_applied_  
_sciences_  
_Article_  
Modulation Linearization Technique for FM/CW SAR Image Processing Using Range Migration  
**[PERSON] \({}^{\star}\)\({}^{\copyright}\) and [PERSON] \({}^{\copyright}\)  
Electrical and Computer Engineering, Southern Polytechnique College of Engineering and Engineering Technology, Kennesaw State University, Kennesaw, GA 30060, USA; [EMAIL_ADDRESS]  
\({}^{\star}\) Correspondence: [EMAIL_ADDRESS]  
**Featured Application: The technique described herein is a method for linearizing the frequency modulation error of a Frequency-Modulated/Continuous Wave (FMCW) radar as part of range migration processing when forming SAR images that results in no additional computation cost. **  
**Abstract: Linear FMCW radar suffers from impairments in range and range rate if there are errors in the modulation rate or phase discontinuities. Often, this is a result of a nonlinearity of the voltage-controlled oscillator that is in the source of the transmit and receive local oscillator. The nonlinearity can be corrected at the source by using a nonlinear control voltage or by processing the received beat frequency. Any signal processing using the later method leads to computation time and energy costs, which can be considerable in some applications. When the range migration algorithm using the Stolt Transform is used for Synthetic Aperture Radar (SAR) image processing, the autofocus linearization technique described here costs nothing in additional hardware or computation time. **  
autofocus; FMCW; radar; range migration; Stolt transform; nonlinear error  
**1. Introduction**  
Much has been written about FMCW radar, especially using linear modulation [1, 2, 3, 4]. These can be seen in block diagram Figure 0(a), and in Figure 0(b), which shows an example waveform where the desired modulation is a linear ramp from 400 MHz to 800 MHz  
The transmitted frequency during a single linear sweep is  
\[\omega_{t}(t)=\frac{d}{dt}\theta_{t}(t)=2\pi\;m_{o}t+\omega_{o} \tag{1}\]  
where  
\[\theta_{t}(t)=\pi\;m_{o}t^{2}+\omega_{o}t+\varphi\]  
is the instantaneous transmit phase, \(m_{o}\) is the modulation slope in Hz/s, \(t\) is time, \(\omega_{o}\) is the frequency offset at the beginning of the ramp in rad/s, and \(\varphi\) is a constant phase angle in radians. The phase of the received signal from a single point scatterer at a delay \(\theta r\), is  
\[\theta_{t}(t)=\pi\;m_{o}(t+\tau)^{2}+\omega_{o}(t+\tau)+\varphi \tag{2}\]  
where \(\theta r(t)\) is the transmit or receive phase as a function of time, and \(\tau\) is the round-trip time delay for a single point target. The received signal is a time delayed version of the transmit signal due to a radar scattering object. The transmit and receive signals are mixed and the baseband difference signal is kept for further processing. \[\theta_{\delta}(t)=\theta_{r}(t)-\theta_{t}(t)=\left(\pi m_{o}\tau^{2}+\omega _{o}\tau\right)+2\pi m_{o}t\tau+\omega_{o}\pi \tag{3}\]The discrete phase as sampled as In phase and Quadrature phase baseband signals can be expressed as  
\[\theta_{\delta}(n)=2\pi m_{o}nT\tau+\omega_{o}\Big{(}\pi m_{o}\tau^{2}+\omega_{o} \tau\Big{)}+\pi \tag{4}\]  
where \(T\) is the sample period and \(n\) is the sample number. The frequency of the baseband signal is  
\[\omega_{\delta}=\frac{d}{dt}\theta_{\delta}(t)=2\pi m_{o}\tau \tag{5}\]  
which would show up as a peak after taking the Fourier Transform of the sampled data. It is also known that nonlinearity in the modulation, i.e., replacing \(m_{o}\) with \(m(t)\), results in a baseband frequency that is not constant over the modulation ramp. This is a common problem in FMCW radar where the signal source is a voltage-controlled oscillator (VCO), and does not have a linear voltage to frequency characteristic. If left uncorrected, defocusing in Synthetic Aperture Radar (SAR) images becomes worse, especially at large squint angles and wide bandwidths. Solutions have been proposed and proven to compensate or correct for nonlinearities.