Patent Publication Number: US-9419584-B2

Title: Antenna sharing device

Description:
This application is a U.S. national stage application of the PCT international application No. PCT/JP2011/000869, filed Feb. 17, 2011. 
     BACKGROUND OF THE INVENTION 
     1. Technical Field 
     The present invention relates to an antenna duplexer having a plurality of filters. 
     2. Background Art 
     In recent years, a cellular phone of a communication system of performing transmission/reception simultaneous communication such as W-CDMA (Wideband Code Division Multiple Access) is being rapidly spread. Accordingly, demand of an antenna duplexer such as a duplexer is increasing. As elements constructing an antenna duplexer, an SAW (Surface Acoustic Wave) element, a boundary elastic wave element, a BAW (Bulk Acoustic wave) element, and the like which are excellent from the viewpoints of small size, low height, and mass production are mainstream. 
     Generally, an antenna duplexer has two filters (a transmission filter and a reception filter) to separate a signal in a transmission band and a signal in a reception band neighboring the high frequency side of the transmission band from each other. Particularly, as the transmission filter, a ladder filter in which series resonators and parallel resonators are connected in a ladder shape is employed. 
     For example, the gap (crossband) between the transmission band and the reception band in Band2 determined in 3GPP (3 rd  Generation Partnership Project) is 20 MHz (1.06% in expression of fractional bandwidth). The gap is narrower as compared with 20 MHz (fractional bandwidth: 2.36%) as the crossband of Band5 often used in conventional antenna duplexers. 
     Therefore, a technique of adding weight on an IDT (Inter Digital Transducer) of a resonator of a transmission filter in order to assure steepness to address the narrow crossband is proposed (refer to, for example, Japanese Translation of PCT Application No. 2001-500697). 
     Various techniques of making the propagation angles of the main elastic waves of resonators in a ladder filter are also proposed (refer to, for example, Japanese Unexamined Patent Publication No. H07-283688 and WO 2005/060094). 
     However, Japanese Translation of PCT Application No. 2001-500697 relates to the technique for narrowing the bandwidth and does not disclose means realizing an antenna duplexer whose bandwidth is wide like 60 MHz in Band2. That is, the transmission filter in the antenna duplexer disclosed in Japanese Translation of PCT Application No. 2001-500697 has a problem such that when steepness is increased to sufficiently assure attenuation in the reception band, the transmission bandwidth is narrowed, and a loss in a wide passband increases. Particularly, the fractional bandwidth of the transmission/reception passband in Band2 is 3.2% and wide, and it is very difficult to maintain a small loss in the wide transmission passband. 
     It is expected that, in future, not only in Band2 but also Band3, Band8, and the like, an antenna duplexer capable of satisfying both a low loss characteristic in a wide band and a steep attenuation characteristic is in demand. 
     Conventionally known techniques, however, have a problem such that when a low loss characteristic is realized in a wide band, sufficient steepness cannot be obtained. 
     SUMMARY OF THE INVENTION 
     The present invention has been achieved in consideration of the problems and an object of the invention is to realize both steepness in the crossband and a low loss characteristic in a passband in an antenna duplexer. 
     An antenna duplexer of the present invention has a first filter passing a signal in a first frequency band and a second filter passing a signal in a second frequency band higher than the first frequency band. The first filter has a ladder filter having a piezoelectric body, a protective film formed on the piezoelectric body, and an electrode which is formed between the piezoelectric body and the protective film and excites a main elastic wave, and formed by connecting a plurality of resonators including the electrode in series and in parallel. The ladder filter has a plurality of series resonators, and the plurality of series resonators include a series resonator having a lowest antiresonance frequency and other resonators other than the series resonator having the lowest antiresonance frequency, and a propagation angle of the main elastic wave of the series resonator having the lowest antiresonance frequency and that of the main elastic wave of the other resonators are made different from each other so that an electromechanical coupling coefficient of the series resonator having the lowest antiresonance frequency becomes smaller than that of the other resonators. 
     An antenna duplexer of the present invention has a first filter passing a signal in a first frequency band and a second filter passing a signal in a second frequency band higher than the first frequency band. The second filter includes a ladder filter having a piezoelectric body, a protective film formed on the piezoelectric body, and an electrode which is formed between the piezoelectric body and the protective film and excites a main elastic wave, and formed by connecting resonators including the electrode in series and in parallel. The ladder filter has a plurality of parallel resonators, the plurality of parallel resonators include a parallel resonator having a highest resonance frequency and other resonators other than the parallel resonator having the highest resonance frequency, and a propagation angle of the main elastic wave of the parallel resonator having the highest resonance frequency and that of the main elastic wave of the other resonators are made different from each other so that an electromechanical coupling coefficient of the parallel resonator having the highest resonance frequency becomes smaller than that of the other resonators. 
     Further, an antenna duplexer of the present invention has a first filter passing a signal in a first frequency band and a second filter passing a signal in a second frequency band higher than the first frequency band. The first filter includes a ladder filter having a piezoelectric body, a protective film formed on the piezoelectric body, and an electrode which is formed between the piezoelectric body and the protective film and excites a main elastic wave, and formed by connecting a plurality of resonators including the electrode in series and in parallel. The ladder filter has a plurality of series resonators, the plurality of series resonators include one series resonator and other series resonators having an antiresonance frequency higher than that of the one series resonator, and a propagation angle of the main elastic wave of the one series resonator and that of the main elastic wave of the other series resonators are made different from each other so that an electromechanical coupling coefficient of the one series resonator becomes smaller than that of the other resonators. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic circuit diagram of an antenna duplexer in a first exemplary embodiment of the present invention. 
         FIG. 2  is a schematic cross section of a first filter in the antenna duplexer in the first exemplary embodiment of the invention. 
         FIG. 3  is a diagram showing an electromechanical coupling coefficient of the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 4  is a diagram illustrating antiresonance frequency, capacitance, propagation angle ψ, and electromechanical coupling coefficient of each of series resonators constructing the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 5  is a diagram showing pass characteristic of the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 6  is a frequency characteristic diagram of a group of other resonators in the case where a piezoelectric body of the first filter in the antenna duplexer according to the first embodiment of the invention is lithium niobate based body having Euler angles (0°, −87.5°, 0°). 
         FIG. 7  is a frequency characteristic diagram of a group of other resonators in the case where a piezoelectric body of the first filter in the antenna duplexer according to the first embodiment of the invention is lithium niobate based body having Euler angles (0°, −90°, 0°). 
