The noise process in (5.44) does not have to be white . In the non-white case, the spectral shape of the noise needs to be estimated and ``divided out'' of the spectrum. That is, a ``pre-whitening filter'' needs to be constructed and applied to the data, so that the noise is made white. Then the previous case can be applied.
Testing for White Noise
Multiple Sinusoids in Additive Gaussian White Noise