# SMD

## PURPOSE [H,Gamma] = SMD(R,maxiter,epsilon,Mu,vers);

## SYNOPSIS function [H,Gamma] = SMD(R,maxiter,epsilon,Mu,SFlag);

## DESCRIPTION ```[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 .

[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:

 S. Redif, S. Weiss, and J.G. McWhirter, "Sequential Matrix Diagonalisation
Algorithms for Polynomial EVD of Parahermitian Matrices," to appear in IEEE
Transactions on Signal Processing.

Please acknowledge this paper if this function is utilised for academic output.```

## CROSS-REFERENCE INFORMATION This function calls:
This function is called by:

## SOURCE CODE ```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 .
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 %   S. Redif, S. Weiss, and J.G. McWhirter, "Sequential Matrix Diagonalisation
0071 %      Algorithms for Polynomial EVD of Parahermitian Matrices," to appear in IEEE
0072 %      Transactions on Signal Processing.
0073 %
0074 %  Please acknowledge this paper if this function is utilised for academic output.```

Generated on Wed 03-Dec-2014 19:38:11 by m2html © 2005