         FIG. 8A  is a diagram showing an electromechanical coupling coefficient (k2) of a fast transverse wave when thickness of a protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 8B  is a diagram showing Q factor (Qs) of resonance of a fast transverse wave when thickness of the protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 8C  is a diagram showing Q factor (Qa) of antiresonance of a fast transverse wave when thickness of the protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 9A  is a diagram showing the electromechanical coupling coefficient (k2) of a fast transverse wave when thickness of the protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 9B  is a diagram showing Q factor (Qs) of resonance of a fast transverse wave when thickness of the protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 9C  is a diagram showing Q factor (Qa) of antiresonance of a fast transverse wave when thickness of the protective film in the first filter in the antenna duplexer according to the first embodiment of the invention is changed. 
         FIG. 10A  is a diagram showing admittance characteristic in the case where φ=−9° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10B  is a diagram showing the admittance characteristic in the case where φ=−6° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10C  is a diagram showing the admittance characteristic in the case where φ=−3° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10D  is a diagram showing the admittance characteristic in the case where φ=0° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10E  is a diagram showing the admittance characteristic in the case where φ=+3° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10F  is a diagram showing the admittance characteristic in the case where φ=+6° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 10G  is a diagram showing the admittance characteristic in the case where φ=+9° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11A  is a diagram showing the admittance characteristic in the case where ψ=−9° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11B  is a diagram showing the admittance characteristic in the case where ψ=−6° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11C  is a diagram showing the admittance characteristic in the case where ψ=−3° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11D  is a diagram showing the admittance characteristic in the case where ψ=0° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11E  is a diagram showing the admittance characteristic in the case where ψ=+3° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11F  is a diagram showing the admittance characteristic in the case where ψ=+6° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 11G  is a diagram showing the admittance characteristic in the case where ψ=+9° in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 12A  is an admittance characteristic diagram of a group of the other resonators in the case where, in a first filter in the antenna duplexer according to the first embodiment of the invention, a piezoelectric body is lithium niobate based body having Euler angles (7°, −87.5°, 8.4°), an electrode is made of copper having thickness of 0.03λ, and a protective film is made of silicon oxide, whose top face is flat, and having a thickness of 0.35λ. 
         FIG. 12B  is an admittance characteristic diagram showing the case where, in the first filter in the antenna duplexer according to the first embodiment of the invention, a piezoelectric body is lithium niobate based body having Euler angles (9°, −87.5°, 10.7°). 
         FIG. 13  is a diagram showing a desirable range of φ and ψ in the Euler angles (φ, θ, ψ) of the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 14  is a diagram showing Q factor of Rayleigh wave of a group of the other resonators in the case where ψ in the Euler angles (φ, θ, ψ) of the piezoelectric body is changed around ψ=1.193φ, in the first filter in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 15  is a diagram showing Q factor of a fast transverse wave of a group of the other resonators in the case where ψ in the Euler angles (φ, θ, ψ) of the piezoelectric body of the first filter is changed around ψ=−2φ, in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 16  is a diagram showing electromechanical coupling coefficient (k2) of Rayleigh wave of a group of the other resonators in the case where θ in the Euler angles (φ, θ, ψ) of the first filter is changed, in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 17  is a diagram showing normalization coupling coefficient of SH wave of a group of the other resonators in the case where θ in the Euler angles (φ, θ, ψ) of the piezoelectric body in the first filter is changed, in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 18  is a diagram showing electromechanical coupling coefficient (k2) of Rayleigh wave of a group of the other resonators in the case where φ in the Euler angles (φ, θ, ψ) of the piezoelectric body in the first filter is changed according to the relation of ψ=1.193φ, in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 19  is a diagram showing normalization coupling coefficient of the SH wave of a group of the other resonators in the case where φ in the Euler angles (φ, θ, ψ) of the piezoelectric body in the first filter is changed according to the relation of ψ=1.193φ, in the antenna duplexer according to the first embodiment of the invention. 
         FIG. 20  is a schematic circuit diagram of an antenna duplexer in a second exemplary embodiment of the invention. 
         FIG. 21  is a schematic cross section of a first filter in the antenna duplexer in the second exemplary embodiment of the invention. 
         FIG. 22A  is an admittance characteristic diagram of a group of the other resonators in the case where a piezoelectric body of the first filter in the antenna duplexer according to the second embodiment of the invention is lithium niobate based body having Euler angles (7°, −87.5°, 8.4°), an electrode is made of aluminum having a thickness of 0.08λ, and a protective film is made of silicon oxide, has a film thickness of 0.35λ, and has a projection having a height T=0.08λ on its top face. 
         FIG. 22B  is an admittance characteristic diagram in the case where the piezoelectric body of the first filter in the antenna duplexer according to the second embodiment of the invention is lithium niobate based body having Euler angles (9°, −87.5°, 10.7°). 
         FIG. 23A  is a diagram showing an example of a method of manufacturing a first filter in an antenna duplexer according to the second embodiment of the invention. 
         FIG. 23B  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23C  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23D  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23E  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23F  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23G  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 23H  is a diagram showing an example of the method of manufacturing the first filter in the antenna duplexer according to the second embodiment of the invention. 
         FIG. 24  is a schematic circuit diagram of an antenna duplexer in a third embodiment of the present invention. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     First Exemplary Embodiment 
     A first exemplary embodiment of the present invention will now be described with reference to the drawings. 
       FIG. 1  is a schematic circuit diagram of antenna duplexer  1  in the first exemplary embodiment of the present invention. 
     Antenna duplexer  1  in the embodiment has first filter  3  as a transmission filter and second filter  4  as a reception filter each connected to antenna terminal  2 . Antenna duplexer  1  has phase shifter  10  connected between first and second filters  3  and  4  to assure isolation between first and second filters  3  and  4 . 
     Antenna duplexer  1  is, for example, for band  2 . First filter  3  passes a signal in a first frequency band (transmission band) of 1850 MHz to 1910 MHz, and second filter  4  passes a signal in a second frequency band (reception band) of 1930 MHz to 1990 MHz higher than the first frequency band. 
     The circuit configuration of each of first and second filters  3  and  4  will now be described specifically. 
     First filter  3  has input terminal  5 , and first series resonator  6 , second series resonator  7 , third series resonator  8 , and fourth series resonator  9  which are connected in order from input terminal  5  and in series to antenna terminal  2 . First filter  3  also has first parallel resonator  11  grounded and connected in parallel between first and second series resonators  6  and  7 , second parallel resonator  12  grounded and connected in parallel between second and third series resonators  7  and  8 , and third parallel resonator  13  grounded and connected in parallel between third and fourth series resonators  8  and  9 . 
