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Small low-cost high-resolution synthetic aperture radar (SAR) systems are made possible by using a linear frequency-modulated continuous-wave (LFM-CW) signal. SAR processing assumes that the sensor is moving in a straight line at a constant speed, but in actuality, an unmanned aerial vehicle (UAV) or airplane will often significantly deviate from this ideal. This nonideal motion can seriously degrade the SAR image quality. In a continuous-wave system, this motion happens during the radar pulse, which means that existing motion compensation techniques that approximate the position as constant over a pulse are limited for LFM-CW SAR. Small aircraft and UAVs are particularly susceptible to atmospheric turbulence, making the need for motion compensation even greater for SARs operating on these platforms. In this paper, the LFM-CW SAR signal model is presented, and processing algorithms are discussed. The effects of nonideal motion on the SAR signal are derived, and new methods for motion correction are developed, which correct for motion during the pulse. These new motion correction algorithms are verified with simulated data and with actual data collected using the Brigham Young University muSAR system.