CN105279350A

CN105279350A - Designing method for near-complete reconstruction non-uniform cosine modulated filter bank

Info

Publication number: CN105279350A
Application number: CN201510853207.8A
Authority: CN
Inventors: 蒋俊正; 江庆; 欧阳缮; 刘庆华; 谢跃雷; 程小磊; 穆亚起; 郭云
Original assignee: Guilin University of Electronic Technology
Current assignee: Guilin University of Electronic Technology
Priority date: 2015-11-30
Filing date: 2015-11-30
Publication date: 2016-01-27
Anticipated expiration: 2035-11-30
Also published as: CN105279350B

Abstract

The invention discloses a method for designing an approximately completely reconstructed non-uniform cosine modulated filter bank. The non-uniform filter bank obtained by directly optimizing the performance of the non-uniform filter bank has smaller reconstruction errors and better overall performance. . Simultaneously, the present invention is by summing up the design problem of the non-uniform filter bank as an unconstrained optimization problem about the prototype filter, wherein the objective function is the weighted sum of the transfer distortion of the non-uniform filter bank and the stopband energy of the prototype filter, and finally utilizes the linear An iterative algorithm is used to solve the optimization problem, which significantly reduces the design cost of the non-uniform filter bank. Therefore, the present invention provides a simple and efficient solution for reducing design complexity and realizing accurate signal reconstruction.

Description

Design Method of Approximately Complete Reconstruction Non-Uniform Cosine Modulated Filter Bank

技术领域technical field

本发明属于滤波器组设计领域，具体涉及一种近似完全重构非均匀余弦调制滤波器组的设计方法。The invention belongs to the field of filter bank design, and in particular relates to a design method of an approximately completely reconstructed non-uniform cosine modulation filter bank.

背景技术Background technique

滤波器组作为多速率信号处理当中的核心内容一直备受关注，其在自适应滤波、语音图像编码和图像处理等方面取得了广泛的应用。设计一般的滤波器组需优化所有分析滤波器和综合滤波器，而设计调制滤波器组则只需优化设计原型滤波器，这大大降低了设计的复杂度。调制滤波器组目前主要有两类，分别是余弦调制滤波器组和离散傅里叶变换调制滤波器组。这两种调制滤波器组相比较而言，由于余弦调制滤波器组是经余弦调制而来，因而更适用于处理实值信号。As the core content of multi-rate signal processing, filter bank has been paid much attention, and it has been widely used in adaptive filtering, speech and image coding and image processing. To design a general filter bank, all analysis filters and synthesis filters need to be optimized, while to design a modulation filter bank, only the prototype filter needs to be optimized, which greatly reduces the complexity of the design. There are currently two main types of modulated filter banks, namely cosine modulated filter banks and discrete Fourier transform modulated filter banks. Compared with these two modulation filter banks, since the cosine modulation filter bank is obtained by cosine modulation, it is more suitable for processing real-valued signals.

均匀滤波器组的各子带滤波器具有相同的频率范围，但是在实际应用中有时需要对信号的频带进行非均匀划分。比如在图像去噪中需要对图像频谱进行非均匀的合理划分从而更有效的实现噪声去除。CoxRV在《IEEETransactionsonAcoustics，SpeechandSignalProcessing》上发表的《Thedesignofuniformlyandnonuniformlyspacedpseudoquadraturemirrorfilter》中创新性地提出通过合并均匀滤波器组来实现非均匀滤波器组的思想，这是非均匀滤波器组设计的一大进步。HoangPQ等人在《IEEEInternationalSymposiumonCircuitsandSystems》上发表的《Nonuniformmultiratefilterbanks:theoryanddesign》中首次提出M通道非均匀正交镜像滤波器组的概念与构造。LiJL等人在《IEEETransactionsonSignalProcessing》上发表的《Asimpledesignmethodfornear-perfect-reconstructionnonuniformfilterbanks》中利用子带合并的方式成功构建了近似完全重构的非均匀余弦调制滤波器组，然而他的设计方法是先获得均匀滤波器组后直接进行子带合并。ManeeshaK等人在《InternationalConferenceonCommunicationsandSignalProcessing》上发表的《AChannelCombinerApproachfortheDesignofNearPerfectReconstructionNonUniformFilterBanks》中利用凯瑟窗函数法获得3dB截止频率为2π/M的原型滤波器后，同样也是通过直接合并的方式来获得非均匀滤波器组。Each sub-band filter of the uniform filter bank has the same frequency range, but in practical applications, it is sometimes necessary to divide the frequency band of the signal non-uniformly. For example, in image denoising, it is necessary to divide the image spectrum reasonably and non-uniformly so as to achieve noise removal more effectively. In "The design of uniformly and nonuniformly spaced pseudo quadrature mirror filter" published by CoxRV on "IEEE Transactions on Acoustics, Speechan and Signal Processing", he innovatively proposed the idea of realizing non-uniform filter banks by merging uniform filter banks, which is a great advancement in the design of non-uniform filter banks. HoangPQ and others first proposed the concept and construction of M-channel non-uniform quadrature filter banks in "Nonuniformmultiratefilterbanks: theoryanddesign" published on "IEEEInternationalSymposiumonCircuitsandSystems". In the "Asimpledesign method for near-perfect-reconstruction nonuniform filter banks" published on "IEEE Transactions on Signal Processing", LiJL et al. successfully constructed an approximately completely reconstructed non-uniform cosine modulation filter bank by using subband merging. However, his design method is to obtain uniform filtering first. The subbands are merged directly after the filter group. In the "AChannelCombinerApproachfortheDesignofNearPerfectReconstructionNonUniformFilterBanks" published by ManeeshaK et al. on "International Conference on Communications and Signal Processing", after using the Kaiser window function method to obtain a prototype filter with a 3dB cut-off frequency of 2π/M, the non-uniform filter bank is also obtained by direct merging.

