scipy.signal.besselap(N, norm='phase')[source]#

Return (z,p,k) for analog prototype of an Nth-order Bessel filter.


The order of the filter.

norm{‘phase’, ‘delay’, ‘mag’}, optional

Frequency normalization:


The filter is normalized such that the phase response reaches its midpoint at an angular (e.g., rad/s) cutoff frequency of 1. This happens for both low-pass and high-pass filters, so this is the “phase-matched” case. [6]

The magnitude response asymptotes are the same as a Butterworth filter of the same order with a cutoff of Wn.

This is the default, and matches MATLAB’s implementation.


The filter is normalized such that the group delay in the passband is 1 (e.g., 1 second). This is the “natural” type obtained by solving Bessel polynomials


The filter is normalized such that the gain magnitude is -3 dB at angular frequency 1. This is called “frequency normalization” by Bond. [1]

Added in version 0.18.0.


Zeros of the transfer function. Is always an empty array.


Poles of the transfer function.


Gain of the transfer function. For phase-normalized, this is always 1.

See also


Filter design function using this prototype


To find the pole locations, approximate starting points are generated [2] for the zeros of the ordinary Bessel polynomial [3], then the Aberth-Ehrlich method [4] [5] is used on the Kv(x) Bessel function to calculate more accurate zeros, and these locations are then inverted about the unit circle.



C.R. Bond, “Bessel Filter Constants”,


Campos and Calderon, “Approximate closed-form formulas for the zeros of the Bessel Polynomials”, arXiv:1105.0957.


Thomson, W.E., “Delay Networks having Maximally Flat Frequency Characteristics”, Proceedings of the Institution of Electrical Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.


Aberth, “Iteration Methods for Finding all Zeros of a Polynomial Simultaneously”, Mathematics of Computation, Vol. 27, No. 122, April 1973


Ehrlich, “A modified Newton method for polynomials”, Communications of the ACM, Vol. 10, Issue 2, pp. 107-108, Feb. 1967, DOI:10.1145/363067.363115


Miller and Bohn, “A Bessel Filter Crossover, and Its Relation to Others”, RaneNote 147, 1998,