Formulas for parameter scaling in the McClellan transform

In using the original 1-D to 2-D McClellan transform
cos{\omega}= F(cos{\omega}_1,cos{\omega}_2)
, scaling of the transform parameters [4,5] is necessary to guarantee that
|F| \leq 1
. In a recent letter [7], we have presented extremely simple formulas for the scaling factors, then proposed a modified transform which not only is scaling-free but also contains one fewer parameter than the original transform. The work in [7] however is restricted only to common elliptical isopotential contours. In this paper, we analyse the problem for more general cases.