WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

The Foundations of Statistics By Prof. Leonard J. Savage. (Wiley Publications in Statistics.) Pp. xv + 294. (New York; John Wiley and Sons, Inc.; London: Chapman and Hall, Ltd., 1954.) 48s. net.

Probability and Statistical Inference

It has been claimed that the empirical mode decomposition (EMD) and its improved version the ensemble EMD (EEMD) are computation intensive. In this study we will prove that the time complexity of the EMD/EEMD, which has never been analyzed before, is actually equivalent to that of the Fourier Transform. Numerical examples are presented to verify that EMD/EEMD is, in fact, a computationally efficient method.

On the computational complexity of the empirical mode decomposition algorithm

A novel approach for bidimensional empirical mode decomposition (BEMD) is proposed in this paper. BEMD decomposes an image into multiple hierarchical components known as bidimensional intrinsic mode functions (BIMFs). In each iteration of the process, two-dimensional (2D) interpolation is applied to a set of local maxima (minima) points to form the upper (lower) envelope. But, 2D scattered data interpolation methods cause huge computation time and other artifacts in the decomposition. This paper suggests a simple, but effective, method of envelope estimation that replaces the surface interpolation. In this method, order statistics filters are used to get the upper and lower envelopes, where filter size is derived from the data. Based on the properties of the proposed approach, it is considered as fast and adaptive BEMD (FABEMD). Simulation results demonstrate that FABEMD is not only faster and adaptive, but also outperforms the original BEMD in terms of the quality of the BIMFs.

/pdf/fast-and-adaptive-bidimensional-empirical-mode-decomposition-4dfu39tcf7.pdf

Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation

Speech pitch determination based on Hilbert-Huang transform

One of the main traditional tools used in scientific and engineering data spectral analysis is the Fourier integral transform and its high performance digital equivalent - the fast Fourier transform (FFT). Both carry strong a-priori assumptions about the source data, such as being linear and stationary, and of satisfying the Dirichlet conditions. A recent development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang transform (HHT), proposes a novel approach to the solution for the nonlinear class of spectral analysis problems. Using a-posteriori data processing based on the empirical mode decomposition (EMD) sifting process (algorithm), followed by the normalized Hilbert transform of the decomposed data, the HHT allows spectral analysis of nonlinear and nonstationary data. The EMD sifting process results in a non-constrained decomposition of a source numerical data vector into a finite set of intrinsic mode functions (IMF). These functions form a nearly orthogonal, derived from the data basis (adaptive basis). The IMFs can be further analyzed for spectrum content by using the classical Hilbert Transform. A new engineering spectral analysis tool using HHT has been developed at NASA GSFC, the HHT data processing system (HHT-DPS). As the HHT-DPS has been successfully used and commercialized, new applications pose additional questions about the theoretical basis behind the HHT EMD algorithm. Why is the fastest changing component of a composite signal being sifted out first in the EMD sifting process? Why does the EMD sifting process seemingly converge and why does it converge rapidly? Does an IMF have a distinctive structure? Why are the IMFs nearly orthogonal? We address these questions and develop the initial theoretical background for the HHT. This will contribute to the development of new HHT processing options, such as real-time and 2D processing using field programmable gate array (FPGA) computational resources, enhanced HHT synthesis, and will broaden the scope of HHT applications for signal processing.

On certain theoretical developments underlying the Hilbert-Huang transform

One of the main heritage tools used in scientific and engineering data spectrum analysis is the Fourier Integral Transform and its high performance digital equivalent - the fast Fourier transform (FFT). The Fourier view of nonlinear mechanics that had existed for a long time, and the associated FFT (fairly recent development), carry strong a-priori assumptions about the source data, such as linearity and of being stationary. Natural phenomena measurements are essentially nonlinear and nonstationary. A development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang transform (HHT) proposes an approach to the solution for the nonlinear class of spectrum analysis problems. Using the empirical mode decomposition (EMD) followed by the Hilbert transform of the empirical decomposition data (HT) as stated in N.E. Huang et al. (1998), N. E. Huang (1999), and N. E. Huang (2001), the HHT allows spectrum analysis of nonlinear and nonstationary data by using an engineering a-posteriori data processing, based on the EMD algorithm. This results in a non-constrained decomposition of a source real value data vector into a finite set of intrinsic mode functions (IMF) that can be further analyzed for spectrum interpretation by the classical Hilbert transform. This paper describes phase one of the development of a new engineering tool, the HHT data processing system (HHTDPS). The HHTDPS allows applying the HHT to a data vector in a fashion similar to the heritage FFT. It is a generic, low cost, high performance personal computer (PC) based system that implements the HHT computational algorithms in a user friendly, file driven environment. This paper also presents a quantitative analysis for a composite waveform data sample, a summary of technology commercialization efforts and the lessons learned from this new technology development.

