Double Conversion Method for Synthesis of Inverse Filters Shahrokh Saeedi1, Student Member, IEEE, Juseop Lee2, Member, IEEE and Hjalti H. Sigmarsson1, Member, IEEE 1: School of Electrical and Computer Engineering, The University of Oklahoma, Norman, OK 73019, United States 2: College of Informatics, Korea University, Seoul 136-701, Korea Abstract — A novel method for the inverse filter synthesis of the same type from an original filter is presented. The proposed procedure utilizes a double conversion, frequency inversion and swapping the transfer and reflection functions, which are commonly used for lowpass to highpass filter transformation. Due to the use of double conversions, the resulting filter always has the same type as the original filter. Using this procedure, the available general Chebyshev synthesis method can be used to synthesize inverse Chebyshev and elliptic filters. As an illustration, the coupling matrix of a third-order inverse Chebyshev bandpass filter is synthesized using the presented method. Then the original filter and its inverse filter are fabricated using microstrip technology. The measured and simulated filter results are shown to be in good agreement. Index Terms — Elliptic, filter synthesis, filter transformation, generalized Chebyshev, inverse Chebyshev. I. INTRODUCTION These days, filters can be found virtually in any RF and microwave communication system (cell phones), navigation system (GPS), object detection system (radar), TV broadcasting, and test and measurement systems. Until the early 1970s, extraction of lumped electrical elements from the characteristic polynomials that mathematically describe the filter response, were mostly used for any microwave filter design [1]. This conventional design technique was adequate for the required applications and available technologies of that time. However, in the early 1970s since the revolution caused by the usage of satellites in telecommunication systems for the first time, more stringent requirements have been put on the specifications of channel filters in terms of both in-band and out-of-band parameters. These requirements spurred the development of technologies for implementation of microwave components at higher frequencies as well as a new method for filter design called the coupling matrix method [2]. Since then, microwave bandpass and bandstop filters are mainly designed using this method. Due to the capabilities of this method, extensive research has been devoted to filter synthesis. Therefore, varieties of direct and indirect coupling matrix synthesis methods for both bandpass and bandstop filters with maximally-flat, generalized Chebyshev, and elliptic responses have been reported in the literature [3]-[6]. However, coupling matrix synthesis for filters with inverse frequency response has not yet been presented. In this paper a novel method to synthesize a filter with inverse (dual) frequency response of the same type from an original filter response is presented. To this end, an original filter is first synthesized using one of the available methods. Fig. 1. An example of the proposed double conversion method for synthesis of fourth order inverse Chebyshev lowpass prototype. Then, a tandem combination of frequency inversion and swapping S21 and S11 is utilized to generate the filtering function of the inverse filter. This is followed by regular coupling matrix extraction. At the end, an optional direct coupling matrix scaling is performed to re-normalize the response of the resultant filter based on the desired corner frequency definition. This procedure for a fourth-order inverse Chebyshev lowpass filter with minimum stopband attenuation of 25 dB is shown in Fig. 1. The filter is generated from its dual filter i.e. a fourth-order Chebyshev lowpass filter with passband return loss of 25 dB. However, this method is quite general and can be used to synthesize any filter type from its inverse filter. This method can be used to synthesize inverse Chebyshev and elliptic filters only using generalized Chebyshev filter synthesis without requiring the inverse Chebyshev and elliptic filter synthesis. Elliptic filter synthesis requires a tedious calculation involving elliptic integrals and the Jacobian elliptic functions. Furthermore, this method can be used to prescribe equi-ripple stopband levels without optimization or iteration of the location of transmission zeros. In order to demonstrate the proposed method, the coupling matrix of a third-order inverse Chebyshev bandpass filter is generated from a thirdorder Chebyshev bandpass filter. The filters are fabricated using microstrip square open-loop resonators and tested. Overall, the presented double conversion method provides a convenient synthesis tool for realizing complex filtering functions. 