# FMCW SAR chirp return from target modelling

I have to Model a chirp signal that was return from a Distance D from target and then multiply with the reference transmitted signal. Which will give me beat frequency signal and then fft to get the range. For this I was searching around and find a pulse LFM SAR implementation.

Nf=2^10; %number of points for freq. range
fc=242.4*1E6; %stopfreq
B=133.5*1E6; %bandwidth
freq=linspace(fc-B,fc,Nf);

function [resp] = my_sinthSISO_planresp( pos,freq,tar,tar_amp )
%MY_SINTHSISO_PLANRESP calculates synthetic SISO at position pos response
%in frequency range freq on target tar(target postion), tar_amp: RCS like.
%Targets and Array must be in the same (image) plane, which XY-coordinate
%are represented as complex values.

if nargin < 4, tar_amp=ones(size(tar)); end % tar_amp = 1 if not given

%speed of light
co=299792458;
%redistribution of input data
tar_dim=numel(tar);
tar=reshape(tar,[1,1,tar_dim]);
tar_amp=reshape(tar_amp,[1,1,tar_dim]);

freq_dim=numel(freq);
freq=reshape(freq,[1,freq_dim,1]);

pos_dim=numel(pos);
pos=reshape(pos,[pos_dim,1,1]);

%% algo
D=abs(repmat(tar,[pos_dim,1,1])-repmat(pos,[1,1,tar_dim]));
D=repmat(D,[1,freq_dim,1]);
%wavenumber
k=2*pi*repmat(freq,[pos_dim,1,tar_dim])/co;
%target amplitude
tar_amp=repmat(tar_amp,[pos_dim,freq_dim,1]);
%SISO response
resp=tar_amp.*exp(-1j*k*2.*D);
resp=sum(resp,3);

end


Question 1: Is resp in a frequency domain? since its dependent on k and k is frequency related. Question 2: if its freq domain then why the inverse fft does not give the desired chirp signal. while its showing: Question 3: If it doesnot represent chirp in frequency domain. Then what it represents?

Question 4: How can I use this data set to model FMCW. Which means multiplication of the chirp at reference and after shift?

I know its quite alot and very basics but I got confused. Thanks.

I am not sure you really understand the processing you do so I will try to explain what you measure. When you use the returned chirp mixed against the reference chirp you get an instantaneous phase measurement. When you do the FFT the frequency domain represents distance - since the phase differences are changing faster as the distance is greater. The inverse FFT of the Distance will give you (assuming high signal to noise ratio) a periodic signal that is the function of the distance you measured. In the graph you show, it is not clear what you show since the phase is not presented.

• Dear Moti, thanks for the ellobration. In the second last sentence if I take the ifft of the previously taken fft then will it not give me the same mixed instantaneous phase signal. Also in the code resp=tar_amp.*exp(-1j*k*2.*D); is representing chirp shifted with 2D distance or range compressed (if range compressed then I didnt got that why it has all frequency) signal in frequency domain. – Mr. Khan Apr 6 '15 at 10:45
• If you take and FFT and than an inverse FFT you should get the same original signal if you took care properly of scaling. It also depends on the length of the FFT and the quantization levels. – Moti Apr 6 '15 at 15:09
• I have modelled the instantaneous phase (mixed signal) with beat time signal, i.e time_fast = [0:1/Fs:pulse_length]; distance_vector is the distance of target from the SAR elements beat_frequencies= (radar_bandwidth*2*distance_vector)/(co*pulse_length); signal_mixed= exp(1i*2*pibeat_frequencies*time_fast); Is this ok if I take fft of this signal to represents the distance. – Mr. Khan Apr 7 '15 at 8:10
• If you talk about SAR application than the received signal contains information from many points, in different distances. Each distance has its own "beat" frequency. Is your approach the legacy approach used with SAR systems? – Moti Apr 7 '15 at 20:03
• "My radar is FMCW SAR, the data collected is through the multiplication of the return with transmitted signal, this provides me with the beat signal in time domain (filtered for removal of higher additive frequency). This beat signal(time domain) is sampled and then put into the row of the matrix this procedure is repeated such the whole length is traversed, So my data is in row wise (beat time sampled signal) coloumn wise (sweep repeatation 'slow time') – Mr. Khan Apr 8 '15 at 15:56