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Title:
MULTI-BAND FILTER APPARATUS AND METHOD OF USE THEREOF
Document Type and Number:
WIPO Patent Application WO/2017/203216
Kind Code:
A1
Abstract:
Multi-band filter apparatus is provided including a housing in which at least one resonating means is provided. The at least one resonating means is capable of resonating at one or more frequencies. Each of said at least one resonating means is arranged such that one or more radio frequency signals in at least two different and distinct passbands are able to pass therethrough in use.

Inventors:
MUSONDA EVARISTO (GB)
PARRY RICHARD (GB)
Application Number:
PCT/GB2017/051403
Publication Date:
November 30, 2017
Filing Date:
May 19, 2017
Export Citation:
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Assignee:
RADIO DESIGN LTD (GB)
International Classes:
H01P1/205; H01P1/20; H01P1/208
Foreign References:
US5410284A1995-04-25
US5349316A1994-09-20
GB2481291A2011-12-21
Other References:
LAIYUN JI ET AL: "Design and performance of dual-band high temperature superconducting filter", SCIENCE CHINA INFORMATION SCIENCES, SP SCIENCE CHINA PRESS, HEIDELBERG, vol. 55, no. 4, 30 May 2011 (2011-05-30), pages 956 - 961, XP035028707, ISSN: 1869-1919, DOI: 10.1007/S11432-010-4173-8
KENNETH S K YEO ET AL: "8 pole high temperature superconductor microstrip dual band bandpass filter design", MICROWAVE SYMPOSIUM DIGEST (MTT), 2011 IEEE MTT-S INTERNATIONAL, IEEE, 5 June 2011 (2011-06-05), pages 1 - 4, XP032006592, ISBN: 978-1-61284-754-2, DOI: 10.1109/MWSYM.2011.5972643
CHEN FU-CHANG ET AL: "Dual-Band Coaxial Cavity Bandpass Filter With Helical Feeding Structure and Mixed Coupling", IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, IEEE SERVICE CENTER, NEW YORK, NY, US, vol. 25, no. 1, 1 January 2015 (2015-01-01), pages 31 - 33, XP011569322, ISSN: 1531-1309, [retrieved on 20150106], DOI: 10.1109/LMWC.2014.2369965
HONG-MING LEE ET AL: "Dual-band coupling and feed structure for microstrip filter design", MICROWAVE SYMPOSIUM DIGEST, 2004 IEEE MTT-S INTERNATIONAL FORT WORTH, TX, USA JUNE 6-11, 2004, PISCATAWAY, NJ, USA,IEEE, vol. 3, 6 June 2004 (2004-06-06), pages 1971 - 1974, XP010728138, ISBN: 978-0-7803-8331-9, DOI: 10.1109/MWSYM.2004.1338997
SARAWUTH CHAIMOOL ET AL: "Resonator-Embedded Four-Pole Cross-Coupled Dual-Band Microstrip Bandpass Filters", COMMUNICATIONS AND INFORMATION TECHNOLOGIES, 2006. ISCIT;06. INT ERNATIONAL SYMPOSIUM ON, IEEE, PI, 1 October 2006 (2006-10-01), pages 1076 - 1079, XP031068326, ISBN: 978-0-7803-9740-8
Attorney, Agent or Firm:
BAILEY WALSH & CO LLP (GB)
Download PDF:
Claims:
Claims

1. Multi-band filter apparatus, said filter apparatus mcluding a housing in which at least one resonating means is provided, said at least one resonating means capable of resonating at one or more frequencies, and wherein each of said at least one resonating means is arranged such that one or more radio frequency signals in at least two different and distinct passbands are able to pass therethrough in use.

2. Multi-band filter apparatus according to claim 1 wherein two or more resonating means are provided and are coupled together via electromagnetic coupling.

3. Multi-band filter apparatus according to claim 1 wherein three or more resonating means are provided and are coupled together via electromagnetic cross coupling.

4. Multi-band filter apparatus according to claim 1 wherein the housing defines at least one resonator cavity and the at least one resonating means is provided in or associated with the at least one resonating cavity, and wherein two or more, or all, the resonating cavities are joined, are continuous with and/ or are in communication with each other.

5. Multi-band filter apparatus according to claim 1 wherein one or more transmission zeroes are located between at least two of the pass bands of the apparatus.

