CN107294512B - Non-uniform filter bank filtering method based on tree structure - Google Patents

Non-uniform filter bank filtering method based on tree structure Download PDF

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CN107294512B
CN107294512B CN201710378787.9A CN201710378787A CN107294512B CN 107294512 B CN107294512 B CN 107294512B CN 201710378787 A CN201710378787 A CN 201710378787A CN 107294512 B CN107294512 B CN 107294512B
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CN107294512A (en
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张春杰
田春雨
杨珑琪
李善双
郝东斌
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Harbin Engineering University
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    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
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    • H03H17/02—Frequency selective networks
    • H03H17/0202—Two or more dimensional filters; Filters for complex signals
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H17/00—Networks using digital techniques
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Abstract

The invention provides a tree structure-based non-uniform filter bank construction method. The method comprises the following steps: a two-channel FIR orthogonal mirror image filter bank is used as a basic module, and a tree structure is adopted to build a non-uniform filter bank. Through the derivation of the reconstruction condition of the whole tree structure non-uniform filter bank, the complete reconstruction condition of the whole system is finally summarized as the design of a prototype filter meeting specific conditions. The designed prototype filter meets the reconstruction condition by adopting the iteration passband cut-off frequency, and in the specific design process of the prototype filter, the invention adopts the discrete weighted square error criterion, and compared with other methods, the criterion can realize better passband flatness characteristic and larger stopband attenuation. The design scheme of the invention effectively reduces the iteration complexity, can obtain larger stop band attenuation and better system reconstruction performance, and has important application value for radar broadband channelized receivers, voice and image signal processing.

Description

Non-uniform filter bank filtering method based on tree structure
Technical Field
The invention belongs to the technical field of filter banks, and particularly relates to a non-uniform filter bank filtering method based on a tree structure.
Background
The multichannel filter bank theory is widely applied to the field of signal processing of radar, voice, images and the like, and the use of the technology effectively reduces the data processing rate requirement, the data storage space, the operation complexity and the like. A filter bank system can divide the frequency band of an input signal through a plurality of filters with different frequency band characteristics in a system front-end analysis module, then extracts the speed reduction, and then processes the sub-band signals of different frequency bands according to actual needs. The subband signals are then reconstructed into the required original signals at the back end of the system as much as possible by interpolation and a corresponding synthesis filter bank, so that the reconstruction performance of the whole filter bank is the research focus in the filter bank theory.
Filter banks are an important part of multirate signal processing and have been gaining attention in recent years. Filter banks are widely used in communications, speech coding, audio coding, and image signal processing. If the output and input of the system differ only by a proportional magnitude and a certain delay, the system is called a fully reconstructed system. And the signal is decomposed into sub-bands for post-processing, so that better effect can be obtained by utilizing the frequency characteristic of the signal. In a system consisting of an analysis filterbank and a synthesis filterbank, it is usually the aim of the filterbank design to make the reconstructed signal at the output identical to the original signal at the input. However, from a practical point of view, a simple design method with less restrictions and high efficiency is more valuable under the condition that distortion control is within a certain range. Multirate signal processing has wide applications in many fields such as communications, image coding, speech coding, radar, etc. The multi-rate technology can effectively reduce the processing complexity of signals, the transmission rate of data and the storage capacity.
