Data processing: measuring – calibrating – or testing – Measurement system – Measured signal processing
Reexamination Certificate
2000-01-18
2003-02-18
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Measured signal processing
C702S189000, C702S190000, C702S198000, C702S199000
Reexamination Certificate
active
06522996
ABSTRACT:
BACKGROUND OF THE INVENTION
The present invention relates to the evaluation of signals, more particularly to the classification of signal attributes of a non-stationary time series signal wherein such classification involves derivation of features thereof.
Non-stationary signals are used in almost every field of science and technology, such as the medical field, geophysical field, automobile manufacturing and submarine silencing (to name but a few). Some examples of non-stationary signals include, but are not limited to, the following: heartbeat electrocardiograms, seismic recordings, automobile vibration, and submarine acoustic transients.
Non-stationary signals are characterized by attributes that vary in some manner with time. It is for this reason that the study and analysis of such signals must involve a process that incorporates time. The most commonly uses process is termed time-frequency analysis. In general, time-frequency analysis is based on the construction of a two-dimensional distribution in time and frequency. In 1966, Leon Cohen developed a unifying formulation for time-frequency distributions with the correct marginals; see L. Cohen, “Generalized Phase-Space Distribution Functions,”
J. Math. Phys.,
vol. 7, no. 5, pp 781-786, 1966, incorporated herein by reference. Cohen's generalized formulation stipulates that all time-frequency representations can be obtained from
Q
(
t
,
f
)
=
1
2
π
∫
∫
∫
{
s
*
(
u
-
1
2
τ
)
s
(
u
+
1
2
τ
)
φ
(
θ
,
τ
)
ⅇ
-
j
θ
t
-
j
τ
2
π
f
+
jθ
u
}
ⅆ
u
ⅆ
τ
ⅆ
θ
(
1
)
where &phgr;(&thgr;, &tgr;) is a two dimensional function, called the kernel, which determines the distribution and its properties. Ideally, such a distribution should be manifestly positive, for a proper interpretation as a joint time-frequency energy density function, and further, should yield the correct marginal densities of time, |s(t)|
2
, and frequency, |S(f)|
2
.
Mathematically, for a time-frequency distribution to be interpreted as a joint time-frequency energy density, it must satisfy the two fundamental properties of nonnegativity and the correct frequency and time marginals:
Q
(
t
,
f
)
≥
0
(
2
)
∫
-
∞
∞
Q
(
t
,
f
)
ⅆ
t
=
&LeftBracketingBar;
S
(
f
)
&RightBracketingBar;
2
(
3
a
)
∫
-
∞
∞
Q
(
t
,
f
)
ⅆ
f
=
&LeftBracketingBar;
s
(
t
)
&RightBracketingBar;
2
(
3
b
)
where
S
(
f
)
=
∫
-
∞
∞
s
(
t
)
ⅇ
-
j2π
f
t
ⅆ
t
is the Fourier transform of the signal. Integrating the expression of (1) with respect to frequency yields the instantaneous power, |s(t)|
2
, provided &phgr;(&thgr;, 0)=1. Likewise, integrating (1) with respect to time yields the energy density spectrum, |S(f)|
2
, provided &phgr;(0, &tgr;)=1. When a time-frequency distribution meets the conditions of (2) and (3) it is termed a “positive time-frequency distribution”.
As it turned out, even though the generalized formulation for all time-frequency distributions was specified as early as 1966, the condition for positivity was not known until several years later when Cohen and Posch, and also Cohen and Zaparovanny, gave the condition and a simple, general formulation for positive time-frequency distributions. See L. Cohen and T. Posch, “Positive Time-Frequency Distribution Functions,”
IEEE Trans. Acoust. Speech Signal Processing
vol. ASSP-33, no. 1, pp 31-38, 1985, incorporated herein by reference; and, L. Cohen and Y. Zaparovanny, “Positive Quantum Joint Distributions,”
J. Math. Phys.,
vol. 21, no. 4, pp 794-796, 1980, incorporated herein by reference. Thus, in 1985, it was known that positive time-frequency distributions could be constructed. However, it remained a secret of how to actually construct them for almost a decade.
It was not until 1994 that a means of constructing a positive time-frequency distribution using minimum cross-entropy was developed by Loughlin et al; see P. Loughlin, J. Pitton and L. Atlas, “Construction of Positive Time Frequency Distributions,” IEEE
Trans. Sig. Proc.,
vol. 42, no. 10, pp 2697-2705, 1994, incorporated herein by reference. The method of Loughlin et al. was certainly a theoretical breakthrough in science; however, the usefulness of the original formulation was limited because of computational constraints. These constraints were alleviated by Groutage; see Dale Groutage, “A Fast Algorithm for Computing Minimum Cross-Entropy Positive Time-Frequency Distributions,”
IEEE Trans. Sig. Proc.,
vol. 45, no. 8, August 1997, pp 1954-1970, incorporated herein by reference.
The positive time-frequency distribution is currently the only way known to quantify the energy density of a non-stationary signal—in particular, to obtain a plausible representation of the energy density with the correct marginals.
Once a two-dimensional time-frequency distribution has been constructed, it is natural from a mathematical viewpoint to construct the statistical moments. These moments can be useful for formulating features that associate with the non-stationary signal s(t). For a signal s(t), the temporal and spectral moments are given by
⟨
t
n
⟩
=
∫
-
∞
∞
t
n
&LeftBracketingBar;
s
(
t
)
&RightBracketingBar;
2
ⅆ
t
and
⟨
f
m
⟩
=
∫
-
∞
∞
f
m
&LeftBracketingBar;
S
(
f
)
&RightBracketingBar;
2
ⅆ
f
(
4
)
where (n,m=1,2,3, . . . ) respectively. The joint time-frequency moments are given by
⟨
t
n
f
m
⟩
=
∫
-
∞
∞
∫
-
∞
∞
t
n
f
m
Q
(
t
,
f
)
ⅆ
t
ⅆ
f
for
n
,
m
=
1
,
2
,
3
,
…
(
5
)
Currently there are numerous methodologies for deriving features from time series records and imagery, but none of these are entirely satisfactory. These methods use a variety of mathematical approaches to derive the features including the joint moments of equation (5). A desirable method would be an approach that derived features, which associate with temporal and spectral moments analogous to those of equation (4). Most of the methods are “ad hoc” and use a so-called “receipt” that searches for salient aspects of the time series record or image. These techniques are rather limited in scope, and search for a priori features, which are thought to be contained within the time series record.
A known method which is not based on searching for a priori features is disclosed by L. M. D. Owsley et al; see L. M. D. Owsley, L. E. Atlas, and G. D. Bernard, “Self-Organizing Feature Maps and Hidden Markov Models for Machine-Tool Monitoring,”
IEEE Transactions on Signal Processing,
vol. 45, no. 11, November 1997, pp 2787-2798, hereby incorporated herein by reference. Owsley et al. disclose the most current methodology not based upon a priori feature search, a methodology involving the so-called “Self Organizing Feature Map.”
Essentially, according to Owsley et al., a “codebook” is generated that contains vectors derived from a series of non-stationary signals from a given class of signals. The codebook contains an address for each vector. For the two-dimensional case, the address of each vector is its location in terms of the row-column that it resides. For example, a 5 by 5 codebook has 5 rows and 5 columns. Location (2,3) is for the vector residing at location occupied by the second row and third col
Hoff Marc S.
Kaiser Howard
Suarez Felix
The United States of America as represented by the Secretary of
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