APPROVED:

Hualiang Zhang, Major Professor Yan Wan, Co-Major Professor Xinrong Li, Committee Member Shengli Fu, Chair of the Department of

Electrical Engineering Dr. Costas Tsatsoulis, Dean of College of

Engineering Mark Wardell, Dean of the Toulouse Graduate

School

SYNTHESIS AND DESIGN OF MICROWAVE FILTERS AND DUPLEXERS

WITH SINGLE AND DUAL BAND RESPONSES

Iman K. Mandal

Thesis Prepared for the Degree of

MASTER OF SCIENCE

UNIVERSITY OF NORTH TEXAS

August 2013

Mandal, Iman K. Synthesis and Design of Microwave Filters and Duplexers with Single

and Dual Band Responses. Master of Science (Electrical Engineering), August 2013, 61 pp., 5

tables, 28 figures, bibliography, 52 titles.

In this thesis the general Chebyshev filter synthesis procedure to generate transfer and

reflection polynomials and coupling matrices were described. Key concepts such as coupled

resonators, non-resonant nodes have been included. This is followed by microwave duplexer

synthesis. Next, a technique to design dual band filter has been described including ways to

achieve desired return loss and rejection levels at specific bands by manipulating the stopbands

and transmission zeros. The concept of dual band filter synthesis has been applied on the

synthesis of microwave duplexer to propose a method to synthesize dual band duplexers.

Finally a numerical procedure using Cauchy method has been described to estimate the filter

and duplexer polynomials from measured responses. The concepts in this thesis can be used to

make microwave filters and duplexers more compact, efficient and cost effective.

Copyright 2013

by

Iman K. Mandal

ii

ACKNOWLEDGEMENTS

I would like to greatly express my gratitude to my advisor, Dr. Hualiang Zhang for

countless academic and professional guidance. He has had enormous patience in guiding me

through a research area which I was completely unacquainted with. I would also like to thank

my co-advisor, Dr. Yan Wan, for her kind help and support through this process.

I want to express my thanks to Dr. Xinrong Li for his support as committee member for

my thesis. I would also like to thank all my professors and friends at my lab which made my

research experience so much fun. I am also highly obliged to Professor G. Macchiarella, Richard

Cameron and Microwave Filter community for providing assistance and advice on numerous

occasions.

Finally, none of what I have done or have been so far would have been possible without

the support of my family, especially my brother Dr. Suman Mandal for being constant source of

inspiration and support.

iii

TABLE OF CONTENTS

ACKNOWLEGEMENTS ..................................................................................................................... iii

TABLE OF CONTENTS....................................................................................................................... iv

LIST OF TABLES ................................................................................................................................ vi

LIST OF FIGURES ............................................................................................................................. vii

CHAPTER 1 SYNTHESIS OF NARROW BAND MICROWAVE FILTERS ................................................ 1

1.1 Introduction........................................................................................................... 1

1.2 Original Contributions ........................................................................................... 2

1.3 Coupled Resonator Filters ..................................................................................... 3

1.4 Filters using Non-Resonant Nodes ........................................................................ 7

1.5 Transfer and Reflection Polynomial Synthesis .................................................... 11

1.6 Evaluation of Coupling Matrix ............................................................................. 12

1.7 Coupling Matrix Reconfiguration ........................................................................ 16

1.8 Example of Synthesis ........................................................................................... 17

CHAPTER 2 SYNTHESIS OF MICROWAVE DUPLEXERS ................................................................... 20

2.1 Introduction......................................................................................................... 20

2.2 General Structure ................................................................................................ 20

2.3 Polynomial Synthesis ........................................................................................... 21

2.4 Example of Synthesis ........................................................................................... 23

CHAPTER 3 SYNTHESIS OF DUAL BAND FILTERS ........................................................................... 27

3.1 Introduction......................................................................................................... 27

3.2 Polynomial Synthesis ........................................................................................... 27

iv

Chapter 4 DESIGN OF DUAL BAND DUPLEXERS ............................................................................ 34

4.1 Introduction......................................................................................................... 34

4.2 Synthesis Procedure ............................................................................................ 34

4.3 Implementation in Cavity Type and Non-Resonating Node Type ....................... 39

CHAPTER 5 MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS ....................................... 45

5.1 Introduction......................................................................................................... 45

5.2 Formulation of Cauchy Method .......................................................................... 45

5.3 Application in General Polynomial Function: ...................................................... 47

5.4 Application in General Microwave Filter Function ............................................. 49

5.5 Application in Microwave Duplexers .................................................................. 51

5.6 Conclusion ........................................................................................................... 53

CHAPTER 6 CONCLUSION AND FUTURE WORK ............................................................................ 54

BIBLIOGRAPHY .............................................................................................................................. 55

v

LIST OF TABLES

Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter. ................................................... 17

Table 2.1 Extracted TX Filter Polynomials ..................................................................................... 25

Table 2.2 Extracted RX Filter Polynomials .................................................................................... 25

Table 2.3 Duplexer Polynomials.................................................................................................... 26

Table 5.1 Low Pass Polynomial Coefficients ................................................................................. 50

vi

LIST OF FIGURES

Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its

network representation. ......................................................................................................... 4

Fig. 1.2 A general coupling arrangement ........................................................................................ 8

Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles

represent capacitance in parallel to frequency invariant susceptance (normalized

resonator); Black lines represent admittance inverters. ...................................................... 10

Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling SLM .

(b) Equivalent circuit of the k th low-pass resonator in the array. ......................................... 13

Fig. 1.5 Series type low pass prototype with inter-resonator couplings. ..................................... 13

Fig. 1.6 2+N canonical coupling matrix for the transversal array. The matrix is symmetric with

respect to the principal diagonal. ......................................................................................... 16

Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain. ............... 18

Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain. ........................... 19

Fig. 1.9 Filter response generated from coupling matrix ............................................................. 19

Fig. 2.1 Duplexer configuration ..................................................................................................... 20

Fig. 2.2 Frequency mapping: 0 1, 2,RX RXf f f= , 2, 1,TX RXB f f= − . ............................................... 21

Fig. 2.3 Synthesized Duplexer polynomial response .................................................................... 24

Fig. 3.1 Effect of narrowing of passbands on return loss levels. .................................................. 30

Fig. 3.2 Narrowing done around upper passband. ....................................................................... 30

Fig. 3.3 Narrowing done around lower passband edges. ............................................................. 31

Fig. 3.4 Effect of additional transmission zero at w = 1.6 ............................................................. 32

vii

Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels

are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines)............... 33

Fig. 4.1 Frequency transformation for dual band duplexers. ....................................................... 35

Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology

with cavity resonators........................................................................................................... 39

Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology

(S21 for RX channel, S31 for TX channel). ............................................................................. 41

Fig. 4.4 Insertion loss at RX channel ( 12S ). .................................................................................... 43

Fig. 4.5 Insertion loss at TX channel ( 13S )...................................................................................... 44

Fig. 4.6 Return loss at Input ( 11S ). ................................................................................................. 44

Fig. 5.1 SNR = 5 dB. ....................................................................................................................... 48

Fig. 5.2 SNR = 20 dB. ..................................................................................................................... 48

Fig. 5.3 SNR = 50 dB. ..................................................................................................................... 49

Fig. 5.4 Example of polynomial estimation for a 6th order filter. ................................................. 51

Fig. 5.5 Exact and estimated polynomial response for duplexers. ............................................... 52

viii

CHAPTER 1

SYNTHESIS OF NARROW BAND MICROWAVE FILTERS

1.1 Introduction

Microwave filters and duplexers are essential components of communication systems.

Frequency spectrum is the most expensive resource among all and to optimize its use filters

and duplexers must be designed to be compact and capable of handling high power at the same

time.

