Patent Publication Number: US-10763825-B2

Title: Radio-frequency filter, radio-frequency front-end circuit, and communication device

Description:
This is a continuation of International Application No. PCT/JP2018/003140 filed on Jan. 31, 2018 which claims priority from Japanese Patent Application No. 2017-020579 filed on Feb. 7, 2017. The contents of these applications are incorporated herein by reference in their entireties. 
    
    
     BACKGROUND OF THE DISCLOSURE 
     Field of the Disclosure 
     The present disclosure relates to a radio-frequency filter having resonators, a radio-frequency front-end circuit, and a communication device. 
     Description of the Related Art 
     Hitherto, tunable radio-frequency filters (tunable filters) have been suggested as radio-frequency filters that support multiband functionality. 
     As such a tunable radio-frequency filter, for example, a first configuration in which two parallel arm resonators having resonant frequencies different from each other are connected in series with each other and a switch is connected in parallel with at least one of these two parallel arm resonators is known (see, for example, FIG. 10 of Patent Document 1). This first configuration is star-delta transformed into a second configuration that a series connection circuit made up of a resonator (correction resonator) and a switch is connected in parallel with two series arm resonators connected via a node to which a parallel arm resonator is connected (see, for example, FIG. 9A of Patent Document 1). 
     Patent Document 1: Japanese Patent No. 5441095 
     BRIEF SUMMARY OF THE DISCLOSURE 
     With the above-described existing configuration, a first attenuation pole at a lower edge of a pass band changes in frequency toward the pass band in response to switching between an on state (conductive state) and an off state (nonconductive state) of the switch. Therefore, the presence or absence of a new third attenuation pole between a second attenuation pole at a higher edge of the pass band and the pass band can be switched. Specifically, when the switch is switched from the on state to the off state, the first attenuation pole on the lower side of the pass band changes in frequency toward the pass band (that is, toward a higher frequency side), and a new third attenuation pole is added between the second attenuation pole on the higher side of the pass band and the pass band. With the above-described existing configuration, the difference from the frequency of the second attenuation pole to the frequency of the third attenuation pole is greater than the variable frequency width of the first attenuation pole. For this reason, there is inconvenience that the loss at the lower edge of the pass band deteriorates (an insertion loss increases) and an attenuation on the higher side of the pass band deteriorates, or the loss at the higher edge of the pass band deteriorates (an insertion loss increases) and an attenuation on the lower side of the pass band deteriorates. 
     It is an object of the present disclosure to provide a tunable radio-frequency filter, radio-frequency front-end circuit, and communication device that are able to improve a loss at a lower edge of a pass band and an attenuation on a higher side of the pass band or improve a loss at a higher edge of the pass band and an attenuation on a lower side of the pass band. 
     To achieve the above object, a radio-frequency filter according to an aspect of the present disclosure includes a series arm circuit provided in a path connecting a first input and output terminal and a second input and output terminal, and a parallel arm circuit connected to a ground and a node in the path. The series arm circuit includes a first series arm resonator provided in the path, and a first variable frequency circuit connected in parallel with the first series arm resonator. The first variable frequency circuit includes a second series arm resonator and a first switch connected in series with the second series arm resonator. The first variable frequency circuit is configured to change at least one of a resonant frequency that provides a pass band of the radio-frequency filter and an anti-resonant frequency that provides an attenuation pole of the radio-frequency filter. A resonant frequency of the second series arm resonator is different from a resonant frequency of the first series arm resonator. 
     With such a configuration, at least one of the resonant frequency that provides the pass band of the radio-frequency filter and the anti-resonant frequency that provides the attenuation pole of the radio-frequency filter is changed for the series arm circuit by switching between an on state and off state of the first switch. Specifically, when the first switch is off, the series arm circuit has a third resonant frequency that is the same as the resonant frequency of the first series arm resonator and a third anti-resonant frequency that lies on a slightly lower frequency side than an anti-resonant frequency of the first series arm resonator. When the first switch is on, the series arm circuit has a first resonant frequency that is the same as the resonant frequency of the first series arm resonator, a second resonant frequency that is the same as the resonant frequency of the second series arm resonator, a first anti-resonant frequency that lies between the resonant frequency of the first series arm resonator and the resonant frequency of the second series arm resonator, and a second anti-resonant frequency that lies on a higher frequency side than the second resonant frequency. 
     In this respect, when the resonant frequency of the second series arm resonator is lower than the resonant frequency of the first series arm resonator, and when the first switch is off, the third resonant frequency provides the pass band, and the third anti-resonant frequency provides the attenuation pole on a higher side of the pass band. In this case, when the first switch is on, the second resonant frequency provides the pass band, the first anti-resonant frequency provides the attenuation pole on a lower side of the pass band, and the second anti-resonant frequency provides the attenuation pole on a higher side of the pass band. In other words, in this case, a new attenuation pole is provided on the lower side of the pass band by turning the first switch from the off state to the on state. 
     On the other hand, when the resonant frequency of the second series arm resonator is higher than the resonant frequency of the first series arm resonator, and when the first switch is off, the third resonant frequency provides the pass band, and the third anti-resonant frequency provides the attenuation pole on a higher side of the pass band. In this case, when the first switch is on, the first resonant frequency provides the pass band, and both the first anti-resonant frequency and the second anti-resonant frequency provide the attenuation poles on a higher side of the pass band. In other words, a new attenuation pole is provided on a higher side of the pass band by turning the first switch from the off state to the on state. 
     Thus, when the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency of the parallel arm circuit, is considered, the following can be said on the relationship between switching the on state and off state of the first switch and the pass band. Specifically, when the first switch is switched from the off state to the on state, a new attenuation pole is provided on one side (for example, a lower side) of the pass band, so the stop band on the one side of the pass band is widened. At this time, since no new attenuation pole is provided on the other side (in this case, a higher side) of the pass band, the deterioration of a loss at the other edge (in this case, a higher edge) of the pass band is reduced, and the deterioration of the attenuation on the other side (in this case, a lower side) of the pass band is reduced. 
     Therefore, according to this aspect, the tunable radio-frequency filter that is able to improve a loss at the higher edge of the pass band and the attenuation on the lower side of the pass band or improve a loss at the lower edge of the pass band and the attenuation on the higher side of the pass band is implemented. 
     The resonant frequency of the second series arm resonator may be lower than the resonant frequency of the first series arm resonator. 
     When the resonant frequency of the second series arm resonator is lower than the resonant frequency of the first series arm resonator in this way, the number of attenuation poles on the lower side of the pass band can be increased or reduced by switching between the on state and off state of the first switch. Thus, the frequencies of the pass band and the frequency of the attenuation pole on the lower side of the pass band can be changed while a loss at the higher edge of the pass band is reduced and the attenuation on the lower side of the pass band is improved. 
     The series arm circuit may include a plurality of the first variable frequency circuits, and the resonant frequencies of a plurality of the second series arm resonators that the plurality of first variable frequency circuits includes may be different from each other. 
     When the resonant frequencies of the plurality of second series arm resonators are different from each other in this way, the pass band and the stop band can be changed according to the first switch selected and set to the on state from among the plurality of first switches that the plurality of first variable frequency circuits includes. In other words, while a loss at one edge of the pass band is reduced, the variable width of the frequencies of the pass band and the variable width of the frequency of the attenuation pole on the other side of the pass band can be minutely set. 
     The resonant frequency of at least one of the plurality of second series arm resonators may be lower than the resonant frequency of the first series arm resonator, and the resonant frequency of another one of the plurality of second series arm resonators may be higher than the resonant frequency of the first series arm resonator. 
     When the resonant frequency of at least one of the plurality of second series arm resonators is lower than the resonant frequency of the first series arm resonator in this way, the frequencies of the pass band and the frequency of the attenuation pole on the lower side of the pass band can be changed while a loss at the higher edge of the pass band is reduced by switching between the on state and off state of the first switch (hereinafter, lower-side variable switch) connected in series with the at least one second series arm resonator. When the resonant frequency of at least another one of the plurality of second series arm resonators is higher than the resonant frequency of the first series arm resonator in this way, the frequencies of the pass band and the frequency of the attenuation pole on the higher side of the pass band can be changed while a loss at the lower edge of the pass band is reduced by switching between the on state and off state of the first switch (hereinafter, higher-side variable switch) connected in series with the at least one second series arm resonator. 
     Therefore, according to this aspect, the frequency of the attenuation pole can be changed while a loss at the edge of the pass band is reduced for each of the lower side and higher side of the pass band by individually switching between the on state and off state of a corresponding one of the lower-side variable switch and the higher-side variable switch. 
     Any of the resonant frequencies of the plurality of second series arm resonators may be lower than the resonant frequency of the first series arm resonator. 
     When any of the resonant frequencies of the plurality of second series arm resonators is lower than the resonant frequency of the first series arm resonator in this way, the pass band and the stop band on the lower side of the pass band can be changed according to the first switch selected and set to the on state from among the plurality of first switches that the plurality of first variable frequency circuits includes. In other words, while a loss at the higher edge of the pass band is reduced, the variable width of the frequencies of the pass band and the variable width of the frequency of the attenuation pole on the lower side of the pass band can be minutely set. 
     The series arm circuit may further include a first impedance element connected in series with the first series arm resonator, and the first variable frequency circuit may be connected in parallel with a first series connection circuit made up of the first series arm resonator and the first impedance element. 
     When the first series arm resonator and the first impedance element are connected in series with each other in this way, the difference between the resonant frequency and anti-resonant frequency of the first series connection circuit can be changed from the frequency difference of the first series arm resonator. Specifically, the frequency difference can be reduced when the first impedance element is a capacitor, and the frequency difference can be increased when the first impedance element is an inductor. Thus, since the difference between the second resonant frequency and second anti-resonant frequency of the series arm circuit can be adjusted by adjusting the circuit constants of the first impedance element as needed, the difference between a cut-off frequency on the higher side of the pass band, determined by the frequency difference, and the attenuation pole on the higher side of the pass band can be adjusted. Therefore, the pass band width and the steepness of an attenuation slope can be selectively set. 
     The parallel arm circuit may include a first parallel arm resonator connected between the node and the ground, and a second variable frequency circuit connected in series with the first parallel arm resonator, the second variable frequency circuit may include a second impedance element and a second switch connected in parallel with the second impedance element, and a resonant frequency of the first parallel arm resonator may be lower than the resonant frequency of the first series arm resonator. 
     As a result of changing the frequency of the attenuation pole by switching between the on state and off state of the first switch, the attenuation can deteriorate. For this reason, by providing the second variable frequency circuit connected in series with the first parallel arm resonator, the resonant frequency that provides the attenuation pole of the radio-frequency filter can be changed for the parallel arm circuit by switching between the on state and off state of the second switch. Thus, when a circuit constant of the second impedance element is adjusted as needed according to the stop band and attenuation required for the radio-frequency filter, the deterioration of the attenuation on the lower side of the pass band resulting from changing of the frequency of the series arm circuit is reduced. 
     The parallel arm circuit may further include a second parallel arm resonator connected in parallel with a second series connection circuit made up of the first parallel arm resonator and the second variable frequency circuit, and a resonant frequency of the second parallel arm resonator may be higher than the resonant frequency of the first parallel arm resonator. 
     When the parallel arm circuit includes the second parallel arm resonator whose resonant frequency is higher than the resonant frequency of the first series arm resonator in this way, a new attenuation pole is provided on the higher side of the pass band by the resonant frequency of the second parallel arm resonator. Therefore, the attenuation on the higher side of the pass band is improved. 
     Since the parallel arm circuit includes the second parallel arm resonator, the parallel arm circuit has two resonant frequencies and two anti-resonant frequencies. In comparison with the difference between the resonant frequency and the anti-resonant frequency in the case where the parallel arm circuit includes no second parallel arm resonator, the difference between the lower frequency-side resonant frequency and the lower frequency-side anti-resonant frequency is reduced. Therefore, according to this aspect, the cut-off frequency on the lower side of the pass band can be shifted toward a lower frequency side, so a loss at the lower edge of the pass band is improved. 
     The second impedance element may be a capacitor. When the first switch is on, the second switch may be off. When the first switch is off, the second switch may be on. 
     Thus, when the second switch is switched from the on state to the off state, the capacitor connected in series with the first parallel arm resonator becomes effective. Thus, since the resonant frequency of the parallel arm circuit shifts toward a higher side, the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency, shifts toward a higher frequency side. Therefore, since the second switch is off when the first switch is on, an attenuation on the lower side of the pass band when the lower side of the pass band is shifted toward a higher frequency side (at the time of higher frequency shift) is improved (increased). 
     A control circuit for the first switch and a control circuit for the second switch can be integrated, so the miniaturization of the radio-frequency filter is possible. 
     The second impedance element may be an inductor. When the first switch is on, the second switch may be on. When the first switch is off, the second switch may be off. 
     Thus, when the second switch is switched from the on state to the off state, the inductor is connected in series with the first parallel arm resonator. Thus, since the resonant frequency of the parallel arm circuit shifts toward a lower side, the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency, shifts toward a lower frequency side. Therefore, since the second switch is on when the first switch is on, the frequency on the lower side of the pass band when the lower side of the pass band is shifted toward a lower frequency side (at the time of lower frequency shift) is decreased. In other words, the variable frequency width on the lower side of the pass band is increased. 
     A control circuit for the first switch and a control circuit for the second switch can be integrated, so the miniaturization of the radio-frequency filter is possible. 
     The first switch may be connected to the first input and output terminal or the second input and output terminal without intervening any of the series arm resonators. 
     Thus, when the series arm resonators are provided on or in a chip for resonators and the first switch is provided on or in another chip, the number of terminals of the chip for resonators is reduced. Specifically, in this case, in the chip for resonators, terminals that are connected to the one input and output terminal and terminals that are connected to the other chip are respectively shared. Therefore, according to this aspect, in comparison with the configuration in which these terminals are individually provided, the number of terminals of the chip for resonators is reduced, so the miniaturization of the radio-frequency filter is achieved. 
     The first switch may be an FET switch made from GaAs or CMOS, or a diode switch. 
     Thus, the resistance of the first switch is reduced, so a loss in the pass band is reduced. In addition, the first switch is miniaturized, so the miniaturization and cost reduction of the radio-frequency filter are possible. 
     A radio-frequency front-end circuit according to an aspect of the present disclosure includes any one of the above-described radio-frequency filters, and a control unit configured to control the first switch between the on state and the off state. 
     Thus, the small-sized radio-frequency front-end circuit that improves a loss at the lower edge of the pass band and the attenuation on the higher frequency side of the pass band or improves a loss at the higher edge of the pass band and the attenuation on the lower side of the pass band is achieved. 
     A communication device according to an aspect of the present disclosure includes an RF signal processing circuit configured to process a radio-frequency signal that is transmitted or received by an antenna element, and the above-described radio-frequency front-end circuit configured to transfer the radio-frequency signal between the antenna element and the RF signal processing circuit. 
     Thus, the small-sized communication device that improves a loss at the lower edge of the pass band and the attenuation on the higher side of the pass band or improves a loss at the higher edge of the pass band and the attenuation on the lower side of the pass band is achieved. 
     With the tunable radio-frequency filter, radio-frequency front-end circuit, and the communication device according to the present disclosure, a loss at the lower edge of the pass band and the attenuation on the higher side of the pass band are improved, or a loss at the higher edge of the pass band and the attenuation on the lower side of the pass band are improved. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 1-1) of a first embodiment. 
         FIG. 1B  shows graphs of various characteristics of the radio-frequency filter according to Example 1-1. 
         FIG. 2  shows a schematic plan view and cross-sectional view of the electrode configuration of an acoustic wave resonator that uses surface acoustic waves. 
         FIG. 3A  is a circuit configuration diagram of a radio-frequency filter according to Comparative Example. 
         FIG. 3B  shows graphs of various characteristics of the radio-frequency filter according to Comparative Example. 
         FIG. 4A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1 and Comparative Example at the time of lower frequency shift. 
         FIG. 4B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1 and Comparative Example at the time of higher frequency shift. 
         FIG. 5  is a graph that shows the influence on filter characteristics when the resonant frequency (fr_s 1   b ) of a series arm resonator is changed in the radio-frequency filter according to Example 1-1. 
         FIG. 6  is a circuit configuration diagram of a radio-frequency filter according to a modification. 
         FIG. 7A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 1-2-1) of the first embodiment. 
         FIG. 7B  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 1-2-2) of the first embodiment. 
         FIG. 7C  shows graphs of various characteristics of the radio-frequency filters according to Example 1-2-1 and Example 1-2-2 in comparison with Example 1-1. 
         FIG. 7D  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1, Example 1-2-1, and Example 1-2-2 at the time of lower frequency shift. 
         FIG. 7E  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1, Example 1-2-1, and Example 1-2-2 at the time of higher frequency shift. 
         FIG. 8A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 1-3) of the first embodiment. 
         FIG. 8B  shows graphs of various characteristics of the radio-frequency filter according to Example 1-3 in comparison with Example 1-1. 
         FIG. 8C  shows graphs of changes in filter characteristics when switches of the radio-frequency filter according to Example 1-3 are individually switched between an on state and an off state. 
         FIG. 9A  shows an equivalent circuit model of a resonator and its resonant characteristics. 
         FIG. 9B  shows an equivalent circuit model when an impedance element is connected in series with a resonator and its resonant characteristics. 
         FIG. 9C  shows an equivalent circuit model of two resonators connected in parallel with each other and its resonant characteristics. 
         FIG. 10A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 2-1) of a second embodiment. 
         FIG. 10B  shows graphs of various characteristics of the radio-frequency filter according to Example 2-1. 
         FIG. 11A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-1 and Example 1-1 at the time of lower frequency shift. 
         FIG. 11B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-1 and Example 1-1 at the time of higher frequency shift. 
         FIG. 12  shows graphs of changes in filter characteristics when switches of the radio-frequency filter according to Example 2-1 are individually switched between an on state and an off state. 
         FIG. 13A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 2-2) of the second embodiment. 
         FIG. 13B  shows graphs of various characteristics of the radio-frequency filter according to Example 2-2. 
         FIG. 14A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-2 and Example 1-1 at the time of lower frequency shift. 
         FIG. 14B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-2 and Example 1-1 at the time of higher frequency shift. 
         FIG. 15  shows graphs of changes in filter characteristics when switches of the radio-frequency filter according to Example 2-2 are individually switched between an on state and an off state. 
         FIG. 16A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 3) of a third embodiment. 
         FIG. 16B  shows graphs of various characteristics of the radio-frequency filter according to Example 3. 
         FIG. 17A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 3 and Example 1-1 at the time lower frequency shift. 
         FIG. 17B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 3 and Example 1-1 at the time of higher frequency shift. 
         FIG. 18A  is an enlarged view of the pass band and its vicinity of the graph shown at the top of  FIG. 17A . 
         FIG. 18B  is an enlarged view of the pass band and its vicinity of the graph shown at the top of  FIG. 17B . 
         FIG. 19  shows graphs of changes in filter characteristics when switches of the radio-frequency filter according to Example 3 are individually switched between an on state and an off state. 
         FIG. 20A  is a circuit configuration diagram of a radio-frequency filter according to Example (Example 4) of a fourth embodiment. 
         FIG. 20B  is a plan view that illustrates the structure of the radio-frequency filter according to Example 4. 
         FIG. 21  is a circuit configuration diagram of a multiplexer according to Example 4. 
         FIG. 22  is a configuration diagram of a communication device according to a fifth embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE DISCLOSURE 
     Hereinafter, embodiments of the present disclosure will be described in detail with reference to Examples and the drawings. Any of the embodiments that will be described below describes a comprehensive or specific example. Numeric values, shapes, materials, elements, disposition and connection modes of the elements, and the like, that will be described below are illustrative, and are not intended to limit the present disclosure. Of the elements in the following embodiments, the elements not included in the independent claims will be described as optional elements. In addition, the size or size ratio of elements shown in the drawings is not necessarily strict. In the drawings, like reference signs denote substantially identical components, and the overlap description may be omitted or simplified. For circuit elements, such as resonators, constants can be adjusted as needed according to the required specifications or others. Therefore, for circuit elements, constants can be different even with the same reference numerals. In the following description, the term “lower edge of a pass band” and the term “higher edge of a pass band” respectively mean a lower edge (lower frequency-side edge) in a pass band and a higher edge (higher frequency-side edge) in a pass band. In the following description, the term “lower side of a pass band” and the term “higher side of a pass band” respectively mean frequencies outside a pass band and lower (lower frequency side) than the pass band and frequencies outside a pass band and higher (higher frequency side) than the pass band. 
     The resonant frequency of a resonator or circuit, unless otherwise specified, is a resonant frequency for forming a pass band of a filter including the resonator or the circuit or an attenuation pole near the pass band, and is a frequency at a resonant point that is a singular point where the impedance of the resonator or the circuit is local minimum (ideally, a point where the impedance is zero). 
     The anti-resonant frequency of a resonator or circuit, unless otherwise specified, is an anti-resonant frequency for forming a pass band of a filter including the resonator or the circuit or an attenuation pole near the pass band, and is a frequency at an anti-resonant point that is a singular point where the impedance of the resonator or the circuit is local maximum (ideally, a point where the impedance is infinite). 
     In the following embodiments, a series arm circuit and a parallel arm circuit are defined as follows. 
     A parallel arm circuit is a circuit disposed between a ground and a node in a path that connects a first input and output terminal and a second input and output terminal. 
     A series arm circuit is a circuit disposed between the first input and output terminal or second input and output terminal and a node in the path to which the parallel arm circuit is connected or a circuit disposed between one node in the path to which one parallel arm circuit is connected and another node in the path to which another parallel arm circuit is connected. 
     First Embodiment 
     Hereinafter, a radio-frequency filter according to the present embodiment will be described by way of Examples (Example 1-1, Example 1-2, and Example 1-3). 
     [1-1. Configuration] 
       FIG. 1A  is a circuit configuration diagram of a radio-frequency filter  10  according to Example 1-1. The radio-frequency filter  10  shown in the diagram includes a series arm circuit  11  and a parallel arm circuit  12 . 
     The series arm circuit  11  is a resonant circuit provided in a path (series arm) connecting an input and output terminal  11   m  (first input and output terminal) and an input and output terminal  11   n  (second input and output terminal). Specifically, the series arm circuit  11  includes a series arm resonator s 1   a  (first series arm resonator) and a variable frequency circuit  11   b  (first variable frequency circuit). The variable frequency circuit  11   b  is made up of a series arm resonator s 1   b  (second series arm resonator) and a switch SWb (first switch). In this way, since the series arm circuit  11  includes the series arm resonators s 1   a , s 1   b  each having a resonant frequency (resonant point) at which the impedance is local minimum and an anti-resonant frequency (anti-resonant point) at which the impedance is local maximum, the series arm circuit  11  has a resonant frequency and an anti-resonant frequency that depend on the resonant frequencies and anti-resonant frequencies of the series arm resonators s 1   a , s 1   b . The details will be described later. 
     The series arm resonator s 1   a  is a first series arm resonator provided in the series arm connecting the input and output terminal  11   m  and the input and output terminal  11   n , and is connected between the input and output terminal  11   m  (first input and output terminal) and the input and output terminal  11   n  (second input and output terminal). 
     The series arm resonator s 1   b  is a second series arm resonator provided in the series arm and in a path that bypasses the series arm resonator s 1   a . One terminal of the series arm resonator s 1   b  is connected to one terminal of the series arm resonator s 1   a , and the other terminal of the series arm resonator s 1   b  is connected to the other terminal of the series arm resonator s 1   a  via the switch SWb. 
     The switch SWb is a first switch connected in series with the series arm resonator s 1   b  (second series arm resonator). The switch SWb is an SPST (single pole single throw) switch element. Examples of the switch SWb include an FET (field effect transistor) switch made from GaAs (gallium arsenide) or CMOS (complementary metal oxide semiconductor), and a diode switch. The switch SWb is configured as, for example, a switch IC (integrated circuit). The switch SWb is not limited to a semiconductor switch formed on or in a semiconductor substrate and may be a mechanical switch made up of MEMS (micro electro mechanical systems). 
     The variable frequency circuit  11   b  (first variable frequency circuit) made up of these series arm resonator s 1   b  and switch SWb changes the resonant frequency that provides the pass band of the radio-frequency filter  10  and the anti-resonant frequency that provides the attenuation pole of the radio-frequency filter  10 . Specifically, the variable frequency circuit  11   b  changes the resonant frequency and the anti-resonant frequency by switching between the on state (conductive state) and off state (nonconductive state) of the switch SWb in accordance with a control signal from a control unit (not shown), such as an RFIC or other like controller, processor, or integrated or discrete circuit. 
     The variable frequency circuit  11   b  just needs to change at least one of the resonant frequency and the anti-resonant frequency and may change only any one of the resonant frequency and the anti-resonant frequency. 
     In this Example, the series arm resonator s 1   b  and the switch SWb that make up the variable frequency circuit  11   b  are connected in this order from the input and output terminal  11   m  side. Alternatively, the series arm resonator s 1   b  and the switch SWb may be connected in reverse order. 
     In this way, the series arm circuit  11  includes the series arm resonator s 1   a  (first series arm resonator) and the variable frequency circuit  11   b  connected in parallel with the series arm resonator s 1   a.    
     The parallel arm circuit  12  is a resonant circuit connected with a ground (reference terminal) and a node x 1  in the path (series arm) connecting the input and output terminal  11   m  and the input and output terminal  11   n . Specifically, the parallel arm circuit  12  includes a parallel arm resonator p 1  connected between the node x 1  and the ground. In this way, since the parallel arm circuit  12  includes the parallel arm resonator p 1  that has a resonant frequency (resonant point) at which the impedance is local minimum and an anti-resonant frequency (anti-resonant point) at which the impedance is local maximum, the parallel arm circuit  12  has a resonant frequency and an anti-resonant frequency that depend on the resonant frequency and anti-resonant frequency of the parallel arm resonator p 1 . 
     In the thus configured radio-frequency filter  10 , each of the resonators (in this Example, the series arm resonators s 1   a , s 1   b , and the parallel arm resonator p 1 ) is, for example, an acoustic wave resonator that uses surface acoustic waves. 
       FIG. 2  shows a schematic plan view and cross-sectional view of the electrode configuration of an acoustic wave resonator  120  that uses surface acoustic waves. The electrode configuration shown in the drawings is the one for illustrating the typical structure of each of the resonators in the radio-frequency filter  10 , and the number of electrode fingers that make up the electrode, the length of each electrode finger, and the like, are not limited thereto. 
     As shown in the drawings, the acoustic wave resonator  120  includes a piezoelectric substrate  101  and an interdigital transducer electrode  111 . The piezoelectric substrate  101  is a substrate at least partially having piezoelectricity. The interdigital transducer electrode  111  is formed on the piezoelectric substrate  101 . 
     The interdigital transducer electrode  111  includes a plurality of electrode fingers  111   a  and a pair of busbar electrodes disposed on opposite sides with the plurality of electrode fingers  111   a  interposed therebetween. The plurality of electrode fingers  111   a  is alternately connected to one and the other of the pair of busbar electrodes. The plurality of electrode fingers  111   a  is formed along a direction perpendicular to a propagation direction of surface acoustic waves, and is periodically formed along the propagation direction. 
     In the thus configured acoustic wave resonator  120 , the wave length of surface acoustic waves to be excited is determined by design parameters of the interdigital transducer electrode  111 , or other parameters. Hereinafter, the design parameters of the interdigital transducer electrode  111  will be described. 
     The wave length of surface acoustic waves is determined by a cycle period λ of the electrode fingers  111   a  connected to one busbar electrode among the plurality of electrode fingers  111   a . An electrode finger pitch (the pitch of the plurality of electrode fingers  111   a , that is, an electrode finger period) P is a half of the cycle period λ. The electrode finger pitch is defined as P=(W+S) where the line width of each electrode finger  111   a  is W and the space width between any adjacent electrode fingers  111   a  is S. An electrode duty (duty ratio) is the occupancy of the line widths of the plurality of electrode fingers  111   a , and is the proportion of the line width to an added value of the line width of each of the plurality of electrode fingers  111   a  and the space width between any adjacent two of the plurality of electrode fingers  111   a , that is, the electrode duty is defined as W/(W+S). In other words, the electrode duty is the ratio of the width of each of the plurality of electrode fingers  111   a  to the electrode finger pitch (the pitch of the plurality of electrode fingers  111   a ), that is, the electrode duty is defined as W/P. The number of pairs is the number of paired electrode fingers  111   a , and is about a half of the total number of the electrode fingers  111   a . For example, where the number of pairs is N and the total number of the electrode fingers  111   a  is M, M=2N+1. The electrostatic capacity C 0  of the acoustic wave resonator  120  is expressed by the following mathematical expression 1. 
     
