Tugnait, J.K. “Validation, Testing, and Noise Modeling”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
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
1999byCRCPressLLC
16
Validation, Testing, and Noise
Modeling
Jitendra K. Tugnait
Auburn University
16.1 Introduction
16.2 Gaussianity, Linearity, and Stationarity Tests
Gaussianity Tests
•
Linearity Tests
•
Stationarity Tests
16.3 Order Selection, Model Validation, and Confidence Intervals
Order Selection
•
Model Validation
•
Confidence Intervals
16.4 Noise Modeling
Generalized Gaussian Noise
•
MiddletonClassANoise
•
Stable


i=0
h(i)q
−i
(16.2)
where q
−1
is the backward shift operator (i.e., q
−1
x(t) = x(t − 1), etc.). If q is replaced with the
complex variable z, then H(z) is the Z-transform of {h(i)}, i.e., it is the system transfer function.
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1999 by CRC Press LLC
Using (16.2), (16.1) may be rewritten as
x(t) = H(q)(t).
(16.3)
Fittinglinear models tothemeasurementrecordrequiresestimationof H(q), orequivalentlyof{h(i)}
(without observing {(t)} ). Typically H(q)is parameterized by a finite number of parameters, say
by the parameter vector θ
(M)
of dimension M. For instance, an AR model representation of order
M means that
H
AR
(q; θ
(M)
) =
1
1 +

xx
(ω) = σ
2

|H(e
jω
)|
2
,σ
2

= E{
2
(t)},
(16.5)
one cannot determine the phase of H(e
jω
) independent of |H(e
jω
)|. Determination
of the true phase characteristic is crucial in several applications such as blind equaliza-
tion of digital communications channels. Use of higher-order statistics allows one to
uniquely identify nonminimum-phase parametric models. Higher-order cumulants of
Gaussian processes vanish, hence, if the data are stationary Gaussian, a minimum-phase
(ormaximum-phase) model isthe“best” that onecan estimate. Therefore, another aspect
considered in this section is testing for non-Gaussianity of the given record.
• If the data are Gaussian, one may fit models based solely upon the second-order statistics
of the data —else useof higher-orderstatistics in addition toor in lieu of the second-order
statistics is indicated, particularly if the phase of the linear system is crucial. In either case,
one typically fits a model H(q; θ

(ω
1
,ω
2
) is defined as [12]
B
xxx
(ω
1
,ω
2
) =
∞

i=−∞
∞

k=−∞
C
xxx
(i, k)e
−j(ω
1
i+ω
2
k)
.
(16.7)
Similarly, its fourth-order cumulant function C
xxxx

−j(ω
1
i+ω
2
k+ω
3
l)
.
(16.9)
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1999 by CRC Press LLC
If {x(t)} obeys (16.1), then [12]
B
xxx
(ω
1
,ω
2
) = γ
3
H(e
jω
1
)H (e
jω
2
)H
∗
(e

j(ω
1
+ω
2
+ω
3
)
)
(16.11)
where
γ
3
= C

(0, 0, 0) and γ
4
= C

(0, 0, 0, 0).
(16.12)
For Gaussian processes, B
xxx
(ω
1
,ω
2
) ≡ 0 and T
xxxx
(ω
1

1
+ ω
2
)
=
γ
3
σ
6

= constant ∀ ω
1
,ω
2
,
(16.13)
and using (16.5) and (16.11),
|T
xxxx
(ω
1
,ω
2
,ω
3
)|
2
S
xx
(ω

.
(16.14)
The above two relations form a basis for testing linearity of a given measurement record. How
the tests are implemented depends upon the statistics of the estimators of the higher-order cumulant
spectra as well as that of the power spectra of the given record.
16.2.1 Gaussianity Tests
Suppose that the given zero-mean measurement record is of length N denoted by {x(t), t =
1, 2,···,N}. Suppose that the given sample sequence of length N is divided into K nonover-
lapping segments each of size N
B
samples so that N = KN
B
.LetX
(i)
(ω) denote the discrete Fourier
transform (DFT) of the ith block {x(t + (i − 1)N
B
), 1 ≤ t ≤ N
B
} (i = 1, 2,···,K)given by
X
(i)
(ω
m
) =
N
B
−1

l=0

n
=
2π
N
B
n) as

B
xxx
(m, n), given by averaging over K blocks

B
xxx
(m, n) =
1
K
K

i=1

1
N
B
X
(i)
(ω
m
)X
(i)
(ω

B

.
(16.18)
Values of

B
xxx
(m, n) outside D can be inferred from that in D.
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1999 by CRC Press LLC