1. Overview¶
In engineering applications one often wants to simulate abrupt changes, such as turning on a light switch or kicking a can. At some level things actually behave continuously. But when they happen over a short time scale, it may be mathematically cleaner to treat the change as discontinuous without having a significant effect on the solution. (Compare this to how complicated Riemann sums are compared to integration by antiderivatives.)
We will encounter two types of abrupt changes: steps, which represent permanent changes in circumstances, and impulses, which represent momentary changes that have lasting effects.
1.1. Glossary¶
- delta function¶
Alternate name for an impulse.
- Heaviside function¶
Alternate name for the Heaviside function.
- impulse¶
Idealized window function of infinitesimal duration and infinite amplitude.
- impulse response¶
Solution of a linear IVP with forcing applied over an infinitesimally short window.
- Laplace transform¶
Integral transformation of a function of time to another function of an abstract variable \(s\).
- pole¶
Zero in the denominator of a rational function, corresponding to an exponential solution when appearing in a transform.
- superposition¶
Addition of partial solutions to a linear problem to get the complete solution.
- transfer function¶
Multiplier of the transform of a forcing function that produces the transform of a particular solution.
- unit step function¶
Function that is zero up to a specified time, then one thereafter.
- window function¶
Function that is one over a specific time interval and zero otherwise; expressible as the difference of two unit step functions.