     Second filter  4  has, between antenna terminal  2  and output terminal  18 , fifth series resonator  14 , sixth series resonator  15 , seventh series resonator  16 , and eighth series resonator  17  connected in order from antenna terminal  2  side and in series. Second filter  4  also has fourth parallel resonator  19  grounded and connected in parallel between fifth and sixth series resonators  14  and  15 , fifth parallel resonator  20  grounded and connected in parallel between sixth and seventh series resonators  15  and  16 , sixth parallel resonator  21  grounded and connected in parallel between seventh and eighth series resonators  16  and  17 , and seventh parallel resonator  22  grounded and connected in parallel between eighth series resonator  17  and output terminal  18 . Second filter  4  may have a multiple-mode elastic wave filter (not shown). 
       FIG. 2  is a schematic cross section of first filter  3  in antenna duplexer  1  in the first exemplary embodiment of the invention. First filter  3  has piezoelectric body  23 , protective film  24  formed on piezoelectric body  23 , and electrode  25  as an IDT (Inter-Digital Transducer) formed between piezoelectric body  23  and protective film  24  and exciting a main elastic wave which is, for example, an SH (Shear Horizontal) wave having a wavelength λ. Each of the above-described resonators is formed by piezoelectric body  23 , protective film  24 , and electrode  25 . 
     Electrode  25  is an electrode having a comb shape and is made of, for example, a single metal such as aluminum, copper, silver, gold, titanium, tungsten, molybdenum, platinum, or chromium, an alloy whose main component is any of the metals, or a stacked body of any of the metals. The metallization ratio (duty) expressed by “electrode finger width/pitch” of electrode  25  is, desirably, equal to or higher than 0.45 and lower than 0.6 from the viewpoint of mass production and lower loss rate. 
     Piezoelectric body  23  is a piezoelectric single crystal substrate made of lithium niobate based body (LiNbO 3 ), lithium tantalate (LiTaO 3 ), or the like. Particularly, in the case where piezoelectric body  23  is made of a lithium niobate based body (LiNbO 3 )-based material, for reasons to be described later, preferably, the Euler angles (φ, θ, ψ) of piezoelectric body  23  in third and fourth series resonators  8  and  9  and first, second, and third parallel resonators  11 ,  12 , and  13  satisfy the relations of −100°≦θ≦−60°, 1.193φ−2°≦ψ≦1.193φ+2°, and ψ≦−2φ−3°, and −2φ+3°≦ψ. Each of φ and θ denotes a cutting angle of piezoelectric body  23 , and ψ denotes the propagation angle of the main elastic wave of third and fourth series resonators  8  and  9  and first, second, and third parallel resonators  11 ,  12 , and  13 . 
     Since piezoelectric body  23  of a lithium niobate based body is trigonal crystal, the Euler angles have the following relation. 
     
       
         
           
             
               
                 
                   
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     Protective film  24  is, for example, a silicon oxide (SiO 2 ) film. Protective film  24  has a temperature characteristic opposite to that of piezoelectric body  23 . In the case where protective film  24  is made of silicon oxide, by making the thickness of the film larger than a predetermined film thickness, the frequency temperature characteristic of first filter  3  can be improved. At this time, the Euler angles (φ, θ, ψ) are changed in third and fourth series resonators  8  and  9  and first, second, and third parallel resonators  11 ,  12 , and  13  other than first and second series resonators  6  and  7 , in piezoelectric body  23  made of lithium niobate based body. Concretely, φ and ψ are set to predetermined angle or larger and changed so as to follow the relation of ψ=1.193φ to a certain degree. By the operation, while suppressing generation of unnecessary spurious signals caused by the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be also suppressed. The unnecessary spurious signal suppression action will be described later. 
       FIG. 3  is a diagram showing an electromechanical coupling coefficient of first filter  3  in antenna duplexer  1  according to the first embodiment of the invention. In the example of  FIG. 3 , it is assumed that piezoelectric body  23  is made of lithium niobate based body having the Euler angles (6°, −87.5°, ψ°), electrode  25  is made of aluminum having a thickness of 0.12λ, and protective film  24  is made of silicon oxide having a thickness of 0.35λ and has, in its top face, a projection above an electrode finger of electrode  25 .  FIG. 3  shows the electromechanical coupling coefficient of the main elastic wave in the case where the propagation angle w is changed. As the electromechanical coupling coefficient, a value standardized with a value at the time of ψ=7.4° is shown. The height of the projection is larger than 0.03λ and is equal to or less than the height of electrode  25 , and the width of a top part of the projection is smaller than that of the electrode finger of electrode  25 . 
     In the specification, the thickness of protective film  24  is defined as a distance from a boundary face between piezoelectric body  23  and protective film  24  to the top face of protective film  24 , in a part where piezoelectric body  23  and protective film  24  are in contact (in the part where electrode  25  is not formed). 
     As shown in  FIG. 3 , when first filter  3  in the embodiment is used, by changing the propagation angle ψ, the electromechanical coupling coefficient of the main elastic wave can be suppressed. The electromechanical coupling coefficient of the main elastic wave in first filter  3  is 9.6% at maximum in the case where the propagation angle ψ=0°. 
     Next, as an example of first filter  3 , the case where piezoelectric body  23  is made of lithium niobate based body having the Euler angles (6°, −87.5°, 7.4°), electrode  25  is made of aluminum having a thickness of 0.12λ, and protective film  24  is made of silicon oxide having a thickness of 0.35λ and has, in its top face, a projection above an electrode finger of electrode  25  will be described. 
       FIG. 4  is a diagram illustrating antiresonance frequency, capacitance, propagation angle ψ, and electromechanical coupling coefficient of each of the series resonators constructing first filter  3  in antenna duplexer  1  according to the first embodiment of the invention. 
     As shown in  FIG. 4 , the antiresonance frequency of second series resonator  7  is lower than that of the other series resonators, that is, first, third, and fourth series resonators  6 ,  8 , and  9 . The electromechanical coupling coefficient of first and second series resonators  6  and  7  including second series resonator  7  is made smaller than that of other third and fourth series resonators  8  and  9 . Consequently, the propagation angle w of the main elastic wave of first and second series resonators  6  and  7  is made different from that of the main elastic wave of third and fourth series resonators  8  and  9 . 