Filters may be classified into categories in several ways. One typical way is to classify

them based on different classes of response functions, defined in terms of the location of the

poles, the insertion-loss function, and the zeros within the passband. The zeros are usually

spaced throughout the passband to give an equiripple or Chebyshev response since this is far

more optimum and superior to the maximally flat or Butterworth response, which is rarely

used. As far as the poles are concerned, the most common type of filter response has all these

poles located at dc or infinity and is often described as an all-pole Chebyshev filter, or simply as

a Chebyshev filter [1]. When one or more poles are introduced into the stopbands at finite

frequencies, the filter is known as a generalized Chebyshev filter or as a pseudo-elliptic filter.

The special case where the maximum number of poles are located at finite frequencies such

that the stopbands have equal rejection level is the well-known elliptic function filter. This is

now rarely used since it has problems in practical realization and is not optimum when specific

stopbands are required—one seldom needs rejection up to infinite frequency. It is almost

always better to place the poles where they are most needed, and also to minimize their

1

number, since each additional finite frequency pole may increase the implementation

complexity and expense.

The above discussion relates equally to the main categories of filters defined in terms of

the general response types of low-pass, bandpass, high-pass, and bandstop.

In general, pseudoelliptic filters (or generalized Chebyshev filters) as the most useful

and powerful filter types are best designed using exact synthesis techniques. Several

techniques were developed to reduce computational complexities [2]. The typical procedure is

to synthesize a low-pass prototype, which is then resonated to form a bandpass filter. The

categories considered are combline, interdigital, parallel-coupled-line bandpass and bandstop,

ring and patch filters [3], and stepped-impedance filters [4]. The several media for

implementation include waveguide, dielectric resonators, coaxial lines, evanescent-mode

filters, and various printed circuit filters using microstrip, stripline, and suspended substrate.

Also frequency tuning is another very important aspect of filter designs [5].

In this chapter the synthesis technique for cross coupled resonator bandpass filters

exhibiting pseudo-elliptic filter response and the concept of non-resonant nodes has been

described, which also lays foundation for the discussion in the next a few chapters. A history of

early filter researches can be found in [6], [7]. A large number of filter concepts discussed in this

thesis can be found in great detail in the classic book on microwave filters by Matthaei, Young

and Jones [8]. Also extensive details on the newer filter synthesis techniques can be found in

[9].

1.2 Original Contributions

There are two significant contributions in this thesis:

2

• Comprehensive study on synthesis of dual band filters, control of return loss

levels using stopband modifications and additional transmission zeros.

• Proposing a method for the synthesis of dual band duplexers, which is based on

duplexer synthesis and dual band filter synthesis techniques.

1.3 Coupled Resonator Filters

Extensive literature is available on the theory of coupled resonator filters [10], [11], [12],

[13], [14], [15], [16], [17], [15], [18], [19], [20]. The equivalent circuit of an n coupled resonator

filter network is shown in Fig. 1.1. The loop equations can be written as:

1 1 1 12 2 11

21 1 2 2 22

1 1 2 2

1

1 0

1 0

n n s

n n

n n n n nn

R jwL i jwL i jwL i ejwC

jwL i jwL i jwL ijwC

jwL i jwL i R jwL ijwC

+ + − − =

− + + − =

− − + + + =

(1.1)

where ij jiL L= represents the mutual inductance between resonators i and j , and all

the loop currents are supposed to have the same direction as shown in Fig. 1.1.

3

Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its

network representation.

In matrix form, the set of equations can be represented as:

1 1 12 11

1

21 2 2 22

1 2

1

10

01

n

s

n

n

n n n nn

R jwL jwL jwLjwC

i ejwL jwL jwL i

jwC

ijwL jwL R jwL

jwC

+ + − − − + − = − − + +

(1.2)

where the n n× matrix is the impedance matrix [ ]Z . For simplicity, let us first consider a

synchronously tuned filter. In this case, all resonators resonate at the same frequency, namely

the midband frequency of filter 0 1w LC= , where 1 2 nL L L L= = = = and

1 2 nC C C C= = = = . The impedance matrix may be expressed by

[ ] 0Z w L FBW Z = ⋅ ⋅ (1.3)

4

where FBW w w= ∆ is the fractional bandwidth and Z is the normalized impedance matrix

which, for synchronously tuned filter is given by:

11 12

0 0 0

221

0 0

1 2

0 0 0

1 1

1 1

1 1

n

n

n n n

LR Lw wp j jw L FBW w L FBW w L FBW

LLw wj p jw L FBW w L FBWZ

L L Rw wj j pw L FBW w L FBW w L FBW

+ − ⋅ − ⋅ − ⋅ − ⋅ = − ⋅ − ⋅ + ⋅

(1.4)

with

0

0

1 wwp jFBW w w

= −

(1.5)

as the complex lowpass frequency variable. Also,

0

1i

ei

Rw L Q

= (1.6)

1eQ and enQ are the external quality factors of the input and output resonators,

respectively. The coupling coefficients are defined as:

ij

ij

LM

L= (1.7)

The normalized coupling coefficients are given by:

ij

ij

Mm

FBW= (1.8)

In case of an asynchronously tuned filter the resonant frequency of each resonator is

different and may be given by 0 1i i iw L C= , the coupling coefficient of asynchronously tuned

filter is defined as:

5

ijij

i j

LM

L L= (1.9)

From the above relations the normalized Z can be derived as:

11 12 11

21 22 2

1 2

1

1

ne

n

n n nnen

p jm jm jmq

jm p jm jmZ

jm jm p jmq

+ − − −

− − − =

− − + −

(1.10)

Each iim account for asynchronous tuning for each resonator, i.e., frequency shifts from

the center frequency. Then the scattering parameters are obtained as:

[ ]

[ ]

121 1

1

111 11

1

12

21

ne en

e

S Aq q

S Aq

−

−

= ⋅⋅

= ± − ⋅

(1.11)

with

[ ] [ ] [ ] [ ]A q p U j m= + −

where [ ]U is the n n× identity matrix, [ ]q is n n× matrix with all elements zero except the

11 11 eq q= and 1nn enq q= , [ ]m is the general n n× coupling matrix. The 2n + coupling

matrix also includes the external couplings from source and load to each resonator. It is very

easy to transform one form of coupling matrix to another [12], [21], [22]. Till now only inductive

coupling has been considered. Coupling can be capacitive too, and the corresponding coupling

coefficients are called electrical coupling coefficients detailed in [11].

6

1.4 Filters using Non-Resonant Nodes

Using non-resonant nodes with admittance inverters provides a useful approach to filter

design, which is often convenient for microstrip implementation. Several literatures are

available on non-resonant nodes [23], [24], [25], [26], [27], [28]. Overall, four types of

components are used in the low pass prototype in this approach.

1) Resonators: These are represented by unit capacitors in parallel with the

frequency-invariant reactances ijb which account for the frequency shifts in

their resonant frequencies from the center frequency.

2) Admittance Inverters iJ : These are identical to the coupling coefficients

between the nodes.

3) Non-resonating Nodes: These are internal nodes connected to ground by

frequency-invariant reactance ijB . These are not in parallel with any capacitor.

4) Input (source) and Output (load): These are normalized conductances,

1S LG G= = .

A resonator that is responsible for an attenuation pole at a normalized frequency

i is jw= is represented by a unit capacitor in parallel with a constant reactance i ijb jw= − .

Such a dangling resonator is only connected to an NRN. For a filter of order N with zN

attenuation poles at finite real frequencies, there are zN number of dangling resonators and N

- zN resonators along the inline path between the input and the output. When zN N< the

choice of arrangement is not unique, but flexible and for the designer to decide.

7

The general topology of a filter with non-resonating node is shown in Fig. 1.2. There are

three kinds of possible couplings here: Resonant-Resonant ( 34J ), Resonant-Non Resonant ( 13J )

and Nonresonant-Nonresonant ( 12J ). A suitable topology is shown in Fig. 1.3.

