I want to compute a covariance matrix of Fourier amplitude spectrum. Paper said Matrix element is $A_{i,j} = \int_\Omega\langle x,e_i \rangle\langle x,e_j \rangle\,|l(x)|dx$
$l(x)$ is the Fourier spectrum. $e_i$ and $e_j$ are members of the canonical basis of the Fourier domain Ω. But What is a canonical basis?
I ask Paper author。He said:"In any vector space, the canonical basis are the vector with only one component equals to one and the others equals to zero. For example, in 3D: e_0 = (1,0,0), e_1 = (0,1,0), e_2 = (0,0,1).
What that means is, in the context of 5D covariance tracing, where the space is (x,y,u,v,t):
\Sigma_0,0 is the variance for the x coordinate. \Sigma_0,1 is the co-variance between the x and y coordinates. \Sigma_0,2 is the co-variance between the x and u coordinate.
But expressed in the corresponding Fourier domain." But in DFT ave negative frequency, inner product will be negative. I do not think canonical basis just ont component is one.