Microwave Multi-Level Band-Pass Filter Using Discrete-Time Yule-Walker Method

Transcription

1 Mcrowave Mult-Level Band-Pass Flter Usng Dscrete-Tme Yule-Walker Method Chng-Wen Hsue, Jer-We Hsu, Yen-Jen Chen Department of Electronc Engneerng, Natonal Tawan Unversty of Scence and Technology 43 Keelung Road, Secton 4, Tape, Tawan, R.O.C. Kuo-Lung Chen Natonal Communcatons Commsson 6F No. 5, Sec., Ren-A Rd., Tape, Tawan, R.O.C. Abstract - The modfed Yule-Walker scheme s employed to defne arbtrary transmsson scatterng parameter of a multple-level band-pass flter n the dscrete-tme doman. Equal-length mcrostrps ncludng sngle- and double-secton stubs are used to mplement the multple-level band-pass flter. Expermental results are presented to llustrate the valdty of ths desgn method. Index Terms - Yule-Walker scheme, multple-level flter, Z doman. I. INTRODUCTION Mcrowave flters [-5] are two-port networks used to allow transmsson of sgnals over the pass-band and reject all sgnals over the stop-band. Typcal frequency-doman flters nclude low-pass, hgh-pass, band-pass and band-stop flters. Conventonal methods to desgn and mplement mcrowave flters begn wth lumped-element prototypes. Rchard s transformaton [2] and Kuroda s denttes [3], are then used to transform lumped-element values to the correspondng transmsson-lne crcuts. Butterworth, Chebyshev and ellptcal flters are three popular prototypes used n flter desgn. Each of three has ts dstnct characterstc that meets specfc applcatons. The motvaton of ths work s to explore the desgn of flter havng arbtrary frequency response. In other words, the goal s to develop a flter desgn method for a gven arbtrary frequency-doman response. The modfed Yule-Walker equaton [6-7] s commonly used n statstcs for estmatng the tme doman parameters of an autoregressve process. But there s stll no flter desgn usng modfed Yule-Walker equaton. The modfed Yule-Walker scheme defnes explctly arbtrary responses of flters. In ths letter, we lmt our focus on the multple-level band-pass flter. We begn wth a multple-level band-pass flter havng arbtrary frequency-doman response. The Yule-Walker scheme converts the frequency-doman response nto the Z-doman response [8]. To mplement a flter havng transfer functon n the Z doman, an equal-length

2 Table I Chan-Scatterng-Parameter Matrces of Basc Transmsson lnes Fg : Defned magntude response of two-level band-pass flter and ts Yule-Walker approxmaton. transmsson-lne confguraton s proposed to emulate the multple-level flter. Both numercal and measured results are presented to llustrate the valdty of ths desgn method. II. TWO-LEVEL BAND-PASS FILTER USING MODIFIED YULE-WALKER EQUATION Fg. depcts a two-level band-pass flter wth ampltude responses as m = [,,,,.37,.37,, ] at the normalzed frequences f = [,.33,.375,.4,.43,.56,.57, ]. As shown n Fg., two pass-bands occur at and In partcular, two pass-bands represent a dfference of db n ampltude response. Upon usng the command statement [8], [b,a] = yulewalk (f,m); we obtan the system functon of the band-pass flter n the Z doman, whch s as follows 9 j= 9 F( z) =, az = bz j j () where {b j, b j 9} = {.269, -.97,.27, -.2,.69,.85, -.5,.25, -.52,.5 } and {a, a 9} = {., , 3.543, , , , , -.83,.4962, }. Fg. also shows the ampltude response of equaton (), whch devates slghtly from the orgnal ampltude response. We may select hgher-order values of (,j) to obtan F(z) so that F(z) wll ft better the orgnal ampltude response. However, hgher-order values of (,j) wll lead to a more complex crcut when F(z) s mplemented wth transmsson lnes. Equaton () represents the system functon of arbtrary two-level band-pass flter. Notce that the ampltude response F(z) n Fg. s symmetrc wth respect to Ω = π wth Ω beng the normalzed frequency. F(z) s also a perodc functon wth a perod of 2π. III. EQUAL-LENGTH TRANSMISSION LINES IN THE Z DOAMIN Table I [5] shows the chan-scatterng parameters matrces of basc transmsson lnes,

