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FFT

FFT (Fast Fourier Transformation), is the fast Fourier transform, discrete fast Fourier transform algorithm, which is the discrete Fourier transform of the odd, even, true, real and other properties of the discrete Fourier transform algorithm to improve access to of. Its Fourier transform theory, and no new


FFT algorithm Figure
Found, but in a computer system or the application of digital systems Discrete Fourier Transform, can be said that a big step.
Let x (n) for the complex sequence of N items from the DFT transform, either X (m) calculation requires N complex multiplications and complex additions N-1 times, and a complex multiplication is equal to four real multiplications and two real addition, a complex addition is equal to two real additions, even if a complex multiplication and one complex addition is defined as an "operator" (four real multiplications and four real additions), then find the complex sequence of N items X (m ), the N-point DFT transform N2 times about the need for operations. When N = 1024 points or more, the need for N2 = 1048576 time operations, the FFT, using the periodicity and symmetry of WN, to an N-item sequence (set N = 2k, k is a positive integer), divided into two N / 2 items in sequence, each N / 2 point DFT transform requires (N / 2) 2 operations a second, then the two operations N times N / 2 point DFT transform to form an N-point DFT transformation. After this transformation, the total number of operations becomes N 2 (N / 2) 2 = N N2 / 2. Continuing the above example, N = 1024, the total number of operations became 525 312 times, saving about 50% of the computation. And if we will this "one divides into two" thinking constantly proceed, until the group is divided into two hundred twenty-one DFT operation unit, the N point DFT transform need only Nlog2N time computing, N in 1024, the operation 10240 times the amount only is the previous 1% of the direct algorithm, the more points, the greater the computational savings, this is the superiority of the FFT.