U.S. patent application number 11/342057 was filed with the patent office on 2007-08-02 for method and apparatus for removing quantization effects in a quantized signal.
Invention is credited to Gregory A. Cumberford, Ramakrishna C. Dhanekula, Kenny C. Gross, Eugenio J. Schuster.
Application Number | 20070179727 11/342057 |
Document ID | / |
Family ID | 38266927 |
Filed Date | 2007-08-02 |
United States Patent
Application |
20070179727 |
Kind Code |
A1 |
Gross; Kenny C. ; et
al. |
August 2, 2007 |
METHOD AND APPARATUS FOR REMOVING QUANTIZATION EFFECTS IN A
QUANTIZED SIGNAL
Abstract
One embodiment of the present invention provides a system that
reconstructs a high-resolution signal from a set of low-resolution
quantized samples. During operation, the system receives a time
series containing low-resolution quantized signal values which are
sampled from the high-resolution signal. Next, the system performs
a spectral analysis on the time series to obtain a frequency series
for the low-resolution quantized signal values. The system next
selects a subset of frequency terms from the frequency series which
have the largest amplitudes. The system then reconstructs the
high-resolution signal by performing an inverse spectral analysis
on the subset of the frequency terms.
Inventors: |
Gross; Kenny C.; (San Diego,
CA) ; Dhanekula; Ramakrishna C.; (San Diego, CA)
; Schuster; Eugenio J.; (Breinigsville, PA) ;
Cumberford; Gregory A.; (Calgary, CA) |
Correspondence
Address: |
SUN MICROSYSTEMS INC.;C/O PARK, VAUGHAN & FLEMING LLP
2820 FIFTH STREET
DAVIS
CA
95618-7759
US
|
Family ID: |
38266927 |
Appl. No.: |
11/342057 |
Filed: |
January 27, 2006 |
Current U.S.
Class: |
702/76 |
Current CPC
Class: |
H03M 1/20 20130101 |
Class at
Publication: |
702/076 |
International
Class: |
G01R 23/16 20060101
G01R023/16; G06F 19/00 20060101 G06F019/00 |
Claims
1. A method for removing quantization effects from a set of
low-resolution quantized samples, the method comprising: receiving
a time series containing a low-resolution quantized signal which is
sampled from a high-resolution signal; performing a spectral
analysis on the time series to obtain a frequency series for the
low-resolution quantized signal; selecting a subset of frequency
terms from the frequency series which have the largest amplitudes;
and reconstructing the high-resolution signal from the
low-resolution quantized signal by performing an inverse spectral
analysis on the subset of the frequency terms, wherein
reconstructing the high-resolution signal facilitates removing the
quantization effects from the low-resolution quantized signal.
2. The method of claim 1, wherein selecting the subset of the
frequency terms involves first sorting the frequency terms in the
frequency series based on amplitudes of the frequency terms in the
frequency series.
3. The method of claim 1, wherein the spectral analysis is a
Discrete Fourier Transform (DFT).
4. The method of claim 1, wherein the inverse spectral analysis is
an inverse Discrete Fourier Transform (IDFT).
5. The method of claim 3, wherein the DFT is computed using a Fast
Fourier Transform (FFT).
6. The method of claim 1, wherein the subset of the frequency terms
is a predetermined number of frequency terms which have the largest
amplitudes from the frequency series.
7. The method of claim 6, wherein the predetermined number is
determined by: identifying a value n that gives rise to a minimum
difference between the high-resolution signal and a reconstructed
signal by iteratively, selecting the first n frequency terms which
have the largest amplitudes from the frequency series;
reconstructing a signal by performing an inverse spectral analysis
on the selected n frequency terms; and computing a difference
between the high-resolution signal and the reconstructed
signal.
8. The method of claim 7, wherein computing the difference between
the high-resolution signal and the reconstructed signal involves
computing a residual signal by subtracting the reconstructed signal
from the high-resolution signal.
9. The method of claim 2, wherein the amplitudes of the frequency
series are determined by a power spectral density (PSD) of the
frequency series.
