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2025-L07 - General Info ECE-2025 Fall-2008 Quiz#1 coming up...

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ECE-2025 Fall-2008 Lecture 7 Fourier Series & Spectrum 12-Sept-2008 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 2 General Info Quiz #1 coming up: Monday, 15 Quiz #1 coming up: Monday, 15 - Sept Sept Calculators OK, One 8.5” X 11” page of handwritten notes More Problems (w/ solutions) are on the SP-First CDROM web site Office Hours: Visit any Prof or TA See t See t - square for matrix of Office Hours square for matrix of Office Hours t - square: square: OFFICIAL ANNOUNCEMENTS Prob Set #4 due next Week HW #1, #2, #3 and #4 will be covered on Quiz #1 Lab #3 due starting on Tuesday 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 3 READING ASSIGNMENTS This Lecture: Fourier Series in Ch 3, Sects 3 Fourier Series in Ch 3, Sects 3 - 4, 3 4, 3 - 5 & 3 5 & 3 - 6 Notation: a k for Fourier Series Other Reading: Next Lecture: Sampling 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 4 LECTURE OBJECTIVES ANALYSIS ANALYSIS via Fourier Series For PERIODIC signals: x ( t+T 0 ) = x ( t ) SPECTRUM SPECTRUM from Fourier Series a k is Complex Amplitude for k-th Harmonic = 0 0 0 0 ) / 2 ( 1 ) ( T dt e t x a t T k j T k π Fundamental period
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09/11/2008 EE-2025 Fall-2008 jMc-BHJ 5 0 100 250 –100 –250 f (in Hz) 3 / 7 π j e 3 / 7 π j e 2 / 4 π j e 2 / 4 π j e 10 Spectrum Diagram Notation Recall Complex Amplitude vs. Freq k k a X = 2 1 X k = A k e j ϕ k 1 2 X k * { } = + + = N k t f j t f j k k e e t x 1 2 2 ) ( π π k a { * k a 0 a 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 6 Harmonic Signal must be Periodic PERIOD/FREQUENCY of COMPLEX EXPONENTIAL: t f k j k k e a t x 0 2 ) ( π −∞ = = ( ) 0 0 0 0 0 1 or 2 2 f T T f = = = π ω π ) ( ) ( 0 t x T t x = + 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 7 Harmonic Signal (3 Freqs) T = 0.1 a 3 a 5 a 1 09/11/2008 EE-2025 Fall-2008 jMc-BHJ 8 STRATEGY: x(t) Æ a k ANALYSIS Get representation from the signal Works for PERIODIC PERIODIC Signals Fourier Series Answer is: an INTEGRAL over one period = 0 0 0 0 ) ( 1 T dt e t x a t k j T k ω
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09/11/2008 EE-2025 Fall-2008 jMc-BHJ 9
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