Bessel analog lowpass filter prototype

`[z,p,k] = besselap(n)`

`[z,p,k] = besselap(n)`

returns
the poles and gain of an order-`n`

Bessel analog
lowpass filter prototype. `n`

must
be less than or equal to 25. The function returns the poles in the
length `n`

column vector `p`

and
the gain in scalar `k`

. `z`

is an empty matrix because there are no zeros. The transfer
function is

$$H(s)=\frac{k}{\left(s-p(1)\right)\left(s-p(2)\right)\cdots \left(s-p(n)\right)}$$

`besselap`

normalizes the poles and gain so
that at low frequency and high frequency the Bessel prototype is asymptotically
equivalent to the Butterworth prototype of the same order [1]. The magnitude of the filter is less
than $$1/\sqrt{2}$$ at the unity cutoff frequency
Ω_{c} = 1.

Analog Bessel filters are characterized by a group delay that is maximally flat at zero frequency and almost constant throughout the passband. The group delay at zero frequency is

$${\left(\frac{\left(2n\right)!}{{2}^{n}n!}\right)}^{1/n}$$

[1] Rabiner, L. R., and B. Gold. *Theory
and Application of Digital Signal Processing.* Englewood
Cliffs, NJ: Prentice-Hall, 1975, pp. 228–230.

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