Patent Publication Number: US-2016240905-A1

Title: Hybrid folded rectangular waveguide filter

Description:
TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to a group of resonators in rectangular waveguide (rectangular waveguide resonators) for use in a rectangular waveguide filter and to a rectangular waveguide filter employing the group of rectangular waveguide resonators. 
     The invention is particularly though not exclusively applicable to microwave filters in the front end of ground and satellite payloads for e.g. telecommunication, radar, Synthetic Aperture Radar (SAR), radiometers, radiolinks, etc. 
     BACKGROUND OF THE INVENTION 
     Microwave filters consisting of sections of rectangular waveguide (also referred to as microwave filters in rectangular waveguide) have been known for more than 50 years. In the most basic “in-line” implementation of such a microwave filter, as illustrated e.g. in  FIG. 14 , rectangular cavity resonators  1410 , i.e. sections of rectangular waveguide having a length corresponding to half a wavelength, are coupled to each other with small sections  1470  of rectangular waveguide below cut-off (inductive coupling windows) located in the input-output walls of each resonator. A discussion of such microwave filters, which are commonly used in the front end of many different types of payloads, including telecommunication, radars, SAR, radiometers, radiolinks, etc. is provided in M. Guglielmi, A. Melcon, Novel Design Procedure for Microwave Filters, Proceedings of the 23 rd  European Microwave Conference, 1993. 
     For all payloads a reduction in size, and in particular a reduction of the so-called “footprint”, which is the area occupied by the filter when seen in projection on a mounting surface, is a very important issue. This is especially the case for mobile applications and space applications, in which the available area of mounting space is severely limited and oftentimes has to be shared by multiple components. 
     Moreover, in many of the technical applications in which microwave filters are commonly used, there is the desire for being able to implement more complex transfer functions that go beyond standard Chebyshev transfer functions, such as transfer functions displaying phase equalization or transmission zeros at finite frequency. Such more complex transfer functions are discussed in R. Cameron, Advanced Filter Synthesis, IEEE Microwave Magazine, October 2011. However, microwave filters consisting of sections of rectangular waveguide as discussed above do not allow for the implementation of couplings between non-adjacent resonators (i.e. non-adjacent along the RF-path) because of their in-line structure. In consequence, such microwave filters do not allow for the implementation of the desired more complex transfer functions. 
     The latter issue has been addressed in the prior art by providing more complex filter designs. In J. R. Montejo-Garai, J. A. Ruiz-Cruz, J. M. Rebollar, M. J. Padilla-Cruz, A. Onoro-Navarro, I. Hidalgo-Carpintero, Synthesis and Design of In-Line N-Order Filters with N Real Transmission Zeros by Means of Extracted Poles Implemented in Low-Cost Rectangular H-Plane Waveguide, IEEE Transactions on Microwave Theory and Techniques, Vol. 53, No. 5, May 2005, additional resonators are added to the microwave filter, while a microwave filter structure is folded in the horizontal plane in J. A. Ruiz-Cruz, K. A. Zaki, J. R. Montejo-Garai, J. M. Rebollar, Rectangular Waveguide Elliptic Filters with Capacitive and Inductive Irises and Integrated Coaxial Excitation, International Microwave Symposium Digest, 2005 IEEE MTT-S. 
     Although both of the above approaches prove to be effective in implementing more complex transfer functions, they clearly fail in reducing the footprint of the filter. In fact, by adding additional resonators or by folding the filter structure in the horizontal plane, the above approaches undertaken in the prior art even tend to increase the footprint of the resulting microwave filter. 
     Moreover, microwave filters designed in accordance with the above prior art approaches may not be manufactured using the so-called clam-shell approach, according to which two matching halves are joined together to form the microwave filter. This configuration is particularly convenient from an electrical performance point of view because the surface defined by the mating of the two halves is not cut by any electrical current. Furthermore, the clam-shell approach enables particularly simple and inexpensive manufacture of microwave filters. As a consequence there is the additional problem in the prior art that manufacturing of filters that implement more complex transfer functions is comparably difficult and expensive. 
     Summarizing, at present there is no viable approach to providing a microwave rectangular waveguide filter that would allow for the implementation of more complex transfer functions and at the same time has a reduced footprint and can be manufactured in a simple manner. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to overcome the limitations of the prior art discussed above. It is another object of the invention to provide a rectangular waveguide filter with reduced size and reduced footprint. It is yet another object of the invention to provide a rectangular waveguide filter that allows for the implementation of more complex transfer functions beyond the standard Chebyshev transfer functions. It is yet another object of the invention to provide a rectangular waveguide filter that may be manufactured in a simple and inexpensive manner. 
     In view of the above objects, the present invention proposes a group of rectangular waveguide resonators and a rectangular waveguide filter comprising the group of rectangular waveguide resonators having the features of the respective independent claims. Preferred embodiments of the invention are described in the dependent claims. 
     In the below summary of aspects of the present invention, it is understood that a resonator in rectangular waveguide has a guide direction which defines a longitudinal direction of the resonator. Conventionally, the z-axis of a coordinate system used to describe the resonator is defined to extend along the longitudinal direction of the resonator. Further, the (transverse) cross section of the resonator perpendicular to the longitudinal direction of the resonator is referred to simply as the cross section of the resonator. An axis extending along the longitudinal direction and intersecting the cross section in its center is referred to as the central axis of the resonator. Walls of the resonator that extend in parallel to the longitudinal direction of the resonator are referred to as the lateral walls of the resonator, and walls that are perpendicular to the longitudinal direction are referred to as end walls. Lateral walls of the resonator that correspond to broad sides (i.e. longer sides) of the cross section are referred to as broad walls, or the top wall and the bottom wall of the resonator. Conventionally, the x-axis of the coordinate system is defined to extend in parallel to the broad sides of the cross section. In other words, the broad walls extend in a plane (referred to as the horizontal plane) spanned by the x-axis and the z-axis. Lateral walls of the resonator that correspond to narrow sides (i.e. shorter sides) of the cross section are referred to as narrow walls, or the side walls of the resonator. Conventionally, the y-axis of the coordinate system is defined to extend in parallel to the narrow sides of the cross section. In other words, the narrow walls extend in a plane spanned by the y-axis and the z-axis. Further, a width direction of the resonator is said to extend in parallel to the broad sides of the cross section (i.e. along the x-axis), and a height direction of the resonator is said to extend in parallel to the narrow sides of the cross section (i.e. along the y-axis). In the resonator as defined above, the electric field component E y  of the TE101 (TE 101 ) resonant mode is oriented along the height direction, while the magnetic field component Hz of the TE101 resonant mode is oriented along the guide direction, and the HX component of the magnetic field of the TE101 resonant mode is oriented along the width direction. Of course, in all filters described below, in addition to the TE101 mode all modes of a rectangular waveguide resonator, namely TE imn  and TM imn , where i, m, n are integers, can be used as well, if found convenient or desirable. 
     According to an aspect of the invention, a group of rectangular waveguide resonators for use in a rectangular waveguide filter is provided, the group comprising a first resonator and a second resonator, wherein the first and second resonators are arranged so that first lateral walls of the first resonator extend in parallel to second lateral walls of the second resonator, the first lateral walls corresponding to broad sides (longer sides) of a first cross section of the first resonator perpendicular to a guide direction of the first resonator and the second lateral walls corresponding to broad sides (longer sides) of a second cross section of the second resonator perpendicular to a guide direction of the second resonator, the first and second resonators are further arranged so that one of the first lateral walls at least partially faces one of the second lateral walls, and the first resonator is electromagnetically coupled to the second resonator through a first aperture in the one of the first lateral walls and a second aperture in the one of the second lateral walls. 
     According to the above configuration, the first and second resonators are arranged so that they at least partially overlap when seen in projection on a mounting surface which extends in parallel to the first lateral walls of the first resonator (i.e. the top and bottom walls, or the broad walls of the first resonator). Since the first and second resonators are overlapping, a length of the group of resonators is reduced. Thus, by employing the inventive group of resonators in a microwave filter, the footprint of the microwave filter can be reduced. 
     Moreover, in the inventive group of rectangular waveguide resonators the second resonator is arranged away from a horizontal plane in which the first resonator is arranged. As a consequence, a third resonator can be arranged next to (i.e. below) the second resonator along the central axis of the first resonator, so that the cross section of the third resonator is aligned with the cross section of the first resonator. The third resonator can then be electromagnetically coupled to the first resonator through apertures in the end walls of the first and third resonators. Accordingly, the present invention enables electromagnetic coupling between non-adjacent resonators (i.e. non-adjacent along the RF-path; here the first resonator and the third resonator are non-adjacent along the RF-path, assuming that the third resonator is also coupled to the second resonator), and non-standard transfer functions can be implemented without having to fold the microwave filter in the horizontal plane, i.e. without increasing the footprint of the microwave filter. 
     Lastly, by virtue of the inventive configuration, a microwave filter can be provided that implements a non-standard transfer function and that is at the same time symmetric with respect to a symmetry plane extending along the guide direction and the height direction of the first resonator (i.e. extending in parallel to the narrow walls of the first resonators, or along the z-axis and the y-axis). Such a microwave filter, due to its symmetry, can be manufactured by the clam-shell approach in which matching halves are manufactured and machined separately, and subsequently joined to form the microwave filter. Accordingly, a microwave filter employing the inventive group of rectangular waveguide resonators can be manufactured in a particularly simple and inexpensive manner. 
     Preferably, the first aperture and the second aperture have identical shape and the first and second resonators are further arranged so that the first and second apertures fall in line with each other. Further preferably, the first aperture has the shape of a rectangle extending over the full width of the first cross section in a width direction of the first resonator, and the second aperture has the shape of a rectangle extending over the full width of the second cross section in a width direction of the second resonator, the width direction of the first resonator being defined by the broad sides of the first cross section and the width direction of the second resonator being defined by the broad sides of the second cross section. 
     The first and second resonators may be further arranged so that the guide direction of the first resonator extends in parallel to the guide direction of the second resonator, lateral walls of the first resonator other than the first lateral walls extend in parallel to lateral walls of the second resonator Other than the second lateral walls, and the second resonator is shifted with respect to the first resonator in the guide direction of the first resonator. 
     In a preferred embodiment, the group of rectangular waveguide resonators further comprises a third resonator, wherein the third resonator is arranged so that a guide direction of the third resonator is aligned with the guide direction of the first resonator and the first cross section is aligned with a third cross section of the third resonator perpendicular to the guide direction of the third resonator (one of end walls of the first resonator faces one of end walls of the third resonator), and the third resonator is electromagnetically coupled to the second resonator. In particular, the third resonator may be further arranged so that one of third lateral walls of the third resonator at least partially faces the one of the second lateral walls, the third lateral walls corresponding to broad sides of the third cross section, and the second resonator is electromagnetically coupled to the third resonator through a third aperture in the one of the second lateral walls, the third aperture being distinct from the second aperture, and a fourth aperture in the one of the third lateral walls. 
     By the above inventive configuration, a third order filter (three pole filter) can be provided that is significantly shorter than a conventional three pole filter and that has significantly smaller footprint than the conventional three pole filter. Moreover, the above group of resonators is symmetric with respect to a symmetry plane extending along the width direction and the height direction, such that the group of resonators can be manufactured using the clam-shell approach, which enables particularly simple and inexpensive manufacture. 
