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

Abstract

The invention discloses a designing method for a near-complete reconstruction non-uniform cosine modulated filter bank. According to the designing method, the reconstruction errors of the non-uniform filter bank obtained through the method of directly optimizing the performance of the non-uniform filter bank are smaller, and the overall performance is better. Meanwhile, the designing problem of the non-uniform filter bank is reduced to an unconstrained optimization problem about a prototype filter, wherein the objective function is the weighted sum of transmission distortion of the non-uniform filter bank and the stopband energy of the prototype filter; finally, the optimization problem is solved by utilizing a linear iterative algorithm, and therefore the designing cost of the non-uniform filter bank is remarkably lowered. Accordingly, the designing method provides a simple and efficient solution for lowering the designing complexity and implementing accurate construction of signals.

Description

Design method for approximate complete reconstruction non-uniform cosine modulation filter bank

Technical Field

The invention belongs to the field of filter bank design, and particularly relates to a design method of an approximate complete reconstruction non-uniform cosine modulation filter bank.

Background

The filter bank has been attracting attention as a core content in multi-rate signal processing, and has been widely applied to adaptive filtering, speech image coding, image processing, and the like. The design of a general filter bank needs to optimize all analysis filters and synthesis filters, and the design of a modulation filter bank only needs to optimize a prototype filter, so that the design complexity is greatly reduced. Currently, there are two main types of modulation filter banks, namely, a cosine modulation filter bank and a discrete fourier transform modulation filter bank. Compared with the two modulated filter banks, the cosine modulated filter bank is more suitable for processing real-valued signals because the cosine modulated filter bank is cosine modulated.

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, non-uniform reasonable division of an image spectrum is required to achieve noise removal more effectively. The idea of combining uniform filter banks to implement a non-uniform filter bank was innovatively proposed by CoxRV in "the design of the non-uniform filter bank and published in" ieee transactions on industries, spechan & d signal processing ", which is a great advance in the design of non-uniform filter banks. The concept and construction of an M-channel non-uniform quadrature mirror filter bank was first proposed by HoangPQ et al in Nonuniformulteriomterfahilbatters, published by IEEEInternationality symposium on circuits and systems. An approximate fully-reconstructed non-uniform cosine modulation filter bank is successfully constructed by LiJL et al in an approximate fully-reconstructed-perfect-reconstruction method published in IEEETransactionsonSignalProcessing, however, the design method is to obtain the uniform filter bank and then directly perform the sub-band combination. After obtaining a prototype filter with a 3dB cutoff frequency of 2 pi/M by using the Kaiser window function method in the AChannelCombiner apparatus for the design of New communications non Unifor Filter Bank published by Maneshak et al in International Conferenceon communications and Signalprocessing, the non-uniform filter bank is also obtained by direct combination.

However, the above-mentioned way of directly combining the subbands of the non-uniform filter bank for designing the non-uniform filter bank has a disadvantage: the performance of the non-uniform filter bank cannot be directly controlled and optimized, so that the performance of the non-uniform filter bank is completely determined by the selected uniform filter bank.

Disclosure of Invention

The invention aims to solve the technical problem that the performance of a non-uniform filter bank cannot be directly controlled and optimized in the existing direct combination method for designing the non-uniform filter bank, and provides a design method for an approximate complete reconstruction non-uniform cosine modulation filter bank.

In order to solve the problems, the invention is realized by the following technical scheme:

the design method of the approximate complete reconstruction non-uniform cosine modulation filter bank comprises the following steps:

step 1, respectively converting an analysis filter bank and a comprehensive filter bank of a non-uniform cosine modulation filter bank into functions related to a prototype filter h according to a cosine modulation theory and an equivalence theory.

Step 2, distortion E of transmission of the non-uniform cosine modulation filter bank_t(h) Into a function for the prototype filter h.

Step 3, the stop band energy E of the prototype filter_s(h) Into a function for the prototype filter h.

Step 4, according to the performance index of the filter bank design, small transmission distortion and low stop band energy can ensure to obtain the filter bank with better overall performance

$\underset{h}{min}\mathrm{\Φ}\left(h\right)={\mathrm{\αE}}_t\left(h\right)+(1-\mathrm{\α})E_s\left(h\right)$ ①

Where Φ (h) represents an objective function, E_t(h) Representing the transfer distortion of a non-uniform cosine modulated filterbank, E_s(h) Represents the stop band energy of the prototype filter, h represents the prototype filter, and α represents the weights.

Step 5, linear iterative algorithm is adopted to solve ①, an initial prototype filter h is given₀After the objective function is converted into a convex quadratic function related to the prototype filter, solving to obtain another prototype filter h_minJudgment | h_min-h₀‖₂< (is a given small positive number, used to control the reconstruction error, | |)₂Representing 2-norm) is satisfied, if so, the iteration is terminated, and h is output_min(ii) a Otherwise let h₀＝(h_min+h₀) And/2, returning to continue the iterative process.

Step 6, according to the optimal prototype filter h obtained in the step 5_minAnd combining the step 1 to obtain the whole non-uniform cosine modulation filter bank.

In step 4 above, the weight α ∈ (0,1) is used to make a trade-off between the transfer distortion and the stopband energy.

In the above step 5, the value range is set to 10^-5～10^-8。

Compared with the prior art, the method for directly optimizing the performance of the non-uniform filter bank has the advantages that the obtained non-uniform filter bank has smaller reconstruction error and more excellent overall performance. Meanwhile, the design problem of the non-uniform filter bank is generalized to be an unconstrained optimization problem about a prototype filter, wherein an 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, and finally, the optimization problem is solved by utilizing a linear iterative algorithm, so that the design cost of the non-uniform filter bank is remarkably reduced. Therefore, the invention provides a simple and efficient solution for reducing the complexity of design and realizing accurate reconstruction of signals.

Drawings

Fig. 1 is a flowchart for designing a non-uniform cosine modulated filter bank according to the present invention.

Fig. 2 shows the basic structure of a non-uniform filter bank.

Fig. 3 shows the magnitude response of prototype filters of example 1 and LiJL method of the present invention.

Fig. 4 shows the magnitude response of the non-uniform analysis filter bank obtained by the LiJL method and example 1 of the present invention.

Fig. 5 is a graph showing the variation of the objective function value with the number of iterations in example 1 of the present invention.

Detailed Description

A method for designing an approximate complete reconstruction non-uniform cosine modulated filter bank, as shown in fig. 1, includes the following steps:

the first step is as follows: FIG. 2 shows the general structure of a K-channel non-uniform filter bank, the output of whichWith respect to the input X (omega) of

$\stackrel{\&OverBar;}{X}\left(\mathrm{\&omega;}\right)=T_0\left(\mathrm{\&omega;}\right)X\left(\mathrm{\&omega;}\right)+\underset{i=0}{\overset{K-1}{\&Sigma;}}\underset{l=1}{\overset{n_i-1}{\&Sigma;}}\frac{1}{n_i}{F}_i\left(\mathrm{\&omega;}\right)H_i\left(\mathrm{\&omega;}-\frac{2\mathrm{\&pi;}l}{n_i}\right)X\left(\mathrm{\&omega;}-\frac{2\mathrm{\&pi;}l}{n_i}\right)---\left(1\right)$

Wherein $T_0\left(\mathrm{\&omega;}\right)=\underset{i=0}{\overset{K-1}{\&Sigma;}}\frac{1}{n_i}{F}_i\left(\mathrm{\&omega;}\right)H_i\left(\mathrm{\&omega;}\right)---\left(2\right)$

Where ω denotes a frequency domain variable, K denotes the number of channels of the non-uniform filter bank, X (ω) denotes an input signal,representing the output signal, H_i(omega) and F_i(omega) analysis filters and synthesis filters, respectively, representing a non-uniform filter bank, n_iWhich is indicative of a sampling factor, is,l＝1,2,…,n_i-1 represents the aliasing distortion term, T₀And (ω) represents the transfer function, with index i being 0,1, …, K-1. Here we only consider the sampling factor as an integer and satisfy the critical sampling condition, i.e.The case (1).

