Misplaced Pages

#679320

44-404: 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 ,

88-535: A , b {\displaystyle a,b} : Reversing the time (i.e. replacing n {\displaystyle n} by N − n {\displaystyle N-n} ) in x n {\displaystyle x_{n}} corresponds to reversing the frequency (i.e. k {\displaystyle k} by N − k {\displaystyle N-k} ). Mathematically, if { x n } {\displaystyle \{x_{n}\}} represents

132-416: A finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point. Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions: One standard approach for proving these identities

176-418: A function whose domain is the set of natural numbers , then a periodic sequence is simply a special type of periodic function . The smallest p for which a periodic sequence is p -periodic is called its least period or exact period . Every constant function is 1-periodic. The sequence 1 , 2 , 1 , 2 , 1 , 2 … {\displaystyle 1,2,1,2,1,2\dots }

220-445: A linear phase e i 2 π N n m {\displaystyle e^{{\frac {i2\pi }{N}}nm}} for some integer m corresponds to a circular shift of the output X k {\displaystyle X_{k}} : X k {\displaystyle X_{k}} is replaced by X k − m {\displaystyle X_{k-m}} , where

264-403: A periodic sequence (sometimes called a cycle or orbit ) is a sequence for which the same terms are repeated over and over: The number p of repeated terms is called the period ( period ). A (purely) periodic sequence (with period p ), or a p- periodic sequence , is a sequence a 1 , a 2 , a 3 , ... satisfying for all values of n . If a sequence is regarded as

308-750: A discrete domain and a continuous domain are not discrete transforms. For example, the discrete-time Fourier transform and the Z-transform , from discrete time to continuous frequency, and the Fourier series , from continuous time to discrete frequency, are outside the class of discrete transforms. Classical signal processing deals with one-dimensional discrete transforms. Other application areas, such as image processing , computer vision , high-definition television , visual telephony, etc. make use of two-dimensional and in general, multidimensional discrete transforms. Periodic sequence In mathematics ,

352-489: Is k {\displaystyle k} cycles per N {\displaystyle N} samples. The normalization factor multiplying the DFT and IDFT (here 1 and 1 N {\displaystyle {\tfrac {1}{N}}} ) and the signs of the exponents are the most common conventions . The only actual requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that

396-417: Is a periodic summation of the x {\displaystyle x} sequence : ( x N ) n   ≜ ∑ m = − ∞ ∞ x ( n − m N ) . {\displaystyle (x_{_{N}})_{n}\ \triangleq \sum _{m=-\infty }^{\infty }x_{(n-mN)}.} Customarily,

440-824: Is a continuous and periodic function. The DFT computes N equally-spaced samples of one cycle of the DTFT. (see Fig.2 and § Sampling the DTFT ) 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}\right\}:=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}\right\}:=X_{0},X_{1},\ldots ,X_{N-1},} which

484-388: Is a linear transform, i.e. if F ( { x n } ) k = X k {\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}} and F ( { y n } ) k = Y k {\displaystyle {\mathcal {F}}(\{y_{n}\})_{k}=Y_{k}} , then for any complex numbers

SECTION 10

#1732779553680

528-527: Is a periodic extension of an N-length y {\displaystyle y} -sequence, which can also be expressed as a circular function : Then the convolution can be written as : which gives rise to the interpretation as a circular convolution of x {\displaystyle x} and y . {\displaystyle y.} It is often used to efficiently compute their linear convolution. (see Circular convolution , Fast convolution algorithms , and Overlap-save ) Similarly,

572-402: Is a primitive N th root of unity . For example, in the case when N = 2 {\displaystyle N=2} , ω N = e − i π = − 1 {\displaystyle \omega _{N}=e^{-i\pi }=-1} , and (which is a Hadamard matrix ) or when N = 4 {\displaystyle N=4} as in

616-560: Is also N {\displaystyle N} -periodic (in index n). In Eq.2 , each X k {\displaystyle X_{k}} is a complex number whose polar coordinates are the amplitude and phase of a complex sinusoidal component ( e i 2 π k N n ) {\displaystyle \left(e^{i2\pi {\tfrac {k}{N}}n}\right)} of function x n . {\displaystyle x_{n}.} (see Discrete Fourier series ) The sinusoid's frequency

