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Generalized Chebyshev Narrowband Bandpass Filter (Closed Form): NBPFCG



NBPFCG models represent lumped-element Generalized Chebyshev (or "Quasi-Elliptic") narrowband bandpass filters. The applicability of this filter type is not limited to narrow ( i.e., less than 5%) bandwidths, as the name would appear to imply. The group delay is flatter than that of a "regular" Generalized Chebyshev 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 is defined by arbitrarily specified transmission zeros. Real-frequency finite transmission zeros can be specified to improve selectivity at the expense of ultimate stopband attenuation and passband group delay, while complex-plane finite transmission zeros can be specified to provide passband group delay equalization at the expense of selectivity and ultimate stopband attenuation. Generalized Chebyshev filters represents a compromise between the simplicity of Chebyshev filters, the optimum amplitude response of more complicated Elliptic filters, and the phase linearity of Bessel and Gaussian filters. Because this type of filter allows one to make explicit design trade-offs between complexity, selectivity, and group delay equalization, it is often used to meet the demanding requirements of modern communications systems.


Name Description Unit Type Default
ID Element ID Text NBPFCG1
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
*PPD Passband parameter description:- Maximum Insertion Loss,- Minimum Return Loss, or- Maximum VSWR. Enumerated Maximum Insertion Loss
PPV Passband parameter value (when Qu is infinite) dB or Scalar 0.1 dB
TZF Real frequency, finite transmission zeros (Real) Frequency {2} GHz
*TZR Real parts of complex finite transmission zeros (Imaginary) Frequency {0} GHz
*RS Source resistance Resistance 50 ohm
*RL Load resistance Resistance 50 ohm
*QU Average unloaded Q of the lumped bandpass resonators.   0

* indicates a secondary parameter

Parameter Details

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 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 (S|11|) zeros.

PPD & PPV. Parameters PPD and PPV work together to specify the characteristic of the filter's passband. PPD is used to indicate what the value of PPV represents. The flexibility these parameters provide eliminates the need to manually convert from the passband specification parameter of one's preference into whatever specific parameter the software was written to accept.

TZF & TZR. List parameters TZF and TZR are used to specify the complex transmission zeros, Z, of the bandpass filter response. Of the N positive-frequency complex transmission zeros, Zi = TZRi+jTZFi, the model allows up to (N-1) to be specified. Each consists of a real part, TZRi, and an imaginary part, the real frequency TZFi. If TZRi is not specified, it is assumed to be zero. You must provide transmission zeros with nonzero real parts in pairs; that is, for each zero A+jB transmission, zero -A+jB must be present. Each unspecified transmission zero is mapped to a normalized lowpass prototype frequency of infinity. Each unspecified transmission zero is mapped to a normalized lowpass prototype frequency of infinity.

Parameter Restrictions and Recommendations

  1. 32 > N > 1.

  2. FP1 > 0.

    FP2 > 0.

    FP1 ≠ FP2.

  3. If PPD = "Maximum Insertion Loss", then PPV > 0.

    If PPD = "Minimum Return Loss", then PPV > 0.

    If PPD = "Maximum VSWR", then PPV > 1.

  4. TZFi > 0 .

    If TZRi is specified, then TZFi must be specified.

    If TZRi is zero, then FP1 >TZFi>FP2 .

    If TZRi≠0 , there must be a TZRk=-TZRi and a TZFk=TZFi .

  5. RS > 0.

    RL > 0.

  6. 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 [1, 2]:

where FN and EN are normalized lowpass prototype polynomials of order N, and

A specified narrowband bandpass transmission zero, Z[i] = TZR[i] + jTZF[i] is mapped to a normalized lowpass prototype transmission zero, z[ i ], using [5]:

And, _FREQ (the variable that represents the project frequency) is mapped to the normalized lowpass prototype radian frequency,ω, using [5]:

while the specified narrowband bandpass resonator unloaded Q, QU, is converted to an equivalent lowpass prototype element unloaded Q, qu, using [5][6]:


is constructed using a doubly recursive algorithm [1][3]:

where i = 2 to N and, employing the normalized lowpass prototype transmission zero, zk, for k = 1 to N:

And, polynomial EN(s) is found by applying the "alternating singularity principle" [1][2][4] to the roots of

. Then, EN and FN are split into complex-even and complex-odd polynomials [2] such that EN= Ee+E0 and FN= Fe+F0, where

and ei and fi (i = 0 to N) are the complex coefficients of EN and FN. Finally [3]:


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

The transmission zeros can be tuned or optimized by assigning variables to the elements of the TZF and/or TZR lists and then tuning or optimizing these variables.

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).

This filter model is non-causal and not usable in transient simulations. An error message is issued if a transient simulation of a circuit containing this model is attempted. (Causality is defined as the response of a circuit following a stimulus--not preceding a stimulus. Non-causal models do not correspond to a physically realizable device.)


[1] Richard J. Cameron, "Fast generation of Chebyshev filter prototypes with asymmetrically-prescribed transmission zeros," ESA J., vol. 6, pp. 83-95, 1982.

[2] Richard J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Tech., vol. 47, no. 4, pp. 433-442, April 1999.

[3] Douglas R. Jachowski, unpublished notes, 1995 and 2002.

[4] J. D. Rhodes and A. S. Alseyab, "The generalized Chebyshev low pass prototype filter," Int. J. Circuit Theory Applicat., vol. 8, pp. 113-125, 1980.

[5] H. J. Blinchikoff and A. I. Zverev, Filtering in the Time and Frequency Domains, (Robert E. Krieger Publishing Co., 1987), pp. 178-186, 272.

[6] George L. Matthaei, Leo Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, (Artech House, 1980), pp. 149-156.

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