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by: Spencer Ondricka
Spencer Ondricka
GPA 3.91

M. Liebling

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M. Liebling
Class Notes
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This 1 page Class Notes was uploaded by Spencer Ondricka on Thursday October 22, 2015. The Class Notes belongs to ECE 130A at University of California Santa Barbara taught by M. Liebling in Fall. Since its upload, it has received 42 views. For similar materials see /class/227036/ece-130a-university-of-california-santa-barbara in ELECTRICAL AND COMPUTER ENGINEERING at University of California Santa Barbara.

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Date Created: 10/22/15
ECE 130A Signal Analysis and Processing Fall 2008 1 Complex Numbers zajl7 1 j lle 11 a2 b2 ej9cos9jsin9 This implies that 399 7 399 399 7 399 81 8 1 e 7e 1 cos9 7 sin9 7 2 2 2 Integration b I b b I f fxgxdxfxgxla7f fxgxdx a a 3 Fourier Series For a periodic waveform xt with fundamental period T and fundamental frequency we 2T the Fourier series coef cients ak satisfy 00 xt Z ak 21km k7oo where 1 7 t jkwotdt ak T Tn e Fourier Series of Real Signals 00 XU Z ak 81km with ak afki k7oo Equivalently 00 xt a0 2 2 Akcoskw0ttpk k1 with ak Akerc and 00 xt a0 2 2 BkcosUcwo t 7 Cksinkw0 U k1 Profi Michael Liebling o Fmal Exam Formulary where ak Bk jcki Fourier Series Example 00 xm Z 6t7 kT k7oo we have ak l for all kt Fourier Series Properties In the following table we assume that both xt and yt have fundamental period T Signal Fourier Series Coef cients xt ak yt 17k axt3yt aakl3bk xt7 t0 ake jkwot j x7t Lk xt afk 00 xtyt Z ka 1700 xt real ak afk yt bkjkw0ak Parseval s relation for Fourier series 1flxml2dt E la l2 l k T T k7oo 4 Impulses Sifting property 00 x t f xI6 t 7 TdT 700 Impulse and stepfunction relationships d Eu 6t t utf 6IdI 700 Electrical amp Computer Engineering Depti University of California Santa Barbara 5 Convolution m X1lelf x1rx2t7rdr 7m 6 Fourier Transform Fourier Transform For an aperiodic signal xt the Fourier transform is de ned as 00 Xjw f my dt 700 Inverse Fourier Transform 1 00 xt 7 XjweW dw 271 700 Fourier Transform Pairs Indicator function I t7utW ut W7 1 mltW W39W 7 0 mgtw Triangle function In I I m lilll llllt1 r1 l 22 7 1212 1212 0y mgt1 Sinc function 1 t 0 sinc t U 7613 mgto Signal Fourier Transform xt Xjw elwot 2716w7w0 coswot n6w7w06ww0 sinwot 6w7w076ww0 xt 1 2716w IHLT m 2T1sinc W 75mg II7WWIW 6t 1 6t7 t0 e JmU um 7I6u 7 t 1 e 1 ut Reagt0 Hm


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