Signals a nd Systems

1. Signals and Systems

2. Chapter 4 the Laplace Transform

3. In the preceding chapters, we have seen that the tools of Fourier analysis are extremely useful in the study of many problems of practical importance involving signals and LTI systems. • The continuous-time FT provide us with a representation for signals as linear combinations of complex exponentials of the form est with s=jw. • However,many consequences of FT apply as well for arbitrary values of s and not only those values are purely imaginary. • This observation leads to a generation of the continuous-time FT, known as the Laplace transform.

4. Introduction • The Laplace transform (拉普拉斯变换) is a generalization of the continuous-time Fourier transform. • The Laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form with s=σ+jω. • With Laplace transform, we expand the application in which Fourier analysis cannot be used.

5. For some signals which have not Fourier transforms, if we preprocess them by multiplying with a real exponential signal , then they may have Fourier transforms. Let s =σ+jω, unilateral Laplace transform bilateral Laplace transform

6. Relationship between Fourier and Laplace Transform if σ= 0 So:

7. LT of • LT of the Laplace Transform for some typical signals

8. LT of Im s-plane Re –a

9. Im s-plane Re –a Example:Consider the signal (a is real) For convergence, we require that Re{s +a} < 0, or Re{s} < –a

10. Im s-plane -2 -1 Re Exercise 1 Consider the signal Exercise 2 Find the LT of Exercise 3 Find the LT of

11. Homework • Page250 #4-1 (1)(8)(11)