Variable fractional delay filter design method and system

Abstract

The embodiment of the invention provides a variable fractional delay filter design method and system. The method is applied to the technical field of communication technology and signal processing, and comprises the following steps: constructing a function relationship between an analog filter and a digital filter through a sampling theory based on a VFD filter principle of a Farrow structure; the analog filter is cut off through a window function method, and a high-order FIR filter is obtained through calculation; the high-order FIR filter coefficient is decomposed into a plurality of different fractional delay filters, and a VFD filter coefficient of a Farrow structure is calculated by using a segmentation fitting method; and designing a variable fractional delay filter through a VFD filter coefficient of a Farrow structure based on the function relationship. In this way, the filter designed by the method has obvious advantages in frequency aliasing resistance, and the designed variable fractional delay filter has better application prospects.

Description

Technical Field

[0001] This disclosure relates to the fields of communication technology and signal processing technology, and in particular to a design method and system for a variable fractional delay filter. Background Technology

[0002] Variable fractional delay (VFD) filters are widely used in sampling rate conversion and other fields due to their flexible and adjustable frequency response. Filters based on the Farrow structure can change the group delay characteristics of the filter by adjusting the delay parameters while keeping the polynomial coefficients fixed, making them highly efficient VFD filters. The key to designing a VFD filter lies in determining the coefficients of the Farrow structure sub-filters. Currently, the commonly used method for designing Farrow structure sub-filter coefficients is to design the optimal filter that meets the requirements through optimization methods. However, this method involves matrix inversion and integration operations based on least squares and minimax methods, resulting in long design times when the filter order is high. Furthermore, the selection of weights affects filter performance. Additionally, optimization methods typically only optimize the passband of the filter, neglecting the stopband performance, which can lead to frequency aliasing problems when the designed filter is used for sampling rate conversion.

[0003] Therefore, there is an urgent need for a design scheme that can effectively improve the complexity and performance of variable fractional delay digital filters. Summary of the Invention

[0004] This disclosure provides a design method and system for a variable fractional time delay filter. Based on the VFD filter principle of the Farrow structure, it establishes a design framework for the coefficients of the Farrow structure sub-filter based on higher-order filters, which at least solves the technical problem of frequency aliasing that occurs when the filter designed by the existing design method is used for sampling rate conversion.

[0005] According to a first aspect of this disclosure, a method for designing a variable fractional time delay filter is provided, comprising the following steps: Based on the principle of VFD filters with Farrow structure, the functional relationship between analog and digital filters is obtained through sampling theory. The coefficients of the higher-order FIR filter are calculated by truncating the analog filter using the window function method. The coefficients of the high-order FIR filter are decomposed into multiple fractional time delay filters, and the coefficients of the VFD filter of the Farrow structure are calculated using a piecewise fitting method. Based on the aforementioned functional relationship, a variable fractional time delay filter is designed using the VFD filter coefficients of the Farrow structure.

[0006] In addition to the aspects and any possible implementations described above, a further implementation is provided, wherein the process of constructing the functional relationship between the analog filter and the digital filter through sampling theory based on the Farrow structure VFD filter principle is as follows: The impulse responses of the analog filter and the digital filter are obtained respectively; Obtain an ideal sampling signal, and based on the ideal sampling signal, obtain the frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter.

[0007] In addition to the aspects and any possible implementations described above, a further implementation is provided in which the process of obtaining the frequency domain relationship between the analog filter impulse response and the digital filter impulse response based on the ideal sampled signal is as follows: Fourier transforms are performed on the impulse response of the analog filter, the impulse response of the digital filter, and the ideal sampled signal, respectively. Construct a first functional relationship between the ideal sampled signal and the Fourier transform result of the ideal sampled signal, and construct a second functional relationship between the Fourier transform result of the impulse response of the analog filter and the Fourier transform result of the ideal sampled signal; The frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter is obtained based on the first functional relationship and the second functional relationship.

