Analog multi-passband filter design method

Abstract

The application provides a design method of an analog multi-passband filter. It is a hybrid method, which obtains a rough prototype of the analog multi-passband filter, and then fine tunes it until the equal ripple is realized in each passband. The rough prototype constructed by the analytic method contains important information of the analog multi-passband filter, effectively reducing the range of optimization and fine tuning. The method has fast convergence speed, and the performance of the designed analog multi-passband filter is flexible and controllable, including the number of passbands, the performance of each passband, the transmission zero point, and the complex response to meet more stringent technical requirements. Based on the filter polynomial derived by the design method of the analog multi-passband filter, various topological coupling matrices can be obtained for actual circuit implementation.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, specifically relating to a design method for an analog multi-passband filter. Background Technology

[0002] Filters are key components in radar, communication, and measurement systems. Their function is to allow signals of certain frequencies to pass smoothly while significantly suppressing signals of other frequencies. Their performance has a significant impact on the overall system performance. Filter technical specifications include passband bandwidth, insertion loss, passband ripple, return loss, stopband rejection, in-band phase linearity, and group delay. Analog filters are divided into lumped parameter analog filters and distributed parameter analog filters. In higher frequency bands such as radio frequency / microwave / optical frequencies, transmission line structures such as microstrip lines, striplines, slot lines, fin lines, coplanar waveguides, coaxial lines, and waveguides are mainly used. These transmission lines exhibit distributed parameter effects, and their electrical characteristics are closely related to their structural dimensions. In higher frequency bands, transmission line filters such as waveguide filters, coaxial line filters, stripline filters, and microstrip line filters are commonly used. Among these, microstrip filters have advantages such as small size, light weight, wide operating bandwidth, high reliability, and low manufacturing cost, making them a widely used type of transmission line filter. Furthermore, with the rapid development of modern communication, new wireless communication technologies such as WCDMA and WLANs are constantly emerging. These wireless communication technologies are all concentrated in the low-frequency bands of radio frequency and microwave frequencies, making spectrum resources particularly congested and highlighting the increasing importance of multi-band communication. Using analog multi-passband filters in multi-band communication systems can effectively reduce the size of the entire system and the complexity of the overall circuitry, thereby simplifying the system and reducing equipment costs. Therefore, researching analog multi-passband filter design methods is of paramount importance. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings and defects in the existing technology and solve the problems existing in the design method of existing analog multi-passband filters, namely, the difficulty in flexibly controlling the number of passbands, the performance of each passband and the out-of-band transmission zeros. A new analog multi-passband filter design method is proposed, namely the coarse prototype + fine-tuning method, abbreviated as RP-FT method.

[0004] First, according to filter theory, an analog filter can be described by the parameters of a two-port network. Among these, the two scattering parameters S... 21 (s) and S 11 (s) is represented as

[0005]

[0006]

[0007] P(s) is the transmission polynomial, F(s) is the reflection polynomial, and E(s) is the common polynomial, collectively referred to as the filtering polynomial. Since directly designing analog multi-passband filters in the practical multi-passband frequency domain (called the f-domain) is difficult, two frequency transformations are used to transform the technical specifications of the analog multi-passband filter to the normalized multi-passband frequency domain (called the Ω-domain) before design. Therefore, Ω is the frequency variable in the Ω-domain, and s = jΩ is the complex frequency variable. The filtering polynomial is further expressed as...

[0008]

[0009]

[0010] E(jΩ)·E(jΩ) * =P(jΩ)·P(jΩ) * +F(jΩ)·F(jΩ) * (5)

[0011] There are M finite frequency transmission zeros, and the m-th finite frequency transmission zero is denoted as jΩ. m There are N reflection nulls, and the nth reflection null is denoted as jΩ. n ε is the ripple coefficient.

