Synthesis and Beyond

De-embedding of filters in multiplexers via rational approximation and interpolation

“…where the dot notation is used to indicate first derivatives with respect to x, and the second equality can be easily proven by observing that F 1,2 appears squared in (1) and vanishes at x c thus nulling any further contribution to _ S 1;1 ðx c Þ other than from F 1,1 (x c ). Equation (2) further indicates that the values and the derivatives of the reflection coefficient of the filter at one of its finite or infinite transmission zeros can be both inferred from those of the system's reflection at the same point.…”

Deembedding of filters in multiplexers via rational approximation and interpolation

“…The system of equations (3) contains 2N equations compared to the 2N + 2 unknowns corresponding to the coefficients of polynomials p and q. It is therefore no surprise that the following holds: Proposition 2.1: All the polynomial pairs (p, q) of degree at most N that solve (3) form a two dimensional space 1 . In particular there exist (p 1 , q 1 ) and (p 2 , q 2 ), such that for all complex numbers (α, β), the pair (p, q) defined as:…”

Minimax principle and lower bounds in H2-rational approximation

“…Special interest attaches to the case where the approximated function extends holomorphically on one side of the curve. In connection with system identification and control, such issues typically arise on the line or the circle where they make contact with extremal problems in Hardy spaces [4,15,33,29,30,32,43]. Our model curve in this paper will be the circle, though everything translates easily to the line.…”