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The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is 1 / N {\displaystyle 1/{\sqrt {N}}} , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem .
This is sometimes known as a generalized DFT (or GDFT), also called the shifted DFT or offset DFT, and has analogous properties to the ordinary DFT: X k = ∑ n = 0 N − 1 x n e − i 2 π N ( k + b ) ( n + a ) k = 0 , … , N − 1. {\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {i2\pi }{N}}(k+b)(n+a)}\quad \quad k=0,\dots ,N-1.}
Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. The Fast Fourier Transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.
If one draws the data-flow diagram for this pair of operations, the (x 0, x 1) to (y 0, y 1) lines cross and resemble the wings of a butterfly, hence the name (see also the illustration at right). A decimation-in-time radix-2 FFT breaks a length-N DFT into two length-N/2 DFTs followed by a combining stage consisting of many butterfly operations.
When the DFT is used for spectral analysis, the {x n} sequence usually represents a finite set of uniformly spaced time-samples of some signal x(t) where t represents time. The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of x ( t ) into a discrete-time Fourier transform (DTFT), which ...
The Fourier transform can therefore be seen to relate the coefficients and the values of a polynomial: the coefficients are in the time-domain, and the values are in the frequency domain. Here, of course, it is important that the polynomial is evaluated at the n th roots of unity, which are exactly the powers of α {\displaystyle \alpha } .
The base cases of the recursion are N=1, where the DFT is just a copy =, and N=2, where the DFT is an addition = + and a subtraction =. These considerations result in a count: 4 N log 2 N − 6 N + 8 {\displaystyle 4N\log _{2}N-6N+8} real additions and multiplications, for N >1 a power of two.
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone multi-frequency signaling (DTMF) tones produced by the push buttons of the keypad of a traditional analog telephone.