signals and systems -...
TRANSCRIPT
Spring 2013
信號與系統Signals and Systems
Chapter SS-2Linear Time-Invariant Systems
Feng-Li LianNTU-EE
Feb13 – Jun13
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-2Outline
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant SystemsCausal Linear Time-Invariant Systems
Described by Differential & Difference EquationsSingularity Functions
h(t)h[n]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-3Chapter 1 and Chapter 2
h(t)h[n]
SystemsSignals
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-4In Section 1.5, We Introduced Unit Impulse Functions
Sample by Unit Impulse For x[n] More generally,
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-5DT LTI Systems: Convolution Sum
Representation of DT Signals by Impulses:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-6DT LTI Systems: Convolution Sum
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]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-7DT LTI Systems: Convolution Sum
DT Unit Impulse Response & Convolution Sum:
Linear System
Linear System
Linear System
Linear System
Linear System
…
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-8DT LTI Systems: Convolution Sum
DT Unit Impulse Response & Convolution Sum:
Linear System
Linear System
Linear System
Linear System
Linear System…
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-11DT LTI Systems: Convolution Sum
If the linear system (L) is also time-invariant (TI)
Then,
Linear System
Linear System
L & TI
L & TI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-12DT LTI Systems: Convolution Sum
Hence, for an LTI system,
• Known as the convolution of x[n] & h[n]• Referred as the convolution sum or superposition sum
Symbolically,
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-13DT LTI Systems: Convolution Sum
Example 2.1m: LTILTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-14DT LTI Systems: Convolution Sum
Example 2.2m: LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-16DT LTI Systems: Convolution Sum
Example 2.1o: LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-17DT LTI Systems: Convolution Sum
Example 2.2o: LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013
Example 2.2o:
DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-18
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-26DT LTI Systems: Convolution Sum
Example 2.5: LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-27DT LTI Systems: Matrix Representation of Convolution Sum
For an LTI system,
The convolution of finite-duration discrete-time signals may be expressed as the product of a matrix and a vector.
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-28DT LTI Systems: Matrix Representation of Convolution Sum
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-29Outline
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant SystemsCausal Linear Time-Invariant Systems
Described by Differential & Difference EquationsSingularity Functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-30CT LTI Systems: Convolution Integral
Representation of CT Signals by Impulses:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-31CT LTI Systems: Convolution Integral
Representation of CT Signals by Impulses:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-32CT LTI Systems: Convolution Integral
Graphical interpretation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-33CT LTI Systems: Convolution Integral
CT Impulse Response & Convolution Integral:
Linear System
Linear System
Linear System
Linear System
Linear System
…
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-34CT LTI Systems: Convolution Integral
CT Impulse Response & Convolution Integral:
Linear System
Linear System
Linear System
Linear System
Linear System…
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-35CT LTI Systems: Convolution Integral
LinearSystem
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-36CT LTI Systems: Convolution Integral
CT Unit Impulse Response & Convolution Integral:
Linear System
Linear System
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-37CT LTI Systems: Convolution Integral
If the linear system (L) is also time-invariant (TI)• Then,
Hence, for an LTI system,
• Known as the convolution of x(t) & h(t)• Referred as the convolution integral
or the superposition integral
Symbolically,
LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-38CT LTI Systems: Convolution Integral
Example 2.6:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-39CT LTI Systems: Convolution Integral
Example 2.7:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-40CT LTI Systems: Convolution Integral
Example 2.8:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-41Convolution Sum and Integral
Signal and System:
LTI: h(t)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-42Outline
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant SystemsCausal Linear Time-Invariant Systems
Described by Differential & Difference EquationsSingularity Functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-43Properties of LTI Systems
Convolution Sum & Integral of LTI Systems:
h(t)
h[n]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-44Properties of LTI Systems
Properties of LTI 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
h[n]
h(t)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-45Properties of LTI Systems
Commutative Property:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-46Properties of LTI Systems
Distributive Property:
h2[n]
+
h1[n]
h1[n]+h2[n]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-47Properties of LTI Systems
Distributive Property:
h[n]+
h[n]
+
h[n]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-49Properties of LTI Systems
Associative Property:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-51In Section 1.6.1: Basic System Properties
Systems with or without memory Memoryless systems
• Output depends only on the input at that same time
Systems with memory
(resistor)
(delay)
(accumulator)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-52Properties of LTI Systems
Memoryless:• A DT LTI system is memoryless if
• The impulse response:
• From the convolution sum:
• Similarly, for CT LTI system:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-53In Section 1.6.2: Basic System Properties
Invertibility & Inverse Systems Invertible systems
• Distinct inputs lead to distinct outputs
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-54Properties of LTI Systems
Invertibility:
h1(t)
Identity System δ(t)
h2(t)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-55Properties of LTI Systems
Example 2.11: Pure time shift
h1(t) h2(t)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-56Properties of LTI Systems
Example 2.12
h1[n] h2[n]
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-57Properties of LTI Systems
Causality:
• 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
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-58Properties of LTI Systems
Convolution Sum & Integral
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-59In Section 1.6.4: Basic System Properties
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?