     In the embodiment, first filter  3  in which the propagation angle w in the Euler angles of piezoelectric body  23 , of first and second series resonators  6  and  7  is made different from that of third and fourth series resonators  8  and  9  will be described. However, the invention is not limited to the case. For example, the electromechanical coupling coefficient of second series resonator  7  having the lowest antiresonance frequency among the plurality of series resonators may be made smaller than that of first, third, and fourth series resonators  6 ,  8 , and  9  as the other resonators. In this case, it is sufficient to make the propagation angle w of the main elastic wave of second series resonator  7  having the lowest antiresonance frequency different from that of the main elastic wave of first, third and fourth series resonators  6 ,  8 , and  9  as the other resonators. 
       FIG. 5  is a diagram showing pass characteristic of first filter  3  in antenna duplexer  1  according to the first embodiment of the invention.  FIG. 5  shows comparison between the pass characteristic of first filter  3  of the embodiment and that of a filter in which the propagation angle ψ of first and second series resonators  6  and  7  is not changed with respect to that of third and fourth series resonators  8  and  9 . 
     As shown in  FIG. 5 , by making the propagation angle of first and second series resonators  6  and  7  larger than that of third and fourth series resonators  8  and  9 , the slope characteristic in a high frequency side (portion surrounded by dotted line) in the passband of first filter  3  can be made steep. 
     As described above, first filter  3  in antenna duplexer  1  in the embodiment has protective film  24  formed on piezoelectric body  23  so as to cover electrode  25 . With such a configuration, by changing the propagation angle ψ (propagation direction) of the main elastic wave in each of the series resonators, the electromechanical coupling coefficient of the main elastic wave in each of the series resonators can be controlled widely. 
     Further, in first filter  3 , by making the electromechanical coupling coefficient of second series resonator  7  having the lowest antiresonance frequency, which exerts large influence on steepness smaller than that of the other resonators, the steepness in the crossband can be improved. 
     In the series resonators (hereinbelow, described as other series resonators) other than second series resonator  7  having the lowest antiresonance frequency, which exerts small influence on sharpness, that is, in the embodiment, by largely assuring the electromechanical coupling coefficient of third series resonator  8  having relatively high antiresonance frequency, the passband width can be widened, and a loss can be suppressed in a wide transmission passband. That is, antenna duplexer  1  of the embodiment can satisfy both steepness in the crossband and lower loss in the transmission passband. 
     As shown in  FIG. 4 , preferably, the capacitance of second series resonator  7  having the lowest antiresonance frequency is set to the largest among the plurality of series resonators included in first filter  3  for the following reason. Since passage loss in the case of connecting resonators of large capacitance in series is small, at the time of changing the propagation angle, a steep slope characteristic can be obtained without accompanying deterioration in loss on the high frequency side of the passband of the filter. 
     The propagation angle ψ in the Euler angles of piezoelectric body  23  in the series resonators whose propagation angle is changed (in the embodiment, first and second series resonators  6  and  7 ) does not always have to satisfy the above-described specific range (the relations of ψ≦−2φ−3°, −2φ+3°≦ψ) for the following reason. Since the antiresonance frequency of the series resonator whose propagation angle is changed is low, the degree of influence of the unnecessary spurious signal on the passage characteristic is low. That is, in first filter  3 , it is sufficient to set the Euler angle w of piezoelectric body  23  in third and fourth series resonators  8  and  9  to the above-described specific range. As a result, even when the Euler angle ψ of piezoelectric body  23  in the series resonator whose propagation angle is changed, for example, first and second series resonators  6  and  7  is set out of the above-described specific range, the unnecessary spurious signal can be suppressed, and steepness in the crossband can be improved. 
     The electromagnetic coupling coefficient of an arbitrary series resonator (one series resonator) other than second series resonator  7  having the lowest antiresonance frequency among the plurality of series resonators and that of another series resonator having an antiresonance frequency higher than that of the one series resonator may be made different from each other. For instance, in the example of  FIG. 4 , the electromechanical coupling coefficient of first series resonator  6  and that of third series resonator  8  having antiresonance frequency higher than that of first series resonator  6  are made different from each other. Specifically, the propagation angle ψ of the main elastic wave of first series resonator  6  and that of the main elastic wave of third series resonator  8  are made different from each other so that the electromechanical coupling coefficient of first series resonator  6  becomes smaller than that of third series resonator  8 . As a result, both of steepness in the crossband and lower loss in the transmission passband can be satisfied. 
     In first filter  3  in the embodiment, in the case where first and second series resonators  6  and  7  are used as the series resonator whose propagation angle is changed, by setting the Euler angle of piezoelectric body  23  in third and fourth series resonators  8  and  9  and first, second, and third parallel resonators  11 ,  12 , and  13  (hereinbelow, described as a group  38  of other resonators) as the resonators other than resonators  6  and  7  in a specific range, an unnecessary spurious signal can be suppressed. This action will be described. 
       FIG. 6  is a frequency characteristic diagram of group  38  of the other resonators in the case where piezoelectric body  23  of first filter  3  in antenna duplexer  1  in the first embodiment of the invention is made of a lithium niobate based body having Euler angles (0°, −87.5°, 0°).  FIG. 7  is a frequency characteristic diagram of group  38  of the other resonators in the case where piezoelectric body  23  is made of a lithium niobate based body having Euler angles (0°, −90°, 0°). In  FIGS. 6 and 7 , the vertical axis indicates normalized admittance with respect to a matching value, and the horizontal axis indicates normalized frequency with respect to frequency of the half of a slow transverse wave (sound velocity 4,024 m/s) generated in group  38  of the other resonators. The axes denote the same in the other characteristic diagrams. 
     In the example shown in  FIG. 6 , electrode  25  of first filter  3  is made of copper having a thickness of 0.03λ, and protective film  24  is made of silicon oxide having a thickness of 0.35λ and the top face of protective film  24  is flat. 
     In the example shown in  FIG. 7 , electrode  25  of the first filter is made of aluminum having a thickness of 0.08λ, and protective film  24  is made of silicon oxide having a thickness of 0.35λ and has a projection in its top face, above the electrode finger of electrode  25 . The height of the projection is larger than 0.03λ and is equal to or less than the height of electrode  25 , and the width of a top part of the projection is smaller than that of the electrode finger of electrode  25 . 
