Fast Fourier Transform - Definition and Speed

Definition and Speed

An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. (In the presence of round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly, as discussed below.)

Let x₀, ...., x_N-1 be complex numbers. The DFT is defined by the formula

Evaluating this definition directly requires O(N2) operations: there are N outputs X_k, and each output requires a sum of N terms. An FFT is any method to compute the same results in O(N log N) operations. More precisely, all known FFT algorithms require Θ(N log N) operations (technically, O only denotes an upper bound), although there is no known proof that a lower complexity score is impossible.

To illustrate the savings of an FFT, consider the count of complex multiplications and additions. Evaluating the DFT's sums directly involves N2 complex multiplications and N(N − 1) complex additions . The well-known radix-2 Cooley–Tukey algorithm, for N a power of 2, can compute the same result with only (N/2) log₂ N complex multiplies (again, ignoring simplifications of multiplications by 1 and similar) and N log₂N complex additions. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (see, e.g., Frigo & Johnson, 2005), but the overall improvement from O(N2) to O(N log N) remains.