[0008] In addition to the aspects and any possible implementations described above, a further implementation is provided in which the frequency domain relationship between the analog filter impulse response and the digital filter impulse response is specifically as follows: ; in, The Fourier transform result of the ideal sampled signal H a () The Fourier transform result of the analog filter. Ω To simulate angular frequency, T1 is the sampling period, and ω is the frequency. H p (jω) This is the Fourier transform result of the digital filter.

[0009] In addition to the aspects and any possible implementations described above, a further implementation is provided in which the process of truncating the analog filter using the window function method to calculate the coefficients of the higher-order FIR filter is as follows: The Kaiser window is selected as the window function, and the Kaiser window coefficients w(k) are calculated by determining the filter coefficients K and the variable β. Obtain the group delay and cutoff frequency of the ideal filter, and calculate the impulse response sequence of the simulated filter. The digital filter impulse response sequence is calculated based on the Kaiser window coefficients and the analog filter impulse response sequence, thus obtaining the higher-order FIR filter coefficients.

[0010] In addition to the aspects and any possible implementations described above, a further implementation is provided in which the process of decomposing the high-order FIR filter coefficients into multiple fractional time delay filters and calculating the VFD filter coefficients of the Farrow structure using a piecewise fitting method is as follows: The coefficients of a high-order FIR filter of order K are decomposed into N segments, where each segment has L values ​​and each segment is fitted by L delay parameters p. Perform polynomial fitting on each segment and construct the polynomial matrix; The coefficients of the Farrow structure subfilter are calculated by performing least squares on the polynomial coefficients in the polynomial matrix.

[0011] In addition to the aspects and any possible implementations described above, a further implementation is provided, wherein the polynomial matrix is ​​specifically: ; Among them, c nM Let h(L) be the coefficients of the nth order Farrow structure sub-filter, M = 1, 2, 3…M, where M is the polynomial order, and h(L) be the coefficients of the Lth fractional delay filter. L Let L be the fractional delay parameter.

[0012] In addition to the aspects and any possible implementations described above, a further implementation is provided, wherein the method further includes: The frequency response of the ideal VFD filter is established, and the impulse response of the ideal VFD filter is approximated by a polynomial fitting method based on the delay parameters. By combining the frequency response and impulse response of the ideal VFD filter, the frequency response of the VFD filter with the Farrow structure is calculated.

[0013] According to a second aspect of this disclosure, a variable fractional delay filter design system is provided, comprising: a function relationship construction module, a high-order FIR filter coefficient calculation module, a VFD filter coefficient calculation module, and a variable fractional delay filter construction module; The functional relationship construction module is used to construct the functional relationship between analog and digital filters based on the VFD filter principle of the Farrow structure and through sampling theory. The higher-order FIR filter coefficient calculation module is used to truncate the analog filter using the window function method to calculate the higher-order FIR filter coefficients. The VFD filter coefficient calculation module is used to decompose the high-order FIR filter coefficients into multiple fractional time delay filters, and to calculate the VFD filter coefficients of the Farrow structure using a piecewise fitting method. The variable fractional delay filter construction module is used to design a variable fractional delay filter based on the functional relationship and the VFD filter coefficients of the Farrow structure.

[0014] Compared with the prior art, the present invention has the following technical effects: This invention elucidates the principle of VFD filters based on the Farrow structure, then derives the relationship between analog and digital filters through sampling theory to reveal the principle of higher-order filters approximating analog filters. Furthermore, by deriving the relationship between higher-order filters and fractional delay filters, a design framework for the coefficients of Farrow structure sub-filters based on higher-order filters is established. The filters designed by the method of this invention have significant advantages in anti-frequency aliasing characteristics, and the designed variable fractional delay filters have better application prospects.

