Description

Technical Field

Embodiments of the invention refer to a method for determining element values of a filter structure, to a method for determining element values of a radio frequency filter structure, to a computer program for performing such methods, when the computer program runs on a computer, a filter designed by such methods, and a filter comprising coupling coefficients defined by such methods.

Background of the Invention

Analogue filters are being used in many applications, including RF radio frontends and signal processing equipment, for separating signals of interest existing in one portion of the RF spectrum from unwanted signals and interfering energy in other portions of the RF spectrum. Typical design objectives for such filters include

• minimizing filter attenuation and ripple in the pass-band region, reducing loss and amplitude distortion for the signal of interest;

• maximizing attenuation in the stop-band region, for maximum suppression of unwanted or interfering signals; and

• providing a steep transition between attenuation in the stop-band and attenuation in the pass-band region, to allow for minimum spectral separation between signal of interest and unwanted or interfering signals (“selectivity”, for improved utilization of the RF spectrum)

Filter structures composed of coupled resonators may be both designed and used for fixed frequency, where the span and center frequency of the pass-band is fixed, or for tunable frequency, where the span and/or the center frequency of the pass-band is adjustable. In the general case of tunable frequency, the coupling factor is frequency dependent; therefore dimensioning needs to consider the tunable frequency range.

Bandpass filters structures composed of inductors, capacitors and/or coupled resonators are known to the state of the art long since. Known methods for designing and dimensioning such filters include; ) Filter design using eigenvalues

On circuit-level, this is a well-known approach (e.g. see, for example,“Handbook of Filter Synthesis”, A.l. Zverev, Wiley Interscience John Wiley & Sons, 2005) based on eigenvalues. Such eigenvalues are usually normalized and pre-calculated for defining a certain filter characteristics, e.g. according to Tchebysheff (Chebyshev), Butterworth or Cauer. The method provides actual component dimensioning values for a pre-defined circuit structure. In order to obtain a filter with band-pass characteristics, a low-pass to band-pass transformation is applied.

Such filters are characterized by having a pre-defined filer response, typically at least some amount of ripple in the pass-band section and a (by design) fixed center frequency and passband bandwidth. ) Filter design by approximation

This method uses a target filter response as starting point and tries to approximate this target response using a coupled structure with known transfer characteristics (see, for example, Cameron, “Coupling Matrix Synthesis for a New Class of Microwave Filter Configuration”, Faugerey, Seyfert, MTT-S 2005, INRIA). The base topology model is composed of a plurality of transformer-coupled resonators (see Fig. 2). The coupling matrix considers all potential couplings (denoted by M _t in Fig. 2) and thus tends to become very large, even for a small number of resonators. Dimensioning of actual component values is usually done by numerical means, approximating the target filter response.

This approach is primarily characterized by the complexity of the coupling matrix and the resulting circuit. Known disadvantages are the required number of components and its sensitivity and potential instability in the presence of component deviation and parasitic effects. Furthermore, the filter response has a (by design) fixed center frequency and passband bandwidth and is not necessarily fully flat in the pass-band section.

Compared to the linear topology with direct coupling (Fig. 1) this filter design method does not only consider but usually tries taking advantage of the additional couplings. This is shown in Fig. 3 for a filter design using 7 directly and partially cross-coupled resonators, approximating a quasi-flat response in the pass-band section (see Fig. 4).

A related filter design approach is taught by, for instance, “A New Class of Reconfigurable Microwave Bandpass Filters”, W.M. Fahtelbab, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS— II: EXPRESS BRIEFS, VOL. 55, NO. 3, MARCH 2008, where the coupling elements are specifically chosen as delay lines, obtaining a pre-defined delay (phase) relation as the signal propagates through the filter. A design example is illustrated in Fig. 5. It should be noted that the filter center frequency may be tuned between 1GHz and 2.5GHz by changing the capacitor shown in Fig. 5 (b), and that the filter response is only approximately flat in the passband region.

3) Filter component dimensioning using J-lnverter

This method specifically applies for filters with 2 coupled resonators and additional coupling between the input and the first resonator and the second resonator and the output. An example is shown in, for example, “A 25-75-MHz RF MEMS Tunable Filter”, K. Entesari, K. Obeidat, A.R. Brown, G.M. Rebeiz, IEEE MTT Vol. 55, No 1 1 , Nov. 2007, with the main objective of adjusting the center frequency of the filter, but not to obtain a maximum flat pass-band region. Filter circuitry is illustrated in Fig. 6. This method according to the cited document focuses on calculating the couplings using J-lnverter and applies the findings to an implementation using MEMS. This method according to the cited document does not consider or target critical coupling and does not teach how to design for a certain target filter properties (e.g. maximum flat pass-band or steep transition).

Regarding filter structures with band-pass characteristics and composed of N coupled resonators (as shown in Fig. 1), the special case of N=2 is well understood and the dimensioning of the then single coupling factor m ₂₁ is discussed at length in technical literature and prior patents.

US 1945427 relates to coupled circuits and particularly to methods of varying the coupling coefficient between two circuits without varying the natural resonant frequencies. For example, it is disclosed in the description that“two circuits tuned to the same frequency and coupled with somewhat more than critical coupling act as band pass filters as the resonance curve of the coupled circuits is not a single sharp peak but comprises a relatively flat-topped curve resulting from the combination of the two spaced resonant curves.”

