Spurious Self-Suppression Method: Application to TM Cavity Filters

This article presents a new approach for improving spurious free bands in filters. The main idea is to control the coupling coefficients of the spurious resonances, in order to force them to cancel each other. This approach is completely different from the classical one where all couplings are minimized in order to excite spurious resonances as less as possible. In fact, in this approach, all spurious couplings are maximized except the central coupling that is instead minimized. This method is first explained in terms of equivalent circuits. Then, in order to show its applicability to real microwave filters, it is applied to the TM cavity filters and used to remove both spurious resonances due to higher order cavity modes and to TE10 mode of coupling irises. Design examples of fourth- and sixth-order TM cavity filters are presented. With respect to the classical TM cavity filter, where the first spurious appears around 11.4 GHz, the TM filters designed by exploiting the self-suppression method present spurious free bands up to 20.2 GHz (20 GHz for sixth-order filter), improving the filter response up to 2.2 times the center frequency of 9.2 GHz. Another advantage is that, in contrast to the classical spurious suppression method that does not allow responses with TZs in TM filters, the self-suppression method allows for a number of TZs equal to N-2 (N being the filter order) and it also allows a certain control of their position.

communication systems. Passband filters often suffer from spurious resonances that limit the wideness of their stopband. Wider stopbands are obtained by cascading a low-pass filter to a passband filter. The advantage of having a passband filter with a wide spurious free stopband is that it allows avoiding the use of low-pass or at least it allows using a less bulky one, thus reducing the weight and volume of overall payload communication system which strongly impacts the overall cost [3], [4].
Different techniques have been proposed for increasing the stopband performance in microwave filters. In [5], wide stopband in waveguide filters is achieved by using stepped impedance resonators (SIRs). Morelli et al. [6] proposed the use of resonators with different transversal dimensions to control spurious resonances in rectangular waveguide filters. In [7], the integration of a low-pass filter with bandpass filter for improving stopband performance is proposed. A compact waveguide filter using capacitively loaded cavity with improved stopband response is proposed in [8]. Mushroomshaped posts are used in [9] and [10] for improving the stopband characteristic in combline filters. The use of capacitive irises to improve stopband performance can be found in [11].
Another classical and intuitive approach consists in avoiding the excitation of the spurious resonances by minimizing (possibly zeroing) their coupling coefficients [12]. The limitation of this approach is that, in some cases, coupling values can be minimized but not avoided (e.g., because of manufacturing tolerances) and this may result in narrowband spurious and stopband performance degradation.
Other different approaches for stopband performance improvement in microwave bandpass filters using microstrip and substrate-integrated waveguide (SIW) technology have been reported in [13], [14], [15], [16], [17], [18], [19], [20], [21], and [22]. In [13], SIR-based approach is used in planar filter configuration to achieve wider stopband. Microstrip bandpass filter with ultrawide stopband is reported in [14]. Jia et al. [15] proposed multilayer SIW filter with proper coupling position to suppress higher order modes. Quite similar approach is adopted in [16], to suppress neighboring mode to improve out-of-band response. A method similar to that presented in [12] for waveguide technology has been reported in [17] and [18] for SIW technology. However, due to lossy nature of SIW technology, it is easy to suppress spurious (due to losses in dielectric material), if it appears as a very narrow parasitic band, as it is generally the case when spurious coupling values are minimized as much as possible.
Recently, some articles studied the way to improve the out-of-band behavior of SIR-based filters, but this was only for wideband application, and it resulted in an increased geometrical complexity [23], [24], [25].
Several articles have been published that exploit the concept of nonresonating nodes (NRNs) along with resonating modes to realize transmission zeros (TZs) and to achieve better selectivity and better stopband performance in waveguide filters [26], [27]. This method has been also applied to transverse magnetic (TM) cavity filters that allow designing filter with reduced length [28]. Indeed, TM cavity filters represent an excellent compromise when very compact structures with reasonably high Q factors are required [29], [30], [31], [32]. A generalized study about TM dual-mode cavity filters has been presented in [33] and [34]. Improvement of the out-ofband behavior in TM dual-mode cavities is presented in [35] by using circular TM cavities and multiple-aperture irises.
In this article, a new method for the self-suppression of spurious resonances through the control of the spurious coupling arrangement is presented. The method is first presented at equivalent circuit level. In order to demonstrate their applicability to real microwave filters, the method has been applied in the improvement of the out-of-band performance in TM cavity filters. The proposed technique consists in forcing the spurious resonances to cancel each other. In contrast to the classical approach, where all the coupling values to spurious resonances are minimized, in the proposed method all spurious couplings except the central one are maximized. The central one is instead minimized (or zeroed when possible). The basic idea of the self-suppression approach has been reported in [36], where it has been applied to a second-order TM cavity filter. This article further extends the study of the proposed method to higher (even) order TM cavity filters. With respect to [36], where only spurious frequencies due to cavity higher order modes have been removed, this article also shows the method for removing spurious frequencies due to the coupling irises. A detailed analysis has been presented in this article for removing spurious frequencies due to irises as well as higher order TM 120 and TM 210 cavity modes. Design examples of fourth-and sixth-order TM cavity filters are presented. Furthermore, the proposed approach for spurious suppression is compared with the classical one [12], [17], [18] in which all couplings to spurious resonances are minimized. Results show the feasibility of the proposed approach, which offers better spurious suppression performance. Moreover, in the specific application to TM filters, the proposed method allows for a better control of the band selectivity. Indeed, in contrast to the classical spurious suppression method where TZs completely disappear, by using the proposed self-suppression method N-2 TZs are still allowed (being N the filter order) and their position can be still controlled. In order to demonstrate the feasibility of the proposed approach, several fourth-and sixthorder TM cavity filters have been designed. Finally, a sixthorder filter has been manufactured and measured. A good agreement between simulation and measured results has been obtained.
This article is organized as follows. In Section II, coupling theory and band suppression of both even and odd order bands are discussed. In Section III, fourth-order TM cavity filter designs exploiting classical design approaches are presented. In Section IV, design examples of fourth-and sixth-order TM cavity filters with the proposed spurious self-suppression approach are presented. In Section V, the measurement result of sixth-order filter is presented. The conclusion is drawn in Section VI.

