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This letter focuses on the efficient estimation of the frequencies and damping factors of a single 2-D damped complex exponential in additive Gaussian noise. We derive the estimators by extending the FFT-based frequency estimator that relies on interpolation on Fourier coefficients to 2-D damped signals. Performance analysis shows that the algorithm can achieve minimum variances at the fixed point when implemented in an interleaved manner for two iterations. Furthermore, we propose linearized version of the estimators that render them more amenable to real-time DSP implementation. We also demonstrate that the iterative implementation of the algorithm combining both versions is both unbiased and accurate.