On a formal measure of Doppler Tolerance

In the context of radar waveforms, there are many references to “Doppler Tolerance” in the literature, but a formal, complete, precise, and reasonable definition has not been forthcoming. We attempt to fill this void in this paper. We revisit existing definitions and demonstrate, that they are either too restrictive for any practical use, incomplete, or imprecise. Our definition uses the ambiguity function as its main ingredient. We emphasize that the Doppler tolerance is a 3D function. The first parameter is a spatial variable which relates to the measures of connectedness of possible disjoint ambiguity function peaks. The second parameter is the time delay at which the Doppler tolerance is itself specified, and the third parameter is similar to a threshold and is related to height of the ambiguity function used in measuring the Doppler tolerance. As a byproduct of our definition, we analytically conclude that for small time bandwidth products the linear frequency modulated (LFM) waveform is only as Doppler tolerant as an unmodulated rectangular pulse. We therefore bust the well known myth that “(all) chirps are Doppler tolerant”.