II. SPURIOUS SELF-SUPPRESSION METHOD
Microwave filters are obtained by cascading resonators that resonate in the filter band. The desired passband is obtained by properly selecting all coupling values. In some cases, however, resonators can produce spurious modes that resonate at different frequencies producing spurious bands. Filter bands and spurious bands are conceptually similar, the only difference is that spurious bands are undesired. In this article, it is shown a new method for eliminating (or better saying to strongly attenuate) the undesired bands. The method is slightly different in case of even and odd order responses. Both of them will be shown in the following.

A. Self-Suppression of Even Order Parasitic Band
The classical method used for the parasitic band cancellation [12] is to minimize all couplings. Of course, this method works very well if we are able to set all couplings to zero. However, in many practical cases, it is not possible and the best thing we can do is to minimize all couplings. In such cases, much better results are obtained by using the self-cancellation method where only the central coupling is minimized, whereas all other are maximized.
In order to show how the self-suppression method works, the four-pole configuration of Fig. 1(a) is considered. The relevant response is shown in Fig. 1(b) in the case of all identical coupling values (M S1 = M 12 = M 23 = 0.5). As shown in Fig. 2, when the central coupling M 23 decreases, the band attenuation increases. In the same figure, the response obtained by reducing all couplings is also shown. As can be seen in that case spurious frequencies have not been eliminated.
This method works for all even order structures (2, 4, 6, 8, 10, 12, . . .) only, and it is based on a sort of self-suppression in which resonances cancel one another. The mechanism allowing the self-cancellation can be explained by using the transversal topology. According to the theory explained in [37], it is possible to transform an inline topology into a transversal topology. In the case of fourth-order filter, the inline structure of Fig. 1(a) can be transformed into the transversal topology of Fig. 3(a). In contrast with the inline topology where all resonators resonate at the same frequency (e.g., Chebyshev response), in the transversal topology resonators must work at different frequencies; otherwise, they cancel one another destroying the response. By the way, such resonance frequencies correspond to the reflection zeros (RZs) of the response, as shown in Fig. 1(b). The cancellation is due to the fact that the output couplings of resonators have alternating signs, as shown in Fig. 3(a). This cancellation is  the one exploited in the proposed self-cancellation method. Indeed, the decrease of the central coupling value in the inline configuration corresponds to a reduction of the frequency separation between the two consecutive resonators f i and f i+1 , with i = 1, 3, 5, . . . For the fourth-order filter, this means that frequency f 1 became closer to f 2 and frequency f 3 became closer to f 4 . This can be clearly seen in Fig. 3(b).
From the point of view of the response this means that when the couple of RZs becomes closer, it passes from imaginary axis to the complex plane thus resulting in a couple of conjugated zeros. This suggests that in case, we would like to synthesize a bandpass exploiting the self-suppression method, we have to impose RZs in the complex plane. This cancellation, explained above at the level of routing scheme, has a very clear physical meaning when a full-wave structure is considered. As an example, the structure of Fig. 4 consisting of cavities connected through irises is considered. When the inline model is considered, each resonator of the routing scheme corresponds to the resonant mode of a single isolated cavity. When instead the transversal model is considered, each resonator of the routing scheme corresponds to a resonant mode (eigenmode) of the whole physical structure, as shown in Fig. 4. Such eigenmodes can be easily found by using a fullwave simulator. According to Fig. 4, all four eigenmodes can be seen as a combination of cavity modes that can have phases equal to 0 • or 180 • . As an example, the mode resonating at the lower frequency f 1 consists in the field configuration where all resonant modes of the cavities have the same phase (0 • ). The second mode resonating at f 2 consists instead of the first two cavity modes with phase 0 • and the last two with phase 180 • etc. Note that all eigenmodes have phases equal to 0 • at the input (corresponding to the source), while the output (corresponding to the load) phase is 0 • for the first and third eigenmode and 180 • for the second and fourth. This is consistent with the sign of the coupling in the routing scheme of Fig. 3(a). These different signs at the output of the eigenmodes allow the cancellation of eigenmodes resonating at f 1 and f 2 and eigenmodes resonating at f 3 and f 4 when the central coupling tends to zero.