3 namely, seres lne, shunt-open stub, shunt-short stub, and shunt-open two-secton stub. We assume that each of fnte sectons has the same electrc length wth β l = ωτ, where ω s the angular frequency and τ s the propagaton delay tme of each fnte lne. The dscrete-tme doman response of transmsson-lne confguraton s obtaned by settng z - = exp (-j2β l ), where Z, β, and l (=, 2, a, b, c) are the characterstc mpedance, propagaton constant, and physcal length, respectvely. Notce that Z s the reference characterstc mpedance, whch s assumed to be 5Ω. The chan scatterng matrx of a cascaded network consstng of seres-shunt transmsson lnes s the sequental multplcaton of the chan scatterng matrx of each fundamental element. It s gven by P T T2 T T 2, T2 T = (2) 22 network = T2 T22 where P s number of the fundamental elements, and T, T 2,T 2, T 22 are the matrx elements representng the -th element. If a network conssts of K two-secton open-crcuted stubs, L seres sectons, M short stubs, and N open stubs, the transfer functon T,network (z) of the overall network s gven as follows n equaton (3), where all a are real and are determned by the characterstc mpedances of all seres and shunted components. When the output port of the cascade network s properly termnated, the transfer functon T(z) of such a network becomes equaton (4), where z -L/2 represents tme delay of seres lnes and A = a L 2 ( Γm ) l=, 2K+L+M+N (5) s a functon of the characterstc mpedances of all components nvolved. Both system functon F(z) of two-level band-pass flter and transfer functon T(z) of transmsson-lne network are expressed n terms of Z parameters. In order to mplement a two-level band-pass flter wth a transmsson lne network, we set F( z) = T( z). (6) Notce that A n T(z) s a functon of characterstc mpedances of transmsson lnes. The task s, therefore, to adjust characterstc mpedances of transmsson lnes so that T(z) s as close to F(z) as 2K+ L+ M+ N az =, network ( z) = K L M N 2 L /2 2 + γkz + z z Γl z + z k= l= m= n= T ( 2 ) ( ) ( ) ( ) (3) T( z) = = z T ( z), network K M N 2 (+ 2 γz + z ) ( z ) ( + z ) L /2 k= m= n= 2K+ L+ M+ N Az = (4)

4 Fg 2: Pole-zero locatons of band-pass flter n the z plane. possble n the sense of least square error [5]. IV. IMPLEMENTATION OF FILTER Fg. 2 shows pole-zero locatons of equaton () n the complex z plane. There are four zeros not located on the unt crcle, whch are z =.5353,.239 and.87959± We rearrange equaton () and the system functon of the two-level flter s as follows n equaton (7). Notce that all zeros located on the unt crcle can be realzed wth shunt stubs [4]. On the other hand, zeros not located on the unt crcle cannot be mplemented wth smple transmsson lnes. In order to remove these zeros from the numerator, we dvde both numerator and denomnator of equaton (7) wth the factor [(-.5353z - ) (-.239z - ) (-.7598z - + Fg 3: Pole-zero locatons after the removal of zeros not located on the unt crcle..8882z -2 )]. In addton, to ntroduce a zero at DC, we multply both numerator and denomnator of equaton (7) wth the factor (-z - ). After the algebrac manpulaton, we obtan the system functon of two-level band-pass flter as follows 6 j= 4 F( z) =, az = bz j j (8) where {b j, b j 6} = {.269, -.42, -.,.,.98,.45, -.268} and {a, a 4} = {., -.43, 2.559, -2.36,.942, -.345,.349, , -.778, -.28, -.595, -.287, -.2, -.8, -.84}. Fg.3 depcts the pole-zero locatons of equaton (8) after the removal of zeros not located on the bz ( ) F( z) = = az ( ) 5 j= bz j j (-.5353z )(-.239z )(-.7598z z ) 9 = az (7)

