corrmtx

Data matrix for autocorrelation matrix estimation

Description

H = corrmtx(x,m) returns an (n+m)-by-(m+1) rectangular Toeplitz matrix H = H such that HH is a biased estimate of the autocorrelation matrix for the input vector x. n is the length of x, m is the prediction model order, and H is the conjugate transpose of H.

example

H = corrmtx(x,m,method) computes the matrix H according to the method specified by method.

[H,r] = corrmtx(___) also returns the (m + 1)-by-(m + 1) autocorrelation matrix estimate r, computed as HH, for any of the previous syntaxes.

Examples

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Generate a signal composed of three complex exponentials embedded in white Gaussian noise. Compute the data and autocorrelation matrices using the 'modified' method.

n = 0:99;
s = exp(i*pi/2*n)+2*exp(i*pi/4*n)+exp(i*pi/3*n)+randn(1,100);
m = 12;
[X,R] = corrmtx(s,m,'modified');

Plot the real and imaginary parts of the autocorrelation matrix.

[A,B] = ndgrid(1:m+1);
subplot(2,1,1)
plot3(A,B,real(R))
title('Re(R)')
subplot(2,1,2)
plot3(A,B,imag(R))
title('Im(R)') Input Arguments

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Input data, specified as a vector.

Prediction model order, specified as a positive real integer.

Matrix computation method, specified as 'autocorrelation', 'prewindowed', 'postwindowed', 'covariance' or 'modified'.

• 'autocorrelation': (default) H 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 model. The matrix can be used to perform autoregressive parameter estimation using the Yule-Walker method. For more details, see aryule.

• 'prewindowed': H 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 model.

• 'postwindowed': H 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 model.

• 'covariance': H is the (nm)-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 model. The matrix can be used to perform autoregressive parameter estimation using the covariance method. For more details, see arcov.

• 'modified': H is the 2(nm)-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 model. The matrix can be used to perform autoregressive parameter estimation using the modified covariance method. For more details, see armcov.

Output Arguments

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Data matrix, returned for autocorrelation matrix estimation. The size of H depends on the matrix computation method specified in method.

Biased autocorrelation matrix, returned as a (m + 1)-by-(m + 1) rectangular Toeplitz matrix.

Algorithms

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

$H=\frac{1}{\sqrt{n}}\left[\begin{array}{ccccc}x\left(1\right)& 0& \cdots & 0& 0\\ x\left(2\right)& x\left(1\right)& \cdots & 0& 0\\ x\left(3\right)& x\left(2\right)& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ x\left(m\right)& x\left(m-1\right)& \cdots & x\left(1\right)& 0\\ x\left(m+1\right)& x\left(m\right)& \cdots & x\left(2\right)& x\left(1\right)\\ x\left(m+2\right)& x\left(m+1\right)& \cdots & x\left(3\right)& x\left(2\right)\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ x\left(n-1\right)& x\left(n-2\right)& \cdots & x\left(n-m\right)& x\left(n-m-1\right)\\ x\left(n\right)& x\left(n-1\right)& \cdots & x\left(n-m+1\right)& x\left(n-m\right)\\ 0& x\left(n\right)& \cdots & x\left(n-m+2\right)& x\left(n-m+1\right)\\ ⋮& ⋮& ⋰& ⋮& ⋮\\ 0& 0& \cdots & x\left(n-1\right)& x\left(n-2\right)\\ 0& 0& \cdots & x\left(n\right)& x\left(n-1\right)\\ 0& 0& \cdots & 0& x\left(n\right)\end{array}\right].$

In the 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 H of corrmtx for each method:

• 'autocorrelation' — (default) H = H.

• 'prewindowed'H is the n-by-(m + 1) submatrix of H whose first row is [x(1) … 0] and whose last row is [x(n) … x(nm)].

• 'postwindowed'H is the n-by-(m + 1) submatrix of H whose first row is [x(m + 1) … x(1)] and whose last row is [0 … x(n)].

• 'covariance'H is the (nm)-by-(m + 1) submatrix of H whose first row is [x(m + 1) … x(1)] and whose last row is [x(n) … x(nm)].

• 'modified'H is the 2(nm)-by-(m + 1) matrix Hmod defined by

${H}_{\mathrm{mod}}=\frac{1}{\sqrt{2\left(n-m\right)}}\left[\begin{array}{ccc}x\left(m+1\right)& \cdots & x\left(1\right)\\ ⋮& ⋰& ⋮\\ x\left(n\right)& \cdots & x\left(n-m\right)\\ {x}^{\ast }\left(1\right)& \cdots & {x}^{\ast }\left(m+1\right)\\ ⋮& \ddots & ⋮\\ {x}^{\ast }\left(n-m\right)& \cdots & {x}^{\ast }\left(n\right)\end{array}\right].$

 Marple, S. Lawrence. Digital Spectral Analysis: With Applications. Prentice-Hall Signal Processing Series. Englewood Cliffs, N.J: Prentice-Hall, 1987.