     As shown in  FIGS. 6 and 7 , when the thickness of protective film  24  made of silicon oxide is set to, for example, 0.35λ in order to improve the temperature characteristic of group  38  of the other resonators, unnecessary spurious signals  26  and  27  are generated around 1.2 times of resonance frequency. It is considered that the unnecessary spurious signals are generated due to a fast transverse wave generated in group  38  of the other resonators. A transverse wave having the highest sound velocity in transverse waves generated in group  38  of the other resonators will be described as a fast transverse wave, and a transverse wave having the lowest sound velocity in the transverse waves generated in group  38  of the other resonators will be described as a slow transverse wave. 
     Next, characteristic changes when, in first filter  3 , piezoelectric body  23  is made of lithium niobate based body having the Euler angles (0°, −87.5°, 0°), electrode  25  is made of copper having a thickness of 0.03λ, protective film  24  is made of silicon oxide and has a flat top face, and the thickness of protective film  24  is changed will be described. 
       FIG. 8A  is a diagram showing an electromechanical coupling coefficient (k2) of a fast transverse wave when thickness of protective film  24  is changed in first filter  3  in the antenna duplexer according to the first embodiment of the invention.  FIG. 8B  is a diagram showing Q factor (Qs) of resonance of a fast transverse wave.  FIG. 8C  is a diagram showing Q factor (Qa) of antiresonance of a fast transverse wave. 
     As shown in  FIG. 8B , when the thickness of protective film  24  is set to be larger than 0.27λ (expressing 27% of wavelength λ, this expression will be similarly used in the following description), the Q factor of resonance of the fast transverse wave increases. As shown in  FIG. 8C , when the thickness of protective film  24  is set to be larger than 0.34λ, the Q factor of antiresonance of the fast transverse wave also increases. 
     Next, characteristic changes when, in first filter  3 , piezoelectric body  23  is made of lithium niobate based body having the Euler angles (0°, −90°, 0°), electrode  25  is made of aluminum having a thickness of 0.08λ, protective film  24  is made of silicon oxide and has the above-described projection in its top face above the electrode finger of electrode  25 , and the thickness of protective film  24  is changed will be described. 
       FIG. 9A  is a diagram showing the electromechanical coupling coefficient (k2) of a fast transverse wave when thickness of protective film  24  is changed in first filter  3  in antenna duplexer  1  according to the first embodiment of the invention.  FIG. 9B  is a diagram showing Q factor (Qs) of resonance of a fast transverse wave.  FIG. 9C  is a diagram showing Q factor (Qa) of antiresonance of a fast transverse wave. 
     As shown in  FIG. 9B , when the thickness of protective film  24  is set to be larger than 0.2λ, the Q factor of resonance of the fast transverse wave increases. As shown in  FIG. 9C , when the thickness of protective film  24  is set to be larger than 0.27λ, the Q factor of antiresonance of the fast transverse wave also increases. 
     Conventionally, there is a problem such that an unnecessary spurious signal is generated due to the fast transverse wave and the characteristic quality of a filter to which a resonator is applied or an antenna duplexer deteriorates. 
     An example of changing φ and ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in order to suppress the unnecessary spurious signal due to the fast transverse wave will now be described. 
       FIGS. 10A to 10G  are diagrams showing the admittance characteristic in the case of changing φ in first filter  3  in antenna duplexer  1  according to the first embodiment of the invention.  FIGS. 11A to 11G  are diagrams showing the admittance characteristic in the case of changing ψ. In  FIGS. 10A to 10G  and  FIGS. 11A to 11G , the admittance characteristic in the case of 1e +02  or larger and that in the case of 1e −02  or less are not shown. 
       FIGS. 10A to 10G  and  FIGS. 11A to 11G  are characteristic diagrams showing the case where piezoelectric body  23  is made of lithium niobate based body having the Euler angles (0°, −90°, 0°), electrode  25  is made of aluminum having a thickness of 0.08λ, and protective film  24  is made of silicon oxide having a thickness of 0.35λ and has the above-described projection in its top face above the electrode finger of electrode  25 . 
     In an upper part of each of  FIGS. 10A to 10G  and  FIGS. 11A to 11G , the Euler angles (φ, θ, ψ) of piezoelectric body  23  are shown. For example, also in the case of changing φ as shown in  FIGS. 10A to 10G  and also in the case of changing ψ as shown in  FIGS. 11A to 11G , the unnecessary spurious signal can be suppressed. However, in those cases, a different unnecessary spurious signal is generated in a frequency band slightly lower than the resonance frequency. It is considered that the unnecessary spurious signal is caused by the Rayleigh wave. 
     Consequently, an examination was made to suppress generation of the unnecessary spurious signal due to the Rayleigh wave and also suppress the unnecessary spurious signal due to the fast transverse wave in the case where the thickness of protective film  24  in first filter  3  is larger than 0.27λ. 
     Concretely, first filter  3  in antenna duplexer  1  has a configuration including piezoelectric body  23  made of a lithium niobate based body material having Euler angles (φ, θ, ψ), electrode  25  provided on piezoelectric body  23  and exciting a main elastic wave having wavelength λ, and protective film  24  provided on piezoelectric body  23  so as to cover electrode  25  and thicker than 0.27λ. It is constructed so that the Euler angles of piezoelectric body  23  satisfy −100°≦θ≦−60°, 1.193φ−2°≦ψ≦1.193φ+2°, ψ≦−2φ−3°, and −2φ+3°≦ψ. 
     As described above, φ and ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  are set to predetermined angles or larger and changed so as to follow the relation of ψ=1.193φ to a certain degree. By the operation, while suppressing generation of unnecessary spurious signals caused by the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be also suppressed. 
     Preferably, the upper limit of thickness of protective film  24  is set to 0.5λ so that the electromechanical coupling coefficient of the fast transverse wave becomes a predetermined level. 
       FIG. 12A  is a characteristic diagram of group  38  of the other resonators in the case where, in first filter  3  in antenna duplexer  1  according to the first embodiment of the invention, piezoelectric body  23  is made of a lithium niobate based body having Euler angles (7°, −87.5°, 8.4°), electrode  25  is made of copper having thickness of 0.03λ, and protective film  24  is made of silicon oxide, whose top face is flat, and having a thickness of 0.35λ.  FIG. 12B  is a characteristic diagram showing the case where piezoelectric body  23  is made of a lithium niobate based body having Euler angles (9°, −87.5°, 10.7°). 
     As shown in  FIGS. 12A and 12B , in group  38  of the other resonators in first filter  3  having such a configuration, while suppressing generation of unnecessary spurious signals due to the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be also suppressed. 
       FIG. 13  is a diagram showing a desirable range of φ and ψ in the Euler angles (φ, θ, ψ) of first filter  3  in antenna duplexer  1  according to the first embodiment of the invention. 
     In  FIG. 13 , desirable ranges of φ and ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  made of a lithium niobate based body are shown by oblique lines. In this example, −100°≦θ≦−60°, the thickness of protective film  24  is set larger than 0.27λ, and electrode  25  is made of copper having normalized thickness of 0.03λ. 
     The straight line of ψ= 1 . 193 φ shown in  FIG. 13  shows the relation of φ and ψ in the case where the unnecessary spurious signal caused by the Rayleigh wave is suppressed most. By setting the range of ψ to be within ±2 degrees around the line as a center, that is, the range of 1.193φ−2°≦ψ≦1.193φ+2°, the unnecessary spurious signal caused by the Rayleigh wave can be suppressed. 
       FIG. 14  is a diagram showing Q factor of Rayleigh wave of group  38  of the other resonators in the case where ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  is changed around ψ=1.193φ, in first filter  3  in antenna duplexer  1  according to the first embodiment of the invention. The vertical axis indicates the Q factor (Qs) of the Rayleigh wave, and the horizontal axis indicates a change amount Δψ from the relation of ψ=1.193φ. As shown in  FIG. 14 , in the range of ±2 degrees of ψ=1.193φ in the Euler angles (φ, θ, ψ) of piezoelectric body  23 , the Q factor of the Rayleigh wave of group  38  of the other resonators can be suppressed to a predetermined level or less. 
     Referring again to  FIG. 13 , the straight line satisfying the relation of ψ=−2φ shows the relation of φ and ψ in the case where the largest unnecessary spurious signal caused by the fast transverse is generated. By setting the range of ψ to be equal to or larger than ±3 degrees around the line as a center, that is, the ranges of ψ≦−2φ−3° and −2φ+3°≦ψ, the unnecessary spurious signal caused by the fast transverse wave can be suppressed. 
       FIG. 15  is a diagram showing the Q factor of the fast transverse wave of group  38  of the other resonators in the case where ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  of first filter  3  is changed around the relation of ψ=−2φ, in antenna duplexer  1  according to the first embodiment of the invention. In  FIG. 15 , ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  is changed around the relation of ψ=−2φ, and Q factors of the fast transverse wave when φ=0°, 0.5°, 1°, 1.5°, 2°, and 2.5°. 
     As shown in  FIG. 15 , in the range of ±3 degrees or larger from ψ=−2φ+3° in the Euler angles (φ, θ, ψ) of piezoelectric body  23 , the Q factor of the fast transverse wave of group  38  of the other resonators can be suppressed to a predetermined level or less. 
       FIG. 16  is a diagram showing electromechanical coupling coefficient (k2) of the Rayleigh wave of group  38  of the other resonators in the case where θ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in first filter  3  is changed, in antenna duplexer  1  according to the first embodiment of the invention. As shown in  FIG. 16 , to suppress the electromechanical coupling coefficient of the Rayleigh wave to 0.01 or less, θ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  has to satisfy the relation of −100°≦θ≦−60°. 
       FIG. 17  is a diagram showing the normalized coupling coefficient of SH wave of group  38  of the other resonators in the case where θ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in first filter  3  is changed, in antenna duplexer  1  according to the first embodiment of the invention. The normalized coupling coefficient is a value obtained by normalizing the value of the electromechanical coupling coefficient with the value of the electromechanical coupling coefficient of the case of θ=−90°. As shown in  FIG. 17 , when θ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  is in the range of −110°≦θ≦−60° including the range of −100°≦θ−60°, the electromechanical coupling coefficient (k2) of the SH wave is equal to or larger than a predetermined value. 
       FIG. 18  is a diagram showing the electromechanical coupling coefficient (k2) of the Rayleigh wave of group  38  of the other resonators in the case where φ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in first filter  3  is changed according to the relation of ψ=1.193φ, in antenna duplexer  1  according to the first embodiment of the invention. As shown in  FIG. 18 , in the range of φ≦20°, the electromechanical coupling coefficient of the Rayleigh wave can be suppressed to about 0.002 or less which is lower than the above-described value 0.01. Also in the case of changing the Euler angles of piezoelectric body  23  in the negative direction with respect to φ, a similar result can be obtained. That is, under the above-described condition, preferably, φ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in first filter  3  satisfies the relation of |φ|≦20°. Consequently, the electromechanical coefficient of the Rayleigh wave can be further suppressed. 
       FIG. 19  is a diagram showing the normalized coupling coefficient of the SH wave of group  38  of the other resonators in the case where φ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  in first filter  3  is changed according to the relation of ψ=1.193φ, in antenna duplexer  1  according to the first embodiment of the invention.  FIG. 19  shows the case where the Euler angles of piezoelectric body  23  with respect to φ are turned in the positive direction. Also in the case of turning the Euler angles of piezoelectric body  23  with respect to φ in the negative direction, a similar result can be obtained. As shown in  FIG. 19 , from the viewpoint of the SH wave as the main elastic wave, when φ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  satisfies the relation of |φ|≦20°, the electromechanical coupling coefficient of the SH wave of the predetermined value or larger can be obtained. 
     In the embodiment, the case of using lithium niobate based body having the Euler angles in the above-described predetermined range as piezoelectric body  23  has been described. The antenna duplexer of the present invention is not limited to the example. For example, lithium niobate based body or lithium tantalate having the Euler angles out of the specific range can be also used. 
     The above-described main elastic wave can be applied as both a surface acoustic wave and a boundary acoustic wave which propagates in the surface of piezoelectric body  23 . For example, when the thickness of protective film  24  is set to λ or larger, the main elastic wave becomes the boundary acoustic wave. 
     The embodiment has been described using the example of making the antiresonance frequency of second series resonator  7  lower than that of first, third, and fourth series resonators  6 ,  8 , and  9 . The present invention, however, is not limited to the example. For example, the antiresonance frequency of first series resonator  6  disposed closest to input terminal  5  may be lower than that of second, third, and fourth series resonators  7 ,  8 , and  9 . With the configuration, the width of the electrode finger of first series resonator  6  closest to the side of input terminal  5  can be widened most. Therefore, antenna duplexer having excellent power durability can be provided. 
     Second Exemplary Embodiment 
     A second exemplary embodiment of the present invention will now be described. 
