Coherence (signal processing)

In signal processing, the 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. In signal processing, the 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 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 0 ≤ C x y ( f ) ≤ 1 {displaystyle 0leq 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: y ( t ) = h ( t ) ∗ x ( t ) {displaystyle y(t)=h(t)*x(t)} , where * denotes convolution. In the Fourier domain this equation becomes Y ( f ) = H ( f ) X ( f ) {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: G y y = | H ( f ) | 2 G x x ( f ) {displaystyle G_{yy}=|H(f)|^{2}G_{xx}(f)} and G x y = H ( f ) G x x ( f ) {displaystyle G_{xy}=H(f)G_{xx}(f)} , and since G x x ( f ) {displaystyle G_{xx}(f)} 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 C x y ≤ 1 {displaystyle C_{xy}leq 1} . 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 1 − C x y {displaystyle 1-C_{xy}} 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: G v v {displaystyle G_{vv}} provides a spectral quantification of the output power that is uncorrelated with noise or other inputs. Here we illustrate the computation of coherence (denoted as γ 2 {displaystyle gamma ^{2}} ) as shown in figure 1.Consider the two signals shown in the lower portion of figure 2.There appears to be a close relationship between the ocean surface water levels and the groundwater well levels. It is also clear that the barometric pressure has an effect on both the ocean water levels and groundwater levels.