Double spectral theorem and two type magnitude-squared 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, the old magnitude-squared coherence function (OMSCF) has been proved identical to 1, so that to cannot be used to extract any coherence information. Here, we will prove a core theorem in frequency domain coherence theories in signal processing, called as the double spectral theorem (DST). Based on the theorem, we presented the two types of new magnitude-squared coherence functions (MSCFs), called as the same type magnitude-squared coherence function (SMSCF) and the difference magnitude-squared coherence function (DMSCF) respectively, which were mathematically derived from DST and the conditions that they are equal to 1 or 0 can be theoretically derived from DST. Here, we further demonstrated that SMSCF and DMSCF could be used to exactly extract the coherence between two signals by each component.