**"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 ω_{2} and a modulation signal
m(t). The modulation signals are different, so we will subscript them m_{1}(t)
for PM and m_{2}(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 m_{1}(t)
is __in phase__ with the FM modulation m_{2}(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 m_{2}(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 (v_{p}(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.