The non-uniform filter bank can divide an input signal into sub-signals with different frequency bandwidths according to actual needs, and has better flexibility. Compared with a uniform filter bank, the non-uniform filter bank is more flexible in frequency spectrum division, so in recent years, the design research of the non-uniform filter bank draws attention of a plurality of scholars, and the scholars make great contribution in the aspects of the theory and design of the non-uniform filter bank. But achieving a complete reconstruction of the non-uniform filter bank has remained a design challenge to date. Because the optimized parameters are more, the design of the completely reconstructed non-uniform filter bank is more difficult, and the completely reconstructed design method is complicated, complex and difficult to realize, a flexible and simple approximate reconstruction design scheme is generally selected in the actual design of the non-uniform filter bank. Nguyen et al, in Signal Processing IEEE Transactionson, propose designing a non-uniform filter bank using a method of combining uniform filter banks. Xie X derives the reconstruction relationship of the non-uniform filter bank directly from the frequency domain in the document of A simple design method of linear-phased non-uniform filter banks with integer determination factors published in Circuits and Systems, and designs the non-uniform filter bank according to the relationship. Soni designs the Non-uniform filter bank with a tree structure in An optimized Design of Non-uniform filter using Blackman Window Family (International journal of Signal & Image Processing), and Kumar designs the Non-uniform filter bank with a tree structure in a Design of near implementation filter bank and a loop of connected equal FIR technology (Applied Soft Computing), and simplifies the iterative objective function. The above non-uniform filter bank construction scheme has the problems of aliasing error, amplitude distortion, or phase distortion, and the like, and the reconstruction performance of the whole system needs to be improved.
Disclosure of Invention
The invention aims to provide a non-uniform filter bank filtering method based on a tree structure, which solves the problem that the existing non-uniform filter bank construction scheme can not design a non-uniform filter system with good reconstruction performance.
The purpose of the invention is realized as follows:
a tree structure-based non-uniform filter bank construction method comprises the following steps:
the first step is as follows: setting specific parameters of the prototype filter, including coefficient length N and passband cut-off frequency wpStop band cut-off frequency wsInitial iteration step and iteration termination error Recei;
the second step is that: prototype filter h design using discrete weighted square error criterionl(n) then solve for hl(n) a value Real of the frequency response at the quadrature mirror point w ═ 0.5 pi;
the third step: judging whether the actual error is smaller than a set termination error Recei, namely whether the following formula is satisfied:
|Real-0.7071|＜Recei (1)
if the above equation is true, the filter h is usedl(n) solving for hh(n) then labeling with two channelsConstructing a tree-structure non-uniform filter bank by the quasi-orthogonal mirror image filter bank; if not, further judging the size between Real and 0.707: if Real > 0.707, wp＝wp-step, step ═ step/2; if Real is less than 0.707, wp＝wp+ step, step being step/2, step becoming half of the original after each iteration; wherein h ish(n) denotes a high-pass filter, hlAnd (n) represents a low-pass filter.
The fourth step: updating the passband cutoff frequency wpThen using the new wpRedesigning the low-pass filter hlAnd (n) sequentially iterating until the error value is smaller than the given error range.
For a non-uniform filter bank filtering method based on a tree structure, a prototype filter H is enabled to be used by iterating the cut-off frequency of the filter passbandl(z) satisfying the derived reconstruction condition; in the specific design process of the prototype filter, the prototype low-pass filter is designed by adopting a discrete weighted square error criterion method.
Wherein Hl(z) is a low-pass filter hl(n) transfer function.
For a non-uniform filter bank filtering method based on a tree structure, the discrete weighted square error criterion method comprises the following specific processes:
defining an error function using a weighted discrete square error criterion
E(w)＝W(w)[A(w)-Ad(w)](2)
Wherein E (w) represents Ad(w) weighted error between (w) and a (w); a. thed(w) represents h to be approximatedd(n) amplitude function, hd(n) represents an ideal filter; a (w) represents the amplitude function of h (n), and h (n) represents the actually designed filter; w (w) ≧ 0 is a weighting function.
The weighted discrete squared error Δ is defined as:
Figure BDA0001304626300000031
wherein (w)mAnd m 1, 2.., L) isL sample points in the frequency domain.
The amplitude function of the FIR filter is represented in the form:
A(w)＝Q(w)G(w) (4)
wherein:
Figure BDA0001304626300000032
where q (w) cos (w/2), K (N-1)/2, and N is the designed filter order.
The intermediate coefficients g (n) (1, 2.. multidot., K) are solved by the optimization idea, the first half of the actual coefficients h (n) are solved according to the following formula, and then all the coefficients h (n) of the actually required FIR filter are solved by symmetry.
Figure BDA0001304626300000033
Wherein, n is 1, 2.
