Abstract:
In a filter having a half-ladder structure comprising an alternating series of series branches and shunt branches, including a signal input and a signal output related to a common ground. at least one series branch or one parallel branch of the filter is configured as a BAW device comprising a first BAW resonator and a second BAW resonator connected either antiparallel or antiseries. The second harmonic emission generated by the first BAW resonator substantially cancels the second harmonic emission of the second BAW resonator and the second harmonic emission of at least one other series branch or one other parallel branch of the filter. By this method, the filter&#39;s second harmonic emissions are reduced.

Description:
BACKGROUND 
       [0001]    There is an increasing demand for communication devices capable of operating across a variety of different frequency bands. In particular, there is an increasing demand for cellular or mobile telephones that can operate in multiple frequency bands. In such devices, separate transmit and receive filters are in general employed for each transmit and receive frequency band. In practice, bulk acoustic wave (BAW) filters are often employed. 
         [0002]    The simplest implementation of a BAW resonator comprises a layer of piezoelectric material arranged between two metal electrodes. Common piezoelectric materials are, for example, aluminum nitride (AIN) or zinc oxide (ZnO). 
         [0003]      FIG. 1  shows an exemplary BAW resonator  10  having a static capacitance C which comprises a layer of piezoelectric material which will be referred to as piezo layer  12  below and is located between a first electrode, or top electrode T, and a second electrode, or bottom electrode B. The designations top electrode and bottom electrode merely serve definition purposes and do not represent any limitation with regard to the spatial arrangement and positioning of the BAW resonator. Rather, the designations top electrode and bottom electrode serve to define the positions of these electrodes in relation to a polarization of the piezoelectric material, as will be explained below, so that the polarization of the respective BAW resonators can be derived from an equivalent circuit diagram designating the T and B electrodes. 
         [0004]    If an electric field is applied between first electrode T and second electrode B of BAW resonator  10 , the reciprocal or inverse piezoelectric effect will cause BAW resonator  10  to mechanically expand or contract, the case of expansion or of contraction depending on the polarization of the piezoelectric material, as has been mentioned above. This means that the opposite case applies if the electric field is inversely applied between the T and B electrodes. In the case of an alternating field, an acoustic wave is generated in piezo layer  12 , and, depending on the implementation of the BAW resonator, this wave will propagate, for example, in parallel with the electric field, as a longitudinal wave, or, as a transversal wave, transverse to the electric field, and will be reflected, for example, at the interface of piezo layer  12 . Whenever the thickness d of piezo layer  12  and of the top and bottom electrodes equals an integer multiple of half the wavelength λ of the acoustic waves, resonance states and/or acoustic resonance vibrations will occur. The fundamental resonance frequency, i.e. the lowest resonance frequency F RES , will then be inversely proportional to total thickness of the resonator. This means that the BAW resonator vibrates at the frequency specified externally. 
         [0005]    The piezoelectric properties and, thus, also the resonance properties of a BAW resonator depend on various factors, e.g. on the piezoelectric material, the production method, the polarization impressed upon the BAW resonator during manufacturing, and the size of the crystals. As has been mentioned above, it is particularly the resonance frequency which depends on total thickness of the resonator. 
         [0006]    As has been mentioned above, BAW resonators exhibit electric polarization. The direction of mechanical deformation, extraction or contraction, of the BAW resonator depends on the direction of the electric field applied to first electrode T and second electrode B and on the direction of polarization of BAW resonator  10 . For example, if the polarization of the BAW resonator and the direction of the electric field are pointing in the same direction, BAW resonator  10  contracts, whereas BAW resonator  10  expands when the polarization of BAW resonator  10  and the direction of the electric field are pointing in the opposite direction. 
         [0007]    BAW resonators can have a variety of configurations. Typically, one differentiates between so-called FBARs (thin film bulk acoustic resonators) and SMRs (solidly mounted resonators). In addition, a BAW resonator may have one piezo layer  12 , or it may have several piezo layers  12 . 
         [0008]    BAW resonators are often employed in filters. 
         [0009]      FIG. 2  shows a circuit diagram of a filter  20 , comprising a first series BAW resonator  48 , a second series BAW resonator  55 , a third series BAW resonator  52 , a fourth series BAW resonator  53 , a first shunt BAW resonator  54 , a second shunt BAW resonator  56 , a third shunt BAW resonator  58  and a fourth shunt BAW resonator  65 . The series BAW resonators  48 ,  55 ,  52 ,  53  are connected in series between input port  44  and output port  46 . First shunt BAW resonator  54  is connected in parallel between input port  44  and electrical ground  42 . Second shunt BAW resonator  56  is connected between a connection node between first series BAW resonator  48  and second series BAW resonator  55 , and electrical ground  42 . Third shunt BAW resonator  58  is connected between a connection node between second series BAW resonator  55  and third series BAW resonator  52 , and electrical ground  42 . Fourth shunt BAW resonator  65  is between a connection node between third series BAW resonator  52  fourth series BAW resonator  53 , and electrical ground  42 . Each of the series and shunt BAW resonators comprises a top electrode T and a bottom electrode B, which are indicated in the equivalent circuit diagram of  FIG. 2  so as to indicate the polarization of each of the BAW resonators. 
         [0010]    One problem with filter circuits employing one or more BAW resonators, such as filter  20 , is non-linear behavior of one or more of the BAW resonators. This problem occurs particularly when a BAW resonator is driven with higher power levels, and can result in the undesirable generation of harmonic content in the output signal. 
         [0011]    One approach to mitigating this issue is to “cascade” the affected BAW resonators. In the following, a “cascade” means a chain, or series connection of elements. That is, a BAW resonator exhibiting static capacitance C is replaced by a cascade of two BAW resonators, each exhibiting a static capacitance 2C, so that the total capacitance of the series combination is again C. In principle, such a cascaded pair of BAW resonators has the same impedance properties as a corresponding individual BAW resonator. A cascaded pair of BAW resonators exhibiting static capacitance 2C is larger, by a factor of 4, than a corresponding individual BAW resonator exhibiting static capacitance C. As a result of the above-mentioned cascading, the energy density is also smaller by a factor of 4, and, thus, non-linear effects are reduced by 6 dB with a cascaded BAW resonator. 
         [0012]    However, the aforementioned cascading arrangement has some drawbacks. A major disadvantage of replacing a BAW resonator by an equivalent cascaded pair of BAW resonators is the increase in size by a factor of 4, as noted above. Accordingly, such cascading considerably increases the size of the filter if it is carried out for all BAW resonators of a filter. As a result, it is generally impractical to utilize cascading for all resonator branches of a filter. 
         [0013]    What is needed, therefore, is a general matching network and method of matching an antenna or other device to a plurality of BAW, SAW, and/or FBAR filters than can alleviate one or more of these shortcomings. 
       SUMMARY 
       [0014]    In an example embodiment, a filter having a half-ladder structure comprising an alternating series of series branches and shunt branches, includes a signal input and a signal output related to a common ground, wherein at least one series branch or one parallel branch of the filter is configured as a BAW device comprising a first BAW resonator and a second BAW resonator connected either antiparallel or antiseries A second harmonic emission generated by the first BAW resonator substantially cancels a second harmonic emission of the second BAW resonator and a second harmonic emission of at least one other series branch or one other parallel branch of the filter. 
