National Instruments 370753C-01

Chapter 5 Classical Feedback Analysis

values at which the power spectral density is to be computed and where

dependent matrices are the input power spectral density matrix at each

frequency. psd( ) computes a cross power spectral density matrix for each

of a user–specified set of frequency values ω, returning them together in the

PDM Yspec:

psd( ) calls freq( ) internally to compute the frequency response, H, of

the system. It uses the frequency range specified by the domain of Uspec.

The power spectral density of the output as a function of frequency, givenin

Ypsd, is obtained from the real parts of the diagonal terms of the dependent

matrices in Yspec.

Some background information on power spectral density may be useful.

Given a time-domain input series U(t), the power spectral density of U(t)

is the Fourier transform of the autocorrelation of U(t). For a system with

qinputs each input spectral density dependent matrix within Uspec is a

square Hermitian matrix of size q. A Hermitian matrix is a square matrix

equal to its complex conjugate transpose. If Uspec is constant for all

frequencies (when the spectrum is white) then Uspec can be specified as

a single matrix.

If you are working with multiple systems which have been cascaded in

series, the output power spectral density of the first system can be used

as the input power density to the second system in a subsequent use of

psd( ).

For an example of how to verify the response of a system to white noise

input, refer to Example 5-7.

Example 5-7 Verifying the Response of a System to White Noise Input

You can easily generate the power spectral density of an input white noise

process.

sys = polynomial(-0.5)/polynomial([0,0,-2,-10]);

w = logspace(0.01,1,50);

Uspec = pdm(ones(w),w);

You then use psd( ) to obtain the output power spectral density and the

cross-spectral density as a function of frequency.

[Ypsd,Yspec] = psd(sys,Uspec);

Yspec Hjw()×Uspecjw×Hjw–()×=