CN107294512A - A kind of non-homogeneous wave filter group filtering method based on tree - Google Patents
A kind of non-homogeneous wave filter group filtering method based on tree Download PDFInfo
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- H—ELECTRICITY
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Abstract
The present invention provides a kind of non-homogeneous wave filter group building method based on tree.Including：The module based on binary channels FIR quadrature mirror filter groups, non-homogeneous wave filter group is built using tree.Derived by the reconstruction condition to the non-homogeneous wave filter group of whole tree, the perfect reconstruction filter bank of whole system is finally attributed to the ptototype filter that design one meets specified conditions.The ptototype filter for making design using iteration cut-off frequecy of passband meets reconstruction condition, during the specific design of ptototype filter, the present invention uses discrete weightings square error criterion, and compared with other method, the criterion can realize more preferable passband flat characteristic and bigger stopband attenuation.The design of the present invention effectively reduces iteration complexity, and can obtain bigger stopband attenuation and better system reconfiguration performance, has important application value for radar broad-band channel receiver, voice, picture signal processing.
Description
Technical field
The invention belongs to wave filter group technical field, and in particular to a kind of non-homogeneous wave filter group filter based on tree
Wave method.
Background technology
Multi-channel filter group theory is widely used in the field of signal processing such as radar, voice, image, the technology
Using significantly reducing PDR requirement, data space, computational complexity etc..One filter bank system can
To carry out frequency band division to input signal by the wave filter of multiple different frequency bands characteristics in system front end analysis module, take out afterwards
Reduction of speed is taken, then the subband signal of different frequency bands is handled according to actual needs.Then System Back-end by interpolation with
And subband signal is reconstructed into required primary signal by corresponding synthesis filter group as much as possible, therefore whole wave filter group
Reconstruction property is the research emphasis in wave filter group theory.
Wave filter group is an important content in multirate signal processing, is earned widespread respect in recent years.Wave filter group
It is widely used in communication, voice coding, audio coding and picture signal processing.If the difference of output and the input of system is only
It is that amplitude is proportional and there is certain delay, the system that this system is thus referred to as Perfect Reconstruction.By signal decomposition into subband
Post processing, is easy to obtain more preferable effect using the frequency characteristic of signal.By analysis filter group and synthesis filter group structure
Into system in, make output end reconstruct signal it is identical with the primary signal of input, typically Design of filter banks pursue
Target.But consider from practical standpoint, under conditions of distortion control within the specific limits, limitation less, efficiency high, easy set
Meter method is more valuable.Multirate signal processing has extensively in many fields such as communication, Image Coding, voice coding, radar
Application.Multi tate technology can effectively reduce the processing complexity, the transfer rate of data and amount of storage of signal.
Non-homogeneous wave filter group can input signal into the subsignal of different frequency bands width according to actual needs, tool
There is more preferable flexibility.Compared with uniform wave filter group, non-homogeneous wave filter group is more flexible due to dividing frequency spectrum, so, in recent years
The concern of numerous scholars, reason of many scholars in non-homogeneous wave filter group are caused come the design studies to non-homogeneous wave filter group
Many contributions are made that by with design aspect.But up to the present, realizing the Perfect Reconstruction of non-homogeneous wave filter group is still
Design challenges.Because the parameter of optimization is more, the non-homogeneous wave filter group of design Perfect Reconstruction is relatively difficult, Perfect Reconstruction
Design method it is cumbersome, complicated and be not easily accomplished, it is general to choose spirit therefore in the actual design of non-homogeneous wave filter group
Living, simple approximate reconstruction design.Nguyen et al. exists《Signal Processing IEEE Transactions
on》On the document delivered《A simple design method for near perfect reconstruction
nonuniform filter banks》In propose non-homogeneous wave filter group designed using the method for merging uniform wave filter group.
Xie X exist《Circuits and Systems》On deliver《A simple design method of linear-phase
nonuniform filter banks with integer decimation factors》Directly derived in document from frequency domain
The Remodeling of non-homogeneous wave filter group, and non-homogeneous wave filter group is designed with this.Soni is in document《An Optimized
Design of Non-uniform Filterbank using Blackman Window Family》(International
Journal of Signal＆Image Processing) it is middle using the non-homogeneous wave filter group of tree design, same Kumar
In document《Design of nearly perfect reconstructed non-uniform filter bank by
constrained equiripple FIR technique》Also tree is used in (Applied Soft Computing)
Non-homogeneous wave filter group is designed, and simplifies iterative target function.There is aliasing in non-homogeneous wave filter group structural scheme above
The problems such as error, amplitude distortion or phase distortion, the reconstruction property of whole system has much room for improvement.
