4.10 Colored gaussian noise  (Page 2/2)

The sufficient statistic computed for each signal can be given two signal processing interpretations in the colored noise case.Both of these rest on considering the quantity $r^TK^{(-1)}{s}_i$ as a simple dot product, but with different ideas on grouping terms. The simplest is to group the kernel with thesignal so that the sufficient statistic is the dot product between the observations and a modified version of the signal $\stackrel{}{s_i}=K^{(-1)}{s}_i$ . This modified signal thus becomes the equivalent to the unit-sample response of the matched filter. In this form,the observed data are unaltered and passed through a matched filter whose unit-sample response depends on both the signal andthe noise characteristics. The size of the noise covariance matrix, equal to the number of observations used by thedetector, is usually large: hundreds if not thousands of samples are possible. Thus, computation of the inverse of the noisecovariance matrix becomes an issue. This problem needs to be solved only once if the noise characteristics are static; theinverse can be precomputed on a general purpose computer using well-established numerical algorithms. The signal-to-noiseratio term of the sufficient statistic is the dot product of the signal with the modified signal $\stackrel{}{s_i}$ . This view of the receiver structure is shown in .

A second and more theoretically powerful view of the computations involved in the colored noise detector emerges whenwe factor covariance matrix. The Cholesky factorization of a positive-definite,symmetric matrix (such as a covariance matrix or its inverse) has the form $K=LDL^T$ . With this factorization, the sufficient statistic can be written as $$r^TK^{(-1)}{s}_i=(D^{-1/2}L^{(-1)}r)^TD^{-1/2}L^{(-1)}{s}_i$$ The components of the dot product are multiplied by the same matrix ( $D^{-1/2}L^{(-1)}$ ), which is lower-triangular. If this matrix were also Toeplitz, the product of this kind between a Toeplitz matrix and a vector would be equivalent to theconvolution of the components of the vector with the first column of the matrix. If the matrix is not Toeplitz (which,inconveniently, is the typical case), a convolution also results, but with a unit-sample response that varies with theindex of the output--a time-varying, linear filtering operation. The variation of the unit-sample response corresponds to thedifferent rows of the matrix $D^{-1/2}L^{(-1)}$ running backwards from the main-diagonal entry. What is the physical interpretation of theaction of this filter? The covariance of the random vector $x=Ar$ is given by $K_x=A{K}_{r}A^T$ . Applying this result to the current situation, we set $A=D^{-1/2}L^{(-1)}$ and $K_r=K=LDL^T$ with the result that the covariance matrix $K_x$ is the identity matrix! Thus, the matrix $D^{-1/2}L^{(-1)}$ corresponds to a (possibly time-varying) whitening filter : we have converted the colored-noise component of the observed data to white noise! Asthe filter is always linear, the Gaussian observation noise remains Gaussian at the output. Thus, the colored noise problemis converted into a simpler one with the whitening filter: the whitened observations are first match-filtered with the"whitened" signal $s_i^{+}=D^{-1/2}L^{(-1)}{s}_i$ (whitened with respect to noise characteristics only) then half the energy of the whitened signal is subtracted( ).