然而，上述设计非均匀滤波器组所采用的直接合并均匀滤波器组子带的方式存在一个不足之处：不能直接控制优化非均匀滤波器组性能，导致非均匀滤波器组性能的好坏由所选取均匀滤波器组完全决定。However, the method of directly merging the subbands of the uniform filter bank used in the above-mentioned design of the non-uniform filter bank has a disadvantage: it cannot directly control and optimize the performance of the non-uniform filter bank, resulting in the performance of the non-uniform filter bank being determined by The chosen uniform filter bank is completely determined.

发明内容Contents of the invention

本发明所要解决的技术问题是现有设计非均匀滤波器组的直接合并方法中无法直接控制优化非均匀滤波器组性能的不足，提供一种近似完全重构非均匀余弦调制滤波器组的设计方法。The technical problem to be solved by the present invention is the inadequacy of the inability to directly control and optimize the performance of non-uniform filter banks in the existing direct combination method for designing non-uniform filter banks, and to provide a design of an approximately completely reconstructed non-uniform cosine modulated filter bank method.

为解决上述问题，本发明是通过以下技术方案实现的：In order to solve the above problems, the present invention is achieved through the following technical solutions:

近似完全重构非均匀余弦调制滤波器组的设计方法，包括如下步骤：A design method for approximately completely reconstructing a non-uniform cosine modulated filter bank, comprising the following steps:

步骤1，根据余弦调制理论和等价理论，将非均匀余弦调制滤波器组的分析滤波器组和综合滤波器组分别转换为关于原型滤波器h的函数。Step 1. According to the cosine modulation theory and the equivalence theory, the analysis filter bank and the synthesis filter bank of the non-uniform cosine modulation filter bank are respectively transformed into functions about the prototype filter h.

步骤2，将非均匀余弦调制滤波器组的传递失真E_t(h)转换为关于原型滤波器h的函数。Step 2, transform the transfer distortion E _t (h) of the non-uniform cosine modulated filter bank into a function about the prototype filter h.

步骤3，将原型滤波器阻带能量E_s(h)转换为关于原型滤波器h的函数。Step 3, converting the stop band energy E _s (h) of the prototype filter into a function of the prototype filter h.

步骤4，根据滤波器组设计的性能指标，小的传递失真和低的阻带能量可以保证获得整体性能较好的滤波器组，本发明将非均匀滤波器组的设计问题归纳为一个关于原型滤波器的无约束优化问题，其目标函数为非均匀滤波器组传递失真和原型滤波器阻带能量的加权和，表示为Step 4, according to the performance index of filter bank design, little transfer distortion and low stop band energy can guarantee to obtain the better filter bank of overall performance, and the present invention summarizes the design problem of non-uniform filter bank as an about prototype In the unconstrained optimization problem of filters, the objective function is the weighted sum of non-uniform filter bank transfer distortion and prototype filter stopband energy, expressed as

$\underset{h}{m i n} Φ (h) = {αE}_{t} (h) + (1 - α) E_{s} (h)$ ① $\underset{h}{m i no} Φ (h) = {αE}_{t} (h) + (1 - α) {E.}_{the s} (h)$ ①

式中，Φ(h)表示目标函数，E_t(h)表示非均匀余弦调制滤波器组的传递失真，E_s(h)表示原型滤波器的阻带能量，h表示原型滤波器，α表示权重。In the formula, Φ(h) represents the objective function, E _t (h) represents the transfer distortion of the non-uniform cosine modulation filter bank, E _s (h) represents the stopband energy of the prototype filter, h represents the prototype filter, α represents Weights.

步骤5，采用线性迭代算法来求解式①。通过先给定一个最初原型滤波器h₀，将目标函数转换为关于原型滤波器的凸二次函数后，求解获得另一个原型滤波器h_min，判断‖h_min-h₀‖₂＜δ(δ是一个给定的小正数，用来控制重构误差，‖·‖₂表示2-范数)是否成立，若成立，则终止迭代，输出h_min；否则令h₀＝(h_min+h₀)/2，返回继续进行迭代过程。Step 5, use linear iterative algorithm to solve formula ①. By first giving an initial prototype filter h ₀ , after transforming the objective function into a convex quadratic function about the prototype filter, another prototype filter h _min is obtained by solving it, judging that ‖h _min -h ₀ ‖ ₂ <δ( δ is a given small positive number used to control the reconstruction error, ‖·‖ ₂ indicates whether the 2-norm) is true, if true, the iteration is terminated, and h _min is output; otherwise, let h ₀ =(h _min + h ₀ )/2, return to continue the iterative process.