According to the equivalence principle, the non-uniform cosine modulated filter bank can be obtained by directly combining the sub-bands of the uniform cosine modulated filter bank, and the combination formula is as follows:

$H_i\left(\mathrm{\&omega;}\right)=\frac{1}{\sqrt{m_i}}\underset{p=l_i}{\overset{l_{i+1}-1}{\&Sigma;}}{H}_p^u\left(\mathrm{\&omega;}\right)---\left(3\right)$

$F_i\left(\mathrm{\&omega;}\right)=\frac{1}{\sqrt{m_i}}\underset{q=l_i}{\overset{1_{i+1}-1}{\&Sigma;}}{F}_q^u\left(\mathrm{\&omega;}\right)---\left(4\right)$

in the formula, $l_i=\left\{\begin{array}{cc}\underset{r=0}{\overset{i-1}{\&Sigma;}}{m}_r,& i=1,\mathrm{...},K-1\\ 0,& i=0\end{array}\right.,$ m_i＝M/n_iand M is a sampling factor n_iI is the least common multiple of 0, …, K-1, i.e. the number of channels of the uniform cosine modulated filter bank; omega denotes the frequency domain variation, K denotes the number of channels of the non-uniform filter bank, H_i(omega) and F_i(ω) an analysis filter and a synthesis filter representing the non-uniform filter bank, respectively;andan analysis filter and a synthesis filter respectively representing an M-channel uniform cosine modulated filter bank are modulated by the cosine of a prototype filterThe time domain modulation formula is as follows:

${h_p}^u\left(n\right)=2h\left(n\right)cos\left(\frac{\mathrm{\&pi;}}{M}\right(p+\frac{1}{2}\left)\right(n-\frac{D}{2})+{\left(-1\right)}^p\frac{\mathrm{\&pi;}}{4})---\left(5\right)$

${f_q}^u\left(n\right)=2h\left(n\right)cos\left(\frac{\mathrm{\&pi;}}{M}\left(q+\frac{1}{2}\right)\left(n-\frac{D}{2}\right)-{\left(-1\right)}^q\frac{\mathrm{\&pi;}}{4}\right)---\left(6\right)$

in the formula, h_p ^u(n) and f_q ^u(N) represents 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 is 0, …, N-1, p, q is 0, …, M-1, D represents the reconstruction delay, and M represents the number of channels of the uniform cosine modulated filter bank.

The functional relationship between the subband filters of the non-uniform cosine modulated filter bank and the prototype filter h (n) is established below. Let h be [ h (0), h (1), …, h (N-1)]^TRepresenting a prototype filter having a frequency response of H (ω) c^T(ω) h, wherein c (ω) ═ 1, …, e^-j(N-1)ω]^TAnd j is an imaginary unit. From equations (3) - (6), we can derive the analysis filter H of the non-uniform cosine modulated filter bank_i(omega) and a synthesis filter F_i(ω) is related to the prototype filter h by

$H_i\left(\mathrm{\&omega;}\right)=\frac{1}{\sqrt{m_i}}\underset{p=l_i}{\overset{l_{i+1}-1}{\&Sigma;}}{C}^T\left(\mathrm{\&omega;}\right)x_{p}h---\left(7\right)$

$F_i\left(\mathrm{\&omega;}\right)=\frac{1}{\sqrt{m_i}}\underset{q=l_i}{\overset{l_{i+1}-1}{\&Sigma;}}{C}^T\left(\mathrm{\&omega;}\right)l_{q}h---\left(8\right)$

Wherein,

$x_p=2diag\{cos\&lsqb;\frac{\mathrm{\&pi;}}{M}\left(p+\frac{1}{2}\right)\left(0-\frac{D}{2}\right)+{\left(-1\right)}^p\frac{\mathrm{\&pi;}}{4}\&rsqb;,...,cos\&lsqb;\frac{\mathrm{\&pi;}}{M}\left(p+\frac{1}{2}\right)\left(N-1-\frac{D}{2}\right)+{\left(-1\right)}^p\frac{\mathrm{\&pi;}}{4}\&rsqb;\}---\left(9\right)$

