Elliptic response vs Chebyshev and inverse Chebyshev

Characteristics of the elliptic (Cauer) response

The elliptic low-pass response is

$$|T(j\omega)|^2 = \frac{1}{1 + \varepsilon^2 R_n^2(\omega)}$$

where $R_n(\omega)$ is a Chebyshev rational function (Jacobi elliptic function).

• Equiripple in both the passband and the stopband.
• Has finite zeros of transmission on the $j\omega$ axis in the stopband, where the attenuation becomes infinite.
• Sharpest transition (narrowest transition band) of all classical approximations for a given order; so for given $\alpha_{max}$, $\alpha_{min}$, $\omega_s/\omega_p$ it needs the lowest order.
• Stopband attenuation does not keep increasing; it ripples down to $\alpha_{min}$ between the zeros.
• Poorest phase linearity: group delay varies strongly near the band edge, causing pulse distortion.
• Design is complex (elliptic integrals); normally done with tables or software.
• Realization needs both poles and finite zeros (e.g. LC ladders with parallel-resonant series arms, or notch biquads).

 |T|
  1 |~v~v~~.
    |       \        elliptic
    |        \
    |         \  /\    /\
    |          \/  \__/  \___
    +--------------------------> w
          wp  ws (zeros at dips)

Comparison

The elliptic filter is chosen when a very narrow transition band is needed with minimum components and phase is not important; Chebyshev/inverse Chebyshev are simpler alternatives.