EE354 Signals and Sysms Inroducion o Fourir ransform Yao Wang Polychnic Univrsiy Som slids includd ar xracd from lcur prsnaions prpard y McClllan and Schafr
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LECURE OBJECIVES Rviw Frquncy Rspons Fourir Sris Dfiniion of Fourir ransform x d Rlaion o Fourir Sris Exampls of Fourir ransform pairs 4/4/28 23, JH McClllan & RW Schafr 3
Fourir Sris: Priodic x x x + 2 2 x a k k / 2 / 2 a k k x k d Fourir Synhsis Fundamnal Frq. 2π / 2πf Fourir Analysis 4/4/28 23, JH McClllan & RW Schafr 4
Squar Wav Signal x x + a k 2 2 a k k / 4 / 4 / 4 k /4 k d πk / 2 πk / 2 2πk 4/4/28 23, JH McClllan & RW Schafr 5 sinπk / 2 πk
Spcrum from Fourir Sris a k sin πk πk / 2 k, ±, ± 3, K k ± 2, ± 4, K 4/4/28 23, JH McClllan & RW Schafr 6
Wha if x is no priodic? Sum of Sinusoids? Non-harmonically rlad sinusoids Would no priodic, u would proaly non-zro for all. Fourir ransform givs a sum acually an ingral ha involvs ALL frquncis can rprsn signals ha ar idnically zro for ngaiv.!!!!!!!!! 4/4/28 23, JH McClllan & RW Schafr 7
Limiing Bhavior of FS 2 4 8 4/4/28 23, JH McClllan & RW Schafr 8
Limiing Bhavior of Spcrum 2 4 Plo a k 8 4/4/28 23, JH McClllan & RW Schafr 9
4/4/28 23, JH McClllan & RW Schafr FS in h LIMI long priod Fourir Synhsis Fourir Analysis π π π d x a x k k k 2 2 2 a d x d x a k k / 2 / 2 a π d 2 lim π k 2 lim lim a k
x Fourir ransform Dfind For non-priodic signals 2 π d Fourir Synhsis x d Fourir Analysis 4/4/28 23, JH McClllan & RW Schafr
Exampl : x a u a d a a + a+ d a + a > a + 4/4/28 23, JH McClllan & RW Schafr 2
4/4/28 23, JH McClllan & RW Schafr 3 Frquncy Rspons Frquncy Rspons is h Fourir ransform of h H u h + u h
4/4/28 23, JH McClllan & RW Schafr 4 Magniud and Phas Plos a H + a H an 2 2 + + a a H H
Exampl 2: x < / 2 > / 2 /2 d d /2 /2 /2 /2 /2 /2 /2 sin / 2 / 2 4/4/28 23, JH McClllan & RW Schafr 5
x < / 2 > / 2 sin / 2 / 2 No h widr is x, h narrowr in w! 4/4/28 23, JH McClllan & RW Schafr 6
4/4/28 23, JH McClllan & RW Schafr 7 Exampl 3: > < x π sin d d x π π 2 2 x π π 2 2
4/4/28 23, JH McClllan & RW Schafr 8 > < x π sin No h widr is w, h narrowr in x!
Exampl 4: x δ δ d Shifing Propry of h Impuls δ d 4/4/28 23, JH McClllan & RW Schafr 9
x δ 4/4/28 23, JH McClllan & RW Schafr 2
Exampl 5: 2πδ x 2π 2πδ d x 2πδ x 2πδ 4/4/28 23, JH McClllan & RW Schafr 2
x cos πδ + πδ + 4/4/28 23, JH McClllan & RW Schafr 22
Wha aou sin? 4/4/28 23, JH McClllan & RW Schafr 23
Fourir ransform of a Gnral Priodic Signal If x is priodic wih priod, x k a k k a k x k d k hrfor, sinc 2πδ k k 2π a k δ k 4/4/28 23, JH McClllan & RW Schafr 24
Squar Wav Signal x x + 2 2 a k / 2 a k k d + k d k k / 2 k / 2 k /2 4/4/28 23, JH McClllan & RW Schafr 25 πk πk
Squar Wav Fourir ransform x x + 2 2 2π a k δ k k 4/4/28 23, JH McClllan & RW Schafr 26
al of Fourir ransforms cos c c c x πδ πδ + + δ x > < x π sin 2 / 2 / sin 2 / 2 / x > < u x + 2 c x c πδ sin c c c x πδ πδ + + x δ
READING ASSIGNMENS his Lcur: Chapr, Scs. - o -4 4/4/28 23, JH McClllan & RW Schafr 28