# DTFT of beat signal with shift due to multiplication

I am simulating a FMCW radar (Ramp signal is used for transmission)and I have created the transmit signal from matlab FMCW model

hwav = phased.FMCWWaveform('SweepTime',sweep_time,'SweepBandwidth',radar_bandwidth,'SampleRate',Fs);
htx = phased.Transmitter('PeakPower',1,'Gain',tx_gain);

x = step(hwav);    % Generate the FMCW signal
xt = step(htx,x);  % Transmitted FMCW signal


After the reflection from the target the reflected wave is : Some of the code delibrately missed.

xt = step(hrx,xt);


and it comes after some delay and then its multiplied with the transmitted signal to get the beat signal.

Now the multiplication is done between the received signal and the transmitted signal to get the beat signal. The matlab uses dechirp method to perform this operation.

radar_return = dechirp((xt),x);


So the output look like:

So I want to ask some questions related to it: I came across the equation of mixed signal (Linear Frequency Modulated tx mixed with LFM Rx ) as (1/2)*cos(2*pi*(fctd+(4B*R/Tp)*t)) fc= center frequency td = delay time of pulse (From target reflection) B = Bandwidth R = Range Tp = pulse duration t = fast time (actual time) Is this equation of the same signal which I have plotted, I am confused due to leading zeros I am getting in my mixed signal but in equation if I put the values of t then no zeros I am getting.

Also If yes the plot and the equation after mixing (matlab dechirp) are same. Then how can I remove leading zeros and make it signal with continous time and phase shift in the equation. In the equation there is a phase shift

2*pi*(fctd)


and In plotted signal received from mixing two Linear chirp signal I am getting zeros and the mix signal. How can I get the signal starting from t=0 with the same phase as in equation, since they both are same.