步骤6，根据步骤5所求出的最优原型滤波器h_min，再结合步骤1就获得了整个非均匀余弦调制滤波器组。Step 6, according to the optimal prototype filter h _min obtained in step 5, combined with step 1, the entire non-uniform cosine modulation filter bank is obtained.

上述步骤4中，权重α∈(0,1)，用于在传递失真和阻带能量之间作折中。In the above step 4, the weight α∈(0,1) is used to make a compromise between transfer distortion and stopband energy.

上述步骤5中，δ的取值范围设为10^-5～10^-8。In the above step 5, the value range of δ is set to 10 ^-5 -10 ^-8 .

与现有技术相比，本发明通过直接优化非均匀滤波器组性能的方法所获得的非均匀滤波器组重构误差更小，整体性能更加优异。同时本发明通过把非均匀滤波器组的设计问题归纳为一个关于原型滤波器的无约束优化问题，其中目标函数是非均匀滤波器组传递失真与原型滤波器阻带能量的加权和，最后利用线性迭代算法求解该优化问题，显著地降低了非均匀滤波器组的设计代价。所以本发明为降低设计的复杂度，实现信号的准确重建提供了简单高效的解决方案。Compared with the prior art, the reconstruction error of the non-uniform filter bank obtained by the method of directly optimizing the performance of the non-uniform filter bank in the present invention is smaller, and the overall performance is more excellent. Simultaneously the present invention is by summarizing the design problem of the non-uniform filter bank into an unconstrained optimization problem about the prototype filter, wherein the objective function is the weighted sum of the transfer distortion of the non-uniform filter bank and the stopband energy of the prototype filter, and finally utilizes the linear An iterative algorithm is used to solve the optimization problem, which significantly reduces the design cost of the non-uniform filter bank. Therefore, the present invention provides a simple and efficient solution for reducing design complexity and realizing accurate signal reconstruction.

附图说明Description of drawings

图1为本发明提供的设计非均匀余弦调制滤波器组的流程图。FIG. 1 is a flow chart of designing a non-uniform cosine modulation filter bank provided by the present invention.

图2为非均匀滤波器组的基本结构。Figure 2 shows the basic structure of a non-uniform filter bank.

图3为本发明的实例1与LiJL方法原型滤波器的幅度响应。Fig. 3 is the magnitude response of the prototype filter of Example 1 of the present invention and the LiJL method.

图4为本发明的实例1与LiJL方法所得到的非均匀分析滤波器组的幅度响应。Fig. 4 is the magnitude response of the non-uniform analysis filter bank obtained by Example 1 of the present invention and the LiJL method.

图5为本发明的实例1中目标函数值随迭代次数的变化曲线。Fig. 5 is a variation curve of the objective function value with the number of iterations in Example 1 of the present invention.

具体实施方式detailed description

一种近似完全重构非均匀余弦调制滤波器组的设计方法，如图1所示，包括如下步骤：A kind of design method of approximate complete reconstruction non-uniform cosine modulation filter bank, as shown in Figure 1, comprises the following steps:

第一步：图2给出了一个K通道非均匀滤波器组的一般结构，其输出与输入X(ω)的关系为The first step: Figure 2 shows the general structure of a K-channel non-uniform filter bank, whose output The relationship with the input X(ω) is

$\overset{&OverBar; &OverBar;}{X x} ((ω ω)) = = {T T}_{00} ((ω ω)) X x ((ω ω)) + + {Σ Σ}_{i i = = 00}^{K K - - 11} {Σ Σ}_{l l = = 11}^{{n no}_{i i} - - 11} \frac{11}{{n no}_{i i}} {F f}_{i i} ((ω ω)) {H h}_{i i} ((ω ω - - \frac{22 π π l l}{{n no}_{i i}})) X x ((ω ω - - \frac{22 π π l l}{{n no}_{i i}})) - - - - - - ((11))$

其中 $T_{0} (ω) = Σ_{i = 0}^{K - 1} \frac{1}{n_{i}} F_{i} (ω) H_{i} (ω) - - - (2)$ in $T_{0} (ω) = Σ_{i = 0}^{K - 1} \frac{1}{{no}_{i}} f_{i} (ω) h_{i} (ω) - - - (2)$

式中，ω表示频域变量，K表示非均匀滤波器组的通道数，X(ω)表示输入信号，表示输出信号，H_i(ω)和F_i(ω)分别表示非均匀滤波器组的分析滤波器和综合滤波器，n_i表示采样因子，l＝1,2,…,n_i-1表示混叠失真项，T₀(ω)表示传递函数，下标i＝0,1,…,K-1。在这里我们只考虑采样因子为整数且满足临界采样条件即的情况。In the formula, ω represents the frequency domain variable, K represents the number of channels of the non-uniform filter bank, X(ω) represents the input signal, Represents the output signal, H _i (ω) and F _i (ω) represent the analysis filter and synthesis filter of the non-uniform filter bank respectively, n _i represents the sampling factor, l=1,2,...,n _i -1 represents the aliasing distortion item, T ₀ (ω) represents the transfer function, and the subscript i=0,1,...,K-1. Here we only consider that the sampling factor is an integer and satisfies the critical sampling condition, namely Case.

