CN118748549B - Group delay optimization design method of two-channel lattice type orthogonal filter group - Google Patents
Group delay optimization design method of two-channel lattice type orthogonal filter group Download PDFInfo
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Abstract
本发明公开了一种两通道格型正交滤波器组的群延迟优化设计方法，该方法首先根据滤波器阶数，给定个格型系数值，获得一个原型，利用梯度迭代算法最小化完美重构正交滤波器组的加权最小二乘目标函数，再利用Lim‑Lee‑Chen‑Yang算法更新目标函数中的加权函数部分，获得一组最优的格型系数。接着使用得到的最优格型系数系数作为初始值，利用泰勒一阶近似的思想，优化由目标函数与群延迟优化的初始目标函数加权得到的目标函数，并且把群延迟的最值差作为一个观察量，当最值差前后相差很小时，认为得到了最优原型低通滤波器。该方法能够有效改善滤波器的非线性相位问题，减少信号误差，满足完美重构特性。
The present invention discloses a group delay optimization design method for a two-channel lattice orthogonal filter bank. The method firstly , given grid coefficient values, and obtain a prototype , using the gradient iteration algorithm to minimize the weighted least squares objective function of the perfect reconstruction orthogonal filter bank , and then use the Lim‑Lee‑Chen‑Yang algorithm to update the objective function The weighted function part in is used to obtain a set of optimal lattice coefficients. Then, the optimal lattice coefficients are used as the initial values, and the Taylor first-order approximation is used to optimize the objective function The initial objective function for group delay optimization is The objective function is obtained by weighting, and the maximum difference of group delay is taken as an observation. When the maximum difference is very small, it is considered that the optimal prototype low-pass filter is obtained. This method can effectively improve the nonlinear phase problem of the filter, reduce signal errors, and meet the perfect reconstruction characteristics.
Description
Technical Field
The invention belongs to the technical field of digital signal processing, relates to coefficient optimization of a filter bank, and in particular relates to a group delay optimization design method of a two-channel lattice type orthogonal filter bank.
Background
The two-channel orthogonal filter bank is the earliest proposed and used filter bank in the digital filter bank, and the orthogonal filter bank with the lattice structure can well meet the aliasing-free condition of signal reconstruction and is widely applied to the fields of spectrum analysis, audio and video decoding, self-adaptive filtering, medical signal processing and the like. The front low-pass analysis filter and the high-pass analysis filter in the two-channel lattice orthogonal filter bank can decompose the signal to be processed into two groups of subband signals, and the subband signals are respectively processed according to processing requirements, so that the complexity of data processing operation is reduced, and the processed signals are reconstructed through the rear comprehensive filter bank to recover the original signals.
Since the two-channel lattice orthogonal filter bank mainly comprises an analysis filter bank and a synthesis filter bank, in order to meet the aliasing-free condition of signals, the synthesis filter bank must be designed into an association form of the analysis filter bank, and the analysis filter bank is composed of a pair of orthogonal low-pass-high-pass filters, the problem of designing the two-channel lattice orthogonal filter bank can be converted into the problem of optimizing the coefficients of a single prototype low-pass filter H 0 (z), and the objective function f of the two-channel lattice orthogonal filter bank is composed of a weighting function and a nonlinear function of the lattice coefficients of the low-pass analysis filter.
In the prior art, firstly, the problem of minimizing an objective function f is solved through a quasi-Newton algorithm, and then a Lim-Lee-Chen-Yang algorithm is used for updating a weighting function B (omega) in the objective function until an optimal perfect reconstruction orthogonal filter bank is obtained. But due toNonlinear phase filter, the filtered signal will generate certain error, which finally results in the filter set not meeting the perfect reconstruction characteristic, but the quasi-Newton algorithm cannot improveNonlinear phase problems of (a) are described.
Disclosure of Invention
Aiming at the defects of the prior art, a group delay optimization design method of a two-channel lattice type orthogonal filter bank is provided, the range of group delay is continuously reduced on the basis of a prototype low-pass filter, the optimized filter coefficient is obtained, and the prototype low-pass filter is improvedA problem of nonlinear phase of the optical element.
