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# Exam2Handouts - I"mu—r ‘T l l nun LINN—U Ul...

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Unformatted text preview: I "mu—r.- ‘T. l l nun LINN—U Ul llll— tUUluLII IllnNLﬂ UHWI , In Section : a 4.3.4 4.3.6 4.3.3 4.3.3 4.3.3 Property Aperiodic signal Fourier transform ., x(t) X(;c_u-) . ya) H. w)- Dainty XQ) 271' %(-w) _ Linearity ax(t) + by(t) aX( co) + bY( (9) Time Shifting x(t — to) ' e"J“”°X( to) Frequency Shifting eja’o‘xa‘) _X(, lo) ~ 600) Conjugation ' x*(t) X*(~ to) Time Reversal x(~z‘)' X C “‘ w). 1 Time and Frequency x(at) WX<~§> Scaling Convolution x(t) * ya) - X( w)Y( co) 1- . +°° Multiplication x(t)y(t) é—i—é )(Cw)*Y(w) 7- %J Xt 6)Y( a) - 6);.d9 Differentiation in Time \$360) I ij( w) t 1 ' Integration I x(t)dt EM w.) + arX(0)6(co) Differentiation in' tx(t) j gig-M 60) ~ to Frequency X( '60) = X*(~V w) I, . Gisezgxg (0)} = (Rae{X(- 60)} Conjugate Symmetry x0) real 5m{X( 02)} = ~9m>{X(— {a} for Real Signals ’ lX( w)! m ]X(—_ to)! <):X( to) = ~—<)ZX(- w) Symmetry for Real and x(t) real [and even X ( to) real and even Even Signals " ;_ Symmetry for Real and x(t) real and odd X ( w) purely imaginary and 0" Odd Signals ' x50) = 8v{x(t)} [x(t) real] Gie{X( w)} EVEN“ Decompc” me = 002mm}- [xm real] 1mm an} 4.3.3 Xm‘iia‘ VAL“) Thnotemsi sition for Real Sig~ na'ls Parseval’s Relation for Aperiodic Signals “mm: = ~51; L m lX(., (0)124?» 0° 3‘} Eggmﬁlw Sec. 4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs 329 TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS Fourier series coefﬁcients Signal Fourier transform . (if periodic) +00 ' +09 Z akejkwo’ 277 Z ak5(w - kwo) ak k=~oo k=~m . = 1 wet 2775(co — mo) ‘1‘ ak = 0, otherwiSe 1 a = a_ = _ cos wot 7r[8(a) — coo) + 8(a) + (00)] I I 2 _ ak = 0, otherw1se 1 , 17' a = —a_ = 'T smwot —_[5(w — (no) — 6(a) + (00)] ' ‘ 21. ak = O, otherwrse a0=1, ak=0,k#0 160) = 1 27T 8(w) this is the Fourier series representation for any choice of T > 0 Periodic square wave 1’ It! < T1 +oo . . x(t) _ {0, T1 < M S g 2 ”Ln :on1 8(a) — km) (”in sinc (kngl ) = —sm :ZOT! +°° 277' +°° 277k 1 n;m8(t~nT) 7 ;w6<w_7> ak = Tforallk . ‘2 x(t):{ 1, It! < T1 23ian. __ re C t (E (l?) Sr in (T 1.) 0, Itl > T1 w T> "J” I | 1’ sin Wt . 1, < W Xow> = { ‘” ——- 7Tl‘ 0, la)! > W 50‘) ' 1 — (r) i + was _ u jw 8(t — to) e‘jw’o —~ _ 1 e ”’u(t), Gise{a} > 0 _ — V a + Ju) te‘mua) (Hae{a} > 0 ———l—-—— _ . ’ (a + 11102 <,:"_]‘)!e—“‘u(r), 1 __ (Rse{a} > O (a + jw)" ,_ game, Fwa'w TVams‘gou/m EGSmH’S ; {hvoMnW/xa Slhc Knuc’n‘ok/g 3 :{TWO Sing {‘undiows muHx'EHQA {m 1‘ng L Sggh<Wkt> 3‘.“ <WZJC> T \ ”+{S‘MW‘J *T ”3%sz . . 4:: . _ u NMVMFL‘LHIGMK \ A Co‘nvobdt‘m ’V‘ +‘IW‘C "9 sh +V Q‘Z‘mr5L “’3' V .V O.) . \t ‘ :Qy "WU/Vt 73:3; 51\$ SHAKW’L: .. 3‘.“ (W ) ...
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Exam2Handouts - I"mu—r ‘T l l nun LINN—U Ul...

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