ECE351: Signals and Systems I. Thinh Nguyen
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1 ECE35: Sigals ad Sysms I Thih Nguy
2 FudamalsofSigalsadSysms x
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7 Fudamals of Sigals ad Sysms co.
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9 Fudamals of Sigals ad Sysms co. x x]
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11 Classificaio of sigals
12 Classificaio of sigals co. x] x x] =xt s =x =Ts,=,, 0,, x x]
13 Classificaio of sigals co.
14 Classificaio of sigals co. x x =x x =a+b = a b { x =a b x =a +b R{x} I{x} x
15 Classificaio of sigals co. CT sigal: x =x + T p, x T p f p =/T p ω p =πf p =π/t p DT sigal: x] =x + N p ], x] N p N p > 0 N p N p F p =/N p Ω p =πf p Ω p No: Codiio for DT sigals o b priodic: x] =xt s
16 Classificaio of sigals co.
17 Classificaio of sigals co.
18 Classificaio of sigals co. x] =A cosπf p + θ x] x =si 0π
19 Classificaio of sigals co. x = T s T s = 4π x = T s T s = 40
20 Classificaio of sigals co. x x T T x =x +x x =cosπ/,x =cosπ/3,x +x
21 { x ] =x + N ] x ] =x + N ] Classificaio of sigals co. x ]+x ] =x + N sum ]+x + N sum ],
22 Classificaio of sigals co. x E = P = lim T P = T x d T T/ T/ T/ T/ x d x d
23 Classificaio of sigals co. x] E = = P = lim N P = N N =0 x ] N + 0 <E< 0 <P < N = N x ] x ]
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25 Basic opraios o sigals y =cx y] =cx] c>: c<: y =x +x y] =x ] +x ] y =x x y] =x ]x ] y = d dx y = xτdτ { y =xa y] =x] a>: >0 a<: y =x y] =x ] { y =x 0 { y] =x 0 ] 0 > 0: 0 > 0: 0 < 0: 0 < 0: y =xa 0 y] =x 0 ]
26 Elmary sigals x =B a a, B a<0: a>0: a =0: x] =Br 0 <r<: r>: r =: x =A cosω + φ x] =A cosω + φ
27 Elmary sigals co. x] ΩN =πm m Ω= πm N x] x + N] =x] Ω N ΩN x ] =si π x ] = 3cos 4π 7 y] =x ]+x ]
28 Elmary sigals co. θ = cosθ+ si θ. B = A φ B ω = A φ ω = A ω+φ = A cosω + φ+asiω + φ { A cosω + φ =R{B ω } A siω + φ =I{B ω }
29 Elmary sigals co. x = A α siω + φ, α>0 x] = Br siω + φ, 0 <r<
30 Elmary sigals co. u = {, > 0 0, < 0 u] = {, 0 0, < 0 u0 u0]=
31 Elmary sigals co. u u] x = { A, , > 0.5 x u x =Au +/ Au / {, 0 9 x] = 0, o.w. x] u] x] =u] u 0]
32 Elmary sigals co.
33 Elmary sigals co. { 0, 0 δ = δd = δd = {, =0 δ] = 0, 0 x Δ Δ /Δ δ = lim Δx Δ Δ 0 δ =δ xδ 0d = x 0 δ = d d u u = δτdτ δa = aδ, a>0 x d d δ 0d = d d x = 0 x d d δ 0 d = d d x =0
