Improved farrow structure fractional delay design method based on non-equidistant segmentation

Abstract

This invention discloses an improved Farrow structure fractional delay design method based on non-equal interval segmentation. Compared with traditional variable fractional delay filter design methods, the proposed method effectively reduces the structural complexity of the variable fractional delay filter and the number of multipliers while maintaining the same variable frequency response error accuracy. The technical solution of this invention is as follows: Based on an improved Farrow structure fractional delay design method with non-equal interval segmentation, and given a known variable frequency response error threshold, a variable fractional delay filter is designed using the equal interval segmentation criterion of the fractional delay domain and the minimax algorithm based on the coefficient relationship structure. This ensures that the peak error of the variable frequency response of the actual variable fractional delay filter does not exceed the error threshold, ultimately obtaining the discrete-time impulse response of the parallel FIR sub-filter.

Description

Technical Field

[0001] This invention relates to the fields of communication, radar, sonar, and navigation technologies, and specifically to an improved Farrow structure fractional delay design method based on non-equal interval segmentation. Background Technology

[0002] Variable fractional delay filters are digital filters with variable fractional phase delays. They are typically implemented using a Farrow architecture, consisting of a set of parallel FIR sub-filters and a delay control unit. Their advantage is that the fractional delay value can be changed online without redesigning the filter. Variable fractional delay filters have been a popular research area in digital signal processing, widely used in sampling rate conversion, channel simulation, speech coding, delay estimation, and digital signal interpolation. Design methods for variable fractional delay filters can be divided into time-domain and frequency-domain algorithms. Time-domain algorithms, based on polynomial interpolation theory such as Lagrange interpolation, Hermite interpolation, and B-spline interpolation, directly yield filter coefficients and exhibit good variable fractional delay response at low frequencies. Frequency-domain algorithms aim to find a set of filter coefficients that minimizes the variable frequency response of the variable fractional delay filter under certain criteria. Depending on the approximation criteria, frequency-domain algorithms can be further categorized into maximum flatness design, weighted least squares design, and minimax design. Among the above methods, the frequency domain design algorithm has attracted widespread attention from scholars at home and abroad since its introduction due to its advantages such as obtaining a larger frequency bandwidth, better performance at high frequencies, and more flexible design.

[0003] In frequency domain design algorithms, the minimax algorithm for designing variable fractional delay filters is considered one of the most classic and widely used methods. The core idea of ​​the minimax algorithm for designing variable fractional delay filters is to continuously optimize the filter coefficients using a Farrow structure to minimize the peak error of the variable frequency response. Compared to frequency domain design algorithms such as maximum flatness design and weighted least squares design, the minimax algorithm achieves a smaller peak error in the variable frequency response with the same filter implementation complexity, and the amplitude-frequency response of the variable fractional delay filter designed using this algorithm has equal ripple. Currently, researchers both domestically and internationally are focusing on two main research directions for designing variable fractional delay filters based on the minimax algorithm: one is to optimize the minimax design process to reduce the algorithm's design complexity according to different design goals; the other is to improve the Farrow structure by optimizing the filter order and the number of coefficients to reduce the filter implementation complexity and the amount of multiplier resources used.

[0004] To reduce the structural complexity of filter implementations, researchers both domestically and internationally have employed various optimization methods for minimax design, effectively reducing resource consumption to varying degrees. Firstly, researchers proposed a bilinear programming-based minimax design algorithm, which decomposes the variable frequency response error of the filter into real and imaginary errors, optimizing each part separately. By generalizing the order of the sub-filters in the Farrow structure, the structural complexity of the filter implementation is effectively reduced. However, the bilinear programming-based minimax algorithm transforms a nonlinear optimization problem into two linear optimization problems, inevitably leading to incomplete optimization results. Therefore, researchers both domestically and internationally proposed a minimax design algorithm based on SOCP (Second-Order Cone Programming), which can directly minimize the peak error of the variable frequency response, obtaining the optimal optimization result. To further reduce the structural complexity of variable fractional delay filters, researchers both domestically and internationally derived the coefficient relationships between adjacent sub-filters in the Farrow structure and, based on these new coefficient relationships, presented a less complex structure that significantly reduces multiplier and coefficient storage resources compared to the traditional Farrow structure. However, the above methods all optimize the complete fractional delay domain [-0.5, 0.5], which is equivalent to imposing a strong constraint on the nonlinear optimization problem. This requires more sub-filters and filter orders to meet the design accuracy requirements, thus greatly increasing the structural complexity of the variable fractional delay filter implementation.