660-914: Is defined by: X k = ∑ n = 0 N − 1 x n ⋅ e − i 2 π k N n {\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cdot e^{-i2\pi {\tfrac {k}{N}}n}}     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} \right\}} or F ( x ) {\displaystyle {\mathcal {F}}\left(\mathbf {x} \right)} or F x {\displaystyle {\mathcal {F}}\mathbf {x} } . Eq.1 can be interpreted or derived in various ways, for example: Eq.1 can also be evaluated outside

704-404: Is even) and [ − N − 1 2 , N − 1 2 ] {\textstyle \left[-{\frac {N-1}{2}},{\frac {N-1}{2}}\right]} (if N {\displaystyle N} is odd), which amounts to swapping the left and right halves of the result of the transform. The inverse transform is given by: Eq.2 .

748-416: Is non-zero at only discrete frequencies (see DTFT § Periodic data ), and therefore so is its product with the continuous function DTFT { x } . {\displaystyle \scriptstyle {\text{DTFT}}\displaystyle \{x\}.}   That leads to a considerable simplification of the inverse transform. where x N {\displaystyle x_{_{N}}}

792-414: Is periodic with least period 2. The sequence of digits in the decimal expansion of 1/7 is periodic with period 6: More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below). The sequence of powers of −1 is periodic with period two: More generally, the sequence of powers of any root of unity is periodic. The same holds true for

836-458: Is real as well. In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to N − 1 {\displaystyle N-1} (instead of roughly − N / 2 {\displaystyle -N/2} to + N / 2 {\displaystyle +N/2} as above), similar to the inverse DFT formula. This interpolation does not minimize

880-682: Is the DFT up to a permutation of coefficients. Since the number of permutations of n elements equals n!, there exists exactly n! linear and invertible maps with the same fundamental property as the DFT with respect to convolution. It can also be shown that : The trigonometric interpolation polynomial where the coefficients X k are given by the DFT of x n above, satisfies the interpolation property p ( n / N ) = x n {\displaystyle p(n/N)=x_{n}} for n = 0 , … , N − 1 {\displaystyle n=0,\ldots ,N-1} . For even N , notice that

924-496: Is to apply De Moivre's formula to the corresponding root of unity . Such sequences are foundational in the study of number theory . A sequence is eventually periodic or ultimately periodic if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as a k + r = a k {\displaystyle a_{k+r}=a_{k}} for some r and sufficiently large k . For example,

SECTION 20

#1732779553680

968-497: The Discrete Fourier transform § Example above, ω N = e − i π / 2 = − i {\displaystyle \omega _{N}=e^{-i\pi /2}=-i} , and The inverse transform is then given by the inverse of the above matrix, With unitary normalization constants 1 / N {\textstyle 1/{\sqrt {N}}} ,

1012-454: The Fourier transform the counterpart is the discrete Fourier transform . In addition to spectral analysis of signals, discrete transforms play important role in data compression , signal detection , digital filtering and correlation analysis. The discrete cosine transform (DCT) is the most widely used transform coding compression algorithm in digital media , followed by the discrete wavelet transform (DWT). Transforms between

1056-613: The Nyquist component X N / 2 N cos ⁡ ( N π t ) {\textstyle {\frac {X_{N/2}}{N}}\cos(N\pi t)} is handled specially. This interpolation is not unique : aliasing implies that one could add N to any of the complex-sinusoid frequencies (e.g. changing e − i t {\displaystyle e^{-it}} to e i ( N − 1 ) t {\displaystyle e^{i(N-1)t}} ) without changing

1100-457: The cross-correlation of x {\displaystyle x} and y N {\displaystyle y_{_{N}}} is given by : As seen above, the discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. It has been shown that any linear transform that turns convolution into pointwise product

1144-500: The discrete-time Fourier transform (DTFT) indicates that a convolution of two sequences can be obtained as the inverse transform of the product of the individual transforms. An important simplification occurs when one of sequences is N-periodic, denoted here by y N , {\displaystyle y_{_{N}},} because DTFT { y N } {\displaystyle \scriptstyle {\text{DTFT}}\displaystyle \{y_{_{N}}\}}

