Data processing: measuring – calibrating – or testing – Measurement system – Measured signal processing
Reexamination Certificate
1999-08-09
2002-08-13
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Measured signal processing
C324S076120
Reexamination Certificate
active
06434515
ABSTRACT:
COMPUTER PROGRAM APPENDIX
The present application includes a compact disc containing Appendix A, which is one example of a source code routine written in the C programming language for performing the Gabor spectrogam according to the present invention. The compact disc contains a single text file calle “Gabor Spectrogram”. The compact disc is in the IBM-PC format and may be viewed with the Windows operating system. The source code listing in the file “Gabor Spectrogram” is called “Fast Gabor Spectrogram function, Gaborspectrogram.c”, has a size of 74 kbytes, and a date of creation of Mar. 5, 1999. The source code listing on the compact disc filed in the application herewith is hereby incorporated by reference as though fully and completely set forth herein.
RESERVATION OF COPYRIGHT
A portion of the disclosure of this patent document, specifically Appendix A, contains material to which the claim of copyright protection is made. The copyright owner has no objection to the facsimile reproduction by any person of the patent document or the patent disclosure, as it appears in the U.S. Patent and Trademark Office file or records, but reserves all other rights whatsoever.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to digital signal processing and joint time frequency analysis; and more particularly, to systems and methods for determining characteristics of signals having frequency components that vary in time.
2. Description of the Related Art
Due to physical limitations, systems (e.g., an engine, or the human body) or natural phenomena can generally only be studied through the signals (e.g., sound, temperature, heartbeat, and blood pressure) that are generated from or by the system under consideration. Therefore, signal analysis plays a fundamental role in our everyday life. By properly applying analysis techniques, a great deal of information can be obtained about systems without physically invasive procedures.
Most current signal analysis techniques characterize the signal in either the time domain or frequency domain. The time waveform, such as the sound of an engine or an electrocardiogram (ECG), illustrates how the signal's magnitude varies with time, and the frequency function, such as the power spectrum, indicates how often such changes take place. In most applications, the time and frequency behaviors of signals are closely related to each other. However, conventional techniques are designed to analyze them separately.
Prior art systems for analyzing signals with frequency components that vary in time have used the so-called short time Fourier transform STFT, also known as the windowed Fourier transform. This algorithm is based on a computationally intensive Fourier transform of a large number of short time windows of the input signal. Transforms of each of the short time windows are combined to generate a time varying spectrum of the input signal.
Commonly, the STFT-based spectrogram, described above, has been performed to map time domain (or frequency domain) signals into the joint time-frequency domain. The STFT is simple, but it suffers from the co-called windowing effect, i.e., the STFT is subject to the length of the analysis window function. A short window yields good time resolution, but poor frequency resolution, and vice versa. Particularly, the STFT-based spectrogram is not suitable for the instantaneous frequency estimation which is one of the most important signal's aspects of interest. The instantaneous frequency estimation from the STFT-based spectrogram is biased, which is subject to the window selection. Finally, the algorithm of the STFT-based spectrogram is not convenient to zoom in to the time-frequency region of interest.
As an alternative, U.S. Pat. No. 5,353,233 to Qian, et al. titled “Method and Apparatus for Time Varying Spectrum Analysis” discloses a technique which may be referred to as a Gabor spectrogram. The Gabor spectrogram is a Gabor expansion based spectrogram. See, Qian, et al., “Discrete Gabor Transform”, IEEE Trans. Signal Processing, Vol.41, No.7, July 1993, pp.2429-2439. Compared to the STFT-based spectrogram, the Gabor spectrogram has better time-frequency resolution and good zoom in capability. The Gabor spectrogram is extremely powerful when the high-resolution time-dependent spectrum is a must.
FIG. 1
illustrates time, frequency, and joint time-frequency (Gabor Spectrogram) plots of the sound associated with an aneurysm in the human body. It has been determined that such sound is due to vibration stimulated by the blood flow inside the aneurysm and nearby blood vessels. The mechanism that emits such sounds is complicated and involves the wall, chamber, surrounding blood vessels, and moving blood, under varying pressure. The sound signal is non-stationary, i.e., its frequency contents change with time. Moreover, such a sound record is usually combined with the biological noises generated by the heart, respiratory system, and even eye movement. The bottom plot shows a time waveform that was directly recorded from an intracranial aneurysm during surgery. As expected, the time waveform at the bottom of the figure is rather noisy and thereby virtually provides no useful information for the diagnostic. The spectrum of the signal is illustrated to the left of the figure. The frequency spectrum shows the range of resonant frequencies (500 to 620 Hz), but does not give valuable diagnostic information. When the time domain sound wave is converted into the joint time-frequency domain (the middle plot), it can be clearly seen how the power spectrum evolves over time. Researchers have found that it is much easier to identify the existence of the aneurysm from the joint time-frequency domain than from the time or frequency domain alone.
However, computing the Gabor spectrogram usually takes a long time. Therefore, improved methods are desired for more efficiently or more quickly computing the Gabor spectrogram.
Theoretical Background of the Gabor Spectrogram
The Gabor spectrogram is developed from the decomposition of the Wigner-Ville distribution (WVD). For a given signal s(t), the corresponding Gabor expansion is
s
(
t
)
=
∑
m
=
-
∞
∞
∑
n
=
-
∞
∞
C
m
,
n
h
m
,
n
(
t
)
(
1
)
where C
m,n
are known as the Gabor coefficients. The Gabor expansion maps the time domain signal s(t) into a two-dimensional lattice C
m,n
. The elementary function h
m,n
(t) is a time- and frequency-shifted Gaussian function, i.e.,
h
m,n
(
t
)
=g
(
t−mT
)
e
−jn&OHgr;t
(2)
where the normalized Gaussian function g(t) has a form
g
(
t
)
=
(
πσ
2
)
-
0.25
exp
{
-
1
2
σ
2
t
2
}
(
3
)
Taking the Wigner-Ville distribution of both sides of Eq.(1) yields
WVD
s
(
t
,
ω
)
=
∑
m
=
-
∞
∞
∑
m
′
=
-
∞
∞
∑
n
=
-
∞
∞
∑
n
′
=
-
∞
∞
C
m
,
n
C
m
′
,
n
′
*
WVD
h
,
h
′
(
t
,
ω
)
(
4
)
where WVD
h,h′
(t,&ohgr;), the WVD of the time- and frequency-shifted Gaussian function, which can be referred to as an “energy atom”, is concentrated, oscillated, and symmetrical. The energy atom has a closed form, and thus can be pre-calculated and stored in a table. If the Wigner-Ville distribution WVD
s
(t,&ohgr;) describes the signal's energy distribution in the joint time-frequency domain, then Eq.(4) shows that the signal energy is a superposition of an infinite number of energy atoms. Note that the energy (average) contained in each individual WVD
h,h′
(t,&ohgr;) is inversely proportional to the distance between corresponding elementary functions h
m,n
(t) and h
m′,n′
(t). Based on their contribution to the entire signal energy, Eq.(4) can be re-grouped to obtain the Gabor spectrogram as
GS
D
(
t
,
ω
)
=
∑
&LeftBracketingBar;
m
-
m
′
&Ri
Conley Rose & Tayon PC
Hoff Marc S.
Hood Jeffrey C.
National Instruments Corporation
Raymond Edward
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