How do I do a point response FMCW SAR simulation?

Question

I am trying to simulate and focus the point response for an FMCW stripmap SAR system. I have specified my target coordinates as targets = [100,100,0;400,100,0] and I understand from several resources that the transmitted signal model for the chirp is $$s_{\text{TX}} = \exp\bigg[j 2\pi \bigg(f_c t + \frac{1}{2} k_r (t - nT)^2\bigg)\bigg]$$ where $n$ is the pulse number, which I have been able to generate.

My approach to simulate the returns is to create a matrix of zeros with dimensions $numRangeBins \times numPulses$ and iterate over the number of pulses then for each pulse and, within that, for each target, accumulate the echo signal into the aforementioned matrix.

How do I calculate the return echo signal $s_{\text{rx}, k}$ for each target? I am primarily stuck on how $r_0$ can be calculated from the target coordinates in this resource. Alternatively, I would appreciate a code snippet for such a simulation. I have tried various the MATLAB simulation code for FMCW SAR but it seems buggy.

Answer 1

Usually, for FMCW systems, only IF signal (beat frequency) is sampled.

The equation for the IF signal would be just the phase subtraction between the transmitted and received signal. Again the received signal will be time delayed version of transmitted signal. You can use the below equation for IF signal.

$$ s_{IF}(t) = exp\{ j2\pi (f_c \tau - \frac{k_r}{2}\tau^2+ k_rt\tau) \} $$ where $ \tau $ is the round trip delay for the transmitted signal to be receieved after reflecting from a target at Range $ R $

simply $ \tau = \frac{2R}{c} $

Basically $ \tau $ for a point target changes when radar is moving

$ R $ can simply be calculated from simple SAR geometry