CN118748549A - A Group Delay Optimization Design Method for Two-Channel Lattice Orthogonal Filter Banks - Google Patents

Abstract

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

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

Technical Field

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

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.

Since the two-channel lattice orthogonal filter bank mainly includes the analysis filter bank and the synthesis filter bank, in order to meet the signal's no-aliasing condition, the synthesis filter bank must be designed as a related form of the analysis filter bank, and the analysis filter bank is composed of a pair of orthogonal low-pass-high-pass filters. Therefore, the design problem of the two-channel lattice orthogonal filter bank can be transformed into the coefficient optimization design problem of a single prototype low-pass filter H ₀ (z), and its objective function f is composed of a weighting function and a nonlinear function of the low-pass analysis filter lattice coefficient.

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

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.

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

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 :

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

;

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

;

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

;

in:

;

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

;

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:

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 :

;

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

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

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

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in:

;

;

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:

;

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.

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:

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 :

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in Indicates taking the imaginary part value, Indicates taking the real part value.

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in:

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;

;

;

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

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

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

;

;

in:

;

;

;

;

;

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:

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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 :

;

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.

The present invention has the following beneficial effects:

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

FIG. 1 is a low-pass analysis filter optimized using a quasi-Newton algorithm in Example 1 Phase-frequency response diagram;

FIG. 2 is a low-pass analysis filter optimized by the method in Example 1. Phase-frequency response diagram;

FIG. 3 is a low-pass analysis filter optimized using the quasi-Newton algorithm in Example 2 Phase-frequency response diagram;

FIG. 4 is a low-pass analysis filter optimized by the method in Example 2. Phase-frequency response diagram.

DETAILED DESCRIPTION

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

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

Step 1: Determine the prototype low-pass filter according to the design requirements The order of , stop band cut-off frequency point , and given a set The prototype low-pass filter is obtained by this set of lattice coefficient values. .

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

;

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

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.

Step 3: Update the weighting function in step 2 using the Lim-Lee-Chen-Yang algorithm 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.

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. .

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 optimization of the same orthogonal filter group.

Embodiment 1

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 using the quasi-Newton algorithm and this method respectively. The phase-frequency response results are shown in Figures 1 and 2.

Embodiment 2

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 using the quasi-Newton algorithm and this method respectively. The phase-frequency response results are shown in Figures 1 and 2.

Comparing Figures 1 and 2 with Figures 3 and 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.

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

Table 1

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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 (7)

1. A group delay optimization design method of two-channel lattice type orthogonal filter bank comprises determining prototype low-pass filter according to design requirementThe order of (2)Stop band cut-off frequency pointAnd give a group ofIndividual lattice coefficient values from which a prototype low-pass filter is obtainedDefining an objective functionAs a criterion for optimizing a perfectly reconstructed 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 factor; the method is characterized in that: minimizing objective functions using gradient iterative algorithmsUpdating the lattice coefficient value, calculating new prototype lattice low-pass filter and objective functionIs a value of (2);

When the objective function When no decrease occurs, the weighting function is updated by using Lim-Lee-Chen-Yang algorithmWhen the updated result meets the set cut-off condition, the currently obtained lattice coefficient value and the low-pass filter are calculatedCorresponding group delay functionAnd objective functionAnd group delay functionConstructing a new function as an overall objective function, performing first-order Taylor approximation on the overall objective function to obtain a group of coefficient error values, updating the lattice coefficient values until the overall objective function value reaches a set condition, ending the cycle, completing the optimization process, and outputting the lattice coefficient values and the low-pass filter at the moment。

2. 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: obtaining a prototype low-pass filterThe method of (1) is as follows:

Define a size as Lattice coefficient vector of (a)The device is used for storing the lattice coefficients; obtaining Xiang Gexing coefficient vectorsIs stored inIndividual lattice coefficient values as initial coefficient values:

；

Using lattice coefficient vectors Obtaining a prototype lattice low-pass filter from initial coefficient values of the filter：

；

Wherein:

；

。

3. 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: minimizing objective functions using gradient iterative algorithms The method of (1) is as follows:

s2.1, define variables For storing the objective function value obtained in the previous cycle, and for adding the objective functionWith respect to lattice coefficient vectorsPerforming first-order Taylor expansion and neglecting higher-order terms to obtain coefficient error vectors：

；

The lattice coefficient vectorThe elements in (a) are lattice coefficients; for objective functionSolving forIs the first order derivative of (2)：

；

Wherein the method comprises the steps ofRepresentation ofIs used for the conjugation of (a),Representing the real part;

s2.2, leading the first order offset The definition is as follows:

；

Wherein:

；

；

; updating lattice coefficient vectors ：

；

Using updated lattice coefficient vectorsCalculation prototype lattice low-pass filter；

S2.3, orderCalculating a current prototype lattice low-pass filterCorresponding objective functionWhen (when)Will currentlyIs passed toRepeating s 2.1-s 2.2 until。

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: the group delay functionThe method comprises the following steps:

；

Wherein the method comprises the steps of Is a constant variable which is a function of the temperature,Representing group delay:

；

；

Wherein the method comprises the steps of The representation takes the value of the imaginary part,Representing a real part value; Representing the phase-frequency response:

；

；

；

；

；

Group delay Is the maximum difference of (2)As an index for group delay optimization:

；

Group delay function Performing first-order Taylor expansion, and ignoring higher-order terms:

；

；

Wherein:

；

；

；

；

。

5. The method for optimizing the group delay of a two-channel lattice type orthogonal filter set according to claim 4, wherein: the objective function is set And group delay functionAnd (3) carrying out weighted sum to obtain a final objective function, 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,、Solving for trusted domain boundary variablesAndUpdating the value of (2)And：

；

Calculating an index for group delay optimizationAnd is matched with the index value at the time of the last round of updatingComparing ifOutput optimal prototype low-pass filterOtherwise letCalculating an objective functionLow pass filterObjective functionFirst order bias guide of (a)And updating the group delay function。

6. The method for optimizing the group delay of a two-channel lattice type orthogonal filter set according to claim 4, wherein: the trust domain boundary variablesWeight variable、The value of (2) is set according to the design requirements.

7. 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 1-6.