A Group Delay Optimization Design Method for Two-Channel Lattice Orthogonal Filter Banks

Abstract

The invention discloses a group delay optimization design method for a two-channel lattice orthogonal filter group. The method first obtains a prototype according to the filter order, a given lattice coefficient value, and a gradient iteration algorithm is used to minimize the weighted least squares objective function of the perfect reconstruction orthogonal filter group, and then the Lim-Lee-Chen-Yang algorithm is used to update the weighted function part in the objective function to obtain a set of optimal lattice coefficients. Then, the optimal lattice coefficient coefficient obtained is used as the initial value, and the idea of ​​Taylor's first-order approximation is used to optimize the objective function obtained by weighting the objective function and the initial objective function optimized by the group delay, and the maximum value difference of the group delay is used as an observation. When the maximum value 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

[0001] The invention belongs to the technical field of digital signal processing, relates to coefficient optimization of a filter group, and specifically relates to a group delay optimization design method for a two-channel lattice orthogonal filter group. Background Art

[0002] The two-channel orthogonal filter group is the earliest proposed and used filter group in the digital filter group. The lattice structure of the orthogonal filter group can well meet the non-aliasing condition of signal reconstruction and is widely used in the fields of spectrum analysis, audio and video decoding, adaptive filtering and medical signal processing. The pre-low-pass analysis filter and high-pass analysis filter in the two-channel lattice orthogonal filter group can decompose the signal to be processed into two groups of sub-band signals, and process the sub-band signals separately according to the processing requirements, thereby reducing the complexity of data processing operations, and then reconstruct the processed signal through the post-comprehensive filter group to restore the original signal.

[0003] Since the two-channel lattice orthogonal filter group mainly includes the analysis filter group and the synthesis filter group, in order to meet the signal's aliasing-free condition, the synthesis filter group must be designed as a related form of the analysis filter group, and the analysis filter group is composed of a pair of orthogonal low-pass and high-pass filters. Therefore, the design problem of the two-channel lattice orthogonal filter group can be transformed into the coefficient optimization design problem of a single prototype low-pass filter H0(z), and its objective function f is composed of a weighting function and a nonlinear function of the lattice coefficients of a low-pass analysis filter.

[0004] The existing technology first solves the problem of minimizing the objective function f by using the quasi-Newton algorithm, and then uses the Lim-Lee-Chen-Yang algorithm to update the weighting function B(ω) in the objective function until an optimal perfect reconstruction orthogonal filter bank is obtained. It is a nonlinear phase filter. The signal after filtering will produce certain errors, which eventually leads to the filter bank not being able to meet the perfect reconstruction characteristics, and the quasi-Newton algorithm cannot improve it. The nonlinear phase problem. Summary of the invention

[0005] Aiming at the shortcomings of the existing technology, a group delay optimization design method for a two-channel lattice orthogonal filter bank is proposed. On the basis of the prototype low-pass filter, the range of group delay is continuously reduced to obtain the optimized filter coefficients, thus improving the prototype low-pass filter. The nonlinear phase problem.

[0006] A group delay optimization design method for a two-channel lattice orthogonal filter bank, the specific steps are as follows:

[0007] Step 1: Determine the prototype low-pass filter according to the design requirements The order of , given a set of lattice coefficient values, obtain the prototype low-pass filter :

[0008] s1.1、Define a size of The lattice coefficient vector , used to store a set of initial values ​​of lattice coefficients:

[0009] ;

[0010] s1.2、Get lattice coefficient values ​​and store them in the lattice coefficient vector , as the initial coefficient value:

[0011] ;

[0012] s1.3. Using Lattice Coefficient Vectors The values ​​in get a prototype lattice low-pass filter :

[0013] ;

[0014] in:

[0015] ;

[0016] ;

[0017] Step 2: Define a weighted least squares objective function as the criterion for optimizing the perfect reconstruction orthogonal filter bank:

[0018] ;

[0019] in Indicates the frequency point, represents the stopband cutoff frequency point, is a weighted function, is a nonlinear function of the lattice coefficients. Use the gradient iteration algorithm to minimize the objective function , the specific steps are as follows:

[0020] s2.1. Relating the objective function to the lattice coefficient Perform a first-order Taylor expansion and ignore its higher-order terms to obtain the coefficient error vector :

[0021] ;

[0022] s2.2. Find the objective function The first derivative of :

[0023] ;

[0024] in express The conjugate of It means taking the real part within the curly braces.

