CN105787204B - The design method of the complete over-sampling DFT modulated filter group of the double prototypes of bidimensional - Google Patents

Abstract

The design method that the present invention discloses a kind of complete over-sampling DFT modulated filter group of double prototypes of bidimensional is designed using the condition of approximate Perfect Reconstruction using the discrete Fourier transform modulated filter group of complete over-sampling.The design problem of two ptototype filters is attributed to a unconstrained optimization problem by the present invention, it is to transmit the weighted sum of distortion and aliased distortion and ptototype filter stopband energy that wherein objective function, which is the overall distortion of filter group, using the gradient vector of objective function, the optimization problem is solved by double iterator mechanisms.In single step iteration, using the equivalent condition of matrix inversion and the fast algorithm of block Teoplitz (Toeplitz) matrix inversion, computation complexity is reduced significantly.The present invention can obtain the better filter group of overall performance, computation complexity is greatly lowered, and can quickly design large-scale bidimensional filter group.

Description

A Design Method for Two-Dimensional Dual-Prototype Fully Oversampled DFT Modulated Filter Banks

technical field

The invention relates to the technical field of filter bank design, in particular to a design method of a two-dimensional dual prototype complete oversampling DFT modulation filter bank.

Background technique

Multi-rate filter banks have been widely used in image processing, audio and video signal processing, digital communications, computer vision and texture recognition and classification. The theory and design method of M-band uniform filter bank in one-dimensional case has reached a fairly mature stage. Compared with the two-dimensional separable filter bank, the two-dimensional inseparable filter bank has better direction selectivity, flexible frequency domain division and more degrees of freedom in the two-dimensional case. Among them, the two-dimensional DFT modulation filter bank has the characteristics of simple design and low implementation cost, showing more and more advantages. Compared with one-dimensional filter banks, two-dimensional filter banks have several difficulties, especially when designing large-scale filter banks, which is more challenging. In the bi-iterative second-order cone regularization (BI-SOCP) algorithm, a method for designing a two-dimensional dual-prototype DFT modulation filter (DMFB) is proposed, and the design problem boils down to a constrained optimization problem. Since the calculation amount of the BI-SOCP algorithm includes the calculation of the coefficient matrix of linear constraints and the solution of SOCP, the former is determined by the number of discrete points in the corresponding closed region, and the latter is determined by the number of optimization variables and constraints, so BI- SOCP is difficult to design two-dimensional large-scale DMFBs. In order to overcome this defect, modified Newton's method and conjugate gradient method for designing two-dimensional DMFBs and a fast design method for block Toeplitz matrix inversion are proposed, but these three methods are all used to design two-dimensional single Prototype filter bank.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is that the existing two-dimensional DFT modulation filter bank design method has the problems of high computational complexity, large distortion of the designed large-scale filter bank and cannot be quickly designed. Prototype fully oversampled DFT modulation filter bank design method.

In order to solve the above-mentioned problems, the present invention is achieved through the following technical solutions:

The design method of two-dimensional dual prototype fully oversampling DFT modulation filter bank includes the following steps:

Step 1, under the condition of complete oversampling, determine the modulation matrix D ₁ , the sampling matrix D ₂ , the spatial support length La of the analytical prototype filter and the spatial support length L _s of the synthetic prototype filter _;

Step 2, design an initial analysis prototype filter h ₀ ;

Step 3: Convert the frequency domain condition of the filter bank without transfer distortion and aliasing distortion into the time domain condition without transfer distortion and aliasing distortion;

Step 4, based on the time domain conditions of step 3, according to the performance index of the filter bank design, the weighted sum of the transfer distortion of the filter bank, the stop-band energy of the analytical prototype filter and the stop-band energy of the comprehensive prototype filter is used as the target. function, and the design problem of analyzing prototype filters and synthetic prototype filters is reduced to an unconstrained optimization problem;

Step 5: Solve the optimization problem of Step 4 through a double iterative algorithm, that is, based on the initial analysis prototype filter h ₀ , obtain a comprehensive prototype filter g; then based on the obtained comprehensive prototype filter g, obtain a new analysis prototype filter h;