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On the Hilbert-Huang transform data processing system development

This paper discusses the process and results of the performance testing of the GPS receiver planned for use on the International Space Station (ISS) and the X-38 Crew Return Vehicle (CRV). The receiver is a Force-19 unit manufactured by Trimble Navigation and Modified in software by the NASA Goddard Space Flight Center (GSFC) to perform navigation and attitude determination in space. The receiver is the primary source of navigation and attitude information for ISS and CRV. Engineers at GSFC have developed and tested the new receiver with a Global Simulation Systems Ltd (GSS) GPS Signal Generator (GPSSG). This paper documents the unique aspects of ground testing a GPS receiver that is designed for use in space. A discussion of the design and tests using the GPSSG, documentation, data capture, data analysis, and lessons learned will precede an overview of the performance of the new receiver. A description of the challenges of that were overcome during this testing exercise will be presented. Results from testing show that the receiver will be within or near the specifications for ISS attitude and navigation performance. The process for verifying other requirements such as Time to First Fix, Time to First Attitude, selection/deselection of a specific GPS satellite vehicles (SV), minimum signal strength while still obtaining attitude and navigation, navigation and attitude output coverage, GPS week rollover, and Y2K requirements are also given in this paper.

/pdf/testing-of-the-international-space-station-and-x-38-crew-4c93qb5pqf.pdf

Testing of the International Space Station and X-38 Crew Return Vehicle GPS Receiver

Sensor Web-based adaptation and sharing of space flight mission resources, including those of the Space-Ground and Control-User communication segments, could greatly benefit from utilization of heritage Internet Protocols and devices applied for Spaceflight (SpacelP). This had been successfully demonstrated by a few recent spaceflight experiments. However, while terrestrial applications of Internet protocols are well developed and understood (mostly due to billions of dollars in investments by the military and industry), the spaceflight application of Internet protocols is still in its infancy. Progress in the developments of SpacelP-enabled instrument components will largely determine the SpacelP utilization of those investments and acceptance in years to come. Likewise SpacelP, the development of commercial real-time and instrument co-located computational resources, data compression and storage, can be enabled on-board a spacecraft instrument and, in turn, support a powerful application to Sensor Web-based design of a spaceflight instrument. These are presently only co-located with the spacecraft Command and Data Handling System (C&DH). Sensor Web-enabled re-configuration and adaptation of structures for hardware resources and information systems on instrument level will commence application of Field Programmable Gate Arrays (FPGA) and other aerospace programmable logic devices for what this technology was intended. These are a few obvious potential benefits of Sensor Web technologies for spaceflight applications on instrument level. However, they are still waiting to be explored. This is because there is a need for a new approach to spaceflight instrumentation in order to make these mature sensor web technologies applicable for spaceflight. In this paper we present an approach in developing related and enabling spaceflight instrument-level technologies based on the new concept of a representative Instrument Sensor Web (ISW). This concept widens the scope of heritage sensor webs and facilitates the application of sensor web technologies to complex representative instruments onboard future spacecrafts.

/pdf/on-representative-spaceflight-instrument-and-associated-1gusgrqfz0.pdf

On Representative Spaceflight Instrument and Associated Instrument Sensor Web Framework

One of the main heritage tools used in scientific and engineering data spectrum analysis is the Fourier Integral Transform and its high performance digital equivalent - the Fast Fourier Transform (FFT). Both carry strong a-priori assumptions about the source data, such as linearity, of being stationary, and of satisfying the Dirichlet conditions. A recent development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang Transform (HHT), proposes a novel approach to the solution for the nonlinear class of spectrum analysis problems. Using a-posteriori data processing based on the Empirical Mode Decomposition (EMD) sifting process (algorithm), followed by the normalized Hilbert Transform of the decomposition data, the HHT allows spectrum analysis of nonlinear and nonstationary data. The EMD sifting process results in a non-constrained decomposition of a source real value data vector into a finite set of Intrinsic Mode Functions (IMF). These functions form a near orthogonal adaptive basis, a basis that is derived from the data. The IMFs can be further analyzed for spectrum interpretation by the classical Hilbert Transform. A new engineering spectrum analysis tool using HHT has been developed at NASA GSFC, the HHT Data Processing System (HHT-DPS). As the HHT-DPS has been successfully used and commercialized, new applications post additional questions about the theoretical basis behind the HHT and EMD algorithms. Why is the fastest changing component of a composite signal being sifted out first in the EMD sifting process? Why does the EMD sifting process seemingly converge and why does it converge rapidly? Does an IMF have a distinctive structure? Why are the IMFs near orthogonal? We address these questions and develop the initial theoretical background for the HHT. This will contribute to the developments of new HHT processing options, such as real-time and 2-D processing using Field Programmable Array (FPGA) computational resources, enhanced HHT synthesis, and broaden the scope of HHT applications for signal processing.

https://ntrs.nasa.gov/api/citations/20050210067/downloads/20050210067.pdf

Semion Kizhner

Papers

On certain theoretical developments underlying the Hilbert-Huang transform

On the Hilbert-Huang transform data processing system development

Testing of the International Space Station and X-38 Crew Return Vehicle GPS Receiver

On Representative Spaceflight Instrument and Associated Instrument Sensor Web Framework

On the Hilbert-Huang Transform Theoretical Developments