       FIG. 20  is a schematic circuit diagram of antenna duplexer  51  in the second exemplary embodiment of the invention. 
       FIG. 21  is a schematic cross section of first filter  53  in antenna duplexer  51  in the second exemplary embodiment of the invention. Antenna duplexer  51  in the embodiment has, like antenna duplexer  1  shown in  FIG. 1 , first filter  53  and second filter  4 . Unless otherwise specially described, the configuration of first filter  53  is similar to that of first filter  3  of the first embodiment. 
     First filter  53  in the embodiment has piezoelectric body  23  made of a lithium niobate based body having the Euler angles (φ, θ, ψ), electrode  25  provided on piezoelectric body  23  and exciting the main elastic wave having wavelength λ, and protective film  24  formed on piezoelectric body  23  so as to cover electrode  25  and having thickness larger than 0.2λ. 
     Protective film  24  has projection  28  above the electrode finger of electrode  25  in a section in the direction orthogonal to the extension direction of the electrode finger of electrode  25 . The width of top part  29  of projection  28  is smaller than that of the electrode finger of electrode  25 . 
     The Euler angles of piezoelectric body  23  satisfy the relations of −100°≦θ≦−60°, 1.193φ−2°≦ψ≦1.193φ+2°, ψ≦−2φ−3°, and −2φ+3°≦ψ. 
     In the case where protective film  24  has projection  28  like the configuration shown in  FIG. 21 , the unnecessary spurious signal caused by the fast transverse wave becomes an issue. The case of setting the thickness of protective film  24  made of, for example, silicon oxide to be larger than 0.2λ in order to improve the frequency temperature characteristic of first filter  53  is assumed. In this case as well, when φ and ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  23  are set to predetermined angle or larger and changed so as to follow the relation of ψ=1.193φ to a certain degree, while suppressing generation of unnecessary spurious signals caused by the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be also suppressed. 
       FIG. 22A  is a characteristic diagram of group  38  of the other resonators in the case where piezoelectric body  23  of first filter  53  in the antenna duplexer according to the second embodiment of the invention is made of lithium niobate based body having Euler angles (7°, −87.5°, 8.4°), electrode  25  is made of aluminum having a thickness of 0.08λ, and protective film  24  is made of silicon oxide having a film thickness of 0.35λ and has a projection having a height T=0.08λ on its top face.  FIG. 22B  is a characteristic diagram in the case where piezoelectric body  23  is lithium niobate based body having Euler angles (9°, −87.5°, 10.7°). 
     As shown in  FIGS. 22A and 22B , in first filter  53  of the embodiment, while suppressing generation of unnecessary spurious signals caused by the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be also suppressed. 
     Preferably, projection  28  in protective film  24  has a curved shape which is projected downward from top part  29  of projection  28  to lowest part  30 . In this case, width L of top part  29 , defined by the distance between points at which the downwardly projected curved lines or extension lines of the curved lines and a straight line parallel to the top face of piezoelectric body  23  including top part  29  cross each other, is set to be smaller than the width of the electrode finger of electrode  25 . Consequently, mass addition of protective film  24  in projection  28  changes continuously and gently. As a result, while suppressing generation of unnecessary reflection caused by the shape of protective film  24 , electric characteristics of first filter  53  can be improved. 
     Preferably, the width of top part  29  of projection  28  is set to be equal to or less than ½ of the width of the electrode finger of electrode  25 . Preferably, a center position of top part  29  substantially corresponds to a position above the center position of the electrode finger. With the configuration, reflectance in the electrode finger is further increased by the mass addition effect, and the electric characteristics of first filter  53  can be improved. 
     Further, when a height of projection  28  is T and a film thickness of electrode  25  is h, preferably, the relation of 0.03λ&lt;T≦h is satisfied. In an examination of the relation between height T from lowest part  30  to top part  29  of projection  28  in protective film  24  and the electric characteristics, when T is set to be larger than 0.03λ, improvement in reflectance is larger than that in the case where the surface of protective film  24  is flat. On the other hand, when height T is set larger than film thickness h of electrode  25 , at the time of manufacturing (which will be described later), a new step to generate protective film  24  has to be added, and a manufacturing method becomes complicated. Consequently, it is desirable to set height T to be equal to or less than film thickness h. 
     A method of manufacturing first filter  53  in the second embodiment of the invention will now be described. 
       FIGS. 23A to 23H  are diagrams showing an example of a method of manufacturing first filter  53  in the antenna duplexer according to the second embodiment of the invention. 
     First, as shown in  FIG. 23A , on the top face of piezoelectric body  31 , electrode film  32  which becomes at least one of an electrode and a reflector is formed by Al or Al alloy by a method such as deposition or sputtering. 
     As shown in  FIG. 23B , resist film  33  is formed on the top face of electrode film  32 . 
     Further, as shown in  FIG. 23C , resist film  33  is processed into a desired shape by using the exposure and development technique or the like. 
     As shown in  FIG. 23D , electrode film  32  is processed in a desired shape such as an IDT electrode, a reflector, or the like by using the dry etching technique or the like. After that, resist film  33  is removed. 
     Next, as shown in  FIG. 23E , protective film  34  is formed by a method such as depositing or sputtering silicon oxide so as to cover electrode film  32 . As a method of obtaining projection  28  in protective film  34 , so-called bias sputtering of forming a film by sputtering while applying bias to the side of piezoelectric body  31  can be used. 
     By sputtering the target of silicon oxide, protective film  34  is deposited on piezoelectric body  31  and, at the same time, by applying bias, a part of protective film  34  on piezoelectric body  31  is sputtered. That is, by etching a part of protective film  34  while depositing protective film  34 , the shape of protective film  34  can be controlled. As means for controlling the shape of protective film  34 , the ratio between the bias and sputtering power applied to piezoelectric body  31  may be changed during deposition of protective film  34  or a film is formed without applying bias to piezoelectric body  31  initially and, from a certain time, bias may be applied simultaneously with film deposition. At this time, temperature of piezoelectric body  31  is also controlled. 
     As shown in  FIG. 23F , resist film  35  is formed on the surface of protective film  34 . 
     As shown in  FIG. 23G , resist film  35  is processed in a desired shape by using the exposure/development technique or the like. 
     As shown in  FIG. 23H , by using the dry etching technique or the like, protective film  34  in unnecessary parts such as a part of pad  36  for taking an electric signal are removed. After that, resist film  35  is removed. 
     By dividing the resultant by dicing, antenna duplexer  51  having first filter  53  can be obtained. 