[0005] To address this issue, researchers both domestically and internationally have proposed a design method for variable fractional delay filters based on equally spaced segmentation. This method, based on the traditional Farrow implementation, divides the fractional delay domain into equally spaced segments. Within each segment, a minimax design based on SOCP is performed, relaxing the constraints of the minimax design and achieving the same design accuracy as conventional methods with fewer multiplier resources. However, the traditional Farrow implementation has high complexity, and directly applying the equally spaced segmentation criterion for the fractional delay domain does not achieve significant optimization results. Furthermore, the equally spaced segmentation optimization algorithm does not consider the variation of the variable frequency response with the fractional delay, requiring more coefficient storage resources to achieve fewer multiplier resources. Currently, research on optimization within the fractional delay domain is rarely reported. Summary of the Invention

[0006] In view of this, the present invention provides an improved Farrow structure fractional delay design method based on non-equal interval segmentation. Compared with the traditional variable fractional delay filter design method, the proposed method effectively reduces the structural complexity of the variable fractional delay filter and the number of multipliers under the same variable frequency response error accuracy.

[0007] To achieve the above objectives, the technical solution of this invention is as follows: an improved farrow structure fractional delay design method based on non-equal interval segmentation. Under the condition that the variable frequency response error threshold is known, a variable fractional delay filter is designed using the equal interval segmentation criterion of the fractional delay domain and the minimax algorithm based on the coefficient relationship structure, so that the peak error of the variable frequency response of the actual variable fractional delay filter does not exceed the error threshold, and finally the discrete-time impulse response of the parallel FIR sub-filter is obtained.

[0008] Furthermore, the improved farrow structure fractional delay design method based on non-equal interval segmentation specifically includes the following steps:

[0009] Step 1: The fractional delay domain of the even-order variable fractional delay filter is p, p∈[-0.5,0.5]. Using the non-equal interval segmentation criterion, p is divided into K segments, where S... k This represents the segmented interval with index k, where k = 0, 1, ..., K-1.

[0010]

[0011] Step 2: Define k p The index of the segmented interval of the decimal delay domain p is represented as... but

[0012]

[0013] in, This indicates rounding down to the nearest integer.

[0014] Step 3: Initialize parameters: segmented interval index k = 0, polynomial order M k =2, optimization error ε0=0;

[0015] Step 4: Calculate parameters: Among them, M ek M represents the number of sub-filters in the Farrow structure whose polynomial order is an even power. ok This represents the number of sub-filters in the Farrow structure whose polynomial order is an odd power.

[0016] Step 5: Initialize the filter order:

[0017]

[0018] in,

[0019] N mk =max{N emk N omk}, m=1,2,...,M ek (7)

[0020] Where, N emk N represents the order of a sub-filter whose polynomial order is an even power and whose index is m. omk This indicates the order of the sub-filter whose polynomial order is an odd power and whose index is m. The index m represents the m-th sub-filter in the Farrow structure.

[0021] Step 6: Let

[0022]

[0023] in N represents k The result of adding 1 to the element with index i.

[0024] Step 7: Based on parameter M ek M ok , The variable frequency response of a practical variable fractional delay filter is H(ω,p):

[0025]

[0026] in,

[0027]

[0028]

[0029]

[0030] Where ω is the normalized angular frequency; b ek Let b be the filter coefficient matrix. emk For b ek row vectors; f k Let f be the real component matrix of the filter response. mk f k row vectors; g k Let g be the imaginary component matrix in the filter response. mk For g k The row vector.

[0031] Step 8: Based on the variable frequency response of the ideal variable fractional delay filter and the variable frequency response of the actual variable fractional delay filter, the variable frequency response error is obtained as e(ω,p):

[0032]

[0033] Where H I (ω,p) represents the ideal variable frequency response; e R (ω,p) represents the real part of the variable frequency response error e(ω,p); -e I (ω,p) is the imaginary part of the variable frequency response error e(ω,p).