[0012] The characteristic function K(jΩ) is defined as follows:

[0013]

[0014] The RP-FT method described in this invention is a hybrid method that obtains a coarse prototype of an analog multi-passband filter and then fine-tunes it until equal ripple is achieved in each passband. Figure 1 The design flowchart of the RP-FT method is given. It includes seven steps: 1) Through the first frequency transformation, the design specifications are converted from the actual multi-passband frequency domain (called the f-domain) to the normalized multi-passband frequency domain (called the Ω-domain). 2) Through the second frequency transformation, the design specifications of each passband in the Ω-domain are mapped to the normalized single-passband frequency domain (called the Ω*-domain). 3) In the Ω*-domain, based on the design specifications of each passband, the Chebyshev filter synthesis method is used to determine the reflection zeros of each passband and map them back from the Ω*-domain to the Ω-domain. 4) In the Ω-domain, the initial characteristic function of the simulated multi-passband filter is constructed using the specified transmission zeros and reflection zeros. 5) An external transmission zero is introduced and its position is fine-tuned to achieve ripple amplitude difference adjustment. 6) The position of the reflection zero is fine-tuned to achieve equal ripple adjustment. 7) The ripple coefficient ε is determined according to the return loss requirements in the design specifications.

[0015] Step 1: Perform the first frequency conversion

[0016] The design specifications of the analog multi-passband filter to be designed are given in the f-domain: the number of passbands is L, and the order of each passband is Od. l Return loss RL l Lower edge cutoff frequency jf ld Upper edge cutoff frequency jf lu The specified transmission zero point jf for each passband T,lm The subscript *l* indicates that the parameter belongs to the *l*-th passband, where *l* = 1, 2, ..., L. The subscript *m* represents the *m*-th designated transmission zero assigned to the *l*-th passband. In deriving the rough prototype, each passband of the simulated multi-passband filter is initially treated as a single-passband filter. Each designated transmission zero will be assigned to the passband adjacent to it based on its location.

[0017] The first frequency transformation converts the specifications of the analog multi-passband filter in the f-domain to the Ω-domain, i.e.

[0018]

[0019] in, It is the center frequency of the total frequency range covered by all passbands of a simulated multi-passband filter. scale = π(f Lu -f 1d () is the scaling factor. The transformation process is as follows: Figure 2 As shown, it represents the total frequency range [f] in the f-domain. 1d f Lu ] is transformed into [-1,1] in the Ω domain.

[0020] Step 2: Perform a second frequency conversion

[0021] The second frequency transformation maps the passband design specifications in the Ω domain to the Ω* domain. The formula for the second frequency transformation is as follows:

[0022]

[0023] Among them, Ω l0 =(Ω) ld +Ω lu ) / 2 is the center frequency of the l-th passband. It is the scaling factor for the l-th passband. The transformation is as follows: Figure 2 As shown. It represents the frequency range of each passband in the Ω domain [Ω]. ld Ω lu The transformation is to [-1, 1] in the Ω* domain.

[0024] Step 3: Determine the zero point of reflection

[0025] In the Ω* domain, each passband is treated as a single-passband filter. Based on its specified transmission zeros and order, the reflection zeros of each passband are determined using Chebyshev filter synthesis. These reflection zeros are then mapped back to the Ω domain using the second frequency transformation formula. At this point, the specified transmission zeros and reflection zeros of each passband in the Ω domain are known.

[0026] Step 4: Construct the initial characteristic function

[0027] In the Ω domain, multiplying the characteristic functions of each passband yields the initial characteristic function of the analog multi-passband filter.

[0028]

[0029] Where, jΩ R,ln It is the nth reflection zero in the l-th passband, and jΩ T,lm This is the m-th designated transmission zero in the l-th passband. The number of reflection zeros and designated transmission zeros in the l-th passband are N and N, respectively. R,l and M T,l .

[0030] The initial characteristic function K0(jΩ) has a problem: the ripple is not equal in each passband. Two measures are used to achieve the desired ripple performance. First, some external transmission zeros are introduced to adjust the ripple amplitude difference between different passbands. This process is called ripple amplitude difference adjustment. Second, the positions of the reflection zeros within the passbands are adjusted to achieve equal ripple in each passband; this process is called equal ripple adjustment.

