1.0
S(f)
0.5
0.0
-0.2
-0.50
-0.25
0.0
0.25
0.50
0
frequency
64
128
0
time
64
0
128
time
16
32
Lag
1.0
S(f)
0.5
0.0
-0.5
-1.0
-0.5
-0.1
0.1
0.5
0
frequency
64
128
0
time
64
0
128
time
16
32
Lag
Figure 18: Top row: example of a purely continuous spectrum (left) and one real-
Figure 19: Top row: realization of process with purely continuous spectrum (left)
ization of length 128 (right).
and sample autocorrelation (right).
Bottom row: example of a purely discrete spectrum (left) and one realization of
Bottom row: realization of process with purely discrete spectrum (left) and sample
length 128 (right).
autocorrelation (right).
19
20
acvs
sdf (dB)
0.0
0.1
0.2
0.3
0.4
0.5
0
10
20
30
40
50
40
50
Lag
acvs
sdf (dB)
f
0.0
0.1
0.2
0.3
0.4
0.5
0
f
10
20
30
0
1
2
Lag
3
4
t
5
6
7
8
Figure 21: Illustration of the aliasing effect. The dotted curves above show cos(t)
Figure 20: Two spectral density functions (left) and their corresponding autocovari-
versus t. The solid curves show cos([1 + 2kπ]t) versus t for (from top to bottom)
ance sequences (right).
k = 1, 2 and 3. The solid black squares show the common value of all four sinusoids
when sampled at t = 0, 1, . . . , 8.
21
22
Figure 22: Examples of MA(1) spectra – when θ1,1 is positive we have a high frequency spectrum and when θ1,1 is negative we have a low frequency spectrum
dB
dB
dB
dB
Fig 21a: The concept of aliasing illustrated for two different sample intervals. In each picture the thin
black line is the continuous spectrum, and the thick black line is the resulting spectrum for the
discrete process (which is periodic, as shown by its continuation with the thick grey line).
23
0.50
0.25
f
0.0
0.50
0.25
f
0.0
e!<"#%
e!"#%
0.50
0.25
f
0.0
0.50
0.25
f
0.0
e!<"#$
e!"#$
0.25
f
0.50
0.0
0.50
0.25
f
0.50
dB
0.25
f
0.50
0.0
0.0
0.25
f
0.50
0.25
f
0.50
q!"<#$%&#$*'<("<#$%)(›("#$*))
dB
q!"#$%<#$*'<("#$%)(›("<#$*)
dB
0.0
0.0
q!<"#%
dB
q!"#%
0.25
f
dB
0.0
q!"<#$%<#$%'<("<#$%)(›("<#$%))
dB
q!"#$%&#$%'<(#$%(›(#$%)
dB
q!<"#$
dB
q!"#$
0.0
0.25
f
0.50
0.0
0.25
f
0.50
Figure 23: Examples of AR(1) spectra – when φ1,1 is positive we have a low frequency
Figure 24: Examples of AR(2) spectra with real characteristic reciprocal roots,
spectrum and when φ1,1 is negative we have a high frequency spectrum
a = r1 and b = r2 , giving AR parameter values of: φ1,2 = r1 + r2 and φ2,2 = −r1 r2
24
25
q!"#$%&'(#)#<'%*+'$#,
q!"#$%$--#)#<'%.*#,
20
10
dB
dB
dB
0
-10
-20
-30
-40
0.0
0.2
0.4
0.0
0.2
f
0.4
f
20
10
q!"#<$%$--#)#<'%.*#,
dB
0
q!"#<$%&'(#)#<'%*+'$#,
-10
-20
-30
dB
dB
-40
20
10
dB
0
0.0
0.2
0.4
0.0
f
0.2
0.4
f
-10
-20
-30
-40
0.0
0.1
0.2
f
0.3
0.4
0.5
Figure 25: Examples of AR(2) spectra with complex characteristic reciprocal roots,
re±i2πf , with r = 0.99 for the plots in the left column and r = 0.7 for the plots in
Figure 26: Inconsistency of the periodogram. The plots show the periodogram (on
the right column, and f = 0.1 for the plots in the first row, and f = 0.4 for the
a decibel scale) of a unit variance white noise process of length (from top to bottom)
plots in the second row, the AR parameter values (as shown in the titles) can be
N = 128, 256 and 1024. The horizontal dashed line indicates the true sdf.
calculated from φ1,2 = 2r cos(2πf ) and φ2,2 = −r2
26
27
10
20
10
dB
dB
0
-10
-20
0
-30
-40
-10
20
10
dB
0
-10
-20
10
-30
dB
-40
0
20
10
dB
0
-10
-10
0.0
-20
0.1
0.2
0.3
0.4
0.5
f
-30
-40
-0.50
-0.25
0.0
0.25
0.50
Figure 28: Bias properties of the periodogram for an AR(2) process with low dyFigure 27: Fejér’s kernel for sample sizes N = 8, 32 and 128
namic range. The thick curves are the true sdf S(f ), while the thin curves are
E{Ŝ (p) (f )} for sample sizes (from top to bottom) N = 16 and 64.
