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Elliptic-Function Narrowband Bandpass Filter (Closed Form): NBPFE



NBPFE models represent lumped-element elliptic-function (Cauer) bandpass filters. The applicability of this filter type is not limited to narrow bandwidths, as the name would appear to imply. The group delay is flatter than that of a "regular" elliptic-function bandpass filter of the same bandwidth, especially for wideband filters. And the passband magnitude displays arithmetic, rather than geometric, symmetry. The insertion loss ripples between zero and a specified maximum in the passband. The stopband attenuation increases rapidly between the passband edges and the stopband edges, and ripples between a specified minimum stopband attenuation and infinity. This type of filter offers optimum selectivity at the expense of increased complexity and poor group delay flatness.


Name Description Unit Type Default
ID Element ID Text NBPFE1
N Order of filter's lowpass prototype.   3
FP1 Passband corner frequency (when Qu is infinite). Frequency 0.5 GHz
FP2 Passband corner frequency (when Qu is infinite). Frequency 1.5 GHz
AP Maximum Passband Insertion Loss (when Qu is infinite). DB 0.1 dB
AS Minimum Stopband Attenuation (when Qu is infinite). DB 20 dB
*FS1 Stopband corner frequency (when Qu is infinite). Frequency 0 GHz
*FS2 Stopband corner frequency (when Qu is infinite). Frequency 0 GHz
*DM Where to put design margin (see below for details).   1
*RS Source resistance. Resistance 50 ohm
*RL Load resistance Resistance 50 ohm
*QU Average unloaded Q of the bandpass resonators.   0

* indicates a secondary parameter

Parameter Details

An elliptic-function narrowband bandpass filter is completely specified by any three of the following four parameters:

  • lowpass prototype filter order, N,

  • maximum passband insertion loss, AP,

  • minimum stopband attenuation, AS, and

  • selectivity.

Filter selectivity, in turn, is completely specified by any three of the following four parameters:

  • frequency of the upper edge of the lower stopband, FS1,

  • frequency of the lower edge of the passband, FP1,

  • frequency of the upper edge of the passband, FP2, and

  • frequency of the lower edge of the upper stopband, FS2.

There are eighteen valid sets of these parameters, and these are listed in the table below. Columns 2 through 9 represent individual parameters of the filter, while the rows represent the valid parameter sets - which are numbered in column 1. The last column indicates which parameter is unspecified, or is incompletely specified, in the set. For each specific parameter, the body of the table indicates whether that parameter is specified, unspecified, or ignored in a particular parameter set. "S" means the parameter value is specified (non-zero), "0" means it has been set to zero (indicating it is intentionally unspecified), and "x" means it is ignored by the model.

Set N AP AS FP1 FP2 FS1 FS2 DM Incompletely Specified Parameter
1 S S S S S x x x Selectivity
2 S S S S 0 S x x Selectivity
3 S S S S 0 0 S x Selectivity
4 S S S 0 S S x x Selectivity
5 S S S 0 S 0 S x Selectivity
6 S S S 0 0 S S x Selectivity
7 S S 0 S S S x x AS
8 S S 0 S S 0 S x AS
9 S S 0 S 0 S S x AS
10 S S 0 0 S S S x AS
11 S 0 S S S S x x AP
12 S 0 S S S 0 S x AP
13 S 0 S S 0 S S x AP
14 S 0 S 0 S S S x AP
15 0 S S S S S x S N
16 0 S S S S 0 S S N
17 0 S S S 0 S S S N
18 0 S S 0 S S S S N

N. In mathematical terms, N is defined as the highest exponent of the complex frequency variable s in the transfer function, S21(s), of the filter's normalized lowpass prototype, or, equivalently, half of the highest exponent of s in the transfer function of the narrowband bandpass filter. In terms of the number of circuit components, N corresponds to the total number of resonances in single-mode or multi-mode direct-coupled-resonator microwave filters; while, for lumped-element filters, N corresponds to the number of lumped-resonant (LC) circuits that produce zeros of attenuation at finite frequencies. And, in terms of a measurable electrical characteristic, N corresponds to the number of positive-frequency passband reflection (|S11|) zeros.

DM. If zero is specified for N, the model will determine N. But, since N must be an integer, there will typically be some design margin available, and this margin must be assigned to some parameter other than N. Consequently, after determining N, the model will compute a new value for one or more of the other parameters. The value of DM determines which parameter, or parameters, will have their values recomputed after a value is synthesized for N. For each valid value of DM, the table below specifies which parameters (X) are effected by the additional design margin and the manner in which they are effected.

