Lesson_300 - EEL 3135: Discrete-Time Signals and Systems...

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EEL 3135: Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Fourier Techniques Lesson Number: 30 [Chap 11-1 to 11:4] Background: Brief Biography of Jean Baptiste Joseph Fourier (1768-1830) Fourier was the son of a tailor and orphaned at the age of eight. Educated at a non-sectarian military college where he excelled in math, he considered entering the priesthood when the French Revolution broke out. Fourier joined the revolution which left France, in its aftermath, without a viable intelligentsia. Fourier himself had several close encounters with the guillotine. The persecution of scientists was finally abated through the intervention of Napoleon. Napoleon, in fact, encouraged the academic production scientists which benefited Fourier who, at the age of twenty-six, was appointed to the faculty of the new Ecole Normale. Napoleon, in 1789, visited Egypt with a group of scientists including Fourier. Fourier was appointed to the new Institut d’Egypte which became famous for its work with the Rosetta stone. Returning to France in 1801, Fourier’s career advanced and retreated along with Napoleon’s exiles and along with his several changes of political loyalties. In 1807, while serving as the Prefect of Grenoble, Fourier claimed the work that would make him famous. His studies in heat transfer were collected in a paper that introduced what we now would consider to be the foundations of orthogonal functions and the Fourier series. As was the custom of the time, papers were judged by the eminent scholars of the day. They were, in this case, Lagrange, Laplace, Legendre, and a fourth scholar. They found the paper innovative but lacking mathematical elegance and rigor. Lagrange, however, did not believe the results were correct and vigorously opposed its publication over the support of the other three judges. Fourier, unfortunately, could not provide a rigorous defense for his work (Dirichlet later completed the proof). The paper would wait fifteen more years to see print in the text Théorie analytique de la chaleur . In Chapter 11, Fourier transform techniques are introduced for continuous-time signals. The continuous-time Fourier transform and its derivatives are intrinsic to the study of continuous- time signals. The continuous-time Fourier transform (CTFT) is defined in terms of the analysis equation : - - = dt x(t)e ) X(j t j ϖ 1 and the synthesis equation : - = d )e X(j x(t) j t 2 where ϖ is called the analog frequency and is measured in radians per second, and X(j ϖ ) is called the spectrum of x(t). The Fourier transform, while be a fundamental scientific tool for centuries, demonstrated a systemic weakness once we entered the digital computer era. The production the spectrum of x(t) is seen to require both a prior knowledge of the signal and its integral over all time. This is an unrealistic task to assign a digital computer from a 1
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Lesson_300 - EEL 3135: Discrete-Time Signals and Systems...

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