[0031] Step 5: Adjust the ripple amplitude difference

[0032] To evaluate the difference in ripple amplitude between passbands, an evaluation index is defined, called the predicted ripple amplitude. The predicted ripple amplitude of the l-th passband is denoted as U. l ,Right now

[0033]

[0034] Each passband of a simulated multi-passband filter may have a different return loss. ΔR l This represents the difference between the return loss of the l-th passband and the maximum return loss in the design specifications. For an analog multi-passband filter with L passbands, the above formula is used to calculate U for each passband. l (l = 1, 2, ..., L). If these U... l The maximum value in is represented by U. max Then the ripple amplitude difference in each passband can be defined as ΔU l ,Right now

[0035] ΔU l =U max-U l (11)

[0036] ΔU l This indicates the ripple amplitude difference that needs to be compensated for in the l-th passband.

[0037] To adjust the ripple amplitude difference, some externally transmitted zeros need to be introduced. The i-th externally transmitted zero is denoted as jΩ. E,i Correcting the characteristic function K E Defined as

[0038]

[0039] This discussion focuses only on ripple amplitude. Using the concept of decibels, the multiplication operation is simplified to an addition operation. The decibel representation of formula (12) is as follows:

[0040]

[0041] In the above formula, the second term on the right side of the equation represents the contribution of the external transmission zero. Figure 3 The response of a single externally applied transmission zero is revealed, namely 20lg(1 / |jΩ-jΩ) E,i The response of |). When Ω is close to Ω E,i At that time, 20lg(1 / |jΩ-jΩ) E,i The function value increases sharply, and the graph marks the function values ​​at some typical points, providing a more intuitive demonstration of the function characteristics. When an external transmission zero is placed near a passband with a large ripple amplitude difference, it will effectively compensate for the ripple amplitude difference between that passband and other passbands. For example, an external transmission zero is placed at a center frequency of jΩ. l0 Near the l-th passband, its compensation for the ripple amplitude difference in the l-th passband is:

[0042]

[0043] To accurately determine the location of the external transmission zero point, optimization and fine-tuning should be performed according to the following criteria.

[0044]

[0045] Among them, U l This represents the estimated ripple amplitude for the l-th passband. After adding an external transmission zero, K0 in the calculation of the estimated ripple amplitude needs to be changed to K. E δ U and E U δ represents the variance and expectation of the predicted ripple amplitude for all passbands, respectively. U When = 0, the position of each external transmission zero can be determined.

[0046] Step 6: Perform equal ripple adjustment

[0047] After adjusting the ripple amplitude difference, the ripple in each passband is still unequal. Therefore, it is necessary to optimize the position of the fine-tuning zero-point of the reflection to achieve equal ripple adjustment. For each passband, a set of frequency points is found, including the lower edge cutoff frequency jΩ of that passband. ld upper edge cutoff frequency point jΩ lu And all ripple maxima within the passband. The p-th ripple maximum in the l-th passband is denoted as jΩ. M,lp It satisfies At all these points |K E (jΩ)|The variance of the function value is δ l , represented as

[0048]

[0049] Among them, E l For these points |K E (jΩ)|Expectation of the function value, P l It is the number of ripple maxima in the l-th passband. The sum of variances is defined as...

[0050]

[0051] When δ = 0, the zero-point position of reflection that achieves equal ripple can be determined.

[0052] After ripple adjustment, the ripple amplitude difference between each passband should be zero, and the ripple within each passband should be equal. If this requirement is not met, one or more rounds of ripple amplitude difference adjustment and equal ripple adjustment can be performed until the technical specifications are met.

