Data PDF of Circular ARMA process

Based on (17.5), (10.9), we may write

$$\log p({\bf x}; {\bf b} ) = - \frac{1}{2} \sum_{k=0}^{N-1} \; \le... ...{\vert X_k\vert^2 \vert A_k\vert^2}{N \sigma^2 \; \vert B_k\vert^2 } \right\} .$$ (10.18)

Note that although (10.18) 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 (10.6).

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

$$\sum_{k=0}^{N-1} \; \log \left({\vert B_k\vert^2 \over \vert A_k\vert^2 } \right) \simeq 0.$$ (10.19)

Problem 1   Show (10.19). Hint: write the time-domain equivalent of (10.18) (similar to section 10.2.3 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.