# Doppler Shift and the Pulse Repetition Interval

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

• It will not effect the PRI because that is something you specify. Lets say that you send out pulses at 1 per second or 1 Hz. Consider a moving target that is half a second away with a very large Doppler due to its speed. Assume the extreme case that the pulse shortens in time so much that it looks like an impulse. The pulse hits the target, compresses, and returns with a delay but you will still wait the remaining half second to send the next pulse because you specified it at such. The equation you are looking for is akin to the third. – Envidia Sep 20 '19 at 5:25
• @Envidia Your answer seems to be completely different from Mohammad M's... – The Dude Sep 25 '19 at 1:10
• Yes it is, because the way I defined the PRI is different. The view he is coming from is if you repeatedly send pulses out and collect the returns as a single signal. In that case, if you were to try and derive the PRI by looking at the pulse return instants, they would be different due to the pulse being shortened in time due to Doppler. – Envidia Sep 25 '19 at 19:54
• I misread your question. I assumed that your signal was a collection of individually collected pulses, sent out at the PRI. I see now that you defined the transmitted waveform as a collection of N pulses in where you will measure the PRI change due to Doppler, which is the case that Mohammad explains. – Envidia Nov 11 '19 at 20:16

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 ).

• This is quite interesting. Thank you for the reference. Interestingly, your answer completely contradicts the one I got in the comments from @Envidia – The Dude Sep 25 '19 at 1:11

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: $$R_{\rm unamb} = c (T-\tau) /2.$$

• This is not what I was asking at all. This is simple copy and paste from any radar textbook, even wikipedia perhaps. – The Dude Sep 25 '19 at 1:11