[0053] Step 7: Determining the ripple coefficient ε

[0054] At this point, the finite-frequency transmission nulls (including specified transmission nulls and externally added transmission nulls) and reflection nulls have been determined. The complete characteristic function K(jΩ) is defined as follows:

[0055]

[0056] Comparing this with the definition of the characteristic function K(jΩ), the numerator and denominator in the above equation correspond to the reflection polynomial F(jΩ) and the transmission polynomial P(jΩ), respectively. Subsequently, the common polynomial E(jΩ) can be determined. After obtaining these filtering polynomials, the scattering parameter S can be determined. 21 (jΩ) and S 11 (jΩ). Substitute the return loss value at the specified frequency point (e.g., the return loss RL1 at Ω = -1) into the following formula.

[0057]

[0058] The ripple coefficient ε can then be determined. At this point, all the filtering polynomials have been determined.

[0059] Coupled matrix synthesis

[0060] The RP-FT method described in this invention can determine the filtering polynomial based on the technical specifications of an analog multi-passband filter. Using these filtering polynomials, a lateral coupling matrix can be synthesized. Processing the lateral coupling matrix yields various topological coupling matrices for practical circuit implementation. Figure 4 The diagram illustrates how the L-branch folded topology can be used to simulate a multi-passband filter. So, Lo and N represent the source, load, and total order of the simulated multi-passband filter.

[0061] For an analog multi-passband filter with L passbands, the diagonal elements (i.e., self-coupling elements) in its transverse coupling matrix can be divided into L groups based on the proximity of their element values. The transverse coupling matrix can be further divided into L submatrices. Each submatrix consists of self-coupling and input / output coupling. The transverse coupling matrix is ​​shown below.

[0062]

[0063] Assume M 11 and M 22 If two elements have similar values, then the part with gray fill can be extracted and formed into a submatrix.

[0064] The following formula gives the l-th submatrix [M] l The matrix can be transformed into a folded coupling matrix using matrix rotation. It corresponds to Figure 4 A branch in the L-branch folded topology. (The rest of the text appears to be incomplete and possibly contains errors.) By changing the sign of certain off-diagonal elements to alter the coupling property, a submatrix is ​​ultimately obtained. The same process is applied to the other submatrices.

[0065]

[0066] L submatrices are obtained Where l = 1, 2, ..., L. Using these submatrices, the resonant frequency, quality factor, and coupling coefficient can be derived. The formula for calculating the resonant frequency is:

[0067]

[0068] Where n = 1, 2, ... and N l f ln It is the resonant frequency of the nth resonator in the l-th branch. yes The nth diagonal element.

[0069] The formula for calculating relative bandwidth is:

[0070] FBW l =(f Lu -f 1d ) / f l (twenty three)

[0071] Among them, FBW l f represents the relative bandwidth of the l-th branch. l Its center frequency is calculated using the following formula:

[0072]

[0073] exist In the middle, use the non-diagonal element located in the first (or last) row (or column). (or The external quality factor Q of the l-th branch can be obtained from this. lS (or Q) lL ).

[0074]

[0075] use Other non-diagonal elements (i = 1, 2, ..., N) l j = 1, 2, ..., N l (i≠j), the coupling coefficient k between the i-th and j-th resonators in the l-th branch can be calculated. l,ij .

[0076]

[0077] The beneficial effects of the RP-FT method described in this invention are as follows: it is based on a hybrid approach and consists of two main parts: coarse prototype construction and optimization / fine-tuning. The coarse prototype constructed using analytical methods contains important information about the analog multi-passband filter, effectively narrowing the scope of optimization / fine-tuning. This method has a fast convergence speed, and the designed analog multi-passband filter has flexible and controllable performance, including the number of passbands, the performance of each passband, and the transmission zeros, enabling complex responses to meet more stringent technical requirements. Based on the filter polynomial derived from the RP-FT method, various topological coupling matrices can be obtained for practical circuit implementation. Attached Figure Description

[0078] Figure 1 : A flowchart of the RP-FT method described in this invention;

[0079] Figure 2 Schematic diagram of the first and second frequency conversions;

[0080] Figure 3 : 20lg(1 / |jΩ-jΩ E,i The function response graph of |).