WO 2016/064309 A1 discloses a bandpass filter, comprising two resonance circuits, with the first resonance circuit having a resonance frequency FR1 connected in series to a coupling circuit, in turn connected to a second resonance circuit, having a resonant frequency FR2. Fig. 3a of the cited document illustrates how the transfer function changes as a result of “tight”,“critical” or“loose” coupling. The relation between“coupling" and the“shape” of the transfer function is also known from prior art, e.g.“Hochfrequenztechnik 1”, Zinke, Brunswig, Springer Verlag, and“Einfuhrung in die Theorie der Hochfrequenz-Bandfilter”, R. Feldtkeller, 5. Auflage, S.Hirzel Verlag Stuttgart, 1961. By convention, cited publication differentiates 3 cases, depending on the value of the coupling factor m ₂₁ :

• m _2i > 1,“tight” coupling:

The transfer function shows two discrete maxima, but also regions of increased attenuation (or“ripple") in between these maxima.

• m _2i = 1,“critical” coupling:

The transfer function shows the intended“maximum flat” behavior, around the one discrete maxima; also shows minimum attenuation (loss) at the maxima.

• m ₂₁ < 1,“loose” coupling:

The transfer function shows a discrete maxima, however some level of attenuation (loss), even at the discrete maxima.

Fig. 7 illustrates the normalized filter transfer function when using 2 coupled resonators for different value ranges for the coupling factor m ₂₁ .

US 2912656 discloses a constant bandwidth coupling system, with the design objective of having a„ideal or desired constant bandwidth", illustrated by the horizontal line 82 in Fig. 5 of the cited document. Furthermore, the cited document discloses that“the ideal flat-topped shape shown at 60 in Fig. 3” is achieved if the end circuits are critically loading. The transfer function illustrated in Fig. 3 of the cited document follows a similar pattern as Fig. 7 of present application, however this behavior is resulting form end circuit loading, not from resonator coupling.

Besides electrical (including inductive or capacitive) coupling, the coupling between the resonators may also rely on alternative coupling mechanisms, e. g., acoustic coupling. For example, US 6720844 B1 relates to a coupled resonator bulk acoustic wave filter. Fig. 8 of the cited document depicts examples of the filter transfer characteristics, for a pair of resonators having various amounts of coupling. Furthermore, Fig. 9 of the cited document depicts a measured transfer characteristics for a filter using a pair of acoustically coupled resonators, electrically connected in series with a second pair of acoustically coupled resonators, while Fig. 11 of the cited document displays and compares the filter transfer characteristics for a first and a second filter, where the first and second filter are the same, except that the thickness of each of the electrodes has been increased by a small amount for the second filter.

It is well known from prior art that filter structures composed of N=2 resonators are limited in their selectivity, i.e., limiting the steepness of the transition between attenuation in the stop- band and attenuation in the pass-band region. It is also well known from literature that cascading filter structures, i.e., increasing the number of resonators (N>2) is beneficial in improving filter selectivity.

US2004/0130414 A1 relates to a system and method for an electronically tunable frequency filter having constant bandwidth and temperature compensation for center frequency, bandwidth and insertion loss and discloses a tunable band-pass filter structure, using a number of resonance circuits tailored to define the center frequency and a number of bandwidth control circuits, tailored to define the bandwidth of the filter. However, it should be noted that this document does not disclose or teach how these circuits shall be dimensioned nor how the variation of an individual (or a group of, or all) resonance circuit numerically affects the center frequency, nor how the variation of an individual (or a group of, or all) bandwidth control circuits numerically affects the bandwidth of the filter.

Further prior art document, “A coupling matrix synthesis for a tunable band pass filter”, A. Jaschke, M. Schiihler, M. Tessema, IEEE EuMW 2017 analyses an exemplary bandpass filter, composed of 3 inductively coupled resonators, assuming an symmetrical arrangement of components, i.e. C _x = C ₃ , L _t = L ₃ and L ₄ = L ₆ .

Using component values normalized to reference impedance R, the impedance matrix Z is derived as Z = M + jRl + R, and with the coupling factors m _i · being the elements of the matrix M and l _c = ^L / _r .

Note that disclosed method of the cited document applies numerical means to further analyze the behavior of the filter in the pass-band region and to derive a dimensioning for l _c and to concludes that m ₂₁ = m ₂₃ = -jo l _c is a resulting property of this exemplary bandpass filter. Furthermore, the method of the cited document derives component values considering specific center frequency and passband bandwidth inputs. In view of the above discussion, there is a need for a concept for design of filter which brings an improved tradeoff between complexity and filter characteristics.

Summary of the Invention

An embodiment according to the present invention creates a method for determining element values, preferably value of inductor L, and value of conductor , in case N=i, of a filter, preferably radio frequency filter, structure having a linear sequence of coupled resonators, preferably directly coupled only between adjacent or subsequent resonators, wherein a coupling between adjacent resonators is determined by one or more coupling factors, for example, coupling factors th _ί} . The method comprises; deriving, for example, in a symbolic form, i.e. , as a parameterized equation, an input-to-output transfer function, for example, in a form one of 5 ₂₁ (M, W), S ₂₁ (M, w), |S ₂₁ (M, il)|, |5 ₂₁ (M, w)|, which is dependent on a plurality of coupling factors, on the basis of a matrix representing one of Kirchhoff’s laws for loops, e.g., an input loop, loop I, one or more inner loops comprising only one resonator and two coupling elements, loop II, and an output loop, loop III, or nodes of the filter structure and including the coupling factors, e.g., as parameters. The matrix representing one of Kirchhoff’s laws is an Impedance matrix or an Admittance matrix, and wherein the matrix is determined as a function of input resistance, output resistance, resonator detuning and as a function of coupling factors. The method further comprises determining one or more conditions, e.g., equations, for the coupling factors such that the input-to-output transfer function, or an absolute value thereof, takes a maximum value with respect to frequency, e.g., with respect to normalized frequency at a center frequency, preferably at which the resonators have a resonance frequency which may be the center frequency of the filter structure; determining the coupling factors using the one or more determined conditions; and deriving element values of the filter structure in dependence on the determined coupling factors.

