technologia: Fast Fourier transform

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A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from…

Evidence: 40 • authority: 80

Fast algorithms for 2-D circular convolutions and number theoretic transforms based on polynomial transforms over finite rings

Evidence: 40 • authority: 72

The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. It works by recursively applying fast Fourier transform (FFT) over the integers modulo 2 n + 1 {\displaystyle 2^{n}+1} . The run-time bit complexity to multiply two n-digit numbers using the algorithm is O ( n ⋅ log ⁡ n ⋅ log ⁡ log ⁡ n ) {\displaystyle O(n\cdot \log n\cdot \log \log n)} in big O notation. The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication, and starts to outper…

Evidence: 33 • authority: 72

In mathematics and computer science, an algorithm ( ) is any well-defined set of instructions that when followed terminates after a finite number of steps that comprise a solution to a given computational problem. Advanced algorithms may utilize loops and involve many conditionals that decide the next step based on the inputs provided, resulting in long sequences of steps before halting, but all algorithms terminate by definition. In contrast to algorithms, heuristics might be applied to problems for which it is not possible for an algorithmic solution to exist, e.g., because the problem is undecidable or there is otherwise no way to define the correct or optimal result. For example, althou…

Evidence: 33 • authority: 72

In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another sequence of the same length, representing the amplitude and phase of different frequency components. In this way, it changes data from a description in terms of sampled values to a description in terms of oscillations. The inverse discrete Fourier transform reverses this process and recovers the original sequence. For data sampled at equally spaced points, the DFT can be understood more precisely as converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore a basic…