Tunable bandpass filter

1. A filter for receiving an input signal and providing a filtered output signal, the filter comprising:

an input coupling adapted to receive the input signal;

an output coupling adapted to provide the filtered output signal;

at least three transmission-line couplings forming a main transmission line;

at least four nodes connected in series between the input coupling and the output coupling, each node being connected to an adjacent node through one of the transmission-line couplings;

at least four resonant elements, each coupled to a different node of the at least four nodes and each having an adjustable resonant frequency; and

at least four non-resonant elements, each connected to a different node of the at least four nodes, wherein:

(i) at least one of the non-resonant elements is one of (a) a capacitor coupled to ground and (b) a structure that is equivalent to a capacitor coupled to ground;

(ii) at least one of the non-resonant elements is one of (a) an inductor coupled to ground and (b) a structure that is equivalent to an inductor coupled to ground;

(iii) at least two of the resonant elements provide at least two transmission zeros at at least two different frequencies in a lower stopband of the filter;

(iv) at least two of the resonant elements provide at least two transmission zeros at at least two different frequencies in an upper stopband of the filter; and

(v) each of the non-resonant elements provides the coupling between one of the nodes along the main transmission line and a corresponding one of the resonant elements, such that the filter has a bandpass filtering characteristic between the lower and upper stopbands.

2. The filter of claim 1 , wherein each node of the at least four nodes is connected to at least one corresponding resonant element and to at least one corresponding non-resonant element.

3. The filter of claim 2 , wherein each node of the at least four nodes is connected to only one corresponding resonant element and to only one corresponding non-resonant element.

4. The filter of claim 2 , wherein each non-resonant element provides electromagnetic coupling between one of the resonant elements and a corresponding one of the nodes.

5. The filter of claim 1 , wherein the at least four non-resonant elements comprises an equal number of capacitive non-resonant elements and inductive non-resonant elements.

6. The filter of claim 1 , wherein each transmission-line coupling has a length that is about equal to the wavelength of a center frequency of a passband of the filter divided by four.

7. The filter of claim 1 , wherein each resonant element is a cavity resonator.

8. The filter of claim 1 , wherein the non-resonant elements are stubs.

9. The filter of claim 1 , wherein each non-resonant element provides electromagnetic coupling between one of the resonant elements and one of the nodes.

10. The filter of claim 1 , wherein each resonant element has an adjustable capacitance, the adjustment of which causes a corresponding adjustment of the resonant frequency of the resonant element.

11. The filter of claim 1 , wherein:

the resonant elements are adapted such that poles of the filter, when normalized to a predetermined bandwidth and plotted in a real-imaginary plane, move substantially parallel to the imaginary axis when the resonant frequencies of the resonant elements are adjusted, without substantial movement parallel to the real axis.

12. The filter of claim 1 , wherein the resonant elements are adapted such that, when poles of the filter are normalized to a predetermined bandwidth and plotted in the real-imaginary plane and the bandwidth of the filter is adjusted in a range between about 8 MHz and about 20 MHz:

(i) one or more of the poles of the filter having a positive imaginary component move in the real-imaginary plane no closer to the imaginary axis than along a curve having a slope that is not greater than about −6 imaginary/real units, and

(ii) one or more of the poles of the filter having a negative imaginary component move in the real-imaginary plane no closer to the imaginary axis than along a curve having a slope that is not less than about +6 imaginary/real units.

13. The filter of claim 1 , wherein the resonant elements are adapted such that a real component of corresponding poles of the filter changes by no more than about 20% when the bandwidth of the filter is adjusted in a range between about 8 MHz and about 20 MHz.

14. The filter of claim 1 , wherein the filter is adapted to have a substantially constant absolute insertion loss when the bandwidth of the filter is adjusted from about 8 MHz to about 20 MHz.

15. The filter of claim 14 , wherein the filter is adapted to have an absolute insertion loss that varies by less than about 0.5 dB for bandwidths from about 4 MHz to about 20 MHz.

16. The filter of claim 14 , wherein the filter is adapted to have an absolute insertion loss that varies by less than about 0.1 dB for bandwidths from about 8 MHz to about 20 MHz.

17. The filter of claim 1 , wherein the filter is adapted to have an absolute selectivity that is constant for bandwidths from about 8 MHz to about 20 MHz.

18. The filter of claim 1 , wherein the parameters of the resonant and non-resonant elements are selected based on polynomial equations F, P, and E of scattering parameters of the filter according to the following equations:

F=F A E B −F* B E* A

P=P A P B , and

E=E A E B −F* A F B ;

wherein:

the roots of P A are upper-stopband, transmission-zero frequencies of a first prototype filter A;

the roots of F A are passband reflection-zero frequencies of the first prototype filter A;

the roots of E A are the poles of the first prototype filter A;

the roots of P B are lower-stopband, transmission-zero frequencies of the second prototype filter B;

the roots of F B are passband reflection-zero frequencies of second prototype filter B; and

the roots of E B are the poles of the second prototype filter B.