Step 6: Check the iteration termination condition, that is, determine whether ||hh ₀ || ₂ ≤η is established or the number of iterations exceeds C; if so, the iteration process stops, and the analysis prototype filter h and comprehensive prototype filter obtained in this iteration are The filter g is the optimal solution; otherwise, update the initial analytical prototype filter h ₀ to 0.5(h ₀ +h), and return to step 5 to continue the iterative process; where η and C are both given positive integers ;

Step 7, the analytical prototype filter h and the comprehensive prototype filter g obtained in step 6, obtain the analytical prototype filter coefficient h _i (n) and the comprehensive prototype filter of each channel filter by the two-dimensional DFT modulation formula coefficients _gi (n), thereby determining the entire two-dimensional DFT modulation filter bank; where i=0, 1, . . . , |D ₁ |-1, D ₁ is the modulation matrix.

In the above-mentioned step 5, firstly, based on the initial analysis prototype filter h ₀ , use the formula ① to obtain the comprehensive prototype filter g; then based on the obtained comprehensive prototype filter g, use the formula ② to obtain a new analysis prototype filter h;

g=Q ₂ B ₁ (I _K +B ₁ ^T Q ₂ B ₁ ) ^-1 b ①

h=Q ₁ B ₂ (I _K +B ₂ ^T Q ₁ B ₂ ) ^-1 b ②

In the formula, g is the comprehensive prototype filter, h is the analysis prototype filter, Q ₁ =(αR _s (h)) ^-1 , Q ₂ =(αR _s (g)) ^-1 , where α is the balance distortion and resistance Weights with energy, R _s (h) and R _s (g) are two block Toeplitz matrices constructed based on the analytical prototype filter and the synthetic prototype filter, respectively, B ₁ =(AG(h ₀ )) ^T , B ₂ =(AG(g)) ^T , where A is a matrix of K×(2L _a +2L _s +1) ² , K is the order of the inverse matrix after reducing the dimension, L _a , L _s are the spatial support lengths of the analysis and synthesis prototype filters, respectively, G(h) is a block matrix composed of the coefficients of the analysis prototype filter h, G(g) is a block matrix composed of the coefficients of the synthesis prototype filter g, I _K is a K×K identity matrix, and b is a K×1 vector that is zero except for the (K-1)/2th element that has a value.

In the above step 4, the unconstrained optimization problem of the analysis prototype filter is:

The unconstrained optimization problem of the synthetic prototype filter is:

In the two formulas, is the part that controls the transfer distortion when the analytical prototype filter is known, is the part that controls the transfer distortion when the synthetic prototype filter is known, A is a matrix of K×(2L _a +2L _s +1) ² , K is the order of the inverse matrix after reducing the dimension, _La , L _s are the spatial support lengths of the analysis and synthesis prototype filters, respectively, b is a K×1 vector with zero values except for the (K-1)/2th element, and g is the synthesis prototype filter, G(h) is the block matrix composed of the coefficients of the analysis prototype filter h, h is the analysis prototype filter, G(g) is the block matrix composed of the coefficients of the synthetic prototype filter g, E(h) is the analysis prototype The stopband energy of the filter, E(g) is the stopband energy of the integrated prototype filter, and α is the weight of the balance distortion and stopband energy.

In the above step 7, the two-dimensional DFT modulation formula is:

In the formula, h _i (n) represents the coefficient of the ith channel analysis prototype filter, g _i (n) is the coefficient of the comprehensive prototype filter of the ith channel, n is the coefficient variable, D ₁ is the modulation matrix, and u _i is the modulation Set definition of matrices.

The present invention utilizes the condition of approximate complete reconstruction, and adopts a fully oversampled discrete Fourier transform (DFT) modulation filter bank to design. In the present invention, the design problem of two prototype filters is reduced to an unconstrained optimization problem, wherein the objective function is the weighted sum of the overall distortion (transfer distortion and aliasing distortion) of the filter bank and the stopband energy of the prototype filter. The gradient vector of the function, the optimization problem is solved by a double iterative mechanism. In single-step iteration, the equivalent conditions for matrix inversion and the fast algorithm for block Toeplitz matrix inversion are used, which significantly reduces the computational complexity. The invention can obtain a filter bank with better overall performance, greatly reduce the computational complexity, and can quickly design a large-scale two-dimensional filter bank.