       
         
           
             
               
                 
                   
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                       Mathematical 
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                         ELECTROSTATIC 
                       
                     
                     
                       
                         
                           CAPACITY 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             0 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           0 
                         
                         · 
                         
                           ɛ 
                           r 
                         
                         · 
                         NUMBER 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       OF 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         PAIRS 
                         · 
                         OVERLAP 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       WIDTH 
                     
                     
                       2 
                       · 
                       
                         ( 
                         
                           1 
                           - 
                           
                             ELECTRODE 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             DUTY 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     ε 0  is the permittivity in a vacuum, and εr is the permittivity of the piezoelectric substrate  101 . 
     [1-2. Characteristics] 
     The radio-frequency filter  10  configured as described above is able to reduce the loss at the higher edge of the pass band and change (shift) frequencies on the lower side of the pass band. Hereinafter, the filter characteristics (bandpass characteristics) of the radio-frequency filter  10  of this Example will be described while impedance characteristics (resonant characteristics) that determine the filter characteristics are discussed. 
     Circuit constants of the radio-frequency filter  10  of Example 1-1 are shown in Table 1. An off capacitance (Coff) that is a capacitive component when the switch SWb is off is 0.2 pF. The details of the off capacitance (Coff) will be described later. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Series Arm 
                 Series Arm 
                 Parallel Arm 
               
               
                   
                 Resonator s1a 
                 Resonator s1b 
                 Resonator p1 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 Resonant Frequency fr 
                 735 
                 702 
                 697 
               
               
                 [MHz] 
               
               
                 Anti-Resonant 
                 794 
                 758 
                 725 
               
               
                 Frequency fa [MHz] 
               
               
                 Electrostatic Capacity 
                 2.0 
                 2.0 
                 2.0 
               
               
                 C 0  [pF] 
               
               
                   
               
            
           
         
       
     
