Does PCA have any advantages or usages in the frequency domain?  My question is about analysis of signals with PCA in the frequency domain.


*

*As frequency analysis offers a powerful tool for signal processing, does Principal Component Analysis (PCA) have any "well-known" advantages or usages in the frequency domain?  

*Can it be stated that PCA is doing a kind of "frequency analysis" as the principal components represent the most important frequencies in the signal?

 A: The advantage of using PCA in the frequency domain is to choose a set of weights by exploiting the cross-correlations between the signals at particular cycles.
For example, (depending on the field of application) the behaviour of the variables under study can be different in the short, medium and long run. Using PCA in the frequency domain will allow to choose weights depending on the frequency. 
The difference between PCA in the time domain and frequency domain can be understood in terms of how the eigenvalues are computed. In time domain, the correlation matrix is used. In the frequency domain, the fourier transform of the correlation matrix or the spectral density matrix is used to compute the eigenvalues. 
For technical applications of using PCA in frequency domain, there is a description in book by Jolliffe,I.T(2002), Principal Component Analysis, 2nd Edition. Here is a link to the relevant page.
Regarding your second question, I have understood PCA by itself to be a method of finding combinations of variables which extract the maximum information in the data by maximizing the variance of the principal components. Therefore, it does not seem to be dealing with any cyclic or frequency information in the data. 
A: There was a paper at the recent ICML by Li and Prakash. It is about a complex linear dynamical system, which turns out to be a model of which Fourier transform and PCA are special cases. Have a look at Time Series Cluster: complex is simpler.
A: If you accept spectroscopy as frequency-domain: PCA is used a lot there. E.g. a pubmed search on principal component analysis and spectroscopy yields more than 2500 results. On the other hand, spectroscopists rarely look at the time domain (Fourier-transform spectroscopy does use a spatial domain as "intermediate" but for data analysis the frequency/wavelength/wavenumber domain is generally used and interpreted). 
If you do a PCA in the frequency domain, the first PCs will tell you which frequencies contribute most to the variance in the data set and moreover which frequencies vary together (with pos. or neg. correlation): they end up in the same PC or independent of each other (end up in different PCs). Whether this coincides with most contributing frequencies depends on whether/where your data is centered (by its nature or by centering).
I'd say whether PCA should be done in time or frequency domain depends on the interpretation of these domains. 


*

*If you want to find things that happen at the same time or times where the same things happen, then the time domain should be appropriate. 

*IF you want to find things that happen with the same frequency or frequencies where the same things happen, then use frequency domain. 

A: Principal component analysis (PCA) is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. This transformation is defined in such a way that the first principal component has as high a variance as possible (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (uncorrelated with) the preceding components. Principal components are guaranteed to be independent only if the data set is jointly normally distributed.
I am not much knowledge about frequency analysis or signal processing. But PCA has application in this field. Check literature. 
A: PCA,  like wave transformations, is an orthogonal transformation. It tries to rotate data to maximize certain metric (i.e. the variance). Interestingly, if the data is sufficiently random (isotropic), no rotation will help to enlarge variance. In this case, PCA will basically turn to a wave transformation.
