[H,Gamma] = SMD(R,maxiter,epsilon,Mu,vers);
Polynomial matrix eigenvalue decomposition (PEVD) algorithm factorising
a parahermitian matrix represented by R using the sequential matrix
diagonalisation (SMD) method [1].
[H,Gamma]=SMD(R) takes as input an MxMx(2L+1) matrix R representing a
parahermitian matrix R(z) of the form
R(z) = RL'z^L + ... + R1' z + R0 + R1 z^{-1} + ... + RLz^{-L}
whereby
R(:,:,1) = RL';
...
R(:,:,L) = R1';
R(:,:,L+1) = R0;
R(:,:,L+2) = R1;
...
R(:,:,2*L+1) = RL;
The function returns a paraunitary matrix H(z) in H which creates an
approximately diagonalised parahermitian Gamma(z)
Gamma(z) = H(z) R(z) H~(z).
The format of Gamma representing Gamma(z) is analogously to R above. For
H(z) = H0 + H1 z^{-1} + H2 z^{-2} + ...
the returned parameter H is
H(:,:,1) = H0;
H(:,:,2) = H1;
H(:,:,3) = H2;
...
SMD will strive of diagonalise and spectrally majorise Gamma(z). The
algorithm stops either after a fixed number of iterations or once a
threshold for the maximum absolute value of off-diagonal elements has
been reached. Default values are outlined below.
[H,Gamma] = SMD(R,maxiter) stops after maxiter iterations. The default
value for the optional parameter maxiter is 400.
[H,Gamma] = SMD(R,maxiter,epsilon) stops either after maxiter iterations
or once the maximum absolute off-diagonal element has a value smaller than
epsilon.
[H,Gamma] = SMD(R,maxiter,epsilon,mu) additionally performs a truncation
of the parahermitian matrix at every iteration such that matrices at outer
lags containing a mu-th of the total power are truncated. This stops
unnecessary growth of the parahermitian matrix, and therefore keeps
computational complexity down and Gamma of sufficiently low order.
[H,Gamma] = SMD(R,maxiter,epsilon,mu,vers) take the optional input vers
to switch between SMD (vers='SMD') and the coding-gain optimised maximum
element SMD (MESMD, vers='MESMD').
Input parameters:
R polynomial covariance matrix
maxiter maximum number of iterations (optional)
default: 400
epsilon stop if largest absolute off-diag element is smaller than
epsilon (optional)
default: 0.0001
mu power ratio in tail of polynomial matrix to be truncated at
every iteration (optional)
default: 0.0 (only truncating true zeroes)
vers SMD version ('SMD' or 'MESMD')
default: 'SMD'
Output parameters:
H paraunitary matrix
Gamma polynomial covariance matrix
Reference:
[1] S. Redif, S. Weiss, and J.G. McWhirter, "Sequential Matrix Diagonali-
sation Algorithms for Polynomial EVD of Parahermitian Matrices," IEEE
Transactions on Signal Processing, 63(1):81-89, January 2015.
Please acknowledge this paper if this function is utilised for academic output.
0001 function [H,Gamma] = SMD(R,maxiter,epsilon,Mu,SFlag); 0002 %[H,Gamma] = SMD(R,maxiter,epsilon,Mu,vers); 0003 % 0004 % Polynomial matrix eigenvalue decomposition (PEVD) algorithm factorising 0005 % a parahermitian matrix represented by R using the sequential matrix 0006 % diagonalisation (SMD) method [1]. 0007 % 0008 % [H,Gamma]=SMD(R) takes as input an MxMx(2L+1) matrix R representing a 0009 % parahermitian matrix R(z) of the form 0010 % R(z) = RL'z^L + ... + R1' z + R0 + R1 z^{-1} + ... + RLz^{-L} 0011 % whereby 0012 % R(:,:,1) = RL'; 0013 % ... 0014 % R(:,:,L) = R1'; 0015 % R(:,:,L+1) = R0; 0016 % R(:,:,L+2) = R1; 0017 % ... 0018 % R(:,:,2*L+1) = RL; 0019 % The function returns a paraunitary matrix H(z) in H which creates an 0020 % approximately diagonalised parahermitian Gamma(z) 0021 % Gamma(z) = H(z) R(z) H~(z). 0022 % The format of Gamma representing Gamma(z) is analogously to R above. For 0023 % H(z) = H0 + H1 z^{-1} + H2 z^{-2} + ... 0024 % the returned parameter H is 0025 % H(:,:,1) = H0; 0026 % H(:,:,2) = H1; 0027 % H(:,:,3) = H2; 0028 % ... 0029 % SMD will strive of diagonalise and spectrally majorise Gamma(z). The 0030 % algorithm stops either after a fixed number of iterations or once a 0031 % threshold for the maximum absolute value of off-diagonal elements has 0032 % been reached. Default values are outlined below. 0033 % 0034 % [H,Gamma] = SMD(R,maxiter) stops after maxiter iterations. The default 0035 % value for the optional parameter maxiter is 400. 0036 % 0037 % [H,Gamma] = SMD(R,maxiter,epsilon) stops either after maxiter iterations 0038 % or once the maximum absolute off-diagonal element has a value smaller than 0039 % epsilon. 0040 % 0041 % [H,Gamma] = SMD(R,maxiter,epsilon,mu) additionally performs a truncation 0042 % of the parahermitian matrix at every iteration such that matrices at outer 0043 % lags containing a mu-th of the total power are truncated. This stops 0044 % unnecessary growth of the parahermitian matrix, and therefore keeps 0045 % computational complexity down and Gamma of sufficiently low order. 0046 % 0047 % [H,Gamma] = SMD(R,maxiter,epsilon,mu,vers) take the optional input vers 0048 % to switch between SMD (vers='SMD') and the coding-gain optimised maximum 0049 % element SMD (MESMD, vers='MESMD'). 0050 % 0051 % Input parameters: 0052 % R polynomial covariance matrix 0053 % maxiter maximum number of iterations (optional) 0054 % default: 400 0055 % epsilon stop if largest absolute off-diag element is smaller than 0056 % epsilon (optional) 0057 % default: 0.0001 0058 % mu power ratio in tail of polynomial matrix to be truncated at 0059 % every iteration (optional) 0060 % default: 0.0 (only truncating true zeroes) 0061 % vers SMD version ('SMD' or 'MESMD') 0062 % default: 'SMD' 0063 % 0064 % Output parameters: 0065 % H paraunitary matrix 0066 % Gamma polynomial covariance matrix 0067 % 0068 % Reference: 0069 % 0070 % [1] S. Redif, S. Weiss, and J.G. McWhirter, "Sequential Matrix Diagonali- 0071 % sation Algorithms for Polynomial EVD of Parahermitian Matrices," IEEE 0072 % Transactions on Signal Processing, 63(1):81-89, January 2015. 0073 % 0074 % Please acknowledge this paper if this function is utilised for academic output.