根据等价原理，非均匀余弦调制滤波器组可以通过直接合并均匀余弦调制滤波器组的子带获得，合并公式为：According to the equivalence principle, the non-uniform cosine modulated filter bank can be obtained by directly combining the subbands of the uniform cosine modulated filter bank, and the combined formula is:

${H h}_{i i} ((ω ω)) = = \frac{11}{\sqrt{{m m}_{i i}}} {Σ Σ}_{p p = = {l l}_{i i}}^{{l l}_{i i + + 11} - - 11} {H h}_{p p}^{u u} ((ω ω)) - - - - - - ((33))$

${F f}_{i i} ((ω ω)) = = \frac{11}{\sqrt{{m m}_{i i}}} {Σ Σ}_{q q = = {l l}_{i i}}^{11_{i i + + 11} - - 11} {F f}_{q q}^{u u} ((ω ω)) - - - - - - ((44))$

式中， $l_{i} = \{\begin{matrix} Σ_{r = 0}^{i - 1} m_{r}, & i = 1, ..., K - 1 \\ 0, & i = 0 \end{matrix},$ m_i＝M/n_i，且M为采样因子n_i,i＝0,…,K-1的最小公倍数，即均匀余弦调制滤波器组的通道数；ω表示频域变量，K表示非均匀滤波器组的通道数，H_i(ω)和F_i(ω)分别表示非均匀滤波器组的分析滤波器和综合滤波器；和分别表示M通道均匀余弦调制滤波器组的分析滤波器和综合滤波器，由原型滤波器余弦调制而来，其时域调制公式为：In the formula, $l_{i} = \{\begin{matrix} Σ_{r = 0}^{i - 1} m_{r}, & i = 1, ..., K - 1 \\ 0, & i = 0 \end{matrix},$ m _i =M/n _i , and M is the least common multiple of the sampling factor n _i , i=0,...,K-1, that is, the number of channels of the uniform cosine modulation filter bank; ω represents the frequency domain variable, and K represents the non-uniform The number of channels of the filter bank, H _i (ω) and F _i (ω) respectively represent the analysis filter and the synthesis filter of the non-uniform filter bank; and Respectively represent the analysis filter and the synthesis filter of the M-channel uniform cosine modulation filter bank, which are cosine modulated by the prototype filter, and the time domain modulation formula is:

${h h}_{p p}^{u u} ((n no)) = = 22 h h ((n no)) c c o o s the s ((\frac{π π}{M m} ((p p + + \frac{11}{22})) ((n no - - \frac{D D.}{22})) + + {((- - 11))}^{p p} \frac{π π}{44})) - - - - - - ((55))$

${f f}_{q q}^{u u} ((n no)) = = 22 h h ((n no)) c c o o s the s ((\frac{π π}{M m} ((q q + + \frac{11}{22})) ((n no - - \frac{D D.}{22})) - - {((- - 11))}^{q q} \frac{π π}{44})) - - - - - - ((66))$

式中，h_p ^u(n)和f_q ^u(n)分别表示均匀滤波器组分析、综合滤波器的单位脉冲响应，h(n)表示长度为N的原型滤波器的单位脉冲响应，n＝0,…,N-1，p,q＝0,…,M-1，D表示重构延迟，M表示均匀余弦调制滤波器组的通道数。where h _p ^u (n) and f _q ^u (n) represent the unit impulse response of the uniform filter bank analysis and synthesis filter respectively, h(n) represents the unit impulse response of the prototype filter with length N, n =0,...,N-1, p,q=0,...,M-1, D represents the reconstruction delay, M represents the channel number of the uniform cosine modulation filter bank.

下面建立非均匀余弦调制滤波器组的子带滤波器与原型滤波器h(n)的函数关系。令h＝[h(0),h(1),…,h(N-1)]^T表示原型滤波器，其频率响应为H(ω)＝c^T(ω)h，其中c(ω)＝[1,…,e^-j(N-1)ω]^T，j为虚数单位。根据公式(3)-(6)，我们可以得出非均匀余弦调制滤波器组的分析滤波器H_i(ω)和综合滤波器F_i(ω)与原型滤波器h的关系式为The functional relationship between the sub-band filter of the non-uniform cosine modulation filter bank and the prototype filter h(n) is established below. Let h=[h(0),h(1),…,h(N-1)] ^T represents a prototype filter whose frequency response is H(ω)=c ^T (ω)h, where c(ω) =[1,...,e ^-j(N-1)ω ] ^T , j is the imaginary unit. According to formulas (3)-(6), we can draw the relationship between the analysis filter H _i (ω) and the synthesis filter F _i (ω) of the non-uniform cosine modulated filter bank and the prototype filter h as