     As shown in Table 1, the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b  (second series arm resonator) is different from the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a  (first series arm resonator) (fr_s 1   b ≠fr_s 1   a ), and, in this Example, is lower than the resonant frequency of the series arm resonator s 1   a  (fr_s 1   b &lt;fr_s 1   a ). The resonant frequency (fr_p 1 ) of the parallel arm resonator p 1  is lower than the resonant frequency of the series arm resonator s 1   a  (fr_p 1 &lt;fr_s 1   a ). 
       FIG. 1B  shows graphs of various characteristics of the radio-frequency filter  10 . Specifically, the top graph shows filter characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off. The middle graph shows the impedance characteristics of the series arm circuit  11  and parallel arm circuit  12 . The bottom graph shows the impedance characteristics of the series arm resonator s 1   a , the series arm resonator s 1   b , and the parallel arm resonator p 1 . For the series arm circuit  11 , the impedance characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off, are shown. 
     The radio-frequency filter  10  according to this Example is a tunable receiving filter that switches an accepting frequency band (that is, a frequency band to be filtered) between Band 29 and Band 12 defined in 3GPP (third generation partnership project). Therefore, as shown in the top graph of  FIG. 1B , the radio-frequency filter  10  switches the pass band between Band 29Rx (717 MHz. to 727 MHz) that is the receiving band of Band 29 and Band 12Rx (729 MHz. to 746 MHz) that is the receiving band of Band 12. 
     First, characteristics when the switch SWb is off in the circuit configuration shown in  FIG. 1A  will be described. 
     In this case, the series arm circuit  11  is placed in a state where the variable frequency circuit  11   b  whose switch SWb is off is connected in parallel with the series arm resonator s 1   a . At this time, since the impedance of the variable frequency circuit  11   b  is extremely high (ideally, infinite), the series arm circuit  11  exhibits substantially the same characteristics (ideally, exactly the same characteristics) as the characteristics of the series arm resonator s 1   a.    
     However, actually, as shown in the middle and bottom graphs in  FIG. 1B , the anti-resonant frequency (fa_s 1 _off) of the series arm circuit  11  is influenced by a combined capacitance of the series arm resonator s 1   b  of the variable frequency circuit  11   b  and the off capacitance (Coff) of the switch SWb. Therefore, the anti-resonant frequency (fa_s 1 _off) slightly shifts (Afa_s 1   a ) toward a lower frequency side from the anti-resonant frequency (fa_s 1   a ) of the series arm resonator s 1   a.    
     The off capacitance (Coff) of the switch SWb is a capacitive component when the switch SWb is off. In other words, the switch SWb is placed in a state of ideally no capacitive component (that is, the impedance is infinite) when the switch SWb is off; however, the switch SWb actually has an off capacitance (Coff) that is a slight capacitive component. Since the off capacitance is sufficiently less than the electrostatic capacity of the series arm resonator s 1   b  (0.2 pF in this Example), a combined capacitance of the series arm resonator s 1   b  and the off capacitance of the switch SWb is sufficiently less than the electrostatic capacity of the series arm resonator s 1   b . Thus, there is a slight frequency difference (Afa_s 1   a ) in the anti-resonant frequency of the series arm circuit  11  between when the switch SWb is off and when the switch SWb is on. 
     On the other hand, since the resonant frequency (fr_s 1 _off) of the series arm circuit  11  is not influenced by the variable frequency circuit  11   b , the resonant frequency (fr_s 1 _off) is equal to the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a.    
     Next, characteristics when the switch SWb is on in the circuit configuration shown in  FIG. 1A  will be described. 
     In this case, the series arm circuit  11  is placed in a state where the variable frequency circuit  11   b  whose switch SWb is on is connected in parallel with the series arm resonator s 1   a . In other words, the series arm circuit  11  becomes a parallel connection circuit of the series arm resonator s 1   a  and the series arm resonator s 1   b.    
     Therefore, as shown in the middle graph in  FIG. 1B , the series arm circuit  11  has two resonant frequencies and two anti-resonant frequencies. Specifically, the series arm circuit  11  has a first resonant frequency (fr 1 _s 1 _on) equal to the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b , a second resonant frequency (fr 2 _s 1 _on) equal to the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a , a first anti-resonant frequency (fa 1 _s 1 _on) that lies between the first resonant frequency (fr_s 1 _on) and the second resonant frequency (fr 2 _s 1 _on), and a second anti-resonant frequency (fa 2 _s 1 _on) that lies between the second resonant frequency (fr 2 _s 1 _on) and the anti-resonant frequency (fa_s 1   a ) of the series arm resonator s 1   a.    
     In this way, in this Example, the anti-resonant frequency and resonant frequency of the series arm circuit  11  shift in response to switching between the on state and off state of the switch SWb. 
     The resonant frequency and anti-resonant frequency of the parallel arm circuit  12  do not shift in response to switching between the on state and off state of the switch SWb. The resonant frequency of the parallel arm circuit  12  is equal to the resonant frequency (fr_p 1 ) of the parallel arm resonator p 2 , and the anti-resonant frequency of the parallel arm circuit  12  is equal to the anti-resonant frequency (fa_p 1 ) of the parallel arm resonator p 2 . 
     Generally, a series arm circuit made up of only a series arm resonator has one resonant frequency (frs) and one anti-resonant frequency (fas). Similarly, a parallel arm circuit made up of only a parallel arm resonator has one resonant frequency (frp) and one anti-resonant frequency (fap). For this reason, in a bandpass ladder filter made up of a series arm circuit and a parallel arm circuit, a pass band is provided by bringing the anti-resonant frequency (fap) of the parallel arm circuit and the resonant frequency (frs) of the series arm circuit close to each other, an attenuation pole on the lower side of the pass band is provided by the resonant frequency (frp) of the parallel arm circuit, and an attenuation pole on the higher side of the pass band is provided by the anti-resonant frequency (fas) of the series arm circuit. 
     Therefore, in this Example, filter characteristics as follows are provided. 
     First, when the switch SWb is off, the pass band is provided by the resonant frequency (fr_s 1 _off=fr_s 1   a ) of the series arm circuit  11  and the anti-resonant frequency (fa_p 1 ) of the parallel arm circuit  12 , the attenuation pole on the lower side of the pass band is provided by the resonant frequency (fr_p 1 ) of the parallel arm circuit  12 , and the attenuation pole on the higher side of the pass band is provided by the anti-resonant frequency (fa_s 1 _off=fa_s 1   a −Δfa_s 1   a ) of the series arm circuit  11 . 
     Thus, when the switch SWb is off, the filter characteristics represented by the dashed line (B29 side (SWb Off)) in the top graph of  FIG. 1B , that is, the filter characteristics having Band 29Rx (717 MHz to 727 MHz) as a pass band, are provided. At this time, the loss (I. Loss: insertion loss) at the lower edge (717 MHz) of the pass band is 0.288 dB, and the loss at the higher edge (727 MHz) of the pass band is 0.265 dB, so the loss in the pass band is reduced. Band 29 is a receive-only band. 
     On the other hand, when the switch SWb is on, the pass band is provided by the second resonant frequency (fr 2 _s 1 _on, that is, fr_s 1   a ) of the series arm circuit  11  and the anti-resonant frequency (fa_p 1 ) of the parallel arm circuit  12 , the attenuation pole on the lower side of the pass band is provided by the resonant frequency (fr_p 1 ) of the parallel arm circuit  12 , and the attenuation pole on the higher side of the pass band is provided by the second anti-resonant frequency (fa 2 _s 1 _on) of the series arm circuit  11 . In addition, a new attenuation pole is provided on the lower side of the pass band by the first anti-resonant frequency (fa 1 _s 1 _on) of the series arm circuit  11 . 
     Thus, when the switch SWb is on, the filter characteristics represented by the continuous line (B12 side (SWb On)) in the top graph of  FIG. 1B , that is, the filter characteristics having Band 12Rx (729 MHz to 746 MHz) as a pass band and having Band 12Tx (699 MHz to 716 MHz) as a stop band, are provided. At this time, the loss at the lower edge (729 MHz) of the pass band is 0.314 dB, and the loss at the higher edge (746 MHz) of the pass band is 0.237 dB, so the loss in the pass band is reduced. The attenuation (Att.: attenuation) at the lower edge (699 MHz) of the stop band is 17.555 dB, and the attenuation at the higher edge (716 MHz) of the stop band is 4.020 dB, so the attenuation in the stop band is ensured. 
     In this way, the radio-frequency filter  10  according to this Example is applicable to a variable frequency filter that switches between Band 29Rx and Band 12Rx. 
     [1-3. Advantageous Effects, and Others] 
     Hereinafter, advantageous effects that are obtained from this Example will be described by way of comparison with a radio-frequency filter according to Comparative Example of the present embodiment. 
     [1-3-1. Comparative Example] 
       FIG. 3A  is a circuit configuration diagram of the radio-frequency filter  10 X according to Comparative Example. The radio-frequency filter  10 X shown in the diagram includes a series arm circuit  11 X made up of a series arm resonator s 1  and a parallel arm circuit  12 X made up of parallel arm resonators p 1   a , p 1   b  and a switch SWx. The parallel arm resonator p 1   a  and the parallel arm resonator p 1   b  are connected in series with each other. The parallel arm resonator p 1   b  and the switch SWx are connected in parallel with each other. 
     The thus configured radio-frequency filter  10 X is also able to change the frequency of the attenuation pole on the lower side of the pass band by switching between an on state and off state of the switch SWx. Hereinafter, the filter characteristics (bandpass characteristics) of the radio-frequency filter  10 X of Comparative Example will be described while impedance characteristics (resonant characteristics) that determine the filter characteristics are also discussed. 
     Circuit constants of the radio-frequency filter  10 X of Comparative Example are shown in Table 2. The off capacitance (Coff) of the switch SWx is 0.2 pF. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Series Arm 
                 Parallel Arm 
                 Parallel Arm 
               
               
                   
                 Resonator s1 
                 Resonator p1a 
                 Resonator p1b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 Resonant Frequency fr 
                 735 
                 697 
                 735 
               
               
                 [MHz] 
               
               
                 Anti-Resonant 
                 794 
                 725 
                 764 
               
               
                 Frequency fa [MHz] 
               
               
                 Electrostatic Capacity 
                 2.0 
                 5.0 
                 3.0 
               
               
                 C 0  [pF] 
               
               
                   
               
            
           
         
       
     