[0034] Step 9: Using the variable frequency response error and the non-equal interval segmentation criterion of the fractional delay domain, the minimax design of the even-order variable fractional delay filter is expressed as follows:

[0035]

[0036] in This represents the peak error.

[0037]

[0038] In the segmented interval S k Minimax design was performed internally to obtain the peak error. and the corresponding filter coefficients b ek .

[0039] Step 10: Repeat steps 6 to 9, traversing from i = 0 to i = M. ek -1, a total of M can be obtained ek One peak error; select the smallest peak error from all results. Right now

[0040]

[0041] and use the corresponding Update N k ,Right now

[0042] Step 11: Determine Is it less than a given error threshold? If yes, proceed to step 13; otherwise, proceed to step 12; set the error threshold. How to set it up: Set it up according to your actual needs.

[0043] Step 12: Calculate parameters

[0044]

[0045] Where δ is the current optimization ratio parameter.

[0046] and use the current Update ε0, that is ε0 is set to 0; the initial value is set to 0 as recorded in step 3; determine whether the current optimization ratio parameter δ is greater than the given optimization ratio threshold Δ, if so, return to step 6; otherwise, let M k Increment by 2, ε0 = 0, and return to step 4.

[0047] Step 13: Set the even-power window function w em (n) Even-power window function w om (n) Continue to optimize the number of coefficients of adjacent sub-filters.

[0048]

[0049] Among them, w em The number of 1s in (n) represents the number of sub-filter coefficients of the polynomial with index m and an even degree, w om The number of 1s in (n) represents the number of sub-filter coefficients of the polynomial with index m and an odd degree.

[0050] Then f mk and g mk Rewritten as

[0051]

[0052] Where m = 1, 2, ..., M ek ;

[0053] Step 14: Let For the currently optimized even-power window function; w ei (n) is the i-th even-power window function.

[0054] And order Set the last non-zero element in the array to 0, and use w em (n)(m≠i), w om (n) Perform the minimax design in step 9, and iterate through step 14 from i=1 to i=M. ek M ek Peak error Select the smallest peak error from all results. Right now

[0055]

[0056] Step 15: Let For the currently optimized odd-power window function; w oi (n) is;

[0057] And order Set the last non-zero element in the array to 0, and use w om (n)(m≠i), w em (n) Perform the minimax design shown, iterating through step 15 from i=1 to i=M. ek M ek Peak error Select the smallest peak error from all results. Right now

[0058]

[0059] Step 16: Let Determine ε min Is it greater than the given error threshold? If so Keep it unchanged and proceed to step 17; otherwise, update according to the following principles.

[0060]

[0061]

[0062] Return to step 14;

[0063] Step 17: For the next segmented interval S k+1 To optimize, let k = k + 1, M k =M k-1 Return to step 4; iterate through steps 4 to 16 from k=0 to k=K-1 to obtain the polynomial order M over all segmented intervals. k Parallel FIR sub-filter order N k Window function w em (n), w om (n) and the corresponding sub-filter coefficients b ek k = 0, 1, ..., K-1.

[0064] Step 18: Through coefficient b ek Given k = 0, 1, ..., K-1, the impulse response a of the even-order variable fractional delay filter is calculated. nm :

[0065] a nm =[a0(n,2m)a1(n,2m)…a K-1 (n,2m)] T (61)

[0066] in,

[0067]

[0068] bemk (n,m) is b emk Medium parameter; k = 0, 1, ..., K-1.

[0069] Furthermore, the value of K is chosen as: K = 4.

[0070] Furthermore, in step 8,

[0071]

[0072] Beneficial effects:

[0073] This invention proposes an improved Farrow fractional delay filter design method based on the coefficient relationship structure and the non-equal interval segmentation criterion of the fractional delay domain. Compared with existing technologies, this invention analyzes from the time domain design perspective by dividing the fractional delay domain into non-equal interval segments, which can better approximate the discrete impulse response of the filter, thereby reducing the polynomial order. Analyzing from the frequency domain design perspective, performing minimax design within each segment interval is equivalent to relaxing the constraints, thereby reducing the polynomial order and the order of the parallel FIR filter.

[0074] This invention utilizes the non-equal interval segmentation criterion of the fractional delay domain to optimize the design of a variable fractional delay filter based on the coefficient relationship structure. The original low-complexity structure is further optimized, requiring fewer multiplier resources.

[0075] This invention segments the fractional delay domain into non-equal intervals, which better reflects the variation of the variable frequency response of the variable fractional delay filter with the fractional delay parameter. Compared with equal interval segmentation, it achieves better optimization and lower structural complexity. Attached Figure Description

[0076] Figure 1 This is a structural diagram of the present invention;

[0077] Figure 2 This is a schematic diagram of the non-equal interval segmentation criterion for the decimal delay domain of the present invention. Detailed Implementation

[0078] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0079] This invention provides an improved Farrow structure fractional delay design method based on non-equal interval segmentation, the basic implementation process of which is as follows:

[0080] The basic idea behind designing a variable-decimal-time delay filter using the minimax design algorithm is to design an actual variable-decimal-time delay filter to approximate an ideal variable-decimal-time delay filter. After decomposition using the Farrow structure principle, the actual variable-decimal-time delay filter consists of multiple parallel FIR sub-filters and a delay control unit. Utilizing a symmetrical coefficient structure, adjacent FIR sub-filters can be implemented using the same filter coefficients, thereby reducing the complexity of the implementation structure. By continuously optimizing the coefficients of each sub-filter, the peak error of the variable frequency response of the actual variable-decimal-time delay filter is minimized.

[0081] The variable frequency response error of a practical variable fractional delay filter can be expressed as:

[0082] e(ω,p)=H(ω,p)-H I (ω,p) (1)

[0083] Among them, H I (ω,p) is the variable frequency response of an ideal even-order variable fractional delay FIR filter.

[0084] H I (ω,p)=e -jωp =cos(ωp)-jsin(ωp) (2)

[0085] H(ω,p) is the actual even-order variable fractional delay FIR filter's variable frequency response.

[0086]

[0087] Where ω∈[0,απ] is the normalized angular frequency, α∈(0,1) represents the passband cutoff frequency απ, p∈[-0.5,0.5] represents the fractional delay parameter, M represents the polynomial order, i.e., the number of parallel FIR sub-filters, each sub-filter has an order of 2N, and the corresponding discrete-time impulse response is a(n,m), n=-N,-N+1,...,N,m=0,1,...,M. This represents the number of sub-filters whose polynomial order is an even power, and their corresponding discrete-time impulse responses are a(n,2m), n=-N,-N+1,...,N,m=1,2,...,M e . This indicates rounding down to the nearest integer.

[0088] The method proposed in this invention, under the condition that the variable frequency response error threshold is known, uses the equal-interval segmentation criterion of the fractional delay domain and the minimax algorithm based on the coefficient relationship structure to design a variable fractional delay filter to approximate the ideal variable fractional delay filter. This ensures that the peak error of the variable frequency response of the actual variable fractional delay filter does not exceed the error threshold, and finally obtains the discrete-time impulse response a(n,2m) of the parallel FIR sub-filter. Specifically, as follows:

[0089] Step 1: The fractional delay domain of the even-order variable fractional delay filter is p∈[-0.5,0.5]. It is divided into K segments using the non-equal interval segmentation criterion, where S... k This represents the segmented interval with index k, where k = 0, 1, ..., K-1.

[0090]

[0091] In this embodiment of the invention, K=4 is selected, which achieves a significant optimization effect.

[0092] Step 2: Define k p This represents the index of the segment interval containing the decimal delay p, i.e. but

[0093]

[0094] in, This indicates rounding down, and k -p =k p .