[0025] s2.3, the first-order partial derivative Defined as:

[0026] ;

[0027] in:

[0028] ;

[0029] ;

[0030] s2.4, order ,definition Used to store the objective function value obtained in the previous cycle. Find the current objective function If the value of , complete the objective function Minimize, go to step 3, otherwise the current The value is passed to , update the lattice coefficient vector Then return to s1.3:

[0031] ;

[0032] Step 3: Update the weighting function using the Lim-Lee-Chen-Yang algorithm , define the cutoff parameter ,definition Used to store the cutoff parameters obtained in the previous cycle Based on the lattice coefficient vector obtained in step 2 , calculated , if satisfied , then go to step 4, otherwise the current The value is passed to , return to s1.3.

[0033] Step 4: Based on the objective function and group delay of the optimized perfect reconstruction orthogonal filter bank, the overall objective function is constructed to solve the filter bank parameters:

[0034] s4.1, based on the above steps to obtain the lattice coefficient vector and low pass filter , calculate the phase-frequency response and group delay :

[0035] ;

[0036] in Indicates taking the imaginary part value, Indicates taking the real part value.

[0037] ;

[0038] in:

[0039] ;

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] s4.2. Define a new function To characterize the initial objective function for group delay optimization:

[0045] ;

[0046] in is a constant variable. The maximum difference of group delay is used as the measurement index for group delay optimization, denoted as :

[0047] ;

[0048] Function Perform a first-order Taylor expansion and ignore higher-order terms:

[0049] ;

[0050] ;

[0051] in:

[0052] ;

[0053] ;

[0054] ;

[0055] ;

[0056] ;

[0057] s4.3, the objective function of optimizing the perfect reconstruction of the orthogonal filter bank The initial objective function for group delay optimization is The weighted sum is used as the final objective function of group delay optimization, and the first-order Taylor approximation is performed to transform the optimization problem into:

[0058] ;

[0059] in, , Respectively express the , The infinite norm of , is the weight variable, , is the trust region boundary variable. Let , set according to design requirements , The value of is solved by using the CVX optimization function toolbox in Matlab software and The value of and :

[0060] ;

[0061] s4.4, return to s4.2 calculation The value is recorded as ,like , output optimal prototype low-pass filter , otherwise let , calculate the objective function , low pass filter And the objective function The first-order partial derivative of and returns s4.1.

[0062] The present invention has the following beneficial effects:

[0063] The group delay of the filter is an important indicator for judging whether the phase of the filter is linear. This method uses the idea of ​​first-order Taylor approximation and takes group delay as part of the optimization target. It avoids the complexity of the quasi-Newton algorithm in calculating the approximate Hessian matrix, greatly reduces the amount of calculation and time, and transforms the original non-convex optimization problem of the filter coefficient into a convex optimization problem of the coefficient increment. The optimal lattice coefficient is obtained by iteration, and then used as the initial value to continue to optimize the group delay. The value of the variable can be flexibly changed according to the design requirements, and finally the nonlinear phase problem of the filter is effectively improved, making it close to linear. Thereby reducing the error introduced by the signal after filtering by the lattice orthogonal filter, so that the filter group can better meet the perfect reconstruction characteristics. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 The low-pass analysis filter optimized by the quasi-Newton algorithm in the first embodiment Phase-frequency response diagram;

[0065] Figure 2 The low-pass analysis filter optimized by the method in Example 1 is Phase-frequency response diagram;