34 Elmary sigals co.
35 Elmary sigals co. r = {, 0 0, < 0 = uτdτ = 0 dτ = u r] = {, 0 0, < 0
36 Sysm propris/characrisics H x H y x y
37 Sysm propris/characrisics co. y = a x a>0 y] = ρ x ] =0
38 Sysm propris/characrisics co. y = 5x+ y] = +5 m= 5 xm] xτdτ y] =x]+x ]
39 Sysm propris/characrisics co. y =Hx x 0 H y 0 x 0 ] H y 0 ]
40 Sysm propris/characrisics co. E:Issysmy]=r x]imivaria? y= a x : y]=u]x]:
41 Sysm propris/characrisics co. x H y x H y ax +bx H ay +by dx d H dy d H xτdτ xm] m= yτdτ H ym] m=
42 Sysm propris/characrisics co. y] =x 3] y =5x + 0 y = x
43 Basics of Malab = p p p = si pi 0.3 = p = 0 : 0.00 : ; % T s =0.00s x = si pi ; % x. si pi
44 Basics of Malab co. A =; +0.7] B = A B = 0.7].. A = ; 3 4] B =4 5; 6 7] C = A. B C = 4 0; 8 8] C = A B C = 6 9; 36 43]./ / A = 3] B = 4]C = A/B C = A./B C = ]. A =.44 B = A B = A =.44.73] B = A B = A. B = 3]
45 Tim-Domai Rprsaio of LTISysms Sysm H is a liar im-ivaria LTI sysm. How o aalyz a sysm. Giv a ipu, fid sysm oupu. Impuls rspos of a LTI sysm H:
46 Covoluio sum x = δ H y =h x] = δ] H y] =h] whr h CT ad h] DT ar h sysm impuls rsposs. h or h] complly characrizs a LTI sysm. By owig h or h], sysm oupu ca b obaid for a arbirary ipu sigal x or x]. How is y/y] rlad o x/x] ad h/h]?
47 Covoluio sum co. W will sar wih DT sysms, ad h aalyz CT sysms. Ay sigal x] ca b xprssd as a sum of im-shifd impulss as show graphically x slid
48 Covoluio sum co.
49 Covoluio sum: x] h] = Propris of covoluio x] h] =h] x] δ] h] =h] δ ] h] =h ] Covoluio sum co. = E: A sysm wih ipu-oupu rlaioship as x]h ] y] =x]+/x ] a Sysm impuls rspos? b Fid y] for x] =, =0 4, =, = 0, o.w.
50 Covoluio sum co.
51 Covoluio sum valuaio procdur L w ] =x]h ]. Th y] is xprssd as y] =x] h] = = w ]. Graph boh x] ad h]. Tim rvrsal h] h ] 3. Tim shif h ] by shifs h ] lf shif 4. For a spcific, form produc x]h ] 5. Sum all sampls of x]h ] y] = = x]h ]
52 Covoluio sum valuaio procdur co. Graphical illusraio of h covoluio sum: a LTI sysm wih impuls rspos h] ad ipu x]. Th dcomposiio of h ipu x] io a wighd sum of im-shifd impulss rsuls i a oupu y] giv by a wighd sum of im-shifd impuls rsposs x slid.
53 Covoluio sum valuaio procdur co.
54 Covoluio sum valuaio procdur co. E: x] =δ]+δ ] + δ ] is applid o a LTI sysm wih impuls rspos h] =4δ]+3δ ]+δ ] + δ 3]. Fid y]. Exrcis: y] =x] h] Vrify his h rsuls usig Malab. x = ]; h = 4 3 ]; y = covx, h;
55 Covoluio sum valuaio procdur co.
56 Covoluio sum valuaio procdur co.
57 Covoluio igral For CT cas. Rcall DT cas: x] = = x]δ ] No: Wighd SUM of im-shifd impulss. Similarly, x = xτδ τdτ No: Wighd suprposiio of im-shifd impulss. x H y
58 Covoluio igral co. { y = H = } xτδ τdτ xτh{δ τ} dτ liar opraors Thus, No: δ τ H h τ y =x h = xτh τdτ x h =h x δ h =h δ 0 h =h 0
59 Covoluio igral valuaio procdur. Graph x ad h. Tim rvrs hτ h τ 3. Tim shif h τ by h τ 4. For a spcific valu of, form produc xτh τ 5. Igra xτh τ = y = xτh τdτ