10. A computer-readable storage medium storing instructions that
when executed by a computer cause the computer to perform a method
for removing quantization effects from a set of low-resolution
quantized samples, the method comprising: receiving a time series
containing a low-resolution quantized signal which is sampled from
a high-resolution signal; performing a spectral analysis on the
time series to obtain a frequency series for the low-resolution
quantized signal; selecting a subset of frequency terms from the
frequency series which have the largest amplitudes; and
reconstructing the high-resolution signal from the low-resolution
quantized signal by performing an inverse spectral analysis on the
subset of the frequency terms, wherein reconstructing the
high-resolution signal facilitates removing the quantization
effects from the low-resolution quantized signal.
11. The computer-readable storage medium of claim 10, wherein
selecting the subset of the frequency terms involves first sorting
the frequency terms in the frequency series based on amplitudes of
the frequency terms in the frequency series.
12. The computer-readable storage medium of claim 10, wherein the
spectral analysis is a Discrete Fourier Transform (DFT).
13. The computer-readable storage medium of claim 10, wherein the
inverse spectral analysis is an inverse Discrete Fourier Transform
(IDFT).
14. The computer-readable storage medium of claim 12, wherein the
DFT is computed using a Fast Fourier Transform (FFT).
15. The computer-readable storage medium of claim 10, wherein the
subset of the frequency terms is a predetermined number of
frequency terms which have the largest amplitudes from the
frequency series.
16. The computer-readable storage medium of claim 15, wherein the
predetermined number is determined by: identifying a value n that
gives rise to a minimum difference between the high-resolution
signal and a reconstructed signal by iteratively, selecting the
first n frequency terms which have the largest amplitudes from the
frequency series; reconstructing a signal by performing an inverse
spectral analysis on the selected n frequency terms; and computing
a difference between the high-resolution signal and the
reconstructed signal.
17. The computer-readable storage medium of claim 16, wherein
computing the difference between the high-resolution signal and the
reconstructed signal involves computing a residual signal by
subtracting the reconstructed signal from the high-resolution
signal.
18. The computer-readable storage medium of claim 11, wherein the
amplitudes of the frequency series are determined by a power
spectral density (PSD) of the frequency series.
19. An apparatus for removing quantization effects from a set of
low-resolution quantized samples, comprising: a receiving mechanism
configured to receive a time series containing a low-resolution
quantized signal which is sampled from a high-resolution signal; a
performing mechanism configured to perform a spectral analysis on
the time series to obtain a frequency series for the low-resolution
quantized signal; a selection mechanism configured to select a
subset of frequency terms from the frequency series which have the
largest amplitudes; and a reconstruction mechanism configured to
reconstruct the high-resolution signal from the low-resolution
quantized signal by performing an inverse spectral analysis on the
subset of the frequency terms, wherein reconstructing the
high-resolution signal facilitates removing the quantization
effects from the low-resolution quantized signal.
20. The apparatus of claim 19, wherein selecting the subset of
frequency terms involves a sorting mechanism configured to sort the
frequency terms in the frequency series based on amplitudes of the
frequency terms in the frequency series.
21. The apparatus of claim 19, wherein the subset of the frequency
terms is a predetermined number of frequency terms which have the
largest amplitudes from the frequency series.
22. The apparatus of claim 21, wherein the apparatus comprises a
determination mechanism which is configured to determine the
predetermined number by: identifying a value n that gives rise to a
minimum difference between the high-resolution signal and a
reconstructed signal by iteratively, selecting the first n
frequency terms which have the largest amplitudes from the
frequency series; reconstructing a signal by performing an inverse
spectral analysis on the selected n frequency terms; and computing
a difference between the high-resolution signal and the
reconstructed signal.
23. The apparatus of claim 22, wherein computing the difference
between the high-resolution signal and the reconstructed signal
involves a computation mechanism configured to compute a residual
signal by subtracting the reconstructed signal from the
high-resolution signal.
Description
BACKGROUND
[0001] 1. Field of the Invention
[0002] The present invention relates to techniques for proactively
detecting impending problems in computer systems. More
specifically, the present invention relates to a method and an
apparatus for removing quantization effects in a quantized signal
which can be subsequently used to detect impending problems in a
computer system.
