Patent Publication Number: US-11036910-B2

Title: Element removal design in microwave filters

Description:
RELATED APPLICATION DATA 
     The present application claims the benefit of and priority as a continuation to U.S. application Ser. No. 16/443,466, filed Jun. 17, 2019; which is a continuation of U.S. application Ser. No. 14/300,124, filed Jun. 9, 2014; which is a continuation of U.S. application Ser. No. 13/939,763, filed Jul. 11, 2013; which claims the benefit under 35 U.S.C. § 119 to U.S. provisional patent application Ser. No. 61/802,114, filed Mar. 15, 2013, which applications are all incorporated herein by reference in their entirety. 
    
    
     FIELD OF THE INVENTION 
     The present inventions generally relate to microwave filters, and more particularly, to acoustic wave microwave filters designed for narrow-band applications. 
     BACKGROUND OF THE INVENTION 
     Electrical filters have long been used in the processing of electrical signals. In particular, such electrical filters are used to select desired electrical signal frequencies from an input signal by passing the desired signal frequencies, while blocking or attenuating other undesirable electrical signal frequencies. Filters may be classified in some general categories that include low-pass filters, high-pass filters, band-pass filters, and band-stop filters, indicative of the type of frequencies that are selectively passed by the filter. Further, filters can be classified by type, such as Butterworth, Chebyshev, Inverse Chebyshev, and Elliptic, indicative of the type of bandshape frequency response (frequency cutoff characteristics). 
     The type of filter used often depends upon the intended use. In communications applications, band pass and band stop filters are conventionally used in cellular base stations, cell phone handsets, and other telecommunications equipment to filter out or block RF signals in all but one or more predefined bands. Of most particular importance is the frequency range from approximately 500-3,500 MHz. 
     With reference to  FIG. 1  a prior art telecommunications system  10  may include a transceiver  12  capable of transmitting and receiving wireless signals, and a controller/processor  14  capable of controlling the functions of the transceiver  12 . 
     The transceiver  12  generally comprises a broadband antenna  16 , a duplexer  18  having a transmit filter  24  and a receive filter  26 , a transmitter  20  coupled to the antenna  16  via the transmit filter  24  of the duplexer  18 , and a receiver  22  coupled to the antenna  16  via the receive filter  26  of the duplexer  18 . 
     The transmitter  20  includes an upconverter  28  configured for converting a baseband signal provided by the controller/processor  14  to a radio frequency (RF) signal, a variable gain amplifier (VGA)  30  configured for amplifying the RF signal, a bandpass filter  32  configured for outputting the RF signal at an operating frequency selected by the controller/processor  14 , and a power amplifier  34  configured for amplifying the filtered RF signal, which is then provided to the antenna  16  via the transmit filter  24  of the duplexer  18 . 
     The receiver  22  includes a notch or stopband filter  36  configured for rejecting transmit signal interference from the RF signal input from the antenna  16  via the receiver filter  26 , a low noise amplifier (LNA)  38  configured for amplifying the RF signal from the stop band filter  36  with a relatively low noise, a tunable bandpass filter  40  configured for outputting the amplified RF signal at a frequency selected by the controller/processor  14 , and a downconverter  42  configured for downconverting the RF signal to a baseband signal that is provided to the controller/processor  14 . Alternatively, the function of rejecting transmit signal interference performed by the stop-band filter  36  can instead be performed by the duplexer  18 . Or, the power amplifier  34  of the transmitter  20  can be designed to reduce the transmit signal interference. 
     It should be appreciated that the block diagram illustrated in  FIG. 1  is functional in a nature, and that several functions can be performed by one electronic component or one function can be performed by several electronic components. For example, the functions performed by the up converter  28 , VGA  30 , bandpass filter  40 , downconverter  42 , and controller/processor  14  are oftentimes performed by a single transceiver chip. The function of the bandpass filter  32  can be into the power amplifier  34  and the transmit filter  24  of the duplexer  18 . 
