i have to generate two correlated random series X and Y from the given Correlation functions. I have auto correlation function for XX and YY. Similarly i have cross correlation functions for XY and YX. These all correlation functions are in Fourier transformed space (omega space). i am generating sequences of values through every correlation function. Every sequence has 3200 values. So in total i have four sequences , two for auto correlation function and two for cross correlation functions. Now i am confused, How to create my correlation matrix in Toeplitz matrix form.
Let me explain with example, let suppose i have four sequences of values generated with help of their related functions
XX={1, 2, 3, 4, 5, 6, 7, 7 8} % auto correlation for X random series
YY={2, 3, 5, 6 ,7 8 9, , 3, 2} % auto correlation for Y random series
XY={a, b ,c , d, e, f, g, e, f, x} % Cross correlation between X and Y
YX= XY(conjugate) % Cross Correlation between Y and X
Now according to Circulant embedding method, How can i handle my whole correlation matrix. My correlation matrix must be a Block Teoplitz matrix. Can any body help me? Thanks in advance
I have figured out how to do this. I want to add the answer to my own question because may be someone else will have the same problem, so my answer might be helpful.
look at the references "Helgason (2011), Fast and exact synthesis of stationary multivariate Gaussian time series using circulant embedding" and
Adam M. Sykulski, Donald B. Percival, "EXACT SIMULATION OF NONCIRCULAR OR IMPROPER COMPLEX-VALUED STATIONARY GAUSSIAN PROCESSES USING CIRCULANT EMBEDDING "
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