H13
HW6 (due Mar. 6).
Exercises 6.1, 6.2 from the textbook.
The
spectral density function
or simply the
spectrum
of a stationary time series,
1
()
2
ih
h
h
fe
ω
ωγ
π
∞
−
=−∞
=
∑
,
(1)
is the counterpart of the covariance function
h
γ
in frequency domain.
Here
h
is assumed to satisfy

h
∞
−∞
< ∞
∑
(i.e.,
h
is
absolutely summable
).
Since
h
is an even function,
1
−
=
h
, (1) can also be written as
0
1
1
2
c
o
s
2
∞
=
⎛⎞
=+
⎜
⎝⎠
∑
h
h
⎟
f
h
.
(2)
The sequence
h
can be recovered from
f
through the
inverse Fourier transform
h
f
ed
ωω
−
=
∫
.
(3)
Setting
in (3), we have
0
h
=
2
0
σω
−
==
∫
X
f
d
, i.e., the total variance of the process can be
decomposed into contributions from different frequencies, so that
f
d
represents the contribution
to the total variance of the components in the frequency range
(,
)
d
+
.
Example 1
The spectrum of a White Noise
2
2
1
,0
,
22
0,
0
Z
Z
hh
h
h
h
σ
ππ
∞
−
=−∞
⎧
=
⎪
=⇒
=
=
−
≤
⎨
≠
⎪
⎩
∑
≤
The spectrum
f
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 Spring '09
 gUR
 Covariance, Variance, spectral density, Autocorrelation, spectral density function

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