Thanks for this simple UWB pulses demonstration with pulse position modulation (PPM).
I use your doublets (second derivative Gaussian pulse) with and without pulses train and then we can see periodic variations on frequencies display - fine.
The very simple correlation separate correctly single pulses from delay pulses like PPM must do !
I was excited about the result and delighted by the simplicity.
You write the equations completely so that we can change at will by mastering what we do well.