     Microwave filters are generally built using two circuit building blocks: a plurality of resonators, which store energy very efficiently at a resonant frequency (which may be a fundamental resonant frequency f 0  or any one of a variety of higher order resonant frequencies f 1 -f n ); and couplings, which couple electromagnetic energy between the resonators to form multiple reflection zeros providing a broader spectral response. For example, a four-resonator filter may include four reflection zeros. For the purposes of this specification, a reflection zero may refer to the roots of a filter&#39;s reflection function where the inductance and capacitance cancel and a minimum amount of power is reflected. The strength of a given coupling is determined by its reactance (i.e., inductance and/or capacitance). The relative strengths of the couplings determine the filter shape, and the topology of the couplings determines whether the filter performs a band-pass or a band-stop function. The resonant frequency f 0  is largely determined by the inductance and capacitance of the respective resonator. For conventional filter designs, the frequency at which the filter is active is determined by the resonant frequencies of the resonators that make up the filter. Each resonator must have very low internal resistance to enable the response of the filter to be sharp and highly selective for the reasons discussed above. This requirement for low resistance tends to drive the size and cost of the resonators for a given technology. 
     The front-end receive filter (e.g., the receive filter  26  illustrated in  FIG. 1 ) preferably takes the form of a sharply defined band-pass filter to eliminate various adverse effects resulting from strong interfering signals at frequencies near the desired received signal frequency. Because of the location of the front-end receiver filter at the antenna input, the insertion loss must be very low so as to not degrade the noise figure. In most filter technologies, achieving a low insertion loss requires a corresponding compromise in filter steepness or selectivity. In practice, most filters for cell phone handsets are constructed using acoustic resonator technology, such as surface acoustic wave (SAW), bulk acoustic wave (BAW), and film bulk acoustic resonator (FBAR) technologies. Such acoustic resonator filters have the advantages of low insertion loss, compact size, and low cost compared to equivalent inductor/capacitor resonators. 
     Design of practical microwave filters may begin with the design of an initial circuit generated, for instance, using image design or network synthesis design. These approaches generally, from the outset, only consider circuits with the fewest possible number of circuit elements. This is generally performed from a desire to minimize losses in the final filter, and may be a common practice in microwave filter design of all types. The initial design may be generated using simplified, idealized circuit element models, which may typically ignore losses and other unwanted characteristics of the physical circuit elements used to make the final filter. Computer optimization may be a critical and necessary step in the design of practical microwave filters. Design tools including Agilent Advanced Design System (ADS), among others, may use numerical optimization methods, such as Monte Carlo, gradient, etc., to improve the “initial circuit design.” This computer optimization step may use increasingly realistic, accurate circuit element models and may restrict circuit element values to those that can be manufactured in accordance with the final filter design. The optimization may search for the combination of circuit element values that best matches the desired filter response. This type of computer optimization may be often used in microwave filter design. Although the optimization may generally produce a significantly improved design that may be realized with physical circuit elements, it generally does not reduce the number of circuit elements in the final circuit design from the number of circuit elements in the initial circuit design, nor does it change one type of circuit element into another. 
     For example, one initial circuit that is typically used in the design of acoustic wave band-pass filters is a ladder filter  50 , which comprises a number of alternating shunt resonators  52   a  and series resonators  52   b , as illustrated in  FIG. 2 . The filter  50  can be considered an N resonator ladder topology (i.e., N equals the number of resonators, and in this case 6). For the purposes of this specification, an acoustic ladder filter may refer to one or more filters using the Mason-type acoustic wave ladder circuit structure comprising alternating series and shunt acoustic wave resonators. 
     Each of the acoustic resonators  52  may be described by a modified Butterworth-Van Dyke (MBVD) model  54 . MBVD models  54  may also describe SAW resonators, which may be fabricated by disposing interdigital transducers (IDTs) on a piezoelectric substrate, such as crystalline Quartz, Lithium Niobate (LiNbO 3 ), Lithium Tantalate (LiTaO 3 ) crystals or BAW resonators. Each MBVD model  54  includes a motional capacitance C m    56 , a static capacitance C 0    58 , a motional inductance L m    60 , and a resistance R  62 . The motional capacitance C m    56  and motional inductance L m    60  may result from the interactions of electrical and acoustical behavior, and thus, may be referred to as the motional arm of the MBVD model  54 . The static capacitance C 0    58  may result from the inherent capacitance of the structure, and thus, may be referred to as the static (non-motional) capacitance of the MBVD model  54 . The resistance R  62  may result from the electrical resistance of the acoustic resonator  52 . The parameters are related by the following equations: 
                       ω   E     =     1         L   m     ⁢     C   m             ;           [   1   ]                     ω   A       ω   R       =       1   +     1   γ           ,           [   2   ]               
where ω R  and ω A  may be the respective resonance and anti-resonance frequencies for any given acoustic resonator, and gamma γ may depend on a material&#39;s property, which may be further defined by:
 
                       C   0       C   m       =     γ   .             [   3   ]               
Typical γ values may range from about 12 to about 18 for 42-degree X Y cut LiTaO 3 .