19. The filter of claim 1 , wherein:

each node of the at least four nodes is directly connected to the adjacent node through one of the transmission-line couplings, without any intervening elements;

each node of the at least four nodes is connected to only one corresponding resonant element and to only one corresponding non-resonant element;

each non-resonant element provides electromagnetic coupling between one of the resonant elements and one of the at least four nodes;

the at least four non-resonant elements comprises an equal number of capacitive non-resonant elements and inductive non-resonant elements;

each transmission-line coupling has a length that is about equal to wavelength of a center frequency of the passband of the filter divided by four;

each resonant element is a cavity resonator;

the non-resonant elements are stubs;

each resonant element has an adjustable capacitance, the adjustment of which causes a corresponding adjustment of the resonant frequency of the resonant element;

the resonant elements are adapted such that poles of the filter, when normalized to a predetermined bandwidth and plotted in a real-imaginary plane, move substantially parallel to the imaginary axis when the resonant frequencies of the resonant elements are adjusted, without substantial movement parallel to the real axis; and

the resonant elements are adapted such that the real component of each pole of the filter changes by no more than about 20% when the bandwidth of the filter is adjusted in a range between about 8 MHz and about 20 MHz.

20. A method of constructing a filter adapted to receive an input signal and provide a filtered output signal, the method comprising:

providing an input coupling adapted to receive the input signal;

providing an output coupling adapted to provide the filtered output signal;

providing at least three transmission-line couplings forming a main transmission line;

providing at least four nodes connected in series between the input coupling and the output coupling, each node being directly connected to an adjacent node through one of the transmission-line couplings;

providing at least four resonant elements, each coupled to a different node of the at least four nodes and each having an adjustable resonant frequency; and

providing at least four non-resonant elements, each connected to a different node of the at least four nodes, wherein:

(i) at least one of the non-resonant elements is one of (a) a capacitor coupled to ground and (b) a structure that is equivalent to a capacitor coupled to ground, and at least one of the non-resonant elements is one of (a) an inductor coupled to ground and (b) a structure that is equivalent to an inductor coupled to ground;

(ii) at least two of the resonant elements provide at least two transmission zeros at at least two different frequencies in a lower stopband of the filter; and

(iii) at least two of the resonant elements provide at least two transmission zeros at at least two different frequencies in an upper stopband of the filter; and

(iv) each of the non-resonant elements provides the coupling between one of the nodes along the main transmission line and a corresponding one of the resonant elements, such that the filter has a bandpass filtering characteristic between the lower and upper stopbands.

21. A method of adjusting a bandwidth of a filter comprising (1) an input coupling adapted to receive the input signal; (2) an output coupling adapted to provide the filtered output signal; (3) at least three transmission-line couplings forming a main transmission line; (4) at least four nodes connected in series between the input coupling and the output coupling, each node being directly connected to an adjacent node through one of the transmission-line couplings; (5) at least four resonant elements, each coupled to a different node and each having an adjustable resonant frequency, and (6) at least four non-resonant elements, each connected to a different node of the at least four nodes, wherein: (i) at least one non-resonant element is one of (a) a capacitor coupled to ground and (b) a structure that is equivalent to a capacitor coupled to ground, and at least one non-resonant element is one of (a) an inductor coupled to ground and (b) a structure that is equivalent to an inductor coupled to ground, (ii) a first resonant element provides a first transmission zero at a first resonant frequency in a lower stopband of the filter, (iii) a second resonant element provides a second transmission zero at a second resonant frequency, greater than the first resonant frequency, in the lower stopband of the filter, (iv) a third resonant element provides a third transmission zero at a third resonant frequency, greater than the second resonant frequency, in an upper stopband of the filter, (v) a fourth resonant element provides a fourth transmission zero at a fourth resonant frequency, greater than the third resonant frequency, in the upper stopband of the filter, and (vi) each of the non-resonant elements provides the coupling between one of the nodes along the main transmission line and a corresponding one of the resonant elements, such that the filter has a bandpass filtering characteristic between the lower and upper stopbands, the method comprising:

adjusting a characteristic of the first resonant element, such that the first resonant frequency is adjusted by a first frequency difference; and

adjusting a characteristic of the fourth resonant element, such that the fourth resonant frequency is adjusted by a negative of about the first frequency difference, wherein the bandwidth of the filter is adjusted without altering a center frequency of the filter;

adjusting a characteristic of the second resonant element, such that the second resonant frequency is adjusted by about a second frequency difference;

adjusting a characteristic of the third resonant element, such that the third resonant frequency is adjusted by a negative of about the second frequency difference, wherein the bandwidth of the filter is adjusted without altering the center frequency of the filter;

the bandwidth of the filter is adjusted without altering an absolute selectivity of the filter.