Fig. 1.2 A general coupling arrangement

The generalized coupling coefficient is defined as:

,,

i ji j

i j

Jk

B B=

⋅ (1.12)

where

,

,

non-resonant susceptance

resonant susceptance NR i

ieq i

BB

B

=

(1.13)

where ,NR iB is the non-resonant susceptance and eq,iB is the resonant susceptance. eq,iB is

defined as

0

,,

12

ris ieq i

w w

BB

w=

∂=

∂ (1.14)

8

( ),ris iB w represents the total susceptance of the i -th resonator. In case of coupling

with the external loads, a generalized external Q is similarly defined:

, 20, 0

iEXT i

i

BQ

J G= (1.15)

where iB is still given by (1.13) and 0G is the external conductance. The parameters k and EXTQ

have the same dimensions. The generalized coupling coefficients are tabulated in [29]. Let ,i jM

be the admittance inverter parameters, kb the frequency-invariant susceptances, and kc be

the capacitances of the filter prototype. Then a resonant node is defined by the parameters

( ),k kb c and NRN is associated with ( ),0kb . Assuming 0nB B f= is the filter fractional

bandwidth, the novel generalized parameters are evaluated as follows:

• Resonant-resonant coupling:

,, ,

i ji j n

i j

Mk B

c c= (1.16)

• Resonant-nonresonant coupling:

,,

i ji j n

i j

Mk B

c b=

⋅ (1.17)

• Nonresonant-nonresonant Coupling:

,

,i j

i j

i j

Mk

b b=

⋅ (1.18)

,EXT iQ can be evaluated as:

9

, 20,

i nEXT i

i

c BQ

M= , node i resonant (1.19)

, 20,

iEXT i

i

bQ

M= , node i nonresonant (1.20)

Also, the resonant frequencies kf and sign of susceptance kb of NRN are related by:

2

0 42

n k n kk

k k

f B b B bf

c c

⋅ ⋅ = − + +

(1.21)

Once these are obtained, the microstrip implementation can be done with procedure

outlined in [30].

Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles

represent capacitance in parallel to frequency invariant susceptance (normalized resonator); Black lines represent admittance inverters.

Synthesis procedures involving non-resonating nodes were found in several literatures

such as [31], [32], [33].

10

1.5 Transfer and Reflection Polynomial Synthesis

Synthesis of these polynomials is outlined in [9], [34], [35]. For any two-port lossless

filter network composed of a series of N intercoupled resonators, the transfer and reflection

functions (scattering parameters) (definitions can be found in [36]) may be expressed as a ratio

of two thN degree polynomials:

( ) ( )( )11

N R

N

F wS w

E wε

= (1.22)

( )21( )

( )N

N

P wS wE w

ε= (1.23)

where w is the real frequency variable related to the more familiar complex frequency variable

s by s jw= . For a Chebyshev Filtering Function, ε is a constant normalizing 21( )S w to the

equiripple level at 1w = ± as follows:

( )( )/10

1

110 1

N

RLN w

P wF w

ε=

= ⋅−

(1.24)

where RL is the prescribed return loss level in decibels and is assumed that all the

polynomials have been normalized such that their highest degree coefficients are unity. 1Rε =

or 2 1

Rεεε

=−

if the function is fully canonical. ( )11S w and ( )21S w share a common

denominator ( )NE w and the polynomial ( )NP w contains fzn transfer function finite position

transmission zeros.

Using the law of energy conservation for a lossless network, 2 211 21 1S S+ = , along with

(1.22) and (1.23) we have,

11

( ) ( ) ( )( ) ( )( )221 2 2

1 11 1 1N N N

S wC w j C w j C wε ε ε

= =+ + −

(1.25)

where

( ) ( )( )

NN

N

F wC w

P w= (1.26)

( )NC w is known as filtering function of degree N and has a form for the general Chebyshev

characteristic:

( ) ( )1

1cosh cosh

N

N nn

C w x−

=

= ∑ (1.27)

where

1

1n

nn

w wxw w−

=−

and n njw s= is the position of the fzn number of finite position transmission zeros in the

complex s plane and the remaining fzN n− transmission zeros at w = ±∞ . For a prescribed set

of TZs that make up the polynomial ( )P w and a given equiripple return loss level, the reflection

numerator polynomial ( )F w may be built using efficient recursive technique [1] and then ( )E w

may be found using the Conservation of Energy principle for lossless networks.

1.6 Evaluation of Coupling Matrix

The second step of the synthesis procedure is to calculate the values of coupling

elements of a canonical coupling matrix from the transfer and reflection polynomials. The

coupling matrix is a very special matrix that is extremely common in the literature on

microwave filters. A coupling matrix, all by itself, can characterize a low pass prototype filter

12

network. Also, a coupling matrix can be modified using similarity transform, which is a purely

mathematical technique, to obtain different configurations which are easy to realize with a

practical circuit.

Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling SLM .

(b) Equivalent circuit of the k th low-pass resonator in the array.

Fig. 1.5 Series type low pass prototype with inter-resonator couplings.

13

A general coupling matrix is called transversal coupling matrix, which is shown in Fig.

1.4. The transversal coupling matrix comprises a series of N individual first-degree low pass

sections, connected in parallel between the source and load terminations but not to each

other. The direct source load coupling inverter SLM is included to allow fully canonical transfer

functions to be realized (according to the minimum path rule, i.e., maxfzn , the maximum number

of finite position transmission zeros that may be realized by the network = minN n− where minn

is the number of resonator nodes in the shortest route through the couplings in the network

between the source and load terminations). In a fully canonical network, min 0n = and so

maxfzn N= , which is the degree of the network.

Each of the N low-pass sections comprises one parallel-connected capacitor kC and one

frequency invariant susceptance kB , connected through admittance inverters of characteristic

admittances SkM and LkM to the source and load terminations, respectively (the values of all

these parameters will be extracted through the synthesis procedure). The circuit of the k th low-

pass section is shown in Fig. 1.4(b).

Now the admittance parameter matrix [ ]NY is derived in two ways. One is from the

scattering parameters and the other is from the circuit elements of the transversal array

network. By comparing them, elements of the coupling matrix can be derived in terms of the

coefficients of the ( )11S w and ( )21S w polynomials.

14

From the derived coefficients the eigenvalues kλ and the associated residues 22kr and

21kr for 1,2,..., Nk = can be calculated using partial fraction expansion. Thus the following

expression for [ ]NY is obtained:

[ ] ( )11 12

1 21 22

0 10

Nk k

Nk k kk

r rKY j

r rK s jλ∞

=∞

= + ⋅ −

∑ (1.28)

Now using ABCD matrices, converting the elements of low-pass resonator prototypes to

individual y - parameter matrices, and then adding them together to form the complete [ ]NY

matrix the second expression is obtained:

[ ] ( )2

21

0 10

NSL Sk Sk Lk

NkSL k k Sk Lk Lk

M M M MY j

M sC jB M M M=

= + ⋅ +

∑ (1.29)

This leads to the following relations:

1kC = , k kk kB M λ= = − , SLM K∞= , 222Lk kM r= and 21Sk Lk kM M r=

Therefore,

2121

22

, , 1, 2,...,kLk k Sk

k

rM r M k Nr

= = = (1.30)

The capacitors kC are all unity and the frequency-invariant susceptances kB ( kλ= − ,

representing the self-couplings 11 NNM M→ ), the input couplings SkM , the output couplings LkM

, and the direct source-load couplings SLM are all defined, thus completing the reciprocal 2N +

transversal coupling matrix M representing the network. With this coupling matrix, SkM are

the N input couplings and they occupy the first row and column of the matrix from positions 1

15

to N . Similarly, LkM are the N output couplings and they occupy the last row and column of

M . All other entries are zero. The resulting coupling matrix is illustrated in Fig. 1.6.

