CR Bound analysis of the exact PDF of an ARMA processes - filtering approach.

The results of section 10.2.6 are still a little bit cumbersome to implement. Using filtering, we may find a much more efficient approach to finding the derivatives of the exact ARMA PDF (10.6). We can rewrite (10.27) as

$$D(a_m) = - \frac{1}{N\sigma^2} \; {\rm Re}\left\{ e^{j 2\pi k m ... ..._k A_k \over B_k }\right) \left( {\bar{X}_k \over \bar{B}_k }\right) \right\}.$$

In the time domain, this suggests a correlation between u and v where

       u = filter(a,b,x)/sqrt(sig2);
       v = filter(1,b,x)/sqrt(sig2);
       for m=1:P,
        da(m)= -sum(u(K+1:N) .* v(K+1-m:N-m));
       end;

where $K$ is the impulse response length of the wihtening filter (Section 10.2.3). Similarly, we can rewrite (10.28) as

$$D(b_m) = \frac{1}{N\sigma^2} \; {\rm Re}\left\{ e^{j 2\pi k m / ... ...r B_k }\right)\left( {\bar{X}_k \bar{A}_k \over \bar{B}_k^2 }\right) \right\}.$$

In the time domain, this suggests a correlation between u and w, where

       u = filter(a,b,x)/sqrt(sig2);
       w=filter(1,b,u);
       for m=1:Q,
         db(m)= sum(u(K+1:N) .* w(K+1-m:N-m));
       end;

Of course, this approach, however, misses the first $K$ elements in the summation. In fact, we can patch it up. If we use Section 10.2.2 for the first $K$ elements only, then add in the above values, we can obtain an exact hybrid approach based on filtering with order $N$ computation. See software/pdf_arma_exact.m.