Two-channel qmfb coefficient sparseness optimization method based on matrix decomposition

Abstract

The application discloses a two-channel QMFB coefficient sparse optimization method based on matrix decomposition. The method utilizes the symmetry and quasi-periodicity of the impulse response sequence of a prototype FIR low-pass filter, performs matrix decomposition on the original coefficients of the filter to realize preliminary dimension reduction, selects a p-norm value, combines the matrix LU decomposition technology and the Lp-norm minimization principle, and further optimizes the filter coefficients after the preliminary dimension reduction to obtain a two-channel QMFB with a new type of sparse coefficient set. The method can guarantee to meet the requirements of stopband design error and reconstruction error, obtain as many 0-value and 1-value coefficients as possible for the filter bank, save the constant multiplier and adder resources required in the implementation of the corresponding structure, and thus reduce the complexity of the whole system.

Description

Technical Field

[0001] This invention belongs to the field of digital signal processing technology and relates to the design and optimization of filter banks, specifically to a sparse optimization method for two-channel QMFB coefficients based on matrix factorization. Background Technology

[0002] A two-channel QMFB (Quadrature Mirror Filter Bank) is a special type of digital filter bank with significant research and application value in fields such as audio and video processing, radar signal processing, and wireless communication. Composed of Finite Impulse Response (FIR) filters, a two-channel QMFB can achieve linear phase and no aliasing distortion. Its basic processing flow is as follows: the signal to be processed is divided into high-frequency and low-frequency bands using high-pass and low-pass filter channels; the signals in each non-overlapping sub-band are processed according to actual needs; and finally, the desired target signal is reconstructed, ensuring signal integrity.

[0003] A two-channel QMFB can be generally divided into three main parts: an analysis filter bank, a processing unit, and a synthesis filter bank. Its performance depends on the prototype FIR filter in the analysis filter bank. When designing a two-channel QMFB, both frequency domain filtering performance and signal reconstruction performance must be considered, namely passband and stopband errors and reconstruction errors. Existing technologies mostly focus on iterative optimization of filter bank coefficients for these two types of indicators, such as coefficient optimization algorithms based on least squares or weighted least squares, and optimization algorithms based on biological neural networks. However, as the application scope of filter banks increases, the frequency domain performance requirements also increase, leading to higher system complexity, specifically manifested in a large number of constant multipliers and adders in the hardware structure. Therefore, improving the low-complexity design of a two-channel QMFB while meeting frequency domain performance requirements is of significant research value. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention proposes a two-channel QMFB coefficient sparsity optimization method based on matrix decomposition. By using matrix decomposition, the proportion of 0-value and 1-value coefficients in the filter coefficients is increased, so that the coefficients of the filter bank are as sparse as possible while meeting performance requirements, reducing the use of constant multipliers and adders, thereby reducing hardware complexity.

[0005] The two-channel QMFB coefficient sparsity optimization method based on matrix factorization includes the following steps:

[0006] Step 1: Determine the relevant initial parameters of the two-channel QMFB according to the design requirements, including the length N of the prototype FIR low-pass filter H0(z), the number of frequency sampling points L, and the normalized stopband cutoff point ω. s The initial coefficients of the prototype filter are obtained using the direct design method. The specific steps are as follows:

[0007] s1.1 Determine the frequency response expression of the prototype FIR low-pass filter H0(z):

[0008]

[0009] Among them, h0=[h0(0),h0(1),h0(2),...,h0(N / 2-1)] T Let N be the set of column vectors containing the effective symmetric coefficients of H0(z), with a length of N / 2. -j·ω·(N-1) / 2 It is the phase component, c T (ω)·h0 is the amplitude component, and c(ω) represents the corresponding point frequency cosine component.

[0010] s1.2 Determine the sampling point matrix U of the prototype FIR low-pass filter across the entire frequency band. t and the sampling point matrix U on the stopband s :

[0011]

[0012] Where, ω i This represents equidistant frequency points within the normalized [0,π] frequency band.

