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How can we model a radar wave's reflection from a target? Will the phase of the return be different from the signal that gets sent out?

For instance, suppose our radar was transmitting the following signal:

$$x_{tx}(t) = e^{j 2 \pi f_{tx}t + \theta_{tx}} $$

It hits a non-moving target and reflects back to the radar. How do we model the return? Could we model it as:

$$x_{rx}(t) = |\alpha| e^{j 2 \pi f_{tx}t + \theta_{tx} + \angle \alpha} $$

where $\alpha$ is the reflection coefficient of the target material. Will the phase change always be $\angle \alpha = 0$? When can we tell?


It seems to me that in general, we can't tell what $\angle \alpha$ will be. However, if that is true, how can coherent radars exist?

Note: I did cross post this to dsp.stackexchange, but I want to know the physics perspective too. I think you guys may have better insight here.

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The only thing is needed to assume coherency is that the reflection phase $\alpha$ be independent of the instantaneous frequency across the bandwidth of interest. As long as that holds the reflection phase can be ignored.

The reflection phase as long as it is constant plays no role in matched filter detection since it shows up as a simple constant multiplier $e^{j\alpha}$ of the complex amplitude of the filter output whose magnitude is used for detection. Since it is constant over the coherent processing interval when the pulses are summed the reflection phase plays no role. But note it can prevent long coherent integration for bistatic radars, because it can vary from one pulse to another (see, scintillation) for a maneuvering target in which case only noncoherent integration can be applied. A complicated target will be "scintillating" but a simple target will have reasonably stable reflection phase; if your target is complicated you may choose to increase the resolution (angle - antenna processing; range-waveform processing) to reduce and/or remove the scintillation.

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  • $\begingroup$ Okay, I can understand the constant part, but why does it need to be independent? Couldn't you take it out anyway? $\endgroup$ Dec 5, 2019 at 18:45
  • $\begingroup$ what do you mean by independent? $\endgroup$
    – hyportnex
    Dec 5, 2019 at 18:46
  • $\begingroup$ independent in the sense of being constant when plotted against the instantaneous EM frequency; a pulse has many frequency components, for coherent integration it is assumed that the reflection phase is the same for the whole pulse as it is for the carrier frequency. $\endgroup$
    – hyportnex
    Dec 5, 2019 at 18:52
  • $\begingroup$ Ah okay, so if it is different for each EM frequency component, you can't take it out? But wouldn't it just add to some different phase value at the end of the sum? Sorry for the basic idiot questions. $\endgroup$ Dec 5, 2019 at 18:53
  • $\begingroup$ exactly; for example, this is what can happen for wideband radars looking at a complicated scenery or target, as the target changes its attitude relative to the radar different parts of the target may appear dominant with different reflection angle. Only an ideal metal sphere has truly frequency and attitude independent reflection phase. $\endgroup$
    – hyportnex
    Dec 5, 2019 at 18:57

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