
Single channel signal 43
FIGURE 2.8: Example of AIC function. a) Signal generated with AR model order
5(a
1
= 0.2,a
2
= −0.5,a
3
= −0.3,a
4
= 0.1,a
5
= 0.2), b) AIC for models of order p
estimated for that signal. Note a minimum at p = 5.
p
∑
i=0
a
i
x
t−i
= ε
t
(2.54)
Applying Z transform to both sides of the above equation and taking into account its
properties (Sect. 1.4.6) we get:
A( f )X( f )=E( f ) (2.55)
multiplying equation (2.55) by A
−1
leads to:
X( f )=A
−1
( f )E( f )=H( f )E( f ) (2.56)
where we defined H( f )=A
−1
( f ). From the definition of power spectrum we obtain:
S( f )=X( f )X
∗
( f )=H( f )VH
∗
( f ) (2.57)
Expressing H(s) in terms of model coefficients:
H(z)=
1
∑