The content of the invention
It can not design and provide it is an object of the invention to provide the existing non-homogeneous wave filter group structural scheme of one kind solution
There is the non-homogeneous wave filter group filtering method based on tree of the non-homogeneous filtering system problem of good reconstruction property.
The object of the present invention is achieved like this：
A kind of non-homogeneous wave filter group building method based on tree, is comprised the following steps：
The first step：The design parameter of ptototype filter is set, including coefficient length N, cut-off frequecy of passband wp, stopband
Cut-off frequency ws, initial iteration step step, iteration ends error Recei；
Second step：Using discrete weightings square error criterion prototype wave filter hl(n) h, is then solvedl(n) frequency
Respond the value Real at orthogonal mirror image point w=0.5 π；
3rd step：Judge whether actual error is less than the termination error Recei of setting, i.e. whether below equation is set up：
| Real-0.7071 | ＜ Recei (1)
If above formula is set up, using wave filter hl(n) h is solvedh(n), then filtered with binary channels normal orthogonal mirror image
Ripple device group builds the non-homogeneous wave filter group of tree；If invalid, determine whether big between Real and 0.707
It is small：If Real ＞ 0.707, wp=wp- step, step=step/2；If Real ＜ 0.707, wp=wp+ step, step=
After step/2, each iteration, step-length step is changed into original half；Wherein, hh(n) high-pass filter, h are representedl(n) represent
Low pass filter.
4th step：Update cut-off frequecy of passband wp, then using new wpLow pass filter h is designed againl(n), change successively
Generation, until error amount is less than given error range.
For a kind of non-homogeneous wave filter group filtering method based on tree, pass through iterative filter passband cutoff frequency
Rate, makes ptototype filter Hl(z) reconstruction condition derived is met；During the specific design of ptototype filter, using from
Dissipate weighted square error criterion method prototype low pass filter.
Wherein, Hl(z) it is low pass filter hl(n) transmission function.
For a kind of non-homogeneous wave filter group filtering method based on tree, described discrete weightings square error is accurate
Then method includes process in detail below：
Using the discrete square error rule definition error function of weighting
E (w)=W (w) [A (w)-Ad(w)] (2)
Wherein, E (w) represents Ad(w) weighted error between A (w)；Ad(w) h that will be approached is representedd(n) amplitude
Function, hd(n) ideal filter is represented；A (w) represents h (n) amplitude function, and h (n) represents the wave filter of actual design；W(w)
>=0 is weighting function.
Discrete square error Δ is weighted to be defined as：
Wherein, (wm, m=1,2 ..., L) it is L sampled point in frequency domain.
The amplitude function of FIR filter is represented using following form, i.e.,：
A (w)=Q (w) G (w) (4)
Wherein：
Wherein, Q (w)=cos (w/2), K=(N-1)/2, N are the filter orders of design.
Middle coefficient g (n) (n=1,2 ..., K) is solved by the thought of optimization, reality is solved further according to below equation
The first half of border coefficient h (n), afterwards by symmetry, solves all coefficient hs (n) for the FIR filter being actually needed.
Wherein, n=1,2 ..., K-1.
Formula (4), (5) are substituted into formula (3), obtained：
Error function Δ is now expressed as matrix form, error vector Λ is defined：
Λ=(Λ1,Λ2,…,ΛL)T (8)
Wherein：
Then overall error function Δ can be expressed as：
Δ=ΛTΛ (10)
Using matrix form, error vector Λ can be expressed as：
Λ=W (QCg-Ad) (11)
Wherein W and Q are L × L matrixes, i.e.,：
C is L × (K+1) matrix, i.e.,：
AdIt is the vector of L element, i.e.,：
Ad=[Ad(w1),Ad(w1),…,Ad(wL)] (15)
,, can be by solving equation from formula (11) because WQC is L × L matrix as L=K+1
WQCg=WAdObtain the null solution of error delta.Because error delta is equal to zero, so weighting matrix W now is to actual design
Wave filter do not play a role.As L ＞ K+1, now the number of equation group be more than unknown quantity number, therefore equation group without
Solution, if now WQC matrixes are sequency spectrum matrixes, the minimal solution of the error function Δ existence anduniquess shown in formula (11).This
When can be by solving equation below, i.e.,：
(WQC)TWQCg=(WQC)TWAdWQC (16)
Middle coefficient vector g is solved, and then solves by formula (6) the unit impulse response h (n) of actual design.
