Chapter2 LinearTimeinvariantSystem 信号与系统教学课件

2LinearTime InvariantSystems 2 1Discrete timeLTIsystem Theconvolutionsum 2 1 1TheRepresentationofDiscrete timeSignalsinTermsofImpulses 2 LinearTime InvariantSystems Ifx n u n then 2LinearTime InvariantSystems 2LinearTime InvariantSystems 2 1 2TheDiscrete timeUnitImpulseResponseandtheConvolutionSumRepresentationofLTISystems 1 UnitImpulse Sample Response UnitImpulseResponse h n 2LinearTime InvariantSystems 2 ConvolutionSumofLTISystem Solution Question n h n n k h n k x k n k x k h n k 2LinearTime InvariantSystems 2LinearTime InvariantSystems 2LinearTime InvariantSystems ConvolutionSum So ory n x n h n 3 CalculationofConvolutionSum TimeInversal h k h k TimeShift h k h n k Multiplication x k h n k Summing Example2 12 22 32 42 5 2LinearTime InvariantSystems 2 2Continuous timeLTIsystem Theconvolutionintegral 2 2 1TheRepresentationofContinuous timeSignalsinTermsofImpulses Define Wehavetheexpression Therefore 2LinearTime InvariantSystems 2LinearTime InvariantSystems or 2LinearTime InvariantSystems 2 2 2TheContinuous timeUnitimpulseResponseandtheconvolutionIntegralRepresentationofLTISystems 1 UnitImpulseResponse 2 TheConvolutionofLTISystem 2LinearTime InvariantSystems A Becauseof So wecanget ConvolutionIntegral ory t x t h t 2LinearTime InvariantSystems B ory t x t h t ConvolutionIntegral 2LinearTime InvariantSystems 2LinearTime InvariantSystems 3 ComputationofConvolutionIntegral TimeInversal h h TimeShift h h t Multiplication x h t Integrating Example2 62 8 2LinearTime InvariantSystems 2 3PropertiesofLinearTimeInvariantSystem Convolutionformula 2LinearTime InvariantSystems 2 3 1TheCommutativeProperty Discretetime x n h n h n x n Continuoustime x t h t h t x t 2LinearTime InvariantSystems 2 3 2TheDistributiveProperty Discretetime x n h1 n h2 n x n h1 n x n h2 n Continuoustime x t h1 t h2 t x t h1 t x t h2 t Example2 10 2LinearTime InvariantSystems 2 3 3TheAssociativeProperty Discretetime x n h1 n h2 n x n h1 n h2 n Continuoustime x t h1 t h2 t x t h1 t h2 t 2LinearTime InvariantSystems 2 3 4LTIsystemwithandwithoutMemory Memorylesssystem Discretetime y n kx n h n k n Continuoustime y t kx t h t k t Implythat x t t x t andx n n x n 2LinearTime InvariantSystems 2 3 5InvertibilityofLTIsystem Originalsystem h t Reversesystem h1 t So fortheinvertiblesystem h t h1 t t orh n h1 n n Example2 112 12 2LinearTime InvariantSystems 2 3 6CausalityforLTIsystem Discretetimesystemsatisfythecondition h n 0forn 0Continuoustimesystemsatisfythecondition h t 0fort 0 2LinearTime InvariantSystems 2 3 7StabilityforLTIsystem Definitionofstability Everyboundedinputproducesaboundedoutput Discretetimesystem If x n B theconditionfor y n Ais 2LinearTime InvariantSystems Continuoustimesystem If x t B theconditionfor y t Ais Example2 13 2LinearTime InvariantSystems 2 3 8TheUnitStepResponseofLTIsystem Discretetimesystem Continuoustimesystem 2LinearTime InvariantSystems 2 4CausalLTISystemsDescribedbyDifferentialandDifferenceEquation Discretetimesystem DifferentialEquationContinuoustimesystem DifferenceEquation 2LinearTime InvariantSystems 2 4 1LinearConstant CoefficientDifferentialEquation AgeneralNth orderlinearconstant coefficientdifferentialequation or andinitialcondition y t0 y t0 y N 1 t0 Nvalues 2LinearTime InvariantSystems 2 4 2LinearConstant CoefficientDifferenceEquation AgeneralNth orderlinearconstant coefficientdifferenceequation or andinitialcondition y 0 y 1 y N 1 Nvalues Example2 15 2LinearTime InvariantSystems 2 4 3BlockDiagramRepresentationsofFirst orderSystemsDescribedbyDifferentialandDifferenceEquation 1 DicretetimesystemBasicelements A AnadderB MultiplicationbyacoefficientC Anunitdelay 2LinearTime InvariantSystems Basicelements 2LinearTime InvariantSystems Example y n ay n 1 bx n 2LinearTime InvariantSystems 2 ContinuoustimesystemBasicelements A AnadderB MultiplicationbyacoefficientC An differentiator integrator 2LinearTime InvariantSystems Basicelements 2LinearTime InvariantSystems Example y t ay t bx t 2Lin