B. Self-Suppression of Odd Order Parasitic Band
According to Fig  As can be seen, once the resonance frequency of the central resonator is increased and it goes out of the parasitic band range, the remaining RZs start attenuating by suppressing one another. The only visible resonance remains that of the central resonator R2, but it can be pushed outside of the spurious free band required by the specifications. To understand how this mechanism works, we can consider that the frequency-shifted resonator behaves as a coupling structure between its previous and its next resonator. Considering the initial equivalent circuit of Fig. 6(a), after detuning R2, the equivalent circuit became that of Fig. 6(b) where the cascade of M 12 , R2, and M 23 have been substituted by a single impedance inverter M' 12 . Of course, this new equivalent circuit allows a good approximation of the response in the parasitic band and around the parasitic band but does not take into account the resonance of R2 far away from the parasitic band. Now the parasitic band is again an even order band, and the self-suppression method can be applied by decreasing M' 12 . The larger the frequency shift of R2, the smaller the M' 12 . This explains the attenuation of the parasitic band of Fig. 5(b). Another way to reduce M' 12 is also by reducing M 12 and M 23 .
In order to demonstrate the feasibility of the proposed methods, in Section IV, the two strategies discussed above have been applied to TM cavity filters for removing their parasitic bands due to higher order modes (even order) and coupling iris resonances (odd order).

III. CLASSICAL TM CAVITY FILTER DESIGN
In single-mode TM cavity filters, TM 110 modes are exploited as resonant modes to generate the filter passband. Higher order cavity resonant modes TM 120 and the resonance of the fundamental mode of coupling irises (TE 10 ) are instead responsible for undesired parasitic bands. The classical TM cavity [28], is sketched in Fig. 7, together with the field distribution of the fundamental mode TM 110 , and of the higher order mode TM 120 . The configuration of four-pole single-mode TM cavity filter is shown in Fig. 8(a), along with the dimensions. The filter is fed through standard WR-90 waveguide. Fig. 8(b) shows the relevant routing scheme including filter  passband (TM 110 ), filter parasitic band due to higher order modes (TM 120 ), and filter parasitic band due to coupling iris resonances (TE 10 ) mode. Considering that all irises are horizontally positioned, the TM 210 mode is not excited, and it does not contribute to the parasitic bands. For the filter passband a routing scheme with NRNs has been considered. This configuration allows taking into account the bypass coupling of each single cavity (dotted lines) generated by the nonresonating mode TE 101 [28], [29]. The total number of TZs of the filter is equal to the number of cavities where the bypass coupling is not zero [28], [29]. Fig. 8(c) shows the response of a classical four-pole TM cavity filter, where no procedures for improving the out-ofband behavior have been considered. The filter band is centered at 9.2 GHz with a fractional bandwidth (FBW) of 1.3%. Four TZs are present at the lower stopband as a consequence of the cross-couplings generated by the nonresonant TE 101 cavity modes [28]. As can be seen, some parasitic bands degrade the out-of-band performance of the filter. In particular, the parasitic band around 12 GHz is due to the coupling iris resonances, while that around 16 GHz is generated by the higher order modes TM 120 . It should be noted that iris 1 (and iris 5) does not produce resonances (at least not at the frequency of the other irises) and it only contributes, together with the first cavity, to the impedance inverter connecting the source to iris 2 resonance (and load to iris 4 resonance in case of iris 5). For that reason, only the three resonances due to irises 2, 3, and 4 are visible in Fig. 8(c).

A. Classical Approach for Removing Higher Order Mode Spurious Frequencies
The classical approach for eliminating parasitic bands consists of the minimization of all possible couplings to higher order modes generating the parasitic band. As an example, this technique has been applied in [12]. In case of TM filters, according to Fig. 8(a), the best way for decreasing as much as possible all the coupling between higher order modes on adjacent cavities is to alternate the iris orientation from horizontal to vertical. This is shown in Fig. 9(a). Indeed, according to Fig. 7, because of the field distribution of the fundamental resonant TM 110 mode, its coupling does not depend on the iris orientation [33]. This means that the filter passband is not influenced by the orientation. Something different happens instead to the higher order modes: the first horizontal iris excites the TM 120 spurious mode, but the second (vertical) iris does not excite the TM 120 mode [33]. Actually, the coupling to TM 120 modes obtained by a vertical iris is very small, but it is not zero. This small residual coupling is responsible for the spurious frequencies around 15/16 GHz shown in the response of Fig. 9(b). Of course, this response is much better than that in Fig. 8(c), where no strategies for removing spurious frequencies were taken into account. However, some spurious frequencies are still present. Actually, vertical irises may also excite the TM 210 mode thus creating additional spurious resonances around 16 GHz. However, in the response of Fig. 9(b), these resonances have been removed by placing vertical irises in a specific position from the cavity center where TM 210 mode has the minimum magnetic field. This corresponds to a distance from the center equal to a quarter of the cavity size (w/4) [33]. In order to reach this new position, the distance of the vertical irises from the center has been increased. This results in an undesired increase of the coupling M 12 that ruins the filter response. In order to recover the original M 12 value, iris size w 2 has been reduced. As a result, two iris resonances initially present around 12 GHz are now shifted to higher frequency of around 17.8 GHz.

IV. FILTERS WITH SPURIOUS SELF-SUPPRESSION APPROACH
According to the theory discussed in Section II, the application of the strategy for the spurious self-suppression to TM filters can be summarized into the following three steps.
1) Central iris orthogonally positioned. When the central iris is orthogonally placed with respect to all other irises, the central coupling coefficient of the higher order mode parasitic band is dramatically reduced. This results in the parasitic band attenuation. 2) Central iris positioned far away from the cavity center.
This allows for the reduction of the central iris width (w3), thus increasing its resonance frequency. This leads to the attenuation of the iris parasitic band. 3) All irises (except the central one) are positioned close to the cavity center. This increases all couplings in the higher mode parasitic band except the central one. This results in the improvement of the suppression of the higher order mode parasitic band. Note that, according to Fig. 10(a), the distance p x of the iris from the cavity center is here used to remove parasitic bands. In [28], the same parameters are used for controlling the TZs. This means that in spurious self-suppression method, the control of TZs is a little bit more limited. However, in contrast to the classical approach of Fig. 9 where TZs completely disappear, in filter designs using spurious self-suppression method, a certain control capability of the TZs still remains, as shown later on.

A. Fourth-Order Filter
The three steps described above are here applied to the fourth-order TM cavity filter of Fig. 8, starting from the central iris rotation.