${H h}_{i i} ((ω ω)) = = \frac{11}{\sqrt{{m m}_{i i}}} {Σ Σ}_{p p = = {l l}_{i i}}^{{l l}_{i i + + 11} - - 11} {C C}^{T T} ((ω ω)) {x x}_{p p} h h - - - - - - ((77))$

${F f}_{i i} ((ω ω)) = = \frac{11}{\sqrt{{m m}_{i i}}} {Σ Σ}_{q q = = {l l}_{i i}}^{{l l}_{i i + + 11} - - 11} {C C}^{T T} ((ω ω)) {l l}_{q q} h h - - - - - - ((88))$

其中，in,

${x x}_{p p} = = 22 d d i i a a g g {{c c o o s the s [[\frac{π π}{M m} ((p p + + \frac{11}{22})) ((00 - - \frac{D D.}{22})) + + {((- - 11))}^{p p} \frac{π π}{44}]],, ... ...,, c c o o s the s [[\frac{π π}{M m} ((p p + + \frac{11}{22})) ((N N - - 11 - - \frac{D D.}{22})) + + {((- - 11))}^{p p} \frac{π π}{44}]]}} - - - - - - ((99))$

${l l}_{q q} = = 22 d d i i a a g g {{c c o o s the s [[\frac{π π}{M m} ((q q + + \frac{11}{22})) ((00 - - \frac{D D.}{22})) + + {((- - 11))}^{q q} \frac{π π}{44}]],, ... ...,, c c o o s the s [[\frac{π π}{M m} ((q q + + \frac{11}{22})) ((N N - - 11 - - \frac{D D.}{22})) + + {((- - 11))}^{q q} \frac{π π}{44}]]}} - - - - - - ((1010))$

式中，H_i(ω)和F_i(ω)分别表示非均匀余弦调制滤波器组的分析滤波器和综合滤波器，ω表示频域变量，h表示原型滤波器，上标^T表示转置，M表示均匀余弦调制滤波器组的通道数，m_i＝M/n_i，n_i表示采样因子，下标i＝0,1,…,K-1，N表示原型滤波器的长度，D表示重构延迟。where H _i (ω) and F _i (ω) represent the analysis filter and synthesis filter of the non-uniform cosine modulated filter bank respectively, ω represents the frequency domain variable, h represents the prototype filter, and the superscript ^T represents the transpose , M represents the number of channels of the uniform cosine modulation filter bank, m _i =M/n _i , n _i represents the sampling factor, the subscript i=0,1,...,K-1, N represents the length of the prototype filter, D Indicates a rebuild delay.

第二步：结合公式(2)、(7)和(8)，我们可以得出非均匀余弦调制滤波器组传递函数T₀(ω)与原型滤波器h的关系式，即Step 2: Combining formulas (2), (7) and (8), we can obtain the relationship between the non-uniform cosine modulation filter bank transfer function T ₀ (ω) and the prototype filter h, namely

T₀(ω)＝h^TR(ω)h(11)其中， $R (ω) = \frac{1}{M} Σ_{i = 0}^{K - 1} Σ_{q = l_{i}}^{l_{i + 1} - 1} Σ_{p = l_{i}}^{l_{i + 1} - 1} l_{q}^{T} c (ω) c^{T} (ω) x_{p} - - - (12)$ 所以非均匀余弦调制滤波器组的传递失真E_t(h)可以表示为T ₀ (ω)=h ^T R(ω)h(11) where, $R (ω) = \frac{1}{m} Σ_{i = 0}^{K - 1} Σ_{q = l_{i}}^{l_{i + 1} - 1} Σ_{p = l_{i}}^{l_{i + 1} - 1} l_{q}^{T} c (ω) c^{T} (ω) x_{p} - - - (12)$ So the transfer distortion E _t (h) of the non-uniform cosine modulated filter bank can be expressed as

${E E.}_{t t} ((h h)) = = {&Integral; &Integral;}_{00}^{22 π π} | | {h h}^{T T} R R ((ω ω)) h h - - {e e}^{- - j j D D. ω ω} {| |}^{22} d d ω ω - - - - - - ((1313))$

式中，E_t(h)表示非均匀余弦调制滤波器组的传递失真，ω代表频域变量，h表示原型滤波器，上标^T表示转置，j表示虚数单位，D表示重构延迟。where E _t (h) represents the transfer distortion of the non-uniform cosine modulated filter bank, ω represents the frequency domain variable, h represents the prototype filter, the superscript ^T represents the transpose, j represents the imaginary unit, and D represents the reconstruction delay.

第三步：阻带能量E_s(h)的表达式如下：Step 3: The expression of the stop band energy E _s (h) is as follows:

${E E.}_{s the s} ((h h)) = = {&Integral; &Integral;}_{{ω ω}_{s the s}}^{π π} | | H h ((ω ω)) {| |}^{22} d d ω ω = = {h h}^{T T} {&Integral; &Integral;}_{{ω ω}_{s the s}}^{π π} {c c}^{* *} ((ω ω)) {c c}^{T T} ((ω ω)) d d ω ω h h = = {h h}^{T T} S S h h - - - - - - ((1414))$

式中，ω_s表示原型滤波器的阻带截止频率，ω表示频域变量，h表示原型滤波器，上标^T表示转置，上标^*表示共轭，D表示重构延迟，j表示虚数单位。where ω _s represents the stopband cut-off frequency of the prototype filter, ω represents the frequency domain variable, h represents the prototype filter, the superscript ^T represents the transpose, the superscript ^* represents the conjugate, D represents the reconstruction delay, and j represents the imaginary number unit.

