Formulation of the Klopfenstein tapered line analysis from generalized nonuniform line theory

The objective of this problem is to formulate the Klopfenstein analysis of the non-uniform line in the generalized integral equation format. The integral equation approach is a general case without any optimization which gives the equation for reflection coefficient for any tapered line. On the other hand, Klopfenstein uses an optimized reflection coefficient and proceeds to obtain an equation for characteristic impedance in terms of circuit variables. In this paper, the aim is to examine the general formulation analysis of Youla for the non-uniform line, and derive Klopfenstein's result for the input reflection coefficient for the line.