The peak of a causal signal with a given average delay

Derives two results concerning the peak of a causal signal with a given average delay. The first result is that, for an average delay of tau , the maximum possible location of the signal peak is of the order of tau ( tau +3)/2. (This bound can also be interpreted as providing the maximum integer at which the most probable value of a discrete nonnegative random variable could occur, given that the random variable has a known mean.) The second result is that the signals that minimize the peak amplitude, subject to unit energy and average delay tau , have a peak value of the order of 1/ square root 2 tau +1. The authors construct causal signals for which the derived bounds are attained for any given real-valued delay. They also compare the derived bounds to the corresponding ones for all-pass signals.<>