[0013] s1.3. Based on the minimization maximization criterion, solve the maximum error minimization problem shown below to obtain the initial effective symmetric coefficients h0 of the prototype FIR low-pass filter:

[0014]

[0015] Among them, ||·|| ∞ This represents finding the maximum absolute value in a set of vectors, where α is a transition variable in solving the optimal design problem. H d (e jω = [1 1 … 1 0 0 … 0] 1×L Representing the ideal frequency response, the number of 0s and 1s in coefficient h0 depends on the normalized stopband cutoff point of the prototype FIR low-pass filter.

[0016] Step 2: Based on the initial coefficients h0 of the prototype FIR low-pass filter obtained in Step 1, determine the symmetric coefficient row vector h' = [h'(0), h'(1), ..., h'(N / 2-1)] = [h0(N / 2), h0(N / 2+1), ..., h0(N-1)]. Based on the quasi-periodicity of the impulse response sequence, and according to the frequency domain performance requirements, retain the first M coefficients to form the retained coefficient vector matrix h. M :

[0017] h M =[h'(0),h'(1),...,h'(M-1)] (4)

[0018] The remaining coefficients to be decomposed are arranged into a coefficient matrix H of dimension d×R:

[0019]

[0020] Where M+R·d=N / 2.

[0021] Preferably, the value of M is determined based on the performance requirements of the filter; the higher the performance of the filter, the larger the value of M.

[0022] Step 3: Complete the matrix decomposition of the coefficient matrix H to achieve preliminary sparsity of the coefficients. The specific steps are as follows:

[0023] s3.1 Perform LU decomposition on the coefficient matrix H:

[0024] H = L·U (6)

[0025]

[0026]

[0027] Wherein, the lower triangular matrix L is the eigenvalue vector matrix, the upper triangular matrix U is the scaling factor vector matrix, and the rank of matrix H is r.

[0028] s3.2. Based on design requirements, select vector dimension e, e≤r, and approximate the coefficient matrix H by decomposition:

[0029]

[0030] The frequency response correlation expression of the prototype FIR filter based on matrix decomposition is then obtained:

[0031]

[0032] Step 4: Use column vector matrix v to represent the retained coefficient matrix h M And the eigencoefficient matrix X and scaling coefficient matrix Y after LU decomposition of the coefficient matrix.T Determine the frequency response of the corresponding prototype FIR low-pass filter:

[0033]

[0034] Determine the reconstruction error e of a two-channel QMFB based on matrix factorization. R (v) and stopband design error H S (v):

[0035]

[0036] H S (v)=[|H0(ω s v)|,|H0(ω s+1 v)|,…,|H0(ω L v)|] T (13)

[0037] Step 5: Based on gradient search, obtain the initial sparse coefficient vector v after iterative optimization. * The specific steps are as follows:

[0038] s5.1, Set the maximum number of iterations to K max The initial iteration count k = 1, and the iteration step size is 1.

[0039] s5.2 Calculate the reconstruction error partial derivative matrix G at the (k-1)th iteration. R (v( k-1) ) and the stopband design error partial derivative matrix G S (v (k-1) ):

[0040]

[0041]

[0042] Based on the idea of ​​Taylor first-order expansion approximation, the change in coefficients during iterative optimization is set as Δv, and the reconstruction error e under the k-th iteration is determined. R (v' (k) ) and stopband design error H S (v' (k) ):

[0043]

[0044]

[0045] s5.3 Solving for the set of numerical changes in the coefficient vector set v, Δv, under the k-th iteration. (k) :

[0046]

[0047] Where δ is the transition variable in the optimization process, and σ is the transition variable in the optimization process. desire It is the expected value of the stopband design error, ε i It is the threshold for the coefficient increment in each iteration.

