To meet the miniature requirement, a frequency modulated continuous wave synthetic aperture radar (FMCW SAR) puts tight constraint on the compactness, which causes the interference of narrow band noise. The aim of this study is to suppress the strong noise for an FMCW SAR. First, the quantitative analysis of the noise is performed. It is found that a strong spurious noise of the analog-to-digital converter (ADC) is introduced from interferences and significantly affects the image quality; the other noise components are sufficiently small, thus having ignorable influences. Then, a Fast Fourier Transform (FFT) based method of noise suppression is proposed to eliminate the ADC strong spurious noise, adopting an ADC and a field programmable gate array (FPGA). Finally, using the real Ku-band FMCW SAR data, the level of the noise components is measured and the effectiveness of the proposed noise suppression method is validated. The results show that the measured noise level coincides with the theoretical noise level, and the proposed noise suppression method effectively eliminates the ADC strong spurious noise.