[0081] Figure 4 : L-branch folded topology diagram;

[0082] Figure 5 Frequency response diagram of Example 1;

[0083] Figure 6 : 4-branch folded topology diagram;

[0084] Figure 7 : Microstrip structure diagram of a simulated four-passband filter;

[0085] Figure 8 Comparison of the theoretical frequency response obtained in Example 1 and the simulated frequency response in Example 3;

[0086] Figure 9 Comparison of simulated frequency response and test frequency response in Example 3. Detailed Implementation

[0087] To demonstrate the inventiveness and novelty of this invention, the implementation and effects of the technical solution are described in detail below with the aid of embodiments. During the analysis, the description will be combined with the accompanying drawings and specific embodiments, but the implementation of this invention is not limited thereto.

[0088] Without loss of generality, Example 1 is an analog four-passband filter, whose design specifications in the f-domain are: number of passbands L = 4; order of each passband Od l =2, where l = 1, 2, 3 and 4; the return loss of each passband is RL l = -17dB; the center frequencies of the four passbands are: 5.4098GHz, 7.3900GHz, 9.8294GHz, and 11.863GHz; the relative bandwidths of the four passbands are 3.7%, 2.4%, 2.2%, and 1.9%. The three specified transmission zeros are f T,21 =6.2410, f T,31 =8.4721 and f T,41 =10.965.

[0089] According to step 1 of the RP-FT method, the first frequency transformation is performed using formula (7). Where f0 = 8.642 and scale = 20.942.

[0090] Following step 2 of the RP-FT method, a second frequency transformation is performed using formula (8). Where Ω 10 = -0.9700, scale1* =0.0302, Ω 20 = -0.3757, scale2 * =0.0266, Ω 30 =0.3562, scale3 * =0.0323, Ω 40 =0.9663, scale4 * =0.0338.

[0091] According to step 3 of the RP-FT method, the zero point of reflection in the Ω domain is obtained as: Ω R,11 = -0.9911, Ω R,12 = -0.9484, Ω R,21 = -0.3949, Ω R,22 = -0.3574, Ω R,31 =0.3328, Ω R,32 =0.3784, Ω R,41 =0.9414, Ω R,42 =0.9890.

[0092] Based on step 4 of the RP-FT method, the initial characteristic function of the four-passband filter in the Ω domain can be constructed using formula (9).

[0093]

[0094] Following step 5 of the RP-FT method, an external transmission zero is added at the 0 frequency for further optimization and fine-tuning, ultimately reaching -j0.0469. The characteristic function K is then modified. E (jΩ) is

[0095]

[0096] According to step 6 of the RP-FT method, the positions of the reflection zeros in each passband are finely adjusted using the optimization criteria given in formula (17) to achieve equiripple oscillations in each passband. The final reflection zeros are -j0.9910, -j0.9481, -j0.3939, -j0.3563, j0.3329, j0.3786, j0.9422, and j0.9900. The complete characteristic function of the analog four-passband filter can then be obtained as follows:

[0097]

[0098] According to step 7 of the RP-FT method, using formula (19), when RL1=-17dB, ε=46.5797 can be calculated.

[0099] After obtaining the filtering polynomial and scattering parameters of Example 1, its response is as follows: Figure 5 As shown, the four passbands cover the desired frequency range, the return loss of each passband is 17dB, and the oscillation within each passband is equal ripple. The transmission zeros are -j0.7204, -j0.0510, -j0.0469, and j0.6970, respectively.

[0100] Example 2 derives the coupling matrix corresponding to the 4-branch folded topology based on the filtering polynomial derived in Example 1. First, the lateral coupling matrix is ​​constructed as follows.