Substituting the formulas (4) and (5) into the formula (3) to obtain:
Figure BDA0001304626300000034
now the error function Δ is represented in matrix form, defining an error vector Λ:
Λ＝(Λ1,Λ2,…,ΛL)T(8)
wherein:
Figure BDA0001304626300000035
the total error function Δ can be expressed as:
Δ＝ΛTΛ (10)
in matrix form, the error vector Λ can be expressed as:
Λ＝W(QCg-Ad) (11)
where W and Q are L×L matrices, i.e.:
Figure BDA0001304626300000041
Figure BDA0001304626300000042
c is a matrix of L× (K +1), i.e.:
Figure BDA0001304626300000043
Adis a vector of L elements, namely:
Ad＝[Ad(w1),Ad(w1),…,Ad(wL)](15)
when L is K +1, because WQC is a L×L matrix, we can understand from equation (11) that WQCg is WAdWhen L > K +1, the number of equations in the set is greater than the number of unknowns, so the set has no solution, if the WQC matrix is a column full rank matrix, then there is a unique minimum solution for the error function Δ shown in equation (11). The following equations can be solved:
(WQC)TWQCg＝(WQC)TWAdWQC (16)
and solving an intermediate coefficient vector g, and further solving a unit impulse response h (n) of the actual design through a formula (6).
The non-uniform filter bank filtering method based on the tree structure of the dual-channel FIR standard orthogonal mirror filter bank comprises the following specific design steps:
step 1: designing a two-channel FIR standard orthogonal mirror filter bank;
step 2: a two-channel FIR standard orthogonal mirror filter bank is used as a basic module, and a non-uniform filter bank is built by combining a tree structure;
and step 3: deducing the reconstruction condition of the non-uniform filter bank designed by the method;
and 4, step 4: the reconstruction condition of the whole non-uniform filter bank is simplified as follows: the frequency response of the prototype FIR filter has an amplitude at the quadrature point w ═ pi/2 that satisfies Hl(ejπ/2)＝0.7071；
And 5: adopting the passband cut-off frequency of the iterative prototype filter to enable the passband cut-off frequency to meet the reconstruction condition in the step 4;
step 6: in the specific design process of the prototype filter, the prototype low-pass filter is designed by adopting a discrete weighted square error criterion method.
For the non-uniform filter bank filtering method based on the tree structure, the non-uniform filter bank with the tree structure built by the two-channel FIR standard orthogonal mirror filter bank comprises an analysis module and a synthesis module.
Low-channel filter H in analysis modulel(z) and a high-pass filter Hh(z) the relational conditions are set as:
Hh(z)＝Hl(-z)(17)
wherein Hl(z)，Hh(z) are transfer functions of the first channel and the second channel, respectively, in the analysis module.
The filter relationship between the synthesis module and the analysis module is set as:
Fl(z)＝Hh(-z),Fh(z)＝-Hl(-z) (18)
wherein, Fl(z)，Fh(z) are transfer functions of the first channel and the second channel, respectively, in the integrated module.
For a non-uniform filter bank filtering method based on a tree structure, the reconstruction condition of the tree structure non-uniform filter bank built by adopting a two-channel FIR standard orthogonal mirror filter bank is as follows:
Figure BDA0001304626300000051
wherein Hk(ejw) Representing the filter frequency response of the kth channel, w is the frequency point, pi is the circumference ratio, M represents the number of channels of the non-uniform filter bank。
The invention has the beneficial effects that: compared with the existing design scheme, the invention simplifies iterative reconstruction conditions, and applies the discrete weighted square error criterion to the design of the non-uniform filter bank built by the tree structure, so that the whole filter system ensures the linear phase of each channel, and simultaneously, the stop band attenuation of each channel and the amplitude distortion of the whole system are further improved, thereby improving the reconstruction performance of the non-uniform filter bank.