       FIG. 3B  shows graphs of various characteristics of the radio-frequency filter  10 X. Specifically, the top graph shows filter characteristics in two states, that is, the case where the switch SWx is on and the case where the switch SWx is off. The middle graph shows the impedance characteristics of the series arm circuit  11 X and parallel arm circuit  12 X. The bottom graph shows the impedance characteristics of the series arm resonator s 1 , the parallel arm resonator p 1   a , and the parallel arm resonator p 1   b . For the parallel arm circuit  12 X, the impedance characteristics in two states, that is, the case where the switch SWx is on and the case where the switch SWx is off, are shown. 
     First, characteristics when the switch SWx is on in the circuit configuration shown in  FIG. 3A  will be described. 
     In this case, the parallel arm circuit  12 X exhibits the same characteristics as the parallel arm resonator p 1   a  since the parallel arm resonator p 1   b  is short-circuited by the switch SWx. That is, the resonant frequency (fr_p 1 _on) of the parallel arm circuit  12 X is equal to the resonant frequency (fr_p 1   a ) of the parallel arm resonator p 1   a , and the anti-resonant frequency (fa_p 1 _on) of the parallel arm circuit  12 X is equal to the anti-resonant frequency (fa_p 1   a ) of the parallel arm resonator p 1   a.    
     Next, characteristics when the switch SWx is off in the circuit configuration shown in  FIG. 3A  will be described. 
     In this case, since the parallel arm circuit  12 X is a circuit in which the parallel arm resonator p 1   a  and the parallel arm resonator p 1   b  are connected in series with each other, the parallel arm circuit  12 X exhibits combined characteristics of the parallel arm resonator p 1   a  and parallel arm resonator p 1   b.    
     In this case, actually, since there is the off capacitance of the switch SWx, the parallel arm circuit  12 X exhibits characteristics slightly different from the combined characteristics. However, since the off capacitance of the switch SWx is small enough (0.2 pF in this Example) as compared to the electrostatic capacities of the parallel arm resonators p 1   a , p 1   b  (5.0 pF, 3.0 pF in this Example), the off capacitance almost does not influence the characteristics of the parallel arm circuit  12 X. Thus, for the sake of succinctness, the description will be made on the assumption that the switch SWx is an ideal switch having no off capacitance. 
     Therefore, as shown in the middle graph of  FIG. 3B , the parallel arm circuit  12 X has two resonant frequencies and two anti-resonant frequencies. Specifically, the parallel arm circuit  12 X has a first anti-resonant frequency (fa 1 _p 1 _off) equal to the anti-resonant frequency (fa_p 1   a ) of the parallel arm resonator p 1   a , a second anti-resonant frequency (fa 2 _p 1 _off) equal to the anti-resonant frequency (fa_p 1   b ) of the parallel arm resonator p 1   b , a first resonant frequency (fr 1 _p 1 _off) that lies between the resonant frequency (fr_p 1   a ) of the parallel arm resonator p 1   a  and the first anti-resonant frequency (fa 1 _p 1 _off), and a second resonant frequency (fr 2 _p 1 _off) that lies between the first resonant frequency (fr 1 _p 1 _off) and the second anti-resonant frequency (fa 2 _p 1 _off). 
     In this way, in Comparative Example, in response to switching between the on state and off state of the switch SWx, the numbers of the anti-resonant frequencies and resonant frequencies of the parallel arm circuit  12 X change, and the resonant frequencies shift. 
     The resonant frequency and anti-resonant frequency of the series arm circuit  11 X do not shift in response to switching between the on state and off state of the switch SWx. The resonant frequency of the series arm circuit  11 X is equal to the resonant frequency (fr_s 1 ) of the series arm resonator s 1 , and the anti-resonant frequency of the series arm circuit  11 X is equal to the anti-resonant frequency (fa_s 1 ) of the series arm resonator s 1 . 
     Thus, in this Comparative Example, when the switch SWx is on, the filter characteristics represented by the dashed line (B29 side (SWx On)) in the top graph of  FIG. 3B , that is, the filter characteristics (B29 side in the graph) having Band 29Rx as a pass band, are provided. At this time, the loss at the lower edge (717 MHz) of the pass band is 0.323 dB, and the loss at the higher edge (727 MHz) of the pass band is 0.269 dB, so the loss in the pass band is reduced. 
     On the other hand, when the switch SWx is off, the filter characteristics represented by the continuous line (B12 side (SWx Off)) in the top graph of  FIG. 3B , that is, the filter characteristics having Band 12Rx as a pass band and having Band 12Tx as a stop band, are provided. At this time, the loss at the lower edge (729 MHz) of the pass band is 0.267 dB, and the loss at the higher edge (746 MHz) of the pass band is 0.904 dB, so the loss at the higher edge of the pass band becomes deteriorated (increased). The attenuation at the lower edge (699 MHz) of the stop band is 4.249 dB, and the attenuation at the higher edge (716 MHz) of the stop band is 0.694 dB, so the attenuation at the higher edge of the stop band is insufficient. 
     Thus, when the radio-frequency filter  10 X according to Comparative Example is applied to a variable frequency filter that switches between Band 29Rx and Band 12Rx, the loss at the higher edge of the pass band deteriorates and the attenuation in the stop band on the lower side of the pass band deteriorates when SWx is off. In this way, in Comparative Example, when the lower side of the pass band is changed toward a higher frequency side, there is inconvenience that the loss at the higher edge of the pass band deteriorates and the attenuation in the stop band on the lower side of the pass band deteriorates. 
     [1-3-2. Comparison with Comparative Example] 
     The inventors of the subject application diligently studied and finally found that, in Comparative Example, when the frequency of the attenuation pole on the lower side of the pass band was changed toward a higher frequency side, the above inconvenience would occur because of a new attenuation pole formed near the higher side of the pass band. The inventors conceived of the fact that the loss at the higher edge of the pass band and an attenuation on the lower side of the pass band would be improved when a new attenuation pole was not formed near the higher side of the pass band and was formed in lower frequencies of the pass band in the case where the frequency of the attenuation pole on the lower side of the pass band was changed toward a higher frequency side. 
       FIG. 4A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1 and Comparative Example at the time of lower frequency shift (B29 side: when the pass band frequencies and attenuation pole on the lower side of the pass band are changed in frequency toward a lower side).  FIG. 4B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1 and Comparative Example at the time of higher frequency shift (B12 side: when the pass band frequencies and attenuation pole on the lower side of the pass band are changed in frequency toward a higher frequency). 
     As is apparent from  FIG. 4A , at the time of lower frequency shift, Example 1-1 and Comparative Example exhibit equivalent characteristics in the pass band (Band 29Rx). 
     As is apparent from  FIG. 4A  and  FIG. 4B , in Comparative Example, when the pass band frequencies and attenuation pole on the lower side of the pass band are changed in frequency toward a higher frequency side by turning the switch SWx from the on state to the off state, in addition to a first attenuation pole P_H 91  that is provided by the anti-resonant frequency (fa_s 1 ) of the series arm circuit  11 X on the higher side of the pass band, a second attenuation pole P_H 92  that is provided by the second resonant frequency (fr 2 _p 1 _off) of the parallel arm circuit  12 X appears. Since the second attenuation pole P_H 92  lies on the lower frequency side than the first attenuation pole P_H 91 , the loss at the higher edge of the pass band when the switch SWx is on deteriorates. On the other hand, at this time, since only one attenuation pole P_L 1  that is provided by the first resonant frequency (fr_p 1 _off) of the parallel arm circuit  12 X appears on the lower side of the pass band, the attenuation is insufficient in the stop band on the lower side of the pass band. 
     In contrast to this, in this Example, even when the pass band frequencies and attenuation pole on the lower side of the pass band are changed in frequency toward a higher frequency side by turning the switch SWb from the off state to the on state, only one attenuation pole (an attenuation pole P_H 1 off when the switch SWb is off, and an attenuation pole P_H 1 on when the switch SWb is on) that is provided by the second anti-resonant frequency (fa 2 _s 1 _on) of the series arm circuit  11  appears on the higher side of the pass band. Therefore, in this Example, as compared to Comparative Example, the deterioration of the loss at the higher edge of the pass band when the frequencies on the lower side of the pass band are changed toward a higher frequency side is reduced. On the other hand, at this time, since two attenuation poles, that is, the attenuation pole P_L 1  that is provided by the resonant frequency (fr_p 1 ) of the parallel arm circuit  12  and the attenuation pole P_L 2  that is provided by the first anti-resonant frequency (fa_s 1 _on) of the series arm circuit  11 , appear on the lower side of the pass band, the attenuation in the stop band on the lower side of the pass band improves. 
     [1-3-3. Summary] 
     As described above, according to this Example, the resonant frequency that provides the pass band of the radio-frequency filter  10  and the anti-resonant frequency that provides the attenuation pole of the radio-frequency filter  10  can be changed for the series arm circuit  11  by switching between the on state (conductive state) and off state (nonconductive state) of the switch SWb (first switch). Specifically, when the switch SWb is off, the series arm circuit  11  has the third resonant frequency (fr 2 _s 1 _off) equal to the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a  (first series arm resonator) and the third anti-resonant frequency (fa_s 1 _off) that lies on a slightly lower frequency side than the anti-resonant frequency (fa_s 1   a ) of the series arm resonator s 1   a  (first series arm resonator). When the switch SWb is on, the series arm circuit  11  has the first resonant frequency (fr 2 _s 1 _on) equal to the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a , the second resonant frequency (fr 1 _s 1 _on) equal to the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b  (second series arm resonator), the first anti-resonant frequency (fa 1 _s 1 _on) that lies between the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a  and the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b , and the second anti-resonant frequency (fa 2 _s 1 _on) that lies on the higher frequency side than the second resonant frequency (fr 1 _s 1 _on). 
     In this regard, in this Example, the resonant frequency of the series arm resonator s 1   b  is lower than the resonant frequency of the series arm resonator s 1   a  (fr_s 1   b &lt;fr_s 1   a ). Thus, when the switch SWb is off, the third resonant frequency (fr 2 _s 1 _off) provides the pass band, and the third anti-resonant frequency provides the attenuation pole on the higher side of the pass band. When the switch SWb is on, the second resonant frequency (fr 1 _s 1 _on) provides the pass band, the first anti-resonant frequency (fa 1 _s 1 _on) provides the attenuation pole on the lower side of the pass band, and the second anti-resonant frequency (fa 2 _s 1 _on) provides the attenuation pole on the higher side of the pass band. That is, in this Example, a new attenuation pole is provided on the lower side of the pass band by turning the switch SWb from the off state to the on state. 
     Thus, when the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency of the parallel arm circuit, is considered, the following can be said on the relationship between switching the on state and off state of the switch SWb and the pass band. Specifically, when the switch SWb is switched from the off state to the on state, a new attenuation pole is provided on the lower side of the pass band, so the stop band on the lower side of the pass band is widened. At this time, since no new attenuation pole is provided on the higher side of the pass band, the deterioration of the loss at the higher edge of the pass band is reduced, and the deterioration of the attenuation on the lower side of the pass band is reduced. 
     Therefore, according to this Example, the tunable radio-frequency filter  10  that reduces the loss at the higher edge of the pass band and that is able to change the pass band frequencies and the frequency of the attenuation pole on the lower side of the pass band while improving the attenuation on the lower side of the pass band can be implemented. 
     In this way, in this Example, since the resonant frequency of the series arm resonator s 1   b  is lower than the frequency of the series arm resonator s 1   a , the number of attenuation poles on the lower side of the pass band can be increased or reduced by switching between the on state and off state of the switch SWb. Thus, the loss at the higher edge of the pass band and the attenuation on the lower side of the pass band are improved. 
     The resonant frequency of the series arm resonator s 1   b  (second series arm resonator) is not limited to the frequency (702 MHz). 
       FIG. 5  is a graph that shows the influence on filter characteristics when the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b  is changed in the radio-frequency filter  10  according to Example 1-1. Specifically, the graph shows filter characteristics when the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b  is changed in steps of 5 MHz in the range of 690 MHz to 710 MHz while the switch SWb is on, and filter characteristics when the switch SWb is off. For the series arm resonator s 1   b  shown in the graph, a band width ratio ((fa−fr)/fr) that is a value obtained by dividing the difference (fa−fr) between the anti-resonant frequency (fa) and the resonant frequency (fr) by the resonant frequency (fr) is fixed. 
     When the resonant frequency (fr_s 1   b ) of the series arm resonator s 1   b  is changed, the first anti-resonant frequency (fa 1 _s 1 _on) of the series arm circuit  11  when the switch SWb is on changes. Therefore, as is apparent from the graph, of the two attenuation poles on the lower side of the pass band, the frequency of the attenuation pole adjacent to the pass band, which is the attenuation pole that is provided by the first anti-resonant frequency (fa 1 _s 1 _on) of the series arm circuit  11 , changes. 
     Thus, the radio-frequency filter may have a plurality of the second series arm resonators whose resonant frequencies are different from each other. In other words, the radio-frequency filter may include a plurality of the first variable frequency circuits. 
       FIG. 6  is a circuit configuration diagram of the thus configured radio-frequency filter  10 A (radio-frequency filter according to a modification). 
     The radio-frequency filter  10 A shown in the diagram includes a plurality of variable frequency circuits  11   b   1  to  11   bn . Each of the plurality of variable frequency circuits  11   b   1  to  11   bn  corresponds to the variable frequency circuit  11   b  (first variable frequency circuit) of Example 1-1 and includes a corresponding one of series arm resonators s 1   b   1  to s 1   bn , corresponding to the series arm resonator s 1   b  (second series arm resonator) of Example 1-1, and a corresponding one of switches SWb 1  to SWbn, corresponding to the switch SWb (first switch) of Example 1-1. 
     The resonant frequencies of the second series arm resonators s 1   b   1  to s 1   bn  (second series arm resonators) are different from each other (fr_s 1   b   1 ≠ . . . ≠fr_s 1   bn ). Specifically, any of the resonant frequencies of the series arm resonators s 1   b   1  to s 1   bn  is lower than the resonant frequency of the series arm resonator s 1   a  (first series arm resonator) (fr_s 1   b   1 &lt;fr_s 1   a , . . . , fr_s 1   bn &lt;fr_s 1   a ). 
     The thus configured radio-frequency filter  10 A is able to change the first anti-resonant frequency (fr 1 _s 1 _on) of the series arm circuit  11  according to the switch selected and set to the on state from among the plurality of switches SWb 1  to SWbn. In other words, when the frequencies of the pass band and the stop band on the lower side of the pass band can be changed in minute steps, the variable width of the frequencies of the pass band and the variable width of the frequency of the attenuation pole on the lower side of the pass band can be minutely set. Thus, filter characteristics that give the minimum loss can be selected for each channel of a band (for example, Band 12Rx) to be used. 
     [1-4. Other Examples] 
     [1-4-1. Series Connection of Capacitor or Inductor with First Series Arm Resonator] 
     The series arm circuit according to the present embodiment may further include a first impedance element connected in series with the first series arm resonator. Hereinafter, such a radio-frequency filter will be described by way of Examples (Example 1-2-1 and Example 1-2-2). 
       FIG. 7A  is a circuit configuration diagram of a radio-frequency filter  10 Ba according to Example 1-2-1. 
     As shown in the diagram, the radio-frequency filter  10 Ba according to this Example includes a series arm circuit  11 Ba that further includes a capacitor C 11  (first impedance element) connected in series with the series arm resonator s 1   a  (first series arm resonator) as compared to Example 1-1. The variable frequency circuit  11   b  is connected in parallel with a series connection circuit  11   aa  (first series connection circuit) made up of the series arm resonator s 1   a  and the capacitor C 11 . 
       FIG. 7B  is a circuit configuration diagram of a radio-frequency filter  10 Bb according to Example 1-2-2. 
     As shown in the diagram, the radio-frequency filter  10 Bb according to this Example has such a configuration that the capacitor C 11  is replaced by an inductor L 11  (first impedance element) in the radio-frequency filter  10 Ba according to Example 1-2-1. 
     Hereinafter, the characteristics of the thus configured radio-frequency filters according to Example 1-2-1 and Example 1-2-2 will be described in comparison with Example 1-1. 
     Circuit constants of the radio-frequency filter  10 Ba of Example 1-2-1 are substantially the same as the circuit constants of Example 1 and differ only in that the resonant frequency (fr) of the series arm resonator s 1   a  (first series arm resonator) is 725 MHz and the anti-resonant frequency (fa) is 783 MHz. The capacitance of the capacitor C 11  is 8 pF. 
     Circuit constants of the radio-frequency filter  10 Bb of Example 1-2-2 are substantially the same as the circuit constants of Example 1 and differ only in that the resonant frequency (fr) of the series arm resonator s 1   a  (first series arm resonator) is 745 MHz and the anti-resonant frequency (fa) is 804 MHz. The inductance of the inductor L 11  is 2 nH. 
       FIG. 7C  shows graphs of various characteristics of the radio-frequency filters according to Example 1-2-1 and Example 1-2-2 in comparison with Example 1-1. Specifically, the left-column graphs show various characteristics of the radio-frequency filter  10  according to Example 1-1, the middle-column graphs show various characteristics of the radio-frequency filter  10 Ba according to Example 1-2-1, and the right-column graphs show various characteristics of the radio-frequency filter  10 Bb according to Example 1-2-2. The top graphs show filter characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off. The middle graphs show the impedance characteristics of the series arm circuits and parallel arm circuit  12 . The bottom graphs show the impedance characteristics of the series arm resonator s 1   a , the series arm resonator s 1   b , and the parallel arm resonator p 1 . For the series arm circuits, the impedance characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off, are shown. 
     As is apparent from the top graphs, even with the configurations of Example 1-2-1 and Example 1-2-2 in which the impedance element is connected in series with the series arm resonator s 1   a , the lower side of the pass band can be changed while the loss at the higher edge of the pass band is reduced, as in the case of Example 1-1. 
     When the capacitor C 11  is provided as the impedance element that is connected in series with the series arm resonator s 1   a  like Example 1-2-1, filter characteristics whose pass band is narrowed can be provided as compared to Example 1-1 in which no impedance element is provided. On the other hand, when the inductor L 11  is provided as the impedance element connected in series with the series arm resonator s 1   a  like Example 1-2-2, filter characteristics whose pass band is widened can be provided as compared to Example 1-1 in which no impedance element is provided. 
     This will be described below with reference to  FIG. 7D  and  FIG. 7E . 
       FIG. 7D  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1, Example 1-2-1, and Example 1-2-2 at the time of lower frequency shift (when the switch SWb is off).  FIG. 7E  shows comparison graphs of characteristics of the radio-frequency filters according to Example 1-1, Example 1-2-1, and Example 1-2-2 at the time of higher frequency shift (when the switch SWb is on). Specifically, in  FIG. 7D , the top graph shows filter characteristics, and the bottom graph shows the impedances of the series arm circuits when the switch SWb is off. In  FIG. 7E , the top graph shows filter characteristics, and the bottom graph shows the impedances of the series arm circuits when the switch SWb is on. 
     First, characteristics when the switch SWb is off, shown in  FIG. 7D , will be described. 
     In this case, since the series arm resonator s 1   a  and the capacitor C 11  are connected in series with each other, the series arm circuit  11 Ba of Example 1-2-1 can reduce the difference between the resonant frequency and the anti-resonant frequency as compared to the series arm circuit  11  of Example 1-1. Thus, when the resonant frequency of the series arm circuit  11 Ba is adjusted to the pass band of the radio-frequency filter  10 Ba, the anti-resonant frequency of the series arm circuit  11 Ba lies on the lower frequency side than the anti-resonant frequency of the series arm circuit  11 . Therefore, according to Example 1-2-1, since the attenuation pole on the higher side of the pass band lies on the lower frequency side as compared to Example 1-1, the pass band is narrowed. 
     In this case, since the series arm resonator s 1   a  and the inductor L 11  are connected in series with each other, the series arm circuit  11 Bb of Example 1-2-2 can increase the difference between the resonant frequency and the anti-resonant frequency as compared to the series arm circuit  11  of Example 1-1. Thus, when the resonant frequency of the series arm circuit  11 Bb is adjusted to the pass band of the radio-frequency filter  10 Bb, the anti-resonant frequency of the series arm circuit  11 Bb lies on the higher frequency side than the anti-resonant frequency of the series arm circuit  11 . Therefore, according to Example 1-2-2, since the attenuation pole on the higher side of the pass band lies on the higher frequency side as compared to Example 1-1, the pass band is widened. 
     Next, characteristics when the switch SWb is on, shown in  FIG. 7E , will be described. 
     In this case, in the series arm circuit  11 Ba of Example 1-2-1, as compared to the series arm circuit  11  of Example 1-1, the first resonant frequency (lower frequency-side resonant frequency), the second resonant frequency (higher frequency-side resonant frequency), and the first anti-resonant frequency (lower frequency-side anti-resonant frequency) are substantially the same frequency, and the second anti-resonant frequency (higher frequency-side anti-resonant frequency) lies on the lower frequency side. Therefore, according to Example 1-2-1, since the attenuation pole on the higher side of the pass band lies on the lower frequency side as compared to Example 1-1, the pass band is narrowed. 
     In this case, in the series arm circuit  11 Bb of Example 1-2-2, as compared to the series arm circuit  11  of Example 1-1, the first resonant frequency (lower frequency-side resonant frequency), the second resonant frequency (higher frequency-side resonant frequency), and the first anti-resonant frequency (lower frequency-side anti-resonant frequency) are substantially the same frequency, and the second anti-resonant frequency (higher frequency-side anti-resonant frequency) lies on the higher frequency side. Therefore, according to Example 1-2-2, since the attenuation pole on the higher side of the pass band lies on the higher frequency side as compared to Example 1-1, the pass band is widened. 
     As described above, since the series arm resonator s 1   a  (first series arm resonator) and the first impedance element (the capacitor C 11  in Example 1-2-1, and the inductor L 11  in Example 1-2-2) are connected in series with each other, the difference between the resonant frequency and anti-resonant frequency of the first series connection circuit (the series connection circuit  11   aa  in Example 1-2-1, and the series connection circuit  11   ab  in Example 1-2-2) can be changed from the frequency difference of the series arm resonator s 1   a . Specifically, the frequency difference is reduced when the first impedance element is a capacitor (in the case of Example 1-2-1), and the frequency difference is increased when the first impedance element is an inductor (in the case of Example 1-2-2). Thus, since the difference between the second resonant frequency and second anti-resonant frequency of the series arm circuit can be adjusted by adjusting the circuit constant of the first impedance element as needed, the difference between the cut-off frequency on the higher side of the pass band, determined by the frequency difference, and the attenuation pole on the higher side of the pass band can be adjusted. Therefore, the pass band width and the steepness of an attenuation slope can be selectively set. 
     The variable width of narrowing or widening the pass band depends on the constant of the first impedance element. For example, when the first impedance element is a capacitor, the variable width narrows as the constant of the capacitor reduces. When the first impedance element is an inductor, the variable width widens as the constant of the inductor increases. For this reason, the constant of the first impedance element can be determined as needed depending on the frequency specifications that are required from the radio-frequency filter. The capacitor may be a variable capacitor, such as a varicap diode and a DTC (digitally tunable capacitor). The inductor may be a variable inductor that uses MEMS (micro electro mechanical systems). 
     [1-4-2. Connecting Variable Frequency Circuit Including Series Arm Resonator Whose Resonant Frequency is High] 
     Up to here, the description is made on the assumption that the variable frequency circuit (first variable frequency circuit) of the series arm circuit includes the second series arm resonator whose resonant frequency is lower than the resonant frequency of the first series arm resonator (the series arm resonator s 1  of the above-described Examples). However, the resonant frequency of the second series arm resonator just needs to be different from the resonant frequency of the first series arm resonator and may be higher than the resonant frequency of the first series arm resonator. 
     Hereinafter, such a radio-frequency filter will be described by way of Example (Example 1-3). 
       FIG. 8A  is a circuit configuration diagram of a radio-frequency filter  10 C according to Example 1-3. 
     The radio-frequency filter  10 C shown in the diagram differs from the radio-frequency filter  10  according to Example 1-1 in that a series arm circuit  11 C further includes a variable frequency circuit  11   c  (first variable frequency circuit). The variable frequency circuit  11   c  is made up of a series arm resonator s 1   c  (second series arm resonator) and a switch SWc (first switch) connected in series with the series arm resonator s 1   c.    
     The resonant frequency (fr_s 1   c ) of the series arm resonator sic is higher than the resonant frequency (fr_s 1   a ) of the series arm resonator s 1   a.    
     In other words, in this Example, the series arm circuit  11 C includes a plurality of the first variable frequency circuits (two variable frequency circuits  11   b ,  11   c ). The resonant frequency of at least one (here, the series arm resonator s 1   b ) of the plurality of second series arm resonators (two second series arm resonators s 1   b , s 1   c ) is lower than the resonant frequency of the series arm resonator s 1   a  (first series arm resonator) (fr_s 1   b &lt;fr_s 1   a ). Of the plurality of second series arm resonators, the resonant frequency of at least another one (here, the series arm resonator s 1   c ) is higher than the resonant frequency of the series arm resonator s 1   a  (fr_s 1   a &lt;fr_s 1   c ). The number of the first variable frequency circuits is not limited to two and may be three or more. 
     The thus configured radio-frequency filter  10 C is, for example, able to individually switch between an on state and off state of each of the switch SWb and the switch SWc. 
     Hereinafter, the characteristics of the radio-frequency filter  10 C according to Example 1-3 will be described. 
     Circuit constants of the radio-frequency filter  10 C of Example 1-3 are shown in Table 3. The off capacitance (Coff) of each of the switches SWb, SWc is 0.2 pF. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                   
                   
                 Series 
                 Parallel 
               
               
                   
                 Series Arm 
                 Series Arm 
                 Arm 
                 Arm 
               
               
                   
                 Resonator 
                 Resonator 
                 Resonator 
                 Resonator 
               
               
                   
                 s1a 
                 s1b 
                 s1c 
                 p1 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Resonant 
                 735 
                 702 
                 1050 
                 697 
               
               
                 Frequency fr [MHz] 
               
               
                 Anti-Resonant 
                 794 
                 758 
                 1134 
                 725 
               
               
                 Frequency fa [MHz] 
               
               
                 Electrostatic 
                 2.0 
                 2.0 
                 2.0 
                 2.0 
               
               
                 Capacity C 0  [pF] 
               
               
                   
               
            
           
         
       
     
