Lesson+24 - For those currently lost in a laptop moment 1...

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Lesson 24 For those currently lost in a laptop moment… 1
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No Challenge Problem Section 16.1, 2, part of 3 Continuous-Time Fourier Series Challenge 24 Lesson 24 2 Lesson 24
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“When I am working on a problem I never think about beauty. I only think about how to solve the problem.” Are we sure this guy is really French? Jean Baptiste Joseph Fourier (1768-1830). “But when I have finished, if the solution is not beautiful, I know it is wrong." 3 Lesson 24
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He was orphaned at age eight and educated at the Convent of St. Mark. He took a prominent part in promoting the French Revolution , and was rewarded by an appointment to the École Normale Supérieure , and subsequently by a chair at the École Polytechnique . Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, and was made governor of Lower Egypt and secretary of the Institut d'Égypte . After the British victories and the capitulation of the French under General Menou in 1801, Fourier returned to France, and was made prefect of Isère , and it was while there that he made his experiments on the propagation of heat. Jean Baptiste Joseph Fourier (1768-1830). 4 Lesson 24
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In 1806 he quit the École Polytechnique because Napoleon sent him to Grenoble . Fourier moved to England in 1816. Later he returned to France, and in 1822 succeeded Delambre as Permanent Secretary of the French Academy of Sciences . Fourier also believed that keeping the body wrapped up in blankets was beneficial to the health. He died in 1830 when he tripped and fell down the stairs at his home. Irony? Jean Baptiste Joseph Fourier (1768-1830). 5 Lesson 24
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Lesson 24 What’s this I hear about: Spectrum? Frequency - domain? Fourier - representation? 6 Lesson 24
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CTFS The continuous-time Fourier series ( CTFS ) is a signal representation technique that applies to periodic signals (only). Remember periodic signals are non-causal!) x p ( t )= x p ( t +k T ) periodic for all time t t n B t n A A t x n n n p 0 1 0 0 sin cos ) ( Synthesis Eq. If the CTFS of c(t) exists, then the Synthesis Equation is: Da man 7 Lesson 24
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But how do I know that a CTFS exists? Dirichlet conditions for the existence of a CTFS 1. x(t) is absolutely integrable over any period 2. x(t) has a finite number of maxima and minima over any period. 3. x(t) has a finite number of discontinuities over any period. 8 Lesson 24
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If the CTFS exists, then: t n B t n A A t x n n n p 0 1 0 0 sin cos ) ( Synthesis Eq. Analysis Eq. T p n T p n T p dt t n t x T B dt t n t x T A dt t x T A 0 0 0 0 0 0 ) sin( ) ( 2 ) cos( ) ( 2 ) ( 1 9 Lesson 24
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10 Lesson 24 Key Observation Trigonometry sin(a) sin(b) = 0.5(cos(a-b) – cos(a+b)) sin(a) cos(b) = 0.5(sin(a-b) – sin(a+b)) cos(a) cos(b) = 0.5(cos(a-b) + cos(a+b)) What if a=b? Trigonometry sin(a) sin(a) = 0.5(cos(0) – cos(2a) = constant - sinusoid sin(a) cos(a) = 0.5(sin(0) – sin(2a)) = 0 - sinusoid cos(a) cos(a) = 0.5(cos(0) + cos(2a)) = constant + sinusoid How can the constant be separated from the sinusoidal terms?
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