Drawings
FIG. 1 is a tree-structured 4-channel non-uniform filter bank of the present invention;
FIG. 2 is an equivalent diagram of tree 4-way of the present invention;
FIG. 3 is a graph showing the comparison of the performance of FIR filters designed by different methods such as window function method, feature filtering method, equiripple method, and discrete weighted square error criterion;
FIG. 4 is a flow chart of the non-uniform filter bank iteration algorithm of the present invention;
FIG. 5 is a graph of the magnitude response of a prototype FIR filter of the present invention;
FIG. 6 is a simulation diagram of a tree-structured 6-channel non-uniform filter bank according to the present invention;
FIG. 7 is a graph of tree-structured 6-channel non-uniform filter bank amplitude distortion in accordance with the present invention.
Detailed Description
The following description, with reference to the drawings, specifically describes the design scheme of the present invention:
step 1: a two-channel FIR standard orthogonal mirror filter set is adopted to build a tree-structured non-uniform filter set, FIG. 1 is a tree-structured 4-channel non-uniform filter set, FIG. 2 is a tree-structured 4-channel equivalent diagram, and in a two-channel filter system, a low-channel filter H in an analysis modulel(z) and a high-pass filter Hh(z) the relational conditions are set as:
Hh(z)＝Hl(-z) (1)
the filter relationship between the synthesis module and the analysis module is set as:
Fl(z)＝Hh(-z),Fh(z)＝-Hl(-z) (2)
the whole system is free from aliasing distortion and phase distortion.
Step 2: the channel filters in fig. 2 can be represented as:
Figure BDA0001304626300000061
Figure BDA0001304626300000062
and is
Figure BDA0001304626300000063
Wherein H0(z)，H1(z)，H2(z)，H3(z) is the transfer function of each channel in FIG. 2.
And step 3: the non-uniform filter bank reconstruction condition in fig. 2 now is
Figure BDA0001304626300000064
Wherein Hk(ejw) Representing the filter frequency response of the k-th channel.
And 4, step 4: substituting equations (3) and (5) into the reconstruction condition (6) can simplify:
|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 the formula (7), the orthogonal frequency point w ═ pi/2 is simplified to obtain:
2|Hl(ejπ/2)|6+|Hl(ejπ/2)|4+|Hl(ejπ/2)|2＝1 (8)
solving a high order equation (8) to obtain:
Hl(ejπ/2)＝0.7071 (9)
the channel number of the non-uniform filter bank formed by the tree structure is popularized to M, and the extraction interpolation rate of each channel is set to be (2)M-1,2M-1,2M-2…,4,2), the channel coefficients in the equivalent non-uniform filter bank analysis module can be expressed as:
Figure BDA0001304626300000071
and is
Figure BDA0001304626300000072
Substituting the coefficient relationships included in the formulas (10) and (11) into the M-channel reconstruction conditional expression (6) can obtain:
Figure BDA0001304626300000073
with ejwReplacing z in the formula, and taking the orthogonal frequency point w as pi/2 to solve the z, obtaining:
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)
solving the higher order equation (13) yields the same solution H as equation (9)l(ejπ/2) 0.7071, the reconstruction condition of the non-uniform filter bank built by combining the two-channel FIR orthonormal mirror filter bank with the tree structure is Hl(ejπ/2)＝0.7071。
Using an iterative prototype filter Hl(z) such that the passband cut-off frequency satisfies the reconstruction condition (9). Prototype filter H designed using discrete weighted square error criterion in specific design processl(z). The method adopts optimization idea to make the actually designed filter frequency response H (e)jw) Infinite near ideal frequency response Hd(ejw) The error between the two is minimized. Compared with the window function method, the equiripple method and the characteristic filter method, the discrete weighted square error criterion can obtain a flatter pass band and a larger stop band attenuation under the condition of the same design parameters. The specific design process is as follows: the error function is first defined using a weighted discrete square error criterion.
E(w)＝W(w)[A(w)-Ad(w)](14)
The weighted discrete squared error Δ is defined as:
Figure BDA0001304626300000081
the number of frequency samples L is used to characterize the passband and stopband performance the idea of this method to design the FIR filter is to minimize the error Δ defined by equation (15) the amplitude function of the FIR filter is expressed as follows:
A(w)＝Q(w)G(w) (16)
wherein:
Figure BDA0001304626300000082
through the optimization idea, the intermediate coefficient g (n) (1, 2.. multidot.k) is solved, then the first half of the actual coefficient h (n) is solved according to the following formula, and then all the coefficients h (n) of the actually required FIR filter are solved through symmetry.
