Coherence (signal processing)

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.

The coherence (sometimes called magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as:

where G_xy(f) is the Cross-spectral density between x and y, and G_xx(f) and G_yy(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 ${\displaystyle 0\leq C_{xy}(f)\leq 1}$. 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: ${\displaystyle y(t)=h(t)*x(t)}$, where * denotes convolution. In the Fourier domain this equation becomes ${\displaystyle Y(f)=H(f)X(f)}$, 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: ${\displaystyle G_{yy}=|H(f)|^{2}G_{xx}(f)}$ and ${\displaystyle G_{xy}=H(f)G_{xx}(f)}$, and since ${\displaystyle G_{xx}(f)}$ is real, the following identity holds,

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