Filter of Bessel

The filter of Bessel , also indicated under the name of filter of Thompson , is a polynomial filter (“any pole”) whose principal characteristic is to offer a constant time in band-width. Concretely, that means that all the pure frequencies, in band, cross it in a rigorously equal time. The filter of Bessel thus makes it possible to minimize the distortion which a complex signal at the time of an operation of filtering undergoes.

Mathematically, the filter of Bessel is with the time what the filter of Butterworth is with the attenuation: If τ (ω) represents the delay undergone by the frequency ω when she traverses the filter, then the filter of Bessel of order N cancels τ (ω = 0) and all its derivative until the order N . One can as imagine as the filter of Bessel is the polynomial approximation of the transfer transfer function corresponding to a constant delay, i.e.: H (p) = e^-p . This polynomial approximation utilizes the polynomials of Bessel, from where the name of the filter.

If this filter proposes a constant transfer time, that is done with the detriment of its selectivity, which is definitely less good than that of the filter of Butterworth and increases only little with the order of the filter. The figure on the left illustrates the case of two filters of order 7. It is seen that the attenuation of the filter of Bessel is at least 30 dB less low than that of the filter of Butterworth; on the other hand, its travel time of group (time) is rigorously flat, contrary to that of the filter of Butterworth.

The answer in transient is the strong point of the filter of Bessel. On the figure of right-hand side, one sees the answer to a train of crenels (square signals) of 1 V of amplitude: in red, the filter of Bessel, in blue that of Butterworth. To compare the presence or the absence of overpressures ( overshoot ) at the time of the transitions.

However, the equality of the transfer time is not preserved at the time of the low-pass traditional transformation towards band pass. The design of filters band pass to constant time must thus be based either on an empirical method of optimization per computer, or on a direct design. Blinchikoff proposed filters band pass of a nature 2 and 4 which have a quasi-constant travel time, at least optimal within the meaning of least squares. The filter of Bessel is essential when signals should be filtered broad band by preserving the phases, which is the case of the majority of the modern modulations HF high-flow (PSK, 8-PSK, OFDM…). On the other hand, its interest in the field of numerical filtering is null, since it analogically approximates what any numerical filter does naturally, namely a constant delay.

See too

Internal bonds

linear Filtre
Filtre of Bessel
Filtre of Butterworth
Filtre combs from there
Filtre of Legendre
Filtre of elliptic Tchebychev
Filtre

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