[0048] s5.4, Define the condition function for iterative optimization in s5.3:

[0049]

[0050]

[0051]

[0052] Among them, e R (v (k) ) represents the actual reconstruction error at the k-th iteration. H represents the approximate reconstruction error. S (v (k) The ) indicates the actual stopband design error. This represents the approximate stopband design error. This is the reduction factor for the incremental threshold. If and only if the numerators and denominators of a(k) and b(k) are both less than 0, let k = k + 1 and return to s5.2 for the next iteration; otherwise, reduce ε. i And return s5.3. Until ε i Reduced to the set minimum threshold or the number of iterations k reaches K. max When the iteration stops, the initial sparse coefficients v are output. * :

[0053] v * =v (k) =v + △v (k) (twenty two)

[0054] As a preferred option, ε is set i The initial value is 1. It is 0.5.

[0055] Step Six: Further sparsity optimization is performed on the coefficients of the two-channel QMFB through the following steps:

[0056] s6.1, From the initial sparsity coefficient v * The coefficients other than those with values ​​of 0 and 1 are selected from the data and used as the set of variable coefficient elements v. NZ Determine the expected value of the reconstruction error δ desire Set the maximum number of iterations J for norm minimization. max The initial value of the iteration is j = 1.

[0057] s6.2, For the set of elements with variable coefficients v NZ Minimize the Lp norm:

[0058]

[0059] Among them, ||·|| p This indicates the calculation of the p-norm; δ PR =ηδ * , represents the approximate reconstruction error threshold during optimization, η is the expansion factor, and δ * It is the initial sparsity coefficient v * The corresponding reconstruction error, ρ i This represents the threshold for the coefficient increment in each norm minimization iteration.

[0060] s6.3, Define the conditional function for minimizing the Lp norm in s6.2:

[0061]

[0062]

[0063]

[0064]

[0065] During the iterative optimization process, it is first necessary to ensure that the minimization of the Lp norm at each step is a positive optimization, i.e., L'(k) < 0; otherwise, the loop should be exited immediately. At the same time, the actual reconstruction error and stopband design error indicators need to meet the expected values, i.e., T'(j) and A'(j) are both negative, and the iterative optimization will continue to advance, i.e., j = j + 1, and return to s6.2; otherwise, the coefficient increment threshold is reduced according to equation (27), and then s6.2 is returned, until the set maximum number of iterations J is reached. max or ρ i Narrow down to a set minimum threshold, and output the set of coefficients whose norm is minimized.

[0066] s6.4. Based on greedy search, for v' NZ The element with the smallest absolute value v' NZ (i) Perform sparse transformation:

[0067] v' NZ (i)=0 (28)

[0068] This further increases the initial sparse coefficient set v. * To further improve frequency domain performance, the number of zero-valued coefficients is determined. Following step five, gradient-based iterative optimization is performed again to output a sparse coefficient set.

[0069] s6.5, Define the termination condition expression for sparse optimization:

[0070]

[0071]

[0072] To obtain more zero-value coefficients and further improve sparsity, [the following is proposed]: The optimization of minimizing the Lp norm is performed, and the iterative optimization from s6.2 to s6.4 is repeated until the sparse optimization termination conditions shown in equations (29) and (30) can no longer be satisfied simultaneously. Then, the final set of sparse optimization coefficients based on matrix factorization is output. Where m is the number of zero-value coefficients added through equation (28).

[0073] As a preferred option, for the set of variable coefficient elements v NZ When minimizing the Lp norm, p is set to either 1 or 2.

[0074] As a preferred option, set ρ i The initial value is 1, and η is 1.1.

[0075] The present invention has the following beneficial effects:

[0076] By leveraging the quasi-periodicity of the impulse response sequence of the prototype FIR low-pass filter in a two-channel QMFB, matrix decomposition of the initial coefficients is performed, and appropriate dimensions are selected to construct a new set of filter coefficients. This initially yields several zero-value and one-value coefficients, thereby reducing the use of constant multipliers and adders in the system structure. Furthermore, by flexibly selecting the p-norm value and iteratively minimizing the Lp norm of the variable coefficients, more zero-value coefficients are obtained. This further achieves coefficient sparsity while satisfying the expected reconstruction error and stopband design error of the filter bank, thus reducing the system complexity. Attached Figure Description

[0077] Figure 1 Here is a flowchart of a two-channel QMFB coefficient sparsity optimization method based on matrix factorization;

[0078] Figure 2 This is a schematic diagram of a two-channel QMFB filter structure based on matrix decomposition.