第四步：在调制滤波器组的设计中通常需要关注滤波器组的重构特性和原型滤波器的频率特性。衡量滤波器组重构特性的指标是重构误差，由传递失真和混叠失真联合决定；衡量原型滤波器频率特性的性能指标有原型滤波器的通带平坦性和阻带衰减。Step 4: In the design of the modulated filter bank, it is usually necessary to pay attention to the reconstruction characteristics of the filter bank and the frequency characteristics of the prototype filter. The index to measure the reconstruction characteristics of the filter bank is the reconstruction error, which is jointly determined by the transfer distortion and aliasing distortion; the performance index to measure the frequency characteristics of the prototype filter includes the passband flatness and stopband attenuation of the prototype filter.

一个完全重构的滤波器组必须满足条件：滤波器组的传递函数为一个纯延迟且混叠失真为零。当上述条件近似成立时，滤波器组是近似完全重构的。在实际应用中，近似完全重构的滤波器组比完全重构的滤波器组拥有更佳的性能，且计算的复杂度更低，因此本发明研究近似完全重构的非均匀余弦调制滤波器组的设计。A fully reconstructed filter bank must satisfy the condition that the transfer function of the filter bank is a pure delay and the aliasing distortion is zero. When the above conditions approximately hold, the filter bank is approximately fully reconstructed. In practical applications, an approximately completely reconstructed filter bank has better performance than a fully reconstructed filter bank, and the computational complexity is lower, so the present invention studies an approximately completely reconstructed non-uniform cosine modulated filter set of designs.

小的传递失真和低的阻带能量可以保证获得整体性能较好的滤波器组。因此本发明将非均匀滤波器组的设计问题归纳为一个无约束的优化问题，目标函数就是非均匀滤波器组传递失真和原型滤波器阻带能量的加权和，表示为Small transfer distortion and low stopband energy can ensure a filter bank with better overall performance. Therefore the present invention summarizes the design problem of the non-uniform filter bank as an unconstrained optimization problem, and the objective function is exactly the weighted sum of the transfer distortion of the non-uniform filter bank and the stopband energy of the prototype filter, expressed as

$Φ Φ ((h h)) = = α α {&Integral; &Integral;}_{00}^{22 π π} | | {h h}^{T T} - - R R ((ω ω)) h h - - {e e}^{- - j j D D. ω ω} {| |}^{22} d d ω ω + + ((11 - - α α)) {h h}^{T T} S S h h - - - - - - ((1515))$

式中，α∈(0,1)为权值，Φ(h)表示关于h的目标函数，ω表示频域变量，h表示原型滤波器，上标^T表示转置，上标^*表示共轭，j表示虚数单位，D表示重构延迟。目标函数的第一项控制传递失真，第二项控制阻带能量。In the formula, α∈(0,1) is the weight, Φ(h) represents the objective function about h, ω represents the frequency domain variable, h represents the prototype filter, the superscript ^T represents the transpose, and the superscript ^* represents the conjugate , j represents the imaginary unit, and D represents the reconstruction delay. The first term of the objective function controls the transfer distortion, and the second term controls the stopband energy.

第五步：从目标函数的表达式可以看出，目标函数Φ(h)是关于原型滤波器h的四次函数，求解较为困难。本发明采用线性迭代算法来求解该问题。首先设计一个初始原型滤波器h₀，然后将初始原型滤波器h₀代入(15)式，目标函数转换为Step 5: It can be seen from the expression of the objective function that the objective function Φ(h) is a quartic function about the prototype filter h, and it is difficult to solve it. The present invention uses a linear iterative algorithm to solve this problem. First design an initial prototype filter h ₀ , and then substitute the initial prototype filter h ₀ into Equation (15), the objective function is transformed into

${Φ Φ}_{{h h}_{00}} ((h h)) = = α α {&Integral; &Integral;}_{00}^{22 π π} | | {h h}_{00}^{T T} - - R R ((ω ω)) h h - - {e e}^{- - j j D D. ω ω} {| |}^{22} d d ω ω + + ((11 - - α α)) {h h}^{T T} S S h h - - - - - - ((1616))$

从上式可以观察出是一个关于原型滤波器h的凸二次函数，该函数存在最小值点为It can be observed from the above Is a convex quadratic function about the prototype filter h, the function has a minimum point as

h_min＝[αP(h₀)+(1-α)S]^-1αb(h₀)(17)式中向量b(h₀)和矩阵P(h₀)的表达式分别为h _min ＝[αP(h ₀ )+(1-α)S] ^-1 αb(h ₀ )(17) In the formula, the expressions of vector b(h ₀ ) and matrix P(h ₀ ) are respectively

$b b (({h h}_{00})) = = Re Re {{{&Integral; &Integral;}_{00}^{22 π π} {e e}^{j j D D. ω ω} {R R}^{T T} ((w w)) d d ω ω}} {h h}_{00} - - - - - - ((1818))$