[0095] The principle behind this step is as follows:

[0096] The purpose of this step is to segment the fractional delay domain into non-equal intervals. The segmentation criterion is a bisection method, that is, dividing the fractional delay domain p∈[0,0.5] into intervals close to 0.5. In the general design of variable fractional delay filters, the filter's variable frequency response exhibits a certain variation with the fractional delay. The closer the fractional delay is to 0.5, the stronger the constraint on the filter's variable frequency response, requiring more parallel FIR sub-filters to achieve the same accuracy. Therefore, compared to segmenting the fractional delay domain using an equal interval method, using a non-equal interval method better reflects the variation of the filter's variable frequency response with the fractional delay, resulting in a more optimized design across the segmented intervals.

[0097] From a time-domain perspective, segmenting the fractional delay domain into unequal intervals can better approximate the ideal discrete-time impulse response, helping to reduce the order of the piecewise polynomial. Furthermore, the unequal-interval segmentation algorithm improves the polynomial approximation accuracy by reducing the number of segments and increasing the segment interval for weakly constrained intervals, and increasing the number of segments and decreasing the segment interval for strongly constrained intervals.

[0098] From a frequency domain perspective, non-equal interval segmentation is equivalent to relaxing the constraints of minimax design to varying degrees. Under the same design accuracy, the design accuracy can be achieved by using fewer parallel sub-filters and filter orders, thereby reducing the amount of multiplier resources used.

[0099] Step 3: Initialize parameters: segmented interval index k = 0, polynomial order M k =2, optimization error ε0=0.

[0100] Step 4: Calculate parameters: Among them, M ek M represents the number of sub-filters in the Farrow structure whose polynomial order is an even power. ok This represents the number of sub-filters in the Farrow structure whose polynomial order is an odd power.

[0101] Step 5: Initialize the filter order:

[0102]

[0103] in,

[0104] N mk =max{N emk N omk}, m=1,2,...,M ek (7)

[0105] Where, N emk N represents the order of a sub-filter whose polynomial order is an even power and whose index is m. omk This indicates the order of the sub-filter whose polynomial order is an odd power and whose index is m. The index m represents the m-th sub-filter in the Farrow structure.

[0106] Step 6: Let

[0107]

[0108] in N represents k The result of adding 1 to the element with index i.

[0109] Step 7: Based on parameter M ek Mok , The variable frequency response of the practical variable fractional delay filter is as follows:

[0110]

[0111] in,

[0112]

[0113]

[0114]

[0115] The principle behind step 7 is as follows:

[0116] The purpose of this step is to derive the variable frequency response of the actual variable fractional delay filter based on the set polynomial order and the order of the parallel sub-filter.

[0117] The variable frequency response of an ideal even-order variable fractional delay FIR filter can be expressed as:

[0118] H I (ω,p)=e -jωp =cos(ωp)-jsin(ωp) (13)

[0119] Where ω∈[0,απ] is the normalized angular frequency, α∈(0,1) represents the passband cutoff frequency απ, and p∈[-0.5,0.5] represents the fractional delay parameter. The corresponding discrete-time impulse response is:

[0120] h I (n,p)=sinc(np),n∈Z (14)

[0121] in,

[0122]

[0123] Due to the ideal impulse response h I (n,p) is an infinitely long signal in the time domain and cannot be directly realized. In practice, a finite-length impulse response h(n,p) is used to approximate h. I (n,p), then the transfer function of the actual even-order variable fractional delay FIR filter is:

[0124]

[0125] Where the filter order is 2N, and h(n,p) represents the discrete-time impulse response of the variable fractional delay filter. Based on the Farrow structure principle and the non-equal-interval piecewise criterion for the domain of fractional delay, h(n,p) can be expressed as a piecewise polynomial of p, with K pieces and an order M, i.e.:

[0126]

[0127] Among them, a k (n,m) represents the piecewise polynomial coefficients, k = 0, 1, ..., K-1.

[0128] Substituting equation (17) into equation (16) yields

[0129]

[0130] in

[0131]

[0132] According to equation (14), the ideal impulse response of the variable fractional delay filter has the following properties:

[0133]

[0134] Therefore, the discrete-time impulse response of the constrained practical fractional delay filter also satisfies the following condition:

[0135]

[0136] Based on the above conditions, the coefficient a k (n,m) satisfies:

[0137]

[0138] To reduce complexity, let:

[0139] a k (n,0)=δ(n),k=0,1,...,K-1 (23)

[0140] Therefore, each FIR filter A k The coefficients of (z,m) are all symmetric: when m is even, a k (n,m) is even-symmetric; when m is odd, a k (n,m) is odd symmetric, and a k (0,m)=0. Meanwhile, when m=0, the FIR filter A… k (z,0)=1,k=0,1,...,K-1.