$l_q=2diag\left\{cos\&lsqb;\frac{\mathrm{\&pi;}}{M}\left(q+\frac{1}{2}\right)\left(0-\frac{D}{2}\right)+{\left(-1\right)}^q\frac{\mathrm{\&pi;}}{4}\&rsqb;,...,cos\&lsqb;\frac{\mathrm{\&pi;}}{M}\left(q+\frac{1}{2}\right)\left(N-1-\frac{D}{2}\right)+{\left(-1\right)}^q\frac{\mathrm{\&pi;}}{4}\&rsqb;\right\}---\left(10\right)$

in the formula, H_i(omega) and F_i(omega) respectively represents an analysis filter and a synthesis filter of the non-uniform cosine modulation filter bank, omega represents a frequency domain variable, h represents a prototype filter, and a superscript^TDenotes transposition, M denotes the number of channels of the uniform cosine modulated filterbank, M_i＝M/n_i，n_iDenotes the sampling factor, the index i is 0,1, …, K-1, N denotes the length of the prototype filter, and D denotes the reconstruction delay.

The second step is that: by combining the equations (2), (7) and (8), we can obtain the non-uniform cosine modulated filter bank transfer function T₀(ω) relation to the prototype filter h, i.e.

T₀(ω)＝h^TR (ω) h (11) wherein, $R\left(\mathrm{\&omega;}\right)=\frac{1}{M}\underset{i=0}{\overset{K-1}{\&Sigma;}}\underset{q=l_i}{\overset{l_{i+1}-1}{\&Sigma;}}\underset{p=l_i}{\overset{l_{i+1}-1}{\&Sigma;}}{l}_q^{T}c\left(\mathrm{\&omega;}\right)c^T\left(\mathrm{\&omega;}\right)x_p---\left(12\right)$ so that the transfer distortion E of the non-uniform cosine modulated filterbank_t(h) Can be expressed as

$E_t\left(h\right)={\&Integral;}_0^{2\mathrm{\&pi;}}|h^{T}R\left(\mathrm{\&omega;}\right)h-e^{-jD\mathrm{\&omega;}}{|}^{2}d\mathrm{\&omega;}---\left(13\right)$

In the formula, E_t(h) Representing the transfer distortion of a non-uniform cosine modulated filter bank, omega representing a frequency domain variable, h representing a prototype filter, and superscript^TDenotes transposition, j denotes an imaginary unit, and D denotes reconstruction delay.

The third step: stop band energy E_s(h) The expression of (a) is as follows:

$E_s\left(h\right)={\&Integral;}_{{\mathrm{\&omega;}}_s}^{\mathrm{\&pi;}}|H\left(\mathrm{\&omega;}\right){|}^{2}d\mathrm{\&omega;}=h^T{\&Integral;}_{{\mathrm{\&omega;}}_s}^{\mathrm{\&pi;}}{c}^{*}\left(\mathrm{\&omega;}\right)c^T\left(\mathrm{\&omega;}\right)d\mathrm{\&omega;}h=h^{T}Sh---\left(14\right)$

in the formula, ω_sRepresents the stopband cutoff frequency of the prototype filter, ω represents the frequency domain variable, h represents the prototype filter, superscript^TIndicating transposition, superscript^*Denotes the conjugate, D denotes the reconstruction delay, and j denotes the imaginary unit.

The fourth step: the reconstruction characteristics of the filter bank and the frequency characteristics of the prototype filter are often of concern in the design of modulated filter banks. The index for measuring the reconstruction characteristic of the filter bank is reconstruction error which is jointly determined by transfer distortion and aliasing distortion; the performance index for measuring the frequency characteristic of the prototype filter comprises the pass band flatness and stop band attenuation of the prototype filter.

A fully reconstructed filter bank must satisfy the condition: the transfer function of the filter bank is a pure delay and the aliasing distortion is zero. When the above conditions are approximately true, the filter bank is approximately fully reconstructed. In practical application, the approximate fully-reconstructed filter bank has better performance than the fully-reconstructed filter bank, and the calculation complexity is lower, so the design of the approximate fully-reconstructed non-uniform cosine modulation filter bank is researched.

The small transfer distortion and low stop-band energy ensure that a filter bank with better overall performance is obtained. Therefore, the invention reduces the design problem of the non-uniform filter bank into an unconstrained optimization problem, and the 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 and is expressed as

$\mathrm{\&Phi;}\left(h\right)=\mathrm{\&alpha;}{\&Integral;}_0^{2\mathrm{\&pi;}}|h^T-R\left(\mathrm{\&omega;}\right)h-e^{-jD\mathrm{\&omega;}}{|}^{2}d\mathrm{\&omega;}+(1-\mathrm{\&alpha;})h^{T}Sh---\left(15\right)$

In the formula, α∈ (0,1) is weight, Φ (h) represents an objective function with respect to h, ω represents a frequency domain variable, h represents a prototype filter, and superscript^TIndicating transposition, superscript^*Denotes the conjugate, j denotes the imaginary unit, and D denotes the reconstruction delay. The first term of the objective function controls the transfer distortion and the second term controls the stop band energy.

The fifth step: as can be seen from the expression of the objective function, the objective function Φ (h) is a quartic function with respect to the prototype filter h, and it is difficult to solve. The present invention employs a linear iterative algorithm to solve the problem. First, an initial prototype filter h is designed₀Then the initial prototype filter h is filtered₀Substituting into equation (15), the objective function is converted into

${\mathrm{\&Phi;}}_{h_0}\left(h\right)=\mathrm{\&alpha;}{\&Integral;}_0^{2\mathrm{\&pi;}}|{h_0}^T-R\left(\mathrm{\&omega;}\right)h-e^{-jD\mathrm{\&omega;}}{|}^{2}d\mathrm{\&omega;}+(1-\mathrm{\&alpha;})h^{T}Sh---\left(16\right)$

From the above formula can be observedIs a convex quadratic function with respect to the prototype filter h, which has a minimum point of

h_min＝[αP(h₀)+(1-α)S]^-1αb(h₀) (17) wherein the vector b (h) is₀) And matrix P (h)₀) Are respectively expressed as

$b\left(h_0\right)=\mathrm{Re}\left\{{\&Integral;}_0^{2\mathrm{\&pi;}}{e}^{jD\mathrm{\&omega;}}{R}^T\right(w\left)d\mathrm{\&omega;}\right\}{h}_0---\left(18\right)$

$P\left(h_0\right)={\&Integral;}_0^{2\mathrm{\&pi;}}-R^T\left(\mathrm{\&omega;}\right)h_0{h_0}^T-R^{*}\left(\mathrm{\&omega;}\right)d\mathrm{\&omega;}---\left(19\right)$

In the formula, Re {. cndot } represents the operation of real, omega represents the frequency domain variable, and the superscript^TDenotes transposition, j denotes an imaginary unit, D denotes reconstruction delay, h₀Representing an initial prototype filter. Where an iterative solution h is required_minUp to h_minVery close to h₀The effect of minimizing the objective function Φ (h) can be achieved. The linear iterative algorithm of the invention comprises the following steps:

1. initialising the prototype filter, i.e. designing a low-pass filter h of length N₀；

2. Using equation (16), h₀Solving by substitution to obtain h_min；

3. Judgment | h_min-h₀‖₂Whether < (is a given small positive number) holds. If yes, terminating the iteration and outputting h_min(ii) a Otherwise let h₀＝(h_min+h₀) And/2, returning to the second step and continuing the iterative process. In the preferred embodiment of the present invention, the value range is set to 10^-5～10^-8。

The linear iterative algorithm adopted by the method is a modified Newton method. It can be shown that the inventive method is convergent.

And a sixth step: according to the optimal prototype filter h obtained in the fifth step_minAnd then combining the formulas (7) and (8) to obtain the whole non-uniform cosine modulation filter bank.

In order to verify the effectiveness of the method of the invention, simulation experiments are carried out, and all simulations are carried out under the same operating environment.

