User`s guide

3 Linear Model Identification
ˆˆ
ˆ
ˆ
ΦΦ
Φ
Φ
vy
yu
u
ωω
ω
ω
()
=
()
−
()
()
2
This equation for the noise spectrum is derived by assum ing the linear
relationship
yt Gqut vt() ( ) () ()=+
,thatu(t) is in dependent of v(t),andthe
following relationships between the spectra:
ΦΦΦ
y
i
uv
Ge() () ()ωωω
ω
=
()
+
2
ΦΦ
yu
i
u
Ge() ()ωω
ω
=
()
where the noise spectrum is given by the following equation:
Φ
vv
iw
Re() ()ωτ
τ
τ
≡
=−∞
∞
−
∑
ˆ
()Φ
yu
ω
is the output-input cross-spectrum and
ˆ
()Φ
u
ω
is the input spectrum.
The algorithms for estimating the frequency response (such as
spa)perform
the following steps:
1 Compute the covariances and cross-covariance from u(t) and y(t),asfollows:
ˆ
ˆ
ˆ
Rytyt
Rutut
R
y
N
t
N
u
N
t
N
yu
N
ττ
ττ
τ
()
=+
()()
()
=+
()()
()
=
=
=
∑
∑
1
1
1
1
1
yyt ut
t
N
+
()()
=
∑
τ
1
3-10