# How do I do a point response FMCW SAR simulation?

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.

• Questions requesting working code written to a specification are off-topic as they are unlikely to benefit anyone else. Instead, describe the problem you're solving and where you're stuck. Oct 17, 2022 at 12:20
• Noted. Updated the question. Oct 17, 2022 at 19:29
• Thanks! This looks interesting. Need to spend a bit more "awake" time on this. Oct 17, 2022 at 20:31
• Awesome. Thank you. Oct 18, 2022 at 6:35
• I've figured it out. I'll post my solution soon. Oct 19, 2022 at 14:05

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

• Thanks for the response. +1 for the approach. I wound up going with a more robust simulation of starting with the transmit chirp then multiplying with the complex conjugate of the range reference signal to get the IF. I find it allows me to take such things as velocity, antenna pattern etc. into account. Apr 17 at 12:31
• That's Great! Thanks for info. Apr 19 at 5:39