stdtf (Frequency Domain Identification Toolbox)

Calculate standard deviations of amplitude and phase values of an identified transfer function, using the covariance matrix of the parameters.

Syntax

[tf,stda,stdph,stdri,rtf] = stdtf(freqv,pdat,cdat)

Description

The approximate standard deviations of the amplitude and phase values of estimated transfer functions are calculated from the covariance matrix of the estimated parameters. freqv is the vector of frequencies at which the standard deviations are to be evaluated. pdat is the parameter vector, usually generated by elis, or the name of a parameter file (see exppar). cdat is the covariance matrix, corresponding to pdat, in the form of an array, as generated by elis, or in the form of a vector (see expcov), or the name of a covariance file.

tf is the column vector of the complex values of the transfer function, calculated from pdat at the given frequencies, stda is the column vector of the standard deviations of the amplitudes, and stdph is the column vector of the standard deviations of the phases, in radians.

If the distribution of the complex error of tf is circularly symmetric, stda also equals the standard deviations of the real and imaginary parts, but these can also be obtained in stdri, which has three columns: standard deviations of the real parts and of the imaginary parts of tf, furthermore the correlation coefficients of the real and imaginary parts of the corresponding tf value.

rtf is the correlation coefficient matrix of the vector of the real and imaginary parts of all the points: the real and imaginary parts of the first point are the first two elements, those of the second point are the third and the fourth, and so on.

The variance of the transfer function can be calculated as

vartf = stda^2 + (abs(tf).*stdph).^2

vartf = stdri(:,1).^2 + stdri(:,2).^2

Examples

[pvect,fit,Cp] = elis('inpchans.ebn');
[tf,stda,stdph,stdri,rtf] = stdtf([400:400:19600],pvect,Cp);

Algorithm

The sensitivity matrices of the amplitudes and phases on the estimated parameters, furthermore of the real and imaginary parts are calculated by direct derivation of the corresponding expressions. The appropriate covariance matrices of the amplitudes and phases are calculated multiplying the covariance matrix of the parameters by the sensitivity matrices from both sides.