Fig. 1.6 2+N canonical coupling matrix for the transversal array. The matrix is symmetric with respect to the principal diagonal.

1.7 Coupling Matrix Reconfiguration

Once the coupling matrix is obtained, series of similarity transformations can be applied

to it to obtain different filter topologies, without affecting the filter response. This is extremely

useful because this allows filter designers to conveniently change the topology to fit with

practical realization. In [35], [37], several configurations such as folded form, arrow canonical

form, wheel form and cul-de-sac form and ways to convert from one form to another have

been explained.

16

1.8 Example of Synthesis

To demonstrate, an 8th order bandpass filter with cutoff frequencies at 885MHz and

934MHz is synthesized. The synthesized filter polynomials (normalized to the unit frequency)

are listed in Table 1.1 with decreasing order of frequency.

Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter.

ε ( )NF w ( )NP w ( )NE w 5.5793 1.0000

-2.5508

0.8225

2.6956

-2.2980

-0.1900

0.6775

-0.1495

-0.0073

1.0000

-4.6509

8.0518

-6.1556

1.7547

1.0000

-2.5508 + 1.9856i

-1.1489 - 5.1480i

7.8891 + 1.2723i

-5.1496 + 6.8045i

-3.1443 - 6.0988i

4.1799 + 0.2094i

-1.0236 + 1.2853i

-0.0517 - 0.3103i

The polynomial response is shown in Fig. 1.7 (note: this is the response at the

normalized low-pass frequency range). The response in bandpass real frequency is obtained by

the following transformation:

( )2 204

2wB wB f

f+ +

= (1.31)

17

where f is the real frequency and B is the bandwidth. The polynomial response in real

frequency is shown in Fig. 1.8. Also the filter response generated from synthesized coupling

matrix is shown in Fig. 1.9.

Correspondingly, the 2N + coupling matrix is obtained as the following:

0 -0.3546 0.5022 -0.4767 0.3871 -0.2372 0.0858 -0.2952 0.2949 0-0.3546 1.2494 0 0 0 0 0 0 0 0.35460.5022 0 0.8472 0 0 0 0 0 0 0.5022-0.4767 0 0 0.0426 0 0 0 0 0 0.47670.3871 0 0 0 -0.6102 0 0 0 0 0.3871-0.2372 0 0 0 0 -0.9184 0 0 0 0.23720.0858 0 0 0 0 0 -0.9966 0 0 0.0858-0.2952 0 0 0 0 0 0 -1.0819 0 0.29520.2949 0 0 0 0 0 0 0 -1.0830 0.2949

0 0 0 0 0 0 0 0 0 0

Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain.

-3 -2 -1 0 1 2 3-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency

Ref

lect

ion,

Rej

ectio

n Lo

ss

S21S11

18

Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain.

Fig. 1.9 Filter response generated from coupling matrix

8 8.5 9 9.5 10x 108

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency

Ref

lect

ion,

Rej

ectio

n Lo

ss

S21S11

-3 -2 -1 0 1 2 3-120

-100

-80

-60

-40

-20

0

Frequency

Ref

lect

ion,

Rej

ectio

n Lo

ss

S21S11

19

CHAPTER 2

SYNTHESIS OF MICROWAVE DUPLEXERS

2.1 Introduction

In addition to microwave filters, microwave duplexers are indispensable microwave

components for communication systems. They are typically used to connect the RX and TX

filters of a transceiver to a single antenna through a suitable three-port junction. The rapid

development of mobile communication systems over the past decade has stimulated the need

for duplexers with compact size as well as high selectivity. To meet these stringent

requirements, different synthesis techniques have been proposed. Specifically a very

convenient synthesis method was discussed in [38]. Here the general procedure of it is

described.

2.2 General Structure

A common duplexer configuration is shown below. A TX and an RX filter are connected

via a transformer with a turn ratio of :1n and a susceptance 0b .

Fig. 2.1 Duplexer configuration

20

2.3 Polynomial Synthesis

A duplexer is a lossless three port network and four polynomials are required to define

its scattering parameters in the low pass normalized domain.

( )( )

( )( )

( )( )

0 0 011 21 31, ,t t r rn N s p P s p P s

S S SD s D s D s

= = = (2.1)

In (2.1), the highest degree coefficients of , , ,t rN D P P are imposed to be equal to 1

together with constants 0 0, ,o t rn p p respectively. The roots of ( )D s represent poles of the

network and roots of ( )N s represent the transmission zeros in the complex plane.

The synthesis of the duplexer is carried out in a normalized frequency domain defined

by the usual low pass to band pass frequency transformation ( )( )0 00f B f f f fΩ = − .

Fig. 2.2 Frequency mapping: 0 1, 2,RX RXf f f= , 2, 1,TX RXB f f= − .

The frequency mapping and definitions of 0f and B are shown in Fig. 2.2. The passband

limits of RX filter are 1,RXf and 2,RXf while those of the TX filter are represented by 1,TXf , 2,TXf

respectively. The two low pass prototype RX and TX filters are characterized through their

characteristic polynomials which are related to their scattering parameters, which are given in

(2.2).

21

( )( )( )( )

( )( )

( )( )( )( )

( )( )

11

021

11

021

TXTX

TX

TX TX TXnTX

TX TX

RXRX

RX

RX RX RXnRX

RX RX

F sS

E s

P s p P sS

E s E s

F sS

E s

P s p P sS

E s E s

=

= =

=

= =

(2.2)

In (2.2), the polynomials TXF and TXE have degree TXnp (order of TX filter) and RXF ,

RXE have degree RXnp (order of RX filter). The highest degree coefficients of these

polynomials are equal to 1. The polynomials TXP and RXP have the highest degree coefficients

given by 0TXp and 0RXp which determine the return loss level at passband limits. The TX and RX

transmission zeros then define the normalized polynomials TXnP and RXnP .

By analyzing the three port network and computing the admittances,

2 2 0y TX RX TX RX RX TXin

y TX RX

N jb S S D D D Sy n nD S S

+ += = (2.3)

where , , ,TX RX TX RXS S D D are given by

2

2

2

2

TX TXTX

RX RXRX

TX TXTX

RX RXRX

E FS

E FS

E FD

E FD

+=

+=

−=

−=

(2.4)

Finally we get the following expressions:

22

( )

( )

( )

( )

22

0

20

0 20

22

0

0 0 20

0 0 20

1

11

1

,1

,1

TX RX TX RXTX RX

TX RX TX RXTX RX

t TXn RX t TX

r RXn TX r RX

D D S SN s S S njn b

jn bnjn b

D D S SD s S S njn b

nP s P S p pjn b

nP s P S p pjn b

+= − − −

=+

+= +

+

= = +

= = +

(2.5)

Using the transmission zeros of TX and RX filters and the reflection zeros at the input

port of the duplexer, the duplexer characteristic polynomials are generated. Then using an

iterative method by using lossless conditions, the polynomials TXP , RXP and also their

coefficients 0TXp and 0RXp are obtained. In this iterative method the goal is to satisfy lossless

condition preserving real transmission zeros and reflection zeros. Once these polynomials are

obtained, the TX and RX filters polynomials are extracted using final values of TXS , RXS , TXD ,

RXD and polynomial fitting.

Once the characteristic polynomials are obtained the individual TX and RX filters and the

junction can be realized using Waveguides, Cavity resonators or Non-Resonant nodes. For the

cavity type design, coupling matrices need to be obtained. While for the non-resonant node

realization, admittance parameters and dangling resonator frequencies are to be determined.

2.4 Example of Synthesis

A duplexer with RX passband from 880MHz to 915MHz and TX passband from 925MHz

to 960MHz is synthesized. RX Filter is of order seven and TX Filter is of order eight. RX finite

23

transmission zeros are taken at 850, 925 and 913 MHz while TX finite transmission zeros have

been selected at 898, 911.4, 915 and 969MHz. The response of the synthesized duplexer

polynomial is shown in Fig. 2.3. Synthesized duplexer polynomials are listed in Table 2.3. Also,

the extracted TX and RX Filter polynomials are given in Table 2.1 and Table 2.2.