The group delay optimization design method of the two-channel lattice type orthogonal filter bank comprises the following specific steps:
Step one, determining a prototype low-pass filter according to design requirements The order of (2)Given a set of lattice coefficient values, a prototype low-pass filter is obtained:
S1.1, define a size ofLattice coefficient vector of (a)The method is used for storing a group of initial values of the lattice coefficients:
;
s1.2, acquisition of The lattice coefficient value is stored in the lattice coefficient vectorAs an initial coefficient value:
;
s1.3 using lattice coefficient vectors Obtaining a prototype lattice low-pass filter:
;
Wherein:
;
;
step two, defining a weighted least square objective function as a criterion for optimizing the perfect reconstruction orthogonal filter bank:
;
Wherein the method comprises the steps of The frequency point is represented by a frequency point,Representing the stop band cut-off frequency point,Is a weighting function that is used to determine the weight of the object,Is a nonlinear function of the lattice coefficients. Minimizing objective functions using gradient iterative algorithmsThe method comprises the following specific steps:
s2.1 relating the objective function to the lattice coefficients Performing first-order Taylor expansion and neglecting higher-order terms to obtain coefficient error vectors:
;
S2.2 solving the objective functionIs the first order derivative of (2):
;
Wherein the method comprises the steps ofRepresentation ofIs used for the conjugation of (a),The real part in the curly brace is shown.
S2.3, leading the first order offsetThe definition is as follows:
;
Wherein:
;
;
s2.4, order Definition ofFor storing the objective function value obtained in the previous cycle. Solving the current objective functionIf the value of (1)Finish the objective functionAnd (3) entering step three, otherwise, the current is to be calculatedThe value is passed toUpdating lattice coefficient vectorsBack s1.3:
;
step three, updating the weighting function by utilizing Lim-Lee-Chen-Yang algorithm Defining cut-off parametersDefinition ofFor storing the cut-off parameters obtained in the previous cycleIs a value of (2). Lattice coefficient vector based on step twoCalculated to obtainIf it meetsStep four is entered, otherwise the current state is to beIs passed toS1.3 is returned.
Fourth, based on the objective function and group delay of the perfectly reconstructed orthogonal filter bank, constructing an integral objective function, and solving parameters of the filter bank:
s4.1, lattice coefficient vector based on the above steps And a low pass filterCalculating phase-frequency responseAnd group delay:
;
Wherein the method comprises the steps ofThe representation takes the value of the imaginary part,Representing the real part value.
;
Wherein:
;
;
;
;
;
s4.2 defining a new function To characterize the initial objective function of group delay optimization:
;
Wherein the method comprises the steps of Is a constant variable. Taking the most significant difference of group delay as a measurement index for measuring group delay optimization, and recording as:
;
Pair functionPerforming first-order Taylor expansion, and ignoring higher-order terms:
;
;
Wherein:
;
;
;
;
;
s4.3, the objective function of the perfectly reconstructed orthogonal filter bank will be optimized Initial objective function optimized with group delayAnd (3) carrying out weighted sum as a final objective function of group delay optimization, and carrying out first-order Taylor approximation on the final objective function to convert the optimization problem into:
;
Wherein, 、Respectively represent the determination、Is a function of the infinite norm of (c),、As a weight variable,、Is a trusted domain boundary variable. Order theSetting according to design requirements、Is a value of (2). CVX optimization function toolbox solution in Matlab softwareAndUpdating the value of (2)And:
;
S4.4, return s4.2 calculationThe value is recorded asIf (if)Output optimal prototype low-pass filterOtherwise letCalculating an objective functionLow pass filterObjective functionFirst order bias guide of (a)And returns s4.1.
The invention has the following beneficial effects:
The method uses the thought of first-order Taylor approximation, takes the group delay as a part of an optimization target, avoids the complexity of calculating an approximate Hessian matrix by a quasi-Newton algorithm, greatly reduces the calculated amount and the calculated time, converts the original non-convex optimization problem of the filter coefficient into the convex optimization problem of the coefficient increment, acquires the optimal lattice coefficient through iteration, takes the optimal lattice coefficient as an initial value, continuously optimizes the group delay, flexibly changes the value of a variable according to the design requirement, and finally effectively improves the problem of the nonlinear phase of the filter to be close to the linearity. Therefore, errors introduced by signals filtered by the lattice type orthogonal filter are reduced, and the filter bank can better meet perfect reconstruction characteristics.
Drawings
FIG. 1 is a low-pass analysis filter optimized using a quasi-Newton algorithm in accordance with an embodimentA phase frequency response diagram;
FIG. 2 is a low-pass analysis filter optimized using the method in embodiment one A phase frequency response diagram;
FIG. 3 is a low-pass analysis filter optimized using a quasi-Newton algorithm in embodiment two A phase frequency response diagram;
FIG. 4 is a low-pass analysis filter optimized using the method in embodiment two Phase-frequency response diagram.