This embodiment is based on the finding that a plurality of coupling factors is dependent on a frequency range and the coupling factors are included in a matrix representing one of Kirchhoff’s law for loops or nodes of the filter structure. Such a matrix can be derived even for large filter structures in a highly systematic and efficient manner. Thus, based on linear matrix operation, it is possible to derive a transfer function including parameters which comprise coupling factors. That is, the relationship between the coupling factors with respect to frequency at a center frequency is expressed as a form of equation which may be represented in a symbolic form (e.g. having the coupling factors as parameters), i.e., the method provides a symbolic solution. Thus, element values of a filter structure can be derived by resolving the determined equation, and therefore, a highly degree of freedom for design is achieved. For example, different combination of coupling factors may be found on the basis of an equation representing a condition for the coupling factors.

The coupling factors are preferably determined such that the input-to-output transfer function, or an absolute value, thereof, fulfils a maximum flatness condition in an environment of the center frequency. The maximum flatness is defined such as the value difference between maximum value and minimum value in the pass-band of frequency is less than a given number, e.g., given threshold. For example, the maximum flatness condition may define that the input-to-output transfer function is monotonously increasing within a passband of the filter structure up to a center frequency and such that the input-to-output transfer function is monotonously decreasing within the passband above the center frequency. The maximum flatness condition may also define that the input-to-output transfer function does not comprise an inflection point within the passband or does not comprise only ripple within the passband.

It is preferred that the Impedance matrix is a combination of a coupling matrix including only diagonal elements and elements describing a coupling only between adjacent resonators, a resonator impedance matrix and a reference impedance matrix. More preferably, the Impedance matrix is defined by

Z (M, W) = M + j ^' /W + R for the filter structure with N resonators,

where M = is a matrix with coupling factors

I is the identity matrix,

R is a diagonal matrix and

W is a column vector with the frequency variables W ,

wherein

where Q _t is the Quality factor of the resonator i. That is, the characteristic of the filter structure could be determined by the equation including a linear matrix and therefore, it is possible to design the filter structure without using eigenvalues.

In a preferred embodiment, in case of parallel resonance, W; is defined as;

and in case of serial resonance, W ₍ is defined as;

where l _t and are the normalized inductances and capacitances of resonator i, with i =1 to N. Thus, it is possible appropriately to design the filter structure depending on desired resonator configuration.

It is preferred that a first line of the impedance matrix describes a relationship between a source voltage and resonator currents of the N resonators, wherein an input loop is considered; wherein a last line of the impedance matrix describes voltage contributions in a closed output loop in dependence on the resonator currents of the N resonators; wherein one or more intermediate lines of the impedance matrix describe voltage contributions in one or more respective closed loops comprising a respective resonator and couplings with respective adjacent resonators in dependence on the resonator currents of the N resonators. That is, the method allows to obtain a symbolic solution (or to obtain a solution using a symbolic equation solver tool) by using a matrix and therefore, contrary to numerical solution, it is possible to intuitively understand filter structure.

The loops comprise at least an input loop, one or more inner loops comprising only one resonator and two coupling elements, and an output loop. According to this method, at least three different loops are included and therefore, for example, a matrix in case N=3 represent each loop.

The input-to-output transfer function is derived as a parameterized equation. More preferably, the input-to-output transfer function is derived as a parameterized representation of a magnitude of a transmission scattering parameter between an input of the filter structure and an output of the filter structure over frequency. That is, the input-to-output transfer function is derived as an equation including a parameter reflecting the coupling factor.

It is preferable that the maximum value is defined with respect to a normalized frequency at a center frequency at which the resonators have a resonance frequency, which is the center frequency of the filter structure. In addition, it is preferable that there is a direct coupling only between adjacent or subsequent resonators in the linear sequence of coupled resonators.

In a preferred embodiment, in the method determining one or more conditions for the coupling factors such that the input-to-output transfer function, or an absolute value thereof, takes a maximum value with respect to frequency at a center frequency is made under the assumption that resonance frequencies of all resonators are equal; and/or wherein determining one or more conditions for the coupling factors is made such that the input-to- output transfer function, or an absolute value thereof, takes a maximum value with respect to frequency only at a center frequency; and/or determining one or more conditions for the coupling factors is made such that all factors of a numerator of a representation of a derivative of the input-to-output transfer function with respect to frequency take a zero value at the resonance frequency. That is, a plurality of coupling factors is defined with respect to the center frequency, and therefore, there is more flexibility to control passband width.

It is preferably, the one or more conditions include one or more additional conditions, wherein one of the additional conditions is a critical coupling. More, preferably, the one or more conditions include one or more additional conditions, wherein one of the additional conditions is a quality factor determining a bandwidth of the filter structure. Therefore, it is possible to design a filter structure which can achieve appropriate passband width.

