信號與系統 Signals and Systems
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1 Sprig 2 信號與系統 Sigals ad Systems Chapter SS- Sigals ad Systems Feg-Li Lia NTU-EE Feb Ju Figures ad images used i these lecture otes are adopted from Sigals & Systems by Ala V. Oppeheim ad Ala S. Willsky, 997
2 Outlie NTUEE-SS-SS-2 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties
3 Sigals & Systems NTUEE-SS-SS-3 Sigals & Systems: Is about usig mathematical techiques to help describe ad aalyze systems which process sigals Sigals are variables that carry iformatio Systems process iput sigals to produce output sigals Iput Sigal System Output Sigal
4 Sigals & Systems: Sigals NTUEE-SS-SS-4 Discrete-Time Sigals: The weekly Dow-Joes stock market idex
5 Sigals & Systems: Sigals NTUEE-SS-SS-5 Discrete-Time Sigals: A moochromatic picture
6 Sigals & Systems: Sigals NTUEE-SS-SS-6 Cotiuous-Time Sigals: Source voltage & capacity voltage i a simple RC circuit Recordig of a speech sigal
7 Sigals & Systems: Sigals NTUEE-SS-SS-7 Graphical Represetatios of Sigals: Cotiuous-time sigals x(t) or x c (t) Discrete-time sigals x[] or x d []
8 Sigals & Systems: Sigals NTUEE-SS-SS-8 Eergy & Power of a resistor: Istataeous power Total eergy over a fiite time iterval Average power over a fiite time iterval
9 Sigals & Systems: Sigals NTUEE-SS-SS-9 Sigal Eergy & Power: Total eergy over a fiite time iterval Time-averaged power over a fiite time iterval
10 Sigals & Systems: Sigals NTUEE-SS-SS- Sigal Eergy & Power: Total eergy over a ifiite time iterval Time-averaged power over a ifiite time iterval
11 Sigals & Systems: Sigals NTUEE-SS-SS- Three Classes of Sigals: Fiite total eergy & zero average power Fiite average power & ifiite total eergy Ifiite average power & ifiite total eergy
12 Outlie NTUEE-SS-SS-2 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties
13 Sigals & Systems: Trasformatio of the Idepedet Variable NTUEE-SS-SS-3 Time Shift:
14 Sigals & Systems: Trasformatio NTUEE-SS-SS-4 Time Reversal:
15 Sigals & Systems: Trasformatio NTUEE-SS-SS-5 Time Scalig:
16 Sigals & Systems: Trasformatio x ( t ) x ( - t + ) NTUEE-SS-SS-6
17 Sigals & Systems: Trasformatio x ( t ) x ( - t + ) NTUEE-SS-SS-7
18 Sigals & Systems: Trasformatio x ( t ) x ( 3/2 t + ) NTUEE-SS-SS-8
19 Sigals & Systems: Trasformatio NTUEE-SS-SS-9 x ( t ) x ( a t - b ) a < : liearly stretched a > : liearly compressed a < : time reversal b > : delayed time shift b < : advaced time shift Problems: P.2 for CT P.22 for DT
20 Sigals & Systems: Trasformatio NTUEE-SS-SS-2 CT & DT Periodic Sigals:
21 Sigals & Systems: Trasformatio NTUEE-SS-SS-2 Periodic Sigals: A periodic sigal is uchaged by a time shift of T or N They are also periodic with period 2T, 3T, 4T, 2N, 3N, 4N, Tor N is called the fudametal period deoted as T or N
22 Sigals & Systems: Trasformatio NTUEE-SS-SS-22 Periodic sigal? Problems: P.25 for CT P.26 for DT
23 Sigals & Systems: Trasformatio NTUEE-SS-SS-23 Eve & odd sigals:
24 Sigals & Systems: Trasformatio NTUEE-SS-SS-24 Eve-odd decompositio of a sigal: Ay sigal ca be broke ito a sum of oe eve sigal ad oe odd sigal
25 Sigals & Systems: Trasformatio NTUEE-SS-SS-25 Eve-odd decompositio of a DT sigal: Problems: P.23 for CT P.24 for DT
26 Sigals & Systems: Trasformatio NTUEE-SS-SS-26 Uiqueess of eve-odd decompositio:
27 Outlie NTUEE-SS-SS-27 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties
28 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-28 Magitude & Phase Represetatio: Euler s relatio:
29 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-29 CT Complex Expoetial Sigals: where C & a are, i geeral, complex umbers
30 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-3 Real expoetial sigals: If C & a are real
31 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-3 Periodic complex expoetial sigals: If a is purely imagiary It is periodic Because let The Hece
32 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-32 Periodic siusoidal sigals:
33 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-33 Period & Frequecy:
34 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-34 Euler s relatio:
35 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-35 Total eergy & average power: Problem: P.3
36 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-36 Harmoically related periodic expoetials
37 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-37 Geeral complex expoetial sigals:
38 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-38 DT complex expoetial sigal or sequece: where C & a are, i geeral, complex umbers
