The delta function is a mathematical idealization of an impulse and one which allows us to handle DE’s with these types of inputs. For example, a sharp blow to a mass will cause its momentum to jump. An impulse is an input that causes a sudden jump in the system. Up to now all inputs to our systems have caused small changes in a small amount of time. The second important idