A whitening transformation (PCA) is simply a rotation into a space in which variables become uncorrelated.
Because a DFT is a transformation into a coordinate space of orthogonal frequency components, a DFT is a also just a rotation. We can obtain the DFT of a signal by creating a matrix of orthogonal basis vectors and solving for their weights. You can test this in Matlab using:
B = 1000; b = 0:B-1; Fs = 1000; t = b/Fs; y = 3*cos(2*pi*1*t) + 2*sin(2*pi*2*t + pi/2); X = [cos(2*pi*1*Fs/B*t)' sin(2*pi*1*Fs/B*t)' cos(2*pi*2*Fs/B*t)' sin(2*pi*2*Fs/B*t)']; Y = (X'*X)\X'*y'; % assuming y = XY, so Y gives the coefficients of the frequency components
We see that the coefficients corresponding to cosine of 1 and 2 Hz are 3 and 2, as expected (since the sin of 2 Hz has a phase of pi/2).
If both of these are just rotations of the original data, does that mean that whitening the original signal is equivalent to whitening its DFT?