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I am a newcomer to digital signal processing. Recently, I have been interested in this question: If we model a reflected radar signal from a target as $$y(t)=h \ast x(t) + n(t),$$

where $h$ is a complex number represented for the channel between the target and radar, $x$ is the transmitted radar signal and $n$ is the AWGN noise,

Can we estimate the target information such as range and velocity based on the estimated value of $h$? If yes, would you please inform me of the reference for this answer?

Thank you!

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  • $\begingroup$ In general maybe. If the channel doesn't degrade SNR low enough where you can't detect it, and it doesn't distort the waveform too badly, then you have a chance! $\endgroup$
    – Envidia
    Commented Apr 16 at 2:53
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    $\begingroup$ Are you trying to extract the target information from $h$, or account for $h$ to get a more accurate representation of $x$? $\endgroup$
    – Baddioes
    Commented Apr 16 at 6:27

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where h is a complex number represented for the channel between the target and radar,

I'm going to stop you there: if you model the channel like that, it can't be used as radar channel model. The whole point of Radar is that the signal experiences a delay between target and receiver, so the minimum sensible channel model would be

$$y(t) = h_0\cdot \delta(t-\tau_0) \,* \, x(t) + n(t),$$

where $h_0$ contains the attenuation through free space loss of the signal propagating the distance two-way and the limited cross-section of the target, as well as the phase, $\tau_0$ is the delay due to the time that radio waves take to travel (at speed of light $c_0$) from transmitter to target to receiver (so, in a monostatic radar with a target at range $d$, $\tau = 2d/c_0$).

$\delta$ is the Dirac delta functional, $*$ is convolution, not just multiplication.

Can we estimate the target information such as range and velocity based on the estimated value of $h$?

If your signal model is really just "received signal is transmitted signal times a scalar plus noise", then that would mean your target sits exactly at the radar, nothing to estimate there. There's no delay between transmission and reception, so the range must be 0.

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  • $\begingroup$ Thank you for your answer. I understand that the received signal is also phase-shifted if we represent the channel as a complex number. Why this phase shift cannot be used for target estimation? $\endgroup$
    – Rayzain
    Commented Apr 17 at 2:50
  • $\begingroup$ @Rayzain as I explained above: says nothing about the range, because there's no delay associated with it. $\endgroup$
    – Marcus Müller
    Commented Apr 17 at 7:28
  • $\begingroup$ @Rayzain The signal is phase-shifted regardless if the channel is a "complex number". That's just one way to model it. The important thing here is the delay induced, which will produce both a time delay and phase shift at the output of $y(t)$. $\endgroup$
    – Envidia
    Commented Apr 17 at 15:59
  • $\begingroup$ So the only assumption we could make is that the more precise the estimation of $h$, the better the target information estimation. Is this correct? $\endgroup$
    – Rayzain
    Commented Apr 18 at 4:56
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    $\begingroup$ @Rayzain no, that is inccorrect. The single complex number $h$ in itself contains no useful target information, no matter how well you estimate it. Can you, in your own words, tell me how a pulse radar estimates the range of a target? $\endgroup$
    – Marcus Müller
    Commented Apr 18 at 9:30

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