1) Central Iris Orthogonally Positioned:
The four-pole filter of Fig. 8, after its central iris has been vertically positioned, is shown in Fig. 10(a) and (b). Its response is shown in Fig. 10(c).
In theory, according to the modal field distribution of Fig. 7(b), the central coupling between fundamental modes TM 110 (M 23 ) does not change when the iris is rotated by 90 • if the same distance p x from the center and the same size w 3 are maintained. However, in practice, a small readjustment is necessary for reobtaining the desired filter passband ( p x = 2.55 mm and w 3 = 10.85 mm).
In theory, according to Fig. 7(c), the central coupling M pm 23 between TM 120 modes of cavities 2 and 3 is zero. In practice, because of the presence of irises 3 and 4 that perturbs the fields in cavities 2 and 3, it is very small, and it allows, according to the spurious self-suppression method, a strong attenuation of the parasitic band due to the TM 120 modes. However, a spurious frequency related to the TM 210 mode is now present at 16.2 GHz, as can be seen in Fig. 10(c). This is because the vertical position of iris 3 allows the excitation of the TM 210 mode. Fig. 11 shows the magnetic field of TM 210 mode calculated at 16.2 GHz. Finally, as a side effect of the central iris rotation, the resonances of irises 2 and 4 are strongly attenuated (about 18 dB). This is because the iris 3 rotation in some way starts the procedure described in Section II-B related to the suppression of parasitic odd order bands when the central resonator  is pushed out-of-band. Indeed, in this case, iris 3 resonates at a slightly higher frequency with respect to irises 2 and 4, and the rotation of iris 3, strongly decreases the coupling between irises 2 and 4 resonances, thus resulting in a strong attenuation of the parasitic band generated by irises 2 and 4. Of course, iris 3 resonance is still present at 12.6 GHz, but according to the method described in Section II-B, it must be pushed at much higher frequency. This is what is done in the next step.

2) Central Iris Positioned Far Away From the Cavity Center:
Considering the field distribution in Fig. 7(b), the higher the distance of the iris from the center, the higher the coupling between resonating modes. This means that, for maintaining the same coupling level M 23 , the iris width w 3 must be decreased when the distance from the center increases. This is illustrated in Fig. 12(a), where the curve shows the value of p x that shall be selected for each value of iris width w 3 to maintain the value of the coupling M 23 = 0.0087 corresponding to the value obtained from the filter synthesis.
In order to remove the odd order parasitic band, we apply the strategy illustrated in Section II-B consisting of increasing the resonant frequency of the central iris (iris 3). This can be done by decreasing the iris width w 3 . By exploiting the graph of Fig. 12(a) we can select the iris position p x that allows maintaining the filter band unchanged. In other words, the application of the strategy illustrated in Section II-B consists in the decreasing of w 3 and in the increasing of p x according to Fig. 12(a).
The last thing to check is the effect of decreasing w 3 and increasing p x in the coupling between higher order modes TM 210 . This is well explained in [33] and [34]. In contrast to what happens to the coupling between fundamental modes TM 110 that increases when p x increases, the coupling between TM 210 modes decreases reaching zero at p x = w/4 = 5.6875 mm where, according to Fig. 11, its H-field is zero. This effect, combined with the iris width reduction, allows a strong reduction of the coupling between TM 210 higher order modes. The coupling M pm 23 between the TM 120 mode in the two central cavities has been strongly decreased by rotating the central iris. However, the reduction of iris width contributes to a further reduction of the residual coupling due to the presence of the iris that breaks the symmetry of the field distribution.
In conclusion, in order to mitigate both the odd order parasitic band due to iris resonances and the parasitic band due to higher order modes TM 120 /TM 210 the iris width w 3 must be decreased and the position p x must be increased according to Fig. 12(a).
The above-described effects can be clearly seen in the fullwave response of Fig. 12(b), where this phenomenon has been shown in four steps (#1, #2, #3, and #4) corresponding to the four curves related to different central iris widths (w 3 = 10.85 mm, 9.15 mm, 7.2 mm, and 6.5 mm). In order to maintain the same response in the filter band, the corresponding values of iris offset p x = 2.55 mm, 3.6 mm, 5.6875 mm, and 7 mm have been selected according to Fig. 12(a). The step #1 (w 3 = 10.85 mm and p x = 2.55 mm) is that of Fig. 10(c). According to Fig. 12(b), decreasing iris 3 width w 3 , its resonance frequency increases, assuming the values of 12.6 GHz, 13.9 GHz, 18.5 GHz, and 20.3 GHz. At the same time, the attenuation of the iris parasitic band at 12.25 GHz is increased, passing from less than 18 dB of Fig. 10(c) to about 35 dB. As can be seen in the zoomed part of the graph in Fig. 12(b), the central iris width reduction also has two additional desired effects: the increment of the attenuation of the TM 120 mode parasitic band (because of the decreasing of It should be noted that position p x changes the resonance frequency of the TM 210 mode. When the resonance frequency of the TM 210 mode reaches the higher parasitic band of TM 120 modes, it cooperates with TM 120 modes in reinforcing that parasitic band. This is what happens in step #2. However, keep decreasing the iris 3 width w 3 (steps #2 and #3) and increasing p x the resonance frequency of TM 210 mode arrives outside the TM 120 mode parasitic band and the problem disappears. Fig. 13 shows the response of the filter obtained with w 3 = 6.5 mm and p x = 7 mm, which shows the iris parasitic band suppression of about 32 dB and higher order TM 120 spurious modes suppression of 25 dB.

