On the peaks of causal signals with a given average delay

Bounds are derived on the maximum location and minimum amplitude of the peak of a causal signal with a given average delay. For an average delay of τ, the authors show that the maximum possible location of the signal peak is on the order of τ(τ+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. Another bound is that the signals that minimize the peak amplitude, subject to unit energy and average delay of τ, have a peak value on the order of (2τ+1)^-1/2. The authors construct causal signals, for which the two derived bounds are attained for any given real-valued delay. The authors also compare the derived bounds to corresponding ones for a particular class of causal signals: the impulse responses of all-pass filters