[0066] Figure 3 The low-pass analysis filter optimized by the quasi-Newton algorithm in the second embodiment Phase-frequency response diagram;

[0067] Figure 4 The low-pass analysis filter optimized by this method in Example 2 Phase-frequency response diagram. DETAILED DESCRIPTION

[0068] The present invention will be further explained below with reference to the accompanying drawings;

[0069] A group delay optimization design method for a two-channel lattice orthogonal filter bank specifically comprises the following steps:

[0070] Step 1: Determine the prototype low-pass filter according to the design requirements The order of , stopband cutoff frequency point , and given a set The prototype low-pass filter is obtained by this set of lattice coefficient values. .

[0071] Step 2: Define a weighted least squares objective function as the criterion for optimizing the perfect reconstruction orthogonal filter bank:

[0072] ;

[0073] in Indicates the frequency point, represents the stopband cutoff frequency point, is a weighted function, is a nonlinear function of the lattice coefficients.

[0074] Perform first-order Taylor approximation on the objective function to obtain a set of coefficient error values, which are used to update the lattice coefficient values. At the same time, record the objective function value. If the calculated objective function value is greater than the previously recorded objective function value, proceed to step three. Otherwise, update the lattice coefficient value and return to step one to calculate a new prototype lattice low-pass filter.

[0075] Step 3: Use the Lim-Lee-Chen-Yang algorithm to update the weighting function in step 2 When the update result meets the set cutoff condition, it is considered that a set of better grid coefficient values ​​and better , go to step 4; otherwise, return to step 1 and use the updated lattice coefficient values ​​to calculate a new prototype lattice low-pass filter.

[0076] Step 4: Calculate the current grid coefficient value and low-pass filter The corresponding group delay function and the weighted least squares objective function and group delay function Construct a new function as the overall objective function, perform first-order Taylor approximation on the overall objective function, and obtain a set of coefficient error values ​​to update the grid coefficient values. Repeat step 4 and record the overall objective function value at the same time. If the calculated overall objective function value is greater than the previously recorded value, end the loop, complete the optimization process, and output the grid coefficient value and low-pass filter at this time. .

[0077] In order to prove the effectiveness of the present method, the following two embodiments show the comparison results of the performance of the optimized filters using the present method and the quasi-Newton algorithm for optimizing the same orthogonal filter group.

[0078] Embodiment 1

[0079] This embodiment sets the prototype low-pass analysis filter The order of , after the stopband cutoff frequency is normalized , after normalization of the passband cutoff frequency , , The group delay is optimized by the quasi-Newton algorithm and this method respectively. The phase-frequency response results are as follows: Figure 1 , 2 shown.

[0080] Embodiment 2

[0081] This embodiment sets the prototype low-pass analysis filter The order of , after the stopband cutoff frequency is normalized , after normalization of the passband cutoff frequency , , The group delay is optimized by the quasi-Newton algorithm and this method respectively. The phase-frequency response results are as follows: Figure 1 , 2 shown.

[0082] contrast Figure 1 , 2 as well as Figure 3 , 4 It can be seen that the prototype low-pass analysis filter optimized by this method The curvature of the phase-frequency response curve is smaller and closer to a straight line, which improves the nonlinear problem.

[0083] The specific optimization result data comparison is shown in Table 1:

[0084] Table 1

[0085] ;

[0086] From the data in Table 1 and the attached figure, it can be seen that compared with the quasi-Newton algorithm, the maximum difference of group delay and the frequency selectivity of the filter are greatly improved, and the curvature of the phase-frequency response curve becomes smaller, that is, the low-pass analysis filter in the two-channel lattice orthogonal filter group is The nonlinearity of the phase has been well optimized.

