Data PDF of Circular ARMA process

Based on (16.5), (9.9), we may write

$\displaystyle \log p({\bf x}; {\bf b} ) = - \frac{1}{2} \sum_{k=0}^{N-1} \; \le...
...{\vert X_k\vert^2 \vert A_k\vert^2}{N \sigma^2 \; \vert B_k\vert^2 } \right\} .$ (9.18)

Note that although (9.18) is written in terms of the frequency domain quantity $ X_k$, it is a PDF of $ {\bf x}$ and integrates to precicely 1 over $ {\bf x}$ in $ {\cal R}^N$. The circular ARMA process is useful for CR bound analysis and as a way to obtain ML ARMA parameter estimates because it is more tractable than (9.6).

A simplification is possible for large $ N$ valid for circular ARMA processes. Note that

$\displaystyle \sum_{k=0}^{N-1} \; \log \left({\vert B_k\vert^2 \over \vert A_k\vert^2 } \right) \simeq 0.$ (9.19)

Problem 1   Show (9.19). Hint: write the time-domain equivalent of (9.18) (similar to section 9.2.3 but with circulant matrix $ {\bf C}$), then note that the determinant of $ {\bf C}$ is approximated by $ -N/2 \log \sigma^2$. From there, deduce the result.

Baggenstoss 2017-05-19