# Indiana University

 dc.contributor.author Vandenberg, Dennis en dc.date.accessioned 2013-09-06T15:48:32Z en dc.date.available 2013-09-06T15:48:32Z en dc.date.issued 2012 en dc.identifier.uri https://hdl.handle.net/2022/16776 en dc.description (Thesis--M.S.) Indiana University South Bend, 2012. en dc.description.abstract This thesis develops an in-depth mathematical description of the cross-ambiguity function. The cross-ambiguity function is a time (𝜏) and frequency (𝜈) analysis technique employed to solve many signal processing problems such as interference mitigation and the location of emitters. The realm of the cross ambiguity function lies predominantly in the field of communications and electrical engineering where systems design is of importance. As such, the mathematical treatment of the cross-ambiguity function is brief, and is often presented with little detail in order to primarily fulfill engineering goals in the literature. This leaves the reader with subtle, but important gaps in understanding, such as, how convolution takes place, differences in the complex envelope and analytic signals, the Fourier series, and the use of complex conjugates. This thesis provides the mathematical foundation and concepts to more completely illustrate the cross-ambiguity function’s characteristics. There are many signal processing problems that can be used to demonstrate the cross-ambiguity function such as the matched filter, system design, noise reduction, and geolocation. This thesis selects collection of an emitter since the inherent geometry of the problem provides the clearest illustration of the function's time and frequency operations. Upon it mathematical concepts such as convolution, correlation, the work of Euler, the complex conjugate, Hilbert transform, the Fourier transform, and advanced integration techniques are presented. Further, the cross-ambiguity function is applied to the case of a square pulse emitted from a signal slow moving emitter and collected from to disparate collectors assumed to be moving at different speeds. This framework sets the stage not only for clarity of the geolocation problem, but a more clear understanding of time and frequency analysis. Finally, important aspects of the cross-ambiguity function are demonstrated in MATLAB. en dc.description.tableofcontents 1. Introduction -- 2. Literature review -- 3. Convolution -- 4. Discrete-time convolution -- 5. The linear time-invariant system -- 6. The basic convolution system -- 7. Discrete-time convolution exercise -- 8. Continuous-time convolution -- 9. The frequency dimension -- 10. From convolution to cross-correlation -- 11. Complex conjugate -- 12. Compute the cross-ambiguity function -- 13. Cross-ambiguity function assessment of the rectangular pulse -- 14. A practical application -- 15. Final comments and recommendations. dc.format.extent 104 pages dc.format.mimetype PDF dc.language.iso en_US en dc.publisher Indiana University South Bend en dc.subject Research Subject Categories::MATHEMATICS en dc.subject Functions -- Special. en dc.subject Convolutions (Mathematics) en dc.subject Signal processing -- Mathematics. en dc.title Mathematical Survey and Application of the Cross-Ambiguity Function en dc.type Thesis en
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