Radar Wave Reflection -- How can we justify coherent radar?

Question

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?

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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.

Answer 1

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.