Double Spectral theorem and Two Type Magnitude Coherence Functions

Studying frequency domain representation for the coherence between two signals is an important basic theoretical problem in the fundamental theories of signal processing. However, magnitude-squared coherence function (MSCFs) could lose some phase information. Based on the core theorem in the frequency domain coherence theories, called as double spectral theorem (DST), we presented the two types of new magnitude coherence functions (MCFs), called as the same type magnitude coherence function (SMCF) and the difference magnitude coherence function (DMCF) respectively, which had been mathematically derived from DST and the conditions that they are equal to 1 or -1 or 0 were theoretically derived from DST. Here, we further demonstrated that SMCF and DMCF could be used to exactly extract the coherence between two signals by each component.