     A particular advantage is achieved if the first resonator is electro-magnetically coupled to the third resonator through opposing apertures in the one of the end walls of the first resonator and the one of the end walls of the third resonator. Therein, the first resonator may be electromagnetically coupled to the third resonator through a ridge resonator interposed between the one of the end walls of the first resonator and the one of the end walls of the third resonator. Alternatively, the first resonator may be electromagnetically coupled to the third resonator through an inductive coupling section interposed between the one of the end walls of the first resonator and the one of the end walls of the third resonator, or through a hybrid coupling section interposed between the one of the end walls of the first resonator and the one of the end walls of the third resonator. Further, a first electrical length of the first resonator in the guide direction of the first resonator may be equal to half of a second electrical length of the second resonator in the guide direction of the second resonator and equal to a third electrical length of the third resonator in the guide direction of the third resonator. 
     By the above inventive configuration, a three pole filter with a non-standard transfer function can be provided that is significantly shorter than a comparable conventional filter, and that has significantly smaller footprint than the conventional filter. Depending on the choice of the coupling section interposed between coupling apertures in the one of the end walls of the first resonator and the one of the end walls of the third resonator, the transfer function of a filter employing the group of resonators features a transmission zero above or below the pass-band. For instance, for an inductive coupling section, and using a TE101 resonant mode for the first, second, and third resonators, a transmission zero of the transfer function above the pass-band of the filter is achieved. On the other hand, a transmission zero of the transfer function below the pass-band of the filter is achieved if a TE102 (TE 102 ) resonant mode is used for the second resonator and a TE101 resonant mode is used for the first resonator and the third resonator, respectively, since in this case the coupling between the first resonator and the third resonator becomes negative. Employing a ridge resonator as the coupling section, the transmission zero of the transfer function can be tuned to lie below or above the pass-band of the filter by adjusting the design parameters of the ridge resonator (i.e. a capacitance of a capacitive section of the ridge resonator and an inductance of an inductive section of the ridge resonator). Moreover, the above group of resonators is symmetric with respect to a symmetry plane extending along the guide direction and the height direction (i.e. extending in parallel to the narrow walls of the first resonator, or along the z-axis and the y-axis), such that the group of resonators can be manufactured using the clam-shell approach, which enables particularly simple and inexpensive manufacture. 
     In a further preferred embodiment, the group of rectangular waveguide resonators further comprises a third resonator and a fourth resonator, wherein the third resonator is arranged so that a guide direction of the third resonator is aligned with the guide direction of the second resonator and the second cross section is aligned with a third cross section of the third resonator perpendicular to the guide direction of the third resonator (one of end walls of the third resonator faces one of end walls of the second resonator), the fourth resonator is arranged so that a guide direction of the fourth resonator is aligned with the guide direction of the first resonator and the first cross section is aligned with a fourth cross section of the fourth resonator perpendicular to the guide direction of the fourth resonator (one of end walls of the first resonator faces one of end walls of the fourth resonator), the third and fourth resonators are further arranged so that third lateral walls of the third resonator extend in parallel to fourth lateral walls of the fourth resonator, the third lateral walls corresponding to broad sides of the third cross section and the fourth lateral walls corresponding to broad sides of the fourth cross section, the third and fourth resonators are further arranged so that one of the third lateral walls at least partially faces one of the fourth lateral walls, the second resonator is electromagnetically coupled to the third resonator through opposing apertures in one of end walls of the second resonator and one of end walls of the third resonator, and the third resonator is electromagnetically coupled to the fourth resonator through a third aperture in the one of the third lateral walls and a fourth aperture in the one of the fourth lateral walls. 
     By the above inventive configuration, a fourth order filter (four pole filter) can be provided that is significantly shorter than a conventional four pole filter and that has significantly smaller footprint than the conventional four pole filter. Moreover, the above group of resonators is symmetric with respect to a symmetry plane extending along the guide direction and the height direction (i.e. extending in parallel to the narrow walls of the first resonator, or along the z-axis and the y-axis), such that the group of resonators can be manufactured using the clam-shell approach, which enables particularly simple and inexpensive manufacture. 
     A particular advantage is achieved if the first resonator is electromagnetically coupled to the fourth resonator through opposing apertures in the one of the end walls of the first resonator and the one of the end walls of the fourth resonator. Therein, the first resonator may be electromagnetically coupled to the fourth resonator through a ridge resonator interposed between the one of the end walls of the first resonator and the one of the end walls of the fourth resonator. Alternatively, the first resonator may be electromagnetically coupled to the fourth resonator through an inductive coupling section interposed between the one of the end walls of the first resonator and the one of the end walls of the fourth resonator. 
     By the above inventive configuration, a four pole filter with a non-standard transfer function can be provided that is significantly shorter than a comparable conventional filter and that has significantly smaller footprint than the conventional filter. Depending on the choice of the coupling section interposed between the one of the end faces of the first resonator and the one of the end faces of the fourth resonator, the transfer function of a filter employing the group of resonators features a transmission zero above and below the pass-band, or phase equalization. For instance, employing the ridge resonator for coupling the first and fourth resonators results in a transmission zero below the pass-band of the filter and a transmission zero above the pass-band. By employing the inductive coupling section and appropriately tuning the width of the inductive coupling section, which is decisive for a strength of the electromagnetic coupling between the first and fourth resonators, phase equalization of the transfer function is achieved. Moreover, the above group of resonators is symmetric with respect to a symmetry plane extending along the guide direction and the height direction (i.e. extending in parallel to the narrow walls of the first resonator, or along the z-axis and the y-axis), such that the group of resonators can be manufactured using the clam-shell approach, which enables particularly simple and inexpensive manufacture. 
     In a further preferred embodiment, the group of rectangular waveguide resonators further comprises a third resonator, wherein the third resonator is arranged so that third lateral walls of the third resonator extend in parallel to the first lateral walls, the third lateral walls corresponding to broad sides of a third cross section of the third resonator perpendicular to a guide direction of the third resonator, the third resonator is further arranged so that one of the third lateral walls at least partially faces the other one of the first lateral walls, and the first resonator is electromagnetically coupled to the third resonator through a third aperture in the other one of the first lateral walls and a fourth aperture in the one of the third lateral walls. 
     In a yet further preferred embodiment, the group of rectangular waveguide resonators further comprises a third resonator and a fourth resonator, wherein the third resonator is arranged so that a guide direction of the third resonator is aligned with the guide direction of the first resonator, the first resonator is electromagnetically coupled to the third resonator, the fourth resonator is arranged so that third lateral walls of the third resonator extend in parallel to fourth lateral walls of the fourth resonator, the third lateral walls corresponding to broad sides of a third cross section of the third resonator perpendicular to the guide direction of the third resonator and the fourth lateral walls corresponding to broad sides of a fourth cross section of the fourth resonator perpendicular to the guide direction of the fourth resonator, the third and fourth resonators are further arranged so that one of the third lateral walls at least partially faces one of the fourth lateral walls, the third resonator is electromagnetically coupled to the fourth resonator through a third aperture in the one of the third lateral walls and a fourth aperture in the one of the fourth lateral walls, and the second resonator and the fourth resonator are arranged on opposite sides of a central axis of the first resonator extending along the guide direction of the first resonator. 
     By the above inventive configurations, microwave filters having customized transfer functions beyond the standard. Chebyshev transfer functions can be provided that are significantly shorter and have a significantly smaller footprint than conventional filters with comparable electrical performances. Moreover, the above groups of resonators are symmetric with respect to a symmetry plane extending along the guide direction and the height direction (i.e. extending in parallel to the narrow walls of the first resonator, or along the z-axis and the y-axis), such that the groups of resonators can be manufactured using the clam-shell approach, which enables particularly simple and inexpensive manufacture of the resulting microwave filters. 
     According to another aspect of the invention, a rectangular waveguide filter comprising the group of rectangular waveguide resonators is provided. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1A  is a perspective view of a rectangular waveguide filter according to a first embodiment of the invention; 
         FIG. 1B  is a sagittal cut through the filter of the first embodiment; 
         FIG. 1C  is a transverse cut through the filter of the first embodiment; 
         FIG. 1D  is a horizontal cut through the filter of the first embodiment; 
         FIG. 1E  illustrates an electrical performance of the filter of the first embodiment; 
         FIG. 2A  is a perspective view of a rectangular waveguide filter according to a second embodiment of the invention; 
         FIG. 2B  is sagittal cut through the filter of the second embodiment; 
         FIG. 2C  illustrates an electrical performance of the filter of the second embodiment; 
         FIG. 3A  is a perspective view of a rectangular waveguide filter according to a third embodiment of the invention; 
         FIG. 3B  is a sagittal cut through the filter of the third embodiment; 
         FIG. 3C  illustrates an electrical performance of the filter of the third embodiment; 
         FIG. 4A  is a perspective view of a rectangular waveguide filter according to a fourth embodiment of the invention; 
         FIG. 4B  is a sagittal cut through the filter of the fourth embodiment; 
         FIG. 4C  illustrates an electrical performance of the filter of the fourth embodiment; 
         FIG. 5A  is a perspective view of a rectangular waveguide filter according to a fifth embodiment of the invention; 
         FIG. 5B  is a sagittal cut through the filter of the fifth embodiment; 
         FIG. 5C  illustrates an electrical performance of the filter of the fifth embodiment; 
         FIG. 6A  is a perspective view of a ridge resonator structure; 
         FIG. 6B  is a horizontal cut through the ridge resonator structure; 
         FIG. 6C  is a sagittal cut through the ridge resonator structure; 
         FIGS. 6D and 6E  illustrate an electrical performance of the ridge resonator structure; 
         FIG. 7A  is a perspective view of a rectangular waveguide filter according to a sixth embodiment of the invention; 
         FIG. 7B  is a sagittal cut through the filter of the sixth embodiment; 
         FIG. 7C  illustrates an electrical performance of the filter of the sixth embodiment; 
         FIG. 8A  is a perspective view of a rectangular waveguide filter according to a seventh embodiment of the invention; 
         FIG. 8B  is a sagittal cut through the filter of the seventh embodiment; 
         FIG. 8C  illustrates an electrical performance of the filter of the seventh embodiment; 
         FIG. 9A  is a perspective view of a rectangular waveguide filter according to an eighth embodiment of the invention; 
         FIG. 9B  is a sagittal cut through the filter of the eighth embodiment; 
         FIG. 9C  illustrates an electrical performance of the filter of the eighth embodiment; 
         FIG. 10A  is a perspective view of a rectangular waveguide filter according to a ninth embodiment of the invention; 
         FIG. 10B  is a sagittal cut through the filter of the ninth embodiment; 
         FIG. 10C  is a first horizontal cut through the filter of the ninth embodiment; 
         FIG. 10D  is a second horizontal cut through the filter of the ninth embodiment; 
         FIG. 10E  illustrates an electrical performance of the filter of the ninth embodiment; 
         FIG. 11A  is a perspective view of a rectangular waveguide filter according to a tenth embodiment; 
         FIG. 11B  is a sagittal cut through the filter of the tenth embodiment; 
         FIG. 11C  illustrates an electrical performance of the filter of the tenth embodiment; 
         FIG. 12A  is a perspective view of a rectangular waveguide filter according to an eleventh embodiment; 
         FIG. 12B  is a sagittal cut through the filter of the eleventh embodiment; 
         FIG. 12C  illustrates an electrical performance of the filter of the eleventh embodiment; 
         FIG. 13A  is a perspective view of a six channel manifold multiplexer according to a twelfth embodiment; 
         FIG. 13B  is a sagittal cut through the multiplexer of the twelfth embodiment; 
         FIG. 13C  illustrates an electrical performance of the multiplexer of the twelfth embodiment; 
         FIG. 14A  is a perspective view of a fourth order rectangular waveguide filter according to the prior art; 
         FIG. 14B  is a sagittal cut through the filter of  FIG. 14A ; 
         FIG. 14C  is a horizontal cut through the filter of  FIG. 14A ; 
         FIG. 14D  illustrates an electrical performance of the filter of  FIG. 14A ; 
         FIG. 15A  is a perspective view of a third order rectangular waveguide filter according to the prior art; 
         FIG. 15B  is a sagittal cut through the filter of  FIG. 15A ; 
         FIG. 15C  is a horizontal cut through the filter of  FIG. 15A ; and 
         FIG. 15D  illustrates an electrical performance of the filter of  FIG. 15A . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Preferred embodiments of the present invention will be described in the following with reference to the accompanying figures, wherein in the figures identical objects are indicated by identical reference numbers. It is understood that the present invention shall not be limited to the described embodiments, and that the described features and aspects of the embodiments may be modified or combined to form further embodiments of the present invention. 
