A nonlinear trajectory of a radar platform in synthetic aperture radar (SAR) may lead to severe coupling between the range and the azimuth, which may make the ambiguity function (AF) analysis complicated. The numerical algorithm-based AF analysis may be computationally expensive, while the existing analytical algorithm-based AF analysis may cause large errors because it does not consider the coupling between the range and the azimuth. By observing that the singular value decomposition (SVD) is good to deal with the coupling problem, in this article, we propose an effective AF analysis based on SVD. The key idea is to first use a small amount of sampling points for SVD of the coupled term in the AF and then the decoupled vectors are fitted to high-order polynomials for the analytical AF calculation. It converts the double integral into the product of two single integrals in the calculation. From the proposed SVD-based AF analysis, three parameters, namely, 3-dB resolution, peak sidelobe ratio (PSLR), and integrated sidelobe ratio (ISLR), are then effectively computed. The simulated results verify the good performance of the proposed SVD-based AF analysis.