Detailed Description
The invention is further explained below with reference to the drawings;
The group delay optimization design method of the two-channel lattice type orthogonal filter bank specifically comprises the following steps:
step one, determining a prototype low-pass filter according to design requirements The order of (2)Stop band cut-off frequency pointAnd give a group ofIndividual lattice coefficient values from which a prototype low-pass filter is obtained。
Step two, defining a weighted least square objective function as a criterion for optimizing the perfect reconstruction orthogonal filter bank:
;
Wherein the method comprises the steps of The frequency point is represented by a frequency point,Representing the stop band cut-off frequency point,Is a weighting function that is used to determine the weight of the object,Is a nonlinear function of the lattice coefficients.
A first order Taylor approximation is performed on the objective function to obtain a set of coefficient error values for updating the lattice coefficient values. And recording the objective function value at the same time, if the calculated objective function value is larger than the objective function value recorded before, entering a step III, otherwise updating the lattice coefficient value, returning to the step I, and calculating a new prototype lattice low-pass filter.
Step three, updating the weighting function in the step two by utilizing a Lim-Lee-Chen-Yang algorithmWhen the updated result meets the set cutoff condition, a set of preferred lattice coefficient values and preferred values are considered to be obtainedAnd step four, if not, returning to the step one, and calculating a new prototype lattice low-pass filter by using the updated lattice coefficient value.
Step four, calculating the currently obtained lattice coefficient value and a low-pass filterCorresponding group delay function and weighting least square objective functionAnd group delay functionA new function is constructed as an overall objective function, and a first-order Taylor approximation is performed on the overall objective function to obtain a set of coefficient error values for updating the lattice coefficient values. Repeating the fourth step, recording the integral objective function value at the same time, if the calculated integral objective function value is larger than the value recorded before, ending the circulation, completing the optimization process, and outputting the lattice coefficient value and the low-pass filter at the moment。
To demonstrate the effectiveness of the method, the following two examples demonstrate the comparison of the performance of filters optimized using the method and the quasi-newton algorithm, respectively, for the same orthogonal filter.
Example 1
The present embodiment sets a prototype low-pass analysis filterThe order of (2)Normalized stop band cut-off frequencyAfter the passband cut-off frequency is normalized,、. The group delay is optimized by a quasi-Newton algorithm and the method respectively, and after the optimizationThe results of the phase-frequency response of (2) are shown in fig. 1 and 2.
Example two
The present embodiment sets a prototype low-pass analysis filterThe order of (2)Normalized stop band cut-off frequencyAfter the passband cut-off frequency is normalized,,. The group delay is optimized by a quasi-Newton algorithm and the method respectively, and after the optimizationThe results of the phase-frequency response of (2) are shown in fig. 1 and 2.
As can be seen by comparing fig. 1, 2 with fig. 3 and 4, the prototype low-pass analysis filter optimized by the methodThe radian of the phase frequency response curve is smaller and more approximate to a straight line, and the nonlinear problem is improved.
Specific optimization result data pairs are shown in table 1:
TABLE 1
;
As can be seen from the data in Table 1 and the accompanying drawings, compared with the quasi-Newton algorithm, the method has the advantages that the maximum difference of group delay and the frequency selectivity of the filter are greatly improved, the radian of the phase-frequency response curve is also smaller, namely, the low-pass analysis filter in the two-channel lattice type orthogonal filter bankThe non-linearity of the phase of (c) has been well optimized.