Fig. 2.3 Synthesized Duplexer polynomial response

700 750 800 850 900 950 1000 1050 1100-160

-140

-120

-100

-80

-60

-40

-20

0

20

Frequency

Sca

tterin

g P

aram

eter

s in

dB

S11

S21

S31

24

Table 2.1 Extracted TX Filter Polynomials

0p ( )F s ( )E s ( )P s

2.5729e-004 1.0000

0.7476 - 4.4213i

-7.8915 - 2.8652i

-4.3972 + 7.2898i

3.7094 + 3.4598i

1.4865 - 1.0193i

-0.1355 - 0.3441i

-0.0394 + 0.0059i

-0.0001 + 0.0017i

1.0000

0.1086 - 4.4213i

-8.1651 - 0.4299i

-0.6960 + 8.1757i

4.8206 + 0.5930i

0.2845 - 1.7009i

-0.3481 - 0.0760i

-0.0103 + 0.0376i

0.0016 + 0.0005i

1.0000

0 - 0.3826i

0.8315

0 + 0.2068i

-0.0130

Table 2.2 Extracted RX Filter Polynomials

0p ( )F s ( )E s ( )P s

7.3187e-004 1.0000

0.1186 + 3.6918i

-5.4723 + 0.3845i

-0.4868 - 4.1685i

1.7339 - 0.3034i

0.0960 + 0.3859i

-0.0416 + 0.0141i

-0.0007 - 0.0017i

1.0000

0.7922 + 3.6918i

-5.1655 + 2.4742i

-2.9047 - 3.3865i

1.0221 - 1.5925i

0.4132 + 0.1065i

0.0038 + 0.0459i

-0.0017 + 0.0007i

1.0000

0 + 1.3574i

0.7503

0 - 0.0776i

25

Table 2.3 Duplexer Polynomials

( )N s ( )D s ( )tP s ( )rP s

1.0000

-1.5000 + 0.7267i

2.6305 - 1.0900i

-3.9458 + 1.8946i

2.5074 - 2.8418i

-3.7611 + 1.7976i

1.0509 - 2.6964i

-1.5763 + 0.7615i

0.1869 - 1.1422i

-0.2804 + 0.1442i

0.0113 - 0.2163i

-0.0169 + 0.0112i

0.0001 - 0.0169i

-0.0001 + 0.0003i

-0.0000 - 0.0005i

0.0000 + 0.0000i

-0.0000 - 0.0000i

1.0000

3.2670 + 0.7267i

6.8423 + 2.3874i

11.1661 + 4.9989i

13.1200 + 8.1490i

13.2342 + 9.5834i

10.0267 + 9.6647i

6.4497 + 7.3658i

3.0628 + 4.7883i

1.1545 + 2.3448i

0.2804 + 0.9428i

0.0391 + 0.2720i

-0.0025 + 0.0611i

-0.0019 + 0.0090i

-0.0004 + 0.0010i

-0.0000 + 0.0001i

-0.0000 + 0.0000i

1.0000

0.4554 - 3.3091i

-3.0748 - 1.2551i

-0.7701 - 1.5344i

-5.2668 - 0.9836i

-1.8198 + 4.5704i

2.0718 + 1.2250i

0.4402 - 0.5458i

-0.0848 - 0.0905i

-0.0105 + 0.0075i

0.0003 + 0.0006i

0.0000 - 0.0000i

1.0000

0.4281 + 3.0639i

-1.2767 + 1.0664i

0.0110 + 6.5595i

-12.5978 + 2.6996i

-3.9035 -10.8536i

5.4041 - 2.7098i

1.0818 + 1.6577i

-0.3157 + 0.2589i

-0.0365 - 0.0361i

0.0023 - 0.0028i

0.0001 + 0.0001i

Once the extracted filter polynomials are obtained, coupling matrices can be evaluated

and the duplexer can be implemented with cavity resonators or waveguides [38], [39], [40],

[41].

26

CHAPTER 3

SYNTHESIS OF DUAL BAND FILTERS

3.1 Introduction

In modern telecommunication systems, the need for devices that can work at multiple

frequency bands simultaneously is becoming increasingly important in order to reduce size and

power requirement. For example, the employment of a dual band filter eliminates the

requirement of using two filters working at different bands by taking care of both bands, which

leads to significant cost reductions. Also, being one unit they are generally smaller than two

filters combined together. Several literatures are available on dual band filter synthesis and

design such as [42], [43], [44], [45], [46], [47]. In this chapter, the procedure to synthesize a

dual band filter will be discussed.

3.2 Polynomial Synthesis

The characteristic polynomials for a dual band filter are defined similar to that of a

single passband filter. However, there exists no direct iterative method to determine the

polynomials given the transmission zeros, passbands and return loss levels in either bands.

Therefore, an alternative method is initially described in [48] and is further studied in this

chapter. The procedure is summarized below:

• First an initial set of values of poles and zeros is assigned in the passbands and stopbands.

• Next initial filter function is constructed:

( ) ( )( )

( )

( )1

1

N

iiM

ii

w pA wC w

B w w z

=

=

−= =

−

∏

∏ (3.1)

27

where p and z denote the initial poles and zeros while N , M denote their numbers.

• The roots of the following expression are obtained:

( ) ( ) ( ) ( ) ( )dC w

B w A w A w B wdw

′ ′= − (3.2)

• The complex roots are discarded and the real roots are arranged including the passband

and stopband edges. Let the roots in the passbands be α and the roots in the stopband

be β . Each zero (pole) should lie between two roots or stopband (passband) edge.

• The zeros and poles are now updated using the following equations:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 0 1 0 1 0 1 0

0 1 0 0 1 1 0

1 0 0 1 1 0 1 0

0 1 0 1 0 1 0

l l l l l l l l ll

l l l l l l l

l l l l l l l l ll

l l l l l l l

p C C C Cp

p C C C C

z C C C Cz

z C C C C

α α α α α α α αα α α α α α

β β β β β β β ββ β β β β β

− − − −

− − −

− − − −

− − −

+ − + ′ =+ − +

+ − + ′ =+ − +

(3.3)

• This procedure is continued until the values of poles and zeros converge.

• Stopband edges are moved in order to bring equal return loss in lower and higher

passbands (this will be explained later).

• Also complex transmission-zeros can be introduced to control return loss levels.

• Once the poles and zeros are found,

( ) ( )

( ) ( )

1

1

N

iiM

kk

F s s jp

P s s jz

=

=

= −

= −

∏

∏ (3.4)

• ( )E s is obtained by taking the roots of ( ) ( )RP s F sε ε+ and mapping right half plane

roots to the left half plane.

28

Once the filter coefficients are obtained, the coupling matrix synthesis is similar to that

of a single band filter. It is noted that the return loss level in the passband depends heavily on

the width of the passbands. Any dual band filter with unequal passbands tends to have a

different return loss level in the two bands. Through our study, this challenge is overcome by

placing suitable transmission zeros in the stopbands or by using complex transmission zeros in

the passband if necessary. These techniques are illustrated below.

Narrowing of Passbands

Effect of narrowing transmission zeros towards passbands on return loss levels is

illustrated in Fig. 3.1. It is to be noted here that in only one of the band edge, value of ε [34] is

computed. Any asymmetry in the filter causes the other band to have a return loss other than

the specified one. The band in which ε is computed is not affected since ε automatically

compensates it. Each of the stopband edge has its own effect (not equal, generally) on how

much it changes the return loss level, and the positions of stopband edges (and transmission

zeros) are not unique in order to get equal return loss responses. Therefore this process is

analogous to a tuning process to get the desired return loss level as well as the steepness of

passband roll-off. Fig. 3.2 and Fig. 3.3 further explain this effect. The positions of stopband

edges (and transmission zeros) are not unique in order to get equal return loss levels.

