Difference between revisions of "NTS Spring 2012/Abstracts"
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− | Abstract: | + | Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C('''Z'''/''p'') of complex valued functions on '''Z'''/''p'' = {0, ..., ''p'' − 1}, the integers modulo a prime number ''p'' ≫ 0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form |
− | + | R(t) = exp{2πiωt/''p''}⋅S(t+τ) + W(t), | |
− | + | where W(t) in H is a white noise, and τ, ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. | |
− | + | Problem (digital radar problem) Extract τ, ω from R and S. | |
− | + | In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. | |
− | |||
− | In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p | ||
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). | I will demonstrate additional applications to mobile communication, and global positioning system (GPS). |
Revision as of 09:27, 27 January 2012
Contents
February 2
Evan Dummit (Madison) |
Title: Kakeya sets over non-archimedean local rings |
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring F_{q}[[t ]], answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. |
March 1
Shamgar Gurevich (Madison) |
Title: Computing the Matched Filter in Linear Time |
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C(Z/p) of complex valued functions on Z/p = {0, ..., p − 1}, the integers modulo a prime number p ≫ 0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t), where W(t) in H is a white noise, and τ, ω in Z/p, encode the distance from, and velocity of, the object. Problem (digital radar problem) Extract τ, ω from R and S. In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^{2}⋅log(p) operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS). This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). The lecture is suitable for general math/engineering audience. |
March 29
David P. Roberts (U. Minnesota Morris) |
Title: tba |
Abstract: tba |
April 12
Chenyan Wu (Minnesota) |
Title: tba |
Abstract: tba |
April 19
Robert Guralnick (U. Southern California) |
Title: tba |
Abstract: tba |
April 26
Frank Thorne (U. South Carolina) |
Title: tba |
Abstract: tba |
May 3
Alina Cojocaru (U. Illinois at Chicago) |
Title: tba |
Abstract: tba |
May 10
Samit Dasgupta (UC Santa Cruz) |
Title: tba |
Abstract: tba |
Organizer contact information
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