and RL is the prescribed inband equiripple return loss level of the Chebyshev function in dB. S11(ω) and S21(ω) share a common denominator E (ω) The polynomials E (ω) and F (ω) are both of degree N, when the polynomial P (ω) carries the nfz transfer function finite-position transmission zeros. For a Chebyshev filtering function, ε is a constant normalizing
S21(ω) to the equiripple level at ω=±1, and (εR = 1 except for fully canonical filters (ie. nfz = N ).
For a prescribed set of transmission zeros that make up the polynomial P (ω) and a given equiripple return loss level, the reflection numerator polynomial F (ω) may be built up with an efficient recursive technique. and then the polynomial E (ω) found from the conservation of energy principle .
An example of this synthesis method is given in  for a 4th degree prototype with 22 dB return loss level and two imaginary axis TZs at s01 = +j1.3127 and s02 = +j1.8082. These are positioned to give two rejection lobes at 30 dB each on the upper side of the passband. Plots of the transfer and rejection characteristics are shown in Fig. 5.
3.2 Construction of the N +2 Transversal Matrix
The second step in the synthesis procedure is to calculate the values of the coupling elements of a canonical coupling matrix from the transfer and reflection polynomials. Three forms of the canonical matrix are commonly used-the folded , transversal  or arrow . The transversal matrix is particularly easy to synthesize, and the other two may be derived from it quite simply by applying a formal series of analytically-calculated similarity transforms.
The transversal coupling matrix comprises a series of N individual 1st degree low pass sections, connected in parallel between the source and load terminations but not to each other (Fig. 6(a)). The direct source-load coupling inverter MSL is included to allow fully canonical transfer functions to be realized according to the “minimum path” rule, i.e. nfzmax, the maximum number of finite-position TZs that may be realized by the network=N-nmin , where nmin is the number of resonator nodes in the shortest route through the couplings of the network between the source and load terminations. In fully canonical networks, nmin = 0 and So nfzmax = N (the degree of the network).
Each N low-pass section comprises one parallel-connected capacitor Ck and one frequency invariant susceptance Bk, connected through admittance inverters of characteristic admittances MSk and MLk to the source and load terminations respectively. The circuit of the k th lowpass section is shown in Fig. 6(b).
The approach employed to synthesize the N+2 transversal coupling matrix is to construct a 2-port short-circuit admittance parameter matrix [YN] for the overall network in two ways: from the coefficients of the rational polynomials of the transfer and reflection scattering parameters S21(s) and S11(s) (which represent the characteristics of the filter to be realized) or from the circuit elements of the transversal array network. By equating the [YN] matrices derived by these two methods, the elements of the coupling matrix associated with the transversal array network can be related to the coefficients of the S21(s) and S11(s) polynomials .
An example of a reciprocal N+2 transversal coupling matrix M representing the network is shown in Fig.7. MSk are the N input couplings, and they occupy the first row and column of the matrix from positions 1 to N. Similarly, MLk are the N output couplings, and they occupy the last row and column of M from positions 1 to N. All other entries are zero.
4 Similarity Transformation and Reconfiguration
The elements of the transversal coupling matrix that result from the synthesis procedure can be realized directly by the coupling elements of a filter structure if it is convenient to do so. However, for most coupled-resonator technologies, the couplings of the transversal matrix are physically impractical or impossible to realize. It becomes necessary to reconfigure the matrix with a sequence of similarity transforms (sometimes called rotations)  until a more convenient coupling topology is obtained. The use of similarity transforms ensures that the eigenvalues and eigenvectors of the matrix M are preserved. Under analysis, the transformed matrix yields exactly the same transfer and reflection characteristics as the original matrix.
There are several more practical canonical forms for the transformed coupling matrix M. Two of the better-known forms are the ‘arrow’ form  and the more generally useful ‘folded’ form . Either of these canonical forms can be used directly if it is convenient to realize the couplings or be used as a starting point for the application of further transforms to create an alternative resonator intercoupling topology optimally adapted to the physical and electrical constraints of the technology with which the filter will eventually be realized. The method for reduction of the coupling matrix to the folded form with a formal sequence of rotations is detailed in . The ‘arrow’ form may be derived using a very similar method.
