User`s guide

3 Linear Model Identification
where W
M
(k) is the lag window, and M is the width of the lag window. The
output covariance R
y
(kT) is given by the following discrete representation:
ˆ
() ( )()RkT
N
ylT kT ylT
y
l
N
=−
=
∑
1
1
Because there is no scaling in a discrete Fourier transform of a vector, the
purpose of T is to relate the discrete transform of a vector to the physically
meaningful transform of the measured signal. This normalization sets the
units of
Φ
y
()ω
as power per radians per unit time, and makes the frequency
units radians per unit time.
The scaling factor of T is necessary to preserve the energy density of the
spectrum after interpolation or decimation.
By Parseval’s theorem, the average energy of the signal m ust equal the
average energy in the estimated spectrum, as follows:
Ey t d
SEyt
Sd
y
T
T
y
T
T
2
2
1
2
1
2
1
2
() ( )
()
()
/
/
/
/
=
≡
≡
−
−
∫
∫
π
ωω
π
ωω
π
π
π
π
Φ
Φ
Tocomparetheleftsideoftheequation(S1)totherightside(S2), enter the
following commands in the MATLAB Command Window:
load iddata1
% Create time-series iddata o bject
y = z1(:,1,[]);
% Define sample interval from the data
T = y.Ts;
% Estimate frequency response
sp = spa(y);
% Remove spurious dimensions
phiy = squeeze(sp.spec);
% Compute average energy from the estimated
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