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DFT Matrix
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. Definition An ''N''-point DFT is expressed as the multiplication X = W x, where x is the original input signal, W is the ''N''-by-''N'' square DFT matrix, and X is the DFT of the signal. The transformation matrix W can be defined as W = \left(\frac\right)_ , or equivalently: : W = \frac \begin 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^ \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^&\omega^&\omega^&\cdots&\omega^ \end , where \omega = e^ is a primitive ''N''th root of unity in which i^2=-1. We can avoid writing large exponents for \omega using the fact that for any exponent x we have the identity \omega^ = \omega^. This is the Vandermonde matrix for the roots of unity, up to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, 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 number, complex Sine wave, 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, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fractional Frequency
A fraction is one or more equal parts of something. Fraction may also refer to: * Fraction (chemistry), a quantity of a substance collected by fractionation * Fraction (floating point number), an (ambiguous) term sometimes used to specify a part of a floating point number * Fraction (politics), a subgroup within a parliamentary party * Fraction (radiation therapy), one unit of treatment of the total radiation dose of radiation therapy that is split into multiple treatment sessions * Fraction (religion), the ceremonial act of breaking the bread during Christian Communion People with the surname * Matt Fraction, a comic book author See also * Algebraic fraction, an indicated division in which the divisor, or both dividend and divisor, are algebraic expressions ** Irrational fraction, a type of algebraic fraction * Faction (other) * ''Frazione'', a type of administrative division of an Italian ''commune'' * Free and Independent Fraction, a Romanian political party * Par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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The Clock And Shift Matrices
''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multidimensional Transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. Multidimensional Fourier transform One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation to a frequency domain representation.Smith, W. Handbook of Real-Time Fast Fourier Transforms:Algorithms to Product Testing, Wiley_IEEE Press, edition 1, pages 73–80, 1995 The discrete-domain multidimensional Fourier transform (FT) can be computed as follows: : F(w_1,w_2,\dots,w_m) = \sum_^\infty \sum_^\infty \cdots \sum_^\infty f(n_1,n_2,\dots,n_m) e^ where ''F'' stands for the multidimensional Fourier transform, ''m'' stands for multidimensional dimension. Define ''f'' as a multidimensional discrete-domain signal. The inverse multidimensional Fourier transform is given by : f(n_1,n_2,\dots,n_m) = \left(\frac\right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Integral Transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Generalized Fourier Series
In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory. Definition Consider a set of square-integrable functions with values in \mathbb = \Complex or \mathbb = \R, \Phi = \_^\infty, which are pairwise orthogonal for the inner product \langle f, g\rangle_w = \int_a^b f(x)\,\overline(x)\,w(x)\,dx where w(x) is a weight function, and \overline\cdot represents complex conjugation, i.e., \overline(x) = g(x) for \mathbb = \R. The generalized Fourier series of a square-integrable function f : , b\to \mathbb, with respect to Φ, is then f(x) \sim \sum_^\infty c_n\varphi_n(x), where the coefficients are given by c_n = . If Φ is a complete set, i.e., an orthogonal basis of the space of all square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fourier Operator
The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it corresponds to the Fourier transform of one-dimensional functions. It is complex-valued and has a constant (typically unity) magnitude everywhere. When depicted, e.g. for teaching purposes, it may be visualized by its separate real and imaginary parts, or as a colour image using a colour wheel to denote phase. It is usually denoted by a capital letter "F" in script font (\mathcal), e.g. the Fourier transform of a function g(t) would be written using the operator as \mathcalg(t). It may be thought of as a limiting case for when the size of the discrete Fourier transform increases without bound while its spatial resolution also increases without bound, so as to become both continuous and not necessarily periodic. Visualization The Fourier operator defines a continuous two-dimensional function that extends along t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Functions of a continuous variable Consider two functions g(x) and h(x) with Fourier transforms G and H: \begin G(f) &\triangleq \mathcal\(f) = \int_^g(x) e^ \, dx, \quad f \in \mathbb\\ H(f) &\triangleq \mathcal\(f) = \int_^h(x) e^ \, dx, \quad f \in \mathbb \end where \mathcal denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically 2\pi or \sqrt) will appear in the convolution theorem below. The convolution of g and h is defined by: r(x) = \(x) \triangleq \int_^ g( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Parseval's Theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Negative Frequency
The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise. The rate is expressed in units such as revolutions (a.k.a. ''cycles'') per second ( hertz) or radian/second (where 1 cycle corresponds to 2''π'' radians). Sinusoids Let be a nonnegative angular frequency with units of radians/second. Then the function has slope , which is called a negative frequency. But when the function is used as the argument of a cosine operator, the result is indistinguishable from . Similarly, is indistinguishable from . Thus any sinusoid can be represented in terms of positive frequencies. The sign of the underlying phase slope is ambiguous. The ambiguity is resolved when the cosine and sine operators can be observed simultaneously, because leads by 1/4 cycle (= ''π''/2 radians) when , and lags by 1/4 cycle when . Similarly, a vector, , rotates coun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matched Filter
In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise. Matched filters are commonly used in radar, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal. Pulse compression is an example of matched filtering. It is so called because the impulse response is matched to input pulse signals. Two-dimensional matched filters are commonly used in image processing, e.g., to improve the SNR of X-ray observations. Matched filtering is a demodulation technique with LTI (linear time invariant) filters to maximize SNR. It was originally also known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |