Exam3Handouts - lrIul—n T l l nun LJllH—U UI ”IL...

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Unformatted text preview: lrIul—n. T. l l nun LJllH—U UI ”IL. IUUIULII lnr'mlUl UH!" Section Property Aperiodic signal Fourier transform x0) Xena) Y“) Y(. (0)- .430 DuaMy _X(t) ____________ 2_Tr_f_%(~w) 4.3.1 Linearity (2.760?) + by(t) aX_( co) + bY-( co) 4.3.2 Time Shifting x(t — to) ' e“""’°X( to) 4.3.6 Frequency Shifting efwo'xo) VXC .60 " 600) 4.3.3 Conjugation ' x*(t) XV" w) 4.3.5 Time Reversal _ x(-t)' XC' 160} 1 4.3.5 Time and Frequency x(at) _ F4412) .- . a] a Scaling 4.4 Convolution x(t) * ya) V X ( w)Y( co) 1~ . +°° 4.5 Multiplication x(t)y(t) é-i—T-i XCUDé‘YW) *3 21,4 X( . 9)Y( a) -- 0);d6 4.3.4 DifferentiationinTime glint) ' ‘ ij< w) : t 1 I . 4.3.4 Integration J x(t)d2‘ E)“ a") + WX(0)5(60) i... ' d 4.3.6 Differentiation in 060‘) J 21;)“ w) Frequency X( '60) = X*(". m) _ _, ‘ Gise{X(‘ w)} = Gie{X(~ an} 4.3.3 Conjugate Syrmnetry x(t)_ real . 9m{X( w)} = ~Hm{X(~ Ca} for Real Signals !X( m)! z |X(-_ (0)] <):X( co) = “(XE co) 4.3.3 Symmetry for Real and x0?) real and even X ( co) real and even Even Signals A . 4.3.3 Symmetry for Real and x0?) real and odd X ( w) purely imaginary and 0" Odd Signals ' 37.10) = 8v{x(z‘)} [x(t) real] (Rse{X( w)} 4.3.3 Even~0dd DecompO— x00?) 2 (961950)}. [M01321] .j9m{X( (0)} sition for Real Sig~ . i na‘ls x(0) = fir£0>§m)dw H“): X“??? linen}; - 2<_<_°_>_=__C:9’i<f§}5*§c __________________ 4.3.7 Parseval’s Relation for Aperiodic Signals _. [MW = 51-7; f:lX(.,w)Izaiw Sec. 4:6 Tabies 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) +50 . +00 2 akejkwo’ 277 Z ak5(w ~ kwo) ak k=~ea =-oo . = 1 6""0’ 277880 — we) “1 (1;, = 0, otherwise I a = a_ = __ cos wot 7r[8(w — me) + 5(a) + 600)] } I 2 . ak = O, otherw1se l a = “(1.. = —, sin wot Elam “ we) * 6(a) + (00)] l I 2’- } ak = 0, otherwrse [10:1, ak=0,k?é0 x(t) = 1 277‘ 5(0)) this is the Fourier series representation for any choice of T > 0 Periodic square wave 1, [ti < Tl x(t) =[ I W 23inkw0T. (DOT; . koni _ sinkonr 0. T. < M s 2 Z k 8(a) km) W smc 'n‘ kw and k—“w x(t + T) = r0) ”"8 2% +008 27rk ~1f ilk 120%") 7,; “"7“ “Pr 0” \ , f = 1, it! < T; 2sian. ( e + s” rx z W _ r c i: «a .___ x()[0, [ti>T. a) e T> < > W sin Wt , 1, a) < W X(Jw) =[ ’ ' — ’7Tt 0, [col > W 80) 1 _ u(t) i + 778(w) —— ja) 6(r — to) e‘fw’o — e_‘”u(t) Older} > O —~1——— ._ ’ _ a + ja) te““‘u(l) emu} > o “—1“— _ V ' (a + jm)2 “(3:1)! e‘“’zt(t). 1 * (Rse{a} > O (a + jw)" Sec. 5.7 Duality 391 TABLE 5.] PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM MM Section Property Aperiodic Signal Fourier Transform x[n] ' . X(e1‘")} periodic with 5.3.2 5.3.3 5.3.3 5.3.4 5.3.6 5.3.7 5.4 5.5 5.3.5 5.3.5 5.3.8 5.3.4 ‘ 5.3.4 5.3.4 5.3.4 5.3.9 y[n] Y(e1'“f‘) period 2w Linearity , ax[n] + by[n] _ (IX-(elm) +‘bl’ (31») Time Shifting xtn - no] ' e-‘wa‘°X(ev"") Frequency Shifting ej”°"x[n] . . X (e"“’f“’°’.) Conjugation x“ [n] 5 X‘(e‘;"") Time Reversal x[—n] , X(e'1"’) . . . _ x[_nlk], if n = multiple of k flan Time Expansxon min] — { 0, if n aé multiple of k‘ X02- ) . Convolution x[n] * yin] X(e’“’)Y (31w) Multiplication x[n]y[n] 5%; I X(ej9)Y(ej(“’-9‘))d0 1' 2w Differencing in Time x[n] - x[n — 1] (1 — e‘j”)X(ej“’) . " 1 . . . 