     By forming protective film  34  under proper film forming conditions by using the bias sputtering method as described above, a desired shape can be obtained. 
     The characteristics of the group of the other resonators of first filter  53  in the embodiment are similar to those of group  38  of the other resonators of first filter  3  in the first embodiment shown in  FIGS. 13 to 19 . For example, the case where thickness of protective film  34  made of, for example, silicon oxide is set to be larger than 0.2λ to improve the frequency temperature characteristics of first filter  53  is assumed. In this case, when φ and ψ in the Euler angles (φ, θ, ψ) of piezoelectric body  31  are set to predetermined angle or larger and changed so as to follow the relation of ψ=1.193φ to a certain degree, while suppressing generation of unnecessary spurious signals caused by the Rayleigh wave, unnecessary spurious signals around a frequency band in which a fast transverse wave is generated can be suppressed. 
     Third Exemplary Embodiment 
     Next, a third exemplary embodiment of the present invention will be described. 
       FIG. 24  is a schematic circuit diagram of antenna duplexer  61  in the third embodiment of the present invention. 
     In the embodiment, the configuration of first filters  3  and  53  described in the first and second exemplary embodiments is applied to second filter  54  of antenna duplexer  61 . 
     Concretely, the resonance frequency of fourth parallel resonator  19  in second filter  54  is set to be higher than that of the other parallel resonators, concretely, fifth, sixth, and seventh parallel resonators  20 ,  21 , and  22 . The electromechanical coupling coefficient of fourth parallel resonator  19  is set to be smaller than that of fifth, sixth, and seventh parallel resonators  20 ,  21 , and  22 . Consequently, the propagation angle ψ of the main elastic wave of fourth parallel resonator  19  is made different from that of the main elastic wave of fifth, sixth, and seventh parallel resonators  20 ,  21 , and  22 . 
     The configuration of piezoelectric body  23 , protective film  24 , electrode  25 , and the like of second filter  54  in antenna duplexer  61  in the third embodiment of the invention are, unless otherwise specially described, similar to that of first filters  3  and  53  in antenna duplexers  1  and  51  in the first and second embodiments. 
     In the embodiment, an example of changing the propagation angle w in the Euler angles of piezoelectric body  23  in fourth parallel resonator  19  of second filter  54  will be described. However, the invention is not limited to the example. The propagation angle of not only fourth parallel resonator  19  but also the other resonators may be changed. For example, the electromechanical coupling coefficient of fourth and fifth parallel resonators  19  and  20  having the highest resonance frequency among the parallel resonators may be set smaller than that of the other parallel resonators, that is, sixth and seventh parallel resonators  21  and  22 . In this case, it is sufficient to make the propagation angle w of the main elastic wave of fourth and fifth parallel resonators  19  and  20  different from that of the main elastic wave of sixth and seventh parallel resonators  21  and  22 . 
     In reception filter  54  in antenna duplexer  61  of the embodiment, by making the propagation angle w of fourth parallel resonator  19  larger than that of fifth, sixth, and seventh parallel resonators  20 ,  21 , and  22 , the slope characteristic on the high frequency side of the passband of second filter  54  can be made steep. 
     As described above, in a manner similar to the description of the first embodiment, second filter  54  in antenna duplexer  61  has protective film  24  formed on piezoelectric body  23  so as to cover electrode  25 . 
     With the configuration, by changing the propagation angle w (propagation direction) of the main elastic wave in the resonator, the electromechanical coupling coefficient of the main elastic wave of the resonator can be largely controlled. 
     In second filter  54  in antenna duplexer  61  according to the embodiment, by setting the electromechanical coupling coefficient of fourth parallel resonator  19  having the highest resonance frequency which exerts large influence on steepness to be smaller than that of the other parallel resonators, steepness in the crossband can be improved. 
     By setting the electromechanical coupling coefficient of the other parallel resonators having relatively high resonance frequency which exerts small influence on steepness, that is, fifth, sixth, and seventh parallel resonators  20 ,  21 , and  22  to be larger than that of fourth parallel resonator  19 , the passband width can be widened, and a loss in a wider transmission pass band can be suppressed. That is, antenna duplexer  61  of the embodiment can satisfy both steepness in the crossband and lower loss in the transmission passband. 
     In addition, in comparison of the electromechanical coupling coefficient between a parallel resonator other than fourth parallel resonator  19  having the highest resonance frequency among the plurality of parallel resonators, for example, fifth parallel resonator  20  and a parallel resonator whose resonance frequency is lower than that of fifth parallel resonator  20 , the electromechanical coupling coefficient of fifth parallel resonator  20  may be set to be smaller than that of sixth parallel resonator  21 . That is, the propagation angle ψ of the main elastic wave of fifth parallel resonator  20  may be made different from that of the main elastic wave of sixth parallel resonator  21 . Consequently, both steepness in the crossband and lower loss in the reception passband can be satisfied. 
     By making the configuration of the transmission filter in antenna duplexer  61  of the embodiment similar to that of transmission filters  3  and  53  described in the first and second embodiments, both steepness in the crossband and lower loss in the transmission/reception passbands can be satisfied. Antenna duplexer  61  of the embodiment, however, is not limited to the example and can use any of various known ladder filters as the transmission filter. 
     INDUSTRIAL APPLICABILITY 
     As described above, the antenna duplexer of the present invention has an effect that both steepness in the crossband and lower loss in the passband can be satisfied, and can be applied to an electronic device such as a cellular phone. 
     REFERENCE MARKS IN THE DRAWINGS 
     
         
           1 ,  51 ,  61  antenna duplexer 
           2  antenna terminal 
           3 ,  53  first filter 
           4 ,  54  second filter 
           5  input terminal 
           6  first series resonator 
           7  second series resonator 
           8  third series resonator 
           9  fourth series resonator 
           10  phase shifter 
           11  first parallel resonator 
           12  second parallel resonator 
           13  third parallel resonator 
           14  fifth series resonator 
           15  sixth series resonator 
           16  seventh series resonator 
           17  eighth series resonator 
           18  output terminal 
           19  fourth parallel resonator 
           20  fifth parallel resonator 
           21  sixth parallel resonator 
           22  seventh parallel resonator 
           23 ,  31  piezoelectric body 
           24 ,  34  protective film 
           25  electrode 
           28  projection 
           29  top part 
           30  lowest part 
           32  electrode film 
           33 ,  35  resist film 
           36  pad 
           38  group of other resonators