Non-homogeneous wave filter group filtering side of the binary channels FIR normal orthogonal mirror filter groups based on tree
Method, specific design step is：
Step 1：Design binary channels FIR normal orthogonal mirror filter groups；
Step 2：The module based on binary channels FIR normal orthogonal mirror filter groups, builds non-equal with reference to tree
Even wave filter group；
Step 3：The non-homogeneous wave filter group reconstruction condition that this method is designed is derived；
Step 4：The reconstruction condition of whole non-homogeneous wave filter group is reduced to：The frequency response of prototype FIR filter exists
Amplitude at orthogonal points w=pi/2s meets Hl(ejπ/2)=0.7071；
Step 5：It is set to meet the reconstruction condition in step 4 using the cut-off frequecy of passband of iterative prototyping wave filter；
Step 6：During the specific design of ptototype filter, using discrete weightings square error Criterion Method prototype
Low pass filter.
For a kind of non-homogeneous wave filter group filtering method based on tree, the filter of binary channels FIR normal orthogonals mirror image
The non-homogeneous wave filter group that ripple device group builds tree includes analysis module and integration module.
Low-channel filter H in analysis modulel(z) with high pass channel filter Hh(z) relation condition is set to：
Hh(z)=Hl(-z) (17)
Wherein, Hl(z), Hh(z) be respectively first passage and second channel in analysis module transmission function.
Integration module is set to the filter relationship in analysis module：
Fl(z)=Hh(-z),Fh(z)=- Hl(-z) (18)
Wherein, Fl(z), Fh(z) be respectively first passage and second channel in integration module transmission function.
For a kind of non-homogeneous wave filter group filtering method based on tree, using binary channels FIR normal orthogonal mirrors
As the reconstruction condition for the non-homogeneous wave filter group of tree that wave filter group is built is：
Wherein, Hk(ejw) represent k-th of passage filter freguency response, w is Frequency point, and π is pi, and M represents non-
The channel number of uniform wave filter group.
The beneficial effects of the present invention are：Compared with existing design, this invention simplifies iterative reconstruction condition, and
And be applied to discrete weightings square error criterion in the design for the non-homogeneous wave filter group that tree is built so that whole filter
Wave system system is while each channel linear phase is ensured, the stopband attenuation of each passage and the amplitude distortion of whole system are obtained for
It is further to improve, and then improve the reconstruction property of non-homogeneous wave filter group.
Brief description of the drawings
Fig. 1 is the non-homogeneous wave filter group of the passage of tree 4 of the present invention；
Fig. 2 is the passage isoboles of tree 4 of the present invention；
Fig. 3 be in the present invention using window function metht, characteristic filtering method, etc. corrugation methods, discrete weightings square error criterion etc.
The performance comparison figure for the FIR filter that distinct methods are designed；
Fig. 4 is non-homogeneous wave filter group iterative algorithm flow chart of the invention；
Fig. 5 is prototype FIR filter amplitude-response curve of the present invention；
Fig. 6 is the non-homogeneous wave filter group analogous diagram of the passage of tree 6 of the present invention；
Fig. 7 is the non-homogeneous wave filter group amplitude distortion figure of the passage of tree 6 of the present invention.
Embodiment
Illustrate below in conjunction with the accompanying drawings, the design in the present invention is specifically introduced：
Step 1：The non-homogeneous wave filter group of tree is built using binary channels FIR normal orthogonal mirror filter groups,
Fig. 1 is the non-homogeneous wave filter group of the passage of tree 4 of the present invention, and Fig. 2 is the passage isoboles of tree 4 of the present invention, in bilateral
In road filtering system, low-channel filter H in analysis modulel(z) with high pass channel filter Hh(z) relation condition is set to：
Hh(z)=Hl(-z) (1)
Integration module is set to the filter relationship in analysis module：
Fl(z)=Hh(-z),Fh(z)=- Hl(-z) (2)
Now whole system is without aliased distortion and phase distortion.
