ECE2025-L08 - ECE2025 Spring 2012 School of Electrical...

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Unformatted text preview: ECE2025 Spring 2012 School of Electrical & Computer Engineering Georgia Institute of Technology LECTURE #8 Fourier Series & Spectrum February 10, 2012 Spring 2012 EE-2025 Spring-2012 jMc-BHJ 2 READING ASSIGNMENTS This Lecture: Fourier Series in Ch 3, Sects 3-4, 3-5 & 3-6 Notation: a k for Fourier Series Other Reading: Next Lecture: Sampling Spring 2012 EE-2025 Spring-2012 jMc-BHJ 3 LECTURE OBJECTIVES ANALYSIS via Fourier Series For PERIODIC signals: x ( t+T ) = x ( t ) SPECTRUM from Fourier Series a k is Complex Amplitude for k-th Harmonic ) / 2 ( 1 ) ( T dt e t x a t T k j T k Fundamental period Spring 2012 EE-2025 Spring-2012 jMc-BHJ 4 100 250 100 250 f (in Hz) 3 / 7 j e 3 / 7 j e 2 / 4 j e 2 / 4 j e 10 Recall Complex Amplitude vs. Freq k k a X 2 1 X k A k e j k 1 2 X k * N k t f j k t f j k k k e X e X X t x 1 2 2 1 2 2 1 ) ( k a { * k a a Spectrum Diagram Notation Spring 2012 EE-2025 Spring-2012 jMc-BHJ 5 PERIOD/FREQUENCY of COMPLEX EXPONENTIAL: t f k j k k e a t x 2 ) ( 1 or 2 2 f T T f ) ( ) ( t x T t x Harmonic Signal IS Periodic Spring 2012 EE-2025 Spring-2012 jMc-BHJ 6 T = 0.1 a 3 a 5 a 1 Harmonic Signal (3 Freqs) Spring 2012 EE-2025 Spring-2012 jMc-BHJ 7 STRATEGY: x(t) a k ANALYSIS Get representation from the signal Works for PERIODIC Signals Fourier Series Answer is: an INTEGRAL over one period ) ( 1 T dt e t x a t k j T k Spring 2012 EE-2025 Spring-2012 jMc-BHJ 8 Integrate over ONE fundamental PERIOD ) / 2 ( T mt T j dt e m 2 T INTEGRAL Property of exp(j k) ) / 2 ( ) / 2 sin( ) / 2 cos( T T T mt T j dt mt T j dt mt T dt e If integer m is not 0 mt T ) / 2 cos( mt T ) / 2 sin( has m full cycles in T Spring 2012 EE-2025 Spring-2012 jMc-BHJ 9 for 1 ) / 2 ( m T dt dt e T T mt T j k k dt e e T T kt T j t T j 1 1 ) / 2 ( ) / 2 ( ORTHOGONALITY of exp(j k) T t for ) / 2 ( k kt T j...
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ECE2025-L08 - ECE2025 Spring 2012 School of Electrical...

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