[0003] 2. Related Art
[0004] Modern server computer systems are typically equipped with a
significant number of sensors which monitor signals during the
operation of the computer systems. Results from this monitoring
process can be used to generate time series data for these signals
which can subsequently be analyzed to determine how a computer
system is operating. One particularly desirable application of this
time series data is for purposes of "proactive fault monitoring" to
identify leading indicators of component or system failures before
the failures actually occur.
[0005] Unfortunately, many of these computer systems use
low-resolution eight-bit analog-to-digital (A/D) converters in all
of their physical sensors to sample the signals. This causes
readings of physical variables such as voltage, current, and
temperature to be highly quantized. Hence, the sampled signal
values from these sensors can only assume discrete values, and no
readings can be reported between these discrete values. For
example, voltages for system board components may be quantized to
the nearest 10 mV; e.g. 1.60 V, 1.61 V, 1.62 V, etc. Hence, if the
true voltage value is 1.6035 V, it can only be reported as one of
the quantized values, 1.60 or 1.61.
[0006] Note that the above-described quantization effects present a
serious problem for proactive fault monitoring. Normally, one can
apply statistical pattern recognition techniques to continuous
signal values to detect if the signals start to drift away from
steady-state values at a very early stage of system degradation.
However, with significant quantization, conventional statistical
pattern recognition techniques cannot be used effectively to detect
the onset of subtle anomalies that might precede component or
system failures.
[0007] To overcome the drawbacks of the low-resolution quantized
signals, researchers have used a technique called "burst sampling."
Essentially, this technique restores high-resolution signals from
low-resolution A/D converter outputs by removing the quantization
effects. Specifically, a large "burst" of samples (typically
hundreds of sample) are retrieved from low-level hardware registers
of the server computer system being monitored. These samples are
then collected through telemetry channels at the highest data rate
that the hardware channels can support (typically at kHz rates).
Next, the samples in the "burst" are averaged to obtain values that
approximate signals sampled with high-resolution data-acquisition
capability.
[0008] Unfortunately, this burst sampling technique can be used
only for a small subset of signals of interest in a large system.
This is because the burst sampling creates a large burst demand for
the bandwidth that is available for delivering telemetry samples
via the system bus. In some large systems over 1000 telemetry
signals are monitored concurrently. However, the burst sampling
technique can consume the entire system bus bandwidth while
delivering only a few tens of these signals.
[0009] What is needed is a method and an apparatus that removes the
quantization effects from low-resolution quantized signals without
the above-described problems.
SUMMARY
[0010] One embodiment of the present invention provides a system
that reconstructs a high-resolution signal from a set of
low-resolution quantized samples. During operation, the system
receives a time series containing low-resolution quantized signal
values which are sampled from the high-resolution signal. Next, the
system performs a spectral analysis on the time series to obtain a
frequency series for the low-resolution quantized signal values.
The system next selects a subset of frequency terms from the
frequency series which have the largest amplitudes. The system then
reconstructs the high-resolution signal by performing an inverse
spectral analysis on the subset of the frequency terms.
[0011] In a variation on this embodiment, the system selects the
subset of the frequency terms by first sorting the frequency terms
in the frequency series based on amplitudes of the frequency terms
in the frequency series.
[0012] In a variation on this embodiment, the spectral analysis is
a Discrete Fourier Transform (DFT).
[0013] In a variation on this embodiment, the inverse spectral
analysis is an inverse Discrete Fourier Transform (IDFT).
[0014] In a further variation on this embodiment, the DFT is
computed using a Fast Fourier Transform (FFT).
[0015] In a variation on this embodiment, the subset of the
frequency terms is a predetermined number of frequency terms which
have the largest amplitudes from the frequency series.
[0016] In a further variation on this embodiment, the system
precomputes the predetermined number. During this precomputation
process, the system receives a high-resolution signal. Next, the
system iteratively: selects the first n frequency terms which have
the largest amplitudes from the frequency series; reconstructs a
signal by performing an inverse spectral analysis on the selected n
frequency terms; and computes a difference between the
high-resolution signal and the reconstructed signal. The system
then determines the predetermined number by identifying a value of
n that gives rise to a minimum difference between the
high-resolution signal and the reconstructed signal.