     In the following detailed description of the invention, it will be referred to microwave filters. Therein, the term microwave filter is considered to indicate a filter suitable for filtering electromagnetic radiation having a frequency range for which use of a rectangular waveguide is appropriate. 
     Moreover, in the figures discussed in the following, the views of waveguide filters relate to an RF-path view, i.e. only the confining faces of the electromagnetic field inside the filters are shown. That is, the actual physical walls of the filters are not shown in the figures. However, it is understood that for each confining face a corresponding wall is present. 
     First, a rectangular waveguide filter  100  according to a first embodiment of the invention will be described with reference to  FIGS. 1A to 1E .  FIG. 1A  is a perspective view of the rectangular waveguide filter  100  according to the first embodiment of the invention,  FIG. 1B  is a sagittal cut (i.e. a cut along the y-z-plane) through the rectangular waveguide filter  100 ,  FIG. 1C  is a transverse cut (i.e. a cut along the x-y-plane) through the rectangular waveguide filter  100 ,  FIG. 1D  is a horizontal cut (i.e. a cut along the x-z-plane) the rectangular waveguide filter  100 , and  FIG. 1E  illustrates the electrical performance of the rectangular waveguide filter  100 . 
     The rectangular waveguide filter  100  comprises a group of resonators of a first resonator  110  and a second resonator  120 , each of which is a rectangular waveguide resonator (a resonator formed by a section of rectangular waveguide, or a resonator in rectangular waveguide), interposed between an input port  160  and an output port  165 . The first resonator  110  is coupled to the input port  160  through a first coupling section  170 , and the second resonator  120  is coupled to the output port  175  through a second coupling section  175 . Exemplarily, inductive coupling sections (inductive coupling windows or inductive coupling irises) are illustrated as the first and second coupling sections  170 ,  175 . However, instead of inductive coupling sections, also alternative coupling sections that are readily apparent to the expert of skill in the art can be used for coupling the first and second resonators  110 ,  120  to the input and output ports  160 ,  165 , respectively, e.g. capacitive coupling sections (capacitive coupling windows or capacitive coupling irises) or hybrid coupling sections (hybrid coupling windows or hybrid coupling irises). 
     An inductive coupling section is understood as a coupling section having a rectangular cross section with a width of the cross section that is smaller than the width of the rectangular waveguide resonators that are coupled to each other by the inductive coupling section. The height of the cross section is equal to the height of the rectangular waveguide resonators. A capacitive coupling section is understood as a coupling section having a rectangular cross section with a height of the cross section that is smaller than the height of the rectangular waveguide resonators that are coupled to each other by the capacitive coupling section. The width of the cross section is equal to the width of the rectangular waveguide resonators. A hybrid coupling section is understood as a coupling section having a rectangular cross section with a width of the cross section that is smaller than the width of the rectangular waveguide resonators that are coupled to each other by the hybrid coupling section, and a height of the cross section that is smaller than the height of the rectangular waveguide resonators. 
     In the above, it is understood that the term “coupling” refers to electromagnetic coupling. Electromagnetic coupling of two resonators is understood to indicate a situation in which electromagnetic fields present in the two resonators can influence each other, i.e. an electromagnetic field can spread over both resonators. 
     Now, referring to  FIGS. 1A to 1D , directions with respect to a resonator of rectangular waveguide will be defined that shall be valid for all resonators throughout the remainder of the description of the present invention. A guide direction (or longitudinal direction) of the resonator is understood to be a direction defined by the longitudinal direction of the section of waveguide forming the respective resonator. In other words, the guide direction of the resonator extends in parallel to the Hz-component of the TE101 mode of the resonator. For instance, in  FIG. 1C , the guide directions of the first and second resonators  110 ,  120  extend in perpendicular to the paper plane. 
       FIG. 1C  illustrates a transverse cut though the rectangular waveguide filter  100  (i.e. a cut perpendicular to the guide directions of the first and second resonators  110 ,  120 ). In this figure, the upper rectangle represents the cross section of the second resonator  120  and the lower rectangle represents the cross section of the first resonator  110 . The view of  FIG. 1C  is from the left in  FIG. 1A . Vertical lines in the upper rectangle represent a coupling aperture through which the second resonator  120  is coupled to an output port. 
     A width direction of the resonator is perpendicular to the guide direction and is defined by the two broad ones (i.e. longer ones) of the four sides of a cross section of the resonator perpendicular to the guide direction (i.e. the transverse cross section, henceforth referred to simply as the cross-section). For instance, in  FIG. 1C , sides  111 A,  112 A of the cross section of the first resonator  110  define a width direction of the first resonator  110  and sides  121 A,  122 A of the cross section of the second resonator  120  define a width direction of the second resonator  120 . 
     A height direction of the resonator is perpendicular to the guide direction and to the width direction and is defined by the two narrow ones (i.e. shorter ones) of the four sides of the cross section. In other words, the height direction extends in parallel to the E y -component of the TE101 mode of the resonator. For instance, in  FIG. 1C , sides  113 A,  114 A of the cross section of the first resonator  110  define a height direction of the first resonator  110  and sides  123 A,  124 A of the cross section of the second resonator  120  define a height direction of the second resonator  120 . Lastly, a center line of the resonator is defined as a line extending in parallel to the guide direction and intersecting the cross section of the resonator in the center of the cross section. 
     The width of the first resonator  110  in its width direction is denoted by a 1  (i.e. the length of sides  111 A,  112 A), and the height of the first resonator  110  in its height direction is denoted by b 1  (i.e. the length of sides  113 A,  114 A). Likewise, the width of the second resonator  120  in its width direction is denoted by a 2  (i.e. the length of sides  121 A,  122 A), and the height of the second resonator in its height direction is denoted by b 2  (i.e. the length of sides  123 A,  124 A). By definition, we have a 1 &gt;b 1  and a 2 &gt;b 2 . Typically, resonators of rectangular waveguide have a ratio of height to width (aspect ratio) of 1:2. However, the present invention is applicable to resonators having arbitrary aspect ratio 1:x with x&gt;1. Further, the electrical length of the first resonator  110  in its guide direction is denoted by l 1  and the electrical length of the second resonator  120  in its guide direction is denoted by l 2 . Typically, resonators of rectangular waveguide have an electrical length that corresponds to an integer multiple of half the wavelength of the desired base mode of the resonator. 
     In the first embodiment, the electrical length l 1  of the first resonator  110  and the electrical length l 2  of the second resonator  120  are design parameters of the rectangular waveguide filter  100 . 
     The first resonator  110  is bounded by four lateral walls  111 ,  112 ,  113 ,  114  and two end walls  115 ,  116  which are all metallic walls. Lateral walls of the first resonator  110  are those walls of the first resonator  110  that extend in parallel to the guide direction of the first resonator  110 , whereas end walls of the first resonator  110  are those walls that extend in a plane perpendicular to the guide direction of the first resonator  110 . Of the four lateral walls  111 ,  112 ,  113 ,  114 , those two corresponding to broad sides (i.e. longer sides) of the cross section of the first resonator  110 , namely sides  111 A,  112 A, are the top wall  111  and bottom wall  112  of the first resonator  110  (first lateral walls, or broad walls of the first resonator). Accordingly, the top and bottom walls  111 ,  112  of the first resonator  110  extend in a plane spanned by the guide direction and the width direction of the first resonator  110  (i.e. spanned by the z-axis and the x-axis). On the other hand, of the four lateral walls  111 ,  112 ,  113 ,  114  those two corresponding to narrow sides (i.e. shorter sides) of the cross section of the first resonator  110 , namely sides  113 A,  114 A, are the left and right walls  113 ,  114  of the first resonator  110  (lateral walls of the first resonator other than the first lateral walls, or narrow walls of the first resonator). 
     Likewise, the second resonator  120  is bounded by four lateral walls  121 ,  122 ,  123 ,  124  and two end walls  125 ,  126  which are all metallic walls. Lateral walls of the second resonator  120  are those walls of the second resonator  120  that extend in parallel to the guide direction of the second resonator  120 , whereas end walls of the second resonator  120  are those walls that extend in a plane perpendicular to the guide direction of the second resonator  120 . Of the four lateral walls  121 ,  122 ,  123 ,  124 , those two corresponding to broad sides (i.e. longer sides) of the cross section of the second resonator  120 , namely sides  121 A,  122 A, are the top wall  121  and bottom wall  122  of the second resonator  120  (second lateral walls, or broad walls of the second resonator). Accordingly, the top and bottom walls  121 ,  122  of the second resonator  120  extend in a plane spanned by the guide direction and the width direction of the second resonator  120  (i.e. spanned by the z-axis and the x-axis). On the other hand, of the four lateral walls  121 ,  122 ,  123 ,  124  those two corresponding to narrow sides (i.e. shorter sides) of the cross section of the second resonator  120 , namely sides  123 A,  124 A, are the left and right walls  123 ,  124  of the second resonator  120  (lateral walls of the second resonator other than the second lateral walls, or narrow walls of the second resonator). 
     As can be seen in  FIGS. 1A and 1C , the first and second resonators  110 ,  120  have substantially identical width and height, i.e. a 1 =a 2  and b 1 =b 2 . Moreover, the first and second resonators  110 ,  120  are arranged so that their guide directions extend in parallel and also their width and height directions, respectively, extend in parallel. Further, the first and second resonators  110 ,  120  are arranged so that the narrow walls (i.e. the left and right walls  113 ,  114 ) of the first resonator  110  are aligned with the respective narrow walls (i.e. the left and right walls  123 ,  124 ) of the second resonator  120 . In other words, the second resonator  120  is shifted with respect to the first resonator  110  in the guide direction and in the height direction, but not in the width direction. Since the guide directions, width direction and height directions of the first and second resonators  110 ,  120 , respectively, extend in parallel to each other, in the following wherever applicable it will be referred simply to the guide direction, the width direction and the height direction without specifying one of the first and second resonators  110 ,  120 . 