5 Advanced Configurations
In this section, some advanced coupling matrix configurations particularly suitable for filters and diplexers in terrestrial telecommunication systems will be considered. An important application is in the cellular telephony industry where strong growth has meant that very stringent out-of-band rejection and in-band linearity specifications have had to be imposed to cope with a crowded frequency spectrum and increasing numbers of channels. At the RF frequencies allocated to mobile systems (L-band, S-band, and sometimes C-band), coaxial or dielectric resonator technology is often used for the filters of the system because of the compact, flexible, and robust construction with flexible layout possibilities that may be achieved together with the ability to realize advanced filtering characteristics and quite high RF power handling.
A microwave filter topology that has found widespread application in both terrestrial and space systems is the ‘trisection.’ The basic trisection may be used as a stand-alone section or be embedded within a higher-degree filter network. But often multiple trisections are merged to form advanced configurations such as cascaded ‘N-tuplets’ or box filters.
A trisection comprises three couplings between three sequentially-numbered nodes of a network (the first and third of which may be source or load terminals) or it might be embedded within the coupling matrix of a higher-degree network . The minimum path rule indicates that trisections are able to realize one transmission zero each. As will be shown later, trisections may be merged using rotations to form higher-order sections; for example, a quartet capable of realizing two TZs can be formed by merging two trisections.
Fig.8 shows four possible configurations. Fig.8(a) is an internal trisection, whilst Figs.8(b) and (c) show ‘input’ and ‘output’ trisections respectively, where one node is the source or load termination. When the first and third nodes are the source and load terminations respectively
(Fig. 8(d)), we have a canonical network of degree 1 with the direct source-load coupling, MSL , providing the single transmission zero. Trisections may also be cascaded with other trisections, either separately or conjoined (Figs. 8(e) and (f)).
Beingable to realize just one transmission zero each, the trisection is very useful for synthesizing filters with asymmetric characteristics. They may exist singly within a network or multiply as a cascade. Rotations may be applied to reposition them along the diagonal of the overall coupling matrix or to merge them to create quartet sections (two trisections) or quintet sections (three trisections). The following is an efficient procedure for synthesizing a cascade of
5.2 Synthesis of the ‘Arrow’ Canonical Coupling Matrix
The folded cross-coupled circuit and its corresponding coupling matrix was previously introduced as one of the basic canonical forms of the coupling matrix. It is capable of realizing N transmission zeros in an N th degree network. A second form was introduced by Bell  in 1982, which later become known as the ‘wheel’ or ‘arrow’ form. Like the folded form, all the main-line couplings are present; and in addition, the source terminal and each resonator node is cross-coupled to the load terminal.
Fig. 9(a) is an example of a coupling and routing diagram for a 5th degree canonical filtering circuit. It shows clearly why this configuration is referred to as the ‘wheel.’ with the main-line couplings forming the (partially incomplete) rim and the cross-couplings and input/output coupling forming the spokes. Fig. 9(b) shows the corresponding coupling matrix where the cross-coupling elements are all in the last row and column, and together with the main line and self couplings on the main diagonals give the matrix the appearance of an arrow pointing downwards towards the lower right corner of the matrix. The arrow matrix may be synthesized from the canonical transversal matrix with a formal sequence of rotations, similar to that of the folded matrix.
The basis of the trisection synthesis procedure relies on the fact that the value the determinant of the self and mutual couplings of the trisection evaluated at ω=ω 0 (the position of the TZ associated with the trisection) is zero:
where k is the number of the middle resonator of the trisection. Knowing the positions of the transmission zeros of the filtering characteristic, the trisections can be generated one by one within the arrow matrix, and shifted to form a cascade between the input and output nodes.
Fig. 10 gives the topology and coupling matrix for the 4th degree filter with 22 dB RL and two transmission zeros at (ω 01=1.8082 and (ω 02=1.3217 that was used as an example above now configured with two trisections (to realize the two TZs). The shaded areas in the matrix indicate the couplings associated with each trisection.
Once the arrow coupling matrix has been formed, the procedure to create the first trisection realizing the first TZ at ω =ω 01 begins with conditioning the matrix with the application of a rotation at pivot [N-1, N ] and an angle (θ 01 to the original arrow matrix M (0). This trisection is then shifted by a series of rotations to the left of the network.