[0141] Based on the aforementioned symmetry, the transfer function of a practical even-order variable fractional delay filter can be rewritten as follows:

[0142]

[0143] The transfer function of the parallel FIR filter is:

[0144]

[0145] Let z = e jω The variable frequency response of the even-order variable fractional delay filter can be obtained as follows:

[0146]

[0147] in, The frequency response F of the parallel FIR sub-filter k (ω,m) and G k (ω,m) are respectively

[0148]

[0149]

[0150] coefficient b ek (n,2m) and b ok (n, 2m-1) are respectively

[0151]

[0152]

[0153] By comparing equations (13) and (26), the following approximate relationship can be obtained:

[0154]

[0155]

[0156] Differentiating both sides of equation (31) with respect to ω, we can derive...

[0157]

[0158] Right now

[0159]

[0160] Comparing equations (34) and (32), the following coefficient relationship can be obtained:

[0161] b ok (n, 2m-1) = nb ek (n, 2m), m = 1, 2, ..., Me (35)

[0162] To reduce the complexity of the filter structure, M is limited to an even number, and the polynomial order and sub-filter order are generalized. The transfer function of an even-order variable fractional delay filter can then be written as:

[0163]

[0164] in,

[0165]

[0166] Therefore, the variable fractional delay filter based on the coefficient relationship structure and the non-equal interval segmentation of the fractional delay domain can be implemented with a less complex structure.

[0167] Accordingly, the variable frequency response of an even-order variable fractional delay filter can be written as:

[0168]

[0169] in,

[0170]

[0171]

[0172] make

[0173]

[0174] Equation (38) can be rewritten as

[0175]

[0176] in,

[0177]

[0178] Step 8: Based on the variable frequency response of the ideal variable fractional delay filter and the variable frequency response of the actual variable fractional delay filter, the variable frequency response error can be obtained as follows:

[0179]

[0180] in,

[0181]

[0182] Step 9: Utilizing the variable frequency response error and the non-equal interval segmentation criterion of the fractional delay domain, the minimax design of an even-order variable fractional delay filter can be expressed as follows:

[0183]

[0184] in,

[0185]

[0186] In the segmented interval S k By performing a minimax design within the internal framework, the peak error can be obtained. and the corresponding filter coefficients b ek .

[0187] The principle behind this step is as follows:

[0188] The approximation criterion for minimax design is to continuously optimize the coefficients to minimize the peak error of the variable frequency response of the variable fractional delay filter. The optimization problem shown in equations (46) and (47) is a SOCP problem because the constraint is a second-order cone, i.e.

[0189]

[0190] Among them, C s Denotes a second-order cone, defined as follows:

[0191] C s ={(x1,x2)∈R×R N-1 |x1≥||x2||2} (49)

[0192] R represents the set of real numbers, R N-1 It represents a real N-1 dimensional vector.

[0193] Simplify the above SOCP problem by letting

[0194]

[0195] Equation (46) can be rewritten as

[0196]

[0197] in,

[0198]

[0199] For parameters ω∈[0,απ] and Sampling was performed, among which The sampling intervals are απ / (L1-1) and 1 / [2K(L2-1)]. When p = 0, equation (46) is automatically satisfied, so this point does not need to be considered. Therefore, there are a total of L1(L2-1) discrete points. For each discrete point All of them have a constraint of the form of equation (51). Therefore, the SOCP problem shown in equation (51) can be solved using nonlinear optimization tools to obtain the vector coefficients y. k Thus, the peak error is obtained. and the corresponding filter coefficients b ek .

[0200] Step 10: Repeat steps 6 to 9, traversing from i = 0 to i = M. ek -1, a total of M can be obtained ek There are several peak errors. The smallest peak error is selected from all results. Right now

[0201]

[0202] and use the corresponding Update N k ,Right now

[0203] Step 11: Determine Is it less than a given error threshold? If it is less than, proceed to step 13; if it is greater than, proceed to step 12.