$P P (({h h}_{00})) = = {&Integral; &Integral;}_{00}^{22 π π} - - {R R}^{T T} ((ω ω)) {h h}_{00} {h h}_{00}^{T T} - - {R R}^{* *} ((ω ω)) d d ω ω - - - - - - ((1919))$

式中，Re{·}表示取实操作，ω表示频域变量，上标^T表示转置，j表示虚数单位，D表示重构延迟，h₀表示初始原型滤波器。在这里需要迭代求解h_min直到h_min非常接近h₀，才能达到最小化目标函数Φ(h)的效果。本发明的线性迭代算法步骤如下：where Re{ } represents the real operation, ω represents the variable in the frequency domain, the superscript ^T represents the transpose, j represents the imaginary unit, D represents the reconstruction delay, and h ₀ represents the initial prototype filter. Here it is necessary to iteratively solve h _min until h _min is very close to h ₀ , in order to achieve the effect of minimizing the objective function Φ(h). The linear iterative algorithm steps of the present invention are as follows:

1、初始化原型滤波器，即设计一个长度为N的低通滤波器h₀；1. Initialize the prototype filter, that is, design a low-pass filter h ₀ with a length of N;

2、利用公式(16)，将h₀代入求解获得h_min；2. Using formula (16), substitute h ₀ into the solution to obtain h _min ;

3、判断‖h_min-h₀‖₂＜δ(δ是一个给定的小正数)是否成立。若成立，则终止迭代，输出h_min；否则令h₀＝(h_min+h₀)/2，返回第二步继续进行迭代过程。在本发明优选实施例中，δ的取值范围设为10^-5～10^-8。3. Determine whether ‖h _min -h ₀ ‖ ₂ <δ (δ is a given small positive number) holds true. If true, terminate the iteration and output h _min ; otherwise set h ₀ =(h _min +h ₀ )/2, and return to the second step to continue the iterative process. In a preferred embodiment of the present invention, the value range of δ is set to 10 ^-5 -10 ^-8 .

本发明方法采用的线性迭代算法是一种修正牛顿法。可以证明，本发明方法是收敛的。The linear iterative algorithm adopted by the method of the present invention is a modified Newton method. It can be proved that the method of the present invention is convergent.

第六步：根据第五步所求出的最优原型滤波器h_min，再结合公式(7)(8)就获得了整个非均匀余弦调制滤波器组。Step 6: According to the optimal prototype filter h _min obtained in step 5, combined with formulas (7) and (8), the entire non-uniform cosine modulation filter bank is obtained.

为验证本发明方法的有效性，进行了仿真实验，所有的仿真都在相同的运行环境下进行。In order to verify the validity of the method of the present invention, simulation experiments are carried out, and all simulations are carried out under the same operating environment.

实例1：Example 1:

设计一个非均匀余弦调制滤波器组：K＝5,n_i＝[2,4,8,16,16]，分别采用LiJL等人提出的直接合并方法和本发明方法设计。在LiJL等人提出的直接合并方法中，通过子带合并的方式将一个均匀滤波器组转换成一个非均匀滤波器组。其中，均匀滤波器组采用ZhangZJ发表在《IEEESignalProcessingLetters》上的《Efficientdesignofcosinemodulatedfilterbanksbasedongradientinformation》中的算法设计，通道数为16，原型滤波器长度为256。在本发明中，其相关参数为M＝16,N＝256,D＝N-1,ω_s＝π/M,δ＝2.5×10^-7，采用直接优化的方式来设计该非均匀滤波器组。为了比较的公平性，本发明方法迭代所用的初始滤波器，与采用LiJL等人提出的方法中用于合并的均匀滤波器组的原型滤波器相同。图3和图4分别给出了两个非均匀余弦调制滤波器组的原型滤波器的幅度响应和分析滤波器组的幅度响应。图5给出了本发明方法的目标函数值随迭代次数的变化曲线。从图5我们可以发现目标函数值在经过几次迭代之后就趋于不变，即验证了本发明方法的快速收敛性。Design a non-uniform cosine modulation filter bank: K=5, n _i =[2, 4, 8, 16, 16], respectively adopt the direct combination method proposed by LiJL et al. and the method of the present invention to design. In the direct combining method proposed by LiJL et al., a uniform filter bank is converted into a non-uniform filter bank by means of subband combining. Among them, the uniform filter bank adopts the algorithm design in "Efficient design of cosine modulated filter banks based on gradient information" published by ZhangZJ on "IEEE Signal Processing Letters", the number of channels is 16, and the length of the prototype filter is 256. In the present invention, its relevant parameters are M=16, N=256, D=N-1, ω _s =π/M, δ=2.5×10 ^-7 , and the non-uniform filter is designed by direct optimization Group. For the fairness of comparison, the initial filter used for iteration in the method of the present invention is the same as the prototype filter used for merging uniform filter banks in the method proposed by LiJL et al. Figure 3 and Figure 4 show the amplitude response of the prototype filter and the analysis filter bank of the two non-uniform cosine modulated filter banks, respectively. Fig. 5 shows the change curve of the objective function value with the number of iterations of the method of the present invention. From Fig. 5, we can find that the objective function value tends to remain unchanged after several iterations, which verifies the rapid convergence of the method of the present invention.

