An Improved Iterative Design Method for Multi-Channel Cosine Modulated Filter Banks

Abstract

This application discloses an improved iterative design method for multi-channel cosine-modulated filter banks. The aim is to improve the performance of the prototype filter in terms of aliasing error control and system response by introducing constraints on the amplitude-frequency characteristics of the passband and transition band of the prototype filter into the optimization objective. Compared with traditional iterative design methods, the method proposed in this application shows superior performance in these performance indicators. Furthermore, based on the closed-form expression of the objective function given in this application, an iterative solution method can be used to obtain the optimization results, further effectively reducing the computational complexity and implementation difficulty of the proposed method.

Description

Technical Field

[0001] This application relates to the field of communication technology, and in particular to an iterative design improvement method for multi-channel cosine modulation filter banks. Background Technology

[0002] Decimation and interpolation are two fundamental operations in digital signal processing used to change the sampling rate of a signal. They are commonly used for signal sampling rate conversion to adapt to different processing needs, such as signal compression, transmission, or storage. Decimation reduces the number of sampling points in a signal without changing its content, typically used to lower the sampling rate; interpolation, on the other hand, is the opposite of decimation, used to increase the number of sampling points, thereby increasing the sampling rate.

[0003] Polyphase filtering is a widely used technique in digital signal processing, primarily used for signal sampling rate conversion. The key to polyphase filtering is decomposing a traditional filter into several "phase" components, each processing a different part of the input signal. Specifically, for a periodic signal, different sampling times can be divided into multiple "phases," each corresponding to a different time delay. Sampling rate conversion typically involves interpolation or decimation operations. By using polyphase filtering, redundant calculations during interpolation and decimation can be reduced, improving signal processing efficiency. Polyphase filtering is particularly suitable for efficient upsampling (increasing the sampling rate) and downsampling (decreasing the sampling rate) operations.

[0004] Approximate Perfect Reconstruction Filter Banks (APRFBs) are filter banks used in signal processing and communication systems for signal decomposition and reconstruction. Their goal is to decompose the input signal into multiple sub-band signals and then reconstruct it to approximately perfectly recover the original signal. Simply put, APRFBs minimize signal distortion and information loss during signal decomposition and reconstruction, achieving near-perfect reconstruction. The signal decomposition process is also known as the analytical filtering process, and the reconstruction process is known as the synthetic filtering process.

[0005] The analysis-synthesis filtering method is a technique that breaks down the channelization process into an analysis process and a synthesis process. In the analysis process, the frequency domain of the input signal is uniformly divided, and sub-channel outputs are extracted. In the synthesis process, the extracted sub-channel signals are inversely processed to reconstruct a single, time-continuous, and frequency-complete broadband signal.

[0006] An orthogonal mirror filter bank is a special type of near-perfect reconstruction filter bank structure characterized by its orthogonality and mirror symmetry in achieving signal decomposition and reconstruction. Orthogonality refers to the orthogonal relationship between the filters in the filter bank, ensuring that each sub-filter can independently process different parts of the signal without mutual interference. This is crucial for signal decomposition and reconstruction. Mirror symmetry means that the frequency responses of the filters have a specific symmetry relationship. An orthogonal mirror filter bank typically consists of a low-pass filter and a high-pass filter, and the frequency responses of these two filters are mirror-symmetric in the frequency domain. To ensure perfect reconstruction of the signal after decomposition by the filter bank, the filter bank design needs to meet the requirements of perfect or near-perfect reconstruction, which is usually achieved through careful design of the frequency and phase responses of the prototype filters. Summary of the Invention

[0007] This application discloses an iterative design improvement method for multi-channel cosine modulation filter banks to improve the performance of prototype filters in multi-channel cosine modulation filter banks.

[0008] The technical solution adopted in this application is as follows:

[0009] An improved iterative design method for multi-channel cosine-modulated filter banks, comprising the following steps:

[0010] Step 1: Design an N-1 order low-pass filter with linear phase and a passband cutoff frequency of . Where N is an even number greater than 2;

[0011] The tap coefficients of the N-1 order low-pass filter are used as the initial values ​​of the initial tap coefficient vector q of the prototype filter, where,

[0012] Step 2, based on the passband cutoff frequency ω of the prototype filter p and stopband start frequency ω s Calculate the three dimensions as The intermediate variables U1 to U3, and the two dimensions are The intermediate variables A(ω) and B(ω);

[0013] Among them, the elements of intermediate variables U1 to U3 They are respectively:

[0014]

[0015] Among them, element index

[0016] The elements a of intermediate variables A(ω) and B(ω) i b i They are respectively:

[0017]

[0018] Wherein, the angular frequency difference Δω=ω s -ω p ;

[0019] Based on the intermediate variables U1~U3, A(ω), and B(ω), construct three sub-objective functions E1, E2, and E3:

[0020] E1 = p T U1p-2p T A(ω)+C1

[0021] E2 = p T U2p-2p T B(ω)+C2

[0022] E3 = p T U3p

[0023] Where C1 and C2; p represents the tap coefficients of the prototype filter, and

[0024] According to the formula Calculate the intermediate variable U4;

[0025] And construct the sub-objective function E4=(U4p-d) T (U4p-d), where d is a column vector with all elements being 1;

[0026] Among them, the intermediate calculation auxiliary quantity H (Ω) q ) = diag[M q (ω q1 ),M q (ω q2 ),…,M q (ω qk )], passband cutoff frequency input quantity Ω q ={ω q1 =0,ω q2 ,…,ω qk =π / K}, that is, Ω q It is an increasing sequence;

[0027] Intermediate calculation auxiliary quantity

[0028] Among them, M q (·) represents the system gain, and its input is defined as the angular frequency ω, then M q (ω)=2q Tc(ω), where the auxiliary quantity

[0029] The objective function E is constructed based on the weighted sum of the four sub-objective functions E1 to E4;

[0030] Step 3: Solve for the extrema of the objective function E to obtain the current optimal solution p for the tap coefficients p. opt ;

[0031] Step 4, determine the current optimal solution p opt Does it meet the preset convergence condition? If so, then the currently solved p... opt The final optimized tap coefficients are used to obtain the optimized prototype filter, and step 5 is then executed.

[0032] Otherwise, based on the current optimal solution p opt After iteratively updating the initial tap coefficient vector q, the intermediate variable U4 is recalculated to obtain the new sub-objective function E4; then the objective function E is solved again.

[0033] Step 5: Based on the optimized prototype filter obtained in Step 4, construct a multi-channel cosine filter bank for multi-channel cosine modulation filtering of the target object.

[0034] Furthermore, the optimal solution p opt Specifically:

[0035]

[0036] Among them, W1, W2, W3, and W4 are the preset weighting coefficients of the four sub-objective functions E1 to E4, respectively.

[0037] Furthermore, the pre-defined convergence condition is: ||p opt -q||2<δ, where δ is a preset threshold, preferably on the order of 10. -4 ~10 -6 .

[0038] Furthermore, regarding the current optimal solution p opt The iterative update of the initial tap coefficient vector q is specifically as follows:

[0039] q=(1-τ)q+τp opt , where τ takes values ​​in the range (0,1).

[0040] The technical solution provided in this application brings at least the following beneficial effects:

[0041] This application improves the performance of the prototype filter by introducing constraints on the amplitude-frequency characteristics of the passband and transition band of the prototype filter into the optimization objective and using a weighted method to make the overall optimization objective more balanced.

[0042] Compared with other direct solution methods, the proposed scheme in this application reduces the processing complexity. In addition, the prototype filter designed using the method in this application has better performance in controlling aliasing error, system response, maximum aliasing error and maximum amplitude distortion. Attached Figure Description

[0043] The above and / or additional aspects and advantages of this application will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0044] Figure 1 This is a schematic diagram of the cosine modulation filter structure in the embodiments of this application;

[0045] Figure 2 This is a comparison diagram of the amplitude-frequency response of the prototype filter in the embodiments of this application;

[0046] Figure 3 This is a schematic diagram of the amplitude-frequency response of the sub-band filter bank in the embodiments of this application;

[0047] Figure 4 This is a comparison diagram of the sub-band filter bank system response in the embodiments of this application;

[0048] Figure 5 This is a comparison diagram of aliasing errors of subband filter banks in the embodiments of this application. Detailed Implementation

[0049] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions of the embodiments of this application will be described in detail and completely below with reference to the accompanying drawings. Obviously, the embodiments described with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application.

[0050] Currently, commonly used filter bank modulation methods are mainly divided into two categories: complex exponential modulation filter banks and cosine modulation filter banks. Existing research has proven that the two modulation methods are equivalent.

[0051] The expressions for the cosine modulation filter bank analysis and synthesis filter are shown in the following equations, and the structure of the cosine filter bank is as follows. Figure 1 As shown.