 
     It can be appreciated from equation [1] that the resonant frequency of each of the acoustic resonators  52  will depend on the motional arm of the MBVD model  54 , whereas the filter characteristics (e.g., bandwidth) will be determined by γ in equation [2]. The Quality factor (Q) for an acoustic resonator  52  may be an important figure of merit in acoustic filter design, relating to the loss of the element within the filter. Q of a circuit element represents the ratio of the energy stored per cycle to the energy dissipated per cycle. The Q factor models the real loss in each acoustic resonator  52 , and generally more than one Q factor may be required to describe the loss in an acoustic resonator  52 . Q factors may be defined as follows for the filter examples. The motional capacitance C m    56  may have an associated Q defined as QC m =1.0E+8; the static capacitance C 0    58  may have an associated Q defined as QC 0 =200; and motional inductance L m    60  may have an associated Q defined as QL m =1000. Circuit designers may typically characterize SAW resonators by resonant frequency ω R , static capacitance C 0 , gamma γ, and Quality factor QL m . For commercial applications, QL m  may be about 1000 for SAW resonators, and about 3000 for BAW resonators. 
     The frequency response for the filter  50  is illustrated in  FIG. 3  which presents the scattering matrices |S21| 2  (insertion loss) and |S11| 2  (return loss) for the filter response in logarithmic units of decibels (dB) versus frequency f. Let the resonance and anti-resonance frequencies of each of the shunt resonators  52   a  be respectively designated as ω rp  and ω ap , and the resonance and anti-resonance frequencies of each of the series resonators  52   b  be respectively designated as ω rs  and ω as . When ω rs  and ω ap  are approximately equal to each other, a passband centered near ω=ω rs , ω ap  is created, and transmission zeroes at ω=ω rp , ω as  defining the passband edges are created, as shown in the filter response illustrated in  FIG. 3 . For the purposes of this specification, a transmission zero may refer to the roots of a filter&#39;s transmission function where a maximum amount of power is reflected. At frequencies f far from the passband center frequency ω p  the resonators act approximately as capacitors, resulting in a |S21| 2  filter response that forms wings that becomes asymptotically constant for large |ω-ω p |, providing the out-of-band rejection. 
     A band pass filter response may be characterized by the return loss (i.e., the value of the |S11| 2  at the center passband frequency ω p ), insertion loss (i.e., the value of |S21| 2  at the center passband frequency ω p , the passband width (PBW), and the out-of-band rejection ε (i.e., 1/|S21| at a large |ω-ω p |). Band pass ladder filters can be designed only over a limited accessible range of these parameters, with the range depending on the material parameter value γ and the number of resonators (termed the filter order). The material parameter values γ for currently widely used materials for SAW and BAW filters are in the range of 12-14, allowing the resonance frequency and antiresonance frequency to be close to the passband center frequency ω p , thereby creating a relatively narrow passband in the |S21| 2  filter response. Materials with a material parameter value γ of 4 are currently under development. A smaller material parameter value γ would enable a wider passband width PBW, decrease return loss RL, or improve the out-of-band rejection ε. 
     For a fixed passband width PBW, as the out-of-band rejection ε increases, the return loss RL decreases. In some cases, passive circuit elements are coupled to the ladder structure to improve filter performance. For example, adding inductors can decrease the effective material parameter value γ, which can increase the passband width PBW, decrease the return loss RL, or improve the out-of-band rejection ε. However, the benefits from the addition of inductors come at the cost of increased insertion loss, size, and cost. The band pass filter parameters are pushed to the limits of the accessible range in order to maximize performance, with tradeoffs between the parameters depending on the system applications and requirements. Higher order filters can achieve greater out-of-band rejections at a given passband return loss RL and passband width PBW. 
