# What is the noise whitening process?

I came across this process when working on OFDM in wireless communication. It's mentioned if we have two copies of a signal resulted from SIMO system (Single Rx Multiple Tx), we can process the noise whitening, after the FFT, in order to improve the quality of the signal.

What does mean the noise whitening, and what's its process?

Here is an example, in the paper.

After taking the DFT in OFDM, we supposed to get frequency response, so how w[k] is obtained?

• en.wikipedia.org/wiki/Whitening_transformation – Marcus Müller Aug 2 '18 at 9:04
• "it's mentioned": Who mentions that, in what context? Cite, and give context. – Marcus Müller Aug 2 '18 at 9:05
• @MarcusMüller .. 1- Why do you usually vote -1 for all my posts,? that really hurts. .. 2- I've read for example here: sndgw.snd.elec.keio.ac.jp/~sanada/TRANSACTION/… in sections 2.1 and similarly in other papers also. but this one which I'm interested in understanding it. – New_student Aug 2 '18 at 9:26
• I don't usually vote -1 on your posts. This one was at least apparently underresearched and didn't give context. That's a severe question quality problem, so I downvoted; I'm sorry if it hurts you, but you might just want to post well-defined questions presenting your own research to avoid that hurt. Please go ahead and edit your question to include that reference. – Marcus Müller Aug 2 '18 at 9:34
• Having read the section and especially the paragraphs after "As already stated, v(t) in Eq. (2) is the filtered noise.", I wonder what your specific question or unclarity is? – Marcus Müller Aug 2 '18 at 9:46

Consider a WSS (wide sense stationary) discrete-time random process (RP) with i.i.d. samples $X[n,s]$ whose auto-correlation sequence (ACF function) is $\phi_{XX}[m]$. We call a random process as white-noise if its ACF is the following:

$$\phi_{XX}[m] = \begin{cases} {\sigma_X^2 ~~~~~, \text{ for } m=0 \\ 0 ~~~~~~~~, \text{ otherwise} }\end{cases}$$

This also means that the RP is uncorrelated. If a RP does not have this property then it's a colored noise. And it means its samples are correlated.

When a RP is colored, it may be possible to find an LTI filter which will try to remove the correlation between the samples of the noise and hence make it whitened. Such a filter is called either as a decorrelator or equivalently as whitening filter.

It's best described in the context of AR-p (Auto-Regressive) random processes generation through all pole filtering, where the whitening filter will be the inverse of that all-pole AR filter which generates the colored noise from an input white noise.

• Hello .. OK, so let me explain the question, when taking the DFT in OFDM demodulator, we got two copies of signal, and it's described that noise whitening is done. how can we perform that noise whitening? is it by that whitening filter or decorrelator ? – New_student Aug 2 '18 at 11:19
• Since almost all communication engineering optimizations are based on the assumption of additive white noise on the received signals, when the noise is not white, their performance degrades from the theoretical maximum. A suggestion to improve the situation is to perform some processing on the received signal to make it whiter. And that's what the noise whitening block tries to achieve... In your case, the algorithm uses fractional oversampling for OFDM receiver which colors the received noise and therefore it's tried to be whitened to improve the reception performance. – Fat32 Aug 2 '18 at 12:34
• Thank you so much .. So, what's the process of noise whitening? do you have any link which explains that OR matlab code example? – New_student Aug 3 '18 at 13:50
• @Eng.Badr the simplest example is the inverse system of an AR process generator... Literature s abundant on this topic. – Fat32 Aug 3 '18 at 14:00
• so, whitening the noise should improve the performance of signal, is that right? and the other thing, noise whitening can include any algorithm, there is not a specific algorithm or process which is called noise whitening .. is it right? – New_student Aug 3 '18 at 15:10