[0079] Figure 3 The amplitude-frequency response diagrams before and after direct optimization of the two-channel QMFB coefficients based on matrix factorization in Example 1 are shown.

[0080] Figure 4 The graph shows the transmission distortion function before and after direct optimization of the two-channel QMFB coefficients based on matrix decomposition in Example 1.

[0081] Figure 5 The amplitude-frequency response diagrams before and after sparse optimization of the two-channel QMFB coefficients based on matrix factorization in Example 1 are shown.

[0082] Figure 6 The graph shows the transmission distortion function before and after sparse optimization of the two-channel QMFB coefficients based on matrix factorization in Example 1.

[0083] Figure 7 The amplitude-frequency response diagrams before and after direct optimization of the two-channel QMFB coefficients based on matrix factorization in Example 2 are shown.

[0084] Figure 8 The graph shows the transmission distortion function before and after direct optimization of the two-channel QMFB coefficients based on matrix decomposition in Example 2.

[0085] Figure 9 The amplitude-frequency response diagrams before and after sparse optimization of the two-channel QMFB coefficients based on matrix factorization in Example 2 are shown.

[0086] Figure 10 The graph shows the transmission distortion function before and after sparse optimization of the two-channel QMFB coefficients based on matrix decomposition in Example 2. Detailed Implementation

[0087] The present invention will be further explained below with reference to the accompanying drawings;

[0088] like Figure 1 As shown, the two-channel QMFB coefficient sparse optimization method based on matrix factorization includes six steps. Based on steps one to three, the following can be obtained: Figure 2 The prototype filter structure in a two-channel QMFB based on matrix factorization is shown. To verify the rationality and effectiveness of this method, software simulation verification experiments were conducted based on the following two embodiments, and compared with six common existing technologies.

[0089] For ease of explanation, the optimization results of this method through steps one to three are denoted as MD, steps one to five are denoted as MD-IGS, and in step six, the result when the norm is minimized and the p value is 1 is denoted as MD-IGS-L1, and the result when the norm is minimized and the p value is 2 is denoted as MD-IGS-L2.

[0090] Example 1

[0091] In this embodiment, the length of the prototype filter H0(z) is set to N = 80, the number of frequency sampling points is L = 8 × N = 640, and the normalized stopband cutoff frequency is ω. s =0.55, optimizing the maximum number of iterations K max =10000, the maximum number of iterations J for minimizing the Lp norm max=10000, number of coefficients retained M=12, matrix LU decomposition parameters d=4, R=7, e=2, expected reconstruction error 0.01dB, expected stopband design error σ desire It is -46.50dB.

[0092] Through MATLAB software simulation and optimization, a comparison of the amplitude-frequency response curves of the two-channel QMFB based on matrix factorization after MD execution and after MD-IGS optimization is obtained, as shown in the figure. Figure 3 As shown, a comparison graph of the corresponding transmission distortion function amplitude is also obtained, such as... Figure 4 As shown. The transmission distortion function is:

[0093] E r (ω)=20·log 10 T(ω) = 20·log 10 [|H0(ω)| 2 +|H0(ω+π)| 2 (31)

[0094] observe Figure 3 and Figure 4 It is evident that direct matrix decomposition results in significant stopband design and reconstruction errors. However, steps four and five can optimize these errors and make the transmission distortion function curve more ripple-like. (Stopband design error) and reconstruction error E pr The expression is:

[0095]

[0096] E pr =max|20·log 10 T(ω)| for 0≤ω≤π (33)

[0097] Figure 5 and Figure 6 The figures show the amplitude-frequency response and transmission distortion function amplitude diagrams of the two-channel QMFB after further coefficient sparsity optimization. (Observation) Figure 5 and Figure 6 It can be seen that after further optimization of coefficient sparsity, the system's stopband design error and reconstruction error can still meet the requirements.