     As briefly discussed above, the filter  50  may have an initial circuit design, which may then be optimized via a suitable computer optimization technique (e.g., Agilent ADS software) to create a final circuit design. For example, the filter  50  may initially be designed with the resonant frequencies ω R  and static capacitances C 0  for each resonator  52  illustrated in  FIG. 4 a   , which when simulated, results in the frequency response illustrated in  FIG. 4 b   . This frequency response is shown characterized by the following markers: M1 of Mag S21=−65.71 dB at frequency=1.770 GHz; M2 of Mag S21=−36.735 dB at frequency=1.830 GHz; M3 of Mag S21=−4.367 dB at frequency=1.850 GHz; M4 of Mag S21=−1.444 dB at frequency=1.879 GHz; M5 of Mag S21=−2.680 dB at frequency=1.910 GHz; M6 of Mag S21=−30.118 dB at frequency=1.930 GHz; and M7 of Mag S21=−62.874 dB at frequency=1.990 GHz. 
     After optimization, the filter  50  may have the resonant frequencies ω R  and static capacitances C 0  for each resonator  52  illustrated in  FIG. 5 a   , which when simulated, results in the frequency response illustrated in  FIG. 5 b   . This frequency response is shown characterized by the following markers: N1 of Mag S21=−46.943 dB at frequency=1.770 GHz; N2 of Mag S21=−29.865 dB at frequency=1.829 GHz; N3 of Mag S21=−1.479 dB at frequency=1.851 GHz; N4 of Mag S21=−0.833 dB at frequency=1.875 GHz; N5 of Mag S21=−1.898 dB at frequency=1.910 GHz; N6 of Mag S21=−41.977 dB at frequency=1.929 GHz; and N7 of Mag S21=−47.182 dB at frequency=1.990 GHz. 
     As can be appreciated from the foregoing, the values of the MBVD models  54  for the resonators  52  have changed with optimization with improvement in the frequency response. However, the type and number of circuit elements remains unchanged, and thus, does not reduce the footprint or cost of the final circuit. Therefore, for microwave filters generally, and especially filter designs that contain passive elements and/or use more complex design techniques, such as modern network theory or image theory with more complex sections, an improved optimization method is needed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings illustrate the design and utility of preferred embodiments of the present invention, in which similar elements are referred to by common reference numerals. In order to better appreciate how the above-recited and other advantages and objects of the present inventions are obtained, a more particular description of the present inventions briefly described above will be rendered by reference to specific embodiments thereof, which are illustrated in the accompanying drawings. Understanding that these drawings depict only typical embodiments of the invention and are not therefore to be considered limiting of its scope, the invention will be described and explained with additional specificity and detail through the use of the accompanying drawings in which: 
         FIG. 1  is a block diagram of a prior art wireless telecommunications system; 
         FIG. 2  is a schematic diagram of a prior art acoustic ladder filter that can be used in the prior art wireless telecommunications system; 
         FIG. 3  is a frequency response plot of the prior art acoustic ladder filter of  FIG. 2 ; 
         FIG. 4 a    is a schematic diagram of an initial circuit design of the acoustic ladder filter of  FIG. 2  that can be optimized using a conventional filter optimization technique; 
         FIG. 4 b    is a frequency response plot of the initial circuit design of  FIG. 4   a;    
         FIG. 5 a    is a schematic diagram of an optimized final circuit design of the acoustic ladder filter of  FIG. 2  resulting from the optimization of the initial filter circuit design of  FIG. 4 a    using the conventional filter optimization technique; 
         FIG. 5 b    is a frequency response plot of the final circuit design of  FIG. 5   a;    
         FIG. 6  is a flow diagram illustrating an Element Removal Design (ERD) technique used to optimize an acoustic ladder filter in accordance with one method of the present inventions; 
         FIG. 7 a    is a schematic diagram of an initial circuit design of an acoustic ladder filter that can be optimized using the ERD technique of  FIG. 6 ; 
         FIG. 7 b    is a frequency response plot of the initial circuit design of  FIG. 7   a;    
         FIG. 8 a    is a schematic diagram of an optimized final circuit design resulting from the optimization of the initial filter circuit design of  FIG. 7 a    using the ERD technique of  FIG. 6 ; 
         FIG. 8 b    is a frequency response plot of the final circuit design of  FIG. 8 a   ; and 
         FIG. 9  is a block diagram of a computerized filter design system that can implement the computerized steps of the ERD technique of  FIG. 6 . 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     The microwave filter design optimization technique optimizes an acoustic wave (AW) microwave filter (such as surface acoustic wave (SAW), bulk acoustic wave (BAW), and film bulk acoustic wave (FBAR) filters), by changing the circuit element values, changing the circuit element type, and/or discretely removing extra or unnecessary circuit elements. These elements may include inductors, capacitors, and acoustic resonators (modeled, e.g., using the modified Butterworth-Van Dyke (MBVD) model). 