[0101] 0 -0.1457 0.1394 -0.1343 0.1425 -0.1355 0.1468 0.1197 -0.1342 0 -0.1457 -1.0185 0 0 0 0 0 0 0 0.1457 0.1394 0 1.0167 0 0 0 0 0 0 0.1394 -0.1343 0 0 0.9303 0 0 0 0 0 0.1343 0.1425 0 0 0 -0.9227 0 0 0 0 0.1425 -0.1355 0 0 0 0 0.4135 0 0 0 0.1355 0.1468 0 0 0 0 0 -0.4017 0 0 0.1468 0.1197 0 0 0 0 0 0 0.3382 0 0.1197 -0.1342 0 0 0 0 0 0 0 -0.3102 0.1342 0 0.1457 0.1394 0.1343 0.1425 0.1355 0.1468 0.1197 0.1342 0 (30)

[0102] Divide it into four sub-matrices, namely [M l (where l = 1, 2, 3, and 4). Converting it to a folded topology, we can obtain the following equation:

[0103]

[0104] In these submatrices, Will The value is set to 0. Then, the signs of individual elements in the matrix are transformed to obtain a submatrix. as follows.

[0105]

[0106] Its corresponding 4-branch folded topology is as follows Figure 6 As shown.

[0107] Using parameters f0 = 8.642 and scale = 20.942, substituting them into formula (22), we can obtain... Figure 6 The resonant frequencies of each resonator in the topology: f 11 =5.4037, f 12 =5.4103, f 21 =7.4418, f 22 =7.4695, f 31 =9.9102, f 32 =9.8792, f 41 =11.8920 and f 42 =11.8813. According to formula (24), the center frequencies of each branch can be obtained: f1 = 5.407, f2 = 7.456, f3 = 9.895 and f4 = 11.887. According to formula (23), the relative bandwidths of each branch can be obtained: FBW1 = 1.2328, FBW2 = 0.8941, FBW3 = 0.6737 and FBW4 = 0.5608. Further, according to formula (25), the quality factor of each branch can be obtained: Q1S =19.5310, Q 1L =19.5401, Q 2S =28.2725,Q 2L =28.5029,Q 3S =45.4302,Q 3L =46.1339,Q 4S =47.5813 and Q 4L = 47.6458. According to formula (26), the coupling coefficients of the two resonators on each branch can be obtained: k 112 =0.0590, k 212 =0.0409, k 312 = -0.0254 and k 412 = -0.0242.

[0108] Example 3 corresponds to a 4-branch folded topology, which is a microstrip implementation of a simulated four-passband filter. Figure 7 A microstrip structure for implementing an analog four-passband filter is presented. It consists of four resonators, providing eight resonant frequencies. The fundamental mode f of resonator ① is... 11 The fundamental mode f of resonator ② 12 They are mutually coupled, forming the first passband with a center frequency of 5.4070 GHz. Their higher-order modes f 41 and f 42 They are mutually coupled, forming the fourth passband with a center frequency of 11.8867 GHz. The fundamental mode f of resonator ③... 21 The fundamental mode f of the resonator ④ 22 They are mutually coupled, forming a second passband with a center frequency of 7.4556 GHz. Their higher-order modes f 31 and f 32 The components are coupled together to form a third passband with a center frequency of 9.8946 GHz. Using the quality factor and coupling coefficient derived in Example 2, and employing the method for extracting the quality factor and coupling coefficient of microstrip structures, the initial values ​​of the structural parameters for Example 3 can be determined. After electromagnetic optimization, the final structural parameters can be determined.

[0109] Figure 8 A comparison is given between the theoretical frequency response obtained in Example 1 and the simulated frequency response in Example 3, and the two are in good agreement. Figure 9 The simulated frequency response and the test frequency response of Example 3 are compared and show a very good match.

