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60 IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 2, FEBRUARY 1999

Arbitrary Filter Design by UsingNonuniform Transmission Lines

Te-Wen Pan, Ching-Wen Hsue, Senior Member, IEEE, and Jhin-Fang Huang

Abstract We develop new formulations of reflection andtransmission coefficients of nonuniform transmission lines havingunequal reference impedances. By using the

A B C D

transmissionmatrix of a transmission line, we express the reflection andtransmission coefficients of a nonuniform line as polynominalratios in

ZZZ

transforms. These formulations, in conjunction withdigital signal processing (DSP) techniques and a reconstructionmethod, lead to the realization of nonuniform lines which satisfyprescribed scattering characteristics in frequency domain. Someexamples are presented to illustrate the validity of this technique.

Index Terms Filter, nonuniform line.

I. INTRODUCTION

NONUNIFORM transmission lines (NTLs) have beenstudied by many authors for decades in both directscattering and inverse scattering problems [1][5]. As far as

the direct scattering is concerned, both computation efficiency

and computation accuracy become major issues. However,

from the point of view of inverse scattering, the format of

scattering parameters plays an important role in facilitating the

inverse problem. Therefore, we may formulate the scattering

parameters of nonuniform line in various forms to satisfy

specific considerations.

In this letter, we express the reflection and transmission

coefficients of an NTL as polynomial ratios in trans-

forms, which are called autoregressive moving average processand autoregressive process in digital signal processing (DSP)

studies. These formulations, in conjunction with digital filterdesign technique [6] and a reconstruction method [4], allow

us to design NTL filters having arbitrary-amplitude responses

in frequency domain.

II. SCATTERING PARAMETERS OF NONUNIFORM LINES

Fig. 1(a) shows a two-port network having terminal voltages

, and currents , at the respective ports. These phasor

quantities are interrelated through the transmission

matrix [5]. Each terminal voltage and current can be divided

into two signal components traveling in opposite directions,as shown in Fig. 1(b), where the superscript represents

forward (backward) traveling wave, and and are the

reference characteristic impedances on the left and right sides

of nonuniform line, respectively. The reflection coefficient

Manuscript received August 27, 1998. This work was supported by theNational Science Council, R.O.C., under Grant NSC88-2213-E011-053.

The authors are with the Department of Electronic Engineering, NationalTaiwan University of Science and Technology, Taipei, Taiwan, R.O.C. (e-mail: cwh@et.ntust.edu.tw).

Publisher Item Identifier S 1051-8207(99)02658-6.

(a)

(b)

Fig. 1. (a) A two-port network and terminal voltagescurrents and (b) thetraveling-wave representation at terminals.

and transmission coefficient are related to

parameters as follows:

(1a)

and

(1b)

where is the angular frequency. When both sides of transmis-

sion line have the same reference impedance, i.e., ,

, and reduce to those shown in [5]. If we set

and , where is the propagation delay,

is the physical length, and is the propagation constant of

the signal line, we then obtain the matrix [5] of a

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62 IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 2, FEBRUARY 1999

not taken into account in the models of transmission and

reflection coefficients in (4). Generally, the discontinuity effect

affects the filter response greatly. We can compensate the

discontinuity effect by adjusting the physical length of each

subsection line [7].

To verify the theoretical result, we show in Fig. 2 the

measurement result of the nonuniform line, which is obtained

by using HP8510C network analyzer. Because the reference

characteristic impedance of HP8510C is 50 , which is

different from our selected values, the measurement result

should be transformed into a reference system by

using the following relationship:

(5)

where , ,, , and

( ) are the measured scattering parameters.

Both numerical values and measurement results are in goodagreement with the original postulated filter.

IV. CONCLUSION

We have developed the formulations of reflection and trans-

mission coefficients of a nonuniform transmission line with

unequal terminal reference impedances. These formulations,

in conjunction with DSP techniques and a reconstruction

technique, can be employed to design NTL filters having

arbitrary-amplitude responses in frequency domain.

REFERENCES

[1] Y. P. Tang, Z. Li, and S. Y. Tang, Transient analysis of taperedtransmission lines used as transformers for short pulses, IEEE Trans.

Microwave Theory Tech., vol. 43, pp. 25732578, Nov. 1995.[2] P. P. Roberts and G. E. Town, Design of microwave filters by inverse

scattering, IEEE Trans. Microwave Theory Tech., vol. 43, pp. 739743,Apr. 1995.

[3] V. P. Meschanov, I. A. Rasukova, and V. D. Tupikin, Stepped trans-formers on TEM-transmission lines, IEEE Trans. Microwave TheoryTech., vol. 44, pp. 793798, June 1996.

[4] C.-W. Hsue and T.-W. Pan, Reconstruction of nonuniform transmissionlines from time-domain reflectometry, IEEE Trans. Micowave TheoryTech., vol. 45, pp. 3238, Jan. 1997.

[5] D. M. Pozar, Microwave Engineering. New York: Addison-Wesley,1990.

[6] A. V. Oppenheim and R. W. Shafer, Discrete-Time Signal Processing.Englewood Cliffs, NJ: Prentice-Hall, 1989.[7] T. Edwards, Foundations For Microstrip Circuit Design. New York:

Wiley, 1991.