华工信号分析与处理课件俞祝良老师1

1、1 Chapter 2 Linear time-invariant systems (LTI) Yu Zhuliang College of Automation Science and Engineering South China University of Technology 2 Unit impulse and Unit Step Functions (In Chapter 1,discrete-time case) nExtremely important signal, unit impulse 3 Unit impulse and Unit Step Functions (In

2、 Chapter 1) nAnother important signal,unit step 4 Unit impulse and Unit Step Functions (In Chapter 1) nFirst difference nRunning sum 5 Unit impulse and Unit Step Functions (In Chapter 1) nAlternative expression nUnit impulse sequence is always used to sample the value of signals at n=0 or n=n0 6 Uni

3、t impulse and Unit Step Functions (In Chapter 1,continuous-time case) nunit step function 7 Unit impulse and Unit Step Functions (In Chapter 1,continuous-time case) nTher unit impulse function is related to the unit step function in a manner analogous to the relationship between discrete-time unit i

4、mpulse and step functions nRunning integral nFirst derivative ? 8 Unit impulse and Unit Step Functions (In Chapter 1,continuous-time case) nApproximation in definition 9 Unit impulse and Unit Step Functions (In Chapter 1,continuous-time case) nUnit impulse function, no duration but unit area. nScale

5、d unit impulse function nWe have 10 Unit impulse and Unit Step Functions (In Chapter 1,continuous-time case) 11 Unit impulse and Unit Step Functions (In Chapter 1,continuous-time case) nUnit impulse function for sampling nFor sufficiently small interval nTherefore 12 Unit impulse and Unit Step Funct

6、ions (In Chapter 1,continuous-time case) nMore generally 13 Introduciton on LTI nLinearity and time invariance play a fundamental role in signal and system analysis qMany physical processes possess these properties and thus can be modeled as linear time-invariant (LTI) systems qLTI systems can be an

7、alyzed in considerable detail, providing both insight into their properties and a set of powerful tools that form the core of signal and system analysis 14 Discrete-time system: Convolution sum nExpress signal in terms of impulse nIt is a weighted sum of shifted unit impulses. nThe response of a LTI

8、 system to the input signal is just the weighted sum of the shifted system response functions. 15 Discrete-time system: Convolution sum nThe response of system is nWhere 16 Discrete-time system: Convolution sum nSince nConvolution Sum 17 Convolution Sum Linearity: Superposition Impulse Response Time

9、 Invariant Signal Decomposition 18 Discrete-time system: Convolution sum Example 2.1 2.2 19 20 Discrete-time system: Convolution sum nMore on convolution sum (Example 2.3) 21 Discrete-time system: Convolution sum nMore on convolution sum (Example 2.3) 22 Convolution Sum Linearity: Superposition Impu

10、lse Response Time Invariant Signal Decomposition 23 nSignal representation in terms of impulse Continuous-time system: Convolution integral 24 Continuous-time system: Convolution integral nSignal representation in terms of impulse 25 Continuous-time system: Convolution integral nResponse signal repr

11、esentation 26 Continuous-time system: Convolution integral nCalculation of convolution integral nExample 2.6 27 Convolution integral and Sum (Properties) nThe characteristics of an LTI system are completely determined by its impulse response. nIt is important to emphasize that this property holds in

12、 general only for LTI system. 28 Convolution integral and Sum (Properties) nThe commutative property nThe distributive property 29 Convolution integral and Sum (Properties) nThe distributive property 30 Convolution integral and Sum (Properties) nThe Associative property 31 Convolution integral and S

13、um (Properties) nLTI system without memory nInvertbility of LTI system 32 Convolution integral and Sum (Properties) nCausality of LTI system nCausality of an LTI system is equivalent to its impulse response being a causal signal 33 Convolution integral and Sum (Properties) 34 Convolution integral an

14、d Sum (Properties) nStability for LTI system nIf the impulse response is absolutely sumable, that is, if then the system is stable. nDeduction: 35 Casual LTI system Described by Differential and Difference Equations nLinear constant-coefficient Differential Equations nLinear constant-coefficient Dif

15、ference Equations 36 Causual LTI system Described by Differential and Difference Equations nParticular solution + Homogeneous solution (Example 2.14) nFinite impulse response (FIR) system nInfinite impulse response (IIR) system 37 Causual LTI system Described by Differential and Difference Equations

16、 38 Causual LTI system Described by Differential and Difference Equations 39 Causual LTI system Described by Differential and Difference Equations 40 Homework nEx 2.1, nEx 2.3, nEx 2.6, nEx 2.8, nEx 2.11, nEx 2.22, nEx 2.23, nEx 2.43, 41 Convolution Sum (Review) Linearity: Superposition Impulse Resp

17、onse Time Invariant Signal Decomposition 42 Convolution Integral (Review) Linearity: Superposition Impulse Response Time Invariant Signal Decomposition 43 Comparison Linearity: Superposition Impulse Response Time Invariant Signal Decomposition Linearity: Superposition Impulse Response Time Invariant Signal Decomposition 44 Convolution integ