DM AP AS FP1 FP2 FS1 FS2 Description
0 X           decrease AP
1   X         increase AS
2     X X     Decrease FP1 Increase FP2
3     X   X   Decrease FS1 & FP1
4     X     X Decrease FP1 & FS2
5       X X   Increase FS1 & FP2
6       X   X Increase FP2 & FS2
7         X X Increase FS1 Decrease FS2
8     X X X X Increase FS1 & FP2 Decrease FP1 & FS2

Parameter Restrictions and Recommendations

  1. If any of the parameters N, AP, AS, FP1, FP2, FS1, or FS2 are set to 0, the model considers them "unspecified." A value of zero for any of these parameters may, or may not, be valid, depending on whether the set of parameters as a whole corresponds to one of the valid parameter sets listed in the table, above.

  2. If N is specified, it must fall within the range 0 < N < 27.

    If N=0, then 0 ≤DM ≤ 8 , otherwise DM is ignored.

  3. If Ap and AS are both specified, 0 < AP < AS, otherwise 0 < AP or 0 < AS.

    Recommend AP greater than or equal to 0.001 dB.

  4. "Selectivity" is fully specified by any three of FS1, FP1, FP2, and FS2.

    If FS1, FP1 & FP2 are specified, 0 < FS1 < FP1 < FP2.

    If FS1, FP1 & FS2 are specified, 0 < FS1 < FP1 < FS2.

    If FS1, FP2 & FS2 are specified, 0 < FS1 < FP2 < FS2.

    If FP1, FP2 & FS2 are specified, 0 < FP1 < FP2 < FS2.

  5. "Selectivity" is partially specified by any two of FS1, FP1, FP2, and FS2.

    If FS1 & FP1 are specified, 0 < FS1 < FP1.

    If FS1 & FP2 are specified, 0 < FS1 < FP2.

    If FS1 & FS2 are specified, 0 < FS1 < FS2.

    If FP1 & FP2 are specified, 0 < FP1 < FP2.

    If FP1 & FS2 are specified, 0 < FP1 < FS2.

    If FP2 & FS2 are specified, 0 < FP2 < FS2.

  6. 0 < RS.

    0 < RL.

  7. QU > 0 specifies a finite unloaded Q (recommend QU ≤ 1e12 ).

    QU = 0 specifies an infinite unloaded Q.

Implementation Details

The model is implemented as a short-circuit admittance matrix,

, whose equivalent normalized lowpass prototype transfer function, S21(s), is:

where Rn is the Elliptic Rational Function, and

where cd is a Jacobian elliptic function, K is Legendre's complete elliptic integral of the first kind, and a narrowband-bandpass-to-lowpass frequency transformation has been applied:


This frequency transformation has good delay preserving properties for wide band filters and produces passband amplitudes with arithmetic symmetry. Note that _FREQ is the variable containing the project frequency, and the frequency parameters FP1, FP2, FS1, and FS2 are related according to: FP1xFP2=FS1xFS2

The seven parameters of the elliptic-function bandpass filter (N, AP, AS, FP1, FP2, FS1, and FS2) are related by "the degree equation":



The admittances are given by:


This element does not have an assigned layout cell. You can assign artwork cells to any element. See “Assigning Artwork Cells to Layout of Schematic Elements” for details.

Recommendations for Use

Note that this model behaves as if it has ideal impedance transformers at its ports, so there is no attenuation due to mismatched source and load impedances. The model expects that the source impedance will equal RS and that the load impedance will equal RL, but RS need not have any special relationship to RL for ideal transmission (as would normally be the case).


[1] Miroslav D. Lutovac, Dejan V. Tosic, and Brian L. Evans, Filter Design For Signal Processing Using MATLAB and Mathematica, (Prentice Hall, 2001), Chapters 6, 12, and 13.

[2] Alexander J. Grossman, "Synthesis of Tchebycheff parameter symmetrical filters," Proceedings of the IRE, pp. 545-473, April 1957.

[3] Dante Youla, "A tutorial exposition of some key network-theoretic ideas underlying classical insertion-loss filter design," Proc. IEEE, vol. 59, no. 5, pp. 760-799, May 1971.

[4] Miroslav Vleck and Rolf Unbehauen, "Degree, ripple, and transition width of elliptic filters," IEEE Trans. Circuits Syst., vol. 36, no. 3, pp. 469-472, March 1989.

[5] H. J. Orchard and Alan N. Willson, Jr., "Elliptic functions for filter design," IEEE Trans. Circuits Syst., I, vol. 44, no. 4, pp. 273-287, April 1997.

[6] Rolf Schaumann, Mohammed S. Ghausi, and Kenneth R. Laker, Design of Analog Filters: Passive, Active RC, and Switched Capacitor, (Prentice-Hall, 1990), pp. 49-51.

[7] Kendall L. Su, Handbook of Tables for Elliptic-Function Filters, (Kluwer Academic, 1990).

[8] Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. Stegun, (U. S. National Bureau of Standards, 1964), Chapters 16 and 17 by L. M. Milne-Thomson.

[9] H. Blinchikoff, "A note on wide-band group delay," IEEE Trans. Circuit Theory, pp. 577-578, Sept. 1971.

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