Given a multivariate random process {x(t)}t=1,…,N\{\textbf{x}({t})\}_{t=1,…,N} where x=(x1,…,xd)\textbf{x}=(x_{1},…,x_{d}) and a univariate random process {y(t)}t=1,…,N\{y(t)\}_{t=1,…,N}, calculate the Cross Spectral Density (CSD).

If x(t)\textbf{x}(t) is of dimension of 1, we could either

(a) estimate the cross-covariance and then apply a Fourier transform or

(b) Apply Fourier transforms to each signal and then multiply one with the complex conjugate of the other.

How do we compute the CSD in the above case? Are there methods equivalent to the above two methods for multivariate signals?

Is this a valid question? Any help would be greatly appreciated.

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