表1Table 1

表1分别陈列了两个非均匀滤波器组的传递失真、混叠失真、重构误差以及其原型滤波器的阻带衰减。从表1可以看出，本发明方法设计所得的非均匀滤波组比采用LiJL等人提出的直接合并方法设计所得的滤波器组具有更好的整体性能，其重构误差降低了一个数量级。Table 1 shows the transfer distortion, aliasing distortion, reconstruction error and the stopband attenuation of the prototype filter of the two non-uniform filter banks respectively. It can be seen from Table 1 that the non-uniform filter bank designed by the method of the present invention has better overall performance than the filter bank designed by the direct combination method proposed by LiJL et al., and its reconstruction error is reduced by an order of magnitude.

实例2：Example 2:

用本发明方法和现有方法分别设计采样因子为[4,4,2]的3通道非均匀滤波器组，然后进行性能分析与比较。本发明方法的相关参数为：K＝3，n_i＝[4,4,2]，M＝4,ω_s＝π/M,D＝N-1。当N＝44时，α＝0.5,δ＝1.0×10^-6；当N＝64时，α＝0.4,δ＝1.0×10^-7。3-channel non-uniform filter banks with sampling factors [4, 4, 2] are designed respectively by using the method of the present invention and the existing method, and then performance analysis and comparison are carried out. The relevant parameters of the method of the present invention are: K=3, n _i =[4,4,2], M=4, ω _s =π/M, D=N-1. When N=44, α=0.5, δ=1.0×10 ^-6 ; when N=64, α=0.4, δ=1.0×10 ^-7 .

表2Table 2

表2给出了本发明方法与现有方法设计所得的非均匀余弦调制滤波器组的重构误差以及其原型滤波器的阻带衰减。通过比较本发明方法与LiJL方法可以看出，本文所采用的直接优化非均匀滤波器的方法比直接合并方法效果更好，所获得的非均匀滤波器组整体性能更佳，重构误差更小。通过比较本发明方法与另外几个方法可以看出，本发明方法获得的非均匀滤波器组重构误差更小，即重构性能更好，可以更准确的恢复原信号。Table 2 shows the reconstruction error of the non-uniform cosine modulation filter bank designed by the method of the present invention and the existing method and the stopband attenuation of the prototype filter. By comparing the method of the present invention with the LiJL method, it can be seen that the method of directly optimizing the non-uniform filter used in this paper is better than the direct combination method, and the overall performance of the non-uniform filter bank obtained is better, and the reconstruction error is smaller . By comparing the method of the present invention with several other methods, it can be seen that the reconstruction error of the non-uniform filter bank obtained by the method of the present invention is smaller, that is, the reconstruction performance is better, and the original signal can be restored more accurately.

Claims

1. the design method of approximate complete reconstruction non-uniform cosine modulation filter bank, it is characterized in that, comprises the steps:

Step 1, converting the analysis filter bank and the synthesis filter bank of the non-uniform cosine modulation filter bank into functions about the prototype filter h respectively;

Step 2, transforming the transfer distortion E _t (h) of the non-uniform cosine modulated filter bank into a function about the prototype filter h;

Step 3, converting the stop band energy E _s (h) of the prototype filter into a function about the prototype filter h;

Step 4, the design problem of the non-uniform filter bank is summarized as an unconstrained optimization problem about the prototype filter, and its objective function is the weighted sum of the transfer distortion of the non-uniform filter bank and the stop-band energy of the prototype filter, namely:

\underset{h}{m i no} Φ (h) = {αE}_{t} (h) + (1 - α) {E.}_{the s} (h)

①

In the formula, Φ(h) represents the objective function, E _t (h) represents the transfer distortion of the non-uniform cosine modulation filter bank, E _s (h) represents the stopband energy of the prototype filter, h represents the prototype filter, α represents Weights;

Step 5, use linear iterative method to solve formula ①; namely:

Step 5.1, first give an initial prototype filter h ₀ ;

Step 5.2, after converting the objective function into a convex quadratic function about the prototype filter, solve to obtain another prototype filter h _min ;

Step 5.3, judge whether ||h _min -h ₀ || ₂ <δ is true, where δ is a given small positive number, and ||·|| ₂ represents the 2-norm; if true, terminate the iteration and output h _min ; otherwise, set h ₀ =(h _min +h ₀ )/2, and return to step 5.2 to continue the iterative process;

Step 6, substituting the optimal prototype filter h _min obtained in step 5 into step 1 to obtain the entire non-uniform cosine modulation filter bank.

2. The method for designing an approximately completely reconstructed non-uniform cosine modulated filter bank according to claim 1, characterized in that, in step 4, the weight α∈(0,1).

3. The method for designing an approximately completely reconstructed non-uniform cosine modulated filter bank according to claim 1, wherein in step 5.3, the value range of δ is set to 10 ^-5 ~ 10 ^-8 .