Elliptic (or Cauer) Filters

Elliptic filters minimize the peak error by distributing it over the passband and the stopband. Equi-ripples in the passband and the stopband characterize the magnitude response of elliptic filters. Compared with the same order Butterworth or Chebyshev filters, the elliptic design provides the sharpest transition between the passband and the stopband. For this reason, elliptic filters are used widely.

The following graph plots the response of a lowpass elliptic filter. Notice that the ripple in both the passband and stopband is constrained by the same maximum tolerable error (as specified by ripple amount in dB). Also, notice the sharp transition edge for even low-order elliptic filters.