© Operation Function Fourier Transform
= Linearity aywy(t) + arw2(t) a\W,(f) + a22(f)
— Time delay w(t — Tu) W(f) e7/oT
—ứ a Scale change w(at) TT W lá) oom Conjugation w*(r) W*(-ƒ) “© Duality W(t) w(—f) oS Rea signal w(t) cos(w.t + 8) 1[£7W(ƒ — ƒ-) +e W(ƒ + #)] translation “<® (w(t) is real] Xe, Complex signal w(t) er! Wf — Sc) C frequency translation
Trang 3Ourier ee eee At cula bién doi tinh ch + ac PROPERTY SEQUENCE POURIER TRANSFORM ata] x(Q) ala] XQ) sla] x,«Q) Perwdhicity ata) X(Q + 22) = X(Q) Linearity @,5,[0]+a,1,[4] @,X,(Q) + 4,X,(Q)
Time shifting x[n — m, xia)
Frequency shifting oO rin] xiQg-Q,) Conjugation a*(n] X¥*(-Q) Time reversal s[-a] x(-Q) rla/m] lÍ n = kh Time scaling x, [4] 0 nO hen XimQ) Frequency differentiation nx[n] =x First difference a{a]—s{a—-1) (I—e"9)X(0) C xịt xX(01:0) + 1 x(Q) Accumulation 2 [4] i \Qiszx Convolution #,[n]* z.{=] *X,0Y.(8) I Multiplicatien #,In]r;[=} 2 x(@061,(0) Real sequence a{o) = x, (a) + x, [a] X(@! = A:Q) + #0) X(- Qì = X”(G@ì
Even component x In] RelX:(Q'] = A:G!
Trang 4Function Time Waveform w(t) Spectrum W(/) Rectangular n() T{Sa(ƒT)]
Triangular Az] T[Sa( xƒT)]
Unit step u(t) Ê [ph ree 15(f) + so +l, r>0 ] Signum sen(1) 4 {* <0 inf Constant I Kf) Impulse at f = fo &(t — to) e 2afts Sinc Sa(27rWr) = n( sp]
Phasor eilew +e} cl? ô(ƒ “ fo)
Sinusoid cos(w.t + ¢) jel* HF — f.) + fed" HF +f)
Trang 5
Tính chất của biên đôi Laplace
tườợn tượng g4 s4sznseseszsxssooecscososs-scvẳvrẳvvrvsv cTseceoeoe.-e-easaaas-nn.ana-nanananaaa-ananan-nnnn=eeesesxe ee aT « + £2242 4664664642S SSE OOOO CCT Ce reer
Property Time Domain Laplace Domain ROC
Linearity a)x\(t)+a2x2(t) a,X\(s)-+a2X2(s) Atleast R}NR2
Time-Domain Shifting x(t —t) e~ 0X (s) R
Laplace-Domain Shifting e™ x(t) X(s— 50) R+Re{so}
Time/Frequency-Domain Scaling (ar) bX (£) aR
Conjugation x*(t) X*(s*) R
Time-Domain Convolution xX) *x2(f) X1(s)X2(s) At least Ri NR2
Time-Domain Differentiation # x(t) sX (s) At least R
Laplace-Domain Differentiation —rx(?) #X(s) R
Time-Domain Integration Ƒ „x(tdrt 1X (s) At least RM {Re{s} > 0}
Property
Initial Value Theorem x(0”) = lim sX (s) Final Value Theorem lim x(t) = lim sX(s) Lan s—+0
4/13/21
Trang 6Mot so cap bién doi Laplace Pair x(t) X(s) ROC 1 ô() 1 Alls 2 u(t) 4 Re{s} >0 3 —u(—t) 1 Re{s} <0 4 t"u(t) in Re{s}>0 5 —t"u(—t) + Re{s}<0
6 e~“ u(t) _ Re{s} > —a
7 —e “u(—t) ta Re{s} < —a
8 f"e~“* #(t) G ram Re{s} > —a
9 —t"e~““u(—t) G Tam Re{s} < —a
10 [cos Wot} u(t) Stal Re{s} >0
Trang 7Tinh chat cua biên đôi z ee (1s) |
Property Time Domain Z Domain ROC
Linearity aixi(n)+aaxa(n) a ,X\(z)+a2X2(z) Atleast Ri NR2
Translation x(n— nọ) z "X(z) R except possbe add tondeleton ot 0
Z-Domain Scaling a"x(n) X(ar}z) la|R e/Monx(n) X (e~/%z) R Time Reversal x(—n) X (1/z) n— Upsampling († M)x(n) X(z") RUM Conjugation x*(n) X*(z*) R Convolution X1 *X2(n) Xj (z)X2(z) At least Ri NR> Z-Domain Diff x(n) —z£X{(z) R Differencing x(n)—x(n—1) = (1—z7!)X(z) At least RN |z| > 0 Accumulation ) a9) = X(z) Atleast RN |z| > 1 Property
Initial Value Theorem x(0) = lim X(z)
Final Value Theorem lim x(n) = lim {(z — 1)X(z)]
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