I have a same noise passing through two different ARMA filter. The autospectrum of first filter is given by S11=G1*G1*S_in where S_in is input spectrum created by set of random numbers.
Similarly the autospectrum of second filter is given by S22=G2*G2*S_in where S_in is input spectrum created by same set of random numbers as the previous one.
The cross spectrum that i expect is S12=G1*G2*S_in.But the amplitude is reduced is this due to phase shift of the filter or the derivation above is incorrect.
Attached is an image1 explaining the above statements.