     As can be seen in  FIGS. 1A and 1B , one of the broad walls of the first resonator  110  (one of the first lateral walls, i.e. one of the top and bottom walls  111 ,  112 ) partially faces one of the broad walls of the second resonator  120  (one of the second lateral walls, i.e. one of the top and bottom walls  121 ,  122 ). Specifically, the top wall  111  of the first resonator  110  partially faces the bottom wall  122  of the second resonator  120 . In other words, when seen along the height direction, the first and second resonators  110 ,  120  are partially overlapping. 
     As can be seen in  FIG. 1B , the top wall  111  of the first resonator  110  has an aperture  111 B (first aperture) and the bottom wall  122  of the second resonator has an aperture  122 B (second aperture). The first and second apertures  111 B,  122 B are of substantial identical shape and size. Specifically, the first and second apertures  111 B,  122 B have the shape of a rectangle that extends over the full width of the top wall  111  of the first resonator  110  and the bottom wall  122  of the second resonator  120 , respectively. The first and second apertures  111 B,  122 B are aligned with each other, i.e. the first and second apertures  111 B,  122 B fall in line with each other when seen along the height direction. In other words, each of connecting walls between corresponding boundaries of the first and second openings  111 B,  122 B would extend in parallel to respective ones of the narrow walls and the end walls of the first and second resonators  110 ,  120 . 
     The first resonator  110  is electromagnetically coupled to the second resonator  120  through the first aperture  111 B and the second aperture  122 B, for which reason the first and second apertures  111 B,  122 B may also be referred to as coupling apertures. In other words, the electromagnetic field present in the first resonator  110  may interact with the electromagnetic field present in the second resonator  120  through the first aperture  111 B and the second aperture  122 B. 
     The amount of shift of the second resonator  120  with respect to the first resonator  110  in the guide direction is a design parameter of the rectangular waveguide filter  100  and of the corresponding group of resonators, respectively. Likewise, the position along the guide direction of the first aperture  111 B in the top wall  111  of the first resonator  110  and the position along the guide direction of the second aperture  122 B in the bottom wall  122  of the second resonator  120  are design parameters of the rectangular waveguide filter  100  and of the corresponding group of resonators, respectively. 
     Between the top wall  111  of the first resonator  110  and the bottom wall  122  of the second resonator  120 , a connecting section  150  is provided, having four connecting walls between corresponding boundaries of the first and second apertures  111 B,  122 B, each of which extends in parallel to respective ones of the narrow walls and the end walls of the first and second resonators  110 ,  120 . That is, each of the four connecting walls extends in a respective plane perpendicular to the top wall  111  of the first resonator  110  and the bottom wall  122  of the second resonator  120 . The connecting walls of the connecting section  150  may simply result from a finite thickness d 1  of the top wall  111  of the first resonator  110  and a finite thickness d 2  of the bottom wall  122  of the second resonator  120 . In this case, a height of the connecting section  150  in the height direction is given by d 1 +d 2 . Alternatively, the connecting section  150  may have a height in the height direction that is larger than d 1 +d 2 . 
     Summarizing the configuration of the microwave filter according to the first embodiment, the first and second resonators  110 ,  120  are coupled to each other via the top and bottom walls rather than the end walls. Accordingly, a length of the resulting microwave filter, and consequently also a size of the projection of the resulting filter on a mounting surface extending in parallel to the top and bottom faces  111 ,  112 ,  211 ,  212  of the first and second resonators  110 ,  120  (i.e. the footprint of the filter) is reduced. On the other hand, the resulting microwave filter is symmetric with respect to a symmetry plane that extends along the guide direction and the height direction (i.e. along the z-axis and the y-axis), so that the microwave filter can be manufactured using the well-known clam-shell approach. According to the clam-shell approach, a filter is cut longitudinally in two symmetrical parts. Each of these parts is machined separately and the filter is realized by assembly of the two parts. Thus, the resulting microwave filter can be manufactured in a particularly simple and inexpensive manner. Also, using the inventive configuration, tuning screws can be included in the center of the resonators without difficulty. 
     Further, since the first and second resonators  110 ,  120  are provided at different levels along the height direction, i.e. shifted with respect to each other along the height direction, the first resonator  110  can be coupled to a third resonator that is arranged below the second resonator  120  and in-line with the first resonator  110 . Thus, couplings between non-adjacent resonators (i.e. non-adjacent along the RF-path, assuming that the third resonator is also coupled to the second resonator  120 ) become possible, which allows implementing more complex transfer functions that go beyond the standard Chebyshev transfer functions, such as transfer functions displaying transmission zeros at finite frequency, without having to fold the filter in the horizontal plane. Examples of rectangular waveguide filters featuring couplings between non-adjacent resonators will be presented below. 
     In the above, the second resonator  120  has been described to be arranged on top of the first resonator  110 . Alternatively, the first resonator  110  may be arranged on top of the second resonator  120 . In this case, the bottom wall  112  of the first resonator  110  would partially face the top wall  121  of the second resonator  120 , the first aperture would be provided in the bottom wall  112  of the first resonator  110 , and the second aperture would be provided in the top wall  121  of the second resonator  120 . 
     Further, in the above description the inventive group of resonators has been described to be interposed between the first and second coupling sections  170 ,  175 , coupling the first and second resonators  110 ,  120  to the input port  160  and the output port  165 , respectively. However, the inventive group of resonators can be used in any other filter configuration, i.e. interposed between further resonators or groups of resonators. Evidently, the advantage of a reduction of the footprint of the filter is likewise achieved if two adjacent resonators that are coupled via their end walls are replaced by the inventive group of resonators in which the resonators are coupled to each other via their top and bottom walls, respectively. In addition, also the advantage of being able to implement more complex transfer functions is achieved if two adjacent resonators that are coupled via their end walls are replaced by the inventive group of resonators. 
     Further filter configurations comprising the inventive group of resonators are discussed below. However, no limitation of the invention is intended by the particular choice of the presented filter configurations. 
       FIG. 1E  illustrates the electrical performance of the rectangular waveguide filter  100  of  FIGS. 1A to 1D . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  100  in units of dB. Graph  191  indicates the S 21 -component of the S-parameter, and graph  192  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  100 . As can be seen from  FIG. 1E , S 11  has two poles in the pass-band indicated by S 21  (in the figure at about 12.3 and 12.5 GHz). In the case of the rectangular waveguide filter  100 , S 21  does not have a transmission zero at finite frequency. 
     The group of resonators  100  shown in  FIGS. 1A to 1D  may be referred to as the basic building block of the invention. This basic building block can be used to implement a number of microwave filters according to the embodiments of the invention described below. Of these, the second embodiment relates to a third order filter comprising the basic building block, and the third embodiment relates to a fourth order filter comprising the basic building block. 
     A rectangular waveguide filter  200  according to the second embodiment of the invention will be described with reference to  FIGS. 2A to 2C .  FIG. 2A  is a perspective view of the rectangular waveguide filter  200 ,  FIG. 2B  is a sagittal cut through the rectangular waveguide filter  200 , and  FIG. 2C  illustrates the electrical performance of the rectangular waveguide filter  200 . 
     The rectangular waveguide filter  200  comprises a group of resonators of a first resonator  210 , a second resonator  220  and a third resonator  230 , each of which is a rectangular waveguide resonator, interposed between an input port  260  and an output port  265 . The first resonator  210  is coupled to the input port  260  through a first coupling section  270 , and the third resonator  230  is coupled to the output port  265  through a second coupling section  275 . Exemplarily, inductive coupling sections are illustrated as the first and second coupling sections  270 ,  275 . However, instead of inductive coupling sections, also alternative coupling sections that are readily apparent to the expert of skill in the art can be used for coupling the first and third resonators  210 ,  230  to the input and output ports  260 ,  265 , respectively, e.g. capacitive coupling sections or hybrid coupling sections. 
     Accordingly, the group of resonators of the second embodiment of the invention differs from the group of resonators of the first embodiment by the presence of the third resonator  230 . 
     For a definition of the walls and the relative arrangement of the first and second resonators  210 ,  220  it is referred to the above description of the first and second resonators  110 ,  120  of the first embodiment. Thus, the second resonator  220  is arranged on top of the first resonator  210  so that the one of the broad walls of the first resonator  210  (one of the first lateral walls of the firsf resonator, i.e. one of the top and bottom walls  211 ,  212 ) partially faces one of the broad walls of the second resonator  220  (one of the second lateral walls of the second resonator, i.e. one of the top and bottom walls  221 ,  222 ). Specifically, the top wall  211  of the first resonator  210  partially faces the bottom wall  222  of the second resonator  220 . Further, the first aperture  211 B is provided in the top wall  211  of the first resonator  210 , the second aperture  222 B is provided in the bottom wall  222  of the second resonator  220 , and the first resonator  210  is electromagnetically coupled to the second resonator  220  through the first aperture  211 B and the second aperture  222 B. 
     As in the first embodiment, the first and second resonators  210 ,  220  have substantially identical width and height, i.e. a 1 =a 2  and b 1 =b 2 . Moreover, the first and second resonators  210 ,  220  are arranged so that their guide directions extend in parallel and also their width and height directions, respectively, extend in parallel. Further, the first and second resonators  210 ,  220  are arranged so that the narrow walls of the first resonator  210  (the lateral walls of the first resonator other than the first lateral walls, i.e. the left and right walls  213 ,  214 ) are aligned with the respective narrow walls of the second resonator  220  (the lateral walls of the second resonator other than the second lateral walls, i.e. the left and right walls  223 ,  224 ). In other words, the second resonator  220  is shifted with respect to the first resonator  210  in the guide direction and in the height direction, but not in the width direction. 
     Summarizing, also in the second embodiment, the first and second resonators  210 ,  220  are provided at different levels along the height direction and coupled to each other via their top and bottom walls rather than their end walls. 
     The third resonator  230  is bounded by four lateral walls  231 ,  232 ,  233 ,  234  and two end walls  235 ,  236  which are all metallic walls. Lateral walls of the third resonator  230  are those walls of the third resonator  230  that extend in parallel to the guide direction of the third resonator  230 , whereas end walls of the third resonator  230  are those walls that extend in a plane perpendicular to the guide direction of the third resonator  230 . Of the four lateral walls  231 ,  232 ,  233 ,  234 , those two corresponding to broad sides (i.e. longer sides) of the cross section of the third resonator  230  are the top wall  231  and bottom wall  232  of the third resonator  230  (third lateral wills, or broad walls of the third resonator). Accordingly, the top and bottom walls  231 ,  232  of the third resonator  230  extend in a plane spanned by the guide direction and the width direction of the third resonator  230  (i.e. spanned by the z-axis and the x-axis). On the other hand, of the four lateral walls  231 ,  232 ,  233 ,  234  those two corresponding to narrow sides (i.e. shorter sides) of the cross section of the third resonator  230  are the left and right walls  233 ,  234  of the third resonator  230  (lateral walls of the third resonator other than the third lateral walls, or narrow walls of the third resonator). 
     As can be seen in  FIG. 2A , the first, second and third resonators  210 ,  220 ,  230  have substantially identical width and height, i.e. a 1 =a 2 =a 3  and b 1 =b 2 =b 3 , wherein the width of the third resonator  230  in its width direction is denoted by a 3 , and the height of the third resonator  230  in its height direction is denoted by b 3 . 