28
29
20
0.3
60
0
40
0.2
20
h
dB
dB
-20
0
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
-20
-40
0.1
-40
-60
60
0.0
-80
0.3
20
40
dB
20
0
0
0.2
-20
h
-40
dB
-20
•• •• ••• ••• •• ••• ••• •• ••• ••• •• ••• ••• •• ••• ••• •• ••• ••• •• ••
•
-40
•
0.1
60
•
•
•
40
20
•
•
•
-80
dB
0.0
-60
•
•
0
0.3
-20
20
-40
0
0.2
60
40
•
•
• •• ••• ••• •• ••• ••• •• ••• ••• •• ••• ••• •• •
•
dB
•
•
•
•
•
0.0
•
•
•
•
0
-40
-60
•
•
••
-40
•
•
-20
•
•
•
0
•
•
•
0.1
20
dB
h
-20
•
32
•
••
-80
64
0.0
0.1
0.2
0.3
0.4
0.5
f
0.0
0.1
0.2
0.3
0.4
0.5
f
Figure 29: Bias properties of the periodogram for an AR(4) process with high
dynamic range. The thick curves are the true sdf S(f ), while the thin curves are
E{Ŝ (p) (f )} for sample sizes (from top to bottom) N = 16, 64, 256 and 1024.
30
Figure 30: Different data tapers (left column) and associated spectral windows H(f )
(right column), for N = 64.
The tapers are a rectangular taper (top), a 20% (middle) and 50% (bottom) split
cosine bell taper.
31
N=64, original data
40
40
20
20
N=64, tapered data
dB
60
dB
60
0
0
-20
-20
-40
-40
0
16
32
48
64
0
16
32
N=256, original data
40
40
20
20
64
dB
60
dB
60
48
N=256, tapered data
0
0
-20
-20
-40
-40
0
64
128
192
256
0
64
N=1024, original data
40
20
20
192
256
N=1024, tapered data
dB
60
40
dB
60
128
0
0
-20
-20
-40
-40
0.0
0.1
0.2
0.3
0.4
0.5
0
0.0
f
0.1
0.2
0.3
0.4
256
512
768
1024
0
256
512
768
1024
0.5
f
Figure 31: Bias properties of direct spectral estimators for an AR(4) process with
high dynamic range, using a 20% (left column) and 50% (right column) split cosine bell taper. The thick curves are the true sdf S(f ), while the thin curves are
E{Ŝ (p) (f )} for sample sizes (from top to bottom) N = 16, 64 and 256.
32
Figure 32: The left column shows simulations from the AR(4) model:
Xt = 2.7607Xt−1 − 3.8106Xt−2 + 2.6535Xt−3 − 0.9258Xt−4 + �t
For (from top to bottom) N = 64, 256 and 1024.
The right column shows {Xt ht } where {ht } is the appropriate length 50% split cosine
bell taper.
33
60
60
N=64, Yule-Walker AR(4)
N=64, Yule-Walker AR(8)
20
20
dB
40
dB
40
0
0
-20
-20
-40
-40
60
60
N=256, Yule-Walker AR(4)
N=256, Yule-Walker AR(8)
20
20
dB
40
dB
40
0
0
-20
-20
-40
-40
60
60
N=1024, Yule-Walker AR(4)
N=1024, Yule-Walker AR(8)
20
20
dB
40
dB
40
0
0
-20
-20
-40
-40
0.0
0.1
0.2
0.3
0.4
0.5
f
0.0
0.1
0.2
0.3
0.4
0.5
f
Figure 33: The thick line shows the spectrum of the AR(4) process associated with
Figure 34: The thick line shows the spectrum of the AR(8) process associated with
the Yule-Walker estimates of φ1,4 , . . . , φ4,4 , for the sequences shown in the left column
the Yule-Walker estimates of φ1,8 , . . . , φ8,8 , for the sequences shown in the left column
of Figure 32 (i.e. untapered). The thin line shows the true spectrum.
of Figure 32 (i.e. untapered). The thin line shows the true spectrum.
34
35
60
N=64, Tapered Yule-Walker AR(4)
40
dB
20
0
-20
-40
60
N=256, Tapered Yule-Walker AR(4)
40
dB
20
0
-20
-40
60
N=1024, Tapered Yule-Walker AR(4)
40
dB
20
0
-20
-40
0.0
0.1
0.2
0.3
0.4
0.5
f
Figure 35: The thick line shows the spectrum of the AR(4) process associated with
the Yule-Walker estimates of φ1,4 , . . . , φ4,4 , for the sequences shown in the right
column of Figure 32 (i.e. tapered). The thin line shows the true spectrum.
36