[0015] It should be understood that the description in the Summary of the Invention is not intended to limit the key or essential features of the embodiments of this disclosure, nor is it intended to restrict the scope of this disclosure. Other features of this disclosure will become readily apparent from the following description. Attached Figure Description

[0016] The above and other features, advantages, and aspects of the embodiments of this disclosure will become more apparent from the accompanying drawings and the following detailed description. The drawings are provided for a better understanding of the invention and are not intended to limit the scope of this disclosure. In the drawings, the same or similar reference numerals denote the same or similar elements, wherein: Figure 1 A schematic flowchart of a variable fractional delay filter design method according to an embodiment of the present disclosure is shown; Figure 2 A schematic diagram of the impulse response decomposition of a high-order filter according to an embodiment of the present disclosure is shown. Figure 3 A schematic diagram of a system structure for a variable fractional delay filter design according to an embodiment of the present disclosure is shown. Figure 4 This diagram illustrates the amplitude error of an all-pass VFD filter designed according to an embodiment of a variable fractional delay filter design method based on this disclosure. Figure 5 This diagram illustrates the group delay error of an all-pass VFD filter designed according to an embodiment of a variable fractional delay filter design method based on this disclosure. Figure 6This diagram illustrates the amplitude error of an anti-aliasing VFD filter designed according to an embodiment of a variable fractional delay filter design method based on this disclosure. Figure 7 A schematic diagram of the group delay error of an anti-aliasing VFD filter designed according to an embodiment of a variable fractional delay filter design method of the present disclosure is shown. Detailed Implementation

[0017] To make the objectives, technical solutions, and advantages of the embodiments of this disclosure clearer, the technical solutions of the embodiments of this disclosure will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this disclosure, and not all embodiments. Based on the embodiments of this disclosure, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this disclosure.

[0018] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0019] Reference Figure 1 As shown, this embodiment provides a design method for a variable fractional delay filter, including the following steps: S101. Based on the Farrow structure, the VFD filter principle is used to construct the functional relationship between analog and digital filters through sampling theory.

[0020] In this embodiment, considering that the digital filter can be viewed as a sampling result of the analog filter, a higher-order filter can be designed to replace the analog filter. This reduces the computational load of the objective function optimization process, enabling the design of an efficient variable fractional delay (VFD) filter. The impulse responses of the analog and digital filters are respectively taken as... h a (t) and h p (n) The sampling period of the digital filter is taken as T 1 Then we have: ; To obtain the frequency domain relationship between the two, an ideal sampling signal is introduced. Its expression is: ; In the formula, δ(t) It is the impulse function. Take, in sequence... h a(t) , h p (n) and h aʹ (t) The Fourier transform is H a (jΩ) , H p (j ω) and H aʹ (jΩ) Then there is a formula.

[0021] (3) (4) (5) In the formula, Ω To simulate angular frequency, Ω=ωT 1 Substituting the formula into the equation, we get: (6) The relationship between the two is: (7) Therefore, we can conclude that: (8) As can be seen from the formula, H p (jω) It is by... H a (jΩ) Extend T 1 The magnitude decreased to 1 / T 1 Then shift 2kπ / T 1 Obtained. Specifically, when the sampling period... T 1 = At 1 o'clock, H p (jω) amplitude and H a (jΩ) Same. Because H p (jω) The group delay is fixed, therefore H p (jω) This only represents one type of variable fractional delay filter. p The characteristics of the value.

[0022] S102. The analog filter is truncated using the window function method to calculate the coefficients of the higher-order FIR filter.

[0023] In this embodiment, due to the flexible and adjustable nature of the Kaiser window, it is selected as the window function w(k). The expression of the Kaiser window in the time domain is: (9) Where I0(•) is the Bessel function and β is a variable.

[0024] For analog filters, this embodiment is denoted as h I ( k The expression is: (10) Where τ is the group delay of the ideal filter, ω c It is the cutoff frequency.