[0017] In a further variation on this embodiment, the system
computes the difference between the high-resolution signal and the
reconstructed signal by computing a residual signal which is
obtained by subtracting the reconstructed signal from the
high-resolution signal.
[0018] In a further variation on this embodiment, the amplitudes of
the frequency series are determined by a power spectral density
(PSD) of the frequency series.
BRIEF DESCRIPTION OF THE FIGURES
[0019] FIG. 1 illustrates a computer system with sensors that
detect signals in different parts of the computer system in
accordance with an embodiment of the present invention.
[0020] FIG. 2 presents a flowchart illustrating a process of
reconstructing a high-resolution signal from a set of
low-resolution quantized samples in a computer system in accordance
with an embodiment of the present invention.
[0021] FIG. 3 presents a flowchart illustrating a process of
precomputing the number for the selected subset of the frequency
terms in accordance with an embodiment of the present
invention.
[0022] FIG. 4 illustrates the process of reconstructing the
high-resolution voltage signal by using only two highest harmonics
of the frequency terms in accordance with an embodiment of the
present invention.
[0023] FIG. 5 illustrates the process of reconstructing the
high-resolution signal by using different number of the highest
harmonics of the frequency series in accordance with an embodiment
of the present invention.
[0024] FIG. 6 illustrates a process of searching for a value M that
achieves a minimum residual sum-of-squares in an associated
synthesized signal in accordance with an embodiment of the present
invention.
DETAILED DESCRIPTION
[0025] The following description is presented to enable any person
skilled in the art to make and use the invention, and is provided
in the context of a particular application and its requirements.
Various modifications to the disclosed embodiments will be readily
apparent to those skilled in the art, and the general principles
defined herein may be applied to other embodiments and applications
without departing from the spirit and scope of the present
invention. Thus, the present invention is not limited to the
embodiments shown, but is to be accorded the widest scope
consistent with the principles and features disclosed herein.
[0026] The data structures and code described in this detailed
description are typically stored on a computer-readable storage
medium, which may be any device or medium that can store code
and/or data for use by a computer system. This includes, but is not
limited to, magnetic and optical storage devices such as disk
drives, magnetic tape, CDs (compact discs) and DVDs (digital
versatile discs or digital video discs).
Quantization Effect
[0027] FIG. 1 illustrates a computer system 100 with sensors that
monitor signals in different parts of the computer system in
accordance with an embodiment of the present invention.
[0028] Computer system 100 comprises multiple processors 102, 104,
and 106. Processors 102, 104, and 106 communicate with memory 108
through data bus 110. Memory 108 can include any type of memory
that can store code and data for execution by the processors 102,
104, and 106. This includes, but is not limited to, static random
access memory (SRAM), dynamic RAM (DRAM), magnetic RAM (MRAM),
non-volatile RAM (NVRAM), flash memory, and read only memory
(ROM).
[0029] Processors 102, 104, and 106 communicate with storage device
112 through data bus 110. Storage device 112 can include any type
of non-volatile storage device that can be coupled to a computer
system. This includes, but is not limited to, magnetic, optical,
and magneto-optical storage devices, as well as storage devices
based on flash memory and/or battery-backed up memory.
[0030] Processors 102, 104, and 106 can include any type of
processor, including, but not limited to, a microprocessor, a
mainframe computer, a digital signal processor, a personal
organizer, a device controller and a computational engine within an
appliance.
[0031] In order to monitor a signal associated with the operation
of computing system 100, several sensors 114, 116, 118, 120, and
122 can be selectively positioned throughout the computing system
100. These sensors (114, 116, 118, 120, and 122) monitor signals,
such as voltage, current, and temperature, within specific
components in the computer system. For example, sensor 122 can be
used to monitor a particular voltage provided to a bank of storage
cells (not shown) in memory 108.
