Elliptic analog lowpass filter prototype

`[z,p,k] = ellipap(n,Rp,Rs)`

`[z,p,k] = ellipap(n,Rp,Rs)`

returns
the zeros, poles, and gain of an order `n`

elliptic
analog lowpass filter prototype, with `Rp`

dB
of ripple in the passband, and a stopband `Rs`

dB down from the peak value in the passband. The zeros
and poles are returned in length `n`

column vectors `z`

and `p`

and
the gain in scalar `k`

. If `n`

is
odd, `z`

is length `n`

- `1`

. The transfer function in factored
zero-pole form is

$$H(s)=\frac{z(s)}{p(s)}=k\frac{(s-{z}_{1})(s-{z}_{2})\dots (s-{z}_{N})}{(s-{p}_{1})(s-{p}_{2})\dots (s-{p}_{M})}$$

Elliptic filters offer steeper rolloff characteristics than Butterworth and Chebyshev filters, but they are equiripple in both the passband and the stopband. Of the four classical filter types, elliptic filters usually meet a given set of filter performance specifications with the lowest filter order.

`ellipap`

sets the passband edge angular
frequency ω_{0} of the elliptic filter to
1 for a normalized result. The *passband edge angular frequency* is
the frequency at which the passband ends and the filter has a magnitude
response of 10^{-Rp/20}.

[1] Parks, T. W., and C. S. Burrus. *Digital
Filter Design*. New York: John Wiley & Sons, 1987,
chap. 7.

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