6. Multi-band filter apparatus according to claim 1 wherein the resonating means includes any or any combination of a combline resonator, a ceramic resonator, a dielectric resonator, a ceramic puck or a ceramic rod.

7. Multi-band filter apparatus according to claim 1 wherein the filter apparatus includes at least one input port for receiving one or more radio frequency signals and at least one output port for outputting one or more radio frequency signals, and wherein the input port and/ or the output port is each coupled to a single resonating means only.

8. Multi-band filter apparatus according to claim 1 wherein the filter apparatus includes at least one input port for receiving one or more radio frequency signals and at least one output port for outputting one or more radio frequency signals, and wherein the input port and/ or the output port is each coupled to two or more resonating means, and at least one of the two or more resonating means is arranged so as to allow one or more radio frequency signals of two or more distinct and different pass bands to pass therethrough.

9. Multi-band filter apparatus according to claim 1 wherein the apparatus includes at least two different filter paths, each filter path including at least one resonating means, a common port joining the at least two different filter paths together at a first end, and separate independent ports provided for each filter path at a second end, and at least one of the at least two filter paths having two or more passbands.

10. Multi-band filter apparatus according to claim 1 wherein the resonating means within the apparatus are provided in a linear or substantially Hnear manner.

11. Multi-band filter apparatus according to claim 1 wherein a radio frequency signal from each passband is able to pass through each resonating means of the filter apparatus.

12. Multi-band filter apparatus according to claim 1 wherein each resonating means includes or consists of a single physical resonator and/or a single resonator mode or section, and said resonator and/or resonating mode or section is arranged to allow one or more radio frequency signals of two or more different and distinct passbands to pass therethrough.

13. Multi-band filter apparatus according to claim 1 wherein a single resonating means includes, comprises or consists of two or more resonant modes or sections within the limits of the two or more different and distinct passbands, a radio frequency signal falling within any of the pass bands will pass through each resonate mode or section.

14. Multi-band filter apparatus according to claim 1 wherein each resonating means is arranged to allow one or more radio frequency signals of two or more different and distinct passbands to pass through the same resonant mode of each resonating means.

15. Multi-band filter apparatus according to claim 1 wherein tuning means are provided on or associated with the apparatus and/or the one or more resonating means to allow the resonating means to be tuned to the required electromagnetic frequency or frequency range that allows one or more radio frequency signals of the at least two different and distinct pass bands to pass therethrough in use.

16. Multi-band filter apparatus according to claim 1 wherein a frequency response or transfer function of the apparatus is calculated and the one or more resonating means and associated ckcuitry are arranged to allow the required frequency response to be achieved.

17. Multi-band filter apparatus according to claim 1 wherein a frequency response or transfer function of the apparatus must have a characteristic function that satisfies the following equation, wherein:

where: = 1 at each turning point and at it's band edges of the characteristic function; b) CN is a constant c) ωZ is a dependent transmission zero d) NnT7 is the number of transmission zeros at DC and where ls infinite e) NFTZ are the transmission zeros that are prescribed and where is infinite f) T is the number of frequencies ( ωγ where there are turning points in the stopband due to the prescribed transmission zeros (T < NFTZ) g) N is the degree of the function and where the number of transmission zeros at infinity is given by N— N0TZ— NFTZ— 1

18. A method of using multi-band filter apparatus, said filter apparatus including a housing in which at least one resonating means is provided, said at least one resonating means capable of resonating at one or more frequencies, and wherein said method includes the step of tuning each of said at least one resonating means such that one or more radio frequency signals in at least two different and distinct passbands are able to pass through each of said resonating means in use.

UPON RECEIPT BLANK

Description:
Multi-Band Filter Apparatus and Method of Use Thereof

This invention relates to a multi-band filter apparatus and to a method of use thereof.

Filter apparatus is typically used in telecommunication systems to compensate for disturbances, such as interference, that may affect one or more transmission signals being sent and/ or received by the telecommunication system. The filter apparatus is designed to remove unwanted components from the transmit and/or receive signals and/or enhance the desired transmit and/or receive signals. Many components of a mobile communication cell site, forming part of a telecommunications system, contain filters or filter apparatus, such as for example, radios, base transceiver stations (BTSs), combiners, multiplexers and tower mounted amplifiers (TMAs). Each filter or filter apparatus is designed such that it allows one or more radio signals at frequencies within their allocated passband to pass through the same but to reject or substantially reject one or more radio signals at frequencies outside of their allocated passband.