[0025] This embodiment takes the design of an FIR digital filter using a window function as an example. The principle is to truncate the analog filter using a window function w(k), as shown in equation (11): (11) in, h(k) and h I (k) The impulse response sequences of the digital filter and the analog filter are of length K+1, respectively. After determining the filter order K and parameter β in equation (9), the Kaiser window coefficients can be obtained, and then the required digital filter coefficients can be obtained.

[0026] In this embodiment, take h(k) To correspond to the impulse response of the analog filter h a (t) The corresponding impulse response of the higher-order digital low-pass filter is as follows: (12) Where K is the filter order.

[0027] S103. Decompose the coefficients of the high-order FIR filter into multiple fractional time delay filters, and calculate the coefficients of the VFD filter of the Farrow structure using a piecewise fitting method.

[0028] In this embodiment, the high-order filter h(k) of order K is divided into N segments, each segment has L values, and the polynomial of each segment is fitted by L delays p, as shown in (13): (13) Where c(n,m) are the coefficients of the Farrow structure sub-filter, M is the polynomial order, h(L) are the coefficients of the Lth fractional delay filter, and p L Let L be the fractional delay parameter.

[0029] Equation (13) can be expressed in matrix form: (14) The polynomial coefficients on the right side of equation (14) can be solved by the least squares method.

[0030] In this embodiment, after obtaining the high-order filter, it can be decomposed into multiple filters with different fractional time delays, such as... Figure 2 As shown. For Figure 2 The high-order filter impulse response in (a) is sampled L times with a sampling period of L, resulting in L fractional delay filter impulse responses, as shown below. Figure 2 As shown in (b). Figure 2 Taking L=3 as an example, Figure 2 (c) represents the group delay response of the L filters after decomposition.

[0031] It can be seen Figure 2 (b) The fractional time delay filters decomposed in the decomposition have a 1 / L shift in the time domain. According to the shift characteristics of the Fourier transform, the result of shifting the filter coefficients h(n) in the time domain is h(n-1 / L), then we have (15) Among them, H L (jω) is the frequency response of the filter after shifting h(n) by 1 / L.

[0032] Therefore, the group delay characteristics of each fractional delay filter will differ by 1 / L. This is also consistent with... Figure 2 (c) is consistent with the group delay characteristics.

[0033] Therefore, by sequentially performing polynomial fitting and coefficient solving on the N curves of the higher-order filter, the coefficients of the Farrow structure sub-filter can be obtained.

[0034] Variable fractional delay digital filters have been widely used in fields such as sampling rate conversion, delay compensation, image interpolation, and radar beamforming due to their flexible and adjustable frequency response. The filter based on the Farrow structure is an efficient variable fractional delay filter. Its principle is to decompose the filter impulse response into multiple piecewise polynomials. On the basis of fixed polynomial coefficients, the group delay characteristic of the filter can be changed by adjusting the delay parameter. Therefore, the key to designing a variable fractional delay filter lies in the determination of the coefficients of the Farrow structure sub-filter.

[0035] The frequency response of an ideal VFD filter is shown in Equation (16): (16) where ω is the normalized angular frequency, a satisfies 0 < a < 1, [0, ap] represents the filter passband range, p is the fractional delay parameter, and p ∈ [-0.5, 0.5].

[0036] Since an ideal low-pass filter cannot be realized in practical applications, its impulse response can be approximately represented by the method of polynomial fitting, that is: (17) In the formula, M is the polynomial order, and c(n, m) is the coefficient of the Farrow structure sub-filter. Further, the impulse response h d(n) The corresponding frequency response is: (18) where N is the filter order. Equation (18) is the frequency response of the VFD filter with the Farrow structure.

[0037] S104. Design a variable fractional delay filter based on the function relationship and the coefficients of the VFD filter with the Farrow structure.