         [0015]    In another example embodiment, a method is provided for reducing a harmonic emission within a BAW filter having a half-ladder structure comprising an alternating series of series branches and shunt branches, including a signal input and a signal output related to a common ground. The method comprising configuring at least one series branch or one parallel branch of the filter as a BAW device comprising a first BAW resonator and a second BAW resonator connected either antiparallel or antiseries, and configuring the first BAW resonator and the second BAW resonator such that a second harmonic emission generated by the first BAW resonator substantially cancels a second harmonic emission of the second BAW resonator and a second harmonic emission of at least one other series branch or one other parallel branch of the filter. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]    The example embodiments are best understood from the following detailed description when read with the accompanying drawing figures. It is emphasized that the various features are not necessarily drawn to scale. In fact, the dimensions may be arbitrarily increased or decreased for clarity of discussion. Wherever applicable and practical, like reference numerals refer to like elements. 
           [0017]      FIG. 1  shows an exemplary BAW resonator; 
           [0018]      FIG. 2  shows a circuit diagram of a filter having a half-ladder structure comprising an alternating series of series branches and shunt branches each including a BAW resonator; 
           [0019]      FIG. 3  shows a BAW device comprising two BAW resonators connected antiparallel; 
           [0020]      FIG. 4  shows a BAW device comprising two BAW resonators connected antiseries. 
           [0021]      FIG. 5  shows one embodiment of a filter having a half-ladder structure comprising an alternating series of series branches and shunt branches, where all series branches and all parallel branches of the filter are configured as a BAW device comprising a first BAW resonator and a second BAW resonator connected antiseries or antiparallel; 
           [0022]      FIG. 6  shows another embodiment of a filter having a half-ladder structure comprising an alternating series of series branches and shunt branches, where at least one series branch or one parallel branch of the filter is configured as a BAW device comprising a first BAW resonator and a second BAW resonator connected antiseries or antiparallel; 
           [0023]      FIG. 7  shows a plot of the second harmonic emission of the filter of  FIG. 2  at its output port as a function of frequency; 
           [0024]      FIG. 8  shows another embodiment of a filter  90  having a half-ladder structure comprising an alternating series of series branches and shunt branches, where only the “last” series branch nearest the output port is configured as a BAW device comprising a first BAW resonator and a second BAW resonator connected antiseries or antiparallel; 
           [0025]      FIG. 9  shows a plot of the second harmonic emission of the filter of  FIG. 8  at its output port as a function of frequency; 
           [0026]      FIG. 10  shows one embodiment of a filter which exhibits reduced harmonic emissions. 
       
    
    
     DETAILED DESCRIPTION 
       [0027]    In the following detailed description, for purposes of explanation and not limitation, example embodiments disclosing specific details are set forth in order to provide a thorough understanding of an embodiment according to the present teachings. However, it will be apparent to one having ordinary skill in the art having had the benefit of the present disclosure that other embodiments according to the present teachings that depart from the specific details disclosed herein remain within the scope of the appended claims. Moreover, descriptions of well-known apparati and methods may be omitted so as to not obscure the description of the example embodiments. Such methods and apparati are clearly within the scope of the present teachings. 
         [0028]      FIG. 3  shows a BAW device  30  comprising a first BAW resonator  72  and a second BAW resonator  74 , connected antiparallel. First BAW resonator  72  comprises a piezo layer  72 P, a first electrode  72 T, and a second electrode  72 B, and second BAW resonator  74  comprises a piezo layer  74 P, a first electrode  74 T, and a second electrode  74 B. First BAW resonator  72  is polarized, from first electrode  72 T, in the direction of second electrode  72 B, and second BAW resonator  74  is also polarized from first electrode  74 T in the direction of second electrode  74 B, and first electrode  72 T of the first BAW resonator  72  is electrically connected to second electrode  74 B of second BAW resonator  74 , and to a first electrical terminal  76  of BAW device  30 , whereas second electrode  72 B of first BAW resonator  72  is electrically connected to first electrode  74 T of second BAW resonator  74 , and to second electrical terminal  78  of BAW apparatus  30 . Arrow  72 R points to an exemplary direction of a polarization of first BAW resonator  72 , and arrow  74 R points to an exemplary direction of a polarization of second BAW resonator  74 . The only thing that is essential for BAW device  30  and/or antiparallel connection of BAW resonators  72 ,  74  is that the polarizations  72 R,  74 R of two BAW resonators  72 ,  74  have the same direction with regard to the first electrode  72 T,  74 T and the second electrode  72 B,  74 B. Alternatively, for example, the polarization of both BAW resonators  72 ,  74  may also be pointing in the direction from second electrode  72 B,  74 B to first electrode  72 T,  74 T. 