       FIG. 8B  shows graphs of various characteristics of the radio-frequency filter  10 C according to Example 1-3 in comparison with Example 1-1. 
     Specifically, the graphs show, in order from the left column to the right column, (i) various characteristics of Example 1-3 and Example 1-1 when the switch SWb is on and the switch SWc is off, (ii) various characteristics of Example 1-3 when the switch SWb is off and the switch SWc is on, (iii) various characteristics of Example 1-3 and Example 1-1 when the switch SWb is off and the switch SWc is off, and (iv) various characteristics of Example 1-3 when the switch SWb is on and the switch SWc is on. The top graphs show the filter characteristics, and the bottom graphs show the impedance characteristics of the series arm circuits and parallel arm resonator (that is, the parallel arm circuit). 
     Various characteristics of Example 1-1 when the switch SWb is on (or off) and the switch SWc is off are various characteristics of Example 1-1 when the switch SWb is on (or off). 
     As is apparent from the graphs, according to Example 1-3, the resonant frequency and anti-resonant frequency of the series arm circuit  11 C can be changed not only by switching between the on state and off state of the switch SWb, but also by switching between the on state and off state of the switch SWc. 
     Specifically, as a result of turning the switch SWc from the off state to the on state, the series arm circuit  11 C further has a resonant frequency having the same frequency as the resonant frequency (fr_s 1   c ) of the series arm resonator s 1   c  and has an anti-resonant frequency on the higher frequency side than the resonant frequency. Thus, in the series arm circuit  11 C, the anti-resonant frequency closest on the lower frequency side than the resonant frequency (fr_s 1   c ) of the series arm resonator s 1   c  is shifted toward a lower frequency side. The anti-resonant frequency provides an attenuation pole on the higher side of the pass band. Therefore, as a result of turning the switch SWc from the off state to the on state, the attenuation pole on the higher side of the pass band is shifted toward a lower frequency side. 
       FIG. 8C  shows graphs of changes in filter characteristics when the switches SWb, SWc of the radio-frequency filter  10 C according to Example 1-3 are individually switched between an on state and an off state. 
     As shown in the graphs, the radio-frequency filter  10 C is able to change the frequencies of the pass band and the frequency of the attenuation pole on the lower side of the pass band while reducing the loss at the higher edge of the pass band by switching between the on state and off state of the switch SWb. The radio-frequency filter  10 C is able to change the frequencies of the pass band and the frequency of the attenuation pole on the higher side of the pass band while reducing the loss at the lower edge of the pass band by switching between the on state and off state of the switch SWc. 
     Therefore, the radio-frequency filter  10 C is able to change the stop band while reducing the loss at the edges of the pass band for both the lower side and higher side of the pass band by individually switching between the on state and off state of each of the switches SWb, SWc. 
     [1-5. Resonance Analysis] 
     The resonant characteristics of the radio-frequency filter according to the embodiment will be described by way of an equivalent circuit. 
     [1-5-1. Resonator Alone] 
     First, the resonant characteristics of a resonator alone will be described. 
       FIG. 9A  shows an equivalent circuit model of one resonator and its resonant characteristics. As shown in the diagram, the resonator is represented by a parallel circuit (parallel connection circuit) of a capacitor C 0  and a series circuit (series connection circuit) of a capacitor C 1  and an inductor L 1 . The capacitor C 0  is the electrostatic capacity of the resonator. 
     In the equivalent circuit, the resonant frequency fr of the resonator is determined by the series circuit of the capacitor C 1  and the inductor L 1  and is a frequency at which the impedance of the equivalent circuit is zero, so the resonant frequency fr is expressed by the mathematical expression 3 when the mathematical expression 2 is solved. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     0 
                     - 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         L 
                         1 
                       
                     
                     + 
                     
                       1 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     f 
                     r 
                   
                   = 
                   
                     1 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The anti-resonant frequency fa of the resonator is a frequency at which the admittance Y of the equivalent circuit is zero, so the anti-resonant frequency fa is expressed by the mathematical expression 5 when the mathematical expression 4 is solved. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       4 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       1 
                       Z 
                     
                     = 
                     
                       0 
                       = 
                       
                         
                           1 
                           
                             1 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 0 
                               
                             
                           
                         
                         + 
                         
                           1 
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 L 
                                 1 
                               
                             
                             + 
                             
                               1 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       5 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     f 
                     r 
                   
                   = 
                   
                     
                       
                         
                           1 
                           + 
                           
                             
                               C 
                               1 
                             
                             
                               C 
                               0 
                             
                           
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       
                         
                           1 
                           + 
                           
                             
                               C 
                               1 
                             
                             
                               C 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Through the mathematical expression 3 and the mathematical expression 5, as shown in the right-side graph of  FIG. 9A , the anti-resonant frequency fa appears on a higher frequency side than the resonant frequency fr. 
     In other words, the resonator has one resonant frequency and one anti-resonant frequency that lies on the higher frequency side than the resonant frequency. 
     [1-5-2. Connecting Impedance Element in Series with Resonator] 
     Next, the resonant characteristics when an impedance element is connected in series with the resonator will be described by way of an equivalent circuit model. 
       FIG. 9B  shows an equivalent circuit model when an impedance element X 1  is connected in series with a resonator and its resonant characteristics. As shown in the diagram, the resonator is represented by a parallel circuit of the capacitor C 0  and the series circuit of the capacitor C 1  and the inductor L 1 . The capacitor C 0  is the electrostatic capacity of the resonator. A parallel circuit of the impedance element X 1  and the switch SW is connected to the resonator. In the equivalent circuit model in resonance analysis, the switch SW is treated as an ideal switch that has no capacitive component (that is, the impedance is infinite) in the off state and that has zero resistive component (that is, the impedance is zero) in the on state. 
     First, the resonant characteristics of the equivalent circuit when the switch SW is on will be described. When the switch SW is on, the impedance element X 1  is short-circuited, so the resonant frequency fr_on and the anti-resonant frequency fa_on are respectively the same as the resonant frequency fr and the anti-resonant frequency fa in  FIG. 9A  and are expressed by the mathematical expression 6 and the mathematical expression 7. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       6 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       f 
                       r 
                     
                     ⁢ 
                     _on 
                   
                   = 
                   
                     1 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       7 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       f 
                       a 
                     
                     ⁢ 
                     _on 
                   
                   = 
                   
                     
                       
                         
                           1 
                           + 
                           
                             
                               C 
                               1 
                             
                             
                               C 
                               0 
                             
                           
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       _on 
                       ⁢ 
                       
                         
                           1 
                           + 
                           
                             
                               C 
                               1 
                             
                             
                               C 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Next, the case where the switch SW is off will be described separately (1) when the impedance element X 1  is a capacitor or (2) when the impedance element X 1  is an inductor. 
     (1) When Impedance Element X 1  is Capacitor Ct A resonant frequency fr_off 1  when the switch SW is off is a frequency at which the impedance Z of the equivalent circuit is zero, so the resonant frequency fr_off 1  is expressed by the mathematical expression 9 when the mathematical expression 8 is solved. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       8 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     0 
                     = 
                     
                       
                         1 
                         
                           
                             1 
                             
                               1 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   0 
                                 
                               
                             
                           
                           + 
                           
                             1 
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     L 
                                     1 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       C 
                                       1 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                             
                           
                         
                       
                       + 
                       
                         1 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       9 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       f 
                       r 
                     
                     ⁢ 
                     _off 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   = 
                   
                     
                       
                         
                           
                             C 
                             0 
                           
                           + 
                           
                             C 
                             1 
                           
                           + 
                           
                             C 
                             t 
                           
                         
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                             ⁢ 
                             
                               C 
                               t 
                             
                           
                           + 
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               C 
                               0 
                             
                             ⁢ 
                             
                               C 
                               t 
                             
                           
                         
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     On the other hand, an anti-resonant frequency fa_off 1  when the switch SW is off is the same as the anti-resonant frequency fa_on when the switch SW is on and is expressed by the mathematical expression 10. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       10 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       f 
                       a 
                     
                     ⁢ 
                     _off 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   = 
                   
                     
                       
                         1 
                         + 
                         
                           
                             C 
                             1 
                           
                           
                             C 
                             0 
                           
                         
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Through the mathematical expression 6, the mathematical expression 7, the mathematical expression 9, and the mathematical expression 10, when the impedance element X 1  is a capacitor, the anti-resonant frequencies fa_on, fa_off 1  coincide with each other regardless of switching between the on state and off state of the switch SW, as shown in the right-side graph of  FIG. 9B . On the other hand, the resonant frequency appears to shift toward a higher frequency side when the switch SW is off (fr_off 1 ) as compared to when the switch SW is on (fr_on). 
     (2) When Impedance Element X 1  Is Inductor Lt A resonant frequency fr_off 2  when the switch SW is off is a frequency at which the impedance of the equivalent circuit is zero, so the resonant frequency fr_off 2  is expressed by the mathematical expression 12 when the mathematical expression 11 is solved. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       11 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     0 
                     = 
                     
                       
                         1 
                         
                           
                             1 
                             
                               1 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   0 
                                 
                               
                             
                           
                           + 
                           
                             1 
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     L 
                                     1 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       C 
                                       1 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                             
                           
                         
                       
                       + 
                       
                         1 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       12 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       _off 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       L 
                     
                     = 
                     
                       
                         
                           
                             
                               - 
                               b 
                             
                             - 
                             
                               
                                 
                                   b 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                   ⁢ 
                                   a 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                 
                               
                             
                           
                           
                             2 
                             ⁢ 
                             a 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       _off 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       H 
                     
                     = 
                     
                       
                         
                           
                             
                               - 
                               b 
                             
                             + 
                             
                               
                                 
                                   b 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                   ⁢ 
                                   a 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                 
                               
                             
                           
                           
                             2 
                             ⁢ 
                             a 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     a 
                     = 
                     
                       
                         L 
                         1 
                       
                       ⁢ 
                       
                         L 
                         t 
                       
                       ⁢ 
                       
                         C 
                         0 
                       
                       ⁢ 
                       
                         C 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     b 
                     = 
                     
                       
                         
                           - 
                           
                             L 
                             1 
                           
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                       - 
                       
                         
                           L 
                           t 
                         
                         ⁢ 
                         
                           C 
                           0 
                         
                       
                       - 
                       
                         
                           L 
                           t 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     1 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In the mathematical expression 12, fr_off 2 L is a lower frequency-side resonant frequency when the switch SW is off, and fr_off 2 H is a higher frequency-side resonant frequency when the switch SW is off. 
     On the other hand, an anti-resonant frequency fa_off 2  when the switch SW is off is the same as the anti-resonant frequency fa_on when the switch SW is on and is expressed by the mathematical expression 13. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       13 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       f 
                       a 
                     
                     ⁢ 
                     _off 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   = 
                   
                     
                       
                         1 
                         + 
                         
                           
                             C 
                             1 
                           
                           
                             C 
                             0 
                           
                         
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             C 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Through the mathematical expression 6, the mathematical expression 7, the mathematical expression 12, and the mathematical expression 13, when the impedance element X 1  is an inductor, the anti-resonant frequencies fa_on, fa_off 2  coincide with each other regardless of switching between the on state and off state of the switch SW, as shown in the right-side graph of  FIG. 9B . On the other hand, the resonant frequency appears to shift toward a lower frequency side when the switch SW is off (fr_off 2 L) as compared to when the switch SW is on (fr_on). 
     [1-5-3. Two Resonators are Connected in Parallel with Each Other] 
     Next, the characteristics when two resonators are connected in parallel with each other will be described by way of an equivalent circuit model. 
       FIG. 9C  shows an equivalent circuit model of two resonators connected in parallel with each other and its resonant characteristics. The drawing shows a model in which resonators res 1 , res 2  are connected in parallel with each other. The resonator res 1  is represented by a parallel circuit of a capacitor C 01  and a series circuit of the capacitor C 1  and the inductor L 1 . The resonator res 2  is represented by a parallel circuit of a capacitor C 02  and a series circuit of a capacitor C 2  and an inductor L 2 . The capacitors C 01 , C 02  are respectively the electrostatic capacities of the resonators res 1 , res 2 . A resonant circuit made up of these two resonators res 1 , res 2  is represented by an equivalent circuit shown at the lower left side of  FIG. 9C . In other words, the resonant circuit is represented by the series circuit of the capacitor C 1  and the inductor L 1 , the series circuit of the capacitor C 2  and the inductor L 2 , and the parallel circuit of the capacitor C 0  (=C 01 +C 02 ). 
     In the equivalent circuit, the resonant frequency fr of the resonator is determined by the series circuit of the capacitor C 1  and the inductor L 1  and is expressed by the mathematical expression 2. 
     In the equivalent circuit, two resonant frequencies are determined. The resonant frequency fr 1  is determined by the series circuit of the capacitor C 1  and the inductor L 1 . The resonant frequency fr 2  is determined by the series circuit of the capacitor C 2  and the inductor L 2 . The resonant frequencies fr 1 , fr 2  are expressed by the mathematical expression 14. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       14 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       1 
                     
                     = 
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               C 
                               1 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         f 
                         r 
                       
                       ⁢ 
                       2 
                     
                     = 
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               C 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     In other words, the two resonant frequencies fr 1 , fr 2  that are represented by the equivalent circuit are respectively substantially equal to the resonant frequency fr_res 1  of the resonator res 1  and the resonant frequency fr_res 2  of the resonator res 2 . 
     The anti-resonant frequency of the equivalent circuit is a frequency at which the admittance Y of the equivalent circuit is zero, so it is found that the equivalent circuit has two anti-resonant frequencies (fa 1 , fa 2 ) as expressed by the mathematical expression 16 when the mathematical expression 15 is solved. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       15 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       1 
                       Z 
                     
                     = 
                     
                       0 
                       = 
                       
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             0 
                           
                         
                         + 
                         
                           1 
                           
                             
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   1 
                                 
                               
                               + 
                               
                                 1 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                         + 
                         
                           1 
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 L 
                                 2 
                               
                             
                             + 
                             
                               1 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Mathematical 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       16 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           f 
                           a 
                         
                         ⁢ 
                         1 
                       
                       = 
                       
                         
                           
                             
                               
                                 - 
                                 b 
                               
                               - 
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     4 
                                     ⁢ 
                                     a 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               a 
                             
                           
                         
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           f 
                           a 
                         
                         ⁢ 
                         2 
                       
                       = 
                       
                         
                           
                             
                               
                                 - 
                                 b 
                               
                               + 
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     4 
                                     ⁢ 
                                     a 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               a 
                             
                           
                         
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     a 
                     = 
                     
                       
                         L 
                         1 
                       
                       ⁢ 
                       
                         L 
                         2 
                       
                       ⁢ 
                       
                         C 
                         0 
                       
                       ⁢ 
                       
                         C 
                         1 
                       
                       ⁢ 
                       
                         C 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     b 
                     = 
                     
                       
                         
                           - 
                           
                             L 
                             1 
                           
                         
                         ⁢ 
                         
                           C 
                           0 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                       - 
                       
                         
                           L 
                           2 
                         
                         ⁢ 
                         
                           C 
                           0 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                       - 
                       
                         
                           L 
                           1 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                       - 
                       
                         
                           L 
                           2 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       
                         C 
                         0 
                       