978-1-4799-8275-2/15/$31.00 ©2015 IEEE II. SYNTHESIS PROCEDURE The transfer and reflection functions of a lossless two-port network with the general Chebyshev filtering function, can be written using characteristic polynomials P(s), F(s), and E(s) as in [4]: = , ε (1) and = , (2) where P(s) and F(s) contain the filter transmission and reflection zeros, respectively. In addition, E(s) is a Hurwitz polynomial and ε and εr are constant values used to normalize the highest degree coefficients of the polynomials to one. The polynomials can be synthesized using the recursive formulas presented in [3], [4]. For a lossless network, the characteristic polynomials are related to each other using conservation of energy, | | = | | | | ε . = . . . . (3) This relationship can be used to find any of the polynomials when the other two are available. Having the polynomials and the constants, the coupling matrix for the lowpass prototype can be extracted [3], [4]. From general filter and network theory, a normalized highpass filter can be designed from lowpass prototype using frequency inversion 1 . (4) Applying the lowpass to highpass transformation, the order of coefficients in the polynomials is reversed. Therefore, the constants, ε and εr, are required to be recalculated to again normalize the highest degree coefficients of the polynomials to one. Through this transformation, the frequency response of the filter is folded back from infinity to zero and vice versa, as shown in the top-right of the Fig. 1. It was proposed in [7] that the frequency inversion is equivalent to coupling matrix inversion. Therefore, instead of performing frequency inversion to generate a highpass coupling matrix from an original lowpass prototype, one can find the highpass coupling matrix by lowpass coupling matrix inversion. However, it should be mentioned that the rank of the (N+2)×(N+2) coupling matrix for any odd-order filter with symmetric frequency response (like maximally-flat or generalized Chebyshev with symmetric transmission zeros) is (N+1) which is one less than the matrix dimension. Therefore, the coupling matrix is singular and cannot be inverted. In other words, the previously reported method is not general and cannot be used in all cases; although for even-order filters and non-symmetric odd-order filters, this method can expedite the coupling matrix generation. . . − . . . . . . − . . . . Fig. 2. A fifth-order elliptic filter with 45 dB equi-ripple stopband attenuation and 54 dB equi-ripple passband return loss. In the next step, in order to convert the highpass prototype to a lowpass prototype, the new transfer and reflection functions are swapped as proposed in [6] and [8] to design a bandstop filter from an original bandpass filter. Swapping of the transfer and reflection functions, converts the highpass prototype into a new lowpass prototype as its reflection and transfer responses become the new filter transfer and reflection responses, respectively as shown in the bottomright of Fig. 1. Through this synthesis procedure, the return loss characteristic of the original filter in the passband, determines attenuation level of the inverse filter in the stopband. Therefore, the prescribed equi-ripple return loss level of an original Chebyshev filter, for instance, becomes the minimum (equi-ripple) attenuation level of the reverse filter in the stopband. Similarly, the transmission characteristic of the original filter in the stopband determines the return loss of the inverse filter in the passband. For example, the monotonic rejection response of a Chebyshev filter provides a reflection response similar to that of maximally-flat filters for the inverse filter with all reflection zeros at zero frequency. The order of applying the lowpass to highpass transformation and swapping the transfer and reflection functions can be switched without any change in the final response. One of the properties of the proposed method is that the resulting filter is always fully-canonical when the original filter is a Chebyshev filter. Finally, although the resulting filter has a normalized corner frequency, if a new corner frequency is desired, the filter needs to be re-normalized. This can be directly applied to the final coupling matrix using the method described in [9]. Fig. 2 978-1-4799-8275-2/15/$31.00 ©2015 IEEE = . . . Fig. 5. Measured responses of the fabricated filters. . − . − . . . . = . . . . − . − . . . . Fig. 3. The synthesized third-order Chebyshev (RL=23 dB) and inverse Chebyshev filters and their coupling matrices MORG and MINV, respectively. a) b) 21mm 0.27 mm 0.15 mm Fig. 4. Fabricated filters