[0052]

[0053] Among them, h k(n) represents the analysis filter, g k p(n) represents the composite filter, k represents the number of filter channels, n is the filter order number, and p(n) is the prototype filter, with parameters for the k-th channel.

[0054] In addition, to eliminate amplitude distortion, the approximate reconstruction filter bank imposes the following requirements on the design of the ideal prototype filter:

[0055] |P(e jω ) 2 |+|P(e j(ω-π / K) ) 2 |=1, 0<ω<π / K (3)

[0056] |P(e jω )|=0, ω>π / K (4)

[0057] Wherein, P(e jω ) is the prototype filter, and ω is the angular frequency.

[0058] In the above equation, the first term ensures the elimination of amplitude distortion in the filter bank, while the second term ensures that there is no aliasing distortion between non-adjacent subbands. The proposed Parks-McClellan method's objective function is based on the two ideal reconstruction conditions proposed in the above equation, namely…

[0059] E=E'1+αE'2 (5)

[0060] in,

[0061]

[0062] Where, ω s This is the stopband start frequency.

[0063] This method lacks optimization for the filter transition band, ultimately leading to non-ideal errors in the amplitude-frequency characteristics of the filter bank. Therefore, to ensure that the transition band also satisfies equation (3) as much as possible, we have

[0064] |P(e jω ) 2 |+|P(e j(ω-π / K) ) 2 |=1, ω p <ω<ω s (8)

[0065] Where, ω p This indicates the passband cutoff frequency.

[0066] Therefore, the prototype filter can be modeled as:

[0067]

[0068] Where D is the reconstruction delay, and the angular frequency difference Δω = ω s -ω p .

[0069] Thus, in the transition zone ω∈[ω p ,ω s ],have

[0070]

[0071] Therefore, after comprehensive consideration, this application proposes a new objective function as follows:

[0072] E=W1E1+W2E2+W3E3+W4E4 (11)

[0073] in,

[0074]

[0075] Among them, W1, W2, W3, and W4 are weighting coefficients in the objective function, and the design of the prototype filter is simplified to the optimization problem of equation (11).

[0076] Since the objective function contains squared terms and integrals, direct solution becomes extremely complex. Therefore, this application employs an iterative solution method to address this problem. Furthermore, only the case where N is even is discussed; the derivation for odd N is similar and will not be repeated here.

[0077] Due to the prototype filter P(e jω () is a linear-phase low-pass filter with a symmetrical impulse response. When N is even, its frequency response can be expressed as:

[0078] P(e jω ) = M p (ω)e -jω(N-1) / 2 (16)

[0079] M p (ω)=2p T c(ω) (17)

[0080] In formula (17)

[0081]

[0082] Among them, M p (ω) represents the system gain, p is the tap coefficient vector, and c(ω) is the computational auxiliary quantity.

[0083] By combining equation (16) and rewriting equation (11), we can obtain...

[0084]

[0085] for It can be further rewritten as:

[0086]

[0087] Similarly, rewrite for

[0088]

[0089] C1 and C2 are constant terms.

[0090] set up Auxiliary quantity U v For elements v = 1, 2, 3, there are

[0091]

[0092] Additionally, let a i With b i Let i be an element in A(ω) and B(ω), and its expression is shown in the following formula, where the range of values ​​for variable i is...

[0093]

[0094] We can first initialize a linear-phase low-pass filter Q, whose passband cutoff frequency is... There is M q (ω)=2q T c(ω), filter M q Substituting (ω) as a known parameter into equation (15) and rewriting E4, we obtain...

[0095]

[0096] Ω q ={ω q1 =0,ω q2 ,…,ω qk =π / K} (30)

[0097]

[0098] H(Ω q ) = diag[M q (ω q1 ),M q (ω q2 ),…,M q (ω qk (32)

[0099]

[0100] d is a column vector with all elements being 1.

[0101] In summary, equation (20) can be simplified to obtain

[0102]

[0103] because Since it is a positive definite matrix, it has a global minimum for the objective function E, which is the optimal solution. The specific expression is as follows:

[0104]

[0105] However, this extreme point p opt This is not the optimal solution to problem (34) either, and further iterations are needed until p... opt It is close enough to q, that is

[0106] ||p opt -q||2<δ (36)

[0107] Where δ is a preset threshold, thus obtaining the prototype filter tap coefficients.