Step 2：Each path filter can be expressed as in Fig. 2：
And
Wherein, H0(z), H1(z), H2(z), H3(z) it is respectively each channel transfer function in Fig. 2.
Step 3：Now the non-homogeneous wave filter group reconstruction condition in Fig. 2 is
Wherein, Hk(ejw) represent k-th of passage filter freguency response.
Step 4：Formula (3), (5) are substituted into reconstruction condition (6) abbreviation and can obtained：
|Hl(ejw)|6+|Hl(ejw)|4|Hl(ej(π-w))|2+|Hl(ejw)|2|Hl(ej(π-w))|2+|Hl(ej(π-w))|2=1 (7)
In formula (7), take orthogonal frequency point w=pi/2s to carry out abbreviation and obtain：
2|Hl(ejπ/2)|6+|Hl(ejπ/2)|4+|Hl(ejπ/2)|2=1 (8)
The equation of higher order (8) is solved, is solved：
Hl(ejπ/2)=0.7071 (9)
The channel number for the non-homogeneous wave filter group that tree is constituted is generalized to M, and each passage extracts interpolation speed
It is set to (2M-1,2M-1,2M-2..., 4,2), it is equivalent after non-homogeneous filter bank analysis module in each channel factor can represent
For：
And
The Relationship of Coefficients that formula (10), (11) are included is substituted into M passage reconstruction condition expression formulas (6), can be obtained：
Use ejwInstead of the z in above formula, and take orthogonal frequency point w=pi/2s to dissolve it, obtain：
2|Hl(ejπ/2)|2(M-1)+Hl(ejπ/2)|2(M-2)+|Hl(ejπ/2)|2(M-3)+…+|Hl(ejπ/2)|4+|Hl(ejπ/2)|2=
1
(13)
The solution equation of higher order (13) is obtained and the same solution H of formula (9)l(ejπ/2)=0.7071, so, using binary channels FIR
The reconstruction condition for the non-homogeneous wave filter group that normal orthogonal mirror filter group combination tree is built is Hl(ejπ/2)=
0.7071。
Using iterative prototyping wave filter Hl(z) cut-off frequecy of passband makes it meet reconstruction condition (9).In specific design mistake
Discrete weightings square error criterion prototype wave filter H is applied in journeyl(z).This method makes actual design using optimization thought
Filter freguency response H (ejw) infinite approach ideal frequency response Hd(ejw), make error between the two minimum.With window function
Method, etc. ripple is tried, feature filters method is compared, in the case of design parameter identical, discrete weightings square error criterion can
To obtain more flat passband and bigger stopband attenuation.Its specific design process is as follows：It is discrete using weighting first
Square error rule definition error function.
E (w)=W (w) [A (w)-Ad(w)] (14)
Discrete square error Δ is weighted to be defined as：
Stepped-frequency signal number L is used for characterizing passband, Stopband Performance.The thought of this method design FIR filter is exactly to make
The error delta that formula (15) is defined is minimum.The amplitude function of FIR filter is represented using following form, i.e.,：
A (w)=Q (w) G (w) (16)
Wherein：
By the thought of optimization, middle coefficient g (n) (n=1,2 ..., K) is solved, is asked afterwards further according to below equation
Actual coefficients h (n) the first half is solved, all coefficients for the FIR filter being actually needed then are solved by symmetry again
h(n)。
Wherein, n=1,2 ..., K-1.
Formula (16), (17) are substituted into formula (15), obtained：
Error function Δ is now expressed as matrix form, error vector Λ is defined：
Λ=(Λ1,Λ2,…,ΛL)T (20)
Wherein：
Then overall error function Δ can be expressed as：
Δ=ΛTΛ (22)
Using matrix form, error vector Λ can be expressed as：
Λ=W (QCg-Ad) (23)
Wherein, W and Q are L × L matrixes, i.e.,：
C is L × (K+1) matrix, i.e.,：
AdIt is the vector of L element, i.e.,：
Ad=[Ad(w1),Ad(w1),…,Ad(wL)] (27)
,, can be by solving equation from formula (23) because WQC is L × L matrix as L=K+1
WQCg=WAdObtain the null solution of error delta.Because error delta is equal to zero, so weighting matrix W now is to actual design
Wave filter do not play a role.As L ＞ K+1, now the number of equation group be more than unknown quantity number, therefore equation group without
Solution, if now WQC matrixes are sequency spectrum matrixes, the minimal solution of the error function Δ existence anduniquess shown in formula (23).This
When can be by solving equation below, i.e.,：
(WQC)TWQCg=(WQC)T WAdWQC (28)
Middle coefficient vector g is solved, and then solves by formula (18) the unit impulse response h (n) of actual design.