         [0029]    Thus, BAW device  30  consists of BAW resonator pair  72  and  74  connected antiparallel. 
         [0030]    As has been described above in the prior art, there are many varied implementations of BAW resonators, including FBAR resonators and SMRs, each of which implementations may be applied to the BAW device  30  of  FIG. 3 . Also, for example, BAW resonators  72 ,  74  alternatively may also comprise several piezo layers, even if  FIG. 3  only shows one piezo layer  72 P,  74 P. In addition, BAW device  30  may be created by interconnecting two separate and/or individual BAW resonators  72 ,  74 , but alternatively, for example, also by interconnecting two BAW resonators  72 ,  74  sharing a common piezo layer. 
         [0031]    If first BAW resonator  72  and second BAW resonator  74  have the same capacitance, e.g. C/2, then BAW device  30  overall acts like a regular or “common” BAW resonator exhibiting capacitance C. However, BAW device  30  will exhibit considerably reduced non-linear properties (e.g., harmonic emissions) compared with a corresponding common BAW resonator exhibiting capacitance C. In particular, the second harmonic emission of BAW device  30  is reduced substantially compared to a second harmonic emission of a common BAW resonator having the same capacitance. For example only, in one particular embodiment, the second harmonic emission of BAW device  30  is reduced by about 30 dB compared to a corresponding common BAW resonator exhibiting capacitance C. 
         [0032]      FIG. 4  shows a BAW device  40  comprising a first BAW resonator  82  and a second BAW resonator  84 , connected antiseries. First BAW resonator  82  comprises a piezo layer  82 P, a first electrode  82 T, and a second electrode  82 B, and second BAW resonator  84  comprises a piezo layer  84 P, a first electrode  84 T, and a second electrode  84 B. First BAW resonator  82  is polarized, from first electrode  82 T, in the direction of second electrode  82 B, and second BAW resonator  84  is also polarized from first electrode  84 T in the direction of second electrode  84 B. First electrode  82 T of the first BAW resonator  82  is electrically connected to a first electrical terminal  86  of BAW device  40 . Second electrode  82 B of the first BAW resonator  82  is electrically connected to second electrode  84 B of second BAW resonator  84 . First electrode  84 T of second BAW resonator  84  is electrically connected to second electrical terminal  88  of BAW apparatus  40 . Arrow  82 R points to an exemplary direction of a polarization of first BAW resonator  82 , and arrow  84 R points to an exemplary direction of a polarization of second BAW resonator  84 . 
         [0033]    Thus, BAW device  40  consists of BAW resonator pair  82  and  84  connected antiseries. 
         [0034]    As has been described above in the prior art, there are many varied implementations of BAW resonators, including FBAR resonators and SMRs, each of which implementations may be applied to the BAW device  40  of  FIG. 4 . Also, for example, BAW resonators  82 ,  84  alternatively may also comprise several piezo layers, even if  FIG. 4  only shows one piezo layer  82 P,  84 P. In addition, BAW device  40  may be created by interconnecting two separate and/or individual BAW resonators  82 ,  84 . 
         [0035]    If first BAW resonator  82  and second BAW resonator  84  have the same capacitance, e.g. 2C, then BAW device  40  overall acts like a regular or “common” BAW resonator exhibiting capacitance C. However, BAW device  40  will exhibit considerably reduced non-linear properties (e.g., harmonic emissions) compared with a corresponding common BAW resonator exhibiting capacitance C. That is, a second harmonic emission of BAW device  40  is reduced substantially compared to a second harmonic emission of a common BAW resonator having the same capacitance. For example only, in one particular embodiment, the second harmonic emission of BAW device  40  is reduced by about 30 dB compared to a corresponding common BAW resonator exhibiting capacitance C. 