3) All Irises (Except the Central One) Positioned Close to the Cavity Center:
For an efficient suppression of the higher order mode parasitic band, in addition to the central coupling minimization obtained through the previous steps, the maximization of all other coupling values can be exploited. The maximization of the couplings between TM 120 modes is obtained by shifting the irises close to the cavity center and at the same time by increasing the iris width in order to recover the original coupling between fundamental TM 110 modes. This can be clearly seen in Fig. 14(b) where different curves for different values of the iris offsets p 1 and p 2 shown in Fig. 14(a) are plotted. The blue curve is the response of the structure suggested by the classical philosophy of reducing as much as possible the couplings to the spurious modes. Indeed, the central coupling has been already minimized by rotating the central iris, all other couplings are instead minimized when irises 1 and 2 are positioned halfway between the cavity center and the cavity side ( p 1 = h c1 /4 = 5.5 mm and p 2 = h c2 /4 = 5.635 mm). In theory, this position guarantees zero coupling between TM 120 modes. In practice, the coupling is very low (but not zero) and the parasitic band due to TM 120 modes is very narrowband but unfortunately, it reaches 0 dB, thus again demonstrating the weakness of the classical approach. The highest suppression of the parasitic band is instead related to the red curve obtained with the lower values of p 1 and p 2 ( p 1 = 3 mm and p 2 = 3 mm) that maximizes couplings to TM 120 spurious modes.
Actually, the attenuation of the higher order mode parasitic band obtained in Fig. 13 is quite good. This is because irises 1 and 2 (4 and 5) are already close to the cavity center ( p 1 = p 2 = 3 mm). However, as shown in Fig. 15, passing from p 1 = p 2 = 3 mm to p 1 = p 2 = 1.6 mm, the parasitic band attenuation passes from about 25 dB to about 30 dB.
The four-pole TM cavity filter structure with wide spurious free range is shown in Fig. 16(a) and its response is shown in Fig. 16(b). This has been obtained after further optimization. The iris resonances have been suppressed up to 25 dB and higher order mode spurious frequencies have been suppressed  up to 33 dB. The iris resonances (irises 2 and 4) initially present in Fig. 13 at around 12.2 GHz are now present at 10.65 GHz in Fig. 16(b).
This frequency shift is because iris width w 2 has been increased (decreasing p 2 ) during the optimization process for increasing the higher order mode parasitic band attenuation. As demonstrated in Fig. 14(b), this is a trade-off between achieving better higher order mode suppression and getting iris resonance close to the passband. An important aspect of TM filters is the presence of the TZs. It is then important to understand what happens to the TZs when the spurious self-suppression method is applied. According to [28], each TM resonator is potentially capable of a TZ that is generated by exploiting the nonresonating mode TE 101 for transporting some power that bypasses the resonant mode TM 110 . Here, in the following, we resume a few very simple rules regarding the TZs in TM cavities [28].
1) A TM cavity having input and output irises orthogonally placed does not allow the TE 101 mode to bypass the cavity and it does not generate a TZ. 2) The closer the iris to the cavity center, the higher the excitation of the TE 101 . A higher excitation of TE 101 results in a TZ closer to the passband. 3) If in a cavity, irises are both above (or both below) the cavity center, the TZ generated by the cavity is in the lower stopband. 4) If in a cavity, irises are one above and the other below the cavity center, the TZ generated by the cavity is in the upper stopband. Rule numbers 3 and 4 are a consequence of the change of sign of the coupling to the TM 110 mode when the position of one of the two irises passes from above to below the cavity center [28].
According to classical TM filter configuration of Fig. 8(a), all irises are aligned. This results in a number of TZs equal to the number of cavities. According to Fig. 9(a), when the classical spurious suppression method is applied, each cavity has irises orthogonally positioned. According to rule number 1, this results in no TZs. This can be clearly seen in Fig. 9(b) where no TZs are present close to the filter band. According to Fig. 16(a), when the proposed spurious selfsuppression is applied, the two central cavities have irises orthogonally positioned, losing the capability of generating the bypass coupling in the two central cavities. In terms of routing scheme, this means that the coupling between NRN A and B (as well as B and C) is equal to zero when the spurious self-suppression method is applied. In all other cavities, irises are instead aligned and the bypass coupling is maintained. This results in a number of TZs equal to N-2, where N is the number of cavities (corresponding to the filter order). This means that for the fourth-order filter of Fig. 16(a) two TZs are expected. Furthermore, considering that input and output irises in the first cavity (corresponding to irises 1 and 2 of the filter) and in the last cavity (corresponding to irises 4 and 5 of the filter) are all placed in the same side with respect to the cavity center, according to rule number 3, the two TZs must be placed in the lower stopband. Finally, considering that the first and last cavities are identical, the two TZs are positioned exactly at the same frequency [28] thus resulting in a double TZ. This can be clearly seen in Fig. 16(b). Fig. 17(a) shows instead the antisymmetric configuration, where irises in the left part of the filter are all below the cavity center, while those in the right part of the filter are all above the cavity center. In practice, this configuration has been obtained by rotating the first cavity (together with the relevant irises and feeding waveguide) of 180 • around the filter longitudinal axes. The response is plotted in Fig. 17(b) and as expected, the double TZ is still placed in the lower stopband; however, the response shows that the level of spike at 10.65 GHz has been slightly improved to 32 dB. For comparison purpose, the response of the classical filter already shown in Fig. 8(c) has been also included. The narrowband response comparison of the filter passband is shown in Fig. 18 along with the final optimized filter dimensions.
It is worth noting that, according to rule number 2, a fine control of the position of the TZs is possible by increasing/decreasing the distance of the irises (except the central one) from the cavity center. This is clearly shown in Fig. 14(b), where different responses with different positions of the TZs are shown as a function of p 1 and p 2 . Of course, the fine positioning is limited by the exigency of having p 1 and p 2 small as required by the spurious self-suppression. This means that it is not possible to put TZs very far away from the filter band.
Besides the precise positioning, it is also possible to change the TZ position from the lower to the upper stopband. This can be done by exploiting rule number 4, leading to the structure of Fig. 19(a). As shown in Fig. 19(b), with this configuration, the double TZs pass from the lower to the upper stopband, thus improving the filter selectivity in the upper part of the band. The additional spikes appearing at 17.4, 18, and 19.3 GHz that reaches the level of −17 dB, are due to the higher order modes present in the coupling irises (TE 20 ) and cavity mode (TM 220 ). However, those spikes are extremely narrowband, and they are visible only because no losses have been considered in the simulation. When metal losses (aluminum) were con- sidered, the spikes almost disappear, as can be seen in the response of Fig. 19(c). According to the author's experience, the simulation of filters with aluminum losses represents quite well the losses in silverplated manufactured filter. Indeed, in manufactured filters, additional sources of losses are present (e.g., roughness, assembling connections, etc.).