[0204] Step 12: Calculate parameters

[0205]

[0206] and use the current Update ε0, that is Determine if parameter δ is greater than the given optimization ratio threshold Δ. If it is, return to step 6; if it is less, let M... k =M k +2, ε0 = 0, and return to step 4;

[0207] Step 13: Set the window function w em (n), w om (n) Continue to optimize the number of coefficients of adjacent sub-filters.

[0208]

[0209] Among them, w em The number of 1s in (n) represents the number of sub-filter coefficients of the polynomial with index m and an even degree, w om The number of 1s in (n) represents the number of sub-filter coefficients of the polynomial with index m and an odd degree.

[0210] Equation (41) can be rewritten as

[0211]

[0212] Where m = 1, 2, ..., M ek .

[0213] Step 14: Let And order Set the last non-zero element in the array to 0, and use w em (n)(m≠i), w om (n) Perform the minimax design as shown in equation (46), iterating this step from i=1 to i=M. ek M can be obtained ek Peak error Select the smallest peak error from all results. Right now

[0214]

[0215] Step 15: Let And order Set the last non-zero element in the array to 0, and use w om (n)(m≠i), w em (n) Perform the minimax design as shown in equation (46), iterating this step from i=1 to i=M. ek M can be obtained ek Peak error Select the smallest peak error from all results. Right now

[0216]

[0217] Step 16: Let Determine ε min Is it greater than the given error threshold? If it is greater than, then If the value remains unchanged, proceed to step 17; if it is less than the specified value, update according to the following principles.

[0218]

[0219]

[0220] Return to step 14.

[0221] Step 17: For the next segmented interval S k+1 To optimize, let k = k + 1, M k =M k-1Return to step 4. Iterate through steps 4 to 16 from k=0 to k=K-1 to obtain the polynomial order M over all segmented intervals. k Parallel FIR sub-filter order N k Window function w em (n), w om (n) and the corresponding sub-filter coefficients b ek k = 0, 1, ..., K-1.

[0222] Step 18: Through coefficient b ek For k = 0, 1, ..., K-1, the impulse response of the even-order variable fractional delay filter can be calculated using equation (29).

[0223] a nm =[a0(n,2m)a1(n,2m)…a K-1 (n,2m)] T (61)

[0224] in,

[0225]

[0226] b emk (n,m) is b emk Medium parameter; k = 0, 1, ..., K-1.

[0227] As can be seen, through steps 1 to 18, based on the known variable frequency response error threshold, this invention utilizes the non-equal interval segmentation criterion of the fractional delay domain to efficiently design a variable fractional delay FIR filter, achieving lower structural complexity.

[0228] Let the cutoff frequency ω c =0.9π, i.e., α = 0.9, number of segments K = 4, optimization ratio threshold Δ = 0.01, upper limit of variable frequency response error of variable fractional delay filter. The parallel FIR filter is initialized to order N. k =[11…1].

[0229] The proposed algorithm is compared with several state-of-the-art even-order variable fractional delay filter design algorithms, including the minimax design algorithm based on bilinear programming, the minimax design algorithm based on SOCP, the minimax design algorithm based on coefficient relationship structure, and the minimax design algorithm based on SOCP and equally spaced segments of the fractional delay domain.

[0230] Table 1. Comparison of the structural complexity of even-order variable fractional delay filters.

[0231]

[0232]