[0038] As Figure 3 shown, this embodiment also provides a variable fractional delay filter design system, including: a function relationship construction module 1, a high-order FIR filter coefficient calculation module 2, a VFD filter coefficient calculation module 3, and a variable fractional delay filter construction module 4; The function relationship construction module 1 is used to construct the function relationship between the analog filter and the digital filter based on the principle of the VFD filter with the Farrow structure through sampling theory; The high-order FIR filter coefficient calculation module 2 is used to truncate the analog filter by the window function method and calculate the high-order FIR filter coefficients; VFD filter coefficient calculation module 3 is used to decompose the coefficients of the high-order FIR filter into multiple fractional time delay filters, and calculate the coefficients of the VFD filter with the Farrow structure using a piecewise fitting method. The variable fractional delay filter construction module 4 is used to design a variable fractional delay filter based on the functional relationship and through the VFD filter coefficients of the Farrow structure.

[0039] Example: To verify the effectiveness of the present invention, this example uses the design of an all-pass filter and an anti-aliasing low-pass filter as examples to verify the effectiveness of the proposed method. At the same time, the designed filter is applied to sampling rate conversion to verify the application effect of the proposed method.

[0040] Considering the wide applications of all-pass filters in time-domain imaging, speech processing, and lossless signal processing, this embodiment selects the least squares method and the window function method to design high-order filters. Using the proposed Farrow structure filter design method based on high-order filters, the corresponding Farrow structure sub-filter coefficients are designed. For convenience, the design process based on the least squares method and the window function method is referred to as Method 1 and Method 2. The high-order filter specification is taken as K=6999, ω... c =π, β=10.88, α=0.9, L=100, I=7. Meanwhile, to evaluate the performance of the designed filter, the following evaluation metrics are used: 1) Normalized root mean square error (19) 2) Maximum amplitude error (20) 3) Maximum group delay error (twenty one) 4) Filter stopband attenuation index (twenty two) Simulation tests were conducted based on the above indicators, and the proposed method was compared with the design results of weighted least squares method and conjugate gradient method. The comparison results are shown in Table 1.

[0041] Table 1 Comparison of Design Results for All-Pass VFD Filters

[0042] As shown in Table 1, the evaluation index ε2 reflects that the overall amplitude error of the all-pass filter designed by the conjugate gradient method is minimized. ∞ The maximum magnitude error of the proposed window function method is minimized, ε gThe weighted least squares method yields the smallest group delay error. In contrast, the all-pass filter designed based on the least squares method for higher-order filters exhibits slightly worse characteristics. Therefore, the all-pass filter designed using the method proposed in this embodiment achieves the same performance as existing methods. To demonstrate the overall trend of the all-pass filter designed using the proposed method, the amplitude error and group delay error of the filter in the range [0, 0.9π] are shown below. Figure 4 and Figure 5 As shown.

[0043] from Figure 4 and Figure 5 It can be seen that the amplitude error and group delay error of the all-pass filter designed by Method 2 in this embodiment are better than those of Method 1. The passband amplitude error of the filter designed by Method 1 is less than -102dB, and the group delay error is less than 1.1×10. -4 Therefore, it is evident that the all-pass filter designed based on a window function high-order filter proposed in this paper is feasible.

[0044] However, the filter stopband attenuation index ε in Table 1 s This reflects that the designed all-pass filter cannot suppress signals with frequencies near ω=π, further leading to the ineffective filtering of the designed Farrow structure filter coefficients when used for sampling rate conversion. For example, taking downsampling as an example, for a signal with an initial sampling rate of 10kHz containing the fundamental, 3rd, and 21st harmonics, the Farrow structure filter coefficients designed based on four methods are downsampled to 2kHz. The results after downsampling are analyzed using the Fast Fourier Transform method, as shown in Table 2.

[0045] Table 2 shows the results of applying the designed all-pass VFD filter to downsampling.

[0046] The results in Table 2 show that the filter designed in this embodiment will produce a 950Hz frequency aliasing signal when applied to downsampling. This is consistent with the filter stopband attenuation index ε in Table 1. s Correspondingly, in order to make the designed Farrow structure filter have anti-aliasing effect, the cutoff frequency of the filter needs to be adjusted.

