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Bessel analog filter design

`[b,a] = besself(n,Wo)`

[z,p,k] = besself(...)

[A,B,C,D] = besself(...)

`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
an `n`

th-order lowpass analog Bessel filter, where `Wo`

is
the angular frequency up to which the filter's group delay is approximately
constant. Larger values of the filter order `n`

produce
a group delay that better approximates a constant up to frequency `Wo`

.

`besself`

returns the filter coefficients in
the length `n+1`

row vectors `b`

and `a`

,
with coefficients in descending powers of *s*, derived
from this transfer function:

$$H(s)=\frac{B(s)}{A(s)}=\frac{\text{b(1)}\text{\hspace{0.17em}}{s}^{n}+\text{b(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{b(n+1)}}{\text{a(1)}\text{\hspace{0.17em}}{s}^{n}+\text{a(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{a(n+1)}}.$$

`[z,p,k] = besself(...)`

returns
the zeros and poles in length `n`

or `2*n`

column
vectors `z`

and `p`

and the gain
in the scalar `k`

.

`[A,B,C,D] = besself(...)`

returns
the filter design in state-space form, where `A`

, `B`

, `C`

,
and `D`

are

$$\begin{array}{l}\dot{x}=\text{A}\text{\hspace{0.17em}}x+\text{B}\text{\hspace{0.17em}}u\\ y=\text{C}\text{\hspace{0.17em}}x+\text{D}\text{\hspace{0.17em}}u.\end{array}$$

and *u* is the input, *x* is
the state vector, and *y* is the output.

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

`besselap`

function.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.

[1] Parks, Thomas W., and C. Sidney Burrus. *Digital
Filter Design*. New York: John Wiley & Sons, 1987.

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