In a preferred embodiment, the coupling factors include additional coupling factors as an input coefficient coupling factor between input of the filter structure and a first resonator, and an output coefficient coupling factor between a last resonator and output of the filter structure. More preferably, the method comprises determining the input coefficient coupling factor m _L and the output coefficient coupling factor m such that m ₂₁ =±(m _L · m ) when m ₂₁ is a coupling factor describing a coupling between a first resonator adjacent to an input of the filter structure and a subsequent second resonator. According to this embodiment, critical coupling is achieved and therefore, the designed filter structure includes possibly less ripple.

In a preferred embodiment, the method comprises as the following steps: deriving an Impedance matrix representing Kirchhoff's voltage law for loops, which include an input loop, one or more inner loops comprising only one resonator and two coupling elements, and an output loop, of the filter structure and including the coupling factors,

wherein

the Impedance matrix Z(M, W) is defined by

Z(M, W) = M + jlil + R for the filter structure with N resonators,

where M = is a matrix with coupling factors (th^),

/ is the identity matrix,

R is a diagonal matrix and

W is a column vector with the frequency variables W*

wherein

where Q is the Quality factor of the resonator i ;

deriving one or more transmission scattering parameters as a function of normalized frequency and as a function of coupling factors based on the derived Impedance matrix; deriving an input-to-output transfer function based on the derived one or more scattering parameters; determining one or more conditions for the coupling factors for which an absolute value of the input-to-output transfer function takes a maximum value with respect to a normalized frequency at a center frequency at which the resonators have a resonance frequency; determining the coupling factors using the one or more determined conditions; and deriving element values of the filter structure in dependence on the determined coupling factors and a quality factor determining a bandwidth of the filter structure. In case of parallel resonance, W* is defined as;

and in case of serial resonance, W ₍ is defined as;

where and are the normalized inductances and capacitances of resonator i, with i =1 to N. According to the present embodiment, the Impedance matrix theory for tuning center frequency is applied, that is, it is possible to determine element values by resolving a derived equation. Hence, the filter structure is designed without considering eigenvalues and therefore, a period for designing filter structure is reduced.

It is preferable that a frequency of the resonator is tunable in dependence on the element values or in combination of the element values. More preferably, a frequency of resonator is tuned to control the center of frequency. According to this embodiment, the frequency of resonator is tunable with respect to the center frequency, and therefore, there is flexibility of design.

In addition, one or more element values of the filter structure are tunable, for example, all resonators may be tunable in the resonant frequency, and all resonators comprise identical elements. Optionally, element values of one or more coupling factors of the filter structure may be tunable. In the latter case, changing the element value of the coupling also change the bandwidth. Thus, it is possible to minimize the number of difference component in the filter, and for changing center frequency of all resonators equally and consistently, using the same value setting for all tunable elements. The above mentioned aspects of the present invention are combined solely or together with further concepts for employing the method of the present invention.

In other words, any of the features, functionalities and details described with respect to one of the methods, could also be used in any of the apparatus, both individually or taken in combination. Also, features, functionalities and details described in the context of an apparatus can be used in combination with the methods. Any of the embodiments mentioned above can optionally be supplemented by any of the features, functionalities and details disclosed herein (also the detailed description of the embodiments).

Brief description of the drawings

Embodiments of the present invention are described in the following with respect to the figures.

Fig. 1 shows a schematic view of a filter composed of N coupled resonators (for N-2);

Fig. 2 shows a diagram of a base circuit model, using N resonators and considering all couplings of a prior art;

Fig. 3 shows a schematic view of a filter topology using 7 directly and partially cross-coupled resonators of a prior art;

Fig. 4 shows a schematic graph of a filter transfer function using 7 directly and partially cross-coupled resonators of a prior art;

Fig, 5 shows a diagram of a filter using 4 coupled resonators of prior art;

Fig. 6 shows a diagram of a filter with 2 coupled resonators and capacitive input and output coupling of prior art;

Fig. 7 shows a schematic chart filter transfer function, depending on coupling factor of prior art;

Fig. 8 shows a diagram of a bandpass filter analyzed in prior art;

Fig. 9 shows a schematic flowchart of an embodiment of the present invention; Fig. 10 shows a schematic chart of a normalized filter transfer function, when using determined coupling factor of an embodiment of the present invention;

Fig. 1 1 shows a schematic chart of a normalized filter transfer function, when using another coupling factor of an embodiment of the present invention;

Fig. 12 shows a schematic chart of a normalized filter transfer function, when varying the ratio of additional coupling factors of an embodiment of the present invention;

Fig. 13 shows a schematic view of an example to illustrate method steps of an embodiment of the present invention;

Fig. 14 shows a diagram of a bandpass filter of an embodiment of the present invention;

Fig. 15 shows a schematic view of a simulated transfer function of an embodiment of the present invention;

Fig. 16 shows a diagram of a simulation setup of an embodiment of the present invention; and

Fig. 17 shows a flow chart of an embodiment in case N-2 according to the present invention; and

Fig. 18 shows a flow chart of an embodiment in case N-3 according to the present invention. Detailed description of embodiments of the invention

The invention is described in detail with regards to Fig. 1 , 9 to Fig. 18. The invention is no way meant to be limited to the shown and described embodiments. The reference signs in the claims and in the detailed description of embodiments of the invention were added merely to improve readability. They are in no way meant to be limiting. The disclosure of document “A coupling matrix synthesis for a tunable band pass filter", A. Jaschke, M. SchCihler, M. Tessema, IEEE EuMW 2017 (hereinafter written as “EUMW17” in the present application) is Incorporated into the disclosure of the present application. An embodiment of the present invention relates to a method for determining element values of a filter structure, for example, filters with band-pass characteristics, composed of N directly coupled resonators arranged in a linear topology. The general structure of such a filter is shown in Fig. 1 in case N-2, where the circles labeled 1 and 2 indicate the N-2 resonators and the arrow labeled with m ₂ 1 indicates the coupling (coupling factor) between the resonators. Additional arrows indicate the coupling (or signal path) between input IN and the first resonator 1 and the second resonator 2 and the output OUT.

