Design of Microwave Filters Using DSPTechnique

Saurabh Shrivastava and Dharmendra Upadhyay.Division of Electronics & Communication Engineering,

Netaji Subhash Institute of Technology,University of Delhi, New Delhi - 110078, India.

E-mail: ur_saurabh2005gyahoo.com

Abstract- In this paper, a novel method for the design of well-known classical functions, i.e. Butterworth, et al [1],[2].microwave filters using digital signal processing technique with However, more sophisticated and general functions likearbitrary frequency response is presented. It is mainly based on Zolotarev have been also explored [3] and specifically, for thethe conversion of the microwave specifications to the digital case of multipassband filters, some limited analyticalspecification, where the digital filter design technique mainly caseo m assban frs , som li anlyticaIIRPNORM algorithm is applied. With the use of digital signal techniques have been proposed [4], although usually directprocessing techniques, the method calculates the poles and zeros optimization methods have been applied for these cases, withfor the analog frequency response that satisfies the given the involved difficulties [5], [6].specifications. By the poles and zeros, the microwave filter can be Now we demonstrate that microwave filters with especiallyrealized using conventional techniques. As an example to demanding magnitude responses can be synthesized by usingdemonstrate the proposed technique, a filter with user-defined the well established and readily available digital filter designspecifications over two independent passbands has been techniques [7]. The proposed methodology rests on theimplemented and this would be tested in microstrip technology. translation of the target specifications from the analog to the

digital domain, where the microwave designer can takeIndex Terms- Digital signal processing technique, arbitrary idigat advaintagero thes sicr atedean cntnus

frequency response, digital filter design, microwave filter. .dev e adigital techniques, wisthut requiing a dedeveloping digital techniques, without requiring a deepunderstanding of the complex mathematics involved. Thedetails of the novel filter design technique will be explained.

I. INTRODUCTION

ILMICROWAVE FILTER DESIGNlviICROWAVE filters with their various responses can This design technique would be achieved by the followingbe fabricated in different forms; examples are band pass or steps as shown in Fig. 1.low-pass filters with non-uniform transmission lines and bandpass or band stop filters with coupled lines. Among Analog Bilinear Digitalconventional procedures for designing microwave filters, r Specifitionseither the image-parameter method or the insertion-loss Specifications ranSfOrmationSmethod provides the configuration of lumped-elementcircuits.Once the values of lumped elements are obtained, Microwave Inverse Bilinear Digital

Richard's transformation is generally used to convert lumped Synthesis lTrnsformation designfilter elements to stubs of the same electrical length. Inaddition, Kuroda's identities can be used to separate the stubs Fig. 1. Proposed Design Techniquewith transmission-line sections of the same electrical lengthas that of the stubs. In recent years, filter design has beenstudied in the z-domain, by replacing the phasor e0j with thedelay operator, and several effective interesting methods have z-planebeen exploited. Actually, when stubs and transmission-linesections are manufactured with the same electrical length andare cascaded orderly, parameters describing the characteristics = Iof the network can also be represented as functions of thedelay operator. Therefore, abundant assortment of well-ldeveloped discrete-domain techniques can be applied to thestudies of such networks. Upto now, the systematic design ofmicrowave filters has been mainly limited to the use of the Fig. 2. Mapping between z-plane and s-plane