[0110] The embodiments listed above fully illustrate that the analog multi-passband filter design method, i.e., the RP-FT method, described in this invention, fully combines the advantages of analytical and optimization methods, possessing significant advantages such as flexible and controllable center frequency, bandwidth, and passband ripple, demonstrating significant technological progress. Those skilled in the art will recognize that the embodiments described herein are for the purpose of helping the reader understand the principles of the invention and should be understood as not limiting the scope of protection of the invention to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the essence of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for designing an analog multi-passband filter, characterized in that: Step 1: Through the first frequency transformation, the design specifications are converted from the actual multi-passband frequency domain (called the f domain) to the normalized multi-passband frequency domain (called the Ω domain). Step 2: Through a second frequency transformation, the design specifications of each passband in the Ω domain are mapped to the normalized single passband frequency domain (called the Ω* domain); Step 3: In the Ω* domain, based on the design specifications of each passband, use the Chebyshev filter synthesis method to determine the reflection zeros of each passband and map them from the Ω* domain back to the Ω domain. Step 4: In the Ω domain, construct the initial characteristic function of the analog multipassband filter using the specified transmission zeros and reflection zeros; Step 5: Introduce an external transmission zero point and fine-tune its position to achieve ripple amplitude difference adjustment; Step 6: Fine-tune the position of the zero reflection point to achieve equal ripple adjustment; Step 7: Determine the ripple coefficient ε based on the return loss requirements in the design specifications; To evaluate the difference in ripple amplitude between passbands, an evaluation index is defined as the predicted ripple amplitude; the predicted ripple amplitude of the l-th passband is denoted as U. l ,Right now Each passband of a simulated multi-passband filter may have a different return loss; ∆R l This represents the difference between the return loss of the l-th passband and the maximum return loss in the design specifications; for an analog multi-passband filter with L passbands, the above formula is used to calculate U for each passband. l (l=1, 2, ..., L); if these U l The maximum value in is represented by U. max Then the ripple amplitude difference in each passband can be defined as ΔU l ,Right now ∆U l This indicates the ripple amplitude difference that needs to be compensated in the l-th passband; To adjust the ripple amplitude difference, some externally transmitted zeros need to be introduced. The i-th externally transmitted zero is denoted as jΩ. E,i ; Corrected characteristic function K E Defined as 。 2. The analog multi-passband filter design method according to claim 1, wherein the design specifications of the analog multi-passband filter to be designed are given in the f domain: the number of passbands is L, and the order of each passband is Od. l RL return loss l Lower edge cutoff frequency jf ld Upper edge cutoff frequency jf lu The specified transmission zero point jf for each passband T,lm The subscript l indicates that the parameter belongs to the l-th passband, where l = 1, 2, ..., L; the subscript m represents the m-th designated transmission zero assigned to the l-th passband; in deriving the rough prototype, each passband of the analog multi-passband filter is first treated as a single-passband filter; each designated transmission zero will be assigned to the passband adjacent to it according to its position; the first frequency transformation transforms the analog multi-passband filter specifications in the f-domain to the Ω-domain, i.e. in, f0=(f 1d +f Lu ) / 2, which is the center frequency of the total frequency range covered by all passbands of the analog multi-passband filter; Scale=2π(f Lu -f 1d ) / 2 is the scaling factor; this transformation process; will change the total frequency range [f] in the f domain. 1d f Lu ] is transformed into [-1,1] in the Ω domain.

3. According to the analog multi-passband filter design method of claim 1, the second frequency transformation maps each passband design index in the Ω domain to the Ω* domain, and the formula for the second frequency transformation is: , l=1,2,… L in, Ω l0 =(Ω ld +Ω lu ) / 2 is the center frequency of the l-th passband, scalel*=(Ω lu -Ω ld ) / 2 is the scaling factor for the l-th passband; this transformation will change the frequency range of each passband in the Ω domain [Ω ld Ω lu The transformation is to [-1, 1] in the Ω* domain.

4. The analog multi-passband filter design method according to claim 1, in the Ω* domain, each passband is regarded as a single passband filter; according to its specified transmission zero and order, the reflection zero of each passband is determined by the Chebyshev filter synthesis method; the second frequency transformation formula is used again to map these reflection zeros back to the Ω domain; at this time, the specified transmission zero and reflection zero of each passband in the Ω domain are known.

