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Basics of Radar Stretching Process

Stretching Processing method converts a received Linear Frequency Modulated (LFM) pulse into a tone. The frequency of the tone is fixed and it is proportional to the time delay's of the target. It's implemented to Pulsed and FMXW Radar systems.

Step 1 Transmit Waveform

Assume there is a LFM wave centered at Intermediate Frequency ($f_1$)

$$x(t) = cos(2{\pi}f_1t + {\pi} \frac{\beta}{\tau}(t- \frac{\tau}{2})^2 + {\theta_1}) where \space 0 ≤ t ≤ {\tau}$$

where $\beta$ is the bandwidth that waveform is swept, $\tau$ is the pulse length, $\theta_1$ is the phase of the oscillator used to generate $f_1$.

There will be a second oscillator that will mix with the IF signal (LFM wave)

$$LO_{2,tx} = cos(2{\pi}{f_2}t+\theta_2) where \space t ≥ 0$$

When they are mixed, the signal equation will be

$$x(t) = cos(2{\pi}(f_1 + f_2)t + {\pi} \frac{\beta}{\tau}(t- \frac{\tau}{2})^2 + {\theta_1} + {\theta_2}) where \space 0 ≤ t ≤ {\tau}$$

Step 2 Receiver and Return

The receive action occurs with a time delay $t_{rcv}$ that is referenced to the center of the range window

When signals are reflected from the radar, they are mixed on receive.

Assume there is a return from a point target located at time $t_d$ (time delay)

$$x(t) = cos(2{\pi}(f_1 + f_2)(t-{t_d} + {\pi} \frac{\beta}{\tau}(t- \frac{\tau}{2}-t_d)^2 + {\theta_1} + {\theta_2}) where \space t_d ≤ t ≤ t_d+{\tau}$$

References:

Melvin, William L.; Scheer, James (ed.): 'Principles of Modern Radar: Advanced techniques' (Radar, Sonar and Navigation, 2012) DOI: IET Digital Library, https://digital-library.theiet.org/content/books/ra/sbra020e