     In the second embodiment, an electrical length l 1  of the first resonator  210 , an electrical length l 2  of the second resonator  220 , and an electrical length l 3  of the third resonator  230  are design parameters of the rectangular waveguide filter  200 . 
     The third resonator  230  is arranged with respect to the first and second resonators  210 ,  220  so that its guide direction extends in parallel to the guide directions of the first and second resonators  210 ,  220 , and also its width direction and height direction, extends in parallel to the width directions and height directions, respectively, of the first and second resonators  210 ,  220 . Since the guide directions, width direction and height directions of the first, second and third resonators  210 ,  220 ,  230 , respectively, extend in parallel to each other, it the following wherever applicable it will be referred simply to the guide direction, the width direction and the height direction without specifying one of the first, second and third resonators  210 ,  220 ,  230 . 
     The third resonator  230  is further arranged so that the narrow walls of the third resonator  230  (the lateral walls of the third resonator other than the third lateral walls, i.e. the left and right walls  233 ,  234 ) are aligned with the respective narrow walls of the first and second resonators  210 ,  220  (the lateral walls of the first and second resonators other than the first and second lateral walls, i.e. the left and right walls  213 ,  214 ,  223 ,  224 ). In other words, the third resonator  230  is shifted with respect to the first resonator  210  and the second resonator  220  in the guide direction and in the height direction, but not in the width direction. Specifically, the third resonator  230  is arranged relative to the first and second resonators  210 ,  220  so that one of the end walls  235 ,  236  of the third resonator  230  faces one of the end walls  215 ,  216  of the first resonator  210 , and so that one of the broad walls of the third resonator  230  (one of the third lateral walls, i.e. one of the top and bottom walls  231 ,  232 ) partially faces the one of the broad walls of the second resonator  220  (the one of the second lateral walls of the second resonator, i.e. the one of the top and bottom walls  221 ,  222 ). Specifically, the top wall  231  of the third resonator  230  partially faces the bottom wall  222  of the second resonator  220 . In other words, the third resonator  230  is arranged so that its cross section is aligned with the cross section of the first resonator  210  and so that it is arranged below the second resonator  220 , i.e. so that when seen along the height direction, the second and third resonators  220 ,  230  are partially overlapping. Thus, the first and third resonators  210 ,  230  are arranged at a first level along the height direction and the second resonator  220  is arranged at a second level along the height direction different from the first level. 
     As can be seen from  FIGS. 2A and 2B , the bottom wall  222  of the second resonator  220  has a third aperture  222 C which is distinct from the second aperture  222 B, and the top wall  231  of the third resonator  230  has a fourth aperture  231 B. The third and fourth apertures  222 C,  231 B are of substantial identical shape and size. Specifically, the third and fourth apertures  222 C,  231 B have the shape of a rectangle that extends over the full width of the bottom wall  222  of the first resonator  220  and the top wall  231  of the third resonator  230 , respectively. The third and fourth apertures  222 C,  231 B are aligned with each other, i.e. the third and fourth apertures  222 C,  231 B fall in line with each other when seen along the height direction. In other words, connecting walls between corresponding boundaries of the third and fourth apertures  222 C,  231 B would extend in parallel to respective ones of the narrow walls and the end walls of the first, second, and third resonators  210 ,  220 ,  230 . 
     The second resonator is electromagnetically coupled to the third resonator through the third aperture  222 C and the fourth aperture  231 B, for which reason the third and fourth apertures  222 C,  231 B may also be referred to as coupling apertures. In other words, the electromagnetic field present in the second resonator may interact with  11 D the electromagnetic field present in the third resonator through the third aperture  222 C and the fourth aperture  231 B. Thus, also the second and third resonators  220 ,  230  are coupled to each other via their top and bottom walls rather than their end walls. 
     In the above, the shift of the second resonator  220  with respect to the first resonator  210  in the guide direction and the shift of the third resonator  230  with respect to the second resonator  220  in the guide direction are design parameters of the rectangular waveguide filter  200  and of the corresponding group of resonators, respectively. Likewise, the position along the guide direction of the first aperture  211 B in the top wall  211  of the first resonator  210 , the position along the guide direction of the second aperture  222 B in the bottom wall  222  of the second resonator  220 , the position along the guide direction of the third aperture  222 C in the bottom wall  222  of the second resonator  220 , and the position along the guide direction of the fourth aperture  231 B in the top wall  231  of the third resonator  230  are design parameters of the rectangular waveguide filter  200  and of the corresponding group of resonators, respectively. 
     Between the top wall  211  of the first resonator  210  and the bottom wall  222  of the second resonator  220 , and between the top wall  231  of the third resonator  230  and the bottom wall  222  of the second resonator  220 , connecting sections  250 ,  255 , respectively, are provided, each connection section  250 ,  255  having four connecting walls between corresponding boundaries of the first and second openings  221 B,  222 B, and the third and fourth openings  222 C,  231 B, respectively, each of which extends in parallel to respective ones of the narrow walls and the end walls of the first, second, and third resonators  210 ,  220 ,  230 . That is, each of the four connecting walls extends in a respective plane perpendicular to e.g. the top wall  221  of the first resonator  210 , the bottom wall  222  of the second resonator  220 , and the top wall  231  of the third resonator  230 . The connecting walls of the connecting sections  250 ,  255  may simply result from a finite thickness d 1  of the top wall  211  of the first resonator  210  and a finite thickness d 2  of the bottom wall  222  of the second resonator  220 , or the finite thickness d 2  of the bottom wall  222  of the second resonator  220  and a finite thickness d 3 =d 1  of the top wall  231  of the third resonator  230 . In this case, a height of the connecting sections  250 ,  255  in the height direction is given by d 1 +d 2 . Alternatively, the connecting sections  250 ,  255  may have a height in the height direction that is larger than d 1 +d 2 . 
     In the above, the inventive group of resonators has been described to be interposed between the input port  260  and the output port  265 . However, the inventive group of resonators can be used in any other filter configuration, i.e. interposed between further resonators or groups of resonators. Analogous statements are understood to apply also to the further embodiments of the invention that will be described below, and will not be repeated. 
       FIG. 2C  illustrates the electrical performance of the rectangular waveguide filter  200  of  FIGS. 2A and 2B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  200  in units of dB. Graph  291  indicates the S 21 -component of the S-parameter, and graph  292  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  200 . As can be seen from  FIG. 2C , S 11  has three poles in the pass-band indicated by S 21  (in the figure at about 12.6, 12.85 and 13.1 GHz). In the case of the rectangular waveguide filter  200 , S 21  does not have a transmission zero at finite frequency. 
     Thus, the rectangular waveguide filter  200  of the second embodiment is a three pole filter (third order filter). A conventional three pole filter  1500  known in the art is illustrated in  FIGS. 15A to 15D , of which  FIG. 15A  is a perspective view of the conventional three pole filter  1500 ,  FIG. 15B  is a sagittal cut through the conventional three pole filter  1500 ,  FIG. 15C  is a horizontal cut through the conventional three pole filter  1500 , and  FIG. 15D  illustrates the electrical performance of the conventional three pole filter  1500 . 
     The conventional three pole filter  1500  comprises a group of three rectangular waveguide resonators  1510  interposed between an input port  1560  and an output port  1565 , and coupled to each other and to the input/output ports  1560 ,  1565  by inductive coupling sections  1570 . 
     In  FIG. 15D , the abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the conventional three pole filter  1500  in units of dB. Graph  1591  indicates the S 21 -component of the S-parameter, and graph  1592  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the conventional three pole filter  1500 . As can be seen from a comparison of  FIG. 2C  and  FIG. 15D , the rectangular waveguide filter  200  and the conventional three pole filter′ 1500  have comparable electrical performances. 
     On the other hand, the rectangular waveguide filter  200  is significantly shorter than the conventional three pole filter  1500 . For a center frequency of the pass-band of about 12.8 GHz, the rectangular waveguide filter  200  has a length of about 28.92 mm, whereas the conventional three pole filter  1500  has a length of about 40.64 mm. Thus, by employing the inventive group of resonators, a length reduction as well as a corresponding reduction of footprint for a three pole filter of about 29% can be achieved. 
     Next, a rectangular waveguide filter  300  according to a third embodiment of the invention will be described with reference to  FIGS. 3A to 3C .  FIG. 3A  is a perspective view of the rectangular waveguide filter  300 ,  FIG. 3B  is a sagittal cut through the rectangular waveguide filter  300 , and  FIG. 3C  illustrates the electrical performance of the rectangular waveguide filter  300 . 
     The rectangular waveguide filter  300  comprises a group of resonators of a first resonator  310 , a second resonator  320 , a third resonator  330 , and a fourth resonator  340 , each of which is a rectangular waveguide resonator, interposed between an input port  360  and an output port  365 . The first resonator  310  is coupled to the input port  360  through a first coupling section  370 , and the fourth resonator  340  is coupled to the output port  365  through a second coupling section  375 . Exemplarily, inductive coupling sections are illustrated as the first and second coupling sections  370 ,  375 . However, instead of inductive coupling sections, also alternative coupling sections that are readily apparent to the expert of skill in the art can be used for coupling the first and fourth resonators  310 ,  340  to the input and output ports  360 ,  365 , respectively, e.g. capacitive coupling sections or hybrid coupling sections. 
     Accordingly, the group of resonators in the third embodiment of the invention differs from the group of resonators in the first embodiment by the presence of the third resonator  330  and the fourth resonator  340 . 
     For a definition of the faces and the relative arrangement of the first and second resonators  310 ,  320  it is referred to the above description of the first embodiment. As in the first and second embodiments, without intended limitation, the second resonator  320  is arranged on top of the first resonator  310 . Thus, one of the broad walls of the first resonator  310  (one of the first lateral walls of the first resonator, i.e. one of the top and bottom walls  311 ,  312 ) partially faces one of the broad walls of the second resonator  320  (one of the second lateral walls of the second resonator, i.e. one of top and bottom walls  321 ,  322 ). Specifically, the top wall  311  of the first resonator  310  partially faces the bottom wall  322  of the second resonator  320 . Further, a first aperture  311 B is provided in the top wall  311  of the first resonator  310 , a second aperture  322 B is provided in the bottom wall  322  of the second resonator  320 , and the first resonator  310  is electromagnetically coupled to the second resonator  320  through the first aperture  311 B and the second aperture  322 B. As regards alignment of directions and walls of the first and second resonators  310 ,  320 , it is referred to the description of the first embodiment. 
     Summarizing, also in the third embodiment, the first and second resonators  310 ,  320  are provided at different levels along the height direction and coupled to each other via their top and bottom walls rather than their end walls. 
     For a definition of the walls of the third resonator  330  it can be referred to the above description of the second embodiment, wherein however in the third embodiment the arrangement of the third resonator  330  with respect to the first and second resonators  310 ,  320  differs from the arrangement in the second embodiment. The arrangement of the third resonator  330  with respect to the first and second resonators  310 ,  320  will be described below. 