5. According to the analog multi-passband filter design method of claim 1, in the Ω domain, multiplying the characteristic functions of each passband yields the initial characteristic function of the analog multi-passband filter, i.e. in, jΩ R,ln It is the nth reflection zero in the l-th passband, and jΩ T,lm It is the m-th designated transmission zero in the l-th passband, and the number of reflection zeros and designated transmission zeros in the l-th passband are N respectively. R,l and M T,l .

6. According to the analog multi-passband filter design method of claim 1, after adjusting the ripple amplitude difference, the ripple in each passband may still be unequal, so it is necessary to optimize the position of the fine-tuning reflection zero point to achieve equal ripple adjustment; for each passband, find a set of frequency points, including the lower edge cutoff frequency jΩ of the passband. ld upper edge cutoff frequency point jΩ lu And all ripple maxima within the passband; the p-th ripple maxima in the l-th passband is denoted as jΩ. M,lp It satisfies dK E (jΩ) / djΩ| jΩ=jΩM,lp =0; at all these points |K E (jΩ)|The variance of the function value is δ l , represented as in, E l For these points |K E (jΩ)|Expectation of the function value, P l It is the number of ripple maxima in the l-th passband; the sum of variances is defined as... When δ=0, the zero-point position of reflection that achieves equal ripple can be determined; After ripple adjustment, the ripple amplitude difference between each passband should be zero, and the ripple within each passband should be equal. If this requirement is not met, one or more rounds of ripple amplitude difference adjustment and equal ripple adjustment can be performed until the technical requirements are met.

7. According to the analog multi-passband filter design method of claim 1, after the finite frequency transmission zeros (including specified transmission zeros and external transmission zeros) and reflection zeros are determined, the complete characteristic function K(jΩ) is defined as follows: Comparing this with the definition of the characteristic function K(jΩ), the numerator and denominator in the above equation correspond to the reflection polynomial F(jΩ) and the transmission polynomial P(jΩ), respectively; subsequently, the common polynomial E(jΩ) can be determined; after obtaining these filtering polynomials, the scattering parameter S can be determined. 21 (jΩ) and S 11 (jΩ); Substitute the return loss value at the specified frequency point (e.g., the return loss RL1 at Ω=-1) into the following formula. The ripple coefficient ε can then be determined; at this point, all the filtering polynomials have been determined.

8. The analog multi-passband filter design method according to claim 1 can construct a coupling matrix applicable to an L-branch folded topology; for an analog multi-passband filter with L passbands, in its transverse coupling matrix, the diagonal elements, i.e., the self-coupling elements, can be divided into L groups according to the proximity of their element values; the transverse coupling matrix can be divided into L sub-matrices [M]. l ], each submatrix [M l It consists of self-coupling and input / output coupling; using matrix rotation, it can be transformed into a folded coupling matrix. It corresponds to a branch in the L-branch folded topology; for By changing the sign of certain off-diagonal elements to alter the coupling property, a submatrix is ​​ultimately obtained. Where l = 1, 2, ..., L; using these submatrices, the resonant frequency, quality factor, and coupling coefficient can be derived; the formula for calculating the resonant frequency is: in, n=1, 2, ... and N l ;f ln It is the resonant frequency of the nth resonator in the l-th branch. yes The nth diagonal element; The formula for calculating relative bandwidth is: Among them, FBW l f represents the relative bandwidth of the l-th branch; l Its center frequency is calculated using the following formula: exist In the text, use the non-diagonal element located in the first (or last) row (or column). (or The external quality factor Q of the l-th branch can be obtained. lS (or Q) lL ),Right now ， use Other non-diagonal elements (i=1, 2, ..., N) l j=1, 2, ..., N l (i≠j), the coupling coefficient k between the i-th and j-th resonators in the l-th branch can be calculated. l,ij ,Right now 。

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