SPFirst-L06

# SPFirst-L06 - Signal Processing First READING ASSIGNMENTS...

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9/8/2003 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 6 Fourier Series Coefficients 9/8/2003 © 2003, JH McClellan & RW Schafer 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 Replaces pp. 62-66 in Ch 3 in DSP First Notation: a k for Fourier Series Other Reading: Next Lecture: More Fourier Series 9/8/2003 © 2003, JH McClellan & RW Schafer 4 LECTURE OBJECTIVES Work with the Fourier Series Integral ANALYSIS via Fourier Series For PERIODIC signals: x(t+T 0 ) = x(t) Later: spectrum from the Fourier Series = 0 0 0 0 ) / 2 ( 1 ) ( T dt e t x a t T k j T k π 9/8/2003 © 2003, JH McClellan & RW Schafer 5 HISTORY Jean Baptiste Joseph Fourier 1807 thesis (memoir) On the Propagation of Heat in Solid Bodies Heat ! Napoleonic era http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html

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9/8/2003 © 2003, JH McClellan & RW Schafer 6 9/8/2003 © 2003, JH McClellan & RW Schafer 7 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 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
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