B. Sixth-Order Filter
In order to demonstrate that spurious self-suppression procedure can be also applied to higher order filters, the design of a six-pole TM cavity filter fed by the standard WR-90 waveguide, centered at 9.2 GHz having FBW of 1.5% is here reported. For comparison purpose, in Fig. 20, the classical TM filter is shown. In particular, Fig. 20(a) shows the classical six-pole TM geometry. Fig. 20(b) shows the routing scheme including filter passband (TM 110 ) with NRNs, filter parasitic band due to higher order modes (TM 120 ), and filter parasitic band due to coupling iris resonances (TE 10 ) mode. Fig. 20(c) shows instead the classical TM filter response with parasitic bands.
Analogously to what happens in the fourth-order filter, irises 1 and 7 resonances are not resonating in the same band of the other irises. This means that only five resonances are visible in the iris parasitic band positioned around 11-12 GHz. This can be seen in Fig. 20(c), where three RZs positioned in the Similarly, to what has been already done with fourth-order filters, the steps described in Section IV-A have also been applied to the sixth-order filters. This consists of the central iris orthogonally positioned far away from the cavity center and all other irises positioned close to the cavity center. After the application of the spurious suppression method, the number of TZs is equal to the filter order −2. This leads to four TZs in case of sixth-order filter. In order to show the flexibility in the positioning of TZs, two sixth-order filters with spurious free band up to 20 GHz has been successfully designed.
The first design of the proposed six-pole TM cavity filter with spurious self-suppression method is presented in Fig. 21(a). In this case, a double TZ is positioned in the lower and the other in the upper stopband. Indeed, according to the rules illustrated during the design of the fourth-order filter, cavity 1 produces a TZ in the lower stopband (as well as cavity 6), while cavity 2 produces a TZ in the upper stopband (as well as cavity 5). The filter response is shown in Fig. 21(b).
The second design is shown in Fig. 22. As can be seen from Fig. 22(a), all cavities capable of a TZ (cavities 1, 2, 5, and 6) are designed to produce TZs in the upper stopband. This is because all those cavities have irises with alternate positions with respect to the cavity center. In particular, because of the symmetry of the structure, cavities 1 and 6 produce a TZ in the same position, as well as cavities 2 and 5. This leads to a couple of double TZs in the upper stopband. Actually, in Fig. 22(b), the response shows that the double TZ at higher frequency has split into two close TZs. This is because NRNs have not been used here and there are some perturbations that can split double TZ [28], [33], [34].
Results show that iris spurious frequency level is −30 dB and higher order mode spurious frequency level is −24 dB, whereas stopband has been increased to 20 GHz. Iris 4 now resonates at 20 GHz along with other not suppressed higher order modes present in the cavity. Fig. 23 shows the final dimensions of the filter.
Two additional spikes at 17.728 and 18.253 GHz due to higher order modes present in the coupling irises (TE 20 ) and cavity mode (TM 220 ) are also present. Similarly, to that of the fourth-order filter, such spikes are extremely narrowband and practically disappear when metal losses are included. This is clearly shown in Fig. 24, where the simulated response includes the metal losses (aluminum).

