Doppler Shift and the Pulse Repetition Interval

Question

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?

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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.

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Note: A similar question is asked in the DSP Stackexchange question Doppler Shift of Phase Coded Pulse Compression Waveform

Answer 1

Actually the PRI also changes. There is a complete explanation about this topic in "Cobbold - Foundations of Biomedical Ultrasound" chapter 10 (Pulsed Methods for Flow Velocity Estimation and Imaging).

Consider a case where PRI of emitted pulse is 1 s and the wave speed is 9 m/s and the object is 10 m away and moving with speed of 1 m/s toward the antenna. The first echo will be received 2 seconds after the emission (after 1 seconds, the object moved 1 meter toward the antenna and placed 9 meter away from the antenna and the transmitted wave traveled 9 meter away from the antenna). For the next pulse the echo will be received 1.8 seconds after the emission (considering the object is moved closer to the antenna during the PRI between first and second pulse). So the first echo received at t=2 and the second echo received at t=2.8 and the PRI would be 0.8 s (the same is true for the next pulses until the object passes over the antenna).

Also the transformation you have here is something like x(t - (x0-vt)/c ).

Answer 2

The Pulse Repetition Interval (PRI) is chosen based on the maximum unambiguous range you want to detect. The waveform of a pulsed radar is usually represented by the pulse duration $\tau$, the pulse repetition interval (PRI) $T$, or the pulse repetition frequency (PRF) $f_p$. PRI is the time interval between two adjacent pulses.

Target range is the distance between the target and the radar. It can be determined by measuring the time required by the pulse to travel to the target and return. The target range is given by:

$$R = c \tau_d/2$$ where $c$ is the speed of light, $\tau_d$ is the time delay, and the factor 2 accounts for the round-trip delay.

If the time delay is smaller than the PRI, the range determined by the above equation is unambiguous. If the PRF is too high or the PRI is too short, the echo pulses from the target might arrive after the transmission of the next pulse, and then the range measured might be ambiguous. The maximum unambiguous range is given by: $$R_{\rm unamb} = c (T-\tau) /2.$$