Example 1:

designing a non-uniform cosine modulation filter bank: k is 5, n_i＝[2,4,8,16,16]The direct merging method proposed by LiJL et al and the method of the present invention are respectively adopted for 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. Wherein the uniform filter bank is published in Effectiontesignomone modulated filters based on IEEESignalProcessLetters using ZhangZJThe algorithm is designed, the number of channels is 16, and the length of the prototype filter is 256. In the present invention, the relevant parameters are M-16, N-256, D-N-1, ω_s＝π/M,＝2.5×10^-7The non-uniform filter bank is designed in a direct optimization way. For comparative fairness, the initial filters used for the inventive method iterations are the same as the prototype filters using the uniform filter bank for merging in the method proposed by LiJL et al. Fig. 3 and 4 show the amplitude response of a prototype filter of two non-uniform cosine modulated filter banks and the amplitude response of an analysis filter bank, respectively. Fig. 5 shows the variation of the objective function value with the number of iterations for the method of the invention. From fig. 5, we can find that the objective function value tends to be unchanged after several iterations, i.e. the fast convergence of the method of the present invention is verified.

TABLE 1

Table 1 lists the transfer distortion, aliasing distortion, reconstruction error for the two non-uniform filter banks, and the stopband attenuation of their prototype filters, respectively. It can be seen from table 1 that the non-uniform filter set designed by the method of the present invention has better overall performance than the filter set designed by the direct merging method proposed by LiJL et al, and the reconstruction error is reduced by an order of magnitude.

Example 2:

the sampling factors are respectively designed to be [4,4,2 ] by the method of the invention and the existing method]And then performing performance analysis and comparison. The relevant parameters of the method are as follows: k is 3, n_i＝[4,4,2]，M＝4,ω_spi/M, D-N-1, when N-44, α -0.5, 1.0 × 10^-6When N is 64, α is 0.4 and 1.0 × 10^-7。

TABLE 2

Table 2 shows the reconstruction error of the non-uniform cosine modulated filter bank designed by the method of the present invention and the existing method, and the stopband attenuation of the prototype filter. 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 employed herein has better effect than the direct combination method, and the obtained non-uniform filter bank has better overall performance and smaller reconstruction error. Compared with other methods, the method provided by the invention has the advantages that the reconstruction error of the non-uniform filter bank obtained by the method is smaller, namely the reconstruction performance is better, and the original signal can be recovered more accurately.

Claims (3)

1. The design method of the approximate complete reconstruction non-uniform cosine modulation filter bank is characterized by comprising the following steps:

step 1, respectively converting an analysis filter bank and a comprehensive filter bank of a non-uniform cosine modulation filter bank into functions related to a prototype filter h;

step 2, distortion E of transmission of the non-uniform cosine modulation filter bank_t(h) Conversion to a function for prototype filter h;

step 3, the stop band energy E of the prototype filter_s(h) Conversion to relate to prototypesA function of a filter h;

and 4, generalizing the design problem of the non-uniform filter bank into an unconstrained optimization problem about the prototype filter, wherein an 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:

$\underset{h}{min}\mathrm{\&Phi;}\left(h\right)={\mathrm{\&alpha;E}}_t\left(h\right)+(1-\mathrm{\&alpha;})E_s\left(h\right)$ ①

where Φ (h) represents an objective function, E_t(h) Representing the transfer distortion of a non-uniform cosine modulated filterbank, E_s(h) Represents the stop band energy of the prototype filter, h represents the prototype filter, α represents the weights;

step 5, solving a formula I by adopting a linear iteration method; namely:

step 5.1, an initial prototype filter h is first given₀；

Step 5.2, after the objective function is converted into a convex quadratic function related to the prototype filter, solving to obtain another prototype filter h_min；

Step 5.3, judge | | | h_min-h₀||₂< is true, where is a given small positive number, | · | | purple₂Represents a 2-norm; if yes, terminating the iteration and outputting h_min(ii) a Otherwise, let h₀＝(h_min+h₀) 2, returning to the step 5.2 to continue the iterative process;

step 6, filtering the optimal prototype obtained in the step 5H tool_minThe whole non-uniform cosine modulated filter bank is obtained by substituting the step 1.

2. The method of claim 1, wherein in step 4, the weight α ∈ (0, 1).

3. The method as claimed in claim 1, wherein the step 5.3 is performed within a range of 10^-5～10^-8。