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corrmtx - Data matrix for autocorrelation matrix estimation

Syntax

X = corrmtx(x,m) X = corrmtx(x,m,'method') [X,R] = corrmtx(...)

Description

X = corrmtx(x,m) returns an (n+m)-by-(m+1) rectangular Toeplitz matrix X, such that X'X is a (biased) estimate of the autocorrelation matrix for the length n data vector x.

X = corrmtx(x,m,'method') computes the matrix X according to the method specified by the string 'method':

'autocorrelation': (default) X is the (n+m)-by-(m+1) rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using prewindowed and postwindowed data, based on an mth order prediction error model.
'prewindowed': X is the n-by-(m+1) rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using prewindowed data, based on an mth order prediction error model.
'postwindowed': X is the n-by-(m+1) rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using postwindowed data, based on an mth order prediction error model.
'covariance': X is the (n-m)-by-(m+1) rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using nonwindowed data, based on an mth order prediction error model.
'modified': X is the 2(n-m)-by-(m+1) modified rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using forward and backward prediction error estimates, based on an mth order prediction error model.

[X,R] = corrmtx(...) also returns the (m+1)-by-(m+1) autocorrelation matrix estimate R, calculated as X'*X.

Examples

randn('state',1); n=0:99;   
s=exp(i*pi/2*n)+2*exp(i*pi/4*n)+exp(i*pi/3*n)+randn(1,100);
X=corrmtx(s,12,'mod');

Algorithm

The Toeplitz data matrix computed by corrmtx depends on the method you select. The matrix determined by the autocorrelation (default) method is given by the following matrix.

In this matrix, m is the same as the input argument m to corrmtx, and n is length(x). Variations of this matrix are used to return the output X of corrmtx for each method:

'autocorrelation': (default) X = X, above.
'prewindowed': X is the n-by-(m+1) submatrix of X that is given by the portion of X above the lower gray line.
'postwindowed': X is the n-by-(m+1) submatrix of X that is given by the portion of X below the upper gray line.
'covariance': X is the (n-m)-by-(m+1) submatrix of X that is given by the portion of X between the two gray lines.
'modified': X is the 2(n-m)-by-(m+1) matrix X_mod shown below.

References

[1] Marple, S.L. Digital Spectral Analysis,Englewood Cliffs, NJ, Prentice-Hall, 1987, pp.216-223.