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

The results of section 9.2.6 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 (9.6). We can rewrite (9.27) as

$\displaystyle 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 9.2.3). Similarly, we can rewrite (9.28) as

$\displaystyle 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 9.2.2 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.

Baggenstoss 2017-05-19