with inset for showing the smallest features. a) Chebyshev, b) inverse Chebyshev. illustrates the application of the proposed method to generate the coupling matrix of a fifth-order, elliptic filter from an original generalized Chebyshev filter. By changing the return loss level of the original filter, a family of elliptic filters with different attenuation levels in the stopband can be obtained. III. FABRICATION AND MEASUREMENTS A third-order inverse Chebyshev filter with equi-ripple attenuation of 23 dB in the stopband was synthesized from an original Chebyshev filter with return loss of 23 dB to demonstrate the practical application of the proposed method. The filters dimensions were then extracted using known coupling mechanisms and implemented in a Rogers Corp. RO3006® microwave substrate using microstrip square openloop resonators. Both filters were designed with a center frequency of 1800 MHz. The bandwidth of the Chebyshev filter was chosen to be 100 MHz. Since the inverse Chebyshev filter has a different corner frequency definition than the Chebyshev, its coupling matrix is scaled with ωc = 2.8 rad/s. This results in two filters with the same 3-dB corner frequency. The synthesized filters along with their coupling matrices are shown in Fig. 3. A photo of the fabricated filters is shown in Fig. 4. The filters were measured using an Agilent N5225A PNA calibrated using an electronic calibration kit. Fig. 5 shows the measured response of the filters. Good agreement between the results is observed. Both filters exhibit an insertion loss of about 1.1 dB considering the fact that measurement includes the connector losses. There is a minor shift in the center frequency of both filters. This shift is also seen in the left transmission zero of the inverse Chebyshev filter. These discrepancies can be attributed to fabrication tolerances, along with the parasitic coupling between resonators one and three in the inverse Chebyshev filter. IV. CONCLUSION This paper presents an elegant method for synthesizing an inverse filter response of the same type from an original filter. The method is based on using lowpass-to-highpass transformation in tandem with swapping the transfer and reflection functions or vice versa. Using only the generalized Chebyshev filter synthesis method, inverse Chebyshev and elliptic filters can be synthesized. To demonstrate the capabilities of the proposed method, the coupling matrix for a third-order order inverse Chebyshev filter was synthesized from an original Chebyshev filter with return loss of 23 dB. Both filters were made and measured. Good agreement between simulated and measured results was achieved. To the authors’ best knowledge, this is the first time 978-1-4799-8275-2/15/$31.00 ©2015 IEEE that this double conversion method has been used to synthesize an inverse filter response. The proposed method can be used to investigate a wide range of new filter responses for which polynomials may not have been considered yet. ACKNOWLEDGMENT This work was supported by the Agency for Defense Development (ADD) Daejeon, Republic of Korea (Contract Number: UD120046FD). REFERENCES [1] G. L. Matthaei, L. Young and E. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures, New York: McGraw-Hill, 1964. [2] A. E. Atia and A. E. Williams, “New Types of Bandpass Filters for Satellite Transponders,” COMSAT Tech. Rev., vol. 1, pp. 2143, Fall 1971. [3] R. J. Cameron, “General Coupling Matrix Synthesis Methods for Chebyshev Filtering Functions,” IEEE Trans. Microwave Theory & Tech, vol. 47, no. 4, pp. 433-442, 1999. [4] R. J. Cameron, “Advanced Coupling Matrix Synthesis Techniques for Microwave Filters,” IEEE Trans. Microwave Theory & Tech, vol. 51, no. 1, pp. 1-10, 2003. [5] S. Amari, U. Rosenberg and J. Bornemann, “Adaptive Synthesis and Design of Resonator Filters With Source/LoadMultiresonator Coupling,” IEEE Trans. Microwave Theory & Tech, vol. 50, no. 8, pp. 1969-1978, 2002. [6] S. Amari and U. Rosenberg, “Direct Synthesis of a New Class of Bandstop Filters,” IEEE Trans. Microwave Theory & Tech, vol. 52, no. 2, pp. 607-616, 2004. [7] E. Corrales, P. d. Paco and O. Menendez, “Direct Coupling Matrix Synthesis of Bandstop Filters,” Progress In Electromagnetics Research Letters, vol. 27, p. 85–91, 2011. [8] R. J. Cameron, “General prototype network synthesis for Microwave filters,” ESA Journal, vol. 6, no. 2, pp. 193-206, 1982. [9] S. Saeedi, J. Lee and H. H. Sigmarsson, “New Technique for Single Layer Substrate-Integrated Evanescent-Mode Cavity Tunable Bandstop Filter Design”; Unpublished, Submitted to IEEE Trans. Microwave Theory & Tech, 2014. 978-1-4799-8275-2/15/$31.00 ©2015 IEEE

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