Projects 1998-2001
Descriptions of Honors Calculus student projects
Department of Mathematical Sciences
Indiana University - Purdue University Fort Wayne (IPFW)
Descriptions of Honors Calculus student projects
Department of Mathematical Sciences
Indiana University - Purdue University Fort Wayne (IPFW)
The Fourier Transform or Fast Fourier Transform is a standard technique employed in various science and engineering fields such as Dynamics/Vibration, Earthquakes, and even in the stock markets when analyzing continuous or discrete time responses. The objective of time series analysis is to determine the statistical characteristics of the original dynamic response function of time by manipulating the series of discrete numbers f(i), i=1,2,3.... The main interest is identifying the frequency content of a signal f(t) and estimating the Fourier spectra sot hat engineers can design safe structures and machinery by avoiding detrimental resonance(s) which often results in a disaster.
In this project, students will learn the fundamental concepts of the Fourier Transform, Fast Fourier Transform, Fourier spectra, andv ibration. Based on these concepts, they will analyze time responses of representative dynamic systems under periodic, quasi-periodic, chaotic, and stochastic excitations. Students will also learn how to employ scientific/engineering software such as Matlab, Mathematica, or Maple to solve those dynamics problems and how to obtain the corresponding Fourier spectra from time responses. Estimated duration: 2 semesters.
Project Supervisor: Dr. Bongsu Kang, Mechanical Engineering
Students will learn that hemodynamic signals contain information that can be extracted using mathematical analysis. In this particular application, the analytical tool they will use is the Fourier Transform.
The students will perform the Fourier analysis of blood pressure signals from the human body. Specifically, the will analyze the blood pressure signals from:
a) Subjects that are healthy.
b) Subjects that have heart disease, but no arterial disease.
c) Subjects that have arterial disease but no heart disease.
The analysis of the same signal will be performed using data collected from different points in the arterial system.
Project Supervisor: Dr. Josue Njock Libii, Engineering
Radio astronomers study the emissions from stars, galaxies, quasars, pulsars, etc., at radio frequencies. Two important parameters are the sensitivity and the resolution of the radio telescopes used in such studies. The resolution refers to the smallest separation between two objects, which can be detected by a telescope. Extremely high resolutions can be achieved by using pairs of antennas separated by large distances. This is the theory of synthetic aperture radio telescopes. In simple terms, when imaging a radio source such as a galaxy, one actually measures the Fourier transform of the brightness distribution of the source. The actual brightness distribution can then be recovered from the measurement data by Fourier transformation. Discrete Fourier transforms are used since the measurements give the Fourier components at discrete intervals. The reconstructed source map suffers from errors introduced by several factors. These include inadequate number of Fourier components measured, missing components, errors in the magnitude and phase of the measured components, and errors introduced in the amplitude and phase introduced by instrumental errors and atmospheric effects, among others.
The project will involve understanding of Fourier transform theory, learning Fourier transform and graphics software such as MATLAB, producing radio maps of radio sources with assumed brightness distribution and then analyzing the effects of phase and amplitude errors, missing data and discretization on the maps so produced. The study is of relevance to the work of all major radio telescopes of the world such as the Very Large Array radio telescope in Soccoro, NM.
Project Supervisor: Dr. Naresh C. Mathur, Electrical Engineering.
The goal of this experiment is to determine whether it is feasible to use diffractive and Fourier optics to cut a specific pattern on a card. To do this, we have to calculate the Fourier transform of the desired pattern, which determines the shape of the mask. We will have to calculate the optical throughput through the mask to determine efficiency of the system. Ultimately we will build several masks and test their cutting capabilities. In the process of this experiment the student will explore optical filtering of images using FT optics.
Project Supervisor: Dr. Mark Masters, Physics
A static FTS splits the light into two portions, and then through divergence of the light, you get a spatial FT of the light. This spatial FT of the light represents the FT of the wavelength dependent intensity of the light. Static Fourier transform spectrometers have the advantage of measuring the spectrum of a light source almost instantaneously. In this project the student will set up the static FTS, test the FTS and determine its capabilities.
