User`s guide

Identifying F requency-Response Models
2 Compute the Fourier transforms of the covariances a nd the
cross-covariance, as follows:
ˆ
()
ˆ
() ()
ˆ
()
ˆ
() ()
Φ
Φ
yy
M
M
M
i
uu
M
M
M
i
RW e
RW e
ωττ
ωττ
τ
ωτ
τ
ωτ
=
=
=−
−
=−
−
∑
∑
ˆˆ
()
ˆ
() ()Φ
yu yu
M
M
M
i
RWeωττ
τ
ωτ
=
=−
−
∑
where
W
M
()τ
is called the lag window with the width M.
3 Compute the frequency-response function
ˆ
Ge
N
iω
()
and the output noise
spectrum
ˆ
()Φ
v
ω
Alternatively, the disturbance v(t) can be described as filtered white noise:
vt H qet() ( ) ()=
where e(t) is the white n o i se with varia n c e
λ
and the noise p ower spectrum
is given by the following equation:
Φ
v
i
He()ωλ
ω
=
()
2
Understanding Spectrum Normalization
The spectrum of a signal is the s quare of the Fourier transform of the
signal. The spectral estimate using the commands
spa, spa fdr,andetfe is
normalized by the sampling interval T:
Φ
yy
kM
M
iwT
M
TRkTeWk() ( ) ()ω=
=−
−
∑
3-11