数字信号处理与DSP器件：Linear Time-Invariant Systems

1、Linear Time-Invariant SystemsContent and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc. Copyright (C) 2005 Gner Arslan351M Digital Signal Processing2Linear-Time Invariant SystemSpecial importance for their mathematical tractabilityMo

2、st signal processing applications involve LTI systemsLTI system can be completely characterized by their impulse responseRepresent any inputFrom time invariance we arrive at convolutionT.n-khknCopyright (C) 2005 Gner Arslan351M Digital Signal Processing3LTI System Example-505012-505012-505012-505012

3、-505012-505024-50500.51-50500.51LTILTILTILTICopyright (C) 2005 Gner Arslan351M Digital Signal Processing4Properties of LTI SystemsConvolution is commutativeConvolution is distributivehnxnynxnhnynh1nxnynh2n+h1n+ h2nxnynCopyright (C) 2005 Gner Arslan351M Digital Signal Processing5Properties of LTI Sys

4、temsCascade connection of LTI systemsh1nxnh2nynh2nxnh1nynh1nh2nxnynCopyright (C) 2005 Gner Arslan351M Digital Signal Processing6Stable and Causal LTI SystemsAn LTI system is (BIBO) stable if and only if Impulse response is absolute summableLets write the output of the system asIf the input is bounde

5、d Then the output is bounded byThe output is bounded if the absolute sum is finiteAn LTI system is causal if and only ifCopyright (C) 2005 Gner Arslan351M Digital Signal Processing7Linear Constant-Coefficient Difference EquationsAn important class of LTI systems of the formThe output is not uniquely

6、 specified for a given inputThe initial conditions are requiredLinearity, time invariance, and causality depend on the initial conditionsIf initial conditions are assumed to be zero, system is linear, time invariant, and causalExampleMoving AverageDifference Equation RepresentationCopyright (C) 2005

7、 Gner Arslan351M Digital Signal Processing8Eigenfunctions of LTI SystemsComplex exponentials are eigenfunctions of LTI systems:Lets see what happens if we feed xn into an LTI system:The eigenvalue is called the frequency response of the systemH(ej) is a complex function of frequencySpecifies amplitu

8、de and phase change of the inputeigenfunctioneigenvalueCopyright (C) 2005 Gner Arslan351M Digital Signal Processing9Properties of LTI SystemsExample2-1 : Moving averageFilter(b, a, x) functiony(n) = b(1)*x(n) + b(2)*x(n-1) + . + b(nb+1)*x(n-nb) - a(2)*y(n-1) - . - a(na+1)*y(n-na)y1 = filter(num1,den

9、1,x);num1 = 0.3 -0.2 0.4;den1 = 1 0.9 0.8;Copyright (C) 2005 Gner Arslan351M Digital Signal Processing10The input-output description of this filtering operation in the -transform domain is a rational transfer function Filter(b, a, x) functionThe filter function is implemented as a direct form II tra

10、nsposed structure,y(n) = b(1)*x(n) + b(2)*x(n-1) + . + b(nb+1)*x(n-nb) - a(2)*y(n-1) - . - a(na+1)*y(n-na)Copyright (C) 2005 Gner Arslan351M Digital Signal Processing11(From Matlab help)Copyright (C) 2005 Gner Arslan351M Digital Signal Processing12Properties of LTI SystemsExample 2-3 LTI verificatio

11、nn = 0:40; D = 10; a = 3.0; b = -2;x = a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n);xd = zeros(1,D) x;num = 2.2403 2.4908 2.2403; den = 1 -0.4 0.75;ic = 0 0; % Set initial conditionsy = filter(num,den,x,ic);yd = filter(num,den,xd,ic);d = y - yd(1+D:41+D);Copyright (C) 2005 Gner Arslan351M Digital Signal Processing13EndCopyright (C) 2005 Gner Arslan351M Digital Signal Processing14Properties of LTI SystemsExample 2-3 LTI verifica