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The spectral coherence is a statistic that can be used to examine the relation between two signals or data sets. It is commonly used to estimate the power transfer between input and output of a linear system. If the signals are ergodic, and the system function linear, it can be used to ''estimate'' the causality between the input and output. ==Definition and Formulation== The coherence (sometimes called magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as:〔J. S. Bendat, A. G. Piersol, ''Random Data'', Wiley-Interscience, 1986〕〔http://www.fil.ion.ucl.ac.uk/~wpenny/course/course.html, chapter 7〕 :: where Gxy(f) is the cross-spectral density between x and y, and Gxx(f) and Gyy(f) the autospectral density of x and y respectively. The magnitude of the spectral density is denoted as |G|. Given the restrictions noted above (ergodicity, linearity) the coherence function estimates the extent to which y(t) may be predicted from x(t) by an optimum linear least squares function. Values of coherence will always satisfy . For an ''ideal'' constant parameter linear system with a single input x(t) and single output y(t), the coherence will be equal to one. To see this, consider a linear system with an impulse response h(t) defined as: , where * denotes convolution. In the Fourier domain this equation becomes , where Y(f) is the Fourier transform of y(t) and H(f) is the linear system transfer function. Since, for an ideal linear system: and , and since is real, the following identity holds, ::. However, in the physical world an ideal linear system is rarely realized, noise is an inherent component of system measurement, and it is likely that a single input, single output linear system is insufficient to capture the complete system dynamics. In cases where the ideal linear system assumptions are insufficient, the Cauchy–Schwarz inequality guarantees a value of . If Cxy is less than one but greater than zero it is an indication that either: noise is entering the measurements, that the assumed function relating x(t) and y(t) is not linear, or that y(t) is producing output due to input x(t) as well as other inputs. If the coherence is equal to zero, it is an indication that x(t) and y(t) are completely unrelated, given the constraints mentioned above. The coherence of a linear system therefore represents the fractional part of the output signal power that is produced by the input at that frequency. We can also view the quantity as an estimate of the fractional power of the output that is not contributed by the input at a particular frequency. This leads naturally to definition of the coherent output spectrum: :: provides a spectral quantification of the output power that is uncorrelated with noise or other inputs. 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「Coherence (signal processing)」の詳細全文を読む スポンサード リンク
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