     The fourth resonator  340  is bounded by four lateral walls  341 ,  342 ,  343 ,  344  and two end walls  345 ,  346  which are all metallic walls. Lateral walls of the fourth resonator  340  are those walls of the fourth resonator  340  that extend in parallel to the guide direction of the fourth resonator  340 , whereas end walls of the fourth resonator  340  are those walls that extend in a plane perpendicular to the guide direction of the fourth resonator  340 . Of the four lateral walls  341 ,  342 ,  343 ,  344 , those two corresponding to broad sides (i.e. longer sides) of the cross section of the fourth resonator  340  are the top wall  341  and bottom wall  342  of the fourth resonator  340  (fourth lateral walls, or broad walls of the fourth resonator). Accordingly, the top and bottom walls  341 ,  342  of the fourth resonator  340  extend in a plane spanned by the guide direction and the width direction of the fourth resonator  340  (i.e. spanned by the z-axis and the x-axis). On the other hand, of the four lateral walls  341 ,  342 ,  343 ,  344  those two corresponding to narrow sides (i.e. shorter sides) of the cross section of the fourth resonator  340  are the left and right walls  343 ,  344  of the fourth resonator  340  (lateral walls of the fourth resonator other than the fourth lateral walls, or narrow walls of the fourth resonator). 
     As can be seen in  FIG. 3A , the first, second, third and fourth resonators  310 ,  320 ,  330 ,  340  have substantially identical width and height, i.e. a 1 =a 2 =a 3 =a 4  and b 1 =b 2 =b 3 =b 4 , wherein the width of the fourth resonator  340  in its width direction is denoted by a 4 , and the height of the fourth resonator  340  in its height direction is denoted by b 4 . 
     In the third embodiment, an electrical length l 1  of the first resonator  310 , an electrical length l 2  of the second resonator  320 , an electrical length l 3  of the third resonator  330 , and an electrical length of the fourth resonator  340  are design parameters of the rectangular waveguide filter  300 . 
     The third and fourth resonators  330 ,  340  are arranged with respect to the first and second resonators  310 ,  320  so that their guide directions extend in parallel to the guide directions of the first and second resonators  310 ,  320 , and also their width directions and height directions, respectively, extend in parallel to the width directions and height directions of the first and second resonators  310 ,  320 . Since the guide directions, width directions and height directions of the first, second, third and fourth resonators  310 ,  320 ,  330 ,  340 , respectively, extend in parallel to each other, in the following wherever applicable it will be referred simply to the guide direction, the width direction and the height direction without specifying one of the first, second, third and fourth resonators  310 ,  320 ,  330 ,  340 . 
     The third and fourth resonators  330 ,  340  are further arranged so that the narrow walls of the third resonator  330  and the narrow walls of the fourth resonator  340  are aligned with the respective narrow walls of the first and second resonators  310 ,  320 . 
     The third resonator  330  is further arranged relative to the first and second resonators  310 ,  320  so that one of the end walls  335 ,  336  of the third resonator  330  faces one of the end walls  325 ,  326  of the second resonator  320 . The fourth resonator  340  is arranged relative to the first, second and third resonators  310 ,  320 ,  330  so that one of the end walls  345 ,  346  of the fourth resonator  340  faces one of the end walls  315 ,  316  of the first resonator  310 , and so that one of the broad walls of the fourth resonator  340  (one of the fourth lateral walls, i.e. one of the top and bottom walls  341 ,  342 ) partially faces one of the broad walls of the third resonator  330  (one of the third lateral walls, i.e. one of the top and bottom walls  331 ,  332 ). Specifically, the top wall  341  of the fourth resonator  340  partially faces the bottom wall  332  of the third resonator  330 . 
     In other words, the third resonator  330  is arranged so that its cross section is aligned with the cross section of the second resonator  320 . The fourth resonator  340  is arranged so that its cross section is aligned with the cross section of the first resonator  310 , and so that it is arranged below the third resonator  330 , i.e. so that when seen along the height direction, the third and fourth resonators  330 ,  340  are partially overlapping. 
     Thus, the third resonator  330  is shifted with respect to the first resonator  310  in the guide direction and in the height direction, but not in the width direction, and with respect to the second resonator  320  in the guide direction, but not in the width direction or the height direction. The fourth resonator  340  is shifted with respect to the first resonator  310  in the guide direction, but not in the width direction or the height direction, and with respect to the second resonator  320  in the guide direction and in the height direction, but not in the width direction. Put differently, the first and fourth resonators  310 ,  330  are arranged at a first level along the height direction and the second and third resonators  320 ,  330  are arranged at a second level along the height direction different from the first level. 
     The third resonator  330  is electromagnetically coupled to the second resonator  320  through an inductive coupling section  385  interposed between an aperture (coupling aperture) in the one of the end walls  325 ,  326  of the second resonator  320  and an aperture (coupling aperture) in the one of the end walls  335 ,  336  of the third resonator  330 . Although an inductive coupling section  385  is exemplarily shown in  FIGS. 3A and 3B , also an alternative coupling section that is readily apparent to the expert of skill in the art can be used for coupling the second and third resonators  320 ,  330 , such as a capacitive coupling section or a hybrid coupling section. 
     As can be seen from  FIGS. 3A and 3B , the bottom wall  332  of the third resonator  330  has a third aperture  332 B, and the top wall  341  of the fourth resonator  340  has a fourth aperture  341 B. The third and fourth apertures  332 B,  341 B are of substantial identical shape and size. Specifically, the third and fourth apertures  332 B,  341 B have the shape of a rectangle that extends over the full width of the bottom wall  332  of the third resonator  330  and the top wall  341  of the fourth resonator  340 , respectively. The third and fourth apertures  332 B,  341 B are aligned with each other, i.e. the third and fourth apertures  332 B,  341 B fall in line with each other when seen along the height direction. In other words, each of connecting walls between corresponding boundaries of the third and fourth apertures  332 B,  341 B would extend in parallel to respective ones of the narrow walls and the end walls of the first, second, third, and fourth resonators  310 ,  320 ,  330 ,  340 . 
     The third resonator is electromagnetically coupled to the fourth resonator through the third aperture  332 B and the fourth aperture  341 B, which for this reason may also be referred to as coupling apertures. In other words, the electromagnetic field present in the third resonator may interact with the electromagnetic field present in the fourth resonator through the third aperture  332 B and the fourth aperture  341 B. 
     Thus, also the third and fourth resonators  330 ,  340  are coupled to each other via their top and bottom walls rather than their end walls. 
     In the above, the shift of the second resonator  320  with respect to the first resonator  310  in the guide direction and the shift of the fourth resonator  340  with respect to the third resonator  330  in the guide direction are design parameters of the rectangular waveguide filter  300  and of the corresponding group of resonators, respectively. Likewise, the position along the guide direction of the first aperture  311 B in the top wall  311  of the first resonator  310 , the position along the guide direction of the second aperture  322 B in the bottom wall  322  of the second resonator  320 , the position along the guide direction of the third aperture  332 B in the bottom wall  332  of the third resonator  330 , and the position along the guide direction of the fourth aperture  341 B in the top wall  341  of the fourth resonator  340  are design parameters of the rectangular waveguide filter  300  and of the corresponding group of resonators, respectively. 
     Between the top wall  311  of the first resonator  310  and the bottom wall  322  of the second resonator  320 , and between the top wall  341  of the fourth resonator  340  and the bottom wall  332  of the third resonator  330 , connecting sections  350 ,  355  are provided, each connecting section  350 ,  355  having four connecting walls between corresponding boundaries of the first and second apertures  321 B,  322 B, and the third and fourth apertures  332 B,  341 B, respectively, each of which extends in parallel to respective ones of the narrow walls and the end walls of the first, second, third, and fourth resonators  310 ,  320 ,  330 ,  340 . That is, each of the four connecting walls extends in a respective plane perpendicular to e.g. the top wall  311  of the first resonator  310 , the bottom wall  322  of the second resonator  320 , the bottom wall  322  of the third resonator  330 , and the top wall  341  of the fourth resonator  340 . The connecting walls of the connecting sections  350 ,  355  may simply result from a finite thickness d 1  of the top wall  311  of the first resonator  310  and a finite thickness d 2  of the bottom wall  322  of the second resonator  320 , or a finite thickness d 3 =d 2  of the bottom wall  332  of the third resonator  330 , and a finite thickness d 4 =d 1  of the top wall  341  of the fourth resonator  340 . In this case, a height of the connecting sections  350 ,  355  in the height direction is given by d 1 +d 2 . Alternatively, the connecting sections  350 ,  355  may have a height in the height direction that is larger than d 1 +d 2 . 
       FIG. 3C  illustrates the electrical performance of the rectangular waveguide filter  300  of  FIGS. 3A and 3B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  300  in units of dB. Graph  391  indicates the S 21 -component of the S-parameter, and graph  392  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  300 . As can be seen from  FIG. 3C , S 11  has four poles in the pass-band indicated by S 21  (in the figure at about 12.25, 12.35, 12.45, and 12.55 GHz). In the case of the rectangular waveguide filter  300 , S 21  does not have a transmission zero at finite frequency. 
     Thus, the rectangular waveguide filter  300  of the third embodiment is a four pole filter (fourth order filter). A conventional four pole filter  1400  known in the art is illustrated in  FIGS. 14A to 14D , of which  FIG. 14A  is a perspective view of the conventional four pole filter  1400 ,  FIG. 14B  is a sagittal cut through the conventional four pole filter  1400 ,  FIG. 13C  is a horizontal cut through the conventional four pole filter  1400 , and  FIG. 14D  illustrates the electrical performance of the conventional four pole filter  1400 . 
     The conventional four pole filter  1400  comprises a group of four rectangular waveguide resonators  1410  interposed between an input port  1460  and an output port  1465 , and coupled to each other and to the input/output ports  1460 ,  1465  by inductive coupling sections  1470 . 
     The electrical performance of the conventional four pole filter  1400  is illustrated in  FIG. 14D , in which the abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the conventional four pole filter  1400  in units of dB. Graph  1491  indicates the S 21 -component of the S-parameter, and graph  1492  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the conventional four pole filter  1400 . As can be seen from a comparison of  FIG. 3C  and  FIG. 14D , the rectangular waveguide filter  300  and the conventional four pole filter  1400  have comparable electrical performances. 
     On the other hand, the rectangular waveguide filter  300  is significantly shorter than the conventional four pole filter  1400 . For a center frequency of the pass-band of about 12.35 GHz, the rectangular waveguide filter  300  has a length of about 41.20 mm, whereas the conventional four pole filter  1400  has a length of about 61.04 mm. Thus, by employing the inventive group of resonators, a length reduction as well as a corresponding reduction of footprint for a four pole filter of about 32% can be achieved. 
     Next, the distribution of electric field intensity in the conventional four pole inductive filter  1400  and the four pole filter  300  of the third embodiment will be described. It turns out that the maximum electric field strength inside the inventive four pole filter  300  is only 15% higher than the maximum electric field strength inside the conventional four pole filter  1400 . This indicates that a similar difference can be expected both in terms of maximum power and insertion loss capabilities. 
     As has been described above, the present invention allows for a reduction of the length and footprint of rectangular waveguide filters with only minimal adverse effects on the electrical performance. Another important advantage of the present invention is that it enables implementation of more complex transfer functions e.g. featuring transmission zeros at finite frequencies that enhance selectivity, or phase equalization. Specific embodiments of the present invention relating to filters with more complex transfer functions will be described next. 