     This optimization technique utilizes several traditional computer optimization methods to enable the improved optimization of more complex circuits in the initial design than may be possible in the prior art. These initial filter circuit designs may be produced using any design method, for instance image design or network synthesis. This optimization technique results in a final filter circuit design with a reduced number elements compared to the initial filter circuit design, while simultaneously improving the frequency response of the filter. For the purposes of this specification, a frequency response improvement may refer to an improvement in desired microwave performance of a filter (e.g., a lower insertion loss, steeper rejection slope, higher out-of-band rejection, lower node voltage, more linear group delay, etc.). Therefore, a smaller footprint, lower cost, lower insertion loss, and greater selectively filter may be achieved, while allowing the use of traditional manufacturing processes and infrastructure. 
     While the microwave filter design optimization technique is described herein with reference to AW filters, it should be appreciated that this technique can be utilized with other types of microwave filters where the resonant elements can be model at some level of accuracy with a MBVD model. The optimization technique may also apply to other technologies with complex resonator elements (e.g., multi-mode dielectric resonators and other similar technologies). 
     Referring now to  FIG. 6 , one Element Removal Design (ERD) technique  100  that can be used to design an acoustic wave filter will be described. During implementation of the ERD technique  100 , a design in process (DIP) is modified from an initial design to an improved final design. The method, in steps, uses computer optimization and decisions to incrementally improve the DIP to reduce circuit element count and improve performance metrics. 
     To this end, the ERD technique  100  first generates an initial filter circuit design based on performance specifications, such as center passband frequency, passband width, return loss, and out-of-band rejection may be generated using and image filter technique or a network synthesis technique (step  102 ). For the purposes of this specification, a circuit element may refer to an inductor, a capacitor, a resonator, switch or a resistor within a network of circuit interconnections. 
     For example, as shown in  FIG. 7 a   , the initial filter circuit design may be a band-pass ladder  200 , which comprises a number of alternating shunt resonant elements  202   a  and  202   b  and series resonant elements  202   c . In the illustrated embodiment, each shunt leg of the ladder filter  200  includes two parallel shunt resonant elements  202   a  and  202   b  that may resonate at different frequencies. Each of the resonant elements  202  can be modeled with the MBVD model  54  illustrated in  FIG. 2 . The ladder filter  200  also comprises a plurality of non-resonant elements  204  in the form of capacitors respectively associated with the series resonant elements  202   c . For the purposes of this specification, a non-resonant element may refer to a passive component in a circuit. Exemplary non-resonant elements may include inductors, capacitors, switches or resistors. Non-resonant elements may have resonances far from the frequency of interest. For instance, an inductor may have a resonance greater than 50% of the passband frequency. Thus, the filter  200  includes nine resonant elements and three non-resonant elements. 
     The filter  200  may initially be designed with the resonant frequencies ω R  and static capacitances C 0  for each resonator  202  illustrated in  FIG. 7 a   , which when simulated, results in the frequency response illustrated in  FIG. 7 b   . This frequency response is shown characterized by the following markers: R1 of Mag S21=24.627 dB at frequency=1.770 GHz; R2 of Mag S21=−64.652 dB at frequency=1.830 GHz; R3 of Mag S21=−3.857 dB at frequency=1.850 GHz; R4 of Mag S21=−0.987 dB at frequency=1.881 GHz; R5 of Mag S21=−3.039 dB at frequency=1.910 GHz; R6 of Mag S21=−87.468 dB at frequency=1.930 GHz; and R7 of Mag S21=−28.429 dB at frequency=1.990 GHz. It is assumed that the circuit elements of the initial filter circuit design  200  have the following parameters: gamma γ=12, QC 0 =200, Qcap=200, QL m =1000, and Rs=0.5 ohms. 
