Does whitening with PCA correspond to power spectrum flattening?

Question

I'm finally making some headway in understanding data whitening, but I have a question now relating what I think of as "data-based" whitening to the situation where there are time-varying signals.

So in the "data-based" world, suppose I have a mean-centered column matrix $X$ with covariance $\Sigma = \mathbb{E}[XX^\top]$. This data matrix can be transformed into a whitened dataset $W$ by computing $W=\Lambda^{-1/2}EX$, where $\Lambda$ and $E$ are the eigenvalues and eigenvectors of $\Sigma$.

So far, so good. What I'm curious about is, what if my data vectors are a bunch of windowed snippets of audio (for example) ? I could use the DFT to compute a mean power spectrum for these snippets, and then rescale each snippet by the inverse of the spectrum at each frequency. If I understand correctly, this would whiten the power spectrum of my dataset.

Does this process equate to whitening in the "data-based" world above ?

If so, are the eigenvalues of the data covariance (which I understand are also referred to as a spectrum) directly related to the mean power spectrum of my signals ? How ? Any pointers to good reading on the subject would be very welcome.

Sorry if this is an elementary question, but I haven't had much signal processing background and find this to be really fascinating.

Answer 1

I believe the answer here is no.

Remember what 'whitening' is really doing: It is ensuring that the power/variance that correspond to each of your basis vectors are the same. Which basis vectors? The ones you chose.

In the DFT, you are choosing basis vectors comprised of complex exponential functions. In PCA, you are determining the basis vectors empirically. What is the relationship between complex exponential bases and empirical bases? You cannot say.

Thus, when you say you whitened in the spectral domain, you are really saying "I made sure the power/variance associated with all my complex exponential basis vectors representing my data are the same".

When you say you whitened in the PCA domain, you are really saying "I made sure the power/variance associated with of all my empirically derived basis vectors are the same".

Are your empirically derived basis vectors complex exponentials? There is no guarantee. Thus, whitening in either domain is not equivalent to whitening in the other.

If so, are the eigenvalues of the data covariance (which I understand are also referred to as a spectrum) directly related to the mean power spectrum of my signals ? How ?

If I understand your question correctly, then the answer to this is a simple yes, for the simple reason of energy conservation. All the eigenvalues together will give you the total energy/variance in your data. The average of this energy, will have to correspond to your mean power in any domain, including the spectral.

Projecting your data onto either the DFT or PCA (empirical) bases dones not add/remove energy, but simply redistributes it. Thus, your average power has to always be the same.

Answer 2

Generally the vector X, is called measurement vector. For a signal, components of this vector are measurements of signal's value at different times. For a wide-sense stationary signal the auto-covariance of X produce a Toeplitz matrix. A special kind of Toeplitz matrices is Circulant matrix. Circulant matrices diagonalized by DFT. When your signal is wide-sense cyclostationary the auto-covariance becomes a circulant matrix.

So considering when we use DFT we implicitly assumed our signal is periodic, it will be justified why our auto-covariance is a circulant matrix. So instead of calculating eigenvectors (or eigenfunctions) empirically we know the Fourier basis will diagonalize the auto-covariance beforehand and the eigenvalues are power spectrum of the signal.