60 Covoluio igral valuaio procdur co. x =u u 3 h=u u y =??
61 Covoluio igral valuaio procdur co. E: RADAR rag masurm: RADAR-Radio Dcio Ad Ragig: { siwc, 0 T Tx: x = 0 0, o.w. Typically, h =αδ β, β>0 y =x h = xτh τdτ
62 Covoluio igral valuaio procdur co.
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66 IrcocioofLTIsysms Giv: H h.. h N H N = form a biggr sysm H Qusio: How is h rlad o h h N? Paralll Cocio
67 Ircocio of LTI sysms co. Disribuio propry of covoluio procss:
68 Cascad Cocio Ircocio of LTI sysms co.
69 Ircocio of LTI sysms co. L η = τ ν, dη = dτ for fixd ν. Th,
70 Ircocio of LTI sysms co. Associaiv Propry Sam for DT Commuaiv Propry Sam for DT E: Exampl., p 30 :Sfigur blow. Fid h impuls rspos h] of h ovrall sysm.
71 ECE35 5 Ircocio of LTI sysms co. h ] =u] h ] =u +] u] h 3 ] =δ ] h 4 ] =α u]
72 ECE35 6 Ircocio of LTI sysms co. E:AircocioofLTIsysmisdpicdihfigurblow.h ]= u+],h ]=δ],adh 3 ]=u ].Fidh impuls rspos h] of h ovrall sysm.
73 LTI SYSTEM PROPERTIES & IMPULSE RESPONSE Sysm propris Chapr Sabiliy BIBO Mmory dpd o curr ipu oly Causaliy dos o dpd o fuur ipus Liariy Tim ivariac LTI Mmorylss, LTI Mmorylss, Sabl, LTI For LTI sysms: h h ] Complly drmi Ipu-oupu bhaviors Thus, sabiliy, mmory, causaliy ar rlad o h/h].
74 a. If a LTI sysm is MEMORYLESS Proof: iff h ] h c ] c
75 DT : h ] b If a LTI sysm is CAUSAL: CT : h 0 for 0 Proof: 0 for 0
76 c If a LTI sysm is BIBO STABLE: Proof: - h ] h d
77 E Firs-ordr auorgrssiv sysm y ] y ] x ], wih h ] 0 for 0
78 STEP RESPONSE: ] ] y x y H x LTI Sp rspos: if h s y u x ] ] ] ] ] Proof:
79 u ] h ] u ] y ]??
80 DIFFERENTIAL & DIFFERENCE EQUATION REPRESENTATIONS OF LTI SYSTEMS 0 0 x d d b y d d a M N Ipu-oupu rlaio ca b dscribd as CT DT ] ] 0 0 x b y a M N Liar cosa-coffici diffrial quaio. Liar cosa-coffici diffrc quaio. ] ] y x y H x cosas b a, Ordr of diffrial/diffrc quaio: N,M Of M, ad ordr is dscribd usig N oly
81 E x y R L x d d y d d L y d d R y c x d y c y d d L Ry diffrias boh sids Dscrib h followig RLC circui by a diffrial quaio. C
82 E d-ordr diffrc quaio,, 4 /,,,, ] ] ] 4 ] ] 0 0 b b a a a x x y y y y] ca b valuad rcursivly:
83 Nd y-], y-]: Iiial codiios. x-] dpds o ipu applid. For xampl, if, h ], ] ], ] y y u x ] 4 0 0] y y
84 SOLVING DIFFERENTIAL AND DIFFEENCE EQUATIONS Jus a rviw, rahr ha i dph: Compl Soluio Soluio forms: Homogous Paricular CT y y h y p DT y ] y h Naural rspos Ipu sigal x or x ] 0 Dpds o iiial codiio! ] y p ] Forcd rspos Iiial rs Dpds o ipu sigal!
85 GENERAL CASE CT sysm: y h is h soluio of h homogous quaio: 0 0 y d d a h N Th homogous soluio is of h form: whr r i ar h N roos of h sysm s characrisic quaio 0 0 N a r N i r i h i c y No: c i : o b drmid lar so ha h compl soluios saisfy h iiial codiios.
