A definition of pulse delay in the presence of distortion

It is proposed that pulse delay, in the presence of distortion, be defined as ∫TX(T) dT/∫X(T) dT where X(T) = ∫|m1(t - T)|²|m2(t)|²dt is the cross correlation of the squares of the magnitudes of the transmitted and received pulse modulation. In the absence of the distortion, the definition reduces to the group delay.