Bessel analog filter design
[b,a] = besself(n,Wo)
besself designs lowpass, analog Bessel filters,
which are characterized by almost constant group delay across the
entire passband, thus preserving the wave shape of filtered signals
in the passband.
besself does not support the design
of digital Bessel filters.
[b,a] = besself(n,Wo) designs
lowpass analog Bessel filter, where Wo is the
frequency up to which the filter's group delay is approximately constant.
Larger values of the filter order (
a group delay that better approximates a constant up to frequency
besself returns the filter coefficients in
n+1 row vectors
with coefficients in descending powers of s, derived
from this transfer function:
[z,p,k] returns the zeros and
poles in length
p and the gain
in the scalar
[A,B,C,D] returns the filter design
in state-space form, where
and u is the input, x is the state vector, and y is the output.
Design a 5th-order analog lowpass Bessel filter with approximately constant group delay up to
rad/s. Plot the magnitude and phase responses of the filter using
[b,a] = besself(5,10000); freqs(b,a)
Design an analog Bessel filter of order 5. Convert it to a digital IIR filter using
bilinear. Display its frequency response.
Fs = 100; % Sampling Frequency [z,p,k] = besself(5,1000); % Bessel analog filter design [zd,pd,kd] = bilinear(z,p,k,Fs); % Analog to digital mapping sos = zp2sos(zd,pd,kd); % Convert to SOS form fvtool(sos) % Visualize the digital filter
Lowpass Bessel filters have a monotonically decreasing magnitude response, as do lowpass Butterworth filters. Compared to the Butterworth, Chebyshev, and elliptic filters, the Bessel filter has the slowest rolloff and requires the highest order to meet an attenuation specification.
For high order filters, the state-space form is the most numerically accurate, followed by the zero-pole-gain form. The transfer function coefficient form is the least accurate; numerical problems can arise for filter orders as low as 15.
besself performs a four-step algorithm:
It finds lowpass analog prototype poles,
zeros, and gain using the
It converts the poles, zeros, and gain into state-space form.
It transforms the lowpass prototype into a lowpass filter that meets the design specifications.
It converts the state-space filter back to transfer function or zero-pole-gain form, as required.
 Parks, T. W., and C. S. Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, chap. 7.