[0032] In one embodiment, physical sensors 114, 116, and 118 which
reside within processors 102, 104, and 106, respectively, are used
to detect and identify a drift in core voltage "vCore" of the
processors. Each sensor 114, 116 and 118 includes an
analog-to-digital (A/D) converter 124, which is shown inside
close-up view 126. A/D converter 124 converts a continuous analog
signal into a series of discrete, digital values. For example, A/D
converter 124 converts the analog vCore signal of processor 102
into a digital format which can be monitored and manipulated by
processor 102.
[0033] Furthermore, A/D converter 124 shown in FIG. 1 is an
eight-bit A/D converter, which means that the digital output
generated by A/D converter 124 is represented using an eight-bit
value. Specifically, it samples the analog signal at pin V.sub.in
at a rate determined by clock signal CLK, and produces an eight-bit
digital output on pins B1 to B8. Note that A/D converter 124 can
generate 2.sup.8, or 256 discrete values. In comparison, an A/D
converter with a 4-bit data output can produce 2.sup.4, or 16
discrete values. Similarly, a 10-bit A/D converter can produce
2.sup.10, or 1024 discrete values.
[0034] The process of representing a continuous, analog signal with
discrete values is known as "quantization." For example, the
eight-bit A/D converter output representing a vCore signal can be
rounded to the nearest 10 mV, with quantized values of 1.60 V, 1.61
V, 1.62 V, 1.63 V, etc. Hence, an analog voltage of 1.614V can be
represented by the eight-bit A/D converter as either 1.61 V or
1.62V. The distance between the adjacent quantized values is
referred to as a "quantization resolution", which is 10 mV in this
example. The larger this difference is, the lower the quantization
resolution. Note that a low quantization resolution can make it
difficult to detect subtle changes which are fractions of the
quantization resolution before a significant drift has taken
place.
Process for Reconstructing a High-Resolution Signal
[0035] FIG. 2 presents a flowchart illustrating a process of
reconstructing a high-resolution signal from a set of
low-resolution quantized samples in a computer system in accordance
with an embodiment of the present invention.
[0036] The process typically begins by receiving a time series
containing the low-resolution quantized signal values which are
sampled from the high-resolution continuous-time signal (step 200).
In one embodiment of the present invent, the low-resolution
quantized signal values are sampled from the high-resolution signal
at a constant time interval .DELTA.t over a specific time duration
T, wherein the resolution of the quantized signal values is
determined by the number of available bits associated with an A/D
converter used for the quantization. The quantized signal values
over the time duration T are then used to construct the time
series.
[0037] Next, the process performs a spectral analysis on the time
series to obtain a frequency series for the low-resolution
quantized signal values in the frequency domain (step 202). In one
embodiment of the present invent, the spectral analysis is a
Discrete Fourier Transform (DFT), the DFT is computed using a Fast
Fourier Transform (FFT) technique.
[0038] For example, the process performs the frequency domain
transformation on a quantized time series X, by first performing a
conventional Fourier series expansion on X, so that: X t = a 0 2 +
m = 1 N / 2 .times. [ a m .times. .times. cos .function. ( .omega.
m .times. t ) + b m .function. ( .omega. m .times. t ) ] , ( 1 )
##EQU1## where a.sub.0/2 is the mean value of the quantized signal
X.sub.t, a.sub.m and b.sub.m are the Fourier coefficients
corresponding to the Fourier frequency
.omega..sub.m=2.pi..times.m/N, and N is the total number of
observations in the time series X.sub.t.
[0039] Let x.sub.j be the value of X.sub.t at the jth time
increment, the approximation to the Fourier transform yields: a m =
2 N .times. .times. j = 0 N - 1 .times. x j .times. .times. cos
.function. ( .omega. m .times. j ) and b m = 2 N .times. j = 0 N -
1 .times. x j .times. sin .function. ( .omega. m .times. j ) ( 2 )
##EQU2## for 0<m<N/2.