Claims

1. A group delay optimization design method for a two-channel lattice orthogonal filter bank. First, according to the design requirements, the order N and the stopband cutoff frequency point ω of the prototype low-pass filter H0(z) are determined. s , and given a set of N lattice coefficient values, the prototype low-pass filter H0(z) is obtained through this set of lattice coefficient values, and the objective function f is defined as the criterion for optimizing the perfect reconstruction orthogonal filter bank: Where ω represents the frequency point, ω s represents the stopband cutoff frequency point, B(ω) is a weighting function, H0(e jω ) is a nonlinear function of the lattice coefficients; it is characterized by: Use the gradient iteration algorithm to minimize the objective function f, update the grid coefficient value, and calculate the new prototype grid low-pass filter and the value of the objective function f; When the objective function f no longer decreases, the weighting function B(ω) is updated using the Lim-Lee-Chen-Yang algorithm. When the update result meets the set cutoff condition, the group delay function g corresponding to the current lattice coefficient value and the low-pass filter H0(z) is calculated. The group delay function g is: 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 the final objective function, which is then approximated by the first-order Taylor approximation to transform the optimization problem into: Among them, norm(Δq,Inf) and norm(Δα,Inf) respectively represent the calculation of Δq, The infinite norm of ; represents the grid coefficient error vector, ε1 and ε2 are weight variables, ξ1 and ξ2 are trust region boundary variables, and solve Δq and The value of the constant variable q and the lattice coefficient vector Calculate the index δ for group delay optimization and compare it with the index value δ in the previous update now For comparison, if |δ now -δ|≤10 -6 , output the optimal prototype low-pass filter H0(z), otherwise let δ=δ now , calculate the objective function f, the low-pass filter H0(z), and the first-order partial derivative of the objective function f The current value of and updates the group delay function g; When the overall objective function value reaches the set condition, the loop ends, the optimization process is completed, and the grid coefficient value and low-pass filter H0(z) at this time are output.

2. A group delay optimization design method for a two-channel lattice orthogonal filter bank as claimed in claim 1, characterized in that: The method to obtain the prototype low-pass filter H0(z) is: Define a lattice coefficient vector of size N×1 Used to store grid coefficients; get grid coefficient vector Store N lattice coefficient values ​​in as initial coefficient values: Using Lattice Coefficient Vectors The initial coefficient values ​​in obtain a prototype lattice low-pass filter H0(z): in:

3. A group delay optimization design method for a two-channel lattice orthogonal filter bank as claimed in claim 1, characterized in that: The method to minimize the objective function f using the gradient iteration algorithm is: s2.

1. Define variable f pre Used to store the objective function value obtained in the previous cycle, and to convert the objective function f into the lattice coefficient vector Perform a first-order Taylor expansion and ignore its higher-order terms to obtain the coefficient error vector The lattice coefficient vector The elements in are the lattice coefficients; the objective function f is The first derivative of in express The conjugate of , Real{·} means taking the real part; s2.2, the first-order partial derivative Defined as: in: 0≤κ≤N-1; Update the lattice coefficient vector Use the updated lattice coefficient vector Calculate the prototype lattice low-pass filter H0(z); s2.3, let N(ω) = 1, calculate the objective function f corresponding to the current prototype lattice low-pass filter H0(z), when f≤f pre , pass the current value of f to f pre , repeat s2.1~s2.2 until f>f pre .

4. A group delay optimization design method for a two-channel lattice orthogonal filter bank as claimed in claim 1, characterized in that: Where Imag(·) means taking the imaginary part value, and Real(·) means taking the real part value; Represents the phase-frequency response: I=Imag(H0(z)) R=Real(H0(z)) The maximum difference δ of group delay groupdelay(ω) is used as an indicator to measure group delay optimization: δ=max(groupdelay(ω))-min(groupdelay(ω)) Perform a first-order Taylor expansion on the group delay function g and ignore higher-order terms: in Represents the lattice coefficient error vector:

5. A group delay optimization design method for a two-channel lattice orthogonal filter bank as claimed in claim 4, characterized in that: The trust region boundary variable ξ1=ξ2=1, and the values ​​of weight variables ε1 and ε2 are set according to design requirements.

6. A computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to execute the method according to any one of claims 1 to 5.

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