                       + 
                       
                         C 
                         1 
                       
                       + 
                       
                         C 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     It is found that the anti-resonant frequencies fa 1 , fa 2  that are obtained from the mathematical expression 16 are different from the anti-resonant frequencies (indicated as fa_res 1 , fa_res 2  in the graph of  FIG. 9C ) of the resonators alone, which are obtained from the mathematical expression 4. The anti-resonant frequency fa 1  that is derived from the mathematical expression 15 is lower than the anti-resonant frequency fa_res 1  of the resonator res 1  alone and is lower than the anti-resonant frequency fa_res 2  of the resonator res 2  alone. 
     [1-5-4. Description of Examples Based on Resonance Analysis] 
     Based on the above-described resonance analysis, the fact that the resonant characteristics of the series arm circuit shift in response to switching between the on state and off state of the first switch in the radio-frequency filter according to the present embodiment will be described. 
     In other words, for example, in Example 1-1, when the switch SWb (first switch) is on, the series arm resonator s 1   a  and the series arm resonator s 1   b  are connected in parallel with each other. Therefore, the series arm circuit  11  has two resonant frequencies (the first resonant frequency and the second resonant frequency) and the two anti-resonant frequencies (the first anti-resonant frequency and the second anti-resonant frequency). In other words, the first resonant frequency and second resonant frequency of the series arm circuit  11  are respectively the same as the resonant frequency of the series arm resonator s 1   a  and the resonant frequency of the series arm resonator s 1   b . The first anti-resonant frequency of the series arm circuit  11  is lower than the anti-resonant frequency of the series arm resonator s 1   a . The second anti-resonant frequency of the series arm circuit  11  is lower than the anti-resonant frequency of the series arm resonator s 1   b.    
     On the other hand, when the switch SWb (first switch) is off, a combined capacitance of the electrostatic capacity of the series arm resonator s 1   b  and the off capacitance of the switch SWb is connected in parallel with the series arm resonator s 1   a . As described above, the off capacitance of the switch SWb is sufficiently less than the electrostatic capacity of the series arm resonator s 1   b , so the combined capacitance of the electrostatic capacity of the series arm resonator s 1   b  and the off capacitance (Coff) of the switch SWb is sufficiently less than the capacitance of the electrostatic capacity of the series arm resonator s 1   b . Therefore, in this case, the third resonant frequency that is the resonant frequency of the series arm circuit  11  is the same as the resonant frequency of the series arm resonator s 1   a , and the third anti-resonant frequency that is the anti-resonant frequency of the series arm circuit  11  shifts slightly toward a lower frequency side than the anti-resonant frequency of the series arm resonator s 1   a.    
     In other words, in response to turning the switch SWb from the off state to the on state, the resonant frequency and the anti-resonant frequency shift toward a lower frequency side, and a new resonant frequency and a new anti-resonant frequency are added. 
     For example, in Example 1-2-1 and Example 1-2-2, the impedance element (the capacitor in Example 1-2-1, and the inductor in Example 1-2-2) is connected in series with the series arm resonator s 1   a . Therefore, the resonant frequency of the first series connection circuit made up of the series arm resonator s 1   a  and the impedance element shifts toward a higher frequency side or a lower frequency side as compared to the resonant frequency of the series arm resonator s 1   a . The anti-resonant frequency of the first series connection circuit is equal to the anti-resonant frequency of the series arm resonator s 1   a.    
     For example, the resonant characteristics of the series arm circuit  11  when both the switches SWb, SWc are on in Example 1-3 will also be described by way of an equivalent circuit model of three resonators connected in parallel with one another. Therefore, in this case, the series arm circuit  11  has three resonant frequencies at which the impedance of the equivalent circuit model (equivalent circuit) is zero and has three anti-resonant frequencies at which the admittance is zero. The description of specific examples of the resonant frequencies and anti-resonant frequencies in this case is omitted. 
     Second Embodiment 
     In the above-described embodiment, the resonant frequency or anti-resonant frequency of the parallel arm circuit is fixed. However, as in the case of the series arm circuit, the resonant frequency or anti-resonant frequency of the parallel arm circuit may be variable. In the present embodiment, such a radio-frequency filter will be described by way of Examples (Example 2-1 and Example 2-2). 
     [2-1. Configuration] 
       FIG. 10A  is a circuit configuration diagram of a radio-frequency filter  20  according to Example 2-1. 
     As shown in the diagram, a parallel arm circuit  22  in this Example further includes a variable frequency circuit  11   d  (second variable frequency circuit) connected in series with the parallel arm resonator p 1  (first parallel arm resonator) as compared to the parallel arm circuit  12  in Examples of the first embodiment. 
     The variable frequency circuit  11   d  is the second variable frequency circuit including a capacitor C 22  (second impedance element) and a switch SWp (second switch) connected in parallel with the capacitor C 22 . In other words, the variable frequency circuit  11   d  is a parallel connection circuit of the capacitor C 22  and the switch SWp and is connected in series with the parallel arm resonator p 1 . The variable frequency circuit  11   d  changes the resonant frequency of the parallel arm circuit  22  as a result of switching between the on state and off state of the switch SWp in accordance with a control signal from a control unit (not shown), such as an RFIC. 
     In this Example, the parallel arm resonator p 1  and the variable frequency circuit  11   d  are connected in this order from the node x 1  side. Alternatively, the parallel arm resonator p 1  and the variable frequency circuit  11   d  may be connected in reverse order. However, when the switch SWp is disposed closer to the node x 1  than the parallel arm resonator p 1 , the loss of the radio-frequency filter  20  deteriorates because of the resistive component (on resistance) of the switch SWp when the switch SWp is on. Therefore, it is desirable that the variable frequency circuit  11   d  be disposed closer to the ground than the parallel arm resonator p 1 . 
     In this Example, in the variable frequency circuit  11   b  and the variable frequency circuit  11   d , when the switch SWb (first switch) is on (conductive state), the switch SWp (second switch) is off (nonconductive state); whereas, when the switch SWb is off, the switch SWp is on. In other words, when one of the switches SWb, SWp is turned from the on state to the off state, the other one is turned from the off state to the on state at the same time. Thus, control circuits that control the on state and off state of each of the switches SWb, SWp can be integrated, so the miniaturization of the radio-frequency filter  20  is possible. 
     The switch SWp is an SPST switch element and is configured similarly to, for example, the switch SWb. 
     [2-2. Characteristics] 
     Next, the filter characteristics (bandpass characteristics) of the radio-frequency filter  20  of this Example will be described while impedance characteristics (resonant characteristics) that determine the filter characteristics are discussed. 
     Of the circuit elements that make up the radio-frequency filter  20  of this Example, the same circuit elements as those of Example 1-1 have the same constants. The capacitance of the capacitor C 22  is 12 pF, and the off capacitance of the switch SWp is 0.2 pF. 
       FIG. 10B  shows graphs of various characteristics of the radio-frequency filter  20 . Specifically, the top graph shows filter characteristics in two states, that is, the case where the switch SWb is on and the switch SWp is off and the case where the switch SWb is off and the switch SWp is on. The middle graph shows the impedance characteristics of the series arm circuit  11  and parallel arm circuit  22 . The bottom graph shows the impedance characteristics of the series arm resonator s 1   a , the series arm resonator s 1   b , and the parallel arm resonator p 1 . For the series arm circuit  11 , the impedance characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off, are shown. For the parallel arm circuit  22 , the impedance characteristics in two states, that is, the case where the switch SWp is on and the case where the switch SWp is off, are shown. 
     First, characteristics when the switch SWb is off and the switch SWp is on (lower frequency shift) in the circuit configuration shown in  FIG. 10A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is off in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  22  exhibits the same characteristics as the parallel arm resonator p 1  since the capacitor C 22  is short-circuited by the switch SWp. 
     Next, characteristics when the switch SWb is on and the switch SWp is off (higher frequency shift) in the circuit configuration shown in  FIG. 10A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is on in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  22  is a circuit in which the parallel arm resonator p 1  and the capacitor C 22  are connected in series with each other when the switch SWp is off, so the parallel arm circuit  22  exhibits the combined characteristics of the parallel arm resonator p 1  and the capacitor C 22 . 
     In this case, actually, there is the off capacitance of the switch SWp, the parallel arm circuit  22  exhibits characteristics slightly different from the combined characteristics. However, since the off capacitance of the switch SWp is small enough (0.2 pF in this Example) as compared to the capacitance (12 pF in this Example) of the capacitor C 22 , the off capacitance almost does not influence the characteristics of the parallel arm circuit  22 . Thus, for the sake of succinctness, the description will be made on the assumption that the switch SWp is an ideal switch having no off capacitance. 
     Therefore, as shown in the middle and bottom graphs of  FIG. 10B , the parallel arm circuit  22  has a resonant frequency that lies on the higher frequency side than the resonant frequency of the parallel arm resonator p 1  and an anti-resonant frequency equal to the anti-resonant frequency of the parallel arm resonator p 1 . 
     Thus, in this Example, when the switch SWb is off and the switch SWp is on, the filter characteristics represented by the dashed line (SWb Off, SWp On) in the top graph of  FIG. 10B  are provided. On the other hand, when the switch SWb is on and the switch SWp is off, the filter characteristics represented by the continuous line (SWb On, SWp Off) in the top graph of  FIG. 10B  are provided. 
     [2-3. Advantageous Effects] 
     Hereinafter, advantageous effects that are obtained from this Example will be described by way of comparison with Example 1-1. 
       FIG. 11A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-1 and Example 1-1 at the time of lower frequency shift.  FIG. 11B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-1 and Example 1-1 at the time of higher frequency shift. Specifically, in these graphs, the top graphs show filter characteristics, the middle graphs show the impedance characteristics of the series arm circuit, and the bottom graphs show the impedance characteristics of the parallel arm circuit. 
     As is apparent from  FIG. 11A , according to Example 2-1, characteristics equivalent to those of Example 1-1 are obtained at the time of lower frequency shift. 
     As is apparent from the top graph of  FIG. 11B , according to Example 2-1, the attenuation on the lower side of the pass band is improved as compared to Example 1-1 at the time of higher frequency shift. Specifically, in Example 2-1, the capacitor C 22  is connected in series with the parallel arm resonator p 1  as a result of turning the switch SWp from the on state to the off state. Thus, the resonant frequency of the parallel arm circuit  22  shifts toward a higher frequency side, so the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency, shifts toward a higher frequency side, and the frequency at the lower edge of the pass band shifts to a higher frequency. Therefore, since the switch SWp is off when the switch SWb is on, the attenuation on the lower side of the pass band when the frequency at the lower edge of the pass band and the attenuation pole on the lower side of the pass band are shifted toward a higher frequency side (at the time of higher frequency shift) is improved (increased). 
     For example, in most of bands that are defined in 3GPP, a Tx band (transmission band) is on the lower frequency side, an Rx band (receiving band) is on the higher frequency side, and the center frequency and the band width vary among the bands. Therefore, a tunable radio-frequency filter that is used for any one of these bands needs to have any one of the Tx band and the Rx band as a pass band and the other as a stop band, and needs to shift the pass band and the stop band at the same time toward a lower frequency side or a higher frequency side. 
     The radio-frequency filter  20  according to this Example is able to improve the attenuation on the lower side of the pass band when the pass band is shifted toward a higher frequency side, so the radio-frequency filter  20  is suitable as, for example, a receiving filter for the bands defined in 3GPP. 
     The switch SWb and the switch SWp may be individually switched between the on state and the off state. 
       FIG. 12  shows graphs of changes in filter characteristics when the switches SWb, SWp of the radio-frequency filter  20  according to Example 2-1 are individually switched between the on state and the off state. 
     As shown in the graphs, the radio-frequency filter  20  is able to change the lower side of the pass band while reducing the loss at the higher edge of the pass band by switching between the on state and off state of the switch SWb. The radio-frequency filter  20  is able to change the attenuation pole on the lower side of the pass band by switching between the on state and off state of the switch SWp. 
     Therefore, the radio-frequency filter  20  is able to provide filter characteristics for the required frequency specifications by switching between the on state and off state of each of the switches SWb, SWp as needed. 
     [2-4. Other Examples] 
     The impedance element that is connected in series with the parallel arm resonator p 1  is not limited to a capacitor and may be an inductor. 
       FIG. 13A  is a circuit configuration diagram of a radio-frequency filter  20 A according to Example 2-2. 
     As shown in the diagram, a parallel arm circuit  22 A in this Example includes an inductor L 22  instead of the capacitor C 22  in Example 2-1. In other words, the parallel arm circuit  22 A in this Example includes a variable frequency circuit  11   e  (second variable frequency circuit) connected in series with the parallel arm resonator p 1  (first parallel arm resonator). The variable frequency circuit  11   e  is the second variable frequency circuit including the inductor L 22  (second impedance element) and the switch SWp (second switch) connected in parallel with the inductor L 22 . 
     In this Example, in the variable frequency circuit  11   b  and the variable frequency circuit  11   e , when the switch SWb (first switch) is on (conductive state), the switch SWp (second switch) is on; whereas, when the switch SWb is off (nonconductive state), the switch SWp is off. In other words, the switches SWb, SWp are on at the same time or are off at the same time. 
     Next, the filter characteristics (bandpass characteristics) of the radio-frequency filter  20 A of this Example will be described while impedance characteristics (resonant characteristics) that determine the filter characteristics are discussed. 
     Of the circuit elements that make up the radio-frequency filter  20 A of this Example, the same circuit elements as those of Example 2-1 have the same constants. The inductance of the inductor L 22  is 3 nH. 
       FIG. 13B  shows graphs of various characteristics of the radio-frequency filter  20 A. Specifically, the top graph shows filter characteristics in two states, that is, the case where the switch SWb is on and the switch SWp is on and the case where the switch SWb is off and the switch SWp is off. The middle graphs show the impedance characteristics of the series arm circuit  11  and parallel arm circuit  22 A. The bottom graphs show the impedance characteristics of the series arm resonator s 1   a , the series arm resonator s 1   b , and the parallel arm resonator p 1 . For the series arm circuit  11 , the impedance characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off, are shown. For the parallel arm circuit  22 A, the impedance characteristics in two states, that is, the case where the switch SWp is on and the case where the switch SWp is off, are shown. 
     First, characteristics when the switch SWb is off and the switch SWp is off (lower frequency shift) in the circuit configuration shown in  FIG. 13A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is off in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  22 A is a circuit in which the parallel arm resonator p 1  and the inductor L 22  are connected in series with each other when the switch SWp is off, so the parallel arm circuit  22 A exhibits the combined characteristics of the parallel arm resonator p 1  and the inductor L 22 . 
     In this case, actually, there is the off capacitance of the switch SWp, and a ripple occurs in the impedance characteristics of the parallel arm circuit  22 A. However, when the off capacitance of the switch SWp is small enough (for example, 0.4 pF or below), the ripple almost does not influence the characteristics in the pass band, and the off capacitance of the switch SWp is 0.2 pF in this Example, so the influence on the characteristics in the pass band is ignored. Thus, for the sake of succinctness, the description will be made on the assumption that the switch SWp is an ideal switch having no off capacitance. 
     Therefore, as shown in the middle and bottom graphs of  FIG. 13B , the parallel arm circuit  22 A has a resonant frequency that lies on the lower frequency side than the resonant frequency of the parallel arm resonator p 1  and an anti-resonant frequency equal to the anti-resonant frequency of the parallel arm resonator p 1 . 
     Next, characteristics when the switch SWb is on and the switch SWp is on (higher frequency shift) in the circuit configuration shown in  FIG. 13A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is on in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  22 A exhibits the same characteristics as the parallel arm resonator p 1  since the inductor L 22  is short-circuited by the switch SWp. 
     Thus, in this Example, when the switch SWb is off and the switch SWp is off, the filter characteristics represented by the dashed line (SWb Off, SWn Off) in the top graph of  FIG. 13B  are provided. On the other hand, when the switch SWb is on and the switch SWp is on, the filter characteristics represented by the continuous line (SWb On, SWp On) in the top graph of  FIG. 13B  are provided. 
     Hereinafter, advantageous effects that are obtained from this Example will be described by way of comparison with Example 1-1. 
       FIG. 14A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-2 and Example 1-1 at the time of lower frequency shift.  FIG. 14B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 2-2 and Example 1-1 at the time of higher frequency shift. Specifically, in these graphs, the top graphs show filter characteristics, the middle graphs show the impedance characteristics of the series arm circuit, and the bottom graphs show the impedance characteristics of the parallel arm circuit. 
     As is apparent from  FIG. 14A , according to Example 2-2, the attenuation pole on the lower side of the pass band can be shifted toward a lower frequency side as compared to Example 1-1 at the time of lower frequency shift. Specifically, in Example 2-1, the inductor L 22  is connected in series with the parallel arm resonator p 1  when the switch SWp is turned from the on state to the off state. Thus, the resonant frequency of the parallel arm circuit  22 A shifts toward a lower frequency side, so the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency, shifts toward a lower frequency side, and the frequency at the lower edge of the pass band shifts to a higher frequency. Therefore, since the switch SWp is off when the switch SWb is on, the attenuation on the lower side of the pass band when the frequency at the lower edge of the pass band and the attenuation pole on the lower side of the pass band are shifted toward a lower frequency side (at the time of lower frequency shift) is improved. 
     As is apparent from  FIG. 14B , according to Example 2-2, characteristics equivalent to those of Example 1-1 are obtained at the time of higher frequency shift. 
     Therefore, according to Example 2-2, the variable width of the frequencies on the lower side of the pass band and the variable width of the frequency at the attenuation pole on the lower side of the pass band are increased while the loss at the higher edge of the pass band is reduced. 
     The switch SWb and the switch SWp may be individually switched between the on state and the off state. 
       FIG. 15  shows graphs of changes in filter characteristics when the switches SWb, SWp of the radio-frequency filter  20 A according to Example 2-2 are individually switched between the on state and the off state. 
     As shown in the graphs, the radio-frequency filter  20 A is able to change the frequency at the lower edge of the pass band and the frequency at the attenuation pole on the lower side of the pass band while reducing the loss at the higher edge of the pass band by switching between the on state and off state of the switch SWb. The radio-frequency filter  20 A is able to change the frequency at the lower edge of the pass band and at the frequency of the attenuation pole on the lower side of the pass band by switching between the on state and off state of the switch SWp. 
     Therefore, the radio-frequency filter  20 A is able to provide filter characteristics for the required frequency specifications by switching between the on state and off state of each of the switches SWb, SWp as needed. 
     Third Embodiment 
     In the above-described first and second embodiments, the parallel arm circuit includes one parallel arm resonator. However, as in the case of the series arm circuit, the parallel arm circuit may include a plurality of parallel arm resonators. In the present embodiment, such a radio-frequency filter will be described by way of Example (Example 3). 
     [3-1. Configuration] 
       FIG. 16A  is a circuit configuration diagram of a radio-frequency filter  30  according to Example 3. 
     As shown in the diagram, a parallel arm circuit  32  in this Example includes a parallel arm resonator p 1   b  (second parallel arm resonator) connected in parallel with a series connection circuit  321  (second series connection circuit) made up of the parallel arm resonator p 1   a  (first parallel arm resonator) and the variable frequency circuit  11   d  (second variable frequency circuit) as compared to the parallel arm circuit  22  in Example 2-1 of the second embodiment. The parallel arm resonator p 1   a  corresponds to the parallel arm resonator p 1  of Example 2-1. 
     The parallel arm resonator p 1   b  is a resonator (second parallel arm resonator) provided in a parallel arm connecting the node x 1  and the ground. One terminal of the parallel arm resonator p 1   b  is connected to the node x 1 , and the other terminal of the parallel arm resonator p 1   b  is connected to the ground. The resonant frequency of the parallel arm resonator p 1   b  is higher than the resonant frequency of the parallel arm resonator p 1   a.    
     In this Example, as in the case of Example 2-1, in the variable frequency circuit  11   b  and the variable frequency circuit  11   d , when the switch SWb (first switch) is on (conductive state), the switch SWp (second switch) is off (nonconductive state); whereas, when the switch SWb is off, the switch SWp is on. In other words, when one of the switches SWb, SWp is turned from the on state to the off state, the other one is turned from the off state to the on state at the same time. 
     [3-2. Characteristics] 
     Next, the filter characteristics (bandpass characteristics) of the radio-frequency filter  30  of this Example will be described while impedance characteristics (resonant characteristics) that determine the filter characteristics are discussed. 
     Circuit constants of the radio-frequency filter  30  of Example 3 are shown in Table 4. The capacitance of the capacitor C 22  is 3 pF, and the off capacitances (Coff) of the switches SWb, SWp both are 0.2 pF. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 Series 
                 Series 
                   
                   
               
               
                   
                 Arm 
                 Arm 
                 Parallel Arm 
                 Parallel Arm 
               
               
                   
                 Resonator 
                 Resonator 
                 Resonator 
                 Resonator 
               
               
                   
                 s1a 
                 s1b 
                 p1a 
                 p1b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Resonant 
                 737 
                 692 
                 697 
                 770 
               
               
                 Frequency fr [MHz] 
               
               
                 Anti-Resonant 
                 796 
                 747 
                 725 
                 801 
               