[0040] Accordingly, the Power Spectral Density (PSD) function for
the approximated Fourier frequency series can be computed as: I m =
N .times. a m 2 + b m 2 2 . ( 3 ) ##EQU3## Note that when there are
N observations in the original time series, the resulting PSD
function I.sub.m comprises (N/2)+1 harmonics (spectral amplitudes)
in the frequency domain.
[0041] The process next selects a subset of frequency terms from
the frequency series which have the largest amplitudes, i.e., the M
highest harmonics, wherein M<N/2+1 (step 204). More
specifically, the process selects the subset of the frequency terms
by first sorting frequency terms in the frequency series based on
amplitudes of the frequency terms in the frequency series, and then
selecting the M frequency terms with the highest amplitudes. In one
embodiment of the present invention, the amplitudes of the
frequency terms are obtained from the PSD function of the frequency
series. For example, the amplitude of the frequency term
.omega..sub.m can be computed using equation (3).
[0042] The process then reconstructs the high-resolution signal by
performing an inverse spectral analysis on the subset of the
frequency terms (step 206). In one embodiment of the present
invent, the inverse spectral analysis is an Inverse Discrete
Fourier Transform (IDFT), wherein the IDFT is computed using an
Inverse Fast Fourier Transform (IFFT).
Process for Computing a Precomputed Number
[0043] In one embodiment of the present invention, the process
determines the number of terms in the selected subset of the
frequency terms in step 204 based on a precomputed number.
[0044] FIG. 3 presents a flowchart illustrating a process of
precomputing the number for the selected subset of the frequency
terms in accordance with an embodiment of the present
invention.
[0045] The process first receives the original continuous-time
high-resolution analog signal from which the low-resolution signal
is obtained (step 300).
[0046] Next, the process iteratively selects the first n frequency
terms which have the largest amplitudes from the frequency series
of the low-resolution signal (step 302). Note that the system can
begin with n=1, and can increment n by 1 for each new
iteration.
[0047] For each iteration of n, the process then reconstructs a
time-domain signal by performing an inverse spectral analysis on
the selected n frequency terms (step 304). In one embodiment of the
present invent, the inverse spectral analysis is an inverse
Discrete Fourier Transform (IDFT), wherein the IDFT is computed
using an inverse Fast Fourier Transform (IFFT).
[0048] The process next computes the difference between the
high-resolution signal and the reconstructed signal (step 306).
Specifically, the process computes the difference between the
high-resolution signal and the reconstructed signal by computing a
residual signal. In one embodiment of the present invention, the
residual signal is obtained by subtracting the reconstructed signal
from the high-resolution signal.
[0049] Next, the process determines the precomputed number by
identifying a value of n that gives rise to the minimum difference
between the high-resolution signal and the reconstructed signal
(step 308). In one embodiment of the present invention, the
difference between the two signals is determined by computing the
sum-of-squares (SS) of the residual signal computed in step
306.
EXAMPLE
[0050] The following example demonstrates the process of
reconstructing a high-resolution signal from a set of
low-resolution quantized samples using a Fourier spectral analysis
technique. The original high-resolution signal is a continuous-time
voltage signal monitored in a server computing system, wherein the
high-resolution signal has been recorded for comparison purpose.
The set of low-resolution samples is a discrete-time quantized
signal which is sampled as a time series from the high-resolution
signal. The quantized signal is then decomposed into a frequency
series by applying the spectral analysis. Next, the frequency
series are sorted in order of the amplitudes of the harmonics.
[0051] FIG. 4 illustrates the process of reconstructing the
high-resolution voltage signal by using only two highest harmonics
of the frequency terms in accordance with an embodiment of the
present invention. Note that there are four subplots in FIG. 4.
Subplot 400 is the original high-resolution voltage signal. Subplot
402 shows the corresponding quantized signal after sampling with a
low-resolution eight-bit A/D converter chip. Note that the
quantized signal is a time series comprising about 500 samples.
[0052] Next, if one performs a frequency-domain Fourier
transformation of the quantized signal 402 and then selects only
the two highest harmonics in amplitudes from the frequency series
(M=2), the reconstructed time-domain signal is shown in subplot 404
of FIG. 4. Note that reconstructed signal 404 is very different
from the original signal 400. The difference between reconstructed
signal 404 (also referred to as a "synthesized signal") and the
high-resolution signal 400 is referred to as a "residual", which is
plotted in subplot 406 on the bottom of FIG. 4. The Sum-of-Squares
(SS) of the residual is used to measure how large the residual is.
