PolyMatAnalyticEigVectors

[Q,chi] = PolyMatAnalyticEigVectors(R,Lambda,Nmax,Th1,Th2)

   Extract approximate analytic eigenvectors from a parahermitian matrix R 
   using the approach described in [4], returning analytic eigenvalues in 
   Lambda. These are used to order the eigenvectors in each bin, extract 1D 
   eigenspaces across algebraic multiplicitities, and obtain an analytic 
   eigenvector in each such 1d subspace through finding the phase in each bin
   that creates the smoothes possible function, or shortest time-domain 
   support.

   Input parameters
       R        MxMxN space-time covariance matrix (time-domain)
       Lambda   MxL analytic eigenvalues (time domain, one per row)
       Nmax     maximum FFT length for iteration (default 256)
       Th1      threshold for orthonormality (default 5e-5)
       Th2      threshold for trimming (default 1e-4)

   Output parameters
       Q        MxMx? matrix of eigenvectors (time domain)
       chi      error in paraunitarity

  References
  [1] S Weiss, J Pestana and IK Proulder: "On the existence and uniqueness of
      the eigenvalue decomposition of a parahermitian matrix," IEEE Trans. on
      Signal Processing, 66(10):2659-2672, May 2018.
  [2] S Weiss, J Pestana, IK Proulder, and FK Coutts: "Corrections to `On the 
      existence and uniqueness of the eigenvalue decomposition of a para-
      hermitian matrix'," IEEE Trans. on Signal Processing, 66(23):6325-6327, 
      Dec. 2018.
  [3] S Weiss, IK Proulder, and FK Coutts: "Parahermitian matrix eigenvalue 
      decomposition: extraction of analytic eigenvalues," IEEE Trans. on 
      Signal Processing, 69:722-737, Jan. 2021.
  [4] S Weiss, IK Proulder, FK Coutts, and F Khattak: "Parahermitian matrix eigenvalue 
      decomposition: extraction of analytic eigenvectors," IEEE Trans. on 
      Signal Processing, submitted, Feb. 2022.

0001 function [Q,chi] = PolyMatAnalyticEigVectors(R,Lambda,Nmax,Thresh1,Thresh2);
0002 %[Q,chi] = PolyMatAnalyticEigVectors(R,Lambda,Nmax,Th1,Th2)
0003 %
0004 %   Extract approximate analytic eigenvectors from a parahermitian matrix R
0005 %   using the approach described in [4], returning analytic eigenvalues in
0006 %   Lambda. These are used to order the eigenvectors in each bin, extract 1D
0007 %   eigenspaces across algebraic multiplicitities, and obtain an analytic
0008 %   eigenvector in each such 1d subspace through finding the phase in each bin
0009 %   that creates the smoothes possible function, or shortest time-domain
0010 %   support.
0011 %
0012 %   Input parameters
0013 %       R        MxMxN space-time covariance matrix (time-domain)
0014 %       Lambda   MxL analytic eigenvalues (time domain, one per row)
0015 %       Nmax     maximum FFT length for iteration (default 256)
0016 %       Th1      threshold for orthonormality (default 5e-5)
0017 %       Th2      threshold for trimming (default 1e-4)
0018 %
0019 %   Output parameters
0020 %       Q        MxMx? matrix of eigenvectors (time domain)
0021 %       chi      error in paraunitarity
0022 %
0023 %  References
0024 %  [1] S Weiss, J Pestana and IK Proulder: "On the existence and uniqueness of
0025 %      the eigenvalue decomposition of a parahermitian matrix," IEEE Trans. on
0026 %      Signal Processing, 66(10):2659-2672, May 2018.
0027 %  [2] S Weiss, J Pestana, IK Proulder, and FK Coutts: "Corrections to `On the
0028 %      existence and uniqueness of the eigenvalue decomposition of a para-
0029 %      hermitian matrix'," IEEE Trans. on Signal Processing, 66(23):6325-6327,
0030 %      Dec. 2018.
0031 %  [3] S Weiss, IK Proulder, and FK Coutts: "Parahermitian matrix eigenvalue
0032 %      decomposition: extraction of analytic eigenvalues," IEEE Trans. on
0033 %      Signal Processing, 69:722-737, Jan. 2021.
0034 %  [4] S Weiss, IK Proulder, FK Coutts, and F Khattak: "Parahermitian matrix eigenvalue
0035 %      decomposition: extraction of analytic eigenvectors," IEEE Trans. on
0036 %      Signal Processing, submitted, Feb. 2022.