User`s guide

Identifying F requency-Response Models
yt Gqut vt() ( ) () ()=+
where u(t) and y(t) are the input and output signals, respectively. G(q) is
called the transfer function of the system—it takes the input to the output
and captures the system dynamics. The G(q)u(t) notation represents the
following operation:
Gqut gkut k
k
()() ()( )=−
=
∞
∑
1
q is the shift operator,defined by t he followin g equation:
Gq gkq q ut ut
k
k
() () () ( )==−
−
=
∞
−
∑
1
1
1
G(q) that is evaluated on the unit circle, G(q=e
iw
),isthefrequency-response
function.
Together, G(q=e
iw
) and the output noise spectrum
ˆ
()Φ
v
ω
comprise the
frequency-domain description of the system.
According to the Blackman-Tukey approach, the estimated frequency-response
function is given by the following equation:
ˆ
ˆ
ˆ
Ge
N
i
yu
u
ω
ω
ω
()
=
()
()
Φ
Φ
In this case, ^ represents approximate quantities. For a derivation of this
equation, see the chapter on nonparametric time- and frequency-dom ain
methods in System Identifi cation: Theory for the User, Second Edition, by
Lennart Ljung, Prentice Hall PTR, 1999.
The output noise spectrum is given by the following equation:
3-9