While Fig. 1 shows a case of only N=2 resonators, the present invention addresses the dimensioning of the m _i; · for the general case of N³2 resonators, with the objective of achieving a“maximum flat” (no ripple) filter response in the pass-band region of the filter, where N may be selected as required to achieve a required filter selectivity and steep transition between the stop-band and the pass-band region.

Fig. 1 a) assumes a direct signal path between input IN and the first resonator and the second resonator and the output OUT. Fig. 1 b) considers a coupling factor m _L1 between input IN and the first resonator and a coupling factor m _Q2 between the second resonator and the output OUT.

Embodiment for a method according to a flowchart of Fig. 9

Element values of a radio frequency filter structure having a linear sequence of coupled resonators are determined according to a method of the present invention. A coupling between adjacent resonators, for example, coupling of two resonators (i.e., N=2 as shown in Fig. 1 (a)), is determined by a coupling factor m ₂₁ (see, for example, Fig. 1 (a)). Element values are derived according to following steps.

Derivation of input-to-output transfer function (S2), i.e., an input-to-output transfer function is derived, e.g., in a symbolic form, i.e., as a parameterized equation and as a form of any one of 5 _2I (M, W), S ₂₁ (M, w), |5 ₂₁ (M, W)|, |S ₂₁ (M, w)|, which is dependent on a plurality of coupling factors, on the basis of a matrix representing one of Kirchhoff’s laws for loops or nodes of the filter structure and including the coupling factors. The matrix representing one of Kirchhoff’s laws is an Impedance matrix or an Admittance matrix, and wherein the matrix is determined as a function of input resistance, output resistance, resonator detuning and as a function of coupling factors. The symbol“S” corresponds to scattering parameter,“M” corresponds to a matrix with coupling factors,“W” corresponds to a column vector with the frequency variables W _£ (i - 1 ... to N, in this case, i = 1 , 2), and“w” is value of W. Determine conditions for the parameters (S4), i.e., one or more conditions, i.e., one or more equations for the coupling factors such that the input-to-output transfer function, or an absolute value thereof, takes a maximum value with respect to frequency at a center frequency are determined. That is, one or more conditions for the parameters, i.e., coupling factors for which the input-to-output transfer function takes a local maximum with respect to frequency at a center frequency, e.g., W _έ =0 at w= _center .

Determine coupling factors (S6), i.e., the coupling factors are determined using the one or more determined conditions, i.e., ambiguity resolution for coupling factors (parameters) and set of coupling factors m _i; ·.

Derivations of element values (S8), i.e., element values of the filter structure are derived in dependence on the determined coupling factors th^. For deriving element values, bandwidth in formation, i.e., quality factor Q may be also used as an additional condition. The element values, e.g., value of inductor Li and value of conductor Q for the filter structure are determined.

Embodiment for a method according to a flowchart of Fig. 17

In the following, some optional details, extensions and improvements of the method described above will be explained with respect to Fig. 17.

In case of the filter structure having an additional input and output coupling, defined by a coupling factor m _L1 between input and the first resonator and a coupling factor m _Q2 between the second resonator and the output as shown in Fig. 1 (b), element values may also be derived as shown in Fig. 17.

Coupling factors m _i ·, (and optionally, also coupling factors m _L1 and m _Q2 ) may, for example be derived according to the present invention as follows:

An Impedance matrix is determined (S10). In this step, the Impedance matrix is determined as a function of (normalized) frequency (e.g., w) and as a function of coupling factors The Impedance matrix is defined as Z(M, W) = M + jlfl + R for the filter structure with N resonators, / ^m u ^m i N \

where M = - - - is a matrix with coupling factors ? n _i ·,

i ... m _NN J

I is the identity matrix,

R is a diagonal matrix and

W is a column vector with the frequency variables W^

Where W* is defined as W _έ = wI _έ - Vo _j q ' ⁿ case of parallel resonance,

or W* is defined as W; = WQ - ^/ _w in case of serial resonance,

and li and c _{ are the normalized inductances and capacitances of resonator i, with i =1 to N.

A transmission S-parameter is derived based on the Impedance matrix (S12). That is, transform the impedance matrix Z(M, W) into the set of S-Parameters, where each of the S- Parameter is a function of M and W . The S-parameters are derived as a function of (normalized) frequency and as a function of a coupling factors, e.g., a form of S ₂₁ (M, W) or

S ₂ I(M, w).

The input-to-output transfer function \S ₂₁ (M, W)| or |S ₂₁ (M, w) |, where the |x| denotes the absolute value of x, is derived (step S14). The input-to-output transfer function is derived as a function of (normalized) frequency and as a function of coupling factors.