1w Accumulation 15:2... x[k] 1 ~ g-iw X(e ) +aa . +7rX(ej°) Z 5(cu -— 2qu) k=—w jw Differentiation in Frequency nxfn} j d3: ) X(ej“’) = X‘(e'j‘”) (Rae{X(ej‘°)} = (Re{X(e'j“’)} Conjugate Symmetry for x[n] real j 9m{X(e1'“’)} = —5Im{X(e‘j“’)} Real Signals ' [X(ej"’)l = [X(e"j“’)i <3:X(ei“’) = —<}:X(e”j‘”) Symmetry for Real, Even fin] real an even ' X(ej“’) real and even Signals ‘ - Symmetry for Real. Odd x[n] real and odd X (elm) purely imaginary and Signals odd Even-9dr! Decomposition xe [n} = 8v{x[n]} [xfn] real] Giae{X(ej“')} 0f Real Signals scorn] = mum} rxrn] real} muss» Parseval’s Relation for Aperiodic Signals i lxinll” = 3— 1%“)?de 277 211' n=~w a duality relationship between the discrete-time Fourier transform and the continuous-time Fourier series. This relation is discussed in Section 5.7.2. 5.7.1 Duality in the Discrete-Time Fourier Series Since the Fourier series coefficients ak of a periodic signal xIn] are themselves a periodic sequence, we can expand the sequence ea. in a Fourier series. The duality property for discrete-time Fourier series implies that the Fourier series coefficients for the periodic se- ,.;_\~L ' " r TABLE 5.2 BASlC DISCRETE-TIME FOURIER TRANSFORM PAIRS Signal Fourier Series Coefficients (if periodic) +00 2 akejkanlmn 21rk ak k=(N) (a) me = —~ . l, k=m,m:N,mi2N,... emu ak = 0, otherwise (b) 529% irrational $ The signal is aperiodic (a) em = 3%”1 +0, 1 k = 1 t i N i cosmon 7rZoo—mo~2wz)+a<w+wo~2vrn} a = 2’ m’ m ' mm“ 1:“, 0, otherwise (b) ‘25"! irrational 2) The signal is aperiodic (a) we = 111 +0: 5‘7, k==r,riN,r'J:2N,... sinwon g—Z{5(m—mo—27rl)~5(w+wo—27Tl)} 0k = “517-. k= “’3 —r:*:N, “V3211... ’='°° 0, otherwise (b) :13 irrational z} The signal is aperiodic 1 2 EM 21) l. k=0,iN,i2N,... = 77' w " ’77 a = x[n] (”a k 0, otherwise Periodic square wave 1 _ ' 2 k/N N + — m] = 1’ M < N' ak = S_————————-'"K 7T. X ‘ 2)], kaéO, 1N, :2N,... 0’ N; < M s N/Z Nsm[27rk/2N] and ak=2NlN+l,k=0.iN,t2Nw. x[n + N] = x[n] +oo 1 Z 5[n— kN] ak = N forailk k=—aa a"u[n}, Iai < 1 m] 1, inl 5 N1 sin[m(N. + g.» _ O, Ini > N, sin(co/2) 1, 0 5 [ml 5 W smWn _ 1V. 3V_n = 77" —1rsmc(_”) X(w) 0, W<lwis7r — 0 < W < 7’ X(co) periodic with period 271' 5[n] —— 1 +°° u[n] ]_ Na + £207er — 277k) — 6[n — no] —— l n W— ——- (n + 1)a u[n], lat < l (1 _ ““ij + — l l £Z—K:__—l)—g)—a”u[n1, Ia! < I ~— 392 ._ game, Fbmflsv Tvams‘gmrm EQSMH’S , ihvo\ux‘h5 5;th Qawcfi'l’ok’g ; ijWO SMQ ‘Pund’xows MMHCEHQA {m ‘Hme ; SEMWNC) sm<wzt> F; \ +€3--~M<W >16? SM’IFCWZJ‘ TTFJC ”Pvt “V TH“ "H’s , , 54 WWL i} ‘ "Mme; . _:; . mmHVFK‘KKflc‘KK t A E Cobvo\0\l’t"K ”X +‘VKC ”:9 {k if" Q‘ZQ‘braL A ., C»? Tm? 3J3; 33WRW7$ m 3 {xx 0M9 ”34¢ Tri‘ ...
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