight,Analysis value of the ptototype filter frequency vector at λ=1, h0Represent analysis ptototype filter system
Number.
3. figure wave filter group Optimization Design according to claim 1 based on two-step method, is characterized in that, the step 6
In, comprehensive ptototype filter coefficient g0For：
In formula, Ep(g0) represent the pass-band flatness of comprehensive ptototype filter, Es(g0) represent the stopband energy of comprehensive ptototype filter
Amount, α represent weight, aT(λi), i=0 ..., N represent the constrained vector of comprehensive ptototype filter, and b represents the element that length is N+1
It is 2 vector entirely, εrRepresent the parameter of control reconfiguration error, g0Represent comprehensive ptototype filter coefficient.
4. figure wave filter group Optimization Design according to claim 1 based on two-step method, is characterized in that, above-mentioned N, εrAnd α
Setting value is, the wherein span of N is set to 100～200, εrSpan be set to 10-5～0-12, the span of α sets
For 0.1～0.001.
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| CN105243241B (en) * | 2015-11-12 | 2018-04-24 | 桂林电子科技大学 | The two passage biorthogonal figure filter set designing methods based on lift structure |
| CN105279350B (en) * | 2015-11-30 | 2018-05-29 | 桂林电子科技大学 | The design method of approximate Perfect Reconstruction Vertical Nonuniform Cosine modulated filter group |
| CN105787204B (en) * | 2016-03-23 | 2019-04-12 | 桂林电子科技大学 | The design method of the complete over-sampling DFT modulated filter group of the double prototypes of bidimensional |
| CN107239623B (en) * | 2017-06-08 | 2020-07-10 | 桂林电子科技大学 | Optimal design method of M-channel oversampling image filter bank based on convex optimization |
| CN107241082B (en) * | 2017-06-09 | 2020-07-10 | 桂林电子科技大学 | Design method of DFT modulation filter bank based on convex optimization relaxation |
| CN107992711B (en) * | 2018-01-18 | 2021-04-13 | 桂林电子科技大学 | Optimization design method of M-channel oversampling modulation diagram filter bank |
| CN112818526B (en) * | 2021-01-20 | 2022-09-30 | 桂林电子科技大学 | Distributed design method of non-uniform graph filter bank |
| CN113630104B (en) * | 2021-08-18 | 2022-08-23 | 杭州电子科技大学 | Filter bank frequency selectivity error alternation optimization design method of graph filter |
| CN114331926B (en) * | 2021-12-29 | 2022-06-10 | 杭州电子科技大学 | Two-channel graph filter bank coefficient design optimization method based on element changing idea |
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Application publication date: 20150408 Assignee: Guangxi wisdom Valley Technology Co.,Ltd. Assignor: GUILIN University OF ELECTRONIC TECHNOLOGY Contract record no.: X2023980046615 Denomination of invention: Optimization design method for graph filter banks based on two-step method Granted publication date: 20170412 License type: Common License Record date: 20231108 |
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