1188-467: The DFT and inverse DFT summations are taken over the domain [ 0 , N − 1 ] {\displaystyle [0,N-1]} . Defining those DFTs as X {\displaystyle X} and Y {\displaystyle Y} , the result is : In practice, the x {\displaystyle x} sequence is usually length N or less, and y N {\displaystyle y_{_{N}}}

1232-412: The DFT becomes a unitary transformation , defined by a unitary matrix: Discrete transform In signal processing , discrete transforms are mathematical transforms , often linear transforms , of signals between discrete domains, such as between discrete time and discrete frequency. Many common integral transforms used in signal processing have their discrete counterparts. For example, for

1276-3565: The DFT of x {\displaystyle \mathbf {x} } using Eq.1 X 0 = e − i 2 π 0 ⋅ 0 / 4 ⋅ 1 + e − i 2 π 0 ⋅ 1 / 4 ⋅ ( 2 − i ) + e − i 2 π 0 ⋅ 2 / 4 ⋅ ( − i ) + e − i 2 π 0 ⋅ 3 / 4 ⋅ ( − 1 + 2 i ) = 2 X 1 = e − i 2 π 1 ⋅ 0 / 4 ⋅ 1 + e − i 2 π 1 ⋅ 1 / 4 ⋅ ( 2 − i ) + e − i 2 π 1 ⋅ 2 / 4 ⋅ ( − i ) + e − i 2 π 1 ⋅ 3 / 4 ⋅ ( − 1 + 2 i ) = − 2 − 2 i X 2 = e − i 2 π 2 ⋅ 0 / 4 ⋅ 1 + e − i 2 π 2 ⋅ 1 / 4 ⋅ ( 2 − i ) + e − i 2 π 2 ⋅ 2 / 4 ⋅ ( − i ) + e − i 2 π 2 ⋅ 3 / 4 ⋅ ( − 1 + 2 i ) = − 2 i X 3 = e − i 2 π 3 ⋅ 0 / 4 ⋅ 1 + e − i 2 π 3 ⋅ 1 / 4 ⋅ ( 2 − i ) + e − i 2 π 3 ⋅ 2 / 4 ⋅ ( − i ) + e − i 2 π 3 ⋅ 3 / 4 ⋅ ( − 1 + 2 i ) = 4 + 4 i {\displaystyle {\begin{aligned}X_{0}&=e^{-i2\pi 0\cdot 0/4}\cdot 1+e^{-i2\pi 0\cdot 1/4}\cdot (2-i)+e^{-i2\pi 0\cdot 2/4}\cdot (-i)+e^{-i2\pi 0\cdot 3/4}\cdot (-1+2i)=2\\X_{1}&=e^{-i2\pi 1\cdot 0/4}\cdot 1+e^{-i2\pi 1\cdot 1/4}\cdot (2-i)+e^{-i2\pi 1\cdot 2/4}\cdot (-i)+e^{-i2\pi 1\cdot 3/4}\cdot (-1+2i)=-2-2i\\X_{2}&=e^{-i2\pi 2\cdot 0/4}\cdot 1+e^{-i2\pi 2\cdot 1/4}\cdot (2-i)+e^{-i2\pi 2\cdot 2/4}\cdot (-i)+e^{-i2\pi 2\cdot 3/4}\cdot (-1+2i)=-2i\\X_{3}&=e^{-i2\pi 3\cdot 0/4}\cdot 1+e^{-i2\pi 3\cdot 1/4}\cdot (2-i)+e^{-i2\pi 3\cdot 2/4}\cdot (-i)+e^{-i2\pi 3\cdot 3/4}\cdot (-1+2i)=4+4i\end{aligned}}} results in X = ( X 0 X 1 X 2 X 3 ) = ( 2 − 2 − 2 i − 2 i 4 + 4 i ) . {\displaystyle \mathbf {X} ={\begin{pmatrix}X_{0}\\X_{1}\\X_{2}\\X_{3}\end{pmatrix}}={\begin{pmatrix}2\\-2-2i\\-2i\\4+4i\end{pmatrix}}.} The DFT

1320-541: The domain k ∈ [ 0 , N − 1 ] {\displaystyle k\in [0,N-1]} , 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 ] {\textstyle \left[-{\frac {N}{2}},{\frac {N}{2}}-1\right]} (if N {\displaystyle N}