Separately below using feature filters method, etc. corrugated design method, THE DESIGN OF WINDOW FUNCTION method, discrete weightings square error it is accurate
Then method designs FIR filter and result is done into comparative analysis.Fig. 3 be window function metht in the present invention, characteristic filtering method, etc. ripple
Method, discrete weightings square error criterion performance comparison figure, design parameter performance are as shown in table 1.
The distinct methods of table 1 construct FIR filter design parameter Character Comparison
By table 1, in the case of design parameter identical, it can be realized more using discrete weightings square error criterion
Big stopband attenuation, and then preferably suppress out of band signal.
Fig. 4 is non-homogeneous wave filter group iterative algorithm flow chart of the invention, and non-homogeneous wave filter group is provided with reference to Fig. 4
Specific design step：
Step 1：
The first step：The design parameter of ptototype filter is set, including coefficient length N, cut-off frequecy of passband wp, stopband
Cut-off frequency ws, initial iteration step step, iteration ends error Recei；
Second step：Using discrete weightings square error criterion prototype wave filter hl(n) h, is then solvedl(n) frequency
Respond the value Real at orthogonal mirror image point w=0.5 π；
3rd step：Judge whether actual error is less than the termination error Recei of setting, i.e. whether below equation is set up：
| Real-0.7071 | ＜ Recei (29)
If above formula is set up, using wave filter hl(n) h is solvedh(n), then with binary channels FIR normal orthogonal mirrors
As wave filter group builds the non-homogeneous wave filter group of tree；If invalid, determine whether between Real and 0.707
Size：If Real ＞ 0.707, wp=wp- step, step=step/2；If Real ＜ 0.707, wp=wp+ step, step=
After step/2, each iteration, step-length step is changed into original half；Wherein, hh(n) high-pass filter is represented.
4th step：Update cut-off frequecy of passband wp, then using new wpWave filter h is designed againl(n), iteration successively,
Until error amount is less than given error range.
In order to verify effectiveness of the invention, emulation experiment has been carried out.Extracted and inserted using one each passage of present invention design
Value speed is respectively the non-homogeneous wave filter group of 6 passages of (16,16,8,4,4,4), and iteration ends error is set into Recei=
10-4To ensure good precision, iteration step length is set to step=0.15 π.And amplitude distortion function Amdis is defined to characterize
The reconstruction property of whole system, i.e.,：
Wherein, Hm(ejw) represent m-th of passage filter freguency response.
Ptototype filter hl(n) design parameter is：Coefficient length N=63, cut-off frequecy of passband are wp=0.41 π, stopband
Cut-off frequency ws=0.65 π.Using discrete weightings square error criterion construction wave filter hl(n), Fig. 5 is prototype FIR filters of the present invention
Ripple device amplitude-response curve, now stopband attenuation is A as can be seen from Figure 5s=-133dB.Fig. 6 is that tree 6 of the present invention leads to
The non-homogeneous wave filter group analogous diagram in road, Fig. 7 is the non-homogeneous wave filter group amplitude distortion figure of the passage of tree 6 of the present invention, now
The maximum of amplitude distortion is max (Amdis)=1.3 × 10-3.The inventive method is contrasted with existing design method,
As shown in table 2.
The different designs method performance comparison of table 2
This paper design methods of table 3 and Kumar design method performance comparisons
As shown in Table 3, compared with Kumar design methods, this paper design method is averagely improved in terms of stopband attenuation
59.6%, 37.6% is averagely improved in terms of amplitude distortion.In summary analysis is understood, by discrete weightings square error criterion
After being applied in the design for the non-homogeneous wave filter group that tree is built, whole filtering system is ensureing each channel linear phase
While position, the stopband attenuation of each passage and the amplitude distortion of whole system are obtained for further improvement, and then improve
The reconstruction property of non-homogeneous wave filter group.