29

Fig. 3.1 Effect of narrowing of passbands on return loss levels.

Fig. 3.2 Narrowing done around upper passband.

30

Fig. 3.3 Narrowing done around lower passband edges.

Effect of narrowing towards the passband on the higher frequency band is shown

above. In Fig. 3.2 the stopband edges have been narrowed towards the upper passband, which

pushes the return loss levels up in the upper passband. On the other hand, in Fig. 3.3 the

stopband edges have been pushed towards lower passband, which should push up the return

loss levels in the lower passband. However it is to be noted that during the synthesis procedure

ε has been evaluated in the lower passband and hence return loss level is fixed in lower

passband. Therefore, the change is again observed in the upper passband return loss level,

which is not restricted by the value of ε and is pushed down.

Additional Transmission Zeros

Addition of a finite transmission zero to a dual band filter with equal return loss levels in

the two bands will make it unequal. In Fig. 3.4 an addition transmission zero at 1.6=w has

been imposed. The two filtering bands have the same order. As we can see, the higher filter

31

band having transmission zero at 1.6 has higher level of return loss in the passband. It can be

concluded the ability to place transmission zeros at finite frequencies gives designers great

flexibility to design filters that have certain rejection level at a pre-specified frequency band.

Fig. 3.4 Effect of additional transmission zero at w = 1.6

Complex Transmission Zeros

Finally we find that a transmission zero at any point brings down the value of the

polynomial P as given in (3.4) at that point. Therefore a complex conjugate transmission zero

pair can be introduced whose real part lies in the passband to adjust the return loss level of

that passband. A smaller imaginary part has more effect while a large imaginary part of the

complex transmission zero has little effect on it. This is illustrated in Fig. 3.5.

32

Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels

are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines).

33

CHAPTER 4

DESIGN OF DUAL BAND DUPLEXERS

4.1 Introduction

In this chapter we propose a method to design a Microwave Duplexer that can work in

two different frequency bands simultaneously [44]. The discussion will involve design of dual

band filters followed by the combination of the dual band filters to form a Dual Band Duplexer.

4.2 Synthesis Procedure

The proposed dual-band microwave duplexer is composed of two dual-band filters with

the two input ports connected through a three-port junction. The two dual-band filters (RX and

TX) can be characterized separately from the duplexer through several techniques [48] [43]

[42]. In general, the dual-band filter employed in the analysis can be seen as a single-band filter

with some of its transmission zeros falling in the passband, separating the passband into two

bands. In our analysis the passband limits of the RX filter are represented by 1RXf , 2RXf , 3RXf ,

4RXf while those of the TX filter are 1TXf , 2TXf , 3TXf , 4TXf .

The duplexer is synthesized in a normalized frequency domain with the suitable lowpass

↔ bandpass frequency transformation ( )( )0 0 0Ω = −f B f f f f as shown in Fig. 4.1, where

0f and B are defined as follows:

0 1 4

4 1

RX TX

TX RX

f f fB f f

=

= −(4.1)

34

Fig. 4.1 Frequency transformation for dual band duplexers.

Once the lowpass band limits are obtained, the procedure described in [3] was used to

obtain the filter polynomials for both the TX and RX filters. The two lowpass prototype filters in

the normalized frequency domain can be characterized separately from the duplexer through

their characteristic polynomials. The characteristic polynomials of the TX and RX filters are

related to their scattering parameters as follows:

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

011 21

011 21

,

,

TX TX TX TXnTX TX

TX TX TX

RX RX RX RXnRX RX

RX RX RX

F s P s p P sS S

E s E s E s

F s P s p P sS S

E s E s E s

= = =

= = =(4.2)

where ( )TXE s , ( )RXE s are polynomials of degree TXnp , RXnp respectively. All of these

polynomials have unity coefficient for the highest degree.

The major difficulty in evaluating the polynomials of dual-band duplexer is to make the

responses of the dual-band RX/TX filters equal ripple in both of their passbands. As we know,

the return loss level in the passbands depends heavily on the bandwidth of the passbands. A

narrower bandwidth generally gives lower return loss level, and vice versa. Therefore, the dual

band filters with unequal passbands tend to have different return loss level in their two bands.

This challenge can be overcome by adjusting the transmission zeros positions located in the

35

stopbands or adding complex transmission zeros within the passbands as discussed in Chapter

3.

• Evaluation of Duplexer Polynomials

The derivation of duplexer polynomials ( )N s , ( )D s , ( )tP s , ( )rP s is done using the

reflection zeros at the input port of the duplexer and the transmission zeros of the TX and RX

Filters. Assuming lossless overall duplexer and unitary condition of the scattering matrix we

have,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2* * * *0 0 0r r r t t tD s D s n N s N s p P s P s p P s P s− = − + − + − (4.3)

( ) ( )*N s N s− depends only on the imposed reflection zeros with 0 1n = .

Now the evaluation is done in the following iterative steps:

1) Initialization: The RX and TX filters are synthesized independently of the duplexer

with general Chebyshev characteristics.( i.e., the polynomials 0TXF , 0

TXE , 0TXP

and 0RXF , 0

RXE , 0RXP are generated given the number of poles ( TXnp , RXnp ), the

return loss in the two channels, and the transmission zeros of the two filters. For

junctions causing additional zero, an approximate zero has to be added. An initial

estimate of TXS and RXS are available from the above polynomials.

2) Iteration begin: tP and rP are evaluated by polynomial convolution

( ),t TXn RXP conv P S= , ( ),r RXn TXP conv P S= .

36

3) Evaluation of 0tp and 0rp : The required return loss in the two channels ( TXRL

and RXRL ) is imposed at the normalized frequencies, s j= ± .

( )( ) ( ) ( )

2

102 2 22 2

0 0

10 TXRL

r r t t

N j

N j p P j p P j−=

+ ⋅ + ⋅(4.4)

( )( ) ( ) ( )

2

102 2 22 2

0 0

10 RXRL

r r t t

N j

N j p P j p P j−

−=

− + ⋅ − + ⋅ −(4.5)

0tp and 0rp are obtained by solving the above system of linear equations. Applying

spectral factorization technique to (4.3) ( )D s is evaluated from its roots.

4) New estimation of TXS and RXS : The roots of the polynomial

( ) ( )( ) ( ) ( )2 TX RXaN s bD s S s S s+ = ⋅ are computed with 201a jn b= − ⋅ and

201b jn b= + ⋅ . The roots are arranged in ascending order of imaginary parts. Let

_RX LBnp , _RX UBnp be the number of poles in lower and upper bands of RX filter

respectively. Similarly, let _TX LBnp , _TX UBnp be the number of poles in lower and

upper bands of TX filter. Then First _RX LBnp roots are assigned to RXS . Next

_TX LBnp roots are assigned to TXS . Next _RX UBnp roots are assigned to RXS and

next _TX UBnp roots are assigned to TXS .

5) Once the new TXS and RXS are obtained, step 2 to step 4 are iterated until

convergence is achieve to a high degree.

37

Computation of RX and TX Filter Polynomials

20

0 0

20

0 0

1

1

TX t

RX r

jn bp p

n

jn bp p

n

+=

+

=

(4.6)

Now the polynomials TXD and RXD defined similarly as in (2.4) are evaluated same as

[38]. Let TXzS and RXzS be the roots of TXS and RXS respectively. The values of ( )TX TXD zS

and ( )RX RXD zS are computed from the following expressions:

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

TXTX TX TX RX TX TX TX

RX TX

RXRX RX RX TX RX RX RX

TX RX

D zSD zS A D zS S zS D zS

A S zS

D zSD zS A D zS S zS D zS

A S zS

= ⋅ ⇒ =⋅

= ⋅ ⇒ =⋅

(4.7)

where A is given by ( )2 201n jn b+ . Now from the knowledge of the degree of RXD and TXD

one can find RXD and TXD using polynomial interpolation.

