OVERVIEW

We discussed in Chapter 4, "Solutions of Differential Equations," the method of convolution as a solution of differential equations. The problem with convolution is that it is a specific solution not a statement of system's behavior. It cannot be used as a model for the system. We need an expression that could describe the outcome of a system in terms of its dependent and independent variables, so design possibilities can be exploited. A plot of a frequency response is the most desirable picture in a digital or analog filter design, and for that, an algebraic form of a differential equation is required. We need to know what the output amplitude and phase change would be, given a set of input frequencies. The amplitude tells us how much gain or loss is impeded in the output and the phase tells us the fraction of a complete cycle elapsed from a specified reference point. In this chapter, we will solve these problems with the help of the Laplace Transform.

We discovered in the previous chapter that the exponentially decaying sinusoids, whether over-damped, under-damped, or critically damped are indeed solutions of linear differential equations, and the solution is only a matter of identifying the appropriate exponential coefficients. If the coefficients are real, the solution is simply an exponential decaying response, but if the coefficients are complex, we have a sinusoidal response with a damped frequency of oscillation. The damping factor determines the exponentially decaying amplitude. Similar to the Fourier Transform that identifies...