2601 Unit Step and Unit Impulse Response Introduction
In real life, we often do not know the parameters of a system (e.g. the spring constant, the mass, and the damping constant, in a spring-mass-dashpot system). We may not even know the order of the system. For example, there may be many interconnected springs or diodes. Instead, we often learn about a system by watching how it responds to various input signals.
In this session we will study the response of a linear time invariant (LTI) system from rest initial conditions to two standard and very simple signals: the unit impulse and the unit step function . Reasonably enough we will call these responses the unit impulse response and the unit step response.
The theory of the convolution integral studied in the next session will give us a method of dertemining the response of a system to any input once we know its unit impulse response.
Because both and are discontinuous at we will have to be careful with our definition of initial conditions. The most sensible mathematical and physical way to do this is to define our initial conditions at . As input an impulse causes a jump when it is applied. This means that the conditions at will be different than those at . To distinguish these two cases we will use the terms pre-initial conditions (at ) and post-initial conditions (at ). We will be able to state precisely the effect of a unit impulse on these conditions.