Coherence (signal Processing) - Definition and Formulation

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:

where G_xy is the cross-spectral density between x and y, and G_xx and G_yy 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 C_xy 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.