Exam2Handouts - I "was. ‘T. l l nun LItItLU Ul...

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Unformatted text preview: I "was. ‘T. l l nun LItItLU Ul llll— IUUIHLII llerVUI 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(; . ya) H. w)- Dumz’vy XQ) 271' %(-w) _ Linearity ax(t) + by(t) aX( to) + bY( (9) Time Shifting x(t — to) ' e"J“”°X( to) Frequency Shifting efwo‘xa) _X(, lo) ~ coo) Conjugation ' x*(t) X*(~ to) Time Reversal x(~z‘)' X C “‘ w)- 1 Time and Frequency x(at) EX Scaling Convolution x(t) * ya) - X( w)Y( co) 1- . +°° Multiplication x(t)y(t) )(Cw)*Y(w) 7- X(, 6)Y( a) - 6);.d9 Differentiation in Time $360) I ij( w) t 1 ' Integration I x(t)dt EM w.) + 77X(0)5(co) Differentiation in' tx(t) j ( 60) ~ to Frequency X( '60) = X*(~V w) I, . (MXQ w)} = (Rae{X(- an} Conjugate Symmetry x0) real 5m{X( 02)} = ~9m>{X(— for Real Signals ’ ’X( w)‘ m [X(_. (0)! <):X( w) = ~—<)ZX(- 0)) Symmetry for Real and x(t) real [and even X ( to) real and even Even Signals " ;_ Symmetry for Real and x(t) real and odd X ( w) purely imaginary and 0" Odd Signals ' x50) = 8v{x(t)} [x(t) real] Gie{X( w)} EVEN“ Decompc” W) = ®d{x(t)}' [xm real] ijsrm{X< an} 4.3.3 Xm‘iiai VAL“) Thememsi sition for Real Sig~ na'ls 0° 96(0) : Eggmfilw Parseval’s Relation for Aperiodic Signals “mm: = ~51; L m lX(., mlzciw Sec. 4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs 329 TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS Fourier series coefficients Signal Fourier transform . (if periodic) +00 ' +09 Z akejkwo’ 277 Z ak5(w - kwo) ak k=—oo k=~m . = 1 wet 2775(co — mo) ‘1‘ ak = 0, otherwiSe 1 a = a_ = _ cos wot 7r[8(a) — (00) + 8(a) + (00)] I I 2 _ ak = 0, otherw1se 1 . 17' a = —a_ = 'T smth —_[5(w — (no) — 6(a) + (00)] ' ‘ 21, ak = O, otherwrse a0=1, ak=0,k#0 160) = 1 27T 5(a)) this is the Fourier series representation for any choice of T > 0 Periodic square wave 1’ < T] +oo . . x“) _ {0, T1 < M S g 2 -——281ninTl 8(a) — km) (1):? sinc (kngl ) = —Sm 12:07"! +°° 277' +°° 277k 1 n;m8(t~nT) 7 ;w6<w——T——> ak = Tforallk . ‘\ I T 9 = 1, Itl<T1 231an. (t i‘ SHA ( 1. t ——-—— —— b -— ——— xoio, Iri>Ti w r“ T>< > “7:7” I | 1’ sin Wt . 1, w < W Xow> = { ——- 7Tt 0, la)! > W 5(t) ' 1 _ (r) i + 775(w) _ u jw 8(t — to) e‘jw’o —~ _ 1 e “’u(t), Gise{a} > 0 _ — V a + Ju) te‘mua) (Hae{a} > 0 ———l—-—— _ . ’ (a + jwy (iffy—“tum, 1 __ (Rse{a} > 0 (a + jw)" ,_ game, Fbmhev Tvams‘gorm EGSMH'S ; {hvoMnW/xa Slhc Knuc’n‘ok/g 3 :{TWO Sing +ung+10w§ muHx'EHQg {m 1‘ng L ,_._...__-a S‘MW} sm<wzt> “Tr ;${w gm 6—» Wt T1: 1 W . . Iii . . . ‘. u Nm‘fix'fph‘ind‘MKK ‘ ' Wk<WL ...
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This note was uploaded on 02/19/2012 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Spring '06 term at Purdue University-West Lafayette.

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Exam2Handouts - I &amp;quot;was. ‘T. l l nun LItItLU Ul...

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