The microwave region of the electromagnetic spectrum is an essential yet finite resource used by both civilian and military applications such as radar, navigation and wireless communications. Mobile radio communications require careful allocation of the spectrum by the relevant national or international authorities to service providers, who in turn may divide their slots into narrower sub bands for allocation to individual operators.

Prior to 2010, mobile network operators generally only used the frequency bands of 900MHz, 1800MHz and/or 2100MHz. As such, filter apparatus used in mobile communication networks typically only needed to allow signals to pass through a filter contained within the filter apparatus at a single passband frequency. However, the number of frequency bands that are being used and that are becoming available to mobile network operators is increasing. In Europe, mobile network operators are now able to use the following frequency bands: 700MHz, 800MHz, 900MHz, 1400MHz, 1800MHz, 2100MHz, 2300MHz and 2600MHz. This has led to a demand in cell site equipment, and particularly filter apparatus, to function in multiple frequency bands.

Traditionally each filter has one passband associated with the same. This results in multiple components being required for a mobile communication cell site so that multiple radio frequency bands can be supported. For example, traditionally multiple BTSs have been provided in a cell site, with each BTS associated with a particular radio frequency band. An example of a filter response from a conventional filter operating in a single passband is shown in figure 1. The return loss and insertion loss/attenuation (Mag dB) are shown on the vertical axes and the frequency (MHz) of the signals passing through the filter are shown on the horizontal axis. The region 2 where the insertion loss is low shows the frequency of the passband of the filter. In the passband the return loss is shown by the regions 4 and at these frequencies a small proportion of an incident signal is reflected by the filter. The network topology of filter apparatus that produces the filter response of figure 1 is shown in figure 2. The filter apparatus typically comprises a conductive housing with a plurality of resonant cavities defined therein and a resonator 6 located in each cavity. Each resonator 6 is an electronic and/or electromechanical component that exhibits resonance for a narrow range of frequencies. The apparatus will have an input port 5 for allowing one or more radio frequency signals to enter the filter apparatus, and an output port 7 for allowing one or more radio frequency signals at the passband of the filter to leave the apparatus. In order to obtain a required frequency response from the filter, it is necessary to have pre-determined electromagnetic couplings 8 between the resonators 6.

In an attempt to accommodate the increasing number of frequency bands, apparatus for mobile cell sites has been designed which accommodate two or more different frequency bands. The filter or filter apparatus used in such cell site apparatus is often referred to as a dual band or multi-band filter. An example of a filter response from a dual band filter is shown in figure 3. The two large central peaks 10, 12 represent the two different and distinct passbands of the filter. The two sets of smaller peaks 14, 16 offset from the centre show the radio frequency signals reflected by the filter apparatus at each passband. An example of the dual band filter apparatus that produces the filter response of figure 3 is shown in figure 4. The apparatus includes an input port 5 for allowing one or more radio frequency signals to enter the apparatus, and an output port 7 allowing one or more radio frequency signals at the passbands of the apparatus to leave the apparatus. Two filters 18, 20 are arranged in parallel between the input port 5 and output port 7. Each filter 18, 20 is diplexed at each end to the other filter 18, 20 at the input and output ports 5, 7. Filter 18 contains a plurality of resonators 6a with electromagnetic couplings between the same. The resonators 6a of filter 18 are tuned so as to have a first passband frequency. As such, only radio frequency signals falling within the first passband frequency will pass through filter path 18. Filter 20 contains a plurality of resonators 6b with electromagnetic couplings between the same. The resonators 6b of filter 20 are tuned so as to have a second passband frequency different to the first passband frequency. As such, only radio frequency signals falling within the second passband frequency will pass through the filter path 20. Radio signals falling outside of the first and second pass band frequencies will be substantially reflected by the filter apparatus. Although this type of filter apparatus allows multiple frequency bands to be filtered, there are a number of problems associated with the same. For example, the diplexing junctions at each end of the filter apparatus are often difficult to physically realise and they take up valuable space, which is particularly problematic when the equipment in which it is used is being located at the top of a cell site mast where space is limited. In addition, the resulting network does not allow the filter designer to have full control over the location of transmission zeroes in the filter response. The transmission zeroes are the frequencies at which the filter completely attenuates incident signals and are required in order to produce filters with rapid roll offs.

It is therefore an aim of the present invention to provide filter apparatus, and particularly multi-band filter apparatus, that overcomes the abovementioned problems.