One or more conditions for the parameters, i.e., one or more equations for the coupling factors are determined (S16) for which the input-to-output transfer function takes a local maximum with respect to frequency at a center frequency (e.g., W _έ =0 at w = () _center .). That is, identify the maxima of |5 ₂₁ ( , W)|, by differentiating this function with respect to W and setting the differentiated function to 0.

Coupling factors are derived based on the determined one or more conditions (S18). For deriving coupling factors, it is assumed that all W _c = W* = · · = P _N , and it is further assumed that all W ₍ = 0. Then, element values, value of inductor Li and value of conductor Ci, are derived based on the derived coupling factors (S20).

When the above steps are applied to a filter structure composed of N coupled resonators as shown in Fig. 1 (a), resolving the N equations resulting in step S18 for will provide the coupling factors m ₂₁ . By construction in step S18, the resulting transfer function will have the intended“maximum flat” behavior as setting W _c = W, = ··· = W _N is a prerequisite to achieve critical coupling and further setting W _£ = 0 results in all normalized resonator center frequencies to constructively overlap at w = 0.

Translation from normalized resonator center frequency to an actual target center frequency is achieved by calculating the de-normalized coupling factor m _i · = ^mi,i /Q, where Q is the quality factor (or “Q factor”) of the resonator, characterizing the resonator’s bandwidth relative to the center frequency.

As intended, this lead to a transfer functions with “maximum flat” (“maximum flatness)" shown in Fig. 10. In case of resolving ambiguity, additional criteria is used for filter response beyond the equation (condition) of step 16 to select appropriate m _i} having possibly less ripples in the passband region, as shown in Fig. 11.

As mentioned above, the coupling factors m _L1 and m _Q2 are derived according to the method of the present embodiment. That is, the method of the present embodiment also applies to a filter structure composed of N coupled resonators as shown in Fig. 1 (b), with 2 additional coupling factors, m _L1 and m _Q2 , between input and the first resonator and between the second resonator and the output, respectively. Resolving the then N+2 equations resulting in step S18 for m _L1 , m _Q2 and m _£; · will provide the additional coupling factors, m _L1 and m _Q2 as well as the coupling factors m _i - . Again, by construction in step S18, critical coupling is achieved, resulting in a transfer function with the intended“maximum flat” behavior.

Considering the special case N=2 and 2 additional coupling factors, m _L1 and m _Q2 , applying the above steps leads to

m ₂ 1 = ±( n _L1 m _Q2 ).

Other than m ₂₁ = 1 required for a structure with N-2 and direct input and output paths, the additional coupling factors, m _L1 and m _Q2 provide for additional design flexibility. Varying the ratio of m _L1 to m _Q2 while keeping the product equal to m ₂₁ results in the different filter response, as shown in Fig. 12, allowing to control the passband bandwidth and/or the steepness of the transition between passband and stopband region.

The above steps may be applied independent of the actual implementation of the filter, e.g. independent of the coupling elements being inductors or capacitors or transformers in an electrical circuit, or being e.g. mechanical, acoustical, surface waves, micro waves or optical coupling in an alternative technology implementation; there exists also flexibility in the arrangement of the coupling elements, e.g. parallel or serial with respect to the resonators.

The resonators may use tunable elements, including capacitors that can be continuously or in discrete steps varied over a certain capacity range and including inductors that can be continuously or in discrete steps varied over a certain inductivity range. Use of such tunable elements may be for the purpose of changing the filter center frequency and/or passband bandwidth in response to changing the value of the tunable element. In addition, such tunable elements may be used for the purpose compensating for component variation or drift, resulting from external impacts like change in temperature or supply voltage.

Regarding use of tunable elements for the construction of a bandpass filter with adjustable center frequency and/or adjustable bandwidth, the steps S10 to S20 may be easily carried out for different parameters across the adjustable center frequency and/or adjustable bandwidth. The sets of coupling factors (one set per parameter setting) may then be post processed, to derive the collectively best set of coupling factors that optimizes filter performance over the full range of adjustable center frequency and/or adjustable bandwidth.

The tunable elements used in the plurality of resonators can be of the same type, having the same variability range. This is advantageous for minimizing the number of different component in the filter, and for changing center frequency of all resonators equally and consistently, using the same value setting for all tunable elements.

Embodiment for a method to a flowchart of Fio. 18

An example illustrating the execution of steps S30 to S40 of Fig. 18 for a filter using N-3 coupled resonators is given below (see Fig. 13). This example is intended to teach the general principles, and can be easily translated to other values of N and to filter designs with additional input and output coupling m _L1 , m _Q2 .

Determination of an impedance matrix Z(M, W) = M + jin + R as a function of (normalized) frequency (e.g., w) and as a function of coupling factors (e.g., m _i7 ) is derived (S30). This is e.g. done by applying Kirchhoff’s law. Using Kirchhoff’s loop rule, this is explained in further details in and section II of incorporated document EUMW17, for the specific circuit structure show in Fig. 1 of the incorporated document EUMW17, providing one possible implementation of the topology from Fig. 13.

Since a line structure m ₃₁ = m ₁₃ = 0 is used, as all three resonators are working at the same resonance frequency = m ₂₂ = m ₃₃ = m ₀ and W ₁ = W ₂ = W ₃ = W (for example, if the resonators are equal). When passive coupling elements where used, the coupling factors are behaving reciprocally. Hence, it is assumed: m ₂₁ = m ₁₂ and m ₃₂ = m ₂₃ .

