Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan V. Oppenheim and Alan S. Willsky, 1997 Outline NTUEE-SS2-LTI-2 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions h[n] h(t)
Chapter 1 and Chapter 2 NTUEE-SS2-LTI-3 h[n] h(t) Signals Systems In Section 1.5, We Introduced Unit Impulse Functions NTUEE-SS2-LTI-4 Sample by Unit Impulse For x[n] More generally,
DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-5 Representation of DT Signals by Impulses: DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-6 Representation of DT Signals by Impulses: More generally, The sifting property of the DT unit impulse x[n] = a superposition of scaled versions of shifted unit impulses δ[n-k]
DT LTI Systems: Convolution Sum DT Unit Impulse Response & Convolution Sum: NTUEE-SS2-LTI-7 DT LTI Systems: Convolution Sum DT Unit Impulse Response & Convolution Sum: NTUEE-SS2-LTI-8
DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-9 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-10
DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-11 If the linear system (L) is also time-invariant (TI) L & TI L & TI Then, DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-12 Hence, for an LTI system, Known as the convolution of x[n] & h[n] Referred as the convolution sum or superposition sum Symbolically,
DT LTI Systems: Convolution Sum Example 2.1m: LTI NTUEE-SS2-LTI-13 LTI DT LTI Systems: Convolution Sum Example 2.2m: NTUEE-SS2-LTI-14 LTI
DT LTI Systems: Convolution Sum Example 2.2m: NTUEE-SS2-LTI-15 DT LTI Systems: Convolution Sum Example 2.1o: LTI NTUEE-SS2-LTI-16
DT LTI Systems: Convolution Sum Example 2.2o: NTUEE-SS2-LTI-17 LTI DT LTI Systems: Convolution Sum Example 2.2o: NTUEE-SS2-LTI-18
DT LTI Systems: Convolution Sum Example 2.2o: NTUEE-SS2-LTI-19 DT LTI Systems: Convolution Sum Example 2.3: NTUEE-SS2-LTI-20
DT LTI Systems: Convolution Sum Example 2.3: NTUEE-SS2-LTI-21 DT LTI Systems: Convolution Sum Example 2.3: NTUEE-SS2-LTI-22
DT LTI Systems: Convolution Sum Example 2.4: NTUEE-SS2-LTI-23 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-24
DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-25 LTI DT LTI Systems: Convolution Sum Example 2.5: NTUEE-SS2-LTI-26 LTI
DT LTI Systems: Matrix Representation of Convolution Sum NTUEE-SS2-LTI-27 For an LTI system, The convolution of finite-duration discrete-time signals may be expressed as the product of a matrix and a vector. DT LTI Systems: Matrix Representation of Convolution Sum NTUEE-SS2-LTI-28
Outline NTUEE-SS2-LTI-29 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-30 Representation of CT Signals by Impulses:
CT LTI Systems: Convolution Integral Representation of CT Signals by Impulses: NTUEE-SS2-LTI-31 CT LTI Systems: Convolution Integral Graphical interpretation: NTUEE-SS2-LTI-32
CT LTI Systems: Convolution Integral CT Impulse Response & Convolution Integral: NTUEE-SS2-LTI-33 CT LTI Systems: Convolution Integral CT Impulse Response & Convolution Integral: NTUEE-SS2-LTI-34
CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-35 Linear System CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-36 CT Unit Impulse Response & Convolution Integral:
CT LTI Systems: Convolution Integral If the linear system (L) is also time-invariant (TI) Then, NTUEE-SS2-LTI-37 LTI Hence, for an LTI system, Known as the convolution of x(t) & h(t) Referred as the convolution integral or the superposition integral Symbolically, CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-38 Example 2.6:
CT LTI Systems: Convolution Integral Example 2.7: NTUEE-SS2-LTI-39 CT LTI Systems: Convolution Integral Example 2.8: NTUEE-SS2-LTI-40
Convolution Sum and Integral NTUEE-SS2-LTI-41 Signal and System: LTI: h(t) Outline NTUEE-SS2-LTI-42 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions
Properties of LTI Systems Convolution Sum & Integral of LTI Systems: NTUEE-SS2-LTI-43 h[n] h(t) Properties of LTI Systems NTUEE-SS2-LTI-44 Properties of LTI Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality h[n] h(t) 7. Stability 8. Unit step response
Properties of LTI Systems Commutative Property: NTUEE-SS2-LTI-45 Properties of LTI Systems Distributive Property: NTUEE-SS2-LTI-46 h 1 [n] h 1 [n]+h 2 [n] + h 2 [n]
Properties of LTI Systems Distributive Property: NTUEE-SS2-LTI-47 h[n] + h[n] + h[n] Properties of LTI Systems Example 2.10 NTUEE-SS2-LTI-48
Properties of LTI Systems Associative Property: NTUEE-SS2-LTI-49 Properties of LTI Systems NTUEE-SS2-LTI-50