It is a further aim of the present invention to provide a method of using filter apparatus, and particularly multi-band filter apparatus, that overcomes the abovementioned problems.

It is a yet further aim of the present invention to provide a mobile communication system that overcomes the abovementioned problems.

According to a first aspect of the present invention there is provided multi-band filter apparatus, said filter apparatus including a housing in which at least one resonating means is provided, said at least one resonating means capable of resonating at one or more frequencies, and wherein each of said at least one resonating means is arranged such that one or more radio frequency signals in at least two different and distinct passbands are able to pass therethrough in use.

By arranging the resonating means within the filter apparatus to allow radio frequency signals of at least two or more different and distinct passbands to pass therethrough, this means that only a single or a reduced number of filter paths need to be provided in the multi-band filter apparatus. This has the advantage that it removes the requirement of complex diplex junctions being needed and therefore the filter apparatus can be realised more easily, and smaller, more compact filter designs, can be produced. In addition, since the filter design allows the introduction of more cross couplings, this allows the location of transmission zeroes to be controlled.

Preferably the apparatus includes at least two and/or a plurality of resonating means. Each of said two and/or plurality of resonating means is arranged such that the radio frequency signals in the at least two different and distinct passbands are able to pass therethrough.

In one embodiment the housing defines at least one resonator cavity. Preferably the at least one resonating means is provided in or associated with the at least one resonator cavity.

Preferably a single resonating means is provided in a single resonator cavity.

In one embodiment two or more resonating means can be provided in a single resonator cavity.

In one embodiment two or more resonator cavities are joined, are continuous with and/ or are in communication or fluid communication with each other.

In one embodiment all the resonator cavities of the filter apparatus are joined, are continuous with and/ or are in communication or fluid communication with each other.

Preferably the one or more resonator cavities are formed from and/ or include at least one outer most layer of electrically conductive material.

The resonating means is typically defined as an object that is capable of exhibiting resonance or resonant behaviour at one or more frequencies within at least the electromagnetic spectrum.

Preferably the resonating means can include or consist of a single physical resonator and/or a single resonator mode or section. Each resonator and/or resonating mode or section is arranged to allow one or more radio frequency signals of two or more different and distinct passbands to pass therethrough.

In one embodiment a single resonator may include, comprise or consist of two or more resonant modes or sections within the limits of the two or more different and distinct pass bands. In this embodiment a radio frequency signal falling within any of the pass bands will pass through each resonant mode or section. In one embodiment each resonating means is arranged to allow one or more radio frequency signals of two or more different and distinct passbands to pass through the same resonant mode of each resonating means.

In one embodiment the resonating means can include any or any combination of a combline resonator, a ceramic resonator, a dielectric resonator, ceramic puck, ceramic rod and/ or the like.

Preferably two or more resonating means are coupled together via electromagnetic coupling.

In one embodiment two or more resonating means are arranged so as to have electromagnetic cross coupling therebetween. The introduction of the cross coupling allows the location of the transmission zeroes of the filter response to be controlled.

Preferably one or more transmission zeroes can be located between at least two of the pass bands.

Preferably the filter apparatus includes at least one input port for receiving one or more radio frequency signals and at least one output port for outputting one or more radio frequency signals.

In one embodiment the input port and/ or the output port is each coupled to a single resonating means only.

In one embodiment the input port and/or the output port is each coupled to two or more resonating means, and at least one of the two or more resonating means is arranged so as to allow one or more radio frequency signals of two or more distinct and different pass bands to pass therethrough.

In one embodiment the filter apparatus has at least two different filter paths. For example, the filter apparatus can be in the form of a diplexer. Each filter path includes at least one resonating means. Preferably the different filter paths have a common port joining the filter paths at one end, and separate independent ports for each filter path at the other ends. At least one of the at least two filter paths has two or more passbands.

Preferably the resonators or resonating means within the filter apparatus are provided in a linear or substantially linear manner with the resonators or resonating means within each linear filter path arranged so as to allow one or more radio frequency signals of two or more different and distinct pass bands to pass therethrough.

Preferably each resonating means is tuned to, or substantially tuned to within the limits of the at least two or more pass bands (such as for example +/-10% of passband frequency).

In one embodiment two or more resonating means can be tuned to different frequencies providing the response of the two or more resonating means together allows one or more radio frequency signals of two different and distinct pass bands to pass through the same.

Preferably a radio frequency signal from each passband is able to pass through each resonating means of the filter apparatus.