86 DT sysm: i N i i h r c y ] 0 0 N N a r ] y h is h soluio of h homogous quaio: 0 ] 0 y a h N Th homogous soluio is of h form: whr r i ar h N roos of h sysm s characrisic quaio No: c i : sam as CT cas. CT ad DT characrisic quaios ar diffr.
87 Exampl Homogous quaio: y ] y ] 4 0 s x]=0 Characrisic quaio: N 0 a r N 0 Soluio of homogous quaio: 4 h y ] c c : o b drmid so ha h compl soluios saisfy h iiial codiios.
88 A paricular soluio is assumd idpd of h homogous soluio Usually obaid assumig ha oupu has h sam form as ipu sigals. Th form of h paricular soluio associad wih commo ipus ar summarizd i h followig abl. Coiuous im Discr im Ipu Paricular soluio Ipu Paricular soluio c c c +c c +c -a c -a c cos+ c cos++ c si+ cos+ c cos++ c si+ INPUT~ PARTICULAR FORM
89 Exampl Assum a paricular soluio of h form p y ] c p for ipu i h form of u]
90 Compl soluio: 0 for 4 ] ] ] c y y y p h c is obaid from h iiial codiio. 0 for, 4 ] 3 3 0] y c c y For a gral s -ordr diffrc quaio giv as Assumig iiial codiio y-] ad causal x] 0 ] ] ] bx ay y 0 0 for ] x
91 * Soluio: i y ] a y ] a bx i] i0 aural rspos forcd rspos CT Cas also s -ordr oly dy d ay bx a a ** Soluio: y y0 bx d 0 0 aural rspos forcd rspos x causal Compar * ad **!
92 FOURIER REPRESENTATION OF SIGNALS & LTI SYSTEMS CT: f cycl/scod Hz DT: F cycls/sampl rads/s rads/sampl Basic sigals as wighd suprposiio of impulss f F h LTI h x h x y x x ] ] ] ]* ] ] ] ] ] suprposiio wigh dlay LTI propry d h x h x y d x x h LTI *
93 Tim-domai wavform rprss how fas sigal chags. Sigals i rms diffr frqucy compos or wighd suprposiios of complx siusoids. CT: Xf or X DT: X] Why sigals rprsd as wighd suprposiios of complx siusoids? DT: x ] h ] y ] y ] h ] x ]
94 h H ] ~ rlad o h] No: a. is NOT a fucio of, oly a fucio of. Is calld h frqucy rspos. b. Sysm modifis h ampliud of ipu by. : magiud rspos. c. Sysm iroducs a phas lag. h boo uss H H H H H arg H H H H d h H H H y x H CT: wih
95 E Exampl R x + C y - Impuls rspos: RC h u RC Fid frqucy rspos.
96 Soluio: Magiud rspos: Phas rspos:
97 : igfucio of h LTI sysm ig valu H H Now, if h ipu o a LTI sysm is xprssd as a wighd sum of M complx siusoids:, M a x M H a y h RC RC RC RC 4 0 arg H H H
98 Fourir rprsaios of four classs of sigals Tim propry Coiuous im Discr im ] Priodic Fourir Sris FS x X ] T X ] x T 0 0, 0 0 d, Discr-Tim Fourir Sris DTFS N 0 X ], 0 0 x ] X ] N N 0 x ] T T: priod 0 N T: priod Nopriodic Fourir Trasform FT x X d X x d Discr-Tim Fourir Trasform DTFT x ] X X x ] d
99 DTFS: x] priodic wih priod N, fudamal frq. DTFS cofficis of x]: X]. Th N ] ] ] ] N N x N X X x Frq-domai rprsaio of x] x] ad X] ar a DTFS pair: ] ] 0 ; X x DTFS No: a. Eihr x] or X] provids a compl dscripio of h sigal. b. Th limis o sums of x] or X] may b chos diffrly from 0 o N-.