In Section 1.6.1: Basic System Properties NTUEE-SS2-LTI-51 Systems with or without memory Memoryless systems Output depends only on the input at that same time (resistor) Systems with memory (accumulator) (delay) Properties of LTI Systems NTUEE-SS2-LTI-52 Memoryless: A DT LTI system is memoryless if The impulse response: From the convolution sum: Similarly, for CT LTI system:
In Section 1.6.2: Basic System Properties NTUEE-SS2-LTI-53 Invertibility & Inverse Systems Invertible systems Distinct inputs lead to distinct outputs Properties of LTI Systems NTUEE-SS2-LTI-54 Invertibility: h1(t) h2(t) Identity System δ(t)
Properties of LTI Systems Example 2.11: Pure time shift NTUEE-SS2-LTI-55 h1(t) h2(t) Properties of LTI Systems Example 2.12 NTUEE-SS2-LTI-56 h1[n] h2[n]
Properties of LTI Systems Causality: NTUEE-SS2-LTI-57 The output of a causal system depends only on the present and past values of the input to the system Specifically, y[n] must not depend on x[k], for k > n It implies that the system is initially rest A CT LTI system is causal if Properties of LTI Systems Convolution Sum & Integral NTUEE-SS2-LTI-58
In Section 1.6.4: Basic System Properties NTUEE-SS2-LTI-59 Stability Stable systems Small inputs lead to responses that do not diverge Every bounded input excites a bounded output Bounded-input bounded-output stable (BIBO stable) For all x(t) < a, then y(t) < b, for all t Balance in a bank account? Properties of LTI Systems NTUEE-SS2-LTI-60 Stability: A system is stable if every bounded input produces a bounded output Stable LTI
Properties of LTI Systems NTUEE-SS2-LTI-61 Stability: For CT LTI stable system: Stable LTI Properties of LTI Systems NTUEE-SS2-LTI-62 Example 2.13: Pure time shift
Properties of LTI Systems NTUEE-SS2-LTI-63 Example 2.13: Accumulator Properties of LTI Systems NTUEE-SS2-LTI-64 Unit Impulse and Step Responses: For an LTI system, its unit impulse response is: DT LTI CT LTI Its unit step response is: DT LTI CT LTI
Outline NTUEE-SS2-LTI-65 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions Singularity Functions NTUEE-SS2-LTI-66 Singularity Functions CT unit impulse function is one of singularity functions
Singularity Functions Singularity Functions NTUEE-SS2-LTI-67 Singularity Functions Example 2.16 NTUEE-SS2-LTI-68
Singularity Functions Example 2.16 NTUEE-SS2-LTI-69 Singularity Functions Defining the Unit Impulse through Convolution: NTUEE-SS2-LTI-70
Singularity Functions Defining the Unit Impulse through Convolution: NTUEE-SS2-LTI-71 Singularity Functions Defining the Unit Impulse through Convolution: NTUEE-SS2-LTI-72
Singularity Functions Unit Doublets of Derivative Operation: NTUEE-SS2-LTI-73 Singularity Functions Unit Doublets of Derivative Operation: NTUEE-SS2-LTI-74
Singularity Functions Unit Doublets of Integration Operation: NTUEE-SS2-LTI-75 Singularity Functions Unit Doublets of Integration Operation: NTUEE-SS2-LTI-76
Singularity Functions Unit Doublets of Integration Operation: NTUEE-SS2-LTI-77 Singularity Functions In Summary NTUEE-SS2-LTI-78
Outline NTUEE-SS2-LTI-79 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-80 Linear Constant-Coefficient Differential Equations e.x., RC circuit RC Circuit Provide an implicit specification of the system You have learned how to solve the equation in Diff Eqn
Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-81 Linear Constant-Coefficient Differential Equations For a general CT LTI system, with N-th order, CT LTI Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-82 Linear Constant-Coefficient Difference Equations For a general DT LTI system, with N-th order, DT LTI
Causal LTI Systems by Difference & Differential Equations Recursive Equation: NTUEE-SS2-LTI-83 Causal LTI Systems by Difference & Differential Equations For example, Example 2.15 LTI NTUEE-SS2-LTI-84
Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-85 Nonrecursive Equation: When N = 0, Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-86 Block Diagram Representations:
Causal LTI Systems by Difference & Differential Equations Block Diagram Representations: NTUEE-SS2-LTI-87 Causal LTI Systems by Difference & Differential Equations Block Diagram Representations: NTUEE-SS2-LTI-88
Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-89 Block Diagram Representations: Example 9.30 (pp.711) Example 10.30 (pp.786) Chapter 2: Linear Time-Invariant Systems Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral NTUEE-SS2-LTI-90 Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions
Flowchart Signals & Systems (Chap 1) LTI & Convolution (Chap 2) NTUEE-SS2-LTI-91 Bounded/Convergent FS (Chap 3) Periodic CT DT FT Aperiodic CT (Chap 4) DT (Chap 5) Unbounded/Non-convergent LT zt CT DT (Chap 9) (Chap 10) Time-Frequency (Chap 6) Communication (Chap 8) CT-DT (Chap 7) Control (Chap 11)