( ) ( )( )( ) ( )( )

, , 1

, , 1TX TX TX TX TX

RX RX RX RX RX

D s polyfit zS D zS np

D s polyfit zS D zS np

= −

= −(4.8)

where , ,RX RX LB RX UBnp np np= + and , ,TX TX LB TX UBnp np np= + .

Finally, the filter polynomials are obtained as:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

TX TX TX

TX TX TX

RX RX RX

TX RX RX

F s S s D s

E s S s D s

F s S s D s

E s S s D s

= −

= +

= −

= +

(4.9)

38

4.3 Implementation in Cavity type and Non-Resonating node type

To verify the dual-band duplexer synthesis technique presented above, a prototype

duplexer is synthesized using both the non-resonating node topology and the cavity-resonator

topology [38]. The specifications of the duplexer are listed as the following:

RX Passbands: 800MHz to 807MHz, 850MHz to 857MHz

TX Passbands: 815MHz to 822MHz, 865MHz to 872MHz

Center Frequency: 835.22 MHz

Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology with cavity resonators.

39

• Dual-band Duplexer with non-resonating node topology

The topology of the dual-band duplexer with non-resonating nodes is shown in Fig.

4.2(a). Both the RX and TX filters are fourth-orders with two attenuation poles. Following the

proposed synthesis procedure, the duplexer characteristic polynomials are given below. The

polynomial coefficients are reported in descending order.

N = [1, -0.08029i, 2.0807, -0.1405i, 1.4401, -0.07284i, 0.3786, 0.01046i, 0.03211];

D = [1, 1.1634-0.0803i, 2.7574-0.1083i, 1.9599-0.2122i, 2.1502-0.1511i, 0.9004-

0.1306i, 0.5544-0.0504i, 0.0999-0.0204i, 0.0327-0.0029i];

rP = [1, 0.2866 - 0.6026i, 1.1758 -0.1140i, 0.1921-0.3167i, 0.4208-0.0278i, 0.0192 +

0.0047i, 0.0214 - 0.0034i]

tP = [1, 0.2950+0.4282i, 1.2800+0.0602i, 0.2089+0.1365i, 0.4556-0.0048i, 0.0177-

0.0543i, 0.0159+0.0018i];

0rp = 0.2439; 0rp = 0.2712

The S-parameters of the synthesized duplexer are shown in Fig. 4.3. The corresponding

extracted J (admittance inverter parameter) and B (susceptance) values are listed below.

A) RX Filter:

Inverter J values: 0.5429, 1.0000, 0.5277, 1.0000, 1.6988, 3.4385 and 0.5431.

Inline Susceptance: 1.4937, 0.7449, -8.6256, -0.9618

Susceptance of dangling resonators: 0.5918, -0.2178

Resonator Frequency (in MHz): 783.2, 841.9, 843.1, 870.06

B) TX Filter:

40

Inverter J values: 0.5357, 1.0000, 0.4973, 1.0000, 1.9835, 3.9943 and 0.5354.

Inline Susceptance: -1.5686, -0.7128, 11.7735, 1.0060

Susceptance of dangling resonators: -0.6052, -0.1371

Resonator Frequency (in MHz): 893.6, 857.3, 830.3, 799.79

With the above extracted parameters, the duplexer can be easily implemented using

transmission lines.

Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology (S21 for RX channel, S31 for TX channel).

Dual-band Duplexer with cavity resonator topology

Similarly, the dual-band duplexer with cavity resonator as given in Fig. 4.2 (b) is also

synthesized following the procedure described previously. The duplexer characteristic

polynomials are given below. Junction capacitance is chosen as 0.7.

700 750 800 850 900 950 1000-100

-80

-60

-40

-20

0

20

Frequency

Scat

terin

g Pa

ram

eter

s (d

B)

S11 S31S21

41

N = [1, -0.7000 - 0.0803i, 2.0807 + 0.0562i, -1.4565 - 0.1405i, 1.4401 + 0.0983i, -

1.0081 - 0.0728i, 0.3786 + 0.0510i, -0.2650 - 0.0105i, 0.0321 + 0.0073i, -0.0225 - 0.0000i];

D = [1, 1.8199 - 0.0803i, 3.4917 - 0.1615i, 3.7966 - 0.2817i, 3.3984 - 0.2985i,

2.3483 - 0.2298i, 1.1349 - 0.1422i, 0.4870 - 0.0541i, 0.1005 - 0.0178i, 0.0232 - 0.0021i];

rP = [1, 0.2764 - 0.5418i, 1.0522 - 0.0820i, 0.1614 - 0.2621i, 0.3347 - 0.0193i,

0.0135 + 0.0001i, 0.0161 - 0.0021i];

tP = [1, 0.2836 + 0.3613i, 1.1371 + 0.0271i, 0.1702 + 0.0974i, 0.3582 - 0.0081i,

0.0117 - 0.0390i, 0.0119 + 0.0010i];

0tp = 0.2780; 0rp = 0.3134

The extracted TX and RX Filters have the following N+2 (including the source and the

load) coupling matrices respectively.

−−−−−−−

=

00.525700000.52570.24500.615800.40220

00.61580.19030.12960.02590000.12960.31740.5744000.40220.02590.57440.16300.657400000.65740

TXM

−−

=

00.532500000.53250.26580.608600.43810

00.60860.12830.16480.01240000.16480.26300.5720000.43810.01240.57200.17230.671600000.67160

RXM

42

To validate the proposed synthesis method for dual-band duplexers, the cavity-

resonator based duplexer presented above is implemented. As shown in Fig. 4.4 and Fig. 4.5

both the RX and TX filters of the duplexer are fourth-orders. The transmission zeros in the

RX/TX channels are obtained with a quadruplet. The RX/TX junction at the input is realized by

adding an extra resonator.

Fig. 4.4, Fig. 4.5, Fig. 4.6 show the simulated insertion losses ( 12S for the RX channel and

13S for the TX channel) and return loss ( 11S at the input) of the cavity resonator based duplexer

with optimized coupling coefficients. For comparison, the synthesized responses of the

duplexer are also plotted in this figure. The simulated results agree well with the synthesis as

expected, confirming the feasibility of the proposed synthesis technique for dual-band

duplexers.

Fig. 4.4 Insertion loss at RX channel ( 12S ).

700 750 800 850 900 950 1000-100

-80

-60

-40

-20

0

Frequency (MHz)

S12

(dB

)

SyntehsizedSimulated

43

Fig. 4.5 Insertion loss at TX channel ( 13S ).

Fig. 4.6 Return loss at Input ( 11S ).

700 750 800 850 900 950 1000-80

-60

-40

-20

0

Frequency (MHz)

S13

(dB

)

SynthesizedSimulated

700 750 800 850 900 950 1000-100

-80

-60

-40

-20

0

Frequency (MHz)

S11

(dB

)

SynthesizedSimulated

44

CHAPTER 5

MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS

5.1 Introduction

Often to meet specific requirements and lack of flexibilities in microwave duplexer

designs lead to very high order of microwave filters. In practice, it is convenient to reduce the

order of such duplexers that matches the original high order duplexer to a high degree in a

specific band of concern. One method to obtain coupling matrix elements from filter response

by genetic algorithm was described in [49].

The Cauchy method deals with estimation of a function by a ratio of two polynomials.

Given the values of the function and its derivatives at a few points the order of the polynomials

and their coefficients can be evaluated. Once the coefficients of the two polynomials have been

estimated, they can be used to generate the parameter over the entire band of interest. It is

particularly useful when the values of the function have been measured and a mathematical

model for the function has to be estimated.

However any measurement is accompanied by addition of noise. It is shown that for low

level of noise, performance of Cauchy method is very good over the entire sample space.