     As a first example, a rectangular waveguide filter  400  according to a fourth embodiment of the invention, which is a three pole filter with a transmission zero above the pass-band, will be described with reference to  FIGS. 4A to 4C .  FIG. 4A  is a perspective view of the rectangular waveguide filter  400 ,  FIG. 4B  is a sagittal cut through the rectangular waveguide filter  400 , and  FIG. 4C  illustrates the electrical performance of the rectangular waveguide filter  400 . 
     The rectangular waveguide filter  400  according to the fourth embodiment corresponds to the rectangular waveguide filter  200  according to the second embodiment with an additional hybrid coupling section  480  interposed between an aperture (coupling aperture) in the one of the end walls  215 ,  216  of the first resonator  210  and an aperture (coupling aperture) in the one of the end walls  235 ,  236  of the third resonator  230 . However, instead of the hybrid coupling section  480 , also alternative coupling sections, such as an inductive coupling section or a capacitive coupling section may be employed to couple the third resonator  230  to the first resonator  210 .  FIGS. 4A and 4B  correspond to  FIGS. 2A and 2B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 4A and 4B . 
       FIG. 4C  illustrates the electrical performance of the rectangular waveguide filter  400  of  FIGS. 4A and 4B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  400  in units of dB. Graph  491  indicates the S 21 -component of the S-parameter, and graph  492  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  400 . As can be seen from  FIG. 4C , S 11  has three poles in the pass-band indicated by S 21  (in the figure at about 12.45, 12.7, and 13.0 GHz). Further, S 21  has a transmission zero above the pass-band at about 13.3 GHz. 
     Next, as a further example, a rectangular waveguide filter  500  according to a fifth embodiment of the invention, which is a three pole filter with a transmission zero below the pass-band, will be described with reference to  FIGS. 5A to 5C .  FIG. 5A  is a perspective view of the rectangular waveguide filter  500 ,  FIG. 5B  is a sagittal cut through the rectangular waveguide filter  500 , and  FIG. 5C  illustrates the electrical performance of the rectangular waveguide filter  500 . 
     The rectangular waveguide filter  500  according to the fifth embodiment corresponds to the rectangular waveguide filter  200  according to the second embodiment with an additional inductive coupling section  580  interposed between an aperture (coupling aperture) in the one of the end walls  215 ,  216  of the first resonator  210  and an aperture (coupling aperture) in the one of the end walls  235 ,  236  of the third resonator  230 . Additionally, the first to third resonators  210 ,  220 ,  230  are configured such that the resonant mode of the second resonator  220  is the TE102 mode, while the resonant mode of first and third resonators  210 ,  230 , is the TE101 mode.  FIGS. 5A and 5B  correspond to  FIGS. 2A and 2B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 5A and 5B . 
     With the above choice of resonant modes for the first to third resonators  210 ,  220 ,  230 , a negative sign of the coupling between the first and third resonators  210 ,  230  is achieved by using a TE012 mode as a second resonance mode, so that the input and output electrical fields (i.e. the electrical fields in the first and third resonators  210 ,  230 ) are naturally out of phase, and a transmission zero of the filter below the pass-band is obtained. 
       FIG. 5C  illustrates the electrical performance of the rectangular waveguide filter  500  of  FIGS. 5A and 5B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  500  in units of dB. Graph  591  indicates the S 21 -component of the S-parameter, and graph  592  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  500 . As can be seen from  FIG. 5C , S 11  has three poles in the pass-band indicated by S 21  (in the figure at about 12.65, 12.75, and 12.9 GHz). Further, S 21  has a transmission zero below the pass-band at about 12.1 GHz. 
     In the rectangular waveguide filters  400 ,  500  according to the fourth and fifth embodiments of the invention, the transmission zeros above or below the pass-band are implemented by introducing an additional coupling between the first and third resonators  210 ,  230 . The location in frequency of these transmission zeros can be adjusted by changing the coupling between the first resonator  210  and the third resonator  230 . Obviously, such a coupling would not be possible for the standard in-line filter structure as illustrated e.g. in  FIGS. 14A and 15A . 
     An additional possibility to implement a negative coupling is to use a resonant coupling element, such as a ridge resonator.  FIGS. 6A to 6E  illustrate a resonator structure  600  comprising a ridge resonator  680  that can be used instead of the hybrid coupling section  480  in the rectangular waveguide filter  400  according to the fourth embodiment, or the inductive coupling section  580  in the rectangular waveguide filter  500  according to the fifth embodiment.  FIG. 6A  is a perspective view of the resonator structure  600 ,  FIG. 6B  is a horizontal cut through the resonator structure  600 ,  FIG. 6C  is a sagittal cut through the resonator structure  600 , and  FIGS. 6D and 6E  illustrate the electrical performance of the resonator structure  600 . 
     In  FIGS. 6A to 6C , the ridge resonator  680  is interposed between a first resonator  610  and a second resonator  620 . The ridge resonator  680  comprises, along its guide direction, a first section  680 A, a second section  680 B and a third section  680 C. The first to third sections  680 A,  680 B,  680 C have identical heights b 8 A, b 8 B, b 8 C. A width a 8 A of the first section  680 A is equal to a width a 8 C of the third section  680 C, whereas a width a 8 B of the second section  680 B is larger than the widths of the first and third sections  680 A,  680 C, i.e. a 8 B&gt;a 8 A=a 8 C. Inside the second section  680 B, a vertical post  680 D is provided that extends along the height direction from a bottom wall of the ridge resonator  680  to a top wall of the ridge resonator  680 . The post  680 D has a gap  680 E in its middle section. 
       FIGS. 6D and 6E  illustrate the electrical performance of the resonator structure  600  of  FIGS. 6A to 6C . The respective abscissa indicates the frequency in units of GHz, the ordinate in  FIG. 6D  indicates the phase of the S-parameter of the resonator structure  600  in units of degrees, and the ordinate in  FIG. 6E  indicates the modulus of the S-Parameter of the resonator structure  600  in units of dB. Graph  691  indicates the modulus of the S 12 -component of the S-parameter, graph  692  indicates the modulus of the S 11 -component of the S-parameter, and graph  693  indicates the phase of the S 12  -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the resonator structure  600 . 
     As can be seen from  FIG. 6D , the phase of S 12  flips sign from negative to positive at the resonant frequency of the ridge resonator  630  at about 11.84 GHz (cf.  FIG. 6E ). Therefore, if used as a coupling element, the ridge resonator  680  will provide a negative coupling below its resonant frequency, and a positive coupling above its resonant frequency. Although this behavior is indeed well-known, the use of a “de-tuned” ridge resonator as a coupling element has not been reported in the prior art. 
     Using a ridge coupling structure in the three pole filter of the second embodiment, a transmission zero above or below the pass-band can be easily obtained. Specific embodiments of the present invention relating to filters with more complex transfer functions employing ridge resonators as coupling structures will be described next. 
     As a first example of such a use of a ridge resonator as a coupling element, a rectangular waveguide filter  700  according to a sixth embodiment of the invention, which is a three pole filter with a transmission zero below the pass-band will be described with reference to  FIGS. 7A to 7C .  FIG. 7A  is a perspective view of the rectangular waveguide filter  700 ,  FIG. 7B  is a sagittal cut through the rectangular waveguide filter  700 , and  FIG. 7C  illustrates the electrical performance of the rectangular waveguide filter  700 . 
     The rectangular waveguide filter  700  according to the sixth embodiment corresponds to the rectangular waveguide filter  200  according to the second embodiment with a ridge resonator  780  interposed between an aperture (coupling aperture) in the one of the end walls  215 ,  216  of the first resonator  210  and an aperture (coupling aperture) in the one of the end walls  235 ,  236  of the third resonator  230  as a coupling section.  FIGS. 7A and 7B  correspond to  FIGS. 2A and 2B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 7A and 7B . 
       FIG. 7C  illustrates the electrical performance of the rectangular waveguide filter  700  of  FIGS. 7A and 7B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  700  in units of dB. Graph  791  indicates the S 21 -component of the S-parameter, and graph  792  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  700 . As can be seen from  FIG. 7C , S 11  has three poles in the pass-band indicated by S 21  (in the figure at about 12.5, 12.8, and 13.15 GHz). Further, S 21  has a transmission zero below the pass-band at about 11.7 GHz. 
     As a second example of the use of a ridge resonator as a coupling element, a rectangular waveguide filter  800  according to a seventh embodiment of the invention, which is a three pole filter with a transmission zero above the pass-band will be described with reference to  FIGS. 8A to 8C .  FIG. 8A  is a perspective view of the rectangular waveguide filter  800 ,  FIG. 8B  is a sagittal cut through the rectangular waveguide filter  800 , and  FIG. 8C  illustrates the electrical performance of the rectangular waveguide filter  800 . 
     The rectangular waveguide filter  800  according to the seventh embodiment corresponds to the rectangular waveguide filter  200  according to the second embodiment with a ridge resonator  880  interposed between an aperture (coupling aperture) in the one of the end walls  215 ,  216  of the first resonator  210  and an aperture (coupling aperture) the one of the end walls  235 ,  236  of the third resonator  230  as a coupling section.  FIGS. 8A and 8B  correspond to  FIGS. 2A and 2B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 8A and 8B . 
     The rectangular waveguide filter  800  according to the seventh embodiment is different from the rectangular waveguide filter  700  according to the sixth embodiment in that the ridge resonator  880  and the ridge resonator  780  are tuned differently, i.e. they differ in their design parameters and have different resonance frequencies. Design parameters of the ridge resonator are the lengths and width of the first to third sections of the ridge resonator as described with reference to  FIGS. 6A to 6C , as well as the width of the post in the second section and the height of the gap in the post. In the sixth embodiment, the ridge resonator  780  is tuned so that its resonance frequency lies above the pass-band of the rectangular waveguide filter  700 , whereas in the seventh embodiment, the ridge resonator  880  is tuned so that its resonance frequency lies below the pass-band of the rectangular waveguide filter  800 . 
       FIG. 8C  illustrates the electrical performance of the rectangular waveguide filter  800  of  FIGS. 8A and 8B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  800  in units of dB. Graph  891  indicates the S 21 -component of the S-parameter, and graph  892  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  800 . As can be seen from  FIG. 8C , S 11  has three poles in the pass-band indicated by S 21  (in the figure at about 12.3, 12.65, and 12.9 GHz). Further, S 21  has a transmission zero above the pass-band at about 13.1 GHz. 
     In the sixth and seventh embodiments, a de-tuned ridge resonator has been employed as the coupling structure in the three pole filter of the second embodiment. The de-tuned ridge resonator can also be used to provide a negative coupling between the first and fourth resonators (1-4 coupling) in the four pole filter of the third embodiment, thus producing, at the same time, transmission zeros below and above the pass-band. 
     With reference to  FIGS. 9A to 9C  now a rectangular waveguide filter  900  according to an eighth embodiment of the invention, which is a four pole filter employing a de-tuned ridge resonator  980  as coupling structure will be described.  FIG. 9A  is a perspective view of the rectangular waveguide filter  900 ,  FIG. 9B  is a sagittal cut through the rectangular waveguide filter  900 , and  FIG. 9C  illustrates the electrical performance of the rectangular waveguide filter  900 . 
     The rectangular waveguide filter  900  according to the eighth embodiment corresponds to the rectangular waveguide filter  300  according to the third embodiment with a ridge resonator  980  interposed between an aperture (coupling aperture) the one of the end walls  315 ,  316  of the first resonator  310  and an aperture (coupling aperture) in the one of end the walls  345 ,  346  of the fourth resonator  340  as a coupling section.  FIGS. 9A and 9B  correspond to  FIGS. 3A and 3B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 9A and 9B . 
