Exact discrete time synthesis

A procedure for the exact synthesis of LDI ladder filters is developed. Starting from any realisable discrete-time z-plane transducer function H(z), irrespective of the s¿z transformation used, a discrete-time loss-less passive ladder prototype is derived whose complex frequency variable is ¿, the LDI variable, and whose terminations are resistive and functions of ¿, where ¿ =(z1/2 + z¿l/2)/2,¿ =(z1/2¿ z¿1/2)/2 and z is the discrete-time complex frequency variable. The proposed synthesis procedure is carried out by introducing a discrete-time frequency transformation which maps the discrete-time z-plane into another discrete-time plane called the ¿-plane, where ¿ = ¿ + jß. The discrete-time physical frequency variable ß describes a straight line axis rather than a unit circle axis. The frequency characteristics of the synthesised discrete-time transducer function H(z) are exactly preserved in the discrete-time ¿-plane.