1364-545: The formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below. If X k {\displaystyle X_{k}} and Y k {\displaystyle Y_{k}} are the DFTs of x n {\displaystyle x_{n}} and y n {\displaystyle y_{n}} respectively then Parseval's theorem states: where

Discrete Fourier transform - Misplaced Pages Continue

1408-473: The input vector x = ( x 0 x 1 x 2 x 3 ) = ( 1 2 − i − i − 1 + 2 i ) . {\displaystyle \mathbf {x} ={\begin{pmatrix}x_{0}\\x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}1\\2-i\\-i\\-1+2i\end{pmatrix}}.} Calculating

1452-504: The interpolation property, but giving different values in between the x n {\displaystyle x_{n}} points. The choice above, however, is typical because it has two useful properties. First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited . Second, if the x n {\displaystyle x_{n}} are real numbers, then p ( t ) {\displaystyle p(t)}

1496-405: The powers of any element of finite order in a group . A periodic point for a function f  : X → X is a point x whose orbit is a periodic sequence. Here, f n ( x ) {\displaystyle f^{n}(x)} means the n -fold composition of f applied to x . Periodic points are important in the theory of dynamical systems . Every function from

1540-442: The product of their normalization factors be 1 N . {\displaystyle {\tfrac {1}{N}}.} An uncommon normalization of 1 N {\displaystyle {\sqrt {\tfrac {1}{N}}}} for both the DFT and IDFT makes the transform-pair unitary. This example demonstrates how to apply the DFT to a sequence of length N = 4 {\displaystyle N=4} and

1584-492: 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

1628-459: The sequence of digits in the decimal expansion of 1/56 is eventually periodic: A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x 1 ,  x 2 ,  x 3 , ... is asymptotically periodic if there exists a periodic sequence a 1 ,  a 2 ,  a 3 , ... for which For example, the sequence is asymptotically periodic, since its terms approach those of

1672-524: The set of N -dimensional complex vectors: where δ k k ′ {\displaystyle \delta _{kk'}} is the Kronecker delta . (In the last step, the summation is trivial if k = k ′ {\displaystyle k=k'} , where it is 1 + 1 + ⋯ = N , and otherwise is a geometric series that can be explicitly summed to obtain zero.) This orthogonality condition can be used to derive

1716-536: The slope, and is not generally real-valued for real x n {\displaystyle x_{n}} ; its use is a common mistake. Another way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as the DFT matrix , a Vandermonde matrix , introduced by Sylvester in 1867, where ω N = e − i 2 π / N {\displaystyle \omega _{N}=e^{-i2\pi /N}}

1760-409: The star denotes complex conjugation . The Plancherel theorem is a special case of Parseval's theorem and states: These theorems are also equivalent to the unitary condition below. The periodicity can be shown directly from the definition: Similarly, it can be shown that the IDFT formula leads to a periodic extension. Multiplying x n {\displaystyle x_{n}} by

1804-450: The subscript is interpreted modulo N (i.e., periodically). Similarly, a circular shift of the input x n {\displaystyle x_{n}} corresponds to multiplying the output X k {\displaystyle X_{k}} by a linear phase. Mathematically, if { x n } {\displaystyle \{x_{n}\}} represents the vector x then The convolution theorem for

Discrete Fourier transform - Misplaced Pages Continue

1848-435: 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 DFT has many applications, including purely mathematical ones with no physical interpretation. But physically it can be related to signal processing as a discrete version (i.e. samples) of the discrete-time Fourier transform (DTFT), which

1892-535: The time domain and the corresponding effects on its DFT X k {\displaystyle X_{k}} in the frequency domain. The vectors u k = [ e i 2 π N k n | n = 0 , 1 , … , N − 1 ] T {\displaystyle u_{k}=\left[\left.e^{{\frac {i2\pi }{N}}kn}\;\right|\;n=0,1,\ldots ,N-1\right]^{\mathsf {T}}} form an orthogonal basis over

1936-533: The vector x then If F ( { x n } ) k = X k {\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}} then F ( { x n ∗ } ) k = X N − k ∗ {\displaystyle {\mathcal {F}}(\{x_{n}^{*}\})_{k}=X_{N-k}^{*}} . This table shows some mathematical operations on x n {\displaystyle x_{n}} in

#679320