Coherence (signal processing)

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 is linear, it can be used to estimate the causality between the input and output.

[citation needed] The coherence (sometimes called magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as:[1][2] where Gxy(f) is the Cross-spectral density between x and y, and Gxx(f) and Gyy(f) the auto spectral density of x and y respectively.

In cases where the ideal linear system assumptions are insufficient, the Cauchy–Schwarz inequality guarantees a value of

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.

as an estimate of the fractional power of the output that is not contributed by the input at a particular frequency.

provides a spectral quantification of the output power that is uncorrelated with noise or other inputs.

Figure 3 shows the autospectral density of ocean water level over a long period of time.

Likewise, the autospectral density of groundwater well levels are shown in figure 4.

It is clear that variation of the groundwater levels have significant power at the ocean tidal frequencies.

Let us assume that there is a linear relationship between the ocean surface height and the groundwater levels.

If the relation (transfer function) between the input and output is nonlinear, then values of the coherence can be erroneous.

Another common mistake is to assume a causal input/output relation between observed variables, when in fact the causative mechanism is not in the system model.

For example, it is clear that the atmospheric barometric pressure induces a variation in both the ocean water levels and the groundwater levels, but the barometric pressure is not included in the system model as an input variable.

In reality it is a combination of hydrological forcing from the ocean water levels and the tidal potential that are driving both the observed input and output signals.

[3] Coherence has been used to measure dynamic functional connectivity in brain networks.

Studies show that the coherence between different brain regions can change during different mental or perceptual states.

[4] Brain coherence during the rest state can be affected by disorders and diseases.

Figure 1: Coherence between ocean water level (input) and groundwater well level (output).
Figure 2: Barometric pressure (black), ocean water levels (red), and groundwater well level (blue) near Lake Worth Beach, Florida , during Hurricane Frances .
Figure 3: Autospectral density of ocean water level. The tidal components are labeled.
Figure 4: Autospectral density of groundwater well level.