[0108] In one embodiment, based on the objective function (Formula (34)) constructed in this application, the process of iteratively solving the prototype filter tap coefficients can be described as follows:

[0109] Step S1: Design an N-1 order low-pass filter with linear phase and a passband cutoff frequency of . The tap coefficients obtained based on this low-pass filter are used as the initial values ​​of the initial tap coefficient vector q of the prototype filter.

[0110] Step S2, when ||p opt If -q||2>=δ, proceed to step S3; otherwise, based on the currently obtained optimal solution p opt (i.e., the extreme values ​​of the tap coefficients of the prototype filter) yield the tap coefficients of the final optimized prototype filter.

[0111] Step S3: Based on the current value of the initial tap coefficient vector q, calculate the auxiliary variable U according to formulas (24) to (28) and formula (31). l l = 1, 2, 3, 4, and A(ω) and B(ω);

[0112] Then, calculate the current optimal solution p of the objective function E according to formula (35). opt ;

[0113] If ||p optIf -q||2<δ, then the initial tap coefficient vector q remains unchanged; otherwise, update the initial tap coefficient vector q according to the formula: q=(1-τ)q+τp opt ;

[0114] Then, based on the current initial tap coefficient vector q, continue to execute step S2.

[0115] Example

[0116] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0117] The above descriptions are merely some embodiments of the present invention. Those skilled in the art can make various modifications and improvements without departing from the inventive concept of the present invention, and these all fall within the scope of protection of the present invention.

[0118] As one possible implementation, the specific steps of the iterative design improvement method for a multi-channel cosine modulation filter bank provided in this embodiment of the invention include:

[0119] Step 1: Design an N-1 order low-pass filter with linear phase and a passband cutoff frequency of . The obtained tap coefficients are then used as the coefficient vector to initialize the variable q, which is the initial tap coefficient of the prototype filter.

[0120] Step 2: Solve for the values ​​U of the intermediate variables respectively. l ,l=1,2,3,4, A(ω), B(ω);

[0121] Step 3: Substitute the intermediate variables into the extreme values ​​of the objective function, i.e., the closed-form solution of the "optimal prototype filter coefficients", to solve for p. opt ;

[0122] Step 4, determine the p to be solved opt Does it meet the preset convergence conditions (e.g., ||p)? opt If -q||2<δ), then the p obtained from the current solution will be... opt The final optimized tap coefficients are used to obtain the optimized prototype filter;

[0123] Otherwise, based on the currently solved p opt Iteratively update the variable q (e.g., q = (1-τ)q + τp) optAfter that, return to step 2;

[0124] Step 5: Based on the optimized prototype filter obtained in Step 4, construct a multi-channel cosine filter bank for multi-channel cosine modulation filtering of the target object.

[0125] To further verify the performance of the method proposed in the embodiments of this application, simulation verification was performed through test experiments. The specific experimental parameters were set as follows: N = 386, K = 16, ω... s =0.0567π, τ=0.5, δ=10 -4 In addition, in this simulation experiment, the weighting coefficients were set to W1 = 5 × 10. -4 W2 = 4 × 10 -4 W3 = 6000, W4 = 5000.

[0126] The amplitude-frequency response of the prototype filter designed based on the method proposed in the embodiments of this application is similar to that of the filter bank. Figure 2 and Figure 3 The method described in this application provides the system response E corresponding to the sub-band filter bank used. T (e jω ) and aliasing error E a (e jω And a comparison chart with the results of the Parks-McClellan algorithm, as shown below. Figure 4 and Figure 5 As shown.

[0127] Among them, the subband filter bank system response E T (e jω ) and aliasing error E a (e jω The definition is as follows, where H is... i (e jω ), G i (e jω To analyze and synthesize the frequency domain expression of the filter,

[0128]

[0129] Through comparative analysis, such as Figure 2 As shown, the iterative algorithm designed in this application significantly improves the attenuation performance of the prototype filter in the stopband, increasing it from approximately -120 dB in Parks-McClellan algorithm simulations to approximately -145 dB. Furthermore, the algorithm proposed in this application optimizes the system response of the subband filter bank to 1.1 × 10⁻⁶. -5Below dB, the performance is improved by approximately 50% compared to the simulation results in the Parks-McClellan algorithm; at the same time, the overall aliasing error is reduced to less than -165dB, an improvement of nearly 50dB.