Now he process can be repeated for the second trisection at ω =ω 02 and so on until a cascade of trisections is formed—one for each of the TZs in the original prototype, as shown in
Fig. 11(a). The trisections may be realized directly if it is convenient to do so; for example, for coupled coaxial resonators. But for other technologies such as dual-mode waveguide, a cascade of quartets may be more suitable. A cascade of quartets is easily achieved by merging adjacent trisections, as illustrated in Fig. 11(b). Fig.11(c) shows a possible coaxial-resonator realization for the two quartets.
This procedure can be extended to form even higher-order sections in cascade; for example, three trisections may be merged to form a quintet section, as illustrated in Fig.12.
6 Box and Extended Box Sections
6.1 Box Sections
The trisection may also be used to create another class of configuration known as the ‘box’ or ‘extended box’ class . The box section is similar to the cascade quartet section, that is, it has four resonator nodes arranged in a square; however the input to and output from the quartet are from opposite corners of the square. Fig. 13(a) shows the conventional quartet arrangement for a 4th degree filtering function with a single transmission zero and realized with a trisection. Fig. 13(b) shows the equivalent box section realizing the same transmission zero but without the need for the diagonal coupling. Application of the minimum path rule indicates that the box section can realize only a single TZ.
The box section is created by the application of a cross-pivot rotation to a trisection that has been synthesized within the overall coupling matrix for the filter. To transform the trisection into a basic box section, the rotation pivot is set to annihilate the second main-line coupling of the trisection in the coupling matrix. ie. pivot = [2,3] annihilating element M23 in the trisection 1-2-3 in the 4th degree example of Fig.13(a) and in its equivalent coupling and routing schematic in Fig.14(a). In the process of annihilating the main-line coupling M23, the coupling M24 is created (Fig.14(b)), and then, by ‘untwisting’ the network, the box section is formed (Fig.14(c)).
In the resultant box section, one of the couplings is always negative, irrespective of the sign of the cross-coupling (M13) in the original trisection. Fig. 15(a) gives the coupling and routing diagram for a 10th degree example with two transmission zeros realized as trisections. Fig. 15(b) shows that each trisection has been transformed into a box section within the matrix by the application of two cross-pivot rotations at pivots ,  and ,. Having no diagonal couplings, this form is suitable for realization in dual-mode technology.
An interesting feature of the box section is that to create the complementary response (i.e. the transmission zero appears on the opposite side of the passband), it is only necessary to change the values of the self couplings to their conjugate values. In practice, this is a process of retuning the resonators of the RF device—no couplings need to be changed in value or sign. This means that the same physical structure can be used for the filters of, for example, a complementary diplexer.
6.2 Extended Box Sections
The basic box section may be extended to enable a greater number of transmission zeros to be realized, but retaining a convenient physical arrangement is shown in (Fig.16) . Here, the basic 4th degree box section is shown and then the addition of pairs of resonators to form 6th, 8th and 10th degree networks. Application of the minimum path rule indicates that a maximum of 1, 2, 3, 4… (N-2)/2 transmission zeros can be realized by the 4th, 6th, 8th , 10th,…N th degree networks respectively. The resonators are arranged in two parallel rows with half the total number of resonators in each row. The input is at the corner of one end and output from the diagonally opposite corner at the other end. Even though asymmetric characteristics may be prescribed, there are no diagonal cross-couplings.
At present, a formal series of rotations to generate an extended box filter is not known. The form can, however, be derived with the software package Dedale-HF, which is accessible on the Internet .
Because of its simplicity, the box filter is useful for the design of transmit/receive diplexers, which are very often found in the base stations of cellular telephony systems. An example of a simple diplexer comprising two complementary-asymmetric 4th degree filters, each with one transmission zero producing a 30 dB rejection lobe over each other’s usable bandwidth. Is shown in Fig. 17(a), and its performance is shown in Fig. 17(b). This diplexer was designed using software that optimizes the length and impedances of the common-port coupling wires as well as the first few elements of each filter nearest to the CP (coupling values, resonator tuning frequencies). In practice, much greater Tx-Rx isolation is usually required, and higher degree filters with more transmission zeros have to be used.