Compared with the prior art, the present invention has higher flexibility in designing a filter bank, has lower computational cost, and can quickly and effectively design a two-dimensional large-scale filter bank.

Description of drawings

Figure 1 shows the basic structure of a two-dimensional DFT modulation filter bank.

FIG. 2 is a flow chart of designing a two-dimensional DFT modulation filter bank according to the present invention.

Figure 3 is the resulting filter responses in Example 1: (a) is the impulse response of the analytical prototype filter, (b) is the magnitude response of the analytical prototype filter, (c) is the impulse response of the synthesized prototype filter, ( d) is the magnitude response of the synthesized prototype filter.

Figure 4 is the normalized magnitude response of the prototype filter when the spatial supports are equal in Example 2: (a) is the magnitude response of the analytical prototype filter, (b) is the magnitude response of the synthetic prototype filter.

Figure 5 is the normalized magnitude response of the spatially supported unequal time prototype filter in Example 2: (a) is the magnitude response of the analytical prototype filter, (b) is the magnitude response of the synthetic prototype filter.

Detailed ways

Figure 1 shows the basic structure of a two-dimensional DFT modulation filter bank, and a design method of a two-dimensional dual prototype complete oversampling DFT modulation filter bank based on the above structure, as shown in Figure 2, specifically includes the following steps:

Step 1: Under the condition of complete oversampling, determine the modulation matrix D ₁ and sampling matrix D ₂ and the spatial domain supports _La and L _s of the prototype filter.

The second step: design an initial analysis prototype filter h ₀ , in order to achieve the final result more effectively, design a h (h obtained by simulation under the same conditions) as the initial analysis prototype filter h ₀ in the algorithm of the present invention.

Step 3: When the transfer function is a pure delay and the aliasing function is zero, the filter bank has no transfer distortion and aliasing distortion, and the frequency domain condition that the filter bank has no transfer distortion and aliasing distortion can be converted into Time-domain conditions with no transfer distortion and no aliasing distortion. Let h and g be the analysis and synthesis prototype filters, respectively, the expression can be expressed as:

where h( ) represents the analysis prototype filter coefficients, _g ( ) represents the synthesis prototype filter coefficients, La and L _s represent the spatial support lengths of the analysis and synthesis prototype filters, respectively, and T represents the transpose, analysis and synthesis The two-dimensional DFT modulation formula of the prototype filter is:

Among them, n is a coefficient variable, h _i (n) represents the coefficient of the ith channel analysis prototype filter, and g _i (n) is the coefficient of the ith channel's comprehensive prototype filter. The modulation matrix D ₁ is a 2×2 non-singular integer matrix, and its set N(D ₁ ^T ) is defined as follows:

N(D ₁ ^T )={u _i ≡ D ₁ ^T x _i ∈ Z ² :x _i ∈[0,1) ² ,i=0,1,...,|D ₁ |-1} (3)

Accordingly, the frequency response of the analyzed and synthesized prototype filter is:

Where, c(w,L)={e ^-j[-L,-L]w ,...,e ^-j[-L,L]w ...,e ^-j[L,L]w } ^T , w=[ω _x , w _y ] ^T represents a two-dimensional frequency domain vector, H(w) represents the frequency response of the analytical prototype filter, G(w) represents the frequency response of the synthetic prototype filter, H _i (w) represents the frequency response of the ith channel analysis prototype filter, and G _i (w) represents the frequency response of the ith channel synthesis prototype filter. The expression for the child signal is:

Among them, Y _i (w) represents the child signal of the i-th channel, the sampling matrix D ₂ is another 2×2 non-singular matrix, and its set N(D ₂ ^T ) is defined as:

N(D ₂ ^T )={u _m ≡ D ₂ ^T x _m ∈ Z ² :x _m ∈[0,1) ² ,m=0,1,...,|D ₂ |-1} (6)

The input-output relationship of the filter bank is:

Among them, the transfer function and aliasing transfer function are:

Similar to the design of the one-dimensional DFT filter bank, the performance indicators of the two-dimensional dual prototype fully oversampling DFT modulation filter bank design mainly include the transfer distortion and aliasing distortion of the filter bank, which determine the weight of the filter bank. structural error. Also included is the stopband energy of the prototype filter bank, which is designed to expect high stopband attenuation. The transfer distortion can be expressed as:

in, A is a K×(2L _a +2L _s +1) ² matrix, expressed as:

G(h) is the block matrix composed of the coefficients of the analytical prototype filter h, and G(g) is the block matrix composed of the coefficients of the synthetic prototype filter g, respectively expressed as:

where G _ij is a (2L _a +2L _s +1)×(2L _s +1) matrix, J _i,j is a (2L _a +2L _s +1)×(2L _a +1) matrix, and:

Λ _k is a diagonal matrix based _on the sampling matrix D2, expressed as:

According to equations (8a), (9a) and (9b), it can be deduced that the only condition for the distortion-free two-dimensional dual prototype DMFB is:

AG(h)g=b or AG(g)h=b (14)

Among them, b is a K × 1 vector with zero values except for the (K-1)/2th element, which is expressed as b=[0,...,0,|D ₂ |/| D ₁ |,0,...,0] ^T .

Additionally, the stopband energy of the analytical and synthesized prototype filter is expressed as:

in, stands for conjugate transpose, R _s (h) and R _s (g) are two block Toeplitz matrices constructed based on the analytical prototype filter and synthetic prototype filter coefficients, respectively, and their inverses can be obtained by Wax- The Kailath algorithm is fast to solve and only needs to be calculated once.

Step 4: Based on the previous analysis, the objective function of the prototype filter is the weighted sum of total distortion and stopband energy, and the design problem boils down to an unconstrained optimization problem, expressed as:

or

Available when the analytical prototype filter h is known to control the transfer distortion, which can be used when the synthesis prototype filter g is known to control the transmission distortion. E(h) and E(g) are the stop-band energy of the analysis and synthesis prototype filters, respectively, and α is the weight to balance the distortion and stop-band energy. When the filter bank has high stopband energy, aliasing distortion is almost negligible.

The optimization problems of equations (16a) and (16b) can be solved using double iterations, and when h is fixed, the objective function is an unconstrained convex quadratic function with respect to the synthetic prototype filter g:

Let the objective function gradient be zero vector, expressed as:

The optimal solution of g can be obtained as:

g=((AG(h)) ^T (AG(h))+αR _s (g)) ^-1 (AG(h)) ^T b (19)

Similarly, when g is fixed, it is required to solve an unconstrained convex quadratic function about the analytical prototype filter h:

Let the objective function gradient be zero vector, expressed as:

The optimal solution for h can be obtained as:

h=((AG(g)) ^T (AG(g))+αR _s (h)) ^-1 (AG(g)) ^T b (22)

Starting with a suitable initial prototype filter h ₀ , the prototype filter can be optimized using equations (19) and (22) in two iterations, and each iteration can be sequentially updated with the following equations:

When the number of channels of the designed filter bank is large and the spatial support of the filter is large, equation (23) involves the inversion of a large matrix, which requires a huge amount of computation. Therefore, in order to reduce the amount of operations for matrix inversion, the following equivalent conditions for matrix inversion can be used:

(BB ^T +A) ^-1 B=A ^-1 B(I+B ^T A ^-1 B) ^-1 (24)

Among them, I represents the identity matrix. From the above formula, formula (23) can be simplified as:

Wherein, Q ₁ =(αR _s (h)) ^-1 , Q ₂ =(αR _s (g)) ^-1 , B ₁ =(AG(h ₀ )) ^T , B ₂ =(AG(g)) ^T .