Preferably the housing of the filter apparatus is an electrically conductive housing, such as for example, a metallic housing, and/or a housing with at least one electrically conductive outer layer or coating provided thereon.

In one embodiment the filter housing comprises a base, side walls and a top. The one or more resonant cavities are defined in the housing between the base, side walls and top and have at least one opening in the top of the housing. A lid or closure means can be provided on or associated with the top in use.

Preferably toning means are provided on or associated with the filter apparatus and/or the one or more resonating means to allow the resonating means to be tuned to the required electromagnetic frequency or frequency range that allows one or more radio frequency signals of at least two different and distinct pass bands to pass therethrough.

The passband of the filter apparatus is typically a frequency or frequency range of radio signals that can pass through the filter apparatus. Radio frequency signals falling outside or significantly outside of this frequency or frequency range are reflected by the filter apparatus and are prevented or substantially prevented from passing through the filter apparatus.

In one embodiment the frequency response or transfer function of the filter apparatus is calculated and the one or more resonating means and associated circuitry are then designed and arranged to allow the required frequency response and/ or transfer function to be achieved.

The transfer function H(j ω) of a two port network, such as a filter, is a function of angular frequency, ω, and states what proportion of the energy entering the network at the input port leaves via the output port. A transfer function can be written in terms of a characteristic function T N :

Characteristic functions that satisfy the following equation will have a dual band response:

where: 1. at each of the function's turning points and at its band edges. therefore provides a function having a value zero at these

turning points and band edges. The band edge zeros are cancelled by the factor in the denominator,

where are the normalised cut-off frequencies, leaving zeros at just the turning points which are necessary tor

2. C N is a constant

3. ω Z is the dependent transmission zero that cannot be prescribed

4. N OT7 is the number of transmission zeros at DC and where ls infinite

5. N FTZ are the transmission zeros that can be prescribed and where is infinite

6. T is the number of frequencies ω r where there are turning points in the stopband due to the prescribed transmission zeros (T < N FTZ )

7. N is the degree of the function and so the number of transmission zeros at infinity is therefore given by N— N 0TZ — N FTZ — 1

The equation can be solved using numerical methods. Circuit optimisation techniques can be employed to arrive at the desired response for the filter apparatus.

According to a second aspect of the present invention there is provided a method of using filter apparatus, said filter apparatus including a housing in which at least one resonating means is provided, said at least one resonating means capable of resonating at one or more frequencies, and wherein said method includes the step of tuning each of said at least one resonating means such that one or more radio frequency signals in at least two different and distinct passbands are able to pass through each of said resonating means in use. According to a third aspect of the present invention there is provided a telecommunication system including multi-band filter apparatus.

Embodiments of the present invention will now be described with reference to the following drawings, wherein:

Figure 1 (PRIOR ART) is a filter response of a conventional single band filter, with the return loss and insertion loss/attenuation shown on the Y-axis (Mag dB) and the radio frequency shown on the X-axis (MHz);

Figure 2 (PRIOR ART) is an example of the resonator arrangement in a conventional single band filter apparatus of a type producing the filter response shown in Figure 1 ;

Figure 3 (PRIOR ART) is a filter response of a conventional dual band filter, with the return loss and insertion loss /attenuation shown on the Y-axis (Mag dB) and the radio frequency shown on the X-axis (MHz);

Figure 4 (PRIOR ART) is an example of the resonator arrangement in a conventional dual band filter apparatus of a type producing the filter response shown in Figure 3;

Figure 5 is an example of a resonator arrangement in multi-band filter apparatus according to an embodiment of the present invention;

Figure 6 is a further example of a resonator arrangement in multi-band filter apparatus according to a further embodiment of the present invention;

Figure 7a (PRIOR ART) is an example of diplexer filter apparatus according to the prior art; and

Figure 7b is an example of diplexer filter apparatus according to the present invention; and Figure 8 is an example of a filter response of the diplexer filter shown in Figure 7b, with the return loss and insertion loss/attenuation shown on the Y-axis (Mag dB) and the radio frequency shown on the X-axis (MHz).