100 E Fid h frq-domai rprsaio of x] giv priod priod Soluio: N 5 Priod 0 N 5 Fudamal frqucy
101 Soluio:
102 E x] N Fid DTFS cofficis X] of priodic sigal x]
103 Soluio:
104 E x]cos3, Fid X].
105 E x ] si 3 8
106 E l DTFS of a Impuls rai: x ] ln Soluio: 0 N N E DTFS of a squar sigal: M Exampl N 8 x ], 0, M M M N M priod
107 Soluio:
108 E O priod of DTFS cofficis X ], 0 9 Drmi x ] assumig N 0 Soluio:
109 F.S. CT, priodic. x: fudamal priod T fudamal frqucy x X ] T X ] T 0 x 0 x ad X] ar a FS pair: 0 d * ; 0 x FS X ] 0 T FS cofficis X] ar a frq-domai rprsaio of x.
110 E x giv as x 4 Soluio:
111 d d T T d x T X T whr ] X] X]
112 E Drmi X] of Soluio: l x 4l T 4 0 4
113 FS cofficis by ispcio. E x 3cos 4 Soluio: Fid X] 0 x X ] X ] /
114 E xsi3si6. Fid X] Soluio: 0 x X ]
115 E Ivrs FS. Fid. 3], 3] ] ] ] 0 x X 4cos3 si ] X x Soluio:
116 E FS of a squar wav. x T T T 0 T 0 0 T 0 T T 0 T Priod is T, so 0 T Soluio:
117
118 DTFT D.T., opriodic ] ] ] DTFT X x x X d X x DTFT of sigal x], also Frq-domai rprsaio of x]. ] X x X
119 E x ] u ], X? Soluio:
120 E rcagular puls Soluio: x, ] 0, M M X?
121 E Ivrs DTFT of a rcagular spcrum, Exampl 3.9, p 34 Soluio:, X 0, W W X is dfid ovr, priodic i
122 E DTFT of ui impuls x ] ] Soluio: Wha abou ivrs DTFT of a ui impuls spcrum? X ], dfid oly o priod Soluio:
123 E x ], 0, 0 o. w. 9 X? Soluio:
124 E X cos, x]? Us ispcio! X
125 FT C.T., opriodic sigals d x X d X x X x FT
126 a E x u. Fid X X x d Soluio:
127 a a d d u X a a a 0 0
128 E Soluio: Rcagular puls: x, 0, T 0 T T 0 0
129
130 E Ivrs FT of a rcagular spcrum: X, 0, W W
131 E Ui impuls: x E Ivrs FT of a impuls spcrum: X
132 Tim propry C.T. D.T. ] PROPERTIES OF FOURIER REPRESENTATIONS Priodic, Fourir Sris FS x X ] T X ] x T 0 0, 0 0 d, N 0 X ], 0 0 T T: priod Discr-Tim Fourir Sris DTFS x ] N X ] N N 0 x ] Discr ] 0 N: priod Nopriodic, Four Trasform FT x X X x d d Discr-Tim Fourir Trasform DTFT x ] X d X x ] Coiuous Nopriodic, Priodic, Frq. propry
133 Liariy ad symmry z z z ] FT ax by Z FS; 0 ax by Z ] DTFT ax ] by ] Z ax by ax ] by ] ax by DTFS; 0 z ] ax ] by ] Z ] ax ] by ] E Exampl 3.30, p55: 3 z x y Fid h frqucy-domai rprsaio of z. Which yp of frq.-domai rprsaio? FT, FS, DTFT, DTFS?
