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The central result of this thesis is deriving systematic methodologies for finding the fast Fourier transform (FFT) optimal implementations. By employing a divide-conquer procedure (decomposition), we break down the initial transform into combinations of different smalller size sub-transforms that can graphically be presented as breakdown trees. We apply the rewrite rules to the Cooley-Tukey formula and generate a set of algorithms and alternative codes for the FFT computation. We obtain a set of all possible implementations by pairing all possible breakdown trees with all code alternatives. For the runtime evaluation of these implementations, we develop the analytical and experimental perfomance models. Based on these models, we derive the optimal search methods -- the dynamic programming and exhaustive search -- and use them to search over the set of implementation runtimes for finding the implementation with the minimal runtime. The test results demonstrate that good algorithms and codes, accurate performance evaluation models, and effective search methods, combined together within the system framework, provide the best FFT implementations.Keywords: Discrete/fast Fourier transform