Project Supervisor: Dr. Mark Masters, Physics
FT's are used extensively on laser imaging through dense media.
Project Supervisor: Dr. Mark Masters, Physics
In this project the student would build a scanning FTS. They will have to do some computer programming. Ultimately they will test the system to see how it works.
Project Supervisor: Dr. Mark Masters, Physics
Frequency Modulation is a concept that has been available for many years. Despite its longevity, FM is finding more and more areas that lend themselves to its properties. This assignment is to analyze frequency modulation and its applications for the future.
Project Supervisor: Dr. Thomas A. Laverghetta, Electrical Engineering Technology
Spread Spectrum is a technology finding more and more usage in the modern commercial communications world. In particular, frequency hopping spread spectrum is an area that needs to be evaluated and analyzed further to determine future applications within the communications area. This is the assignment, to analyze frequency hopping spread spectrum and project it into the 21st century.
Project Supervisor: Dr. Thomas A. Laverghetta, Electrical Engineering Technology
In the first semester, the student will how to prepare media, sterilize it, and pour it in petri plates. and also learn how to sterilize seeds and plant them on the solidified media in the plates. The media will contain a selective herbicide that will inhibit the growth of the sensitive plants but will not affect the growth of the resistant plants.The student will plant the F2 seed of a cross between herbicide-resistant (hairy leaves) and the herbicide sensitive (hairless leaves). The herbicide-resistant F2 seedlings will be selected then the segregation of hairy leaves and hairless leaves will be scored. The numbers produced will be used to calculate the recombination frequency between the gene encoding herbicide resistance and the hairy-leaves phenotype. The product method (in repulsion phase) of Immer (Formulae & Tables for Calculating Linkage Intensities, 1930, Genetics 15:81-98) will be used for calculating the recombination frequency (linkage intensity) between the two genes. Then the Kosambi mapping function as mentioned in Koorneef et al., (Linkage map of Arabidopsis thaliana, 1983), Journal of Heredity 74:265-272) will be used to calculate the genetic distance, in centiMorgan cM, between the gene for hairy leaves and the gene for herbicide resistance.
Project Supervisor: Dr. George S. Mourad, Biology
Here a well-known reaction to prepare [FeH_6]^{4-} will be followed according to our modification of published directions. The product is characterized by FTNMR and electronic spectroscopy. Assuming there is good success here we will try a new reaction to prepare [MnH_4]^{2-} in solution using techniques learned above. Here the compound is expected to contain unpaired electrons which will make FTNMR less useful. Here electronic spectroscopy, electron paramagnetic spectroscopy and single crystal X-ray diffraction techniques would be more useful.
Project Supervisor: Dr. Donald E. Linn, Chemistry
The hydride [FeH_6]^{4-}, is a good reducing agent but the scope of its use has not been ascertained. The reagent is rather cheap and environmentally benign. The question is whether it has useful properties as a reducing agent. The properties are anticipated to mimic in some regards the borohydride reducing agents. In our iron hydride molecule, however, since iron has a large number of hydrides attached, more extensive hydrogenations (reductions) of a given molecule appear feasible.
Thus far aromatic compounds are found to be reduced to cyclic hydrocarbons. Here the student would test the reagent by reactions with various small organic molecules to see what functional groups are affected.If stoichiometric reactions are found to work, it would then be possible to test these reactions under catalytic conditions.
This exercise would rely heavily on H-1 and C-13 FTNMR to determine the stereochemistry of the products of these reactions. Quantification of the products would also be done by gas chromatography mass spectrometry (GCMS).
It would also be possible to mix elements of both these projectst ogether to suit the taste of the student. The first project is more organometallic and inorganic chemistry in orientation, the second is more organic chemistry.
