100
50
0
–50
–100
0 500 1000
Time
Flow
1500 2000
(b) Stationary, Periodic
600
500
400
300
200
100
1950 1952 1954 1956
Time
AirPassengers
1958 1960
(c) Non-sta tionary
FIGURE 7.6
Examples of stationary and non-stationary time-series.
0 20 40 60 80
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Samples
Amplitude
True
Estimated
FIGURE 10.2
Comparing the signal estimate us ing the inverse FT method and the true signal in Example
10.1.
0 0.2 0.4 0.6 0.8
−1
−0.5
0
0.5
1
Time
Amplitude
Fourier Decomposition of a Square Wave
Original
Fundamental
3
rd
harmonic
5
th
harmonic
(b) Fourier decomposition
FIGURE 10.5
Power spectral a nd Fourier decomposition of the square wave in Example 10.5.
ˆ
θ
θ
0
Objective
(Loss) Function
True value
Estimate
Condence region
Model/
Constraints
Estimator
Known
information set
Z
FIGURE 12.1
Schematic illustrating generic estimation.
−10 −5 0 5 10
0
0.005
0.01
0.015
0.02
0.025
0.03
θ
f(θ|y)
Prior and Posterior p.d.f.s
Prior
After N = 1
N = 10
N = 100
FIGURE 15.1
Prior and posterior p.d.f.s of mean in Example 15.2.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Normalized Length
w[k]
Hanning
Bartlett
Blackman
Flat top
Rectangular
(a) Time-domain profiles
−10 −8 −6 −4 −2 0 2 4 6 8 10
−120
−100
−80
−60
−40
−20
0
Frequency Oset (bins)
dB (normalized)
Hanning
Bartlett
Blackman
Flat top
Rectangular
(b) Normaliz ed magnitude (in dB), |W (ω )|/N
FIGURE 16.5
Window functions and the magnitudes of their Fourier transforms.
0 1 2 3
0.2
0.4
0.6
0.8
Single Realization: N = 250
ω
PSD
0 1 2 3
0.2
0.4
0.6
0.8
1
1.2
Single Realization: N = 2000
ω
PSD
0 1 2 3
0.1
0.15
0.2
ω
PSD
0 1 2 3
0.1
0.15
0.2
ω
PSD
0 5 10 15 20
0
200
400
600
800
PSD (ω = 0)
Count
0 5 10 15 20
0
200
400
600
800
PSD(ω = 0.4 π)
Count
Averaged
True
Averaged
True
FIGURE 16.8
(Top panel) Periodograms from a single realization with N = 250 and N = 2000 obser-
vations; (Middle panel) Averaged periodo grams from 1000 rea lizations; (Bottom panel)
Distribution of ˆγ(0)/γ(0) and 2ˆγ(0.4π)/γ(0.4π) obtained from 1000 realizations.
Input
Procss
Model
Noise
Model
Stochastic
eect
White-noise
(ctitious)
Observed
output
u[k] y[k]
e[k]
v[k]
+
+
FIGURE 17.1
Input-output representation of adeterministic-plus-stochastic LTI system.

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