[0098] The final optimization results of this embodiment are shown in Table 1:

[0099]

[0100] Table 1

[0101] Here, sparsity represents the proportion of zero-value coefficients in the prototype filter coefficient set of the designed two-channel QMFB. As can be seen from the data in Table 1, compared with the existing technology, this method can improve the coefficient sparsity of the corresponding two-channel QMFB while achieving smaller reconstruction error and stopband design error, realize coefficient sparsification, reduce the use of constant multipliers and adders, and thus reduce system complexity.

[0102] The final sparse optimization coefficients of this embodiment are shown in Table 2:

[0103]

[0104] Table 2

[0105] Example 2

[0106] In this embodiment, the length of the prototype filter H0(z) is set to N = 120, the number of frequency sampling points is set to L = 10 × N = 1200, and the normalized stopband cutoff frequency is set to ω. s =0.6, optimizing the maximum number of iterations K max =10000, the maximum number of iterations J for minimizing the Lp norm max =10000, number of coefficients retained M=46, matrix LU decomposition parameters d=2, R=7, e=1, expected reconstruction error 0.01dB, expected stopband design error σ desire It is -135.00dB.

[0107] Through MATLAB software simulation and optimization, a comparison of the amplitude-frequency response curves of the two-channel QMFB based on matrix factorization after MD execution and after MD-IGS optimization is obtained, as shown in the figure. Figure 7 As shown, a comparison graph of the corresponding transmission distortion function amplitude is also obtained, such as... Figure 8 As shown. Figure 9 and Figure 10 The above are the amplitude-frequency response diagram and transmission distortion function amplitude diagram of the two-channel QMFB after further coefficient sparsity optimization.

[0108] Depend on Figures 7 to 10 As can be seen, after optimization by this method, the design error and reconstruction error of the corresponding two-channel QMFB stopband are greatly reduced, and equiripple is achieved across the entire frequency band.

[0109] The final optimization results of this embodiment are shown in Table 3:

[0110]

[0111] Table 3

[0112] As shown in Table 3, this method can achieve a reconstruction error far smaller than that of existing technologies when the stopband design error of a two-channel QMFB meets the desired standard. It also obtains 10 sparse coefficients with zero values, structurally reducing the number of multipliers by 6 and adders by 13. Although the reconstruction error is slightly sacrificed while meeting the stopband design error expectation, this method further sparsifies the coefficients of the matrix factorization-based two-channel QMFB, achieving a sparsity of up to 30%, significantly reducing the system complexity of the two-channel QMFB.

[0113] The prior art in the two embodiments above are as follows:

[0114] Prior art 1: Jain V, Crochiere R. Quadrature mirror filter design in the time domain [J]. IEEE transactions on acoustics, speech, and signal processing, 1984, 32(2): 353-361.

[0115] Prior art 2: Jou Y D.Design of two-channel linear-phase quadraturemirror filter banks based on neural networks[J].Signal Processing, 2007, 87(5):1031-1044.

[0116] Prior art 3: Chen CK, Lee J H. Design of quadrature mirror filters with linear phase in the frequency domain [J]. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 1992, 39 (9): 593-605.

[0117] Prior Art 4: Kuldeep B, Singh V K, Kumar A, et al. Design of two-channel filter bank using nature inspired optimization based fractional derivative constraints[J].ISA transactions, 2015, 54:101-116.

[0118] Prior Art 5: Baderia K, Kumar A, Singh G K. Design of quadrature mirror filter bank using polyphase components based on optimal fractional derivative constraints[J].AEU-International Journal of Electronics and Communications, 2015, 69(9):1254-1264.

[0119] Prior Art 6: Dhabal S, Venkateswaran P. An efficient gbest-guided Cuckoo Search algorithm for higher order two channel filter bank design[J].Swarm and Evolutionary Computation, 2017, 33:68-84.

