Review of Signal and System Theory

1. Review of Signal and System Theory

2. signal — a function (typically of time), typically corresponding to a voltage or a current. system — something that processes signals (or analyzes signals). System y(t) x(t)

3. Systems are typically described by differential equations that relate the input signal to the output signal. For example This equation relates the input x(t) to the output y(t). In this equation, given y(t), it is easy to find x(t). It is not so easy to find y(t) given x(t). Finding y(t) given x(t) is called solving the differential equation.

4. There are various methods of solving differential equations. We will restrict ourselves (primarily) to using Laplace transforms. The Laplace transformation is an operation on a function [e.g., x(t) or y(t)] which produces a new function [e.g., X(s) or Y(s)]. The independent variable of the Laplace transform is s.

5. The Laplace transforms of some common functions are shown in the table below:

6. The most important property of the Laplace transformation is that the Laplace transform of a derivative is equal to s times the Laplace transform of the original function. [There is also an additional term corresponding to the initial value of y(t), but we’ll ignore that for now.]

7. Using this derivative property, we can transform the previous differential equation: Solving for Y(s), we have

8. This last expression is a direct relationship between X(s) and Y(s) in the form of a ratio. This ratio is called the transfer functionfor the system and is denoted by H(s):

9. So, given X(s), we can easily find Y(s): To find y(t), we find the inverse Laplace transform of Y(s).

10. In summary, to solve the differential equation using Laplace transforms and transfer functions, we do the following:

11. Exercise:Suppose we input a step function x(t)= u(t) into our system with transfer function Use the method on the previous slide to find the output function y(t). (Hint: A partial fraction expansion needs to be done.)

12. Looking at the diagram, we may wonder whether or not there is a direct route from x(t) to y(t): ?

13. The answer may be found by applying the definition of the Laplace transformation: Applying this definition to X(s), Y(s) and H(s) we have

15. Since we must have

16. This operation is the direct route from x(t) to y(t). The name of this operation is convolution.The notation for the convolution operation is * :

17. In summary,

18. Exercise:Find the output of our system When the input is x(t)= u(t) using convolution. You must first find the inverseLaplace transform of H(s). (You should get the same answer.)

19. There is one very useful property of the convolution operation.

20. In other words, the convolution of a function with a delta function is the original function itself. The only “trick” in this derivation is the step where we go from to This step is true because the product of a delta function with any other function is zero except when the argument of the delta function is zero: t = t .

21. Exercise: Evaluate the following convolutions: