Suppose that I have a signal $x(t)$ consisting of $N$ pulses at a given frequency $\omega_{c}$. I space them $T$ seconds apart, ie the Pulse Repetition Interval (PRI) for the waveform is $T$.
Now this signal bounces off a target and experiences a large Doppler shift, such that the compression of the pulses are evident. My question is -- does the PRI change as well?
In other words, are each of the waveforms still $T$ seconds apart, or does the PRI scale with the Doppler shift?
I guess the question really is, should the model of the second pulse be:
$$ x(t-T) \underset{v}{\rightarrow} x(\frac{t -T - \tau}{v}) $$
or
$$ x(t-T) \underset{v}{\rightarrow} x(\frac{t -T}{v} - \tau) $$
or
$$ x(t-T) \underset{v}{\rightarrow} x(\frac{t}{v} - T - \tau) $$
Where $\tau$ is the target delay and $\frac{1}{v}$ is the Doppler shift.
Note: A similar question is asked in the DSP Stackexchange question Doppler Shift of Phase Coded Pulse Compression Waveform