[0233] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An improved farrow structure fractional delay design method based on non-equal interval segmentation, characterized in that, Given the known variable frequency response error threshold, a variable fractional delay filter is designed using the non-equal interval segmentation criterion of the fractional delay domain and the minimax algorithm based on the coefficient relationship structure. This ensures that the peak error of the variable frequency response of the actual variable fractional delay filter does not exceed the error threshold, and finally, the discrete-time impulse response of the parallel FIR sub-filter is obtained. The method specifically includes the following steps: Step 1: The fractional delay domain of the even-order variable fractional delay filter is... , Using the non-equidistant segmentation criterion to p Divide into K segments, where Indicates that the index is The segmented intervals, ,Right now (4) Step 2: Definition denotes the fractional delay definition domain the index of the segment interval of the fractional delay definition domain, i.e. then (5) wherein denotes the floor function; Step 3: Initialize parameters: Segmented range index polynomial order Optimize error ; Step 4: Calculate parameters: , ;in, This indicates the number of sub-filters in the Farrow structure whose polynomial order is an even power. This indicates the number of sub-filters in the Farrow structure whose polynomial order is an odd power; Step 5: Initialize the filter order: (6) in, (7) in, Indicates that the index is The order of the sub-filter whose polynomial order is an even power is... Indicates that the index is The order of the sub-filter is the polynomial order raised to an odd power, and the index is... Represents the first in the Farrow structure Sub-filters; Step 6: Let (8) wherein denotes the element at index plus 1; Step 7: Based on parameters , , The variable frequency response of the actual variable fractional delay filter is: : (9) in, (10) (11) (12) in It is the normalized angular frequency; Here is the filter coefficient matrix. for The row vector; Let be the matrix of real components in the filter response. for The row vector; Let be the imaginary component matrix in the filter response. for The row vector; Step 8: Based on the variable frequency response of the ideal variable fractional delay filter and the variable frequency response of the actual variable fractional delay filter, the variable frequency response error is obtained as follows: : (44) in For an ideal variable frequency response; For variable frequency response error The real part; For variable frequency response error The imaginary part; Step 9: Using the variable frequency response error and the non-equal interval segmentation criterion of the fractional delay domain, the minimax design of the even-order variable fractional delay filter is expressed as follows: (46) wherein is the peak error; (47) In segmented intervals Minimax design was performed internally to obtain the peak error. and the corresponding filter coefficients ; Step 10: Repeat steps 6 through 9, from... Traversal to A total of One peak error; select the smallest peak error from all results. ,Right now (53) with the corresponding update i.e. ; Step 11: Determine Is it less than a given error threshold? If yes, proceed to step 13; otherwise, proceed to step 12; set an error threshold. How to set it up: Set it up according to your actual needs. Step 12: Calculate parameters (54) in The current optimized ratio parameter; exist After the calculation is complete, use the current renew ,Right now Step 3 is documented; determine the current optimization ratio parameter. Is it greater than the given optimization ratio threshold? If yes, return to step 6; otherwise, let Increment by 2, And return to step 4; Step 13: Set the even-power window function Even-power window function Continue to optimize the number of coefficients in adjacent sub-filters; (55) in, The number of 1s in the index indicates the index. The number of sub-filter coefficients whose polynomial order is an even power. The number of 1s in the index indicates the index. The number of sub-filter coefficients whose polynomial order is an odd power; Then and is rewritten as (56) in, ; Step 14: Let , This is the currently optimized even-power window function; Let i be the even-power window function; And order Set the last non-zero element in the array to 0, and use , , Perform the Minimax design in step 9, and remove step 14 from... Traversal to ,get Peak error Select the smallest peak error from all results. ,Right now (57) Step 15: Let , This is the currently optimized odd-power window function; for; And order Set the last non-zero element in the array to 0, and use , , Perform the Minimax design as shown, and move step 15 from... Traversal to ,get Peak error Select the smallest peak error from all results. ,Right now (58) Step 16: Let ;judge Is it greater than the given error threshold? If so , Keep it unchanged and proceed to step 17; otherwise, update according to the following principles. , : (59) (60) Return to step 14; Step 17: For the next segment interval Optimize to make , Return to step 4; transfer steps 4 through 16 from... Traversal to This yields the polynomial order over all segmented intervals. Parallel FIR sub-filter order Window function , and the corresponding sub-filter coefficients , ; Step 18: Through coefficients , The impulse response of the even-order variable fractional delay filter was calculated. : (61) in, (62) To Parameters in the middle; .

2. The non-uniformly spaced segment based modified farrow structure fractional delay design method of claim 1, wherein, The value of K is selected as follows: .

3. The improved farrow structure fractional delay design method based on non-equal interval segmentation as described in claim 1, characterized in that, In step 8, (45)。

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