     Next, the ERD technique  100  inputs the DIP (in this case the initial filter design) into a computerized filter optimizer (e.g., Agilient ADS) (step  104 ). For the purposes of this specification, computer optimization of a DIP may refer to improving the frequency response by changing values of the circuit elements in a computer-based circuit simulator. The circuit simulator may use goals in order to compare the simulated response against a desired result. It should be appreciated that the DIP to be computer optimized is the initial circuit design at this point, the ERD technique  100  may be implemented to improve a DIP during any point of a filter design process. At any event, the resulting DIP has the same number of circuit elements as the number of circuit elements in the input DIP. 
     The ERD technique  100  then determines whether any of the non-resonant elements  204  in the DIP has become insignificant (“vanishes”) (step  106 ). For the purposes of this specification, a circuit element tends to vanish if a series element reactance value and/or shunt element susceptance value becomes very small during computer optimization at step  104  and/or the circuit element may be removed from a filter design without a significant impact on filter performance. A non-resonant element  204  may tend to vanish according to its kind and placement within the filter circuit design. 
     For example, if the non-resonant element  204  is a series inductor or a shunt capacitor, it will tend to vanish as its absolute impedance value becomes small; for instance, less than 0.1 nH (inductor) or 0.1 pF (capacitor). In this case, the non-resonant element  204  will be determined to be insignificant if its absolute impedance value is less than the threshold value. In contrast, if the non-resonant element  204  is a shunt inductor or a series capacitor, it will tend to vanish as its absolute impedance value becomes large; for instance, greater than 100 nH (inductor) or 50 pF (capacitor). In this case, the non-resonant element  204  will be determined to be insignificant if its absolute impedance value is greater than a threshold value. 
     As another example, a non-resonant element  204  will tend to vanish as its relative value (impedance or susceptance) becomes small (e.g., less than 10%) compared to other circuit elements of the same type (series or shunt) connected to them. Thus, if the non-resonant element  204  is a series circuit element, it will be determined to be insignificant if a percentage of its absolute value relative to an absolute value of another series non-resonant element  204  in the DIP is less than a threshold value. Similarly, if the non-resonant element  204  is a shunt circuit element, it will be determined to be insignificant if a percentage of its absolute value relative to an absolute value of another shunt non-resonant element  204  in the DIP is less than a threshold value. 
     As still another example, if the non-resonant element  204  is a series circuit element, it will tend to vanish as its impedance is less than a percentage (e.g., 10%) of the impedance seen in either direction from the non-resonant element  204 . In this case, the non-resonant element  204  will be determined to be insignificant if a percentage of the absolute impedance value of the non-resonant element  204  relative to an impedance seen in either direction from the non-resonant element  204  is less than the threshold value. In contrast, if the non-resonant element  204  is a shunt circuit element, it will tend to vanish as its susceptance is less than a percentage (e.g., 10%) of the susceptance seen in either direction from the non-resonant element  204 . In this case, the non-resonant element  204  will be determined to be insignificant if a percentage of the absolute susceptance value of the non-resonant element  204  relative to a susceptance seen in either direction from the non-resonant element  204  is less than the threshold value. 
     As yet another example, a non-resonant element  204  may tend to vanish when removing it results in less than a percentage degradation change (e.g., 10%) in a performance parameter of the filter circuit (e.g., insertion loss, rejection slope, out-of-band rejection, node voltage, group delay flatness, etc.). In this case, the non-resonant element  204  will be determined to be insignificant by removing the non-resonant element  204  from the DIP if the value of the performance parameter without the non-resonant element  204  degrades the value of the performance parameter with the non-resonant element  204  is less than a threshold value. 
     If it is determined that one of the non-resonant elements  204  in the DIP has become insignificant at step  106 , the ERD technique  100  determines whether the sign of the insignificant non-resonant element  204  has been previously changed (i.e., whether the non-resonant element  204  has been transformed from an inductance to a capacitance, or from a capacitance to an inductance) (step  108 ). Of course, in the case where DIP is generated for the first time, there will be no such previous transformation. Notably, the ERD technique  100  makes this inquiry to ensure that removal of the insignificant non-resonant element  204  is preferable over transformation of the non-resonant element  204 . 