[0047] To address the insufficient stopband attenuation in the design of all-pass VFD filters, this embodiment implements an anti-aliasing filter design by lowering the filter cutoff frequency. The high-order filter specification is chosen as K=6999, ω... c =0.9π, β=10.88, α=0.8, L=100, I=7. Anti-aliasing VFD filters were designed using four methods, and the designed filters were evaluated using evaluation metrics. The results are shown in Table 3.

[0048] Table 3 Comparison of Anti-aliasing VFD Filter Design Results

[0049] It can be seen that all four methods can achieve the design of anti-aliasing filters by adjusting the filter cutoff frequency. Evaluation metrics ε² and ε... ∞ The filter designed using the conjugate gradient method exhibits the smallest amplitude error, while the index ε... g and ε s The filter exhibits the best group delay characteristics and stopband attenuation characteristic of the window function method. To demonstrate the overall trend of the anti-aliasing filter designed by the proposed method, the amplitude error and group delay error of the filter in the range [0, 0.8π] are plotted as follows. Figure 6 and Figure 7 As shown.

[0050] from Figure 6 and Figure 7 It can be seen that the amplitude errors of the anti-aliasing VFD filters designed by Method 1 and Method 2 in this embodiment are quite similar, both being less than -105dB. However, the group delay error of Method 2 is better than that of Method 1, with both having a group delay error of less than 1.2 × 10⁻⁶. -4 Meanwhile, the proposed method achieves a stopband attenuation index of ε for the filter. s The proposed method outperforms weighted least squares and conjugate gradient methods, demonstrating its advantages.

[0051] To verify the effectiveness of the designed filter in sampling rate conversion, the filter coefficients designed by the four methods were also applied to downsampling, and the results after downsampling are shown in Table 4.

[0052] Table 4 shows the results of applying the designed anti-aliasing VFD filter to downsampling.

[0053] It can be seen that although the weighted least squares method and the conjugate gradient method can design anti-aliasing filters by lowering the cutoff frequency, they cannot effectively filter out frequency aliasing signals when applied to downsampling. In contrast, the filter designed by the proposed method can prevent frequency aliasing.

[0054] In summary, compared with existing methods, the VFD filter design method proposed in this invention has strong generality and can design high-performance VFD filters according to requirements through methods such as window function method and least squares method. At the same time, the filter can achieve better results when applied to sampling rate conversion.

[0055] It should be noted that, for the sake of simplicity, the foregoing method embodiments are all described as a series of actions. However, those skilled in the art should understand that this disclosure is not limited to the described order of actions, because according to this disclosure, some steps can be performed in other orders or simultaneously. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are all optional embodiments, and the actions and modules involved are not necessarily essential to this disclosure.

[0056] It should be understood that the various forms of processes shown above can be used to rearrange, add, or delete steps. For example, the steps described in this disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this disclosure can be achieved, and this is not limited herein.

[0057] The specific embodiments described above do not constitute a limitation on the scope of protection of this disclosure. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this disclosure should be included within the scope of protection of this disclosure.

Claims

1. A design method for a variable fractional time delay filter, characterized in that, Includes the following steps: Based on the principle of VFD filters with Farrow structure, the functional relationship between analog and digital filters is obtained through sampling theory. The coefficients of the higher-order FIR filter are calculated by truncating the analog filter using the window function method. The coefficients of the high-order FIR filter are decomposed into multiple fractional time delay filters, and the coefficients of the VFD filter of the Farrow structure are calculated using a piecewise fitting method. Based on the aforementioned functional relationship, a variable fractional time delay filter is designed using the VFD filter coefficients of the Farrow structure.

2. The variable fractional delay filter design method according to claim 1, characterized in that, The process by which the functional relationship between analog and digital filters is constructed through sampling theory, based on the Farrow structure VFD filter principle, is as follows: The impulse responses of the analog filter and the digital filter are obtained respectively; Obtain an ideal sampling signal, and based on the ideal sampling signal, obtain the frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter.

