- #1

- 68

- 14

I don't take range information here. I am only concerned about the velocity as in all ranges I consider the velocity to be constant.

My approach:

I consider first when the radar is not rotating and it is directed at ϕ=0 (exactly the same direction of the motion of the target). In that case, the time domain data looks like the following.

[tex]

s(t) = e^{j \frac{4\pi}{\lambda }\mu t}

[/tex]

Where [itex] \mu [/itex] is the velocity of the target. The term [itex] \lambda [/itex] is the wavelength of the radar.

If I observe the phase of this signal carefully, it is just an integral over the velocity with respect to time.

$$

Phase(s(t)) \propto \int \mu dt

$$

This way I did my thought experiment. If I see the target from a rotating radar, the radial velocity I receive from different directions in space would have a cosine dependence.

$$

v_r(t) = cos(\phi(t)) \mu

$$

Here [itex] v_r [/itex] is the radial velocity that radar receives. The term [itex] \phi(t) [/itex] is a function of [itex] t [/itex] and the rotation speed [itex] \Omega [/itex] . This is basically the angle subtended by the radar beam with the target motion direction at a given time.

$$

\phi(t) = \phi_{target} - (\Omega t + \phi_0)

$$

Where [itex] \phi_0 [/itex] is the initial angle of observation and [itex] \phi_{target} [/itex] is the target motion direction.

Keeping this in mind I performed the following integral

$$

\int cos(\phi(t)) \mu dt

$$

Which is

$$

\frac{-1}{\Omega} sin(\beta_{target} - (\Omega t + \phi_0))

$$

So, my time-domain data becomes,

$$

s(t) = e^{j \frac{-4\pi}{\lambda \Omega} sin(\beta_{target} - (\Omega t + \phi_0))}

$$

I did one simulation with the target motion direction as [itex] \phi_{wind} = 0 [/itex] and [itex] \phi_0 = 0 [/itex] . I have data for the entire rotation and I plot a micro-doppler spectrogram. I have [itex] 167\times360 [/itex] points for one rotation. So, I do Doppler FFT processing for every one degree of [itex] \phi [/itex] (167 points DFT) and plot it. Figure attached.

Although it looks correct, I need a second eye who verifies my process here and if I am doing it alright physically. Physically radar receives the radial velocity and with that in mind, I formulated the phase of the time domain signal.