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# scipy.signal.besselap

`scipy.signal.``besselap`(N, norm='phase')

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

  Parameters:

  N : int

  > The order of the filter.

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

  > Frequency normalization:

  > `phase`

  >   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. [R199]

  >   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.

  > `delay`

  >   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

  > `mag`

  >   The filter is normalized such that the gain magnitude is -3 dB at angular frequency 1. This is called “frequency normalization” by Bond. [R194]

  > New in version 0.18.0.

  Returns:

  z : ndarray

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

  p : ndarray

  > Poles of the transfer function.

  k : scalar

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

  See also

  `bessel`
    Filter design function using this prototype

  Notes

  To find the pole locations, approximate starting points are generated [R195] for the zeros of the ordinary Bessel polynomial [R196], then the Aberth-Ehrlich method [R197] [R198] is used on the Kv(x) Bessel function to calculate more accurate zeros, and these locations are then inverted about the unit circle.

  References

  [R194] | (1, 2) C.R. Bond, “Bessel Filter Constants”, http://www.crbond.com/papers/bsf.pdf

  [R195] | (1, 2) Campos and Calderon, “Approximate closed-form formulas for the zeros of the Bessel Polynomials”, arXiv:1105.0957.

  [R196] | (1, 2) 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.

  [R197] | (1, 2) Aberth, “Iteration Methods for Finding all Zeros of a Polynomial Simultaneously”, Mathematics of Computation, Vol. 27, No. 122, April 1973

  [R198] | (1, 2) 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

  [R199] | (1, 2) Miller and Bohn, “A Bessel Filter Crossover, and Its Relation to Others”, RaneNote 147, 1998, http://www.rane.com/note147.html