For M=2 the residuals SS is about 0.0199, which are relatively
large.
[0053] FIG. 5 illustrates the process of reconstructing the
high-resolution signal by using different number of the highest
harmonics of the frequency series in accordance with an embodiment
of the present invention. Subplots 500, 502, 504 and 506 in FIG. 5
represent the reconstructed the high-resolution signal using M=3,
11, 57, and 10,000 highest harmonics of the frequency series,
respectively.
[0054] Note that when M=3, synthesized signal 500 is slightly
closer to the high-resolution signal than synthesized signal 404
for M=2, which is also measured as a decrease of the residual SS
from 0.0199 to 0.0149. As more harmonics are added while
reconstructing the high-resolution signal, the synthesized signal
begins to look increasingly closer to the original high-resolution
signal, while the residual SS continues to decrease. For example,
subplot 502 shows the synthesized signal for M=11, and the residual
SS which has dropped to about 0.00097.
[0055] Even though the residual SS initially decreases with an
increasing M value, note that when M becomes too large, the
synthesized signal approaches the quantized signal 402, which is
undesirable. This means that for sufficiently large M values, the
residual SS increases again. Hence, there exists an M value which
achieves a minimum residual SS while generating a synthesized
signal that near-perfectly reconstructs the high-resolution
signal.
[0056] FIG. 6 illustrates a process of searching for a value M that
achieves a minimum residual SS in an associated synthesized signal
in accordance with an embodiment of the present invention. The
horizontal axis of FIG. 6 shows the M values (600) used in the
reconstruction, and the vertical axis of FIG. 6 shows the
corresponding residual SS (602) for each M value. Using a simple
gradient descent method, one can determine that in this example,
the lowest residual SS is obtained for M=57.
[0057] Referring back to subplot 504 in FIG. 5 which corresponds to
M=57, note that when the optimal value of M=57 is used in the
reconstruction process, the difference between the synthesized
signal 504 and the original high-resolution signal 400 is very
small, as predicted in FIG. 6. This is also indicated by a residual
SS value of only 0.00016.
[0058] As mentioned previously, when M value continues to increase
towards a very large number, the synthesized signal asymptotically
approaches the low-resolution quantized signal 402. This effect is
demonstrated in subplot 504 in FIG. 5 for M 10,000, wherein the
signal waveform is a perfect copy of the one in subplot 402.
[0059] Note that it is desirable to pick a relatively small value
for M for computational efficiency. FIG. 6 shows that one can pick
an M value anywhere between 40 and 100 and still attain an
excellent accuracy in the synthesis of the original high-resolution
signal. Hence, in the process of reconstruction, one can choose an
M value over a broad range, preferably smaller than the optimized M
value that achieves a minimum Residual SS, thereby saving
computational time.
[0060] Note that the spectral synthesis technique introduced herein
consumes only a small fraction of the bandwidth on the system bus.
Hence, it can be applied concurrently to a large number of physical
telemetry signals being monitored within a large system, thereby
removing quantization effects in the low-resolution telemetry
signals and obtaining high-resolution signals. The reconstructed
high-resolution signals can then be used to detect problems in
these signals by applying the conventional statistical pattern
recognition techniques.
[0061] Also note that the spectral synthesis technique introduced
herein is a "preprocessing" approach which is designed to "undo"
the quantization effects at the source of the signals. In one
embodiment of the present invention, the technique is performed on
the same system that is being monitored.
[0062] The foregoing descriptions of embodiments of the present
invention have been presented only for purposes of illustration and
description. They are not intended to be exhaustive or to limit the
present invention to the forms disclosed. Accordingly, many
modifications and variations will be apparent to practitioners
skilled in the art. Additionally, the above disclosure is not
intended to limit the present invention. The scope of the present
invention is defined by the appended claims.
* * * * *