Now Z(M, W) is rewritten to

Thus, Z(M,W) is a 3x3 Matrix. For example, impedance matrix Z(M, W may represent Kirchhoff’s law for loops I, II and III as shown in Fig. 8. That is, a first line of the above impedance matrix describes a relationship between a source voltage and resonator currents of the 3 (N) resonators, wherein an input loop is considered; a second line (intermediate line) of the impedance matrix describes voltage contributions in closed loop comprising a respective resonator and couplings with respective adjacent resonators in dependence on the resonator currents of the 3 resonators; and a third line (last line) of the impedance matrix describes voltage contributions in a closed output loop in dependence on the resonator currents of the 3 resonators.

A set of (forward) transmission scattering parameter (S-parameter) as a function of (normalized) frequency and as a function of coupling factors is derived (S32). That is, the impedance matrix Z(M, W) is transformed into the set of S-Parameters as S ₂₁ (M, W) or 5 ₂₁ (M, w). In order to receive S-Parameters from Z-Matrix, known definition of “Microstrip Filters for RF Microwave Applications” , S.-J. Hong, M.J. Lancaster, Chapter 8, John Wiley & Sons Inc. 2001 is used.

S _2i — 2 Z ₃

Thus, S _2i is a complex value and can be written to

An input-to-output transfer function as a function of (normalized) frequency and as a function of coupling factors is derived (S34). That is, the input-to-output transfer function |S ₂₁ (M, W)| or \S ₂₁ (M, w)| is derived. In order to analyze or plot the transfer function, the normalized form has to be considered. Therefore, m ₀ = 0 has been set and the absolute value of S _2i has been formed.

Therefore S ₂₁ with m ₀ = 0 is written as:

Now the absolute value is built by using the well-known definition:

One or more conditions for the parameters (coupling factors) for which the input-to-output transfer function takes a local maximum with respect to frequency at a center frequency are determined (S36). That is, to identify the maxima of \S ₂₁ (M, W)|, by differentiating this function with respect to W (or with respect to w) and setting the differentiated function to 0

To solve the maxima in W, it has to be set = 0. Hence, it is written

dfi By solving the equation to W, it is achieved three maxima at:

For example, the parameters, i.e., the coupling factors, are chosen such that the derivation of |S ₂₁ | has a zero of multiplicity higher than one at the center frequency (e.g., W=0). Thus, it is possible, for example, to achieve maximum flatness in an environment of the center frequency. Coupling factors are derived based on the one or more determined conditions, for example, the parameters (coupling factors) will be chosen such that and/or to resolve an ambiguity (S38). That is, assume all = W,- = ··· = £t _N , and it is further assumed all W ₍ = 0 (at 5 u>=(A)ce _nt er)· For example, the parameters will be chosen such that and/or to resolve an ambiguity. The three maxima will be joined to the point of the normalized center frequency at = W ₂ = W ₃ = 0. Hence, with W ₂ and W ₃ , it is possible to calculate a definition of m (S38). In order to simplify the circuit design, it is set m ₂₁ = m ₃₂ and has to be solved and achieve the definition of m _2i for critical coupling with:

In order to build/design a filter circuit, it is required to transform the topology in to a circuit.

15

The incorporated document EUMW17 shows how to transform the topology into a circuit and how to de-normalize the matrix into lumped component values. The relation between coupling factor and circuits coupling element is given by: m = ± ¹ l _x

20

where X is the normalized reactance (for example, at the center frequency).

Element values, value of inductor Li and value of conductor Ci are derived based on the derived coupling factors (S40). An example (see Fig. 1 of the incorporated document 25 EUMW17) is given as Fig. 14 for a bandpass filter circuit with three coupled resonators, ( - _L , C _x ); ( L ₂ , C ₂ ) and (L ₃ , C ₃ ), the coupling elements L ₄ and L ₅ and with direct signal paths at input (U ₀ , R ₁ ) and output R ₂ .

For a maximum flat and lossless pass band at f _c = 470 MHz and applying the above sizing 30 of m ₂₁ = m ₃₂ = ± ^= , the coupling elements can be calculated to L ₄ = L _s = 5,38 nH as follows: • Assuming a quality factor Q = 11.235 for a pass-band bandwidth of approximately f _BW = 41.8 MHz, it is calculated

The calculation for the resonators values are given in the incorporated document EUMW17. The forward transfer function of the obtained circuit is depicted in Fig.14, together with the circuit schematic used for simulation in Fig. 15 using lumped elements. Marker ml in Fig. 14 shows the passband with losses of -0.00000731 dB, which can be assumed 0 dB within the numerical precision in the simulation. The pass band can be seen as lossless and maximal flat. Note that, as a result of de-normalization, the de-normalized transfer function (Fig. 14) of the filter circuit is not necessarily identical to the normalized transfer function obtained for the filter topology (see, e.g., Fig. 9).

As already described, a certain structure of the coupling matrix is used, namely

That is, the elements of the main diagonal are all equal (m ₀ ) and there are N-1 further elements in the first secondary diagonal (m _;i , wherein i=1 to N-1 , j=2 to N, j=i+1 ). The 'upper' and 'lower' secondary diagonals are identical. All other elements are 0. Thus, in total there are N elements unequal to 0, or N degrees of freedom.