Initially a good deal of supervision will be required, due to the rigor of the techniques used to handle these hydrides, 99.9995% nitrogen and 10^{-6} torr atmosphere.
Project Supervisor: Dr. Donald E. Linn, Chemistry
This project covers the introduction of basic MATLAB commands and programming techniques. We start with linear algebra applications(matrices, determinants, eigenvalues, etc.), then we will cover visualization (basics about 2D and 3D plots), aspects of numerical computation using MATLAB, and we end with information on convolution,Fast-Fourier transforms, and correlation. Students will work on projects, individually or in groups. Estimated duration: 4 months.
Project Supervisor: Dr. Dan Coroian, Mathematical Sciences
This project covers the introduction of basic MAPLE commands and programming techniques. Students will learn how to use MAPLE to plot two and three dimensional functions; one or more functions in the same window; Fourier series; functions given by parametric equations. They will learn how to use MAPLE to find limits, derivatives, integrals, extremal values of functions, etc..., that is, they will learn to do all of the techniques and manipulations that calculus requires.Students will work on projects, individually or in groups. Estimated duration: 2 semesters.
Project Supervisor: Dr. Peter Hamburger, Mathematical Sciences
Digital electronic computing is mainly constrained by limited interconnection and high propagation delay of the electronic devices. In addition, the carry/borrow generated during an addition/subtraction operation further reduces the computation speed. Carry look ahead addition technique may be used to enhance the computation speed. However, as the operand length increases, the size of the corresponding digital system increases almost linearly the thereby increasing its cost and hardware complexity. Optical computing techniques have shown remarkable promise to alleviate these constraints since huge amounts of data can be processed in parallel almost at the speed of light.
The modified signed-digit (MSD) representation has been widely studied for implementing parallel optical arithmetic. Recently, a higher-order MSD technique for parallel arithmetic using symbolic substitution has been proposed. This technique performs carry-free addition and borrow-free subtraction by checking a pair of reference digit from the next lower order bit position. However, MSD representation requires negative representation of a literal. Most recently, a redundant binary number representation has been proposed for parallel arithmetic using only two literals, 0 and 1. For long bit strings, this scheme doubles the operand length. To overcome the aforementioned problems and at the same time enhance the processing speed, we proposed the recoded trinary arithmetic processing technique which performs multibit carry-free addition and borrow-free subtraction in constant time by employing a two-step symbolic substitution scheme. In the first step, a set of symbolic substitution rules are applied to the pair of numbers to be added into an intermediate pair. Subsequent application of the symbolic substitution rules to the intermediate pair of numbers yields the result. This technique also leads to a compact design by incorporating more information in fewer digits using a higher radix number system. A content-addressable memory (CAM) based implementation will be considered.
Project Supervisor: Dr. Mohammad Showkat-Ul Alam, Engineering
Pattern recognition deals with the detection and identification of a desired pattern or target in an unknown input scene which may or may not contain the target and the determination of the spatial location of any target which is present. Traditional digital pattern recognition techniques require massive computation and are relatively slow. In contrast, optical techniques inherently provide parallelism, ultrahigh processing speed, non-interfering communication, and massive interconnection capability and offer a significantly better alternative to the conventional digital pattern recognition approach. When an optical correlator processes a scene using the inherent Fourier transform properties of lenses, it optically compares all pixels in the input scene to all pixels in the reference image simultaneously. This parallel processing capability of the optical correlators provides a tremendous processing speed advantage over their digital counterparts where the input scene must be processed sequentially.