Claims

1. A two-channel QMFB coefficient sparse optimization method based on matrix factorization, characterized in that: The specific steps are as follows: Step 1: Determine the relevant initial parameters of the two-channel QMFB according to the design requirements, including the length N of the prototype FIR low-pass filter H0(z), the number of frequency sampling points L, and the normalized stopband cutoff point ω. s ; Step 2: Based on the symmetry and quasi-periodicity of the impulse response sequence of the prototype FIR low-pass filter, optimize the coefficients of the first N / 2 terms h'=[h'(0),h'(1),...,h'(N / 2-1)]=[h0(N / 2),h0(N / 2+1),...,h0(N-1)]; retain the first M terms in h' to form the retained coefficient vector matrix h M : h M =[h'(0),h'(1),...,h'(M-1)] (1) Then, the remaining N / 2-M coefficients are arranged into a coefficient matrix H of size d×R: Where M + R·d = N / 2; Step 3: Perform matrix decomposition on the coefficient matrix H to achieve preliminary sparsity of the coefficients. The specific steps are as follows: s3.1 Perform LU decomposition on the coefficient matrix H: H = L·U (3) Wherein, the lower triangular matrix L is the eigenvalue vector matrix, the upper triangular matrix U is the scaling factor vector matrix, and the rank of matrix H is r; s3.

2. Select vector dimension e, e≤r, and approximate the coefficient matrix H by decomposition: The frequency response correlation expression of the prototype FIR filter based on matrix decomposition is then obtained: Step 4: Use column vector matrix v to represent the retained coefficient matrix h M And the eigencoefficient matrix X and scaling coefficient matrix Y after LU decomposition of the coefficient matrix. T Determine the frequency response of the corresponding prototype FIR low-pass filter: Determine the reconstruction error e of a two-channel QMFB based on matrix factorization. R (v) and stopband design error H S (v): H S (v)=[|H0(ω s v)|,|H0(ω s+1 v)|,…,|H0(ω L v)|] T (10) Step 5: Based on gradient search, obtain the initial sparse coefficient vector v after iterative optimization. * The specific steps are as follows: s5.1, Set the maximum number of iterations to K max The initial iteration count k = 1, and the iteration step size is 1. s5.2 Calculate the reconstruction error partial derivative matrix G at the (k-1)th iteration. R (v (k-1) ) and the stopband design error partial derivative matrix G S (v (k-1) ): Based on the idea of ​​Taylor first-order expansion approximation, the change in coefficients during iterative optimization is set as Δv, and the reconstruction error e under the k-th iteration is determined. R (v' (k) ) and stopband design error H S (v' (k) ): s5.3 Solving for the set of numerical changes in the coefficient vector set v, Δv, under the k-th iteration. (k) : Where δ is the transition variable in the optimization process, and σ is the transition variable in the optimization process. desire It is the expected value of the stopband design error, ε i This is the threshold for the coefficient increment in each iteration; the condition function for iterative optimization is set as follows: Among them, e R (v (k) ) represents the actual reconstruction error at the k-th iteration. H represents the approximate reconstruction error. S (v (k) The ) indicates the actual stopband design error. This represents the approximate stopband design error. It is the reduction factor of the incremental threshold; if and only if the numerator and denominator of a(k) and b(k) are both less than 0, let k = k + 1, return to s5.2 for the next iteration, otherwise reduce ε. i , and repeat s5.3; until ε i Reduced to the set minimum threshold or the number of iterations k reaches K. max When the iteration stops, the initial sparse coefficients v are output. * : v * =v (k) =v+△v (k) (19) Step 6: Based on Lp norm minimization, perform optimization on the coefficient matrix v. * Perform sparsity optimization: s6.1, From the initial sparsity coefficient v * The coefficients other than those with values ​​of 0 and 1 are selected from the data and used as the set of variable coefficient elements v. NZ Determine the expected value of the reconstruction error δ desire Set the maximum number of iterations J for norm minimization. max The initial value of the iteration is j = 1; s6.2, For the set of variable coefficient elements v NZ Minimize the Lp norm: Among them, ||·|| p This indicates the calculation of the p-norm; δ PR =ηδ * , represents the approximate reconstruction error threshold during optimization, η is the expansion factor, and δ * It is the initial sparsity coefficient v * The corresponding reconstruction error, ρ i This represents the threshold for the coefficient increment in each norm minimization iteration; s6.3, Define the conditional function for minimizing the Lp norm in s6.2: During the iterative optimization process, it is first necessary to ensure that the minimization of the Lp norm at each step is a positive optimization, i.e., L'(k) < 0; otherwise, the loop should be exited immediately. At the same time, the actual reconstruction error and stopband design error indicators need to meet the expected values, i.e., T'(j) and A'(j) are both negative, and the iterative optimization will continue to advance, i.e., j = j + 1, and return to s6.2; otherwise, the coefficient increment threshold is reduced according to equation (24), and then s6.2 is returned until the set maximum number of iterations J is reached. max or ρ i Narrow down to a set minimum threshold, and output the set of coefficients whose norm is minimized. s6.