Figure BDA0001304626300000083
Wherein, n is 1, 2.
Substituting the formulas (16) and (17) into the formula (15) to obtain:
Figure BDA0001304626300000084
now the error function Δ is represented in matrix form, defining an error vector Λ:
Λ＝(Λ1,Λ2,…,ΛL)T(20)
wherein:
Figure BDA0001304626300000085
the total error function Δ can be expressed as:
Δ＝ΛTΛ (22)
in matrix form, the error vector Λ can be expressed as:
Λ＝W(QCg-Ad) (23)
where W and Q are L×L matrices, i.e.:
Figure BDA0001304626300000091
Figure BDA0001304626300000092
c is a matrix of L× (K +1), i.e.:
Figure BDA0001304626300000093
Adis a vector of L elements, namely:
Ad＝[Ad(w1),Ad(w1),…,Ad(wL)](27)
when L is K +1, since WQC is a L×L matrix, we can understand from equation (23) that WQCg is WAdWhen L is greater than K +1, the number of the equation set is greater than the number of the unknowns, so the equation set has no solution, if the WQC matrix is a column full rank matrix, then the error function Delta shown in the formula (23) has the only maximumAnd (5) solving the small solution. This can be done by solving the following equation:
(WQC)TWQCg＝(WQC)TWAdWQC (28)
the intermediate coefficient vector g is solved, and then the unit impulse response h (n) of the actual design is solved through the formula (18).
Then, an FIR filter is designed by respectively adopting a characteristic filter method, an equiripple design method, a window function design method and a discrete weighted square error criterion method, and the results are compared and analyzed. FIG. 3 is a graph comparing the performance of the window function method, the feature filtering method, the equiripple method and the discrete weighted square error criterion in the present invention, and the specific parameter performance is shown in Table 1.
TABLE 1 comparison of specific parameter characteristics of FIR filters constructed by different methods
Figure BDA0001304626300000101
As can be seen from table 1, under the condition of the same design parameters, a larger stopband attenuation can be achieved by using the discrete weighted square error criterion, so as to better suppress out-of-band signals.
FIG. 4 is a flowchart of an iterative algorithm of the non-uniform filter bank of the present invention, and the following steps are given in conjunction with FIG. 4 to design the non-uniform filter bank:
step 1:
the first step is as follows: setting specific parameters of the prototype filter, including coefficient length N and passband cut-off frequency wpStop band cut-off frequency wsInitial iteration step and iteration termination error Recei;
the second step is that: prototype filter h design using discrete weighted square error criterionl(n) then solve for hl(n) a value Real of the frequency response at the quadrature mirror point w ═ 0.5 pi;
the third step: judging whether the actual error is smaller than a set termination error Recei, namely whether the following formula is satisfied:
|Real-0.7071|＜Recei (29)
if the above equation is true, the filter h is usedl(n) obtainingIs solved out of hh(n), then, constructing a tree-structure non-uniform filter bank by using a two-channel FIR standard orthogonal mirror filter bank; if not, further judging the size between Real and 0.707: if Real > 0.707, wp＝wp-step, step ═ step/2; if Real is less than 0.707, wp＝wp+ step, step being step/2, step becoming half of the original after each iteration; wherein h ishAnd (n) represents a high-pass filter.
The fourth step: updating the passband cutoff frequency wpThen using the new wpRedesign of the filter hlAnd (n) sequentially iterating until the error value is smaller than the given error range.
To verify the effectiveness of the present invention, simulation experiments were performed. The invention designs a 6-channel non-uniform filter bank with (16,16,8,4,4,4) extraction interpolation rates of all channels, and sets an iteration termination error to 10-4To ensure good accuracy, the iteration step is set to step 0.15 pi. And defining an amplitude distortion function Amdis to characterize the reconstruction performance of the whole system, namely:
Figure BDA0001304626300000111
wherein Hm(ejw) Representing the filter frequency response of the mth channel.