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-60Properties of LTI Systems
Stability:• A system is stable
if every bounded input produces a bounded output
Stable LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-61Properties of LTI Systems
Stability:• For CT LTI stable system:
Stable LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-62Properties of LTI Systems
Example 2.13: Pure time shift
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-63Properties of LTI Systems
Example 2.13: Accumulator
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-64Properties of LTI Systems
Unit Impulse and Step Responses:• For an LTI system, its unit impulse response is:
• Its unit step response is:
DT LTI CT LTI
DT LTI CT LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-65Outline
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant Systems1. Commutative property2. Distributive property3. Associative property4. With or without memory5. Invertibility6. Causality7. Stability8. Unit step response
Causal Linear Time-Invariant Systems Described by Differential & Difference Equations
Singularity Functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-66Singularity Functions
Singularity Functions• CT unit impulse function is one of singularity functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-70Singularity Functions
Defining the Unit Impulse through Convolution:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-71Singularity Functions
Defining the Unit Impulse through Convolution:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-72Singularity Functions
Defining the Unit Impulse through Convolution:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-73Singularity Functions
Unit Doublets of Derivative Operation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-74Singularity Functions
Unit Doublets of Derivative Operation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-75Singularity Functions
Unit Doublets of Integration Operation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-76Singularity Functions
Unit Doublets of Integration Operation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-77Singularity Functions
Unit Doublets of Integration Operation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-79Outline
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant Systems1. Commutative property2. Distributive property3. Associative property4. With or without memory5. Invertibility6. Causality7. Stability8. Unit step response
Causal Linear Time-Invariant Systems Described by Differential & Difference Equations
Singularity Functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-80Causal LTI Systems by Difference & Differential Equations
Linear Constant-Coefficient Differential Equations• e.x., RC circuit
• Provide an implicit specification of the system• You have learned how to solve the equation in Diff Eqn
RC Circuit
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-81Causal LTI Systems by Difference & Differential Equations
Linear Constant-Coefficient Differential Equations• For a general CT LTI system, with N-th order,
CT LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-82Causal LTI Systems by Difference & Differential Equations
Linear Constant-Coefficient Difference Equations• For a general DT LTI system, with N-th order,
DT LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-83Causal LTI Systems by Difference & Differential Equations
Recursive Equation:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-84Causal LTI Systems by Difference & Differential Equations
For example, Example 2.15 LTI
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-85Causal LTI Systems by Difference & Differential Equations
Nonrecursive Equation:• When N = 0,
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-86Causal LTI Systems by Difference & Differential Equations
Block Diagram Representations:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-87Causal LTI Systems by Difference & Differential Equations
Block Diagram Representations:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-88Causal LTI Systems by Difference & Differential Equations
Block Diagram Representations:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-89Causal LTI Systems by Difference & Differential Equations
Block Diagram Representations:
Example 9.30 (pp.711) Example 10.30 (pp.786)
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-90Chapter 2: Linear Time-Invariant Systems
Discrete-Time Linear Time-Invariant Systems• The convolution sum
Continuous-Time Linear Time-Invariant Systems• The convolution integral
Properties of Linear Time-Invariant Systems1. Commutative property2. Distributive property3. Associative property4. With or without memory5. Invertibility6. Causality7. Stability8. Unit step response
Causal Linear Time-Invariant Systems Described by Differential & Difference Equations
Singularity Functions
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS2-LTI-91
Periodic Aperiodic
FS CTDT
(Chap 3)FT
DT
CT (Chap 4)
(Chap 5)
Bounded/Convergent
LTzT DT
Time-Frequency
(Chap 9)
(Chap 10)
Unbounded/Non-convergent
CT
CT-DT
Communication
Control
(Chap 6)
(Chap 7)
(Chap 8)
(Chap 11)
Signals & Systems LTI & Convolution(Chap 1) (Chap 2)
Flowchart