Claims (6)
1. The group delay optimization design method of the two-channel lattice orthogonal filter bank comprises the steps of firstly determining an order N of a prototype low-pass filter H 0 (z) and a stop band cut-off frequency point omega s according to design requirements, giving a set of N lattice coefficient values, obtaining the prototype low-pass filter H 0 (z) through the set of the lattice coefficient values, and defining an objective function f as a criterion for optimizing a perfect reconstruction orthogonal filter bank:
Wherein ω represents a frequency point, ω s represents a stop band cut-off frequency point, B (ω) is a weighting function, and H 0(ejω) is a nonlinear function with respect to lattice coefficients, characterized in that the gradient iterative algorithm is used to minimize the objective function f, update the lattice coefficient values, and calculate the values of the new prototype lattice low-pass filter and the objective function f;
When the objective function f is not reduced any more, updating the weighting function B (omega) by utilizing a Lim-Lee-Chen-Yang algorithm, and when the updating result meets a set cut-off condition, calculating a group delay function g corresponding to a currently obtained lattice coefficient value and a low-pass filter H 0 (z), wherein the group delay function g is as follows:
g=groupdelay(ω)-q
where q is a constant variable, groupdelay (ω) represents the group delay;
the objective function f and the group delay function g are weighted and summed to obtain a final objective function, and the final objective function is subjected to first-order Taylor approximation to convert the optimization problem into:
Wherein norm (Δq, inf) and norm (Δα, inf) respectively represent the determination of Δq, Is an infinite norm of (2); representing the error vector of the lattice coefficient, epsilon 1、ε2 is a weight variable, zeta 1、ξ2 is a trust domain boundary variable, and solving deltaq and delta q Updating the values of constant variable q and lattice coefficient vector
Calculating an index delta of group delay optimization, comparing the index delta with an index delta now at the time of the previous round of updating, outputting an optimal prototype low-pass filter H 0 (z) if delta now-δ|≤10-6 is the index delta, otherwise, let delta=delta now, and calculating an objective function f, a low-pass filter H 0 (z) and a first-order partial derivative of the objective function fAnd updating the group delay function g;
And (3) ending the cycle until the integral objective function value reaches the set condition, and finishing the optimization process and outputting the lattice coefficient value and the low-pass filter H 0 (z) at the moment.
2. The method for optimizing the group delay of a two-channel lattice type orthogonal filter set as claimed in claim 1, wherein the method for obtaining the prototype low-pass filter H 0 (z) is as follows:
Defining a lattice coefficient vector of size Nx 1 For storing lattice coefficients, obtaining Xiang Gexing coefficient vectorsN lattice coefficient values are stored as initial coefficient values:
using lattice coefficient vectors Obtaining a prototype lattice low-pass filter H 0 (z):
Wherein:
3. the method for optimizing the group delay of a two-channel lattice type orthogonal filter bank as set forth in claim 1, wherein the method for minimizing the objective function f by using the gradient iterative algorithm is as follows:
s2.1, a definition variable f pre is used for storing the objective function value obtained in the previous cycle, and the objective function f is related to the lattice coefficient vector Performing first-order Taylor expansion and neglecting higher-order terms to obtain coefficient error vectors
The lattice coefficient vectorThe elements in the method are lattice coefficients, and the objective function f is calculatedIs the first order derivative of (2)
Wherein the method comprises the steps ofRepresentation ofIs used for the conjugation of (a), real {.
S2.2, leading the first order offsetThe definition is as follows:
Wherein:
K is more than or equal to 0 and less than or equal to N-1, and updating lattice coefficient vector
Using updated lattice coefficient vectorsCalculating a prototype lattice low-pass filter H 0 (z);
s2.3, let N (ω) =1, calculate the corresponding objective function f of the current prototype lattice low-pass filter H 0 (z), when f is less than or equal to f pre, transmit the value of current f to f pre, repeat s 2.1-s 2.2 until f > f pre.
4. The method for optimizing the group delay of the two-channel lattice type orthogonal filter set according to claim 1, wherein the method comprises the following steps:
wherein Imag (·) represents taking the imaginary value, real (·) represents taking the Real value; Representing the phase-frequency response:
I=Imag(H0(z))
R=Real(H0(z))
The maximum difference delta of the group delay groupdelay (omega) is used as an index for measuring the optimization of the group delay:
δ=max(groupdelay(ω))-min(groupdelay(ω))
Performing first-order taylor expansion on the group delay function g, and ignoring higher-order terms:
Wherein the method comprises the steps of Representing a lattice coefficient error vector:
5. The method for optimizing group delay of two-channel lattice type orthogonal filter set as claimed in claim 4, wherein the trust domain boundary variable ζ 1＝ξ2 =1 and the weight variable ε 1、ε2 are set according to design requirements.
6. A computer readable storage medium having stored thereon a computer program which, when executed in a computer, causes the computer to perform the method of any of claims 1-5.
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| CN201586640U (en) * | 2009-11-05 | 2010-09-22 | 江南大学 | A multi-station surface acoustic filter device testing and sorting control system |
| CN103152012A (en) * | 2013-03-04 | 2013-06-12 | 河海大学常州校区 | Far-infrared (FIR) filter designing method capable of reducing pass-band group delay error |
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| CN201586640U (en) * | 2009-11-05 | 2010-09-22 | 江南大学 | A multi-station surface acoustic filter device testing and sorting control system |
| CN103152012A (en) * | 2013-03-04 | 2013-06-12 | 河海大学常州校区 | Far-infrared (FIR) filter designing method capable of reducing pass-band group delay error |
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