Analytic signal について

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Analytic signal ：ウィキペディア英語版

Analytic signal

In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.〔``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html(1:07:57 PM )〕 The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilbert transform.
The analytic representation of a real-valued function is an ''analytic signal'', comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as single-sideband. As long as the manipulated function has no negative frequency components (that is, it is still ''analytic''), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the phasor concept:〔Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p269〕 while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.
==Definition==

If

s(t)

is a ''real-valued'' function with Fourier transform

S(f)

, then the transform has Hermitian symmetry about the

f = 0

axis:
:

S(-f) = S(f)^*,

which is the complex conjugate of

S(f)

.
The function:
:

\beginS_\mathrm(f) &\stackrel2S(f), &\text\ f > 0,\\S(f), &\text\ f = 0,\\0, &\text\ f < 0\end\\&= \underbrace_S(f) = S(f) + \sgn(f)S(f),\end

where:
* $\operatorname(f)$ is the Heaviside step function,
* $\sgn(f)$ is the sign function,

contains only the ''non-negative frequency'' components of

S(f)

. And the operation is reversible, due to the Hermitian symmetry of

S(f)

:
:

).\end

The analytic signal of

s(t)

is the inverse Fourier transform of

S_\mathrm(f)

:
:

)\\&= \underbrace\}_ + \overbrace\}_} * \underbrace\}_}^\\&= s(t) + j\underbrace_\\&= s(t) + j\hat(t),\end

where
*

\hat(t) \stackrel}()

is the Hilbert transform of

s(t)

;
*

*

is the convolution symbol;
*

j

is the imaginary unit.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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