Extension to cross-spectra

The same procedure as the one described above can be applied to cross-spectra. Let us first rewrite (1) as:

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

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) (10)

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 (10) as:

 

a_t b_t-s=b_t(a_t-a_t+1)+a_t(b_t-b_t+1) (11)

Now we can again use Burg's observation (1998, personal communication) and use (11) to factorize cross-spectra $F(Z)=A(Z) \bar B(1/Z)$ written as polynomials in $Z=e^{i\omega \Delta t}$: 

$$S(Z)-A_t(Z)\bar B_t(1/Z)= A_t( Z)[\bar B_{t+1}(1/Z)-\bar B_t(1/Z)]+ \bar B_t(1/Z)[ A_{t+1}( Z)- A_t( Z)]$$ (12)

After dividing both sides by $A_t(Z)\bar B_t(1/Z)$, 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:  

$${\bar B_{t+1}(1/Z) \over \bar B_t(1/Z)} \ +\ {A_{t+1}(Z) \over A_t(Z)} = 1 \ +\ {S(Z) \over \bar B_t(1/Z)\ A_t(Z)}$$ (13)