               
                 Frequency fa [MHz] 
               
               
                 Electrostatic 
                 2.0 
                 2.0 
                 4.0 
                 0.5 
               
               
                 Capacity C 0  [pF] 
               
               
                   
               
            
           
         
       
     
       FIG. 16B  shows graphs of various characteristics of the radio-frequency filter  30 . Specifically, the top graph shows filter characteristics in two states, that is, the case where the switch SWb is on and the switch SWp is off and the case where the switch SWb is off and the switch SWp is on. The middle graph shows the impedance characteristics of the series arm circuit  11  and parallel arm circuit  32 . The bottom graphs show the impedance characteristics of the series arm resonator s 1   a , the series arm resonator s 1   b , the parallel arm resonator p 1   a , and the parallel arm resonator p 1   b . For the series arm circuit  11 , the impedance characteristics in two states, that is, the case where the switch SWb is on and the case where the switch SWb is off, are shown. For the parallel arm circuit  32 , the impedance characteristics in two states, that is, the case where the switch SWp is on and the case where the switch SWp is off, are shown. 
     First, characteristics when the switch SWb is off and the switch SWp is on (lower frequency shift) in the circuit configuration shown in  FIG. 16A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is off in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  32  is in a state where the parallel arm resonator p 1   a  and the parallel arm resonator p 1   b  are connected in parallel with each other since the capacitor C 22  is short-circuited by the switch SWp. Therefore, the parallel arm circuit  32  exhibits combined characteristics of the parallel arm resonator p 1   a  and the parallel arm resonator p 1   b . Thus, as shown in the middle graph of  FIG. 16B , the parallel arm circuit  32  has two resonant frequencies and two anti-resonant frequencies. Specifically, the parallel arm circuit  32  has a first resonant frequency equal to the resonant frequency of the parallel arm resonator p 1   a , a second resonant frequency equal to the resonant frequency of the parallel arm resonator p 1   b , a first anti-resonant frequency that lies between the first resonant frequency and the second resonant frequency, and a second anti-resonant frequency that lies between the second resonant frequency and the anti-resonant frequency of the parallel arm resonator p 1   b.    
     Next, characteristics when the switch SWb is on and the switch SWp is off (higher frequency shift) in the circuit configuration shown in  FIG. 16A  will be described. 
     In this case, the characteristics of the series arm circuit  11  are similar to the characteristics when the switch SWb is on in Example 1-1, so the detailed description is omitted. 
     In this case, the parallel arm circuit  32  is in a state where the series connection circuit  321  of the parallel arm resonator p 1   a  and the capacitor C 22  and the parallel arm resonator p 1   b  are connected in parallel with each other. Therefore, the parallel arm circuit  32  exhibits combined characteristics of the parallel arm resonator p 1   a  and the capacitor C 22  and the parallel arm resonator p 1   b . Thus, as shown in the middle graph of  FIG. 16B , the parallel arm circuit  32  has two resonant frequencies and two anti-resonant frequencies. 
     Specifically, the parallel arm circuit  32  has a third resonant frequency that lies on the higher frequency side than the resonant frequency of the parallel arm resonator p 1   a , a fourth resonant frequency equal to the resonant frequency of the parallel arm resonator p 1   b , a third anti-resonant frequency that lies between the first resonant frequency and the second resonant frequency, and a fourth anti-resonant frequency that lies between the second resonant frequency and the anti-resonant frequency of the parallel arm resonator p 1   b . In other words, the parallel arm circuit  32  has a third resonant frequency higher in frequency than the first resonant frequency, a fourth resonant frequency equal to the second resonant frequency, a third anti-resonant frequency higher in frequency than the first anti-resonant frequency, and a fourth anti-resonant frequency substantially equal to the second anti-resonant frequency. 
     Therefore, as shown in the middle graph of  FIG. 16B , in the parallel arm circuit  32 , when the switch SWp is turned from the on state to the off state, the lower frequency-side resonant frequency shifts toward a higher frequency side from the first resonant frequency to the third resonant frequency, and the lower frequency-side anti-resonant frequency shifts toward a higher frequency side from the first anti-resonant frequency to the third anti-resonant frequency. 
     Thus, in this Example, when the switch SWb is off and the switch SWp is on, the filter characteristics represented by the dashed line (SWb Off, SWp On) in the top graph of  FIG. 16B  are provided. On the other hand, when the switch SWb is on and the switch SWp is off, the filter characteristics represented by the continuous line (SWb On, SWp Off) in the top graph of  FIG. 16B  are provided. 
     [3-3. Advantageous Effects] 
     Hereinafter, advantageous effects that are obtained from this Example will be described by way of comparison with Example 1-1. 
       FIG. 17A  shows comparison graphs of characteristics of the radio-frequency filters according to Example 3 and Example 1-1 at the time of lower frequency shift.  FIG. 17B  shows comparison graphs of characteristics of the radio-frequency filters according to Example 3 and Example 1-1 at the time of higher frequency shift. Specifically, in these graphs, the top graphs show filter characteristics, the middle graphs show the impedance characteristics of the series arm circuit, and the bottom graphs show the impedance characteristics of the parallel arm circuit. 
     As is apparent from the top graphs of  FIG. 17A  and  FIG. 17B , according to Example 3, the attenuation on the higher side of the pass band is improved both at the time of lower frequency shift and at the time of higher frequency shift as compared to Example 1-1. Specifically, the parallel arm circuit  32  of Example 3 includes the parallel arm resonator p 1   b  (second parallel arm resonator) as compared to the parallel arm circuit  12  of Example 1-1, so the parallel arm circuit  32  has a second resonant frequency equal to the resonant frequency of the parallel arm resonator p 1   b . Thus, a new attenuation pole is provided on the higher side of the pass band by the resonant frequency of the parallel arm resonator p 1   b , so the attenuation on the higher side of the pass band is improved. 
     As is apparent from the top graph of  FIG. 17B , according to Example 3, the attenuation on the lower side of the pass band is improved at the time of higher frequency shift as compared to Example 1-1. Specifically, in Example 3-1, when the switch SWp is turned from the on state to the off state, the capacitor C 22  connected in series with the parallel arm resonator p 1   a  becomes effective. Thus, the first resonant frequency of the parallel arm circuit  32  shifts toward a higher frequency side, so the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency, shifts toward a higher frequency side. Therefore, since the switch SWp is off when the switch SWb is on, the attenuation on the lower side of the pass band when the lower side of the pass band is shifted toward a higher frequency side (at the time of higher frequency shift) is improved. 
       FIG. 18A  is an enlarged view of the pass band and its vicinity of the graph shown at the top of  FIG. 17A .  FIG. 18B  is an enlarged view of the pass band and its vicinity of the graph shown at the top of  FIG. 17B . 
     As is apparent from  FIG. 18A  and  FIG. 18B , according to Example 3, the loss at the lower edge of the pass band is improved (reduced) both at the time of lower frequency shift and at the time of higher frequency shift as compared to Example 1-1. Specifically, the parallel arm circuit  32  of Example 3 includes the parallel arm resonator p 1   b  and, therefore, has two resonant frequencies and two anti-resonant frequencies. As compared to the difference between the resonant frequency and anti-resonant frequency of the parallel arm circuit  12  of Example 1-1, the parallel arm circuit  32  is able to reduce the difference between the lower frequency-side resonant frequency and the lower frequency-side anti-resonant frequency. Therefore, according to Example 3, the cut-off frequency on the lower side of the pass band can be shifted toward a lower frequency side as compared to Example 1-1, so the loss at the lower edge of the pass band is improved. 
     The switch SWb and the switch SWp may be individually switched between the on state and the off state. 
       FIG. 19  shows graphs of changes in filter characteristics when the switches SWb, SWp of the radio-frequency filter  30  according to Example 3 are individually switched between the on state and the off state. 
     As shown in the graphs, the radio-frequency filter  30  is able to change the lower side of the pass band while reducing the loss at the higher edge of the pass band by switching between the on state and off state of the switch SWb. The radio-frequency filter  30  is able to change the attenuation pole on the lower side of the pass band by switching between the on state and off state of the switch SWp. 
     Therefore, the radio-frequency filter  30  is able to provide filter characteristics for the required frequency specifications by switching between the on state and off state of each of the switches SWb, SWp as needed. 
     Fourth Embodiment 
     The configuration of the above-described radio-frequency filter may be applied to a configuration having a plurality of series arm resonators. In the present embodiment, such a radio-frequency filter will be described by way of Example (Example 4). 
     [4-1. Configuration] 
       FIG. 20A  is a circuit configuration diagram of a radio-frequency filter  40  according to Example 4. 
     The radio-frequency filter  40  shown in the diagram is a ladder filter circuit made up of a plurality of series arm circuits (four series arm circuits  210   s ,  220   s ,  230   s ,  240   s  in this Example) and one or more parallel arm circuits (three parallel arm circuits  210   p ,  220   p ,  230   p  in this Example). The plurality of series arm circuits is provided in the path connecting the input and output terminal  11   m  (first input and output terminal) and the input and output terminal  11   n  (second input and output terminal). Specifically, the radio-frequency filter  40  is a variable frequency band pass filter that has a plurality of bands as pass bands, and includes six series arm resonators s 1   a , s 1   b , s 2 , s 3 , s 4   a , s 4   b , fifth parallel arm resonators p 1   a , p 1   b , p 2   a , p 2   b , p 3 , four switches SW 1   b , SW 4   b , SW 1   p , SW 2   p , and two capacitors C 41 , C 42 . The number of the series arm circuits and the number of the parallel arm circuits are not limited to the above-described numbers. 
     Of the series arm circuits  210   s ,  220   s ,  230   s ,  240   s , the series arm circuit  210   s  provided closest to the input and output terminal  11   m  and the series arm circuit  240   s  provided closest to the input and output terminal  11   n  correspond to the series arm circuit  11  of the radio-frequency filter according to any one of the above-described first to third embodiments (here, the first embodiment). Thus, the series arm resonators s 1   a , s 4   a  correspond to the series arm resonator s 1   a  of the series arm circuit  11 , the series arm resonators s 1   b , s 4   b  correspond to the series arm resonator s 1   b  of the series arm circuit  11 , and the switches SW 1   b , SW 4   b  correspond to the switch SW 1   b  of the series arm circuit  11 . The parallel arm circuits  210   p ,  220   p  correspond to the parallel arm circuit  32  of the radio-frequency filter  30  according to Example 3, and the parallel arm circuit  230   p  corresponds to the parallel arm circuit  12  of the radio-frequency filter according to the first embodiment (Example 1-1, the modification, Examples 1-2-1, 1-2-2, 1-3). Therefore, the parallel arm resonators p 1   a , p 2   a  correspond to the parallel arm resonator p 1   a  of the parallel arm circuit  32 , the parallel arm resonator p 3  corresponds to the parallel arm resonator p 1  of the parallel arm circuit  12 , the capacitors C 41 , C 42  correspond to the capacitor C 22  of the parallel arm circuit  32 , the parallel arm resonators p 1   b , p 2   b  correspond to the parallel arm resonator p 1   b  of the parallel arm circuit  32 , and the switches SW 1   p , SW 2   p  correspond to the switch SWp of the parallel arm circuit  32 . That is, in the radio-frequency filter  40 , the configuration corresponding to the radio-frequency filter  30  according to Example 3 is provided closest to the input and output terminal  11   m , and the configuration corresponding to the radio-frequency filter according to the first embodiment is provided closest to the input and output terminal  11   n.    
     The configuration of the radio-frequency filter  40  is not limited thereto. For example, at least one of the plurality of series arm circuits just needs to correspond to the series arm circuit of any one of the above-described first to third embodiments, only the series arm circuit provided closest to one of the input and output terminals may correspond to the series arm circuit of any one of the first to third embodiments, or only the series arm circuit different from the series arm circuit provided closest to any one of the input and output terminals may correspond to the series arm circuit of any one of the first to third embodiments. 
     A parallel arm circuit that connects a ground and a node in a path connecting any one of the input and output terminals (the input and output terminal  11   m  or the input and output terminal  11   n ) to the series arm circuit provided closest to the any one of the input and output terminals may be provided. 
     The thus configured radio-frequency filter  40  includes the configuration of the radio-frequency filter of any one of the above-described first to third embodiments, so a variable frequency radio-frequency filter that improves the loss at the lower edge of the pass band and the attenuation on the higher side of the pass band is implemented. 
     [4-2. Structure] 
       FIG. 20B  is a plan view that illustrates the structure of the radio-frequency filter  40  according to Example 4. 
     As shown in the diagram, in this Example, the resonators (the series arm resonators s 1   a , s 1   b , s 2 , s 3 , s 4   a , s 4   b  and the parallel arm resonators p 1   a , p 1   b , p 2   a , p 2   b , p 3 ) are formed in one package  41  (chip) for resonators, the other elements (the switches SW 1   b , SW 4   b , SW 1   p , SW 2   p , and the capacitors C 41 , C 42 ) are formed in packages  42   a  to  42   c  other than the package  41  for resonators, and these packages  41 ,  42   a  to  42   c  are mounted on a circuit board  43 . In other words, the resonators and the switches are formed in different packages. 
     The packages  41 ,  42   a  to  42   c  have surface electrodes (the circle marks in  FIG. 20B ; also referred to as lands or pads) at their bottom surfaces for being mounted on the circuit board  43 . In  FIG. 20B , for the sake of succinctness, the circuit elements and wires formed on or in the packages are schematically shown, and the surface electrodes on the bottom surfaces are shown through the insides of the packages  41 ,  42   a  to  42   c.    
     The circuit board  43  has external connection electrodes (the circle marks in  FIG. 20B ) that are respectively the input and output terminal  11   m  and the input and output terminal  11   n . The external connection electrodes are, for example, the surface electrodes for mounting the circuit board  43  on a mother board or the like, the connectors that connect the circuit board  43  to another electronic component, or, when another electronic component is mounted on the circuit board  43 , a part of the pattern wires that connect the other electronic component to the package  42   a  or the package  42   c.    
     As is apparent from the diagram, in the package  42   a , one terminal of the switch SW 1   b  is connected to a common terminal  421  and is connected to the input and output terminal  11   m  via the common terminal  421 . The other terminal of the switch SW 1   b  is connected to a first terminal  422  of the package  42   a . The common terminal  421  and a second terminal  423  of the package  42   a  are short-circuited. 
     In the package  42   c , one terminal of the switch SW 4   b  is connected to a common terminal  431  and is connected to the input and output terminal  11   n  via the common terminal  431 . The other terminal of the switch SW 4   b  is connected to a first terminal  432 . The common terminal  431  and a second terminal  433  of the package  42   c  are short-circuited. 
     In the package  41  for resonators, the resonators are connected as follows. 
     First, the connection relation among the series arm resonators (series arm resonators s 1   a , s 1   b , s 2 , s 3 , s 4   a , s 4   b ) will be described. One terminal of the series arm resonator s 1   a  is connected to a first terminal  411  of the package  41 , and the other terminal is connected to a first node N 1 . One terminal of the series arm resonator s 1   b  is connected to a second terminal  412  of the package  41 , and the other terminal is connected to the first node N 1 . One terminal of the series arm resonator s 2  is connected to the first node N 1 , and the other terminal is connected to a second node N 2 . One terminal of the series arm resonator s 3  is connected to the second node N 2 , and the other terminal is connected to a third node N 3 . One terminal of the series arm resonator s 4   a  is connected to the third node N 3 , and the other terminal is connected to a fourth terminal  414  of the package  41 . One terminal of the series arm resonator s 4   b  is connected to the third node N 3 , and the other terminal is connected to a third terminal  413  of the package  41 . 
     Subsequently, the connection relation among the parallel arm resonators (parallel arm resonators p 1   a , p 1   b , p 2   a , p 2   b , p 3 ) will be described. One terminal of the parallel arm resonator p 1   a  is connected to the first node N 1 , and the other terminal is connected to the terminal of the package  42   b  via the terminal of the package  41  and further connected to the ground terminal of the package  42   b  via a parallel circuit of the switch SW 1   p  and the capacitor C 41 , formed in the package  42   b . One terminal of the parallel arm resonator p 1   b  is connected to the first node N 1 , and the other terminal is connected to the ground terminal of the package  41 . One terminal of the parallel arm resonator p 2   a  is connected to the second node N 2 , and the other terminal is connected to the terminal of the package  42   b  via the terminal of the package  41  and further connected to the ground terminal of the package  42   b  via a parallel circuit of the switch SW 2   p  and the capacitor C 42 , formed in the package  42   b . One terminal of the parallel arm resonator p 2   b  is connected to the second node N 2 , and the other terminal is connected to the ground terminal of the package  41 . One terminal of the parallel arm resonator p 3  is connected to the third node N 3 , and the other terminal is connected to the ground terminal of the package  41 . 
     In this way, in the case where one terminal of the switch provided in the series arm is connected to one of the input and output terminals  11   m ,  11   n , when divided into the packages  42   a ,  42   c  each including the switch and the package  41  including the resonators, the switch is located closer to one of the input and output terminals than the resonators. Therefore, the number of terminals of the package  41  for resonators and the packages  42   a ,  42   c  for a switch is reduced, so the miniaturization of the radio-frequency filter  40  is achieved. 