The fifth step: using the initial analysis prototype filter h ₀ obtained in the second step, using the double iterative method of formula (25) to first obtain the comprehensive prototype filter g, then use the obtained g, and finally update to obtain a new analysis prototype filter h.

Step 6: Check the iteration termination condition, and judge whether ||hh ₀ || ₂ ≤ η (η is a given small positive number) or whether the number of iterations exceeds a given number C (C=20 in the test) , if satisfied, the iterative process will stop, h and g are the optimal solutions, otherwise, update the initial analytical prototype filter to 0.5(h ₀ +h) and return to the fifth step to continue the iterative process.

The seventh step: according to the analytical prototype filter h and the comprehensive prototype filter g obtained in the sixth step, obtain the coefficients h _i (n) and g _i ( n), thereby determining the entire two-dimensional DFT modulation filter bank.

Example 1:

Consider designing a two-dimensional fully oversampled DFT modulation filter bank, the modulation matrix, sampling matrix and spatial support are respectively La = 8, L _{s =} 8, η = 1×10 ^-8 . In this example, the relevant parameters are: α=1×10 ⁻² , and the amplitude response of the obtained prototype filter is shown in FIG. 3 . At the same time, the second-order cone programming algorithm (BI-SOCP) is used to design, and the relevant parameters are ε _sbe = 0.01, ε _tbe = 10, ε _pdf = 2, where ε _sbe , ε _tbe , ε _pdf are the stopband energy, Tolerances for transition band energy and pass band flatness. Running under the same environment, Table 1 shows the performance and time comparison of the filter banks obtained by the two methods. It can be seen from Table 1 that the stopband attenuation of the analytical prototype filter designed by the present invention is reduced by 12.27dB , the stop-band attenuation of the integrated prototype filter is reduced by 7.78dB, so the filter bank obtained by the present invention has better frequency characteristics. And the computational complexity is lower, and the time spent is greatly reduced.

Table 1 Comparison of performance and time between the algorithm of the present invention and the BI-SOCP algorithm

Example 2:

Design a two-dimensional large-scale DFT modulation filter bank to satisfy the following parameter settings: L _a =50, L _s =50, α=1×10 ⁻³ , η=1×10 ⁻⁵ . The filter bank has 800 subbands from the modulation matrix D1, and the length _of the space support shows that the coefficient of the prototype filter rises to 10201. Comparing the existing algorithm with the present invention, the normalized amplitude response of the prototype filter obtained in the present invention is shown in FIG. 4 . Table 2 shows the performance comparison of the two algorithms, and the CPU consumption time of the present invention is 45.40 seconds in 8 iterations. At the same time, because the present invention adopts the dual prototype filter bank to design, the analysis and synthesis prototype filters can be adjusted to be different spatial supports. If the analysis prototype filter is kept unchanged, the spatial support of the comprehensive prototype filter is increased to L _s =55, and we get: The magnitude response of the prototype filter is shown in Figure 5. Table 3 shows the performance indicators of the filter when the airspace support is changed. The CPU time consumed in 8 iterations is 69.34 seconds. It can be seen from Table 2 that the stopband attenuation and transfer distortion of the integrated prototype filter are reduced. , and the frequency selection of the filter designed by the present invention is more flexible, and is more suitable for the rapid design of a large-scale two-dimensional dual-prototype filter bank.

Table 2 The performance comparison between the algorithm of the present invention and the existing algorithm

Table 3 Performance comparison when changing the spatial support of the synthetic prototype filter

Claims (2)