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1) Pole/Zero Pot

~~~~~~~~~~~~~~4j ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~,

A I------ ---- -- - --->-s-- ----- - -n- - -- - - -1-- --- -- ------.

AJ~~~~~~~~~~~~~~~~~~~~~~~0

L---------7--------7------ ,.---

0 , , , , ,i 0,2-M --- -- r-- - -- -- T-- - -- -- r-- - -- -- T-- - -- -- T-- - -- - - ------ --- - - - ---n - -- -- ---, ------

4m-3 -- -- -- -L-- -- - -- -'- -- -- - - -- -- - - --- -- - - - -- -- - - - -- -- - -- - ,-- -- - - - --------'----- - --A

=1 > d C ~ ~ ~ -0, 0 ,5

NormalIized frequency(*n rad/sample) Real PortFig.3. Digital Frequency Response using IIRPNORM Algorithm Fig.5. Pole-Zero Plot of Frequency Response

A. From Analog Domain to Digital Domain B. Digital DesignThe desired microwave filter is characterized by a set of There are many sophisticated and continuously developing

analog specifications. From them, the digital design techniques to design a digital filter for the digitalspecifications are obtained by means of a bilinear specifications obtained in the above subsection. Bytransformation [7]. IIRPNORM algorithm, available in MATLAB, which

s=a () minimizes the error function defined as the p-norm of theZ+1 weighted difference between the target filter frequency

27r.fp ~~~~~~response and the designed one, is shown in Fig.3. The valuewhere a 2wfp of p is increased step-by-step so that the resulting designtan (fs) tends to minimize the maximum modulus of the error

function. The algorithm provides some parameters that allowEquation (1) is an algebraic transformnation between the us to control the design process with high flexibility. TheanlgLaplace-transform s-plane and the digital z-transform most relevant ones are the filter order, the weighting vectorpanalog t eaeth nlgfeqec Hz,wt h and the range of values for p. As a result, the poles and zeros

digital frequency (oz [rad /sample], has to be valued at s= jQ, oftedgalitraeobindsshwinFg5where Q = 27X.f and z =e j [7]. The bilinear transformationemploys two parameters: 1) the sampling frequency fs, which C. Digital to Analog conversionmust satisfy the Nyquist criterion (fs .2. fra,being faxthe The bilinear transformation used in Section I1-A has beenmaximum frequency in the specifications with significant also typically employed to transform classical analog filtersspectral content) and 2) the pre-warping frequency fp. which into recursive digital filters. Using this transformation in thecan be fixed to make the digital requirements more relaxed, as opposite way, it will transform the digital filter into theit will be shown later. To demonstrate the proposed design pursued analog filter. This gives rise to the inverse bilinearmethodology, an I Ith order filter is being designed which transformation, as defined in the following equation:would be implemented in micro strip technology. Once theanalog specifications are chosen pass bands from dc to 1GHz

z

a+s (2)and from 4 to 5.5GHz , with return losses around 15dB; and a-sstop bands from 2 to 3GHz with rejection level around 10dB,from 7 to 9GHz with rejection level better than 40dB. With By using the above formula the analog poles and zeros arefs=2.fmax,,,l4GHz, fp, fixed at 2.4GHz to make the digital obtained from the digital poles and zeros.specification more relaxed at the first stop band.

The inverse bilinear transformation has the followingL2 L4 LB LB LI 0 advantageous features:

C C3C C ~~~~~~~~~~1)Mapping Properties.- Inspecting (2), three importantciTT3c T7 c9 T regions can be distinguished as depicted in Fig. 2: the shaded

, T T T T wie eini h -ln smpe otesae wie,~~~~~~~~~~~~~~rgo in th -ln, whl th untcrl ntezpaei

_ ~~~ ~ ~ ~ ~ ~ ~ ~ ~~mpe to th imgnr axi in th s-lae As a _-.Fi.4 adrntoko updeeet osqec,i a esonta asl an tbe dgita

208Itrainl Sypsu nCmuiaiosadIfrainTcnoois(SI 08 4

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filter is transformed into a causal and stable analog filter as -r- dB[S(1,1)1 v dB[S(2,1)Jintended.2) Warping Effect. As it can be easily inferred, the analytical 10 .-..-.-.. .....-..-...-.-..-.-..-..-......-....-..-10relationship that holds between the digital and the analogfrequencies is not linear. This produces the so-called warping Ieffect. However, since the whole design method includes also -- -the bilinear transformation in Section II-A, the total effect 40

-0cancels out completely, providing a distortion-free procedure.

-50 -50

-60 ''' . -60

D. Synthesis of the Microwave Filter-70 .......-70

From the result in Section II-C, it is converted into atransfer function S21 (s) of desired response of analog filter. 0The ladder network of lumped elements that implements the -90, 1 2 _ 4 5 6--900 1 2 3 4 5 6 7 9 9frequency response is shown in Fig.4. The values of the Frequencyelements L, C are easily obtained by using the classicalcontinuous fraction expansion method [8]. The capacitance Fig.6. Simulated Parameter of Lumped-Element Ladder Networkvalues in pF are: C1=1.428, C2=2.389, C5=0.691, C7=0.709,C9=2.25, C11=1.29. The inductance values in nH are:L2=1.079, L4=2.429, L6=6.419, L8=2.381 and L10=1.16. REFERENCEThe microwave filter will be implemented as a stepped- [1] W.K. Chen, The Circuits and Filters Handbook. Orlando, FL;impedance filter in micro strip technology after deriving the CRC, 1995.associated ladder network of lumped elements. However, it is [2] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters,important to stress that other technology (even non-planar) Impedance-Matching Networks, and Coupling Structures.and/or implementation schemes could be used [2]. It can be Norwell, MA: Artech House, 1980.[3] R. Levy, "Generalized rational function approximation in finiteimplement as a stepped impedance filter in two wire intervals using Zolotarev functions," IEEE Trans. Microw. Theorytransmission line. Tech., vol. 18, no. 12, pp. 1052-1064, Dec. 1970.The frequency response of the lumped-element network has [4] G. Macchiarella and S. Tamiazzo, "Design techniques for dual-

been simulated using Zeland IE3D) as shown in Fig.6. As it pass-band filters," IEEE Trans. Microw. Theroy Tech., vol. 53, no.I11, pp. 3265-3271, Nov. 2005.can be seen, it matches with the desired analog response. [5] S. Bila et al, "Direct electromagnetic optimization of microwaveAfter that the microwave filter will be finally implemented as filters," IEEE Microw Mag., vol. 2, no. 1, pp. 46-51, Mar. 2001.a stepped-impedance filter in micro strip technology using a [6] C. Charalambous, "A unified review of optimization," IEEE

Trans. Mi-crow. Theory Tech., vol. 22, no. 3, pp. 289-300, Mar.CUCLAD-250 LX substrate as transmission line alternating 1974.high impedance and low impedance sections [9]. [7] A. Antoniou, Digital Signal Processing. New York: Mcraw-Hill,

2005.[8] R. Schaumann, M. S. Ghausi, and K. R. Laker, Design ofAnalog

III. CONCLUSION Filters.: Passive, Active RC, and Switched Capacitor. EnglewoodCliffs, NJ: Prentice-Hall, 1990.We have simulated a novel design technique to design [9] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines

and Slotlines, 2nd ed. Norwell, MA: Artech House, 1996.microwave filters with arbitrary frequency response based on 'digital signal processing technique. The novel techniqueallows the microwave designer to take immediate advantageof the efficient and sophisticated digital filter design methods.This technique is successfully implemented in two wiretransmission line and stepped impedance filter technology. Itcan also be implemented in other technologies and gives abetter solution to this competitive world.

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