     If the insignificant non-resonant element  204  has been previously transformed (indicating that the non-resonant element  204  vanished as both a capacitor and an inductor) at step  108 , the ERD technique  100  generates a reduced filter circuit design by removing the insignificant non-resonant element  204  from the DIP (step  110 ). If the non-resonant element  204  has not been previously transformed (indicating that the non-resonant element  204  vanished as one of a capacitance and an inductance, but not yet as the other of the capacitance and inductance) at step  108 , the ERD technique  100  modifies the DIP by changing the sign of the non-resonant element  204  (i.e., changing it from a capacitance to an inductance) (to the extent that the non-resonant element  204  is initially a capacitance) or from an inductance to a capacitance (to the extent that the non-resonant element  204  is initially a capacitance)) (step  112 ). 
     The ERD technique  100  then returns to step  104  to again optimize the DIP by inputting either the reduced filter circuit design generated in step  110  or the transformed filter circuit design generated in step  112  into the computerized filter optimizer, and then determines whether any of the remaining non-resonant elements  204  in the DIP has become insignificant at step  108 . If it is determined that one of the non-resonant elements  204  in the DIP has become insignificant, the ERD technique again determines if the insignificant non-resonant element  204  has been previously transformed at step  108 . In the case where the insignificant non-resonant element  204  has been previously transformed, then it is deemed that the insignificant non-resonant element  204  should have been removed from previous DIP, and thus, it is removed at step  110 . In the case where the insignificant non-resonant element  204  has not been previously transformed, then the ERD technique  100  changes it at step  112 . 
     If it is determined that none of the non-resonant elements  204  in the DIP (whether it is the first or a subsequently generated DIP) has become insignificant at step  106 , the ERD technique then determines whether any of the resonant elements  202  in the DIP has become insignificant (“vanishes”) (step  114 ). A resonant element  202  may tend to vanish when the associated transmission zero associated with the resonant element  202  is relatively far from all passbands and stopbands; for instance, when the resonant frequency ω R  and anti-resonant frequency ω A , as given in equations [1] and [2], move to more than 10% from the edge frequency of the nearest passband or stopband. 
     If it is determined that one of the resonant elements  202  in the DIP has become insignificant at step  114 , the ERD technique modifies the DIP by replacing the insignificant resonant element  202  with a static capacitance C 0 , which preferably has a value equal to the value of the static capacitance of the insignificant resonant element  202  (step  116 ). Notably, a resonant element  202  that becomes insignificant will still affect the circuit because of its static capacitance, and so is better replaced by a capacitor than removed. 
     The ERD technique  100  then returns to step  104  to again optimize the DIP and then determines again whether any of the non-resonant elements  204  in the DIP (including any static capacitance C 0  transformed from an insignificant resonant element  202 ) has become insignificant at step  106 . 
     If it is determined that one of the non-resonant elements  204  in the DIP has become insignificant, the ERD technique again determines if the insignificant non-resonant element  204  has been previously transformed at step  108  and proceeds as discussed above. In the case where the insignificant non-resonant element  204  is a static capacitance C 0  that has replaced an insignificant resonant element  202 , then it is confirmed that the insignificant circuit element (which previously was a resonant element  202 , but is now a non-resonant element  204 ) should be entirely removed, and thus, is so removed at step  110 . 
     If it is determined that at step  106  that none of the non-resonant elements  204  in the DIP has become insignificant, the ERD technique  100  again determines whether any of the resonant elements  202  in the DIP has become insignificant at step  114 . If it is determined that one of the resonant elements  202  in the DIP has become insignificant, the ERD technique replaces the insignificant resonant element  202  with a static capacitance C 0  at step  116 , and proceeds as discussed above. 