Optical pattern recognition involves the use of either a matched filter based correlator or a joint Fourier transform correlator (JFTC). A matched filter based correlator is not suitable for real-time operation since a complex filter must be fabricated for each new input. On the other hand, a JFTC is inherently suitable for real time matching and tracking operations since no complex filter is needed. A classical JFTC is found to suffer from poor correlation discrimination, strong zero-order peak and low optical efficiency. A binary JFTC yields superior correlation performance when compared to a classical JFTC but involves computation-intensive binarization of joint power spectrum and may yield spurious correlation peaks. A chirp-encoded JFTC requires multiple input and multiple output planes which significantly increases the space-bandwidth product.Recently, a number of nonlinear techniques have been proposed for optical pattern recognition. Among these nonlinear techniques, the fringe-adjusted JFTC has been found to be particularly attractive. This technique employs a family of real-valued filters called fractional power fringe-adjusted filter (FPFAFs). By suitably adjusting two parameters used in the FPFAF formulation, we can obtain optimum correlation performance in both noise-free and noisy environments. Various types of architectures will be explored to effectively implement the nonlinear Fourier domain processing for the FPFAF based JFTC through analytical modeling and computer simulation. This technique can be used for various practical applications such as machine parts recognition for industrial automation, optical character recognition, finger print identification, and military target detection and tracking.
Project Supervisor: Dr. Mohammad Showkat-Ul Alam, Engineering
Many of the major and great earthquakes are associated with minor quakes (originating from the same source region) before the main quake.These minor earthquakes are called fore shocks. Similarly, the main earthquake is usually followed by a large number of quakes of decreasing intensity with time. These are known as after shocks. Theoretical studies on the frequency content of the fore shocks compared to the main and after shocks have been inconclusive. Actual determination of the frequency content was not possible in the past because the seismographs were band-limited, i.e. recorded only a narrow range of frequencies. More recently, a new type of digital seismograph has been developed. They can essentially record all the frequencies of the earthquake waves arriving at a particular seismograph. This is known as Broad-Band seismograph. Also, a large number of broad-band seismographs have now been installed around the world, so that worldwide data are now available.
We would like to select a few large earthquakes that have foreshocks as well as after shocks. FFT the digital data of the fore shocks, main shock and some of the after shocks. Correct the spectra for the frequency dependent instrument response and analyze if these shocks are characterized by different spectral content.
If we are able to obtain any characteristic spectral signature for each shock, the results will be of extreme importance and may be used for earthquake prediction purposes.
Project Supervisor: Dr. Dipak Chowdhury, Geology
The proposed student project would involve the analysis of a complex natural waveform through the application of Fourier techniques. Specifically, the student would construct a simple [non-filtered] harmonic analysis computer code, extract a digital signal from the natural waveform, measure the waveform's power spectra, and interpret the results. This project would involve an introduction to applied mathematics, as well as notions of fractal power spectra and noise analysis. The waveform to be studied is a geological feature known as a stylolite, taken from rocks of southern Indiana.
Project Supervisor: Dr. Carl N. Drummond, Geology
The project is an analysis of both vocal and instrumental frequencies and their relationship regarding being ``in tune'' or ``out of tune.''
One of the problems with many choirs and instrumental groups is to keep them in tune. That is, when more than one person or more than one instrument play the same note, are they producing the same frequency, or are they in tune. If they are not there will be a complex wave produced that can be analyzed by means of the Fourier Series.
In this project, a variety of samples using both choirs and instrumental groups will be recorded and analyzed. Arrangements have been made with Sweetwater Sound to aid in the recording and data collection for this project.
Project Supervisor: Dr. Thomas S. Laverghetta, Electrical Engineering Technology
In this project choral intonation will be examined. It will be determined mathematically what fundamental frequencies cause the most difficulties for choral intonation. In tuning a choir there is difficulty in tuning on the "third", an intermediate tone. It is needed to examine various choral works, using Fourier Analysis. The question is which frequencies and sound spectrum contribute to a well tuned choir. Using sound synthesis, the frequencies that contribute to poor choir tuning will be investigated; what factors contribute to the "unpleasantness" of a mistuned choir. This will involve studying the process of hearing. Fourier Analysis will be used to determine what frequencies are added to a fundamental tone when someone "pushes". This project can be completed using a reasonably fast computer equipped with a sound card.
Project Supervisor: Dr. Mark Masters, Physics