39 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-39 Real expoetial sigals: If C & a are real
40 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-4 DT Complex Expoetial & Siusoidal Sigals If b is purely imagiary (or a = )
41 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-4 Euler s relatio: Ad,
42 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-42 Geeral complex expoetial sigals:
43 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-43 Periodicity properties of DT complex expoetials: The sigal with frequecy ω is idetical to the sigals with frequecies Oly eed to cosider a frequecy iterval of legth 2π Usually use The low frequecies are located at The high frequecies are located at
44 Sigals & Systems: Expoetial & Siusoidal Sigals CT expoetial sigals NTUEE-SS-SS-44 - cos(*pi*t) - - cos(/8*pi*t) - - cos(/4*pi*t) - - cos(/2*pi*t) - - cos(*pi*t) - - cos(3/2*pi*t) - - cos(7/4*pi*t) - - cos(5/8*pi*t) - - cos(2*pi*t) -
45 Sigals & Systems: Expoetial & Siusoidal Sigals CT expoetial sigals NTUEE-SS-SS-45 - cos(2*pi*t) - - cos(7/8*pi*t) - - cos(9/4*pi*t) - - cos(5/2*pi*t) - - cos(3*pi*t) - - cos(7/2*pi*t) - - cos(5/4*pi*t) - - cos(3/8*pi*t) - - cos(4*pi*t) -
46 Sigals & Systems: Expoetial & Siusoidal Sigals CT & DT expoetial sigals cos(*pi*t) - t cos(/8*pi*t) - cos(/4*pi*t) cos(*pi*) - cos(/8*pi*) - cos(/4*pi*) cos(/2*pi*t) - cos(*pi*t) - cos(3/2*pi*t) NTUEE-SS-SS-46 cos(/2*pi*) cos(*pi*) - cos(3/2*pi*) -
47 Sigals & Systems: Expoetial & Siusoidal Sigals CT & DT expoetial sigals cos(7/4*pi*t) - cos(5/8*pi*t) - cos(2*pi*t) cos(7/4*pi*) - cos(5/8*pi*) - cos(2*pi*) cos(7/8*pi*t) - cos(9/4*pi*t) - cos(5/2*pi*t) NTUEE-SS-SS-47 cos(7/8*pi*) - cos(9/4*pi*) - cos(5/2*pi*) -
48 Sigals & Systems: Expoetial & Siusoidal Sigals CT & DT expoetial sigals cos(3*pi*t) cos(3*pi*) cos(3/8*pi*t) cos(7/2*pi*t) cos(7/2*pi*) cos(4*pi*t) cos(5/4*pi*t) cos(5/4*pi*) NTUEE-SS-SS-48 cos(3/8*pi*) cos(4*pi*) -
49 Sigals & Systems: Expoetial & Siusoidal Sigals Periodicity properties of DT expoetial sigals cos(*pi*) cos(/2*pi*) cos(7/4*pi*) cos(/8*pi*) cos(*pi*) cos(5/8*pi*) NTUEE-SS-SS-49 - cos(/4*pi*) cos(3/2*pi*) cos(2*pi*)
50 Sigals & Systems: Expoetial & Siusoidal Sigals Periodicity properties of DT expoetial sigals cos(2*pi*) cos(5/2*pi*) cos(5/4*pi*) cos(7/8*pi*) cos(3*pi*) cos(3/8*pi*) NTUEE-SS-SS-5 - cos(9/4*pi*) cos(7/2*pi*) cos(4*pi*)
51 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-5 Periodicity properties of DT expoetial sigals Periodicity of N > Problem: P.35 That is, Hece,
52 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-52 Periodicity properties of DT expoetial sigals
53 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-53 Harmoically related periodic expoetials Oly N distict periodic expoetials i the set
54 Sigals & Systems: Expoetial & Siusoidal Sigals NTUEE-SS-SS-54 Compariso of CT & DT sigals: CT DT DT CT Problem: P.36
55 Outlie NTUEE-SS-SS-55 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties
56 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-56 DT Uit Impulse & Uit Step Sequeces Uit impulse (or uit sample) Uit step
57 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-57 Relatioship Betwee Impulse & Step First differece Ruig sum
58 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-58 Relatioship Betwee Impulse & Step Alteratively,
59 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-59 Sample by Uit Impulse For x[] More geerally,
60 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-6 CT Uit Impulse & Uit Step Fuctios Uit step fuctio Uit impulse fuctio
61 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-6 Relatioship Betwee Impulse & Step Ruig itegral First derivative But, u(t) is discotiuous at t =, hece, ot differetiable Use approximatio
62 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-62 Relatioship Betwee Impulse & Step Use approximatio
63 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-63 Relatioship Betwee Impulse & Step
64 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-64 Sample by Uit Impulse Fuctio For x(t) More geerally,
65 Sigals & Systems: Uit Impulse & Uit Step Fuctios NTUEE-SS-SS-65 Example.7:
66 Outlie NTUEE-SS-SS-66 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties
67 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-67 Physical Systems & Mathematical Descriptios Examples of physical systems are sigal processig, commuicatios, electromechaical motors, automotive vehicles, chemical-processig plats A system ca be viewed as a process i which iput sigals are trasformed by the system or cause the system to respod i some way, resultig i other sigals or outputs
