Optimal Time Delay Extraction for Transmission Line Modeling

The frequency-dependent transmission line modeling by the traveling wave method requires approximating the propagation function with a delayed rational function. Some approaches are based on modal decomposition where scalar functions are fitted with a rational model plus a single time delay. The delay is calculated from the modal velocity and the minimum-phase-shift angle that can be reconstructed from the magnitude function. This paper shows that the accuracy in the phase reconstruction as calculated by Bode's magnitude-phase integral relation can be greatly improved by removal of a singularity in the integrand and by prediction of out-of-band samples for the magnitude derivative. It is further shown that the time delay giving the smallest rms-error in the final rational approximation is often substantially larger than the mps induced delay. An improved estimation is calculated via an auxiliary magnitude function and used for determining a bracketing interval for the true optimum that is identified by searching.