4. Based on greedy search, for v' NZ The element with the smallest absolute value v' NZ (i) Perform sparse transformation: v' NZ (i)=0 (25) This further increases the initial sparse coefficient set v. * To further improve frequency domain performance, the number of zero-valued coefficients is determined. Following step five, gradient-based iterative optimization is performed again to output a sparse coefficient set. s6.5, Define the termination condition expression for sparse optimization: To obtain more zero-value coefficients and further improve sparsity, [the following is proposed]: The optimization of minimizing the Lp norm is performed, and the iterative optimization from s6.2 to s6.4 is repeated until the sparse optimization termination conditions shown in equations (26) and (27) can no longer be satisfied simultaneously. Then, the final set of sparse optimization coefficients based on matrix factorization is output. Where m is the number of zero-value coefficients added through equation (25).

2. The coefficient sparsity optimization method for a two-channel QMFB based on matrix factorization as described in claim 1, characterized in that: The initial coefficients h0 of the prototype filter H0(z) are obtained using the direct design method. The specific steps are as follows: s1.1 Determine the frequency response expression of the prototype FIR low-pass filter H0(z): Among them, h0=[h0(0),h0(1),h0(2),...,h0(N / 2-1)] T Let H0(z) be the set of column vectors of effective symmetric coefficients, with a length of N / 2; e -j·ω·(N-1) / 2 It is the phase component, c T (ω)·h0 is the amplitude component, and c(ω) represents the corresponding point frequency cosine component; s1.2 Determine the sampling point matrix U of the prototype FIR low-pass filter across the entire frequency band. t and the sampling point matrix U on the stopband s : Where, ω i This represents equidistant frequency points within the normalized [0,π] frequency band. s1.

3. Based on the minimization maximization criterion, solve the maximum error minimization problem shown below to obtain the initial effective symmetric coefficients h0 of the prototype FIR low-pass filter: Among them, ||·|| ∞ H represents finding the maximum absolute value in a set of vectors, where α is a transition variable in solving the optimal design problem; d (e jω = [1 1 … 1 0 0 … 0] 1×L Representing the ideal frequency response, the number of 0s and 1s in coefficient h0 depends on the normalized stopband cutoff point of the prototype FIR low-pass filter.

3. The coefficient sparsity optimization method for a two-channel QMFB based on matrix factorization as described in claim 1, characterized in that: The value of M is determined based on the performance requirements of the filter; the higher the performance of the filter, the larger the value of M.

4. The coefficient sparsity optimization method for a two-channel QMFB based on matrix factorization as described in claim 1, characterized in that: In step five, the incremental threshold ε for gradient search is set. i The initial value is 1, and the reduction factor is... It is 0.

5.

5. The coefficient sparsity optimization method for a two-channel QMFB based on matrix factorization as described in claim 1, characterized in that: For the set of elements with variable coefficients v NZ When minimizing the Lp norm, p is set to either 1 or 2.

6. The coefficient sparsity optimization method for a two-channel QMFB based on matrix factorization as described in claim 1 or 5, characterized in that: Set the threshold ρ for the coefficient increment in the norm minimization iteration. i The initial value is 1, and the expansion factor η is 1.1.

Images