     If it determined that none of the resonant elements  202  in the DIP has become insignificant, the ERD technique  100  deems the DIP to be the improved final filter circuit design (step  118 ), which in the exemplary embodiment, has the resonant frequencies ω R  and static capacitances C 0  for the remaining resonant elements  202  and non-resonant elements  204  illustrated in  FIG. 8 a   , which when simulated, results in the frequency response illustrated in  FIG. 8 b   . This frequency response is shown characterized by the following markers: P1 of Mag S21=−30.080 dB at frequency=1.770 GHz; P2 of Mag S21=−34.193 dB at frequency=1.830 GHz; P3 of Mag S21=−1.394 dB at frequency=1.850 GHz; P4 of Mag S21=−0.761 dB at frequency=1.872 GHz; P5 of Mag S21=−1.406 dB at frequency=1.910 GHz; P6 of Mag S21=−45.227 dB at frequency=1.930 GHz; and P7 of Mag S21=−45.227 dB at frequency=1.990 GHz. 
     As can be seen from a comparison between  FIG. 7 a    and  FIG. 8 a   , the improved final filter circuit design includes fewer circuit elements, and in particular, two less resonant elements  202  and one less non-resonant element  204 . As can be seen from a comparison between  FIG. 8 a    and  FIG. 8 b   , the final filter circuit design yields a flatter frequency response and loss at the pass-band. It should also be noted that a conventional optimization technique (i.e., without removing circuit elements) was performed on the initial filter circuit  200  illustrated in  FIG. 7 a   , resulting in a final filter circuit having a frequency response performance that was not as good as the frequency response performance yield by the ERT technique illustrated in  FIG. 8 b   . Thus, contrary to conventional wisdom, the removal of circuit elements using the ERT technique not only decreased the cost and size of the microwave filter, it improves the frequency response performance of the microwave filter over prior art microwave filters that have more circuit elements. 
     Notably, the ERD technique  100  may analyze the frequency response of DIP resulting from step  104  in order to determine whether a previous step should be undone or redone. 
     For example, if the frequency response performance of the DIP is not better than the frequency response performance of the initial filter circuit design, this means that the initial filter circuit design is not acceptable, and thus, a different initial filter circuit design may be considered, which may then be inputted into the computerized filter optimizer at step  104 . 
     As another example, if the frequency response performance of the DIP as it existed after a non-resonant element  204  has been transformed (from a capacitance to an inductance, or from an inductance to a capacitance) is worse by a threshold amount relative to the frequency response performance of the DIP as it existed before the non-resonant element  204  was transformed, the ERD technique  100  may simply return to the previous DIP and remove the non-resonant element  204  from that design. 
     As still another example, if the frequency response performance of the DIP as it existed after a resonant element  202  has been transformed into a static capacitance C 0  is worse by a threshold amount relative to the frequency response performance of the DIP as it existed before the resonant element  202  was transformed, the ERD technique may simply return to the previous DIP and restore the resonant element  204  back into that design. 
     Once the improved final filter circuit design is achieved, the ERD technique  100  manufactures an actual microwave filter based on the final filter circuit design (step  120 ). Preferably, the circuit element values of the actual microwave filter will match the corresponding circuit element values in the improved final filter circuit design. 
     Referring first to  FIG. 9 , a computerized filter design system  300  may be used to design a microwave filter using the ERD technique  100 . The computerized filter design system  300  generally comprises a user interface  302  configured for receiving information and data from a user (e.g., parameter values and filter specifications) and outputting an optimized filter circuit design to the user; a memory  304  configured for storing filter design software  308  (which may take the form of software instructions, which may include, but are not limited to, routines, programs, objects, components, data structures, procedures, modules, functions, and the like that perform particular functions or implement particular abstract data types), as well as the information and data input from the user via the user interface  302 ; and a processor  306  configured for executing the filter design software. The filter design software program  308  is divided into sub-programs, in particular, a conventional network design synthesizer  310  (which can be used to generate the initial filter circuit design at step  102 ), a conventional filter optimizer  312  (which can be used to generate the DIP at step  104 ), and an element removal design engine  314  that controls the network design synthesizer  88  and filter optimizer  90  in accordance with filter circuit design aspects of the ERD technique  100  in order to generate the optimized final circuit design. 
     Although particular embodiments of the present invention have been shown and described, it should be understood that the above discussion is not intended to limit the present invention to these embodiments. It will be obvious to those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the present invention. For example, the present invention has applications well beyond filters with a single input and output, and particular embodiments of the present invention may be used to form duplexers, multiplexers, channelizers, reactive switches, etc., where low-loss selective circuits may be used. Thus, the present invention is intended to cover alternatives, modifications, and equivalents that may fall within the spirit and scope of the present invention as defined by the claims.