Prototype filter hlThe specific parameters of (n) are as follows: coefficient length N is 63 and passband cut-off frequency is wp0.41 pi, stop band cut-off frequency ws0.65 pi. Construction of a filter h using a discrete weighted square error criterionl(n), FIG. 5 is the amplitude response curve of the prototype FIR filter of the present invention, and it can be seen from FIG. 5 that the stop band attenuation is now AsFig. 6 is a simulation diagram of a tree-structured 6-channel non-uniform filter bank of the present invention, fig. 7 is an amplitude distortion diagram of the tree-structured 6-channel non-uniform filter bank of the present invention, where the maximum value of the amplitude distortion is max (amdis) ═ 1.3 × 10-3. The method of the present invention was compared to existing design methods as shown in table 2.
TABLE 2 comparison of Performance between different design methods
Figure BDA0001304626300000112
Figure BDA0001304626300000121
TABLE 3 comparison of Performance between the design method herein and the Kumar design method
Figure BDA0001304626300000122
As can be seen from table 3, the design method herein has an average increase in stopband attenuation of 59.6% and an average increase in amplitude distortion of 37.6% compared to the Kumar design method. By combining the analysis, after the discrete weighted square error criterion is applied to the design of the non-uniform filter bank built by the tree structure, the whole filter system ensures the linear phase of each channel, and simultaneously the stop band attenuation of each channel and the amplitude distortion of the whole system are further improved, so that the reconstruction performance of the non-uniform filter bank is improved.

Claims (6)

1. A non-uniform filter bank filtering method based on a tree structure is characterized in that:
the first step is as follows: setting specific parameters of the prototype filter, including coefficient length N and passband cut-off frequency wpStop band cut-off frequency wsInitial iteration step and iteration termination error Recei;
the second step is that: prototype filter h design using discrete weighted square error criterionl(n) then solve for hl(n) a value Real of the frequency response at the quadrature mirror point w ═ 0.5 pi;
the third step: judging whether the actual error is smaller than a set termination error Recei, namely whether the following formula is satisfied:
|Real-0.7071|＜Recei (1)
if the above equation is true, the filter h is usedl(n) solving for hh(n), then, building a tree-structure non-uniform filter bank by using a two-channel standard orthogonal mirror filter bank; if not, further judging the size between Real and 0.707: if Real > 0.707, wp＝wp-step, step ═ step/2; if Real is less than 0.707, wp＝wp+ step, step being step/2, step becoming half of the original after each iteration; wherein h ish(n) denotes a high-pass filter, hl(n) represents a low-pass filter;
the fourth step: updating the passband cutoff frequency wpThen using the new wpRedesigning the low-pass filter hl(n) sequentially iterating until the error value is smaller than a given error range;
by iterating the filter passband cut-off frequency, the prototype filter Hl(z) satisfying the derived reconstruction condition; in the specific design process of the prototype filter, designing the prototype low-pass filter by adopting a discrete weighted square error criterion method;
wherein Hl(z) is a low-pass filter hl(n) a transfer function;
the discrete weighted square error criterion method comprises the following specific processes:
(1) defining an error function using a weighted discrete square error criterion
E(w)＝W(w)[A(w)-Ad(w)](2)
Wherein E (w) represents Ad(w) weighted error between (w) and a (w); a. thed(w) represents h to be approximatedd(n) amplitude function, hd(n) represents an ideal filter; a (w) represents the amplitude function of h (n), and h (n) represents the actually designed filter; w (w) ≧ 0 is a weighting function;
(2) the weighted discrete squared error Δ is defined as:
Figure FDA0002494591090000011
wherein (w)mM 1, 2.., L) is L sample points in the frequency domain;
(3) the amplitude function of the FIR filter is represented in the form:
A(w)＝Q(w)G(w) (4)
wherein:
Figure FDA0002494591090000021
wherein q (w) cos (w/2), K ═ N-1)/2, and N is the designed filter order;
(4) solving intermediate coefficients g (n) (1, 2,., K) through an optimization idea, solving the first half of actual coefficients h (n) according to the following formula, and then solving all coefficients h (n) of the actually needed FIR filter through symmetry;
Figure FDA0002494591090000022
wherein n is 1, 2., K-1;
(5) substituting the formulas (4) and (5) into the formula (3) to obtain:
Figure FDA0002494591090000023
now the error function Δ is represented in matrix form, defining an error vector Λ:
Λ＝(Λ1,Λ2,…,ΛL)T(8)
wherein:
Figure FDA0002494591090000024
the total error function Δ can be expressed as:
Δ＝ΛTΛ (10)
in matrix form, the error vector Λ can be expressed as:
Λ＝W(QCg-Ad) (11)
where W and Q are L×L matrices, i.e.:
Figure FDA0002494591090000025
Figure FDA0002494591090000026
c is a matrix of L× (K +1), i.e.:
Figure FDA0002494591090000031
Adis a vector of L elements, namely:
Ad＝[Ad(w1),Ad(w1),…,Ad(wL)](15)
when L is K +1, because WQC is a L×L matrix, we can understand from equation (11) that WQCg is WAdWhen L is more than K +1, the number of the equation set is more than the number of the unknown quantity, so that the equation set has no solution, if the WQC matrix is a column full rank matrix, a unique minimum solution exists in the error function delta shown in the formula (11), and the following equations can be solved:
(WQC)TWQCg＝(WQC)TWAdWQC (16)
and solving an intermediate coefficient vector g, and further solving a unit impulse response h (n) of the actual design through a formula (6).
2. The tree-based non-uniform filter bank filtering method according to claim 1, wherein: the specific design steps are as follows:
step 1: designing a two-channel FIR standard orthogonal mirror filter bank;
step 2: a two-channel FIR standard orthogonal mirror filter bank is used as a basic module, and a non-uniform filter bank is built by combining a tree structure;
and step 3: deducing the reconstruction condition of the non-uniform filter bank designed by the method;
and 4, step 4: the reconstruction condition of the whole non-uniform filter bank is simplified as follows: the frequency response of the prototype FIR filter has an amplitude at the quadrature point w ═ pi/2 that satisfies Hl(ejπ/2)＝0.7071；
And 5: passband cutoff frequency w using an iterative prototype filterpMaking it meet the reconstruction condition in step 4;
step 6: in the specific design process of the prototype filter, the prototype low-pass filter is designed by adopting a discrete weighted square error criterion method.
3. The tree-based non-uniform filter bank filtering method according to claim 2, wherein: the non-uniform filter bank with the tree structure built by the two-channel FIR standard orthogonal mirror filter bank comprises an analysis module and a synthesis module.
4. A tree based non-uniform filter bank filtering method according to claim 3, characterized in that: low-channel filter H in non-uniform filter bank analysis module with tree-shaped structure built by two-channel FIR (finite impulse response) standard orthogonal mirror filter bankl(z) and a high-pass filter Hh(z) the relational conditions are set as:
Hh(z)＝Hl(-z) (17)
wherein Hl(z)，Hh(z) are transfer functions of the first channel and the second channel, respectively, in the analysis module.
5. The tree-based non-uniform filter bank filtering method according to claim 2, wherein: the filter relationship between the synthesis module and the analysis module is set as:
Fl(z)＝Hh(-z),Fh(z)＝-Hl(-z) (18)
wherein, Fl(z)，Fh(z) are transfer functions of the first channel and the second channel, respectively, in the integrated module.
6. The tree-based non-uniform filter bank filtering method according to claim 2, wherein: the reconstruction condition of the tree-structure non-uniform filter bank built by adopting the two-channel FIR standard orthogonal mirror filter bank is as follows:
Figure FDA0002494591090000041
wherein Hk(ejw) Represents the filter frequency response of the kth channel, w is the frequency point, pi is the circumferential ratio, and M represents the number of channels of the non-uniform filter bank.
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