According to an embodiment of the present invention there is provided multi- band filter apparatus including a housing in which a plurality of resonant cavities are defined (not shown). Resonating means in the form of a resonator is provided in each cavity. A plurality of resonators 22 are electromagnetically coupled together via electromagnetic couplings 24 in series, as shown in the resonator arrangement in figure 5. An input port 26 is provided at the start of the line of resonators and is coupled to the first resonator 22' in the line. An output port 28 is provided at the end of the line of resonators and is coupled to the last resonator 22" in the line. One or more radio frequency signals enter the filter apparatus via input port 26 pass along all the resonators 22 and are output from the filter apparatus via output port 28.

In accordance with the present invention, each resonator 22, 22'; 22" is arranged such that it allows radio frequency signals of two different and distinct passbands to pass through each of them. Thus, each resonator is tuned to a frequency or frequency range which allows signals from both passbands to pass through the same.

The ability to do this using a single path of resonators joined to the input and output ports removes the requirement for complicated diplex junctions to be provided, an which also allows more compact filter designs to be produced. Transmission zeroes can be created which are located between the pass bands.

The filter response of the filter apparatus of the present invention looks similar to the filter response shown in figure 3 but the filter apparatus is arranged in a different manner to that of Figure 4. Referring to figure 6, there is illustrated a further example of an arrangement of resonators in filter apparatus according to an embodiment of the present invention. In this example, some of the electromagnetic couplings between the resonators 22 are cross couplings. Electromagnetic couplings 24 are shown by solid lines and electromagnetic cross couplings 30 are shown by dotted lines.

Referring to figure 7b, there is illustrated a further embodiment of the present invention in the form of a diplexer filter apparatus, wherein two filter paths 30, 32 are provided. Each filter path 30, 32 contains a plurality of resonators 34, 36 respectively. The filter paths 30, 32 each have a common port 38 and an independent port 40, 42 respectively. The resonators 34 within filter path 30 allow one or more radio frequency signals of two or more different and distinct passbands (i.e. first and second pass bands) to pass therethrough. The resonators 36 within filter path 32 allow one or more radio frequency signals of two or more different and distinct passbands (i.e. third and fourth pass bands) to pass therethrough. The first and second pass bands are different to the third and fourth pass bands. For example, transmit signals for two different passbands can pass through filter path 30 and receive signals for two different pass bands can pass through filter path 32 in use.

The arrangement is different to a conventional diplexer arrangement, an example of which is shown in figure 7a. In the prior art arrangement, one common port 44 is coupled to four different resonator paths 46, 48, 50, 52, each resonator path allowing radio frequency signals of a single pass band to pass therethrough. For example, resonator path 46 could be for transmission signals at a first radio frequency and resonator path 48 could be for transmission signals at a second radio frequency. Resonator path 50 could be for receive signals at the first radio frequency and resonator path 52 could be receive signals at the second radio frequency. A first port 54 is coupled to two of the different resonator paths 46, 48, and a second port 56 is coupled to the other two different resonator paths 50, 52. Figure 8 is an example of a filter response from the filter arrangement shown in figure 7b. The frequency peaks 58 and 60 represent the two receive passbands of filter path 32 and the frequency peaks 62 and 64 represent the two transmit passbands of filter path 30.

Conventionally single passband filters are often designed to have an equiripple response. The return loss of such a filter has a number of ripples in the passband of equal size. The response can be enhanced with the introduction of finite frequency transmission zeroes giving an excellent trade off between passband insertion loss and stop band attenuation. The present invention extends this technique to provide an equiripple filter response with two or more passbands.

The transfer function Η(jώ) of a two port network, such as a filter, is a function of angular frequency, ω, and states what proportion of the energy entering the network at the input port leaves via the output port. A transfer function can be written in terms of a characteristic function T N :

Characteristic functions that satisfy the following equation will have a dual band response:

where:

1. at each of the function's turning points and at it's bandedges. therefore provides a function having a value zero at these turning points and bandedges. The bandedge zeros are cancelled by the factor in the denominator, where are the normalised cut-off frequencies, leaving

r

zeros at just the turning points which are necessary tor

2. C N is a constant

3. ω Z is the dependent transmission zero that cannot be prescribed

4. N 0T7 is the number of transmission zeros at DC and where ls infinite

5. N FTZ are the transmission zeros that can be prescribed and where

infinite

6. T is the number of frequencies ω r where there are taming points in the stopband due to the prescribed transmission zeros (T < N FTZ )

7. N is the degree of the function and so the number of transmission zeros at infinity is therefore given by N— N 0TZ — N FTZ — 1

The equation can be solved using numerical methods, and circuit optimisation techniques can be employed to arrive at the desired response for the filter apparatus.




 
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