邬震宇-信号与系统 9.pptVIP

* signal bilateral Laplace Transform Unilateral Laplace transform If f(t) is not causal signal, Fu(s)≠F(s). * Differentiation for unilateral Laplace Transform … … Many of the LT properties are satisfied for unilateral Laplace Transform, as displayed in Table 9.3,P.717, except differentiation in time and time-shifting. * Proof: Table 9.3 P717 * 9.7.1. Using LT to solve CT LTI responses By using LT convolution property An example. * Example9.7.1 A causal LTI system differential equation input signal ，determine the system response 。 Solution：Taking Laplace Transform on both sides * The Laplace transform of the System response Partial fraction expansion Left side signals Right side signals * The differentiation property of ULT is: * (2) The zero input response . (3) The zero state response . (1) The system total response . Example9.7.2 A casual LTI system differential equation The initial conditions are Please determine: * By substituting initial conditions Partial fraction expansion solution: (1) system response Taking Unilateral Laplace transform on both sides * Suppose the input f(t)=0， (2) The zero input response . * Suppose the initial states all are zeros (3) The zero state response . Note that * * Example9.8 An LTI stable system differential equation is: the input signals are （1） ；（2） ；（3） Please find the system responses. Solution: Being careful, the LT’s of the 2 signals do not exist! The system function is (stable system) * （1）For signal is in the ROC . The system response is: （2）signal is not in the ROC. The system response does not exist。 * （3） can be described as Obviously, complex is in the ROC，