     Not limited to the above configuration, the switch (first switch; here, the switches SW 1   b , SW 4   b ) provided in the series arm just needs to be connected to the input and output terminal  11   m  (first input and output terminal) or the input and output terminal  11   n  (second input and output terminal) without intervening any series arm resonator. With such a configuration, when the series arm resonators are provided on or in a chip (here, the package  41 ) for resonators and the switch is provided on or in another chip (here, the packages  42   a ,  42   c ), the number of terminals of the chip for resonators is reduced. Specifically, in this case, in the chip for resonators, terminals that are connected to any one of the input and output terminals and terminals that are connected to the other chip are respectively shared. Therefore, in comparison with the configuration in which these terminals are individually provided, the number of terminals of the chip for resonators is reduced, so the miniaturization of the radio-frequency filter is achieved. 
     For example, in the circuit configuration shown in  FIG. 20A , at least one of the capacitors C 41 , C 42  may be incorporated in the package  41  for resonators. 
     At least one of the parallel arm resonators p 1   a , p 1   b , p 2   a , p 2   b , p 3  may be provided separately from the package  41  for resonators, or the switches SW 1   b , SW 4   b , SW 1   p , SW 2   p  and the capacitors C 41 , C 42  may be packaged with a combination different from the above combination. 
     [4-3. Application Examples to Multiplexer] 
     The above-described radio-frequency filter  40  is applicable to a multiplexer. 
       FIG. 21  is a circuit configuration diagram of a multiplexer (duplexer)  200  according to Example 4. The multiplexer  200  shown in the diagram includes a transmission filter  60 , a receiving filter  50 , and a matching inductor  70 . The transmission filter  60  is connected to an input terminal  200 T and a common terminal  200   c . The receiving filter  50  is connected to the common terminal  200   c  and an output terminal  200 R. 
     The transmission filter  60  is a band pass filter having a transmission band as a pass band, and its circuit configuration is not specifically limited. 
     The receiving filter  50  corresponds to the radio-frequency filter  40  according to Example 4. The receiving filter  50  is a tunable band pass filter having a plurality of bands as pass bands. 
     With the above configuration, in a tunable duplexer that is applied to a system that selects from among a plurality of frequency bands as needed, a high-performance duplexer that is able to improve the loss at the higher edge of the pass band and the attenuation on the lower side of the pass band is achieved. Since a corresponding filter is not disposed for each frequency band and a single filter circuit having a switch is able to support a plurality of frequency bands, the multiplexer is miniaturized. 
     The receiving filter  50  is not limited to the configuration of Example 4 and may be the configuration of any one of Examples of the first to third embodiments. The radio-frequency filters according to Examples are not limited to application to receiving filters and may be applied to transmission filters. These radio-frequency filters are not limited to application to duplexers and may be applied to multiplexers including a plurality of transmission filters or a plurality of receiving filters. 
     Fifth Embodiment 
     The radio-frequency filters and multiplexers described in the first to fourth embodiments are applicable to a radio-frequency front-end circuit that supports a system having many service bands. In the present embodiment, such a radio-frequency front-end circuit and a communication device will be described. 
       FIG. 22  is a configuration diagram of a communication device  300  according to a fifth embodiment. 
     As shown in the diagram, the communication device  300  includes a switch group  310  made up of a plurality of switches, a filter group  320  made up of a plurality of filters, transmission switches  331 ,  332 , receiving switches  351 ,  352 ,  353 , transmission amplifier circuits  341 ,  342 , receiving amplifier circuits  361 ,  362 , an RF signal processing circuit (RFIC), a baseband signal processing circuit (BBIC), and an antenna element (ANT). The antenna element (ANT) needs not be incorporated in the communication device  300 . 
     The switch group  310  connects the antenna element (ANT) to a signal path that supports the predetermined band in accordance with a control signal from a control unit (not shown). The switch group  310  is made up of, for example, a plurality of SPST switches. The signal path to be connected to the antenna element (ANT) is not limited to one and may be multiple. That is, the communication device  300  may support carrier aggregation. 
     The filter group  320  is, for example, made up of a plurality of filters (including a duplexer) having the following bands as pass bands. Specifically, the bands include (i) a transmission band of Band 12, (ii) a transmission band of Band 13, (iii) a transmission band of Band 14, (iv) a transmission band of Band 27, (v) a transmission band of Band 26, (vi) a receiving band of Band 29 and Band 14 (or Band 12, Band 67, and Band 13), (vii-Tx) a transmission band of Band 68 (or Band 28a or Band 28b), (vii-Rx) a receiving band of Band 68 (or Band 28a or Band 28b), (viii-Tx) a transmission band of Band 20, (viii-Rx) a receiving band of Band 20, (ix-Tx) a transmission band of Band 27 (or Band 26), (x-Tx) a transmission band of Band 8, and (x-Rx) a receiving band of Band 8. 
     The transmission switch  331  is a switch circuit that includes a plurality of selection terminals connected to a plurality of low-band transmission signal paths and a common terminal connected to the transmission amplifier circuit  341 . The transmission switch  332  is a switch circuit that includes a plurality of selection terminals connected to a plurality of high-band transmission signal paths and a common terminal connected to the transmission amplifier circuit  342 . These transmission switches  331 ,  332  are switch circuits that are provided upstream (here, upstream in the transmission signal paths) of the filter group  320  and whose connection statuses are switched in accordance with control signals from a control unit (not shown). Thus, radio-frequency signals (here, radio-frequency transmission signals) amplified by the transmission amplifier circuits  341 ,  342  are outputted to the antenna element (ANT) via predetermined filters of the filter group  320 . 
     The receiving switch  351  is a switch circuit that includes a plurality of selection terminals connected to a plurality of low-band reception signal paths and a common terminal connected to the receiving amplifier circuit  361 . The receiving switch  352  is a switch circuit that includes a common terminal connected to a reception signal path of the predetermined band (here, Band 20) and two selection terminals connected to a common terminal of the receiving switch  352 . The receiving switch  353  is a switch circuit that includes a plurality off selection terminals connected to a plurality of high-band reception signal paths and a common terminal connected to the receiving amplifier circuit  362 . These receiving switches  351  to  353  are provided downstream (here, downstream in the reception signal paths) of the filter group  320  and whose connection statuses are switched in accordance with control signals from the control unit (not shown). Thus, radio-frequency signals (here, radio-frequency reception signals) inputted to the antenna element (ANT) are amplified by the receiving amplifier circuits  361 ,  362  via predetermined filters of the filter group  320  and outputted to the RF signal processing circuit (RFIC). An RF signal processing circuit (RFIC) that supports a low band and an RF signal processing circuit (RFIC) that supports a high band may be individually provided. 
     The transmission amplifier circuit  341  is a power amplifier that amplifies the electric power of a low-band radio-frequency transmission signal. The transmission amplifier circuit  342  is a power amplifier that amplifies the electric power of a high-band radio-frequency transmission signal. 
     The receiving amplifier circuit  361  is a low-noise amplifier that amplifies the electric power of a low-band radio-frequency reception signal. The receiving amplifier circuit  362  is a low-noise amplifier that amplifies the electric power of a high-band radio-frequency reception signal. 
     The RF signal processing circuit (RFIC) is a circuit that processes a radio-frequency signal that is transmitted or received by the antenna element (ANT). Specifically, the RF signal processing circuit (RFIC) processes a radio-frequency signal (here, radio-frequency reception signal) inputted from the antenna element (ANT) via the reception signal path by down conversion, or the like, and outputs the processed and generated reception signal to the baseband signal processing circuit (BBIC). The RF signal processing circuit (RFIC) also processes a transmission signal inputted from the baseband signal processing circuit (BBIC) by up conversion, or the like, and outputs the processed and generated radio-frequency signal (here, radio-frequency transmission signal) to the transmission signal path. 
     The thus configured communication device  300  includes the radio-frequency filter according to any one of the first to fourth embodiments as at least one of a filter having (vi) a receiving band of Band 29 and Band 14 (or Band 12, Band 67, and Band 13) as a pass band, a filter having (vii-Tx) a transmission band of Band 68 (or Band 28a or Band 28b) as a pass band, a filter having (vii-Rx) a receiving band of Band 68 (or Band 28a or Band 28b) as a pass band, and a filter having (ix-Tx) a transmission band of Band 27 (or Band 26) as a pass band. That is, the filter switches the pass band in accordance with a control signal. 
     In the communication device  300 , the switch group  310 , the filter group  320 , the transmission switches  331 ,  332 , the receiving switches  351 ,  352 ,  353 , the transmission amplifier circuits  341 ,  342 , the receiving amplifier circuits  361 ,  362 , and the control unit make up the radio-frequency front-end circuit. 
     Although the control unit is not shown in  FIG. 22 , the RF signal processing circuit (RFIC) may include the control unit, or the control unit may make up a switch IC together with the switches that the control unit controls. 
     With the thus configured radio-frequency front-end circuit and communication device  300 , since the radio-frequency filter according to any one of the above-described first to fourth embodiments is provided, the loss at the higher edge of the pass band and the attenuation on the lower side of the pass band are improved. That is, for example, a high-performance radio-frequency front-end circuit and communication device that are able to switch the pass band and the stop band while reducing the loss at the higher edge of the pass band are achieved. The number of filters is reduced as compared to the case where a filter is provided for each band, so miniaturization is achieved. 
     With the radio-frequency front-end circuit according to the present embodiment, the transmission switches  331 ,  332  and the receiving switches  351  to  353  (switch circuits) provided upstream or downstream of the filter group  320  (the plurality of radio-frequency filters) are provided. Thus, parts of signal paths through which a radio-frequency signal is transferred can be integrated. Thus, for example, the transmission amplifier circuits  341 ,  342  or the receiving amplifier circuits  361 ,  362  (amplifier circuits), each of which corresponds to a plurality of radio-frequency filters, can be integrated. Therefore, miniaturization and cost reduction of the radio-frequency front-end circuit are possible. 
     At least one of the transmission switches  331 ,  332  or at least one of the receiving switches  351  to  353  just needs to be provided. The number of the transmission switches  331 ,  332  and the number of the receiving switches  351  to  353  are not limited to the above-described numbers. For example, one transmission switch and one receiving switch may be provided. The number of the selection terminals, or the like, of the transmission switches or receiving switches is not limited to the present embodiment and may be two of each. 
     Other Embodiments, and Others 
     The radio-frequency filter, radio-frequency front-end circuit, and communication device according to the embodiments of the present disclosure are described by way of the first to fifth embodiments; however, the radio-frequency filter, radio-frequency front-end circuit, and communication device of the present disclosure are not limited to the above-described embodiments. The present disclosure also encompasses other embodiments implemented by combining selected elements of the above-described embodiments, modifications obtained by applying various alterations that are conceived of by persons skilled in the art to the above-described embodiments without departing from the purport of the present disclosure, and various devices that include the radio-frequency filter, radio-frequency front-end circuit, or communication device of the present disclosure. 
     The radio-frequency filters according to the first to fourth embodiments are not limited to application to a system that exclusively switches among the frequency bands close to one another and may also be applicable to a system that exclusively switches among a plurality of channels close to one another and allocated to within one frequency band. 
     In the radio-frequency filters according to the first to fourth embodiments, the series arm resonator(s) and the parallel arm resonator(s) are not limited to acoustic wave resonators that use surface acoustic waves and may be, for example, acoustic wave resonators that use bulk waves or boundary acoustic waves. Each of the series arm resonator and the parallel arm resonator is not limited to one acoustic wave resonator and includes a plurality of split resonators into which one acoustic wave resonator is, for example, serially divided. 
     In the radio-frequency filters according to the first to fourth embodiments, the series arm circuit includes at least one second series arm resonator lower in resonant frequency than the first series arm resonator; however, the series arm circuit just needs to include a second series arm resonator having a resonant frequency different from that of the first series arm resonator. Therefore, the series arm circuit may be made up of one first series arm resonator and one first variable frequency circuit, and the one first variable frequency circuit may be made up of a second switch and a second series arm resonator higher in resonant frequency than the first series arm resonator. With the thus configured radio-frequency filter, the higher side of the pass band can be changed (shifted in frequency) while the loss at the lower edge of the pass band is reduced. 
     Specifically, when the resonant frequency (fr_s 1   a ) of the first series arm resonator is higher than the resonant frequency (fr_s 1   b ) of the second series arm resonator (fr_s 1   b &gt;fr_s 1   a ), both the first anti-resonant frequency (fa 1 _s 1 _on) that lies between the resonant frequency (fr_s 1   a ) of the first series arm resonator and the resonant frequency (fr_s 1   b ) of the second series arm resonator and the second anti-resonant frequency (fa 2 _s 1 _on) that lies on the higher frequency side than the first anti-resonant frequency provide attenuation poles on the higher side of the pass band. That is, a new attenuation pole is provided on the higher side of the pass band by turning the first switch from the off state (nonconductive state) to the on state (conductive state). 
     Thus, when the attenuation pole on the lower side of the pass band, which is provided by the resonant frequency of the parallel arm circuit, is considered, the following can be said on the relationship between switching the on state and off state of the first switch and the pass band. Specifically, when the first switch is turned from the on state to the off state, a new attenuation pole is provided on the higher side of the pass band, so the stop band on the higher side of the pass band is widened. At this time, since no new attenuation pole is provided on the lower side of the pass band, the deterioration of the loss at the lower edge of the pass band is reduced, and the deterioration of the attenuation on the higher side of the pass band is reduced. In other words, the tunable radio-frequency filter that improves the loss at the lower edge of the pass band and the attenuation on the higher side of the pass band is achieved. 
     In the radio-frequency filters according to the first to fourth embodiments, the parallel arm circuit may be not an acoustic wave resonator and may be an LC resonant circuit. The parallel arm circuit is not limited to a resonant circuit and may be an inductance element or a capacitance element. 
     The configuration of each of the series arm circuit described in any one of the first to third embodiments is applicable to a ladder filter circuit that includes a plurality of series arm circuits including the series arm circuit and one or more parallel arm circuits. In such a filter circuit, the configuration of the series arm circuit different from the series arm circuits described in the first to third embodiments is not specifically limited. The series arm circuit may be, for example, a resonant circuit, such as a longitudinally-coupled resonator or an LC resonant circuit or may be an inductance element or a capacitance element. Therefore, the configuration of the series arm circuit just needs to be selected as needed according to the required specifications and may be a longitudinally coupled resonator, for example, when attenuation reinforcement, or the like, is required. 
     In the above-described fourth embodiment, the configuration in which the series arm circuit described in any one of the first to third embodiments is provided closest to the input and output terminal  11   m  or the input and output terminal  11   n  is described as the configuration of a ladder filter circuit that includes a plurality of series arm circuits, including the series arm circuit described in the first to third embodiments, and one or more parallel arm circuits. However, the configuration of the ladder filter circuit is not limited thereto. For example, a series arm circuit different from the series arm circuit described in any one of the first to third embodiments may be provided closest to the input and output terminal  11   m  or the input and output terminal  11   n . That is, the series arm circuit described in any one of the first to fourth embodiments may be provided other than an end portion of the plurality of series arm circuits. 
     In the radio-frequency filters, radio-frequency front-end circuit, and communication device according to the first to fifth embodiments, an inductance element or a capacitance element may be further connected between each input and output terminal and the common terminal. An inductance component caused by wires that connect the circuit elements may be provided. 
     The present disclosure is widely usable in communication equipment, such as cellular phones, as a small radio-frequency filter, radio-frequency front-end circuit, and communication device that simultaneously or exclusively use a plurality of close bands and that are applicable to multiband and multimode systems.
           10 ,  10 A,  10 Ba,  10 Bb,  10 C,  10 X,  20 ,  20 A,  30 ,  40  radio-frequency filter     11 ,  11 A,  11 Ba,  11 Bb,  11 C,  11 X,  210   s ,  220   s ,  230   s ,  240   s  series arm circuit     11   aa ,  11   ab ,  321  series connection circuit     11   b ,  11   b   1 ,  11   bn ,  11   c ,  11   d  variable frequency circuit     11   m  input and output terminal     11   n  input and output terminal     12 ,  22 ,  22 A,  32 ,  210   p ,  220   p ,  230   p  parallel arm circuit     41 ,  42   a ,  42   b ,  42   c  package     43  circuit board     50  receiving filter     60  transmission filter     70  matching inductor     101  piezoelectric substrate     111  interdigital transducer electrode     111   a  electrode finger     120  acoustic wave resonator     200  multiplexer (duplexer)     200   c  common terminal     200 R output terminal     200 T input terminal     300  communication device     310  switch group     320  filter group     331 ,  332  transmission switch     341 ,  342  transmission amplifier circuit     351  to  353  receiving switch     361 ,  362  receiving amplifier circuit     411 ,  422 ,  432  first terminal     412 ,  423 ,  433  second terminal     413  third terminal     414  fourth terminal     421 ,  431  common terminal   ANT antenna element   BBIC baseband signal processing circuit   C 11 , C 22 , C 41 , C 42  capacitor   L 11 , L 22  inductor   N 1  first node   N 2  second node   N 3  third node   p 1 , p 1   a , p 1   b , p 2   a , p 2   b , p 3  parallel arm resonator   s 1 , s 1   a , s 1   b , s 1   b   1 , s 1   bn , s 2 , s 3 , s 4   a , s 4   b  series arm resonator   SW 1   b , SW 1   p , SW 2   p , SW 4   b , SWb, SWb 1 , SWbn, SWp, SWx switch   x 1  node