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c23 - 18.03 Class 23 April 7 Step and delta Two additions...

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Step and delta. Two additions to your mathematical modeling toolkit. - Step functions [Heaviside] - Delta functions [Dirac] [1] it is Model of on/off process: a light turns on; first it is dark, then light. The basic model is the Heaviside unit step function u(t) = 0 for t < 0 1 for t > 0 Of course a light doesn't reach its steady state instantaneously; it takes a small amount of time. If we use a finer time scale, you can see what happens. It might move up smoothly; it might overshoot; it might move up in fits and starts as different elements come on line. At the longer time scale, we don't care about these details. Modeling the process by u(t) lets us just ignore those details. One of the irrelevant details is the light output at exactly t = 0. In fact as a matter of realism, you rarely care about the value of a function at any single point. What you do care about is the average value nearby that point; or, more precisely, you care about lim_{t-->a} f(t) The function is continuous if that limit IS the value at t=a. You will also often care about the values just to the left of t=a, or just to the right. These are captured by f(a-) = lim_{t-->a from below} f(t) f(a+) = lim_{t-->a from above} f(t) For example, u(0-) = 0 , u(0+) = 1. A function is continuous at t=a if f(a) = f(a-) = f(a+) . A good class of functions to work with is the "piecewise continuous" functions, which are continuous except at a scattering of points and such that all the one-sided limits exist. So u(t) is piecewise continuous but 1/t is not. The unit step function is a useful building block:-- u(t-a) turns on at t = a Q1: What is the equation for the function which agrees with f(t) between a and b ( a < b ) and is zero outside this window? (1) (u(t-b) - u(t-a)) f(t)

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c23 - 18.03 Class 23 April 7 Step and delta Two additions...

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