       FIG. 9C  illustrates the electrical performance of the rectangular waveguide filter  900  of  FIGS. 9A and 9B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  900  in units of dB. Graph  991  indicates the S 21 -component of the S-parameter, and graph  992  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  900 . As can be seen from  FIG. 9C , S 11  has four poles in the pass-band indicated by S 21  (in the figure at about 12.29, 13.37, 14.46, and 12.53 GHz). Further, S 21  has transmission zeros above and below the pass-band at about 11.88 and 12.83 GHz. 
     In the eighth embodiment, a de-tuned ridge resonator has been employed as coupling structure in the four pole filter of the third embodiment, thus producing, at the same time, transmission zeros below and above the pass-band. Replacing the de-tuned ridge resonator by an appropriately tuned inductive coupling section (1-4 coupling), a four pole self-equalized bandpass filter can be realized. 
     A rectangular waveguide filter  1000  according to a ninth embodiment of the invention, which is a self-equalized four pole filter will now be described with reference to  FIGS. 10A to 10E .  FIG. 10A  is a perspective view of the rectangular waveguide filter  1000 ,  FIG. 10B  is a sagittal cut through the rectangular waveguide filter  1000 ,  FIG. 10C  is a first horizontal cut through the rectangular waveguide filter  1000 ,  FIG. 10D  is a second horizontal cut through the rectangular waveguide filter  1000 , and  FIG. 10E  illustrates the electrical performance of the rectangular waveguide filter  1000 . 
     The rectangular waveguide filter  1000  according to the ninth embodiment corresponds to the rectangular waveguide filter  300  according to the third embodiment with an inductive coupling section  1080  interposed between an aperture (coupling aperture) in the one of the end walls  315 ,  316  of the first resonator  310  and an aperture (coupling aperture) in the one of the end walls  345 ,  346  of the fourth resonator  340 .  FIGS. 10A and 10B  correspond to  FIGS. 3A and 3B , so that reference signs indicating the walls of the respective resonators are omitted in  FIGS. 10A and 10B . Reference signs indicating the walls are also omitted in  FIGS. 10C and 10D . 
     The length of the inductive coupling section  1080  in the guide direction is determined by the arrangement of the first to fourth resonators  310 ,  320 ,  330 ,  340 , wherein the shifts in the guide direction between the first and second resonators  310 ,  320  and between the third and fourth resonators  330 ,  340 , respectively, are design parameters of the rectangular waveguide filter  1000 . As can be seen from  FIGS. 10B to 10D , wherein  FIG. 10C  is a horizontal cut through the first and fourth resonators  310 ,  340  and the inductive coupling section  1080 , and  FIG. 10D  is a horizontal cut through the second and third resonators  320 ,  330  and the inductive coupling section  385 , the width of the inductive coupling section  1080  is smaller than the width of the inductive coupling section  385  between the second and third resonators  320 ,  330 . In particular, the width of the inductive coupling section in the width direction is below cut-off, so that there is no propagation of the base mode of the resonator inside the inductive coupling section  1080 . However, since the base mode decays exponentially in the inductive coupling section  1080 , there is nevertheless small electromagnetic coupling between the first and fourth resonators  310 ,  3400  (1-4 coupling), the coupling strength of which depends on the width of the inductive coupling section  1080 . By appropriately choosing said width, equalization of the group delay in the rectangular waveguide filter  1000  can be achieved. 
       FIG. 10E  illustrates the electrical performance of the rectangular waveguide filter  1000  of  FIGS. 10A to 10D . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the group delay of the S-parameter of the rectangular waveguide filter  1000  in units of nanoseconds (ns). Graph  1094  indicates the group delay of the S 12 -component of the S-parameter. As can be clearly seen from the graph, the group delay performance shows a typical self-equalized filter performance. 
     An additional feature of the family of filters according to the present invention is that they allow for the introduction of transmission zeros via an “interference” mechanism that does not require additional cross-couplings. The tenth embodiment described below relates to a two pole structure that introduces a transmission zero above the pass-band, and the eleventh embodiment described below relates to a two pole structure that introduces a transmission zero below the pass-band. 
     The two pole filter  1100  of the tenth embodiment of the invention will now be described with reference to  FIGS. 11A to 11C .  FIG. 11A  is a perspective view of the rectangular waveguide filter  1100  according to the tenth embodiment,  FIG. 11B  is a sagittal cut through the rectangular waveguide filter  1100 , and  FIG. 11C  illustrates the electrical performance of the rectangular waveguide filter  1100 . 
     The rectangular waveguide filter  1100  comprises a group of resonators having a first resonator  1110  and a second resonator  1120 . Like the first and second resonators  110 ,  120  in the first embodiment, the first and second resonators  1110 ,  1120  are coupled to each other through first and second apertures  11118 ,  1122 B (coupling apertures) in their top and bottom walls  1111 ,  1122 , respectively. 
       FIG. 11C  illustrates the electrical performance of the rectangular waveguide filter  1100  of  FIGS. 11A and 11B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  1100  in units of dB. Graph  1191  indicates the S 21 -component of the S-parameter, and graph  1192  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  1100 . As can be seen from  FIG. 11C , S 11  has two poles in the pass-band indicated by S 21  (in the figure at about 20.3 and 20.5 GHz). Further, S 21  has a transmission zero above the pass-band at about 24 GHz. 
     The interference generating the transmission zero is due to the signal path in the second resonator  1120 . More specifically, the signal entering the second resonator  1120  from the second aperture  1122 B in the bottom wall  1122  of the second resonator  1120  generates two paths, one to the left and one to the right of the second aperture  1122 B. The signal travelling to the left reaches the wall at the end of the second resonator  1120  and is reflected back. When the reflected signal reached the aperture  1122 B between the first and the second resonators  1110 ,  1120 , it interferes with the signal traveling to the right and thus creates the transmission zero shown in  FIG. 11C  above the filter pass-band. 
     As already mentioned above, the same mechanism can be used to generate a transmission zero below the filter pass-band by increasing the length travelled by the interfering signal. 
     The two pole filter  1200  of the eleventh embodiment of the invention, which has a transmission zero below the pass-band will now be described with reference to  FIGS. 12A to 12C .  FIG. 12A  is a perspective view of the rectangular waveguide filter  1200  according to the eleventh embodiment of the invention,  FIG. 12B  is a sagittal cut through the rectangular waveguide filter  1200 , and  FIG. 12C  illustrates the electrical performance of the rectangular waveguide filter  1200 . 
     The rectangular waveguide filter  1200  comprises a group of resonators having a first resonator  1210  and a second resonator  1220 . Like the first and second resonators  110 ,  120  in the first embodiment, the first and second resonators  1210 ,  1220  are coupled to each other through first and second apertures  1211 B,  1222 B (coupling apertures) in their top and bottom walls  1211 ,  1222 , respectively. 
     The rectangular waveguide filter  1200  differs from the rectangular waveguide filter  1100  according to the tenth embodiment in that the second aperture  1222 B in the bottom wall  1222  of the second resonator  1220  is at a different position along the guide direction of the second resonator  1220 . As can be seen from a comparison of  FIGS. 11A, 11B, 12A, and 12B , a ratio between a length of the path from the second aperture  1222 B to the left and a length of the path from the second aperture  1222 B to the right in the eleventh embodiment is larger than the respective ratio in the tenth embodiment. By tuning the value of this ratio, a frequency at which destructive interference as required for the transmission zero occurs, can be shifted. 
       FIG. 12C  illustrates the electrical performance of the rectangular waveguide filter  1200  of  FIGS. 12A and 12B . The abscissa indicates the frequency in units of GHz, and the ordinate indicates the S-parameter of the rectangular waveguide filter  1200  in units of dB. Graph  1291  indicates the S 21 -component of the S-parameter, and graph  1292  indicates the S 11 -component of the S-parameter. For reasons of symmetry, S 11 =S 22  and S 21 =S 12  hold for the rectangular waveguide filter  1200 . As can be seen from  FIG. 12C , S 11  has two poles in the pass-band indicated by S 21  (in the figure at about 20.6 and 20.7 GHz). Further, S 21  has a transmission zero below the pass-band at about 19.5 GHz. 
     Also in this case the interference generating the transmission zero is due to the signal path in the second resonator  1220 . More specifically, the signal entering the second resonator  1220  from the aperture  1222 B in the bottom wall  1222  of the second resonator  1220  generates two paths, one to the left and one to the right of the aperture  1222 B. The signal travelling to the left reaches the wall at the end of the second resonator  1220  and is reflected back. When the reflected signal reaches the aperture  1222 B between the first and the second resonators  1210 ,  1220 , it interferes with the signal traveling to the right and thus creates the transmission zero shown in  FIG. 12C  below the filter pass-band. 
     Another advantage of the family of filters described in the present disclosure is that a number of similar filter structures can be assembled together very easily in a waveguide manifold configuration, including also more conventional rectangular waveguide filters if necessary, maintaining all the electrical characteristics described above and enabling the low cost clam-shell manufacturing approach for the complete manifold structure. One example of such a manifold configuration is the six channel manifold multiplexer  1300  according to the twelfth embodiment, which is illustrated in  FIGS. 13A to 13C .  FIG. 13A  is a perspective view of the six channel manifold multiplexer  1300 ,  FIG. 13B  is a sagittal cut through the six channel manifold multiplexer  1300 , and  FIG. 13C  illustrates the electrical performance of the six channel manifold multiplexer  1300 . 
     The six channel manifold multiplexer  1300  comprises six rectangular waveguide filters  1310  to  1360 , one end of each being attached to a central waveguide manifold  1370 . An input port of the central waveguide manifold is to the right in  FIGS. 13A and 13B , while the left end of the central manifold  1370  is terminated with a short circuit. Six output ports are provided at the respective other ends of the six rectangular waveguide filters  1310  to  1360 . 
     All filters according to the present invention as described above are symmetric with respect to a vertical symmetry plane extending along the guide direction and the height direction of the respective filter (i.e. the y-z-plane). Thus, for all filters according to the present invention, a common approach for manufacturing is to cut the hardware longitudinally in two identical parts. Each individual part can be machined separately and the filter is realized by assembly the two parts. Several different technologies can be used for the actual mechanical realization of the filter parts depending on the required accuracy. If necessary, tuning screws could also be included in the center of the resonators of the respective filters without major difficulties. 
     Summarizing, the present application invention relates to a new family of rectangular waveguide bandpass filters based on a new resonator geometry referred to by the inventor as Hybrid Folded (HF) rectangular waveguide resonators. The new resonator structure allows for a reduction of filter footprint while providing slightly reduced insertion loss and power performance with respect to standard inductive rectangular waveguide resonator filters. Furthermore, it allows for the implementation of advanced filter transfer functions including both asymmetric and symmetric transmission zero implementations, as well as phase equalization. This new type of filter can be employed in practical applications both in ground and space systems especially for applications above the Ku Band. 
     Features, components and specific details of the structures of the above-described embodiments may be exchanged or combined to form further embodiments optimized for the respective application. As far as those modifications are readily apparent for an expert skilled in the art, they shall be disclosed implicitly by the above description without specifying explicitly every possible combination, for the sake of conciseness of the present description.