[0130] Define parameters Used to characterize maximum amplitude distortion and This is used to characterize the maximum aliasing error. To better illustrate the optimization effect of the method proposed in this application, the data of the method proposed in this application and the Parks-McClellan algorithm were compared, and the specific data are shown in Table 1:

[0131] Table 1 Comparison of performance parameters

[0132]

[0133] Both the Parks-McClellan algorithm and the method proposed in this application belong to the category of iterative coefficient optimization problems. The Parks-McClellan algorithm imposes constraints on the stationarity of the system response and stopband attenuation in the objective function. This application further constrains the amplitude-frequency response characteristics of the prototype filter's passband and transition band based on the Parks-McClellan algorithm, and adds weighting coefficients before each optimization sub-objective, resulting in a certain priority among the sub-objectives. As shown in Table 1, the improved method proposed in this application achieves optimizations in both maximum amplitude distortion and maximum aliasing error, and its performance is superior to that of the compared references.

[0134] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the application, and are not intended to limit them; although the application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the application.

Claims

1. An improved iterative design method for multi-channel cosine-modulated filter banks, characterized in that, Includes the following steps: Step 1: Design an N-1 order low-pass filter with linear phase and a passband cutoff frequency of . Where N is an even number greater than 2; The tap coefficients of the N-1 order low-pass filter are used as the initial values ​​of the initial tap coefficient vector q of the prototype filter, where, Step 2, based on the passband cutoff frequency ω of the prototype filter p and stopband start frequency ω s Calculate the three dimensions as The intermediate variables U1 to U3, and the two dimensions are The intermediate variables A(ω) and B(ω); Among them, the elements of intermediate variables U1 to U3 They are respectively: Among them, element index The elements a of intermediate variables A(ω) and B(ω) i b i They are respectively: Wherein, the angular frequency difference Δω=ω s -ω p ; Based on the intermediate variables U1~U3, A(ω), and B(ω), construct three sub-objective functions E1, E2, and E3: E1=p T U1p-2p T A(ω)+C1 E2=p T U2p-2p T B(ω)+C2 E3=p T U3p Where C1 and C2 are constant terms in the expression; p represents the tap coefficients of the prototype filter, and According to the formula Calculate the intermediate variable U4; And construct the sub-objective function E4=(U4p-d) T (U4p-d), where d is a column vector with all elements being 1; Among them, the intermediate calculation auxiliary quantity H (Ω) q )=diag[M q (ω q1 ),M q (ω q2 ),…,M q (ω qk )], passband cutoff frequency input quantity Ω q ={ω q1 =0,ω q2 ,…,ω qk =π / K}; Intermediate calculation auxiliary quantity Among them, M q (·) represents the system gain, and its input is defined as the angular frequency ω, then M q (ω)=2q T c(ω), where the auxiliary quantity The objective function E is constructed based on the weighted sum of the four sub-objective functions E1 to E4; Step 3: Solve for the extrema of the objective function E to obtain the current optimal solution p for the tap coefficients p. opt ; Step 4, determine the current optimal solution p opt Does it meet the preset convergence condition? If so, then the currently solved p... opt The final optimized tap coefficients are used to obtain the optimized prototype filter, and step 5 is then executed. Otherwise, based on the current optimal solution p opt After iteratively updating the initial tap coefficient vector q, the intermediate variable U4 is recalculated to obtain the new sub-objective function E4; then the objective function E is solved again. Step 5: Based on the optimized prototype filter obtained in Step 4, construct a multi-channel cosine filter bank for multi-channel cosine modulation filtering of the target object.

2. The method as described in claim 1, characterized in that, Optimal solution p opt Specifically: Among them, W1, W2, W3, and W4 are the preset weighting coefficients of the four sub-objective functions E1 to E4, respectively.

3. The method as described in claim 1, characterized in that, The preset convergence condition is: ||p opt -q||2<δ, where δ is a preset threshold.

4. The method as described in claim 3, characterized in that, The threshold δ is on the order of 10. -4 ~10 -6 .

5. The method as described in claim 1, characterized in that, At the current optimal solution p opt The iterative update of the initial tap coefficient vector q is specifically: q = (1-τ)q + τp opt , where τ takes values ​​in the range (0,1).

6. The method as described in claim 1, characterized in that, A multi-channel cosine filter bank includes analysis filters and synthesis filters, and is represented as follows: Among them, h k (n) represents the analysis filter, g k p(n) represents the composite filter, k represents the number of filter channels, n is the filter order number, and p(n) is the prototype filter, with parameters for the k-th channel. 0≤k≤K-1, where K is the number of channels set.

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