Linear Filtering

Let ${\bf y}$ be an $M$-by-1 output vector from a linear filtering operation

$${\bf y}={\bf A}\; {\bf x},$$

where ${\bf A}$ is an $M$-by-$N$ filter matrix and ${\bf x}$ is a real $N$-by-1 input vector. Let ${\bf z}$ be the $M$-by-1 vector of squared values of the elements of ${\bf y}$. For example, let

$${\bf A} = \left[ \begin{array}{cccccc} a_2 & a_1 & a_0 & 0 & 0 & ... ...0 & 0 & a_2 & a_1 & a_0 & 0 \\ 0 & 0 & 0 & a_2 & a_1 & a_0 \end{array}\right].$$ (9.10)

Then, we may write the elements of ${\bf z}$ as

$$z_m = {\bf x}^\prime \; {\bf A}^\prime \; {\bf U}_m \; {\bf A}\; {\bf x}, \;\;\; 1\leq m \leq M,$$

where ${\bf U}_m$ is an $M$-by-$M$ matrix of all zeros except a single one on the main diagonal in location $m,m$.

We now test the SPA solution against a Gaussian mixture approximation. We used a filter length of 3 as shown in equation (9.10) and a feature size of $M=3$. The filter coefficients were ${\bf a}= [1\; 1\; 1]$. A Gaussian mixture approximation of $p({\bf z})$ under the WGN assumption was obtained using 5000 samples and 20 mixture components. Next, 100 new samples were generated and the log PDF from the SPA was compared with the mixture approximation.

The results of the experiment are shown in Figure 9.2. On the X-axis is the output of software/pdf_quadspa.m, and on the Y-axis is the Gaussian mixture approximation. Light-blue points are computed using usual saddlepoint. Black points are comouted using saddlepoint plus correction term (next term in Taylor series). See software/module_quadspa_test.m with task='filter'. The correction term can be invoked with level = 1.

Figure 9.2: Comparison of Saddlepoint Approximation with Gaussian mixture.