%iddemo4

%   Lennart Ljung
%   Copyright 1986-2001 The MathWorks, Inc.
%   $Revision: 1.6 $  $Date: 2001/04/06 14:21:46 $

echo on
%       In this demo we consider spectrum estimation, using Marple's
%       test case (The complex data in L. Marple: S.L. Marple, Jr, 
%       Digital Spectral Analysis with Applications, Prentice-Hall,
%       Englewood Cliffs, NJ 1987.)

load marple

%       Most of the routines in the SITB support complex data.
%       For plotting we examine the real and imaginary parts of
%       the data separately, however.

%       First, take a look at the data:
pause   % Press a key for plot.
subplot(211),plot(real(marple)),title('Real part of data.')
subplot(212),plot(imag(marple)),title('Imaginary part of data.')

pause   % Press a key to continue

%       Let's first check the periodogram of the data;

per = etfe(marple);

ffplot(per)    

pause   % Press a key to continue.

%       The spectrum can also be plotted with logarithmic frequency scale
%       as a bodeplot:

bode(per)   

pause   % Press a key to continue.

%       Since the data record is only 64 samples, and the periodogram is
%       computed for 128 frequencies, we clearly see the oscilla-
%       tions from the narrow frequency window. We therefore apply some
%       smoothing to the periodogram (corresponding to a frequency resolution
%       of 1/32 Hz):

sp = etfe(marple,32);

ffplot(per,sp)  

pause   % Press a key to continue.

%       Let's now try the Blackman-Tukey approach to spectrum estimation:

ssm = spa(marple);

ffplot(sp,'b',ssm,'g')    
        % Blue: Smoothed periodogram.
        % Green: Blackman-Tukey estimate.

pause   % Press a key to continue.

%       The default window length gives a very narrow lag window for this
%       small amount of data. We can choose a larger lag window by

ss20 = spa(marple,20);

ffplot(sp,ss20)
        % Blue/solid: Smoothed periodogram.
        % Green/dashed: Blackman-Tukey estimate.

pause   % Press a key to continue.

%       A parametric 5-order AR-model is computed by

t5 = ar(marple,5);

%       Compare with the periodogram estimate:

ffplot(sp,'b',t5,'g')   
        % Blue Smoothed periodogram.
        % Green 5th order AR estimate.
pause   % Press a key to continue.

%       The AR-command in fact covers 20 different methods for
%       spectrum estimation.  The above one was what is known
%       as 'the modified covariance estimate' in Marple's book.
%       Some other well known ones are obtained with:

tb5 = ar(marple,5,'burg');      % Burg's method
ty5 = ar(marple,5,'yw');        % The Yule-Walker method
 

ffplot(t5,tb5,ty5)
        % blue: Modified covariance
        % green: Burg
        % red: Yule-Walker
pause   % Press a key to continue.

%       AR-modeling can also be done using the Instrumental
%       Variable approach:

ti = ivar(marple,4);
 
ffplot(t5,ti)
        % blue: Modified covariance
        % green: Instrumental Variable        
pause   % Press a key to continue.


%       Furthermore, the SITB covers ARMA-modeling of spectra:

ta44 = armax(marple,[4 4]);   % 4 AR-parameters and 4 MA-parameters
  
ffplot(t5,ta44)
       % blue: Modified covariance
       % green: ARMA
pause
echo off
set(gcf,'NextPlot','replace');