"Doppler distortion" vs. "Phase Modulation"

My reference for this comparison is Viterbi, Principles of Coherent Communication, McGraw Hill 1966, Section 2.6 Phase and Frequency Modulation.

For both phase modulation (PM) and frequency modulation (FM) we can write the modulated signal s(t) as a function

Assume we have a carrier frequency ω2 and a modulation signal m(t). The modulation signals are different, so we will subscript them m1(t) for PM and m2(t) for FM.

Then for the PM signal

and for the FM signal

The key thing to note here is that if the phase modulation m1(t) is in phase with the FM modulation m2(t) then there is a 90 degree phase difference in the signals s(t), due to the derivative in equation (3). However if the modulations are 90 degrees out of phase, them the two signals are in phase. That is exactly the case for PM produced by cone excursion (the cone displacement from its equilibrium position) and FM produced by cone velocity; the modulations are 90 degrees out of phase, but the resulting signals are in fact identical, as the next equations show

Define the PM case, as done in the approximate Doppler distortion analysis, by the equation

where x(t) is the cone excursion. Cone velocity is the derivative of x(t) with respect to time. What FM modulation would produce exactly the same signal? Differentiating equation (2) and substituting equation (4)

So the modulation m2(t) is exactly the Doppler shift produced by the cone velocity. The "minor qualification" to this statement is that the Doppler shift as represented in equation (5) is in fact the first term of an infinite series. Terms involving powers of (vp(t)/c) of two and higher are neglected. So there is a small divergence between the two approaches, but it is quite small for realistic piston velocities.

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