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In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, ''u''(''t''), and produces a function, ''H''(''u'')(''t''), with the same domain. The Hilbert transform is important in the field of signal processing where it is used to derive the analytic representation of a signal ''u''(''t''). This means that the real signal ''u''(''t'') is extended into the complex plane such that it satisfies the Cauchy–Riemann equations. For example, the Hilbert transform leads to the harmonic conjugate of a given function in Fourier analysis, aka harmonic analysis. Equivalently, it is an example of a singular integral operator and of a Fourier multiplier. The Hilbert transform was originally defined for periodic functions, or equivalently for functions on the circle, in which case it is given by convolution with the ''Hilbert kernel''. More commonly, however, the Hilbert transform refers to a convolution with the ''Cauchy kernel'', for functions defined on the real line R (the boundary of the upper half-plane). The Hilbert transform is closely related to the Paley–Wiener theorem, another result relating holomorphic functions in the upper half-plane and Fourier transforms of functions on the real line. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the Riemann–Hilbert problem for holomorphic functions. == Introduction == The Hilbert transform of ''u'' can be thought of as the convolution of ''u''(''t'') with the function ''h''(''t'') = 1/(''t''). Because ''h''(''t'') is not integrable, the integrals defining the convolution do not converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by p.v.). Explicitly, the Hilbert transform of a function (or signal) ''u''(''t'') is given by: : provided this integral exists as a principal value. This is precisely the convolution of ''u'' with the tempered distribution p.v. 1/''t'' (due to ; see ). Alternatively, by changing variables, the principal value integral can be written explicitly as: : When the Hilbert transform is applied twice in succession to a function ''u'', the result is negative ''u'': : provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is −''H''. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of ''u''(''t'') (see Relationship with the Fourier transform, below). For an analytic function in upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if ''f''(''z'') is analytic in the plane Im ''z'' > 0 and ''u''(''t'') = Re ''f''(''t'' + 0·''i'' ) then Im ''f''(''t'' + 0·''i'' ) = ''H''(''u'')(''t'') up to an additive constant, provided this Hilbert transform exists. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert transform」の詳細全文を読む スポンサード リンク
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