Minimum Phase/All-Pass Decomposition of LPDA Transfer Functions

It is well-known that the conventional log-periodic dipole array (LPDA) has an impulse response with high levels of ringing compared to other antennas, such as ridged horns or Vivaldi antennas, with comparable frequency domain bandwidth. The ridged horn and Vivaldi antennas are known to have transfer functions that are minimum phase if a suitable spatial origin is chosen. In this paper we first show that the LPDA does not satisfy the minimum phase property, and then decompose the LPDA transfer function into the product of a minimum phase function and an all-pass function. The impulse response corresponding to the minimum phase function has a well defined main pulse and greatly reduced ringing. The all-pass function accounts for virtually all of the long term ringing, and is directly related to the number of transposed transmission line sections joining the elements. The results demonstrate that the LPDA cannot be systematically equalized over an unlimited bandwidth using a passive network.