         [0036]    Antiparallel BAW device  30  and antiseries BAW device  40  may be employed in one or more series or shunt branches of a filter having a half-ladder structure to improve the non-linear performance of the filter, particularly to reduce a second harmonic emission of the overall filter. 
         [0037]      FIG. 5  shows one embodiment of a filter  50  having a half-ladder structure comprising an alternating series of series branches and shunt branches, where each of the series and shunt branches is configured as a BAW device  57  comprising a first BAW resonator and a second BAW resonator connected antiparallel, as shown in  FIG. 3 . 
         [0038]      FIG. 6  shows another embodiment of a filter  60  having a half-ladder structure comprising an alternating series of series branches and shunt branches, where each of the series and shunt branches is configured as a BAW device  67  comprising a first BAW resonator and a second BAW resonator connected antiseries, as shown in  FIG. 4 . 
         [0039]    As noted above, the filters  50  and  60  can exhibit substantially improved non-linear performance compared to a filter  20  of  FIG. 2  having the same capacitance values in each series and shunt branch. In particular, the second harmonic emission performance of filter  50  or filter  60  may be several tens of decibels better than the performance of a filter  20  of  FIG. 2  having the same capacitance values in each series and shunt branch. 
         [0040]    However, there are drawbacks to filters  50  and  60 . 
         [0041]    In filter  50 , each BAW device  57  comprising a first BAW resonator having value C/2 and a second BAW resonator having value C/2 connected antiparallel exhibits a substantially reduced quality factor (i.e., “Q”) compared to an equivalent common BAW resonator having the same capacitance value C. As a result, filter  50  has a greatly diminished quality factor, meaning it is lossier and less efficient than the equivalent filter  20 . In many applications, this reduced quality factor is not acceptable. 
         [0042]    Meanwhile, in filter  60 , each BAW device  67  comprising a first BAW resonator having value 2C and a second BAW resonator having value 2C connected antiseries exhibits a substantially increased size (4×) compared to an equivalent common BAW resonator having the same capacitance value C. As a result, filter  60  requires a greatly increased size than the equivalent filter  20 . In many applications, this increased size is not acceptable. 
         [0043]    Accordingly, other solutions to reducing harmonic emissions are sought. 
         [0044]      FIG. 7  shows a plot of the second harmonic emission of filter  20  of  FIG. 2  at output port  46  as a function of frequency. It can be seen that the plot has two “peaks”—a “negative phase” peak at a lower frequency that is dominated by the second harmonic emission of shunt branches of filter  20 , and a “positive phase” peak at a higher frequency that is dominated by the second harmonic emission of series branches of filter  20 . 
         [0045]    It can be shown that at output port  46 , the main sources second harmonic emissions are from the fourth series branch including BAW resonator  53  and the fourth shunt branch including BAW resonator  65 —i.e., the “last” series and shunt branches nearest output port  46 . Indeed, for example, at output port  46  the contribution of second harmonic emissions from third series BAW resonator  52  and third shunt BAW resonator  58  are about 10 dB less than the contribution of second harmonic emissions from fourth series BAW resonator  53  and fourth shunt BAW resonator  65 . The contributions of first and second series BAW resonators  48  and  55 , and first and second shunt BAW resonators  54  and  56  is even less. 
         [0046]    Accordingly,  FIG. 8  shows another embodiment of a filter  80  having a half-ladder structure comprising an alternating series of series branches and shunt branches, where only the “last” series branch nearest the output port  86  is configured as a BAW device  85  comprising a first BAW resonator and a second BAW resonator connected either antiparallel, as shown in  FIG. 3 , or antiseries, as shown in  FIG. 4 . 