5 Fg 6: Photograph of two-level band-pass flter. Fg. 4: Ampltude responses of F(z) and T(z) for the two-level flter. Fg 5: Confguraton of two-level band-pass flter. unt crcle. Notce that the ampltude response of equaton (8) s the same as that of equaton (7). Fg. 4 shows the ampltude response of equaton (8) when the normalzng frequences s GHz. The modfed system functon of equaton (8) s employed to mplement the two-level band-pass flter by employng equal-length transmsson lnes. Fg. 5 shows the confguraton of transmsson-lne network used to synthesze the flter, of whch K = 2, L = 5, M = 3, and N = 3. There are two two-secton open stubs, namely, (Z 2 (), Z ()) and (Z 2 (2), Z (2)), whch are used to mplement transmsson zeros at 2.43 GHz and 6.6 GHz, respectvely [4]. Three one-secton short stubs and three one-secton open stubs are used to produce transmsson zeros at DC and GHz, respectvely. Upon usng the requrement set by equaton (6) and optmzaton process [5], we obtan the characterstc mpedances of transmsson lnes. The characterstc mpedances of short stubs, from left sde to rght sde n Fg. 5, are 6, , (a) (b) Fg. 7: (a) F(z) and measured S 2 and (b) measured group delay of the band-pass flter n Fg. 6. and 6 Ω. Three open stubs have mpedance profle, from left sde to rght sde, as 6, 6, and Ω. The characterstc mpedances of two two-secton stubs are Z 2 ()=6 Ω, Z ()=24.64 Ω, Z 2 (2)=42.6 Ω, and Z (2)=5.54

6 Ω. In addton, the characterstc mpedances of ffteen seres lnes are 74.83, 25.68, 42.3,., 9.46, 5.3, 22.2, 97.59, 7.49, 6, 38.76, 44.26, 7.55, 43.4, and 35.9 Ω. Fg. 4 shows the smulaton result of frequency response T(z) when the normalzng frequency s GHz,.e., each fnte secton of transmsson lne has a electrcal length of 9 o at the normalzng frequency of GHz. Fg. 6 shows the photograph of two-level band-pass flter, whch s bult on the UL2 substrate wth thckness of 3 ml (.762 mm), relatve delectrc constant of 2.45, and loss tangent of.25. The total length of the flter excludng reference lnes on both szes s 85. mm. Fg. 7(a) shows both measured result and F(z) of the band-pass flter. The nserton losses for band 3.62 GHz- 4.7 GHz and band 4.64 GHz-5.34 GHz are.8 db and db, respectvely. The measured result devates slghtly from postulated two-level flter F(z). Fg. 7(b) depcts the group delay of two-level flter. The group delay s about.2 ns n the transmsson band. V. CONCLUSION A method to desgn the mcrowave flter wth multple-level response was proposed n ths letter. The modfed Yule-Walker scheme was employed to defne arbtrary transmsson scatterng parameter of a mult-level band-pass flter n the dscrete-tme doman. A two-level band-pass flter wth db dfference n ampltude response was mplemented to llustrate the valdty of ths desgn method. REFERENCES [] Davd M. Pozar, Mcrowave Engneerng, New York: Wley (99). [2] P. I. Rchard, Resstor-Transmsson Lne Crcuts, Proc. IRE, Tech 36 (948), [3] K. Kuroda, General Propertes and Synthess of Transmsson-Lne Networks, Mcrowave Flters and Crcuts, A. Matsumoto, Ed, Tech 22 (97). [4] C.-W. Hsue, C.-W. Lng, and W.-T. Hung, Dscrete-tme notch flter and ts applcaton to mcrowave flter, Mcrowave and Optcal Technology Letters, Tech 5 (28), [5] D.-C Chang and C-W Hsue, Wde-Band Equal-Rpple Flters n Nonunform Transmsson Lnes, IEEE Transactons on Mcrowave Theory and Technques, Tech 49 (22), 4-9. [6] Fredlander, B. and B. Porat, The Modfed Yule-Walker Method of ARMA Spectral Estmaton, IEEE Transactons on Aerospace Electronc Systems, AES-2, (984), [7] Mchael Jachan, Gerald Matz, and Franz Hlawatsch, Tme-Frequency- Autoregressve Random Process: Modelng and Fast Parameter Estmaton, Proc. IEEE ICASSP-3, Tech 6 (23), [8] D. Hanselman and B. Lttlefeld, Masterng MATLAB 5. Englewood Clffs, NJ: Prentce Hall (998). ACKNOWLEDGEMENT Ths work was supported by the Natonal Scence Councl, R.O.C., under Grant NSC E--22-MY2.