Claims (8)
1. a kind of non-homogeneous wave filter group filtering method based on tree, it is characterised in that：
The first step：The design parameter of ptototype filter is set, including coefficient length N, cut-off frequecy of passband wp, stopband cut-off
Frequency ws, initial iteration step step, iteration ends error Recei；
Second step：Using discrete weightings square error criterion prototype wave filter hl(n) h, is then solvedl(n) frequency response
Value Real at orthogonal mirror image point w=0.5 π；
3rd step：Judge whether actual error is less than the termination error Recei of setting, i.e. whether below equation is set up：
| Real-0.7071 | ＜ Recei (1)
If above formula is set up, using wave filter hl(n) h is solvedh(n), then with binary channels normal orthogonal mirror filter
Group builds the non-homogeneous wave filter group of tree；If invalid, the size between Real and 0.707 is determined whether：If
Real ＞ 0.707, then wp=wp- step, step=step/2；If Real ＜ 0.707, wp=wp+ step, step=step/
2, after each iteration, step-length step is changed into original half；Wherein, hh(n) high-pass filter, h are representedl(n) low pass filtered is represented
Ripple device.
4th step：Update cut-off frequecy of passband wp, then using new wpLow pass filter h is designed againl(n), iteration successively,
Until error amount is less than given error range.
2. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 1, it is characterised in that：
By iterative filter cut-off frequecy of passband, make ptototype filter Hl(z) reconstruction condition derived is met；In ptototype filter
Specific design during, using discrete weightings square error Criterion Method prototype low pass filter.
Wherein, Hl(z) it is low pass filter hl(n) transmission function.
3. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 2, it is characterised in that：
Described discrete weightings square error Criterion Method includes process in detail below：
(1) using the discrete square error rule definition error function of weighting
E (w)=W (w) [A (w)-Ad(w)] (2)
Wherein, E (w) represents Ad(w) weighted error between A (w)；Ad(w) h that will be approached is representedd(n) amplitude function,
hd(n) ideal filter is represented；A (w) represents h (n) amplitude function, and h (n) represents the wave filter of actual design；W (w) >=0 is
Weighting function.
(2) discrete square error Δ is weighted to be defined as：
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Wherein, (wm, m=1,2 ..., L) it is L sampled point in frequency domain.
(3) amplitude function of FIR filter is represented using following form, i.e.,：
A (w)=Q (w) G (w) (4)
Wherein：
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Wherein, Q (w)=cos (w/2), K=(N-1)/2, N are the filter orders of design.
(4) middle coefficient g (n) (n=1,2 ..., K) is solved by the thought of optimization, reality is solved further according to below equation
The first half of border coefficient h (n), afterwards by symmetry, solves all coefficient hs (n) for the FIR filter being actually needed.
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</mrow>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>)</mo>
</mrow>
<mo>/</mo>
<mn>4</mn>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>&lsqb;</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>/</mo>
<mn>4</mn>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>&lsqb;</mo>
<mn>2</mn>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>/</mo>
<mn>4</mn>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, n=1,2 ..., K-1.
(5) formula (4), (5) are substituted into formula (3), obtained：
<mrow>
<mi>&Delta;</mi>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>L</mi>
</munderover>
<msup>
<mrow>
<mo>{</mo>
<mi>W</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>K</mi>
</munderover>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mi>w</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>A</mi>
<mi>d</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>}</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Error function Δ is now expressed as matrix form, error vector Λ is defined：
Λ=(Λ1,Λ2,…,ΛL)T (8)
Wherein：
<mrow>
<msub>
<mi>&Lambda;</mi>
<mi>m</mi>
</msub>
<mo>=</mo>
<mi>W</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>K</mi>
</munderover>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mi>w</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>A</mi>
<mi>d</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Then overall error function Δ can be expressed as：
Δ=ΛTΛ (10)
Using matrix form, error vector Λ can be expressed as：
Λ=W (QCg-Ad) (11)
Wherein W and Q are L × L matrixes, i.e.,：
<mrow>
<mi>W</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>W</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mi>W</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<mi>W</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>L</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>Q</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>L</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
C is L × (K+1) matrix, i.e.,：
<mrow>
<mi>C</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>Kw</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msub>
<mi>w</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>Kw</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>w</mi>
<mi>L</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msub>
<mi>w</mi>
<mi>L</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>Kw</mi>
<mi>L</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
AdIt is the vector of L element, i.e.,：
Ad=[Ad(w1),Ad(w1),…,Ad(wL)] (15)
,, can be by solving equation WQCg=from formula (11) because WQC is L × L matrix as L=K+1
WAdObtain the null solution of error delta.Because error delta is equal to zero, so filtering of the weighting matrix W to actual design now
Device is not played a role.As L ＞ K+1, now the number of equation group is more than the number of unknown quantity, therefore equation group is without solution, if
Now WQC matrixes are sequency spectrum matrixes, then the minimal solution of the error function Δ existence anduniquess shown in formula (11).It can now lead to
Solution equation below is crossed, i.e.,：
(WQC)TWQCg=(WQC)TWAdWQC (16)
Middle coefficient vector g is solved, and then solves by formula (6) the unit impulse response h (n) of actual design.
4. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 1, it is characterised in that：
Specific design step is：
Step 1：Design binary channels FIR normal orthogonal mirror filter groups；
Step 2：The module based on binary channels FIR normal orthogonal mirror filter groups, non-homogeneous filter is built with reference to tree
Ripple device group；
Step 3：The non-homogeneous wave filter group reconstruction condition that this method is designed is derived；
Step 4：The reconstruction condition of whole non-homogeneous wave filter group is reduced to：The frequency response of prototype FIR filter is orthogonal
Amplitude at point w=pi/2s meets Hl(ejπ/2)=0.7071；
Step 5：Using the cut-off frequecy of passband w of iterative prototyping wave filterpIt is set to meet the reconstruction condition in step 4；
Step 6：During the specific design of ptototype filter, using discrete weightings square error Criterion Method prototype low pass
Wave filter.
5. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 4, it is characterised in that：
The non-homogeneous wave filter group that binary channels FIR normal orthogonal mirror filter groups build tree includes analysis module and comprehensive mould
Block.
6. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 5, it is characterised in that：
Binary channels FIR normal orthogonal mirror filter groups build low channel filtering in the non-homogeneous filter bank analysis module of tree
Device Hl(z) with high pass channel filter Hh(z) relation condition is set to：
Hh(z)=Hl(-z) (17)
Wherein, Hl(z), Hh(z) be respectively first passage and second channel in analysis module transmission function.
7. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 5, it is characterised in that：
Integration module is set to the filter relationship in analysis module：
Fl(z)=Hh(-z),Fh(z)=- Hl(-z) (18)
Wherein, Fl(z), Fh(z) be respectively first passage and second channel in integration module transmission function.
8. a kind of non-homogeneous wave filter group filtering method based on tree according to claim 4, it is characterised in that：
Use the reconstruction condition for the non-homogeneous wave filter group of tree that binary channels FIR normal orthogonal mirror filter groups build for：
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>M</mi>
</munderover>
<mo>|</mo>
<msub>
<mi>H</mi>
<mi>k</mi>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mi>w</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo><</mo>
<mi>w</mi>
<mo>&le;</mo>
<mfrac>
<mi>&pi;</mi>
<mi>M</mi>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, Hk(ejw) represent k-th of passage filter freguency response, w is Frequency point, and π is pi, and M represents non-homogeneous
The channel number of wave filter group.
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| CN109116312A (en) * | 2018-07-20 | 2019-01-01 | 电子科技大学 | Radar signal classification method based on QMFB and convolutional neural networks |
| CN111010146A (en) * | 2019-12-13 | 2020-04-14 | 哈尔滨工程大学 | Signal reconstruction structure based on fast filter bank and design method thereof |
| CN111780780A (en) * | 2020-06-16 | 2020-10-16 | 贵州省人民医院 | Step counting method and device based on filter bank |
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| CN108492271B (en) * | 2018-03-26 | 2021-08-24 | 中国电子科技集团公司第三十八研究所 | Automatic image enhancement system and method fusing multi-scale information |
| CN109116312A (en) * | 2018-07-20 | 2019-01-01 | 电子科技大学 | Radar signal classification method based on QMFB and convolutional neural networks |
| CN111010146A (en) * | 2019-12-13 | 2020-04-14 | 哈尔滨工程大学 | Signal reconstruction structure based on fast filter bank and design method thereof |
| CN111010146B (en) * | 2019-12-13 | 2023-08-01 | 哈尔滨工程大学 | Signal reconstruction structure based on fast filter bank and design method thereof |
| CN111780780A (en) * | 2020-06-16 | 2020-10-16 | 贵州省人民医院 | Step counting method and device based on filter bank |
| CN111780780B (en) * | 2020-06-16 | 2022-06-03 | 贵州省人民医院 | Step counting method and device based on filter bank |
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