Log Bilinear (module_bilinear.m)

The log-bilinear transformation for data in the range [-1, 1] (excluding endpoints) is useful in time-series analysis for conditioning of the reflection coefficients (RCs). Although RCs have the nice property of being fairly uncorrelated, they do have one characteristic that makes PDF estimation difficult: they tend to be limited to the region [-1,1]. Thus, the features tend to have a sharp discontinuities complicating PDF estimation. A simple nonlinear transformation can improve things dramatically. The transformation

$$K^\prime_i= \log\left({1-K_i\over 1+K_i}\right)$$

produces values with a Gaussian-like distribution. These are called log area ratio (LAR) coefficients [25]. Let the input vector be

$${\bf k}= [r_0, K_1, K_2 \ldots K_P]^\prime$$

contain the zero-lag autocorrelation (ACF) estimate (i.e. variance) as well as the $P$ RCs. The module software/module_bilinear.m implements the transformation on ${\bf k}$. In addition, the module also takes the log of $r_0$. The syntax is

   [z,jout]=module_bilinear(k,jin);

where variable k is the input vector ${\bf k}$. Input and output are both of dimension $(P+1)$-by-$M$, where $M$ is the number of segments.

The inversion of the LAR coefficients is accomplished by the module_bilinear_synth function.