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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) = ∫|m<inf>1</inf>(t - T)|<sup>2</sup>|m<inf>2</inf>(t)|<sup>2</sup>dt 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.