V. MEASUREMENT OF THE PROPOSED SIXTH-ORDER TM CAVITY FILTER
In order to show the feasibility of the proposed approach, the sixth-order filter has been manufactured and measured. Photographs of the assembled filter and disassembled filter are shown in Fig. 25(a) and (b), respectively. One tuning screw of radius 1 mm is inserted in each cavity for the tuning of the resonance frequencies. No tuning screws for the coupling have been used. To support the manufacturing process, sharp  corners of cavities and irises have been changed into rounded corners. The thickness of the coupling slot is 0.5 mm, and this contributes to maintain the filter very compact. Fig. 26 shows the comparison between simulated and measured responses in the range of the passband. A slight frequency shift is present between measurement and simulation. The in-band return loss is 14 dB in the measured results, while it is 23 dB in the simulation. This difference is probably due to the manufacturing tolerances. It should be noted that the theoretical Q-factor of the final cavity considering aluminum as a lossy metal is about 3900. This corresponds to a realistic value of 2700 considering an efficiency of 70%. This leads to an insertion loss of 0.9 dB. The measured insertion loss is 1.4 dB. Of course, the filter performance can be improved by  using postprocessing techniques such as brazing, silverplating, etc. Some of such techniques have already been successfully applied to this kind of filters [38], [39]. However, this is beyond the scope of the paper that consists in demonstrating that the theory of the spurious self-suppression works, and it is applicable to real filters. This is well demonstrated by Fig. 27, where the wideband measured response of the filter is compared with the simulated response. Measurements are in good agreement with the simulations.

VI. CONCLUSION
In this article, a new technique called spurious selfsuppression method has been introduced. In contrast to the classical approach for suppressing higher order modes, where all couplings to spurious resonances are minimized as much as possible, the proposed strategy is instead based on a different approach. It is indeed shown that better results are obtained, when only the central coupling to the spurious resonances is minimized, whereas all others are maximized.
In order to demonstrate the practical applicability of such a technique, this method has been used for increasing the outof-band behavior of TM cavity filters of even order.
The practical application of self-suppression method to TM filters consists of three actions: 1) all irises are oriented in the same direction except the central one that is instead orthogonally positioned with respect to all others; 2) the central iris is positioned far away from the cavity center; and 3) all other irises are positioned close to the cavity center. By following those very simple rules, fourth-and sixth-order TM cavity filters centered at 9.2 GHz have been successfully designed with a spurious free band up to 20 GHz.
The sixth-order filter has been manufactured and measured. Measurements confirm the wide spurious free band up to 20 GHz, thus demonstrating the feasibility of the proposed approach.