In the example, it seems that the equation _s j -l+m ₂₁ + m| ₂ = 0 is ambiguous. It is characteristic for the method that the boundary condition m ₂₁ = m ₃₂ is met. This boundary condition results from the desire for a critical coupling. Thus, the solution (except for the sign) is clearly determined, based on the degree of freedom 1 , i.e. The selection of W ₁ =W _£ =···=W _h means that all normalized resonators have an identical center frequency and Is a fact that is also characteristic of the method. However, the resonators are not necessarily identical, since W; = - U _w£: . That is, by selecting I, and q to be different, e.g., the slope of the filter function may be varied. However, only one of I, and may be freely selected in this case. The respective other parameter must then be selected in such a way that the desired W _έ results.

According to described aspect of the present invention, it is possible to minimize additional coupling by using the conductor structure. This has the advantage that the circuit may be implemented to be more robust against interference effects. Couplings influence the filter characteristic and are very sensitive parameters. In addition, use of the conductor structure with respect to the adjustability of the center frequency. That is, couplings are frequency- dependent variables and influence the tuning range if the filter is tuned in, e.g., the center frequency. By avoiding feedback and/or cross-coupling, an adjustable RF filter may be realized in a simpler and more robust manner.

Furthermore, the bandwidth of a bandpass may be varied by changing the coupling factors, taking into account described equations, which result from the calculation scheme. If the conditions from the solution approach are adhered to, the bandwidth may be adjusted as flat as possible and lossless in the passband. The following has to be taken into account: by using transformable coupling, the center frequency remains constant; and by using capacitive or inductive coupling, the center frequency varies as the bandwidth changes.

Due to the filter design, eigenvalues from tables are not necessary to design the filter circuit according to the method of the present invention. Therefore, the calculation time/working time of the circuit design is reduced. By using the method of the present invention, the selection of the coupling element (capacitance or inductance) is freely configurable. Other known coupling methods (for example, methods disclosed in the cited documents mentioned in the background of the present description) are also freely selectable.

According to the method of the present invention, the filter circuit may be minimized, since additional circuit elements (e.g. matching networks etc.) are not required. Additional components may be saved and associated costs may therefore be reduced.

According to the method of the present invention, topological arrangement is used. If additional circuit components such as matching networks are used, a filter characteristic with a maximum flat passband may also be realized by, e.g., a different topological arrangement. A lossless passband could also be possible by optimization algorithms. In addition, a circuit realization may be realized by calculating the circuit elements by means of the low-pass bandpass transformation. Implementation alternatives

Some or all of the method steps may be executed by (or using) a hardware apparatus, like for example, a microprocessor, a programmable computer or an electronic circuit. In some embodiments, some one or more of the most important method steps may be executed by such an apparatus.

Depending on certain implementation requirements, embodiments of the invention can be implemented in hardware or in software. The implementation can be performed using a digital storage medium, for example a floppy disk, a DVD, a Blu-Ray, a CD, a ROM a PROM, an EPROM, an EEPROM or a Flash memory, having electronically readable control signals stored thereon, which cooperate (or are capable of cooperating) with a programmable computer system such that the respective method is performed. Therefore, the digital storage medium may be computer readable.

Some embodiments according to the invention comprises a data carrier having electronically readable control signals, which are capable of cooperating with a programmable computer system, such that one of the methods described herein is performed.

Generally, embodiments of the present invention can be implemented as a computer program product with a program code, the program code being operative for performing one of the methods when the computer program products runs on a computer. The program code may for example be stored on a machine readable carrier.

Other embodiments comprise the computer program for performing one of the methods described herein, stored on a machine readable carrier.

In other words, an embodiments of the invention method is, therefore, a computer program having a program code one of the methods described herein, when the computer program runs on a computer.

A further embodiment of the invention method is, therefore, a data carrier (or a digital storage medium, or a computer-readable medium) comprising, recorded thereon, the computer program for performing one of the methods described herein. The data carrier, the digital storage medium or the recoded medium are typically tangible and/or bib-transitionary. A further embodiment of the invention method is, therefore, a data stream or a sequence of signal representing the computer program for performing one of the methods described herein. The data stream or the sequence of signals may for example be configured to be transferred via a data communication connection, for example via the Internet.

A further embodiment comprises a processor means, for example a computer, or a programmable logic device, configured to or adapted to perform one of the methods described herein.

A further embodiment comprises a computer having installed thereon the computer program for performing one of the methods described herein.

A further embodiment according to the invention comprises an apparatus or a system configured to transfer (for example, electronically or optically) a computer program for performing one of the methods described herein to a receiver. The receiver may, for example be a computer, a mobile device, a memory device or the like. The apparatus or system may, for example, comprise a filter server for transferring the computer program to the receiver.

In some embodiments, a programmable logic device (for example a field programmable gate array) may be used to perform some or all of the functionalities of the methods described herein. In some embodiments, a field programmable gate array may cooperate with a microprocessor in order to perform one of the methods described herein. Generally, the methods are preferably performed by any hardware apparatus.

The apparatus described herein may be implemented using a hardware apparatus, or using a computer, or using a combination of a hardware apparatus and a computer.

The method described herein may be implemented using a hardware apparatus, or using a computer, or using a combination of a hardware apparatus and a computer.

The above described embodiments are merely illustrative for the principles of the present invention. It is understood that modifications and variations for the arrangements and the details described herein will be apparent to others skilled in the art. It is the intent, therefore, to be limited only by the scope of the impending patent claims and not by the specific details presented by way of description and explanation of the embodiments herein.