68 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-68 Simple examples of CT systems RC circuit
69 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-69 Simple examples of CT systems Automobile
70 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-7 Simple examples of DT systems Balace i a bak accout
71 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-7 Simple examples of DT systems Digital simulatio of differetial equatio
72 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-72 Itercoectios of Systems: Audio System: Microphoe or Tape Bass Treble Equalizer Speaker
73 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-73 Itercoectios of Systems Series or cascade itercoectio of 2 systems > e.x. radio receiver + amplifier Parallel itercoectio of 2 systems > e.x. audio system with several microphoes or speakers
74 Sigals & Systems: CT & DT Systems NTUEE-SS-SS-74 Itercoectios of Systems Series-parallel itercoectio Feedback itercoectio > e.x. cruise cotrol, electrical circuit
75 Chapter : Sigals ad Systems NTUEE-SS-SS-75 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties Systems with or without memory Ivertibility & Iverse Systems Causality Stability Time Ivariace Liearity
76 Sigals & Systems: Basic System Properties NTUEE-SS-SS-76 Systems with or without memory Memoryless systems Output depeds oly o the iput at that same time (resistor) Systems with memory (accumulator) (delay)
77 Sigals & Systems: Basic System Properties NTUEE-SS-SS-77 Ivertibility & Iverse Systems Ivertible systems Distict iputs lead to distict outputs
78 Sigals & Systems: Basic System Properties NTUEE-SS-SS-78 Causality Causal systems Output depeds oly o iput at preset time & i the past No-causal systems
79 Sigals & Systems: Basic System Properties NTUEE-SS-SS-79 Stability Stable systems Small iputs lead to resposes that do ot diverge Every bouded iput excites a bouded output Bouded-iput bouded-output stable (BIBO stable) For all x(t) < a, the y(t) < b, for all t Balace i a bak accout?
80 Sigals & Systems: Basic System Properties NTUEE-SS-SS-8 Example.3: Stability
81 Sigals & Systems: Basic System Properties NTUEE-SS-SS-8 Time Ivariace Time-ivariat systems Behavior & characteristics of system are fixed over time A time shift i the iput sigal results i a idetical time shift i the output sigal
82 Sigals & Systems: Basic System Properties NTUEE-SS-SS-82 Time Ivariace Example of time-ivariat system (Example.4)
83 Sigals & Systems: Basic System Properties NTUEE-SS-SS-83 Time Ivariace Example of time-varyig system (Example.6)
84 Sigals & Systems: Basic System Properties NTUEE-SS-SS-84 Liearity Liear systems If a iput cosists of the weighted sum of several sigals, the the output is the superpositio of the resposes of the system to each of those sigals
85 Sigals & Systems: Basic System Properties NTUEE-SS-SS-85 Liearity Liear systems I geeral, OR,
86 Sigals & Systems: Basic System Properties NTUEE-SS-SS-86 Liearity Example.7:
87 Sigals & Systems: Basic System Properties NTUEE-SS-SS-87 Liearity Example.8:
88 Sigals & Systems: Basic System Properties NTUEE-SS-SS-88 Liearity Example.2:
89 Sigals & Systems: Basic System Properties NTUEE-SS-SS-89 Liearity Example.2: However,
90 Chapter : Sigals ad Systems NTUEE-SS-SS-9 Itroductio Cotiuous-Time & Discrete-Time Sigals Trasformatios of the Idepedet Variable Time Shift Time Reversal Time Scalig Periodic Sigals Eve & Odd Sigals Expoetial & Siusoidal Sigals The Uit Impulse & Uit Step Fuctios Cotiuous-Time & Discrete-Time Systems Basic System Properties Systems with or without memory Ivertibility & Iverse Systems Causality Stability Time Ivariace Liearity
91 Flowchart Sigals & Systems (Chap ) LTI & Covolutio (Chap 2) NTUEE-SS-SS-9 Bouded/Coverget Periodic Aperiodic FS (Chap 3) CT DT FT CT (Chap 4) DT (Chap 5) Ubouded/No-coverget LT zt CT DT (Chap 9) (Chap ) Time-Frequecy (Chap 6) Commuicatio (Chap 8) CT-DT (Chap 7) Cotrol (Chap )
92 Sigals & Systems: Expoetial & Siusoidal Sigals Problem.26 (Page 6) NTUEE-SS-SS-92 L = 25; = -L:L; x = cos( pi/8 * (.^2) );.5.5 figure() stem(, x, 'o' ); hold o; axis([-l L -.5.5])
93 Sigals & Systems: Expoetial & Siusoidal Sigals Problem.27 (Page 62) NTUEE-SS-SS-93
94 Sigals & Systems: Expoetial & Siusoidal Sigals Problem.27 (Page 62) NTUEE-SS-SS-94
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