5.2 Formulation of Cauchy Method

The Cauchy method approximated a system function H(s) with a ratio of two

polynomials. The procedure was described in [50] and is discussed below.

( ) ( )( )

0

0

Pk

kk

Qk

kk

a sA sH s

B s b s

=

=

≈ =∑

∑(5.1)

45

Here the information is assumed to be N measured values of the function (H) at

frequency points , j 1, 2,3...Njs = .

In such case, the Cauchy Problem is,

Given ( )jH s for j=1, 2, 3…N, find P, Q, ak, k = 0, 1, 2,..P and bk, k=0, 1, 2..Q

Then from (5.1),

( ) ( ) ( )( ) ( ) ( ) 0

j j j

j j j

A s H s B s

A s H s B s

=

− =(5.2)

Now using the polynomial expansions for ( )A s and ( )B s ,

2 10 1 2 0 1 0P Q

j j P j j j j j Q ja a s a s a s H b H b s H b s+ + + + − − − − = (5.3)

for 1,2,3j N= .

Writing in matrix form,

[ ] 0a

Cb

=

(5.4)

where

[ ]1 1 1 1 1 1 1

2 2 2 2 2 2 2

11

1

P Q

P Q

P QN N N N N N N

s s H H s H ss s H H s H s

C

s s H H s H s

− − − − − − =

− − −

(5.5)

where

[ ] [ ]1 2 3, , , , TPa a a a a= (5.6)

and

[ ] 1 2 3, , , ,T

Qb b b b b = (5.7)

46

The order of matrix C is ( )2N P Q× + + .

A singular value decomposition (SVD) gives us the required coefficients for a and b .

[ ][ ][ ] 0TU VΣ = (5.8)

According to the Theory of Total Least Squares,

( ) [ ] 2. *

P Q

aconst V

b + +

=

(5.9)

That is, the elements in the last column of V give us the solution.

5.3 Application in General Polynomial Function:

Cauchy method has been applied on a function as below:

( )

4

05

0( 1)

k

k

k

k

ksH s

k s

=

=

=+

∑

∑(5.10)

This ratio is evaluated to find the exact values at 21 points in the range of s=2 to s=4.

Different levels of noise are added to this exact value to see its effect on the estimation. The

estimation accuracy is observed in the following figures. It can be seen that with low noise

levels Cauchy method gives excellent estimation of this kind of functions.

47

Fig. 5.1 SNR = 5

Fig. 5.2 SNR = 20

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency

Val

ue o

f H

Reference HEstimated H without noiseEstimated H with Noise

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency

Val

ue o

f H

Reference HEstimated H without noiseEstimated H with Noise

48

Fig. 5.3 SNR = 50 dB.

5.4 Application in General Microwave Filter function

As mentioned in previous chapters, a general two port network can be characterized by

a set of two parameters of its scattering matrix, namely the transmission and reflection

coefficients, 11S and 21S . These parameter share the same poles. Therefore to generate the

polynomial models for this case it is necessary to ensure the use of a common denominator.

The filter response is then modeled as,

( ) ( )11 1 110 0

n nk k

k kk k

S s a s b s S s= =

= ≈ ∑ ∑ (5.11)

( ) ( )21 2 210 0

Zn nk k

k kk k

S s a s b s S s= =

= ≈

∑ ∑ (5.12)

where n is the filter order and Zn is the number of finite transmission zeros.

This can be represented in a matrix equation:

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency

Val

ue o

f H

Reference HEstimated H without noiseEstimated H with Noise

49

[ ] [ ]

( ) ( )

111

221

11 11

21 21

00

0Z

Z

n N n n

N n n n

i

i

aV S V aa X

V S V bb

S diag S s

S diag S s

×

×

− = = − =

=

(5.13)

V is a Vandermonde matrix defined as,

[ ]

1 21 1 1 1

1 22 2 2 2

1 2

11

1

m m

m m

m

m mN N N N

s s s ss s s s

V

s s s s

−

−

−

=

(5.14)

The Exact polynomial model used for simulation is tabulated below and the exact

polynomial response and estimated response are shown in Fig. 5.4. More details on the

procedure can be found in [51].

Table 5.1 Low Pass Polynomial Coefficients

k ( )1ka ( )2

ka kb

0

1

2

3

4

5

6

- 0.0244 – 0.0110j

- 0.0453 – 0.2532j

- 0.0745 – 0.1796j

-0.1619 – 1.5168j

0.6856 – 0.2242j

-0.0139 – 1.3188j

1.0000

0.1133 + 0.0545j

0.1123 – 0.1158j

0.6002 – 0.5472j

1.3800 – 1.3009j

2.4933 – 1.9912j

2.4082 – 1.5359j

1.9356 – 1.3007j

1.0000

50

Fig. 5.4 Example of polynomial estimation for a 6th order filter.

5.5 Application in Microwave Duplexers

Application of the Cauchy Method on Microwave Duplexer was detailed in [52]. The

parameters 11S , 21S and 31S can be measured from Lab Setups for a diplexer. After

measurement of the scattering parameters, the filters can be approximated using Cauchy

Method.

( ) ( )( ) ( )

( ) ( )( ) ( )

11 01

21 0

11 02

31 0

TX RX

TX RX

TX RX

TX RX

n n kkk

TX m m kkk

n n kkk

RX m m kkk

a sS sK s

S s c s

a sS sK s

S s c s

+

=+

=

+

=+

=

= =

= =

∑∑∑∑

(5.15)

Equation (5.15) can be written in matrix form as:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-60

-50

-40

-30

-20

-10

0

10

Frequency

Sca

tterin

g P

aram

eter

s in

dB

Filter Scattering Parameter Extraction

|S11

|

Estimated |S11

|

|S21

|

Estimated |S21

|

51

( )

( )

[ ] ( )

( )

21 11 1 1

31 11 2 2

00

0TX RX TX RX

TX RX RX TX

n n m n

n n m n

a aS V S V

c M cS V S V

c c

+ +

+ +

−

= = −

(5.16)

where V is the Vandermonde Matrix defined as:

[ ]

1 21 1 1 1

1 22 2 2 2

1 2

11

1

m m

m m

m

m mN N N N

s s s ss s s s

V

s s s s

−

−

−

=

(5.17)

As an example, the exact and estimated polynomial responses are shown in Fig. 5.5,

which agree with each other very well.

Fig. 5.5 Exact and estimated polynomial response for duplexers.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-120

-100

-80

-60

-40

-20

0

20

Frequency

Sca

tterin

g P

aram

eter

s in

dB

Diplexer Scattering Parameter Extraction

|S11

|

Estimated |S11

|

|S21

|

Estimated |S21

|

|S31

|

Estimated |S31

|

52

5.6 Conclusion

As we can see, Cauchy Method is an excellent mathematical procedure that uses the

concept of Total Least Squares to estimate the same or reduced order polynomial models of

measured parameters. Even with noisy data, the models obtained do not deviate far from exact

models.

53

CHAPTER 6

CONCLUSION AND FUTURE WORK

In this work methods to synthesize microwave filters and duplexers have been

discussed. Especially, a method to synthesize a dual band microwave duplexers is proposed. In

addition to using dual band filters to synthesize dual band duplexer, a multiband filter can be

used to synthesize a multi band duplexer. However this will add the complexity of the synthesis

and require high degree of optimization at later stages of design, which can be studied in future

work.

Also, waveguide is another way to design microwave filters and duplexers. Synthesized

dual band duplexers can be designed using waveguides which can sustain high power.

Once the designs of filters and duplexers are complete, measurements can be done on

the prototypes and the scattering parameters can be measured with vector network analyzer at

different frequencies. These measurements can be used to obtain a lower order polynomial

model using Cauchy method in order to obtain filters of desired orders that roughly behave the

same as the original ones.

54

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