Extension to cross-spectra

The same procedure as the one described above can be applied to cross-spectra. Let us first rewrite () as:

f(a_t)=s+f'(a_t)(a_t-a_t+1) (231)

This is clearly nothing but a first order Taylor expansion around (s,a_t+1). We can write a similar relation for a two-dimensional function as:

 

f(a_t,b_t)=s+f^(a)(a_t,b_t)(a_t-a_t+1)+f^(b)(a_t,b_t)(b_t-b_t+1) (232)

where f^(a) and f^(b) denote the partial derivative with respect to a and b.

If we now consider f(a,b)=ab, we can write () as:

 

a_t b_t-s=b_t(a_t-a_t+1)+a_t(b_t-b_t+1) (233)

Now we can again use Burg's observation (1998, personal communication) and use () to factorize cross-spectra written as polynomials in :  After dividing both sides by , we obtain the equation that enables us to find both the causal and the anticausal part of a cross-spectrum with the Wilson-Burg algorithm: