# Is it a good idea to whiten a colored noise in order to apply formulas for AWGN?

I have posted a question before regarding the detection of a signal under colored Gaussian noise: Energy detection in presence of colored Gaussian noise

Theoretical formulas for setting the detection threshold are widely used for AWGN.

Is it a good idea to just perform a pre-whitening to the colored noise and then apply the well-know formulas for energy detection under white Gaussian noise?

Do you have any happy experience using this approach in practice?

The short answer is yes, whitening noise is usually recomended (if you know the spectrum of the noise) but whitening also whitens your signal. Detecting random signals in colored noise is a well known problem is sonar signal processing. There are a number of derivation that I like particularly:

Nielsen, Richard O. Sonar signal processing. Artech House, Inc., 1991.

and Burdic

@book{RN17,
author = {Burdic, William S.},
title = {Underwater acoustic system analysis},
publisher = {Prentice Hall},
ISBN = {0139476075},
year = {1991},
type = {Book}
}


Also the updated

@book{van2004detection,
title={Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory},
author={Van Trees, Harry L},
year={2004},
publisher={John Wiley \& Sons}
}


in section 4.3

and if you can find it, in the old volume 3

If you look, a copy of Eckart's classic paper can be found at Scripts.

Assuming we are talking about a linear system (i.e. within its dynamic range its behavior does not change with the strength of the incoming energy), then it is never worth adding extra noise energy in hopes of improving signal detection. Consider that after removing any out-of-band noise with a simple filter, any reasonable technique beyond that must operate on in-band noise. A short snapshot of in-band noise is almost always colored, anyway. So a "good" AWGN technique must, as a rule, also be a good colored noise technique.

• Do you have a reference?
– user28715
Apr 10 '18 at 6:08
• I've done no research. Certainly we would get a gain by using the higher SNR spots over the lower ~if~ the signal were uncorrelated across the whole band (i.e. multiple narrow-band signals at separate frequencies). But proper modulation should take advantage of Shannon-Hartley and correlate across the band. In that case, changing the spectrum would essentially be a change in modulation. At a minimum that would distort the demodulated information, and at a maximum make it unusable. Apr 10 '18 at 10:53
• Now if we specifically pick a modulation scheme that ~can~ have its spectrum changed without distorting the data, then we are welcome to flatten the spectrum if we want, but such a modulation, by definition, doesn't take full advantage of the band, because to take full advantage of the band is to also use the signal spectrum to convey information. Apr 10 '18 at 11:02
• There is no mention of modulation in the question or the prior question that if referred to. I down voted your answer. Your argument may be the correct answer, but I don’t believe to this question.
– user28715
Apr 10 '18 at 12:18
• If there is no mention of modulation, particularly when there are several text book treatments of energy detection in colored noise), I think we should assume that modulation isn’t relevant. I’ve got no problem with Shannon Theory but it is about bounds and doesn’t say much about how to achieve those bounds. The question has an answer if the noise spectrum is known. There is a better answer if both signal and noise spectra are known. The use of the term “energy detection” is relevant.
– user28715
Apr 10 '18 at 13:11