3. The variable fractional delay filter design method according to claim 1, characterized in that, The process of obtaining the frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter based on the ideal sampled signal is as follows: Fourier transforms are performed on the impulse response of the analog filter, the impulse response of the digital filter, and the ideal sampled signal, respectively. Construct a first functional relationship between the ideal sampled signal and the Fourier transform result of the ideal sampled signal, and construct a second functional relationship between the Fourier transform result of the impulse response of the analog filter and the Fourier transform result of the ideal sampled signal; The frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter is obtained based on the first functional relationship and the second functional relationship.

4. The variable fractional delay filter design method according to claim 3, characterized in that, The frequency domain relationship between the impulse response of the analog filter and the impulse response of the digital filter is as follows: ； Among them, H aʹ (jΩ) represents the Fourier transform result of the ideal sampled signal, H a () represents the Fourier transform result of the analog filter, Ω is the analog angular frequency, T1 is the sampling period, ω is the frequency, and H p (jω) represents the Fourier transform result of the digital filter.

5. The variable fractional delay filter design method according to claim 1, characterized in that, The process of truncating the analog filter using the window function method and calculating the coefficients of the higher-order FIR filter is as follows: The Kaiser window is selected as the window function, and the Kaiser window coefficients w(k) are calculated by determining the filter coefficients K and the variable β. Obtain the group delay and cutoff frequency of the ideal filter, and calculate the impulse response sequence of the simulated filter. The digital filter impulse response sequence is calculated based on the Kaiser window coefficients and the analog filter impulse response sequence, thus obtaining the higher-order FIR filter coefficients.

6. The variable fractional delay filter design method according to claim 1, characterized in that, The process of decomposing the coefficients of a high-order FIR filter into multiple fractional time delay filters and calculating the coefficients of a Farrow-structured VFD filter using a piecewise fitting method is as follows: The coefficients of a high-order FIR filter of order K are decomposed into N segments, where each segment has L values ​​and each segment is fitted by L delay parameters p. Perform polynomial fitting on each segment and construct the polynomial matrix; The coefficients of the Farrow structure subfilter are calculated by performing least squares on the polynomial coefficients in the polynomial matrix.

7. The variable fractional delay filter design method according to claim 6, characterized in that, The polynomial matrix is ​​specifically: ； Among them, c nM Here, h(L) represents the coefficients of the nth-order Farrow structure sub-filter, M is the polynomial order, and h(L) represents the coefficients of the Lth fractional delay filter. L Let L be the fractional delay parameter.

8. The variable fractional delay filter design method according to claim 1, characterized in that, The method further includes: The frequency response of the ideal VFD filter is established, and the impulse response of the ideal VFD filter is approximated by a polynomial fitting method based on the delay parameters. By combining the frequency response and impulse response of the ideal VFD filter, the frequency response of the VFD filter with the Farrow structure is calculated.

9. A variable fractional time delay filter design system, implemented using the variable fractional time delay filter design method as described in any one of claims 1-8, characterized in that, include: Functional relation construction module (1), high-order FIR filter coefficient calculation module (2), VFD filter coefficient calculation module (3), and variable fractional delay filter construction module (4). The function relationship construction module (1) is used to construct the function relationship between analog filters and digital filters based on the VFD filter principle of Farrow structure through sampling theory; The higher-order FIR filter coefficient calculation module (2) is used to truncate the analog filter using the window function method and calculate the higher-order FIR filter coefficients. The VFD filter coefficient calculation module (3) is used to decompose the high-order FIR filter coefficients into multiple fractional time delay filters and calculate the VFD filter coefficients of the Farrow structure using a piecewise fitting method. The variable fractional delay filter construction module (4) is used to design a variable fractional delay filter based on the functional relationship and through the VFD filter coefficients of the Farrow structure.

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