# Whitening signal vs. whitening its DFT

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?