134 Soluio:
135 Symmry: W will dvlop usig coiuous, o-priodic sigals. Rsuls for ohr cass may b obaid i a similar way. * a Assum x ral x x If x is ral X is couga symmric Proof:
136 If x is ral ad v X is ral. Proof:
137 If x is ral ad odd X is purly imagiary. Proof:
138 If x is purly imagiary Ral par of X has odd symmry Imagiary par of X has v symmry
139 Covoluio: Applid o o-priodic sigals. FT y x h Y X H Proof:
140 E L x h si b ipu o a sysm wih impuls rspos si. Fid h sysm oupu y Soluio:
141 E FT x X 4 si. Fid x. Soluio:
142 Th sam covoluio propris hold for discr-im, o-priodic sigals. y ] DTFT x ] h ] Y X H
143 Diffriaio ad igraio: Applicabl o coiuous fucios: im or frqucy or FT, ad DFTF d d FT Diffriaio i im: x X Proof:
144 d d E a Fid FT of u, a 0 Soluio:
145 E Fid x if X, 0, Soluio:
146 If x is priodic, frqucy-domai rprsaio is Fourir Sris FS: d d x 0 x X ] FS ; 0 X ] 0
147 Diffriaio i frqucy: Proof: FT d x X d
148 E A Gaussia puls is giv as : g. Soluio: Fid is FT.
149 Igraio: FT x d X X 0 E Drmi h Fourir rasform of u. Soluio:
150 E Fid x, giv Soluio: X
151 E?. u d d u u X u d d x FT FT FT X d d x FT X x d d FT
152 Tim ad frqucy shif Tim shif: 0 Z X Proof: No: Tim shif phas shif i frqucy domai. Phas shif is a liar fucio of Magiud spcrum dos o chag.
153 ] ] ] ] ; 0 0 ; 0 0 X x X x X x X x DTFS DTFT FS FT E Fid Z si si 0 0 T Z z T X x T x z T FT FT
154 E X 4. Fid x Soluio:
155 Frqucy shif: Proof: FT x X FT X x
156 No: Frqucy shif im sigal muliplid by a complx siusoid. Carrir modulaio. ] ] ] ] 0 ; 0 ; X x X x X x X x DTFS DTFT FS FT E Fid Z. 0,, 0 z
157 Soluio:
158 d d 3 E x u u. Fid X. Soluio:
159 Muliplicaio Z X Y z x y FT CT, o-priodic DT, o-priodic ] ] ] DTFT Z X Y z x y * Priodic covoluio: d Z X Z X * CT, priodic ] ] ] / ; Z X Y z x y T FS 0 / ; ] ] ] ] ] ] ] ] ] ] N m N DTFS m m Z X Z X Z X Y z x y * * DT, priodic
160 Scalig x a X / a a Proof:
161 E x 0 y 0 Fid Y Soluio:
162 S u s FT W ow Tim scalig: Tim shif: Diffriaio: 3 / 3 / / 3 3 d d v s z v Z s z FT FT FT Thus, 3 u s z v x 3 3 Thus,. 3 No: 3 u x u u / 0 0 X x X x a X a x d d FT FT a FT E 3 / if Fid d d X x
163 Parsval s rlaioship: : o - priodicsigal CT, of Ergy x d X X d d x X d d X x W d X x x d x x d x W x d X d x W x i frq.domai rgy i im domai Ergy b spcrum rgy : No :a X
164 T N N X d x x T X x N d X x ] FS: ] ] DTFS: ] DTFT : E / ] 0,, si ] si Drmi. si ] W d d X x E W W X W x W E W x W W x DTFT x
165 Tim-badwidh produc Comprssio i im domai xpasio i frqucy domai Badwidh: Th x of h sigal s sigifica cos. I is i gral a vagu dfiiio as sigifica is o mahmaically dfid. I pracic, dfiiios of badwidh iclud absolu badwidh x% badwidh firs-ull badwidh. If w dfi T d B w / x d : RMS duraio of a rgy sigal x d / X d : RMS badwidh, h X d T d B w /
166 Dualiy FT f F F FT f
167
168 E Fid X if x Dualiy W ow u f X f F F u f FT FT d d d X x Chc 0 0 E Fid x if Xu
169 x x u X u X FT 0 X X d x FT TFT ad FS do o say i hir ow class! ] ] DTFT ad FS: ] ] ] ] DTFS: ; / ; / ; D x X X x x X X x FS DTFT N N DTFS N DTFS
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