Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. X k = ∑ n = 0 N − 1 x n ⋅ e − i 2 π N k n = ∑ n = 0 N − 1 x n ⋅ [ cos ⁡ ( 2 π k n / N ) − i ⋅ sin ⁡ ( 2 π k n / N ) ] , {displaystyle {egin{aligned}X_{k}&=sum _{n=0}^{N-1}x_{n}cdot e^{-{frac {i2pi }{N}}kn}\&=sum _{n=0}^{N-1}x_{n}cdot ,end{aligned}}}     (Eq.1) x n = 1 N ∑ k = 0 N − 1 X k ⋅ e i 2 π k n / N , n ∈ Z , {displaystyle x_{n}={frac {1}{N}}sum _{k=0}^{N-1}X_{k}cdot e^{i2pi kn/N},quad nin mathbb {Z} ,,}     (Eq.2) x n = 1 N ∑ k = 0 N − 1 X k ⋅ e i 2 π k n / N {displaystyle x_{n}={frac {1}{N}}sum _{k=0}^{N-1}X_{k}cdot e^{i2pi kn/N}}     (Eq.3) In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms 'FFT' and 'DFT' are often used interchangeably. Prior to its current usage, the 'FFT' initialism may have also been used for the ambiguous term 'finite Fourier transform'. The discrete Fourier transform transforms a sequence of N complex numbers { x n } := x 0 , x 1 , … , x N − 1 {displaystyle left{mathbf {x_{n}} ight}:=x_{0},x_{1},ldots ,x_{N-1}} into another sequence of complex numbers, { X k } := X 0 , X 1 , … , X N − 1 , {displaystyle left{mathbf {X_{k}} ight}:=X_{0},X_{1},ldots ,X_{N-1},} which is defined by where the last expression follows from the first one by Euler's formula. The transform is sometimes denoted by the symbol F {displaystyle {mathcal {F}}} , as in X = F { x } {displaystyle mathbf {X} ={mathcal {F}}left{mathbf {x} ight}} or F ( x ) {displaystyle {mathcal {F}}left(mathbf {x} ight)} or F x {displaystyle {mathcal {F}}mathbf {x} } . Eq.1 can also be evaluated outside the domain k ∈ [ 0 , N − 1 ] {displaystyle kin } , and that extended sequence is N {displaystyle N} -periodic. Accordingly, other sequences of N {displaystyle N} indices are sometimes used, such as [ − N 2 , N 2 − 1 ] {displaystyle left} (if N {displaystyle N} is even) and [ − N − 1 2 , N − 1 2 ] {displaystyle left} (if N {displaystyle N} is odd), which amounts to swapping the left and right halves of the result of the transform. Eq.1 can be interpreted or derived in various ways, for example: The normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1 / N {displaystyle 1/N} .  A normalization of 1 / N {displaystyle scriptstyle {sqrt {1/N}}} for both the DFT and IDFT, for instance, makes the transforms unitary. A discrete impulse, x n = 1 {displaystyle x_{n}=1} at n = 0 and 0 otherwise; might transform to X k = 1 {displaystyle X_{k}=1} for all k (use normalization factors 1 for DFT and 1/N for IDFT). A DC signal, X k = 1 {displaystyle X_{k}=1} at k = 0 and 0 otherwise; might inversely transform to x n = 1 {displaystyle x_{n}=1} for all n {displaystyle n} (use 1 / N {displaystyle 1/N} for DFT and 1 for IDFT) which is consistent with viewing DC as the mean average of the signal.