Image-parameter approximations for designing linear-phase filters

The stopband loss of a filter with arbitrary loss poles and an equal-ripple passband can be approximated extremely accurately by the loss of a related image-parameter filter. When the actual loss is greater than about 1 Np, its difference
\alpha_{d}
from the image loss is almost completely independent of frequency and is a function only of the passband ripple size. This approximation has been widely used in filter design since it was described by Darlington in 1939. The passband delay can also be approximated by the delay of the same image-parameter filter, but the difference
\tau_{d}
between the two delays is, unfortunately, not a constant but a fairly complicated function of frequency. The paper shows that
\tau_{d}
is the minimum delay corresponding to
\alpha_{d}
and that an approximation to
\tau_{d}
can be derived by taking the Hilbert transform of a spline function which approximates
\alpha_{d}
over the whole stopband. The sum of the image delay and this approximation to
\tau_{d}
represents the actual delay very accurately; both functions and their partial derivatives with respect to the poles can be computed quickly and simply. The two approximations, to loss and delay, can be used as the basis of a fast iterative program for designing high-degree linear-phase filters meeting an arbitrary specification on the stopband loss.