         [0047]    As explained above, in the case of the antiparallel configuration, if first BAW resonator and second BAW resonator have the same capacitance, e.g. C/2, then BAW device  85  overall acts like a regular or “common” BAW resonator exhibiting capacitance C, except the harmonic emission is substantially reduced—in particular, the second harmonic emission level of BAW device  85  is 30 dB less that of an equivalent common BAW resonator. Similarly, in the case of the antiseries configuration, if first BAW resonator and second BAW resonator have the same capacitance, e.g. 2C, then BAW device  85  overall acts like a regular or “common” BAW resonator exhibiting capacitance C, except the harmonic emission is substantially reduced—again, in particular, the second harmonic emission of BAW device  85  is 30 dB less than the equivalent common BAW resonator. 
         [0048]    In comparison to filters  50  and  60 , because filter  80  only replaces a single common BAW resonator with a BAW device  85  comprising a first BAW resonator and a second BAW resonator connected either antiparallel or antiseries, then the size is only increased slightly or its quality factor is only diminished slightly—in contrast to filters  50  and  60 . 
         [0049]      FIG. 9  shows a plot of the second harmonic emission of filter  80  at output port  86  as a function of frequency. Compared to  FIG. 7 , it can be seen that the “negative phase” peak at a lower frequency that is dominated by the second harmonic emissions of shunt branches of filter  90  remains, and in fact, worsens at lower frequencies. This could be addressed by replacing the “last” shunt resonator of filter  80 , nearest output port  86 , with a BAW device comprising a first BAW resonator and a second BAW resonator connected either antiparallel, as shown in  FIG. 3 , or antiseries, as shown in  FIG. 4 . However, it is also seen that the “positive phase” peak at a higher frequency that is dominated by the second harmonic emission of series branches of filter  80  is only reduced about 10 dB compared to filter  20 , even though the second harmonic emission of BAW device  85  is 30 dB less than the equivalent common BAW resonator. 
         [0050]    Accordingly,  FIG. 10  shows one embodiment of a filter  1100  which exhibits reduced harmonic emissions. Filter  1000  has a half-ladder structure comprising an alternating series of series branches  1010 - i  and shunt branches  1020 - i , where each of the series and shunt branches  1010   i / 1020   i  includes a corresponding BAW resonator  1030   i  having the shown polarity, except for a “last” series branch  1010 - 4  and a last shunt branch  1020 - 4 . The last series branch nearest output port  46  includes BAW device  1050 , and the last shunt branch nearest output port  46  includes a BAW device  1060 . In the embodiment of  FIG. 10 , BAW device  1050  includes a first BAW resonator  1052  and a second BAW resonator  1054  connected antiparallel. Meanwhile, BAW device  1060  includes a first BAW resonator  1062  and a second BAW resonator  1064  connected antiparallel. 
         [0051]    Beneficially, BAW device  1050  is configured such that BAW resonator  1052  and second BAW resonator  1054  do not have the same capacitance (e.g., C/2) as each other and are not equivalent to each other. In one particular exemplary implementation, BAW resonator  1052  and BAW resonator  1054  are selected so that: (1) the parallel combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of BAW resonator  1052  substantially equals the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emission of series BAW resonator  1010 - 3 , but in opposite phase. In that case, since BAW resonator  1052  has an opposite polarity than that of BAW resonator  1054  series BAW resonator  1010 - 3 , the 2 nd  harmonic emission of BAW resonator  1052  substantially cancels the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emission of series BAW resonator  1010 - 3 . In this context, substantial cancellation means that the resultant 2 nd  harmonic emission is about 10 dB less than the 2 nd  harmonic emission that would have been produced by the combination of BAW resonator  1052 , BAW resonator  1054 , and series BAW resonator  1010 - 3  if BAW resonator  1052  and BAW resonator  1054  had the same capacitance (e.g., C/2) as each other, such as in filter  90 . 
