Phase lag changes cross spectrum 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.
Derivation seems to be wrong. I am guessing from derivation that it is power spectral density that you are talking about. Your derivation seems to be wrong, cross density would be $$S_{xy}(x)S_in=G1(x)G2\times(G2^*)(x)S_in$$ or $$S_{yx}(x)S_in=G1\times(G1^*)(x)G2(x)S_in.$$ If you take magnitude spectrum both will be same, phase spectrum will be of 180 shift.