Post date: Jun 25, 2017 9:0:23 PM
The detection problem [1] is
where and are the observation, the known signal, and the zero-mean, colored Gaussian noise with covariance matrix , respectively.
+ The generalized matched-filter approach
Karhunnen-Loeve Decomposition (KLD) (for which the principal component analysis (PCA) is based on) states that the random noise can be decomposed into a linear combination of its covariance matrix's eigenvectors, i.e.
Hence the detection problem becomes
For which the optimal test is
where
where is the distorted signal.
+ The pre-whitening approach
Let
be the Cholesky matrix factorization of
, then is a white Gaussian noise with unit variance.
[1] Levy, B. C. (2008). Principles of signal detection and parameter estimation. Springer Science & Business Media.