1. the design method of two-dimensional double prototype complete oversampling DFT modulation filter bank, is characterized in that, comprises the steps: Step 1, under the condition of complete oversampling, determine the modulation matrix D ₁ , the sampling matrix D ₂ , the spatial support length La of the analytical prototype filter and the spatial support length L _s of the synthetic prototype filter _; Step 2, design an initial analysis prototype filter h ₀ ; Step 3: Convert the frequency domain condition of the filter bank without transfer distortion and aliasing distortion into the time domain condition without transfer distortion and aliasing distortion; Step 4, based on the time domain conditions of step 3, according to the performance index of the filter bank design, the weighted sum of the transfer distortion of the filter bank, the stop-band energy of the analytical prototype filter and the stop-band energy of the comprehensive prototype filter is used as the target. function, and reduces the design problem of analytical prototype filters and synthetic prototype filters to an unconstrained optimization problem: The unconstrained optimization problem of the analytical prototype filter is: The unconstrained optimization problem of the synthetic prototype filter is: In the two formulas, A is a K×(2L _a +2L _s +1) ² matrix, K is the order of the inverse matrix after reducing the dimension, and L _a and L _s are the spatial domains of the analytical and synthetic prototype filters, respectively. Support length, b is a K × 1 vector with zero values except for the (K-1)/2th element, g is the comprehensive prototype filter, and G(h) is the value of the analysis prototype filter h. The block matrix composed of coefficients, h is the analysis prototype filter, G(g) is the block matrix composed of the coefficients of the comprehensive prototype filter g, E(h) is the stopband energy of the analysis prototype filter, E(g) is Synthesize the stopband energy of the prototype filter, α is the weight of balanced distortion and stopband energy; In step 5, the optimization problem of step 4 is solved by a double-iterative algorithm, and in a single-step iteration, the equivalent conditions for matrix inversion and the fast algorithm for block Toeplitz matrix inversion are used, that is, based on the initial analysis prototype filter h ₀ , use the formula ① to obtain the comprehensive prototype filter g; then based on the obtained comprehensive prototype filter g, use the formula ② to obtain the new analytical prototype filter h; g=Q ₂ B ₁ (I _K +B ₁ ^T Q ₂ B ₁ ) ^-1 b ③ h=Q ₁ B ₂ (I _K +B ₂ ^T Q ₁ B ₂ ) ^-1 b ④ In the formula, g is the comprehensive prototype filter, h is the analysis prototype filter, Q ₁ =(αR _s (h)) ^-1 , Q ₂ =(αR _s (g)) ^-1 , where α is the balance distortion and resistance Weights with energy, R _s (h) and R _s (g) are two block Toeplitz matrices constructed based on the analytical prototype filter and the synthetic prototype filter, respectively, B ₁ =(AG(h ₀ )) ^T , B ₂ =(AG(g)) ^T , where A is a matrix of K×(2L _a +2L _s +1) ² , K is the order of the inverse matrix after reducing the dimension, _La and L _s are respectively The spatial support length of the analysis and synthesis prototype filter, G(h) is the block matrix composed of the coefficients of the analysis prototype filter h, G(g) is the block matrix composed of the coefficients of the synthesis prototype filter g, and _IK is K×K identity matrix, b is a K×1 vector with zero values except for the (K-1)/2th element; Step 6: Check the iteration termination condition, that is, determine whether ||hh ₀ || ₂ ≤η is established or the number of iterations exceeds C; if so, the iteration process stops, and the analysis prototype filter h and comprehensive prototype filter obtained in this iteration are The filter g is the optimal solution; otherwise, update the initial analytical prototype filter h ₀ to 0.5(h ₀ +h), and return to step 5 to continue the iterative process; where η and C are both given positive integers ; Step 7, the analytical prototype filter h and the comprehensive prototype filter g obtained in step 6, obtain the analytical prototype filter coefficient h _i (n) and the comprehensive prototype filter of each channel filter by the two-dimensional DFT modulation formula coefficients _gi (n), thereby determining the entire two-dimensional DFT modulation filter bank; where i=0, 1, . . . , |D ₁ |-1, D ₁ is the modulation matrix. 2. the design method of two-dimensional dual prototype complete oversampling DFT modulation filter bank according to claim 1, is characterized in that: in step 7, The two-dimensional DFT modulation formula is: In the formula, h _i (n) represents the coefficient of the ith channel analysis prototype filter, g _i (n) is the coefficient of the comprehensive prototype filter of the ith channel, n is the coefficient variable, D ₁ is the modulation matrix, and u _i is the modulation Set definition of matrices.