         [0052]    If one plots the second harmonic emission of filter  1000  versus frequency, one would observe that the “positive phase” peak at a higher frequency, which is dominated by the second harmonic emission of series branches of filter  1000 , is reduced about 20 dB compared to filter  20 . 
         [0053]    In similar fashion to that explained above, beneficially first BAW resonator  1062  and second BAW resonator  1064  are selected so that: (1) the parallel combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of BAW resonator  1062  substantially cancels the 2 nd  harmonic emission of BAW resonator  1064  plus the 2 nd  harmonic emission of shunt BAW resonator  1020 - 3 . 
         [0054]    Furthermore, in another particular exemplary implementation, BAW resonator  1052  and BAW resonator  1054  are selected so that: (1) the parallel combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of BAW resonator  1052  plus the 2 nd  harmonic emission of series BAW resonator  1010 - 2  substantially equals the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emission of series BAW resonator  1010 - 3 , but in opposite phase. In that case, since BAW resonators  1052  and series BAW resonator  1010 - 3  have an opposite polarity than that of BAW resonator  1054  series BAW resonator  1016 , the 2 nd  harmonic emission of BAW resonator  1052  substantially cancels the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emission of series BAW resonator  1010 - 3  and the 2 nd  harmonic emission of series BAW resonator  1010 - 2 . In this context, substantial cancellation means that the resultant 2 nd  harmonic emission is about 10 dB less than the harmonic emission that would have been produced by the combination of BAW resonator  1052 , BAW resonator  1054 , and series BAW resonators  1010 - 2  and  1010 - 3  if BAW resonator  1052  and BAW resonator  1054  had the same capacitance (e.g., C/2) as each other, such as in filter  90 . 
         [0055]    If one plots the second harmonic emission of filter  1000  versus frequency, one would observe that the “positive phase” peak at a higher frequency, which is dominated by the second harmonic emission of series branches of filter  1000 , is reduced more than 20 dB compared to filter  20 . 
         [0056]    Conceptually, in general it is possible to select BAW resonator  1052  and BAW resonator  1054  are selected so that: (1) the parallel combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of BAW resonator  1052  substantially cancels the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emissions of all series BAW resonators  1010 - i  in filter  1000 . 
         [0057]    In similar fashion to that explained above, in general it is possible to select BAW resonator  1062  and second BAW resonator  1064  so that: (1) the parallel combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of BAW resonator  1052  substantially cancels the 2 nd  harmonic emission of BAW resonator  1054  plus the 2 nd  harmonic emissions of all shunt BAW resonators  1020 - i  in filter  1000 . 
         [0058]    Various combinations are possible, including the including of only BAW device  1050  or BAW device  1060  in place of a common BAW resonator of equivalent capacitance. 
         [0059]    Furthermore, although  FIG. 10  shows an embodiment where BAW devices  1050  and  1060  each comprise and antiparallel combination of BAW resonators, one or both BAW devices may instead comprise an antiseries combination of BAW resonators, where: (1) the series combination produces the desired capacitance C; and (2) the 2 nd  harmonic emission of a first BAW resonator substantially cancels the 2 nd  harmonic emission of a second BAW resonator plus the 2 nd  harmonic emission of one or more remaining series or shunt BAW resonators of filter  1000 . 
         [0060]    While example embodiments are disclosed herein, one of ordinary skill in the art appreciates that many variations that are in accordance with the present teachings are possible and remain within the scope of the appended claims. For example, although the descriptions and figures above illustrate an exemplary case where a matching network multiplexes signals to and from an antenna and a plurality of filters, the matching network is not limited to use with an antenna. In general, any appropriate device, such as a broadband amplifier or filter, can be passively multiplexed with the plurality of filters using the matching network as described above. The embodiments therefore are not to be restricted except within the scope of the appended claims.