FMCW FFT frequency is not correct

Hello, I am doing a project using FMCW. I generate a test signal with the code below. However, my "Estimated Frequency Difference" is always two-fold of the theoretical value. What is the problem?? I have been stuck for days.

clc;close all
fs=1e6;
f0=0;
f1=0.2e6;
chirp_freq=1e2;
chirp_period=1/chirp_freq;

delay=3000;
fft_samps=4096;
downsample_factor = 1;
fft_len= fft_samps/downsample_factor;

FFT_Resolution = (fs/downsample_factor)/(fft_len);

fprintf("One chirp has %d points \n",chirp_period*fs);
fprintf("FFT Resolution: %f Hz\n",FFT_Resolution);
fprintf("One point delay means %f Hz\n",((1/fs)/chirp_period)*(f1-f0));
fprintf("Points delay: %d\n",delay);
fprintf("Frequency delay: %f Hz\n",((1/fs)/chirp_period)*(f1-f0)*delay);
fprintf("\n");

repeat_cnt=1024*2;
freq=linspace(f0,f1,chirp_period*fs);

for i=1:log2(repeat_cnt)
   freq = [freq freq];
end

t=linspace(0, repeat_cnt*chirp_period, repeat_cnt*chirp_period*fs);
data=zeros(1, repeat_cnt*chirp_period*fs);

for i=1:repeat_cnt*chirp_period*fs
    data(i) = cos(2*pi*freq(i)*t(i)); 
end

%plot(t,data)

p_data=data(100:100+fft_samps-1);
dp_data=data((100:100+fft_samps-1)+delay);
fft_input_data = downsample( p_data.*dp_data,downsample_factor ); 
% Calculate the frequency difference Cos(f1)*Cos(f2) => Cos(f1-f2)
fft_result=abs(fft(fft_input_data));

figure;
plot(fftshift(fft_result));
[M,I] = max(fft_result(1:(length(fft_result)/2+1)));
fprintf("Maximum occurs at index: %d \n",I);
fprintf("Maximum occurs at frequency: %d Hz\n",I*FFT_Resolution);
fprintf("Theoratical time delay: %d s\n",delay*1/fs);
fprintf("Estimated time delay: %d s\n",I*FFT_Resolution/(f1-f0)*chirp_period);

Answer 1

[David Tu]

I think I know the problem. Here is the mathematical derivation:

f(t) = deltaF * t

cos(2*pi*f(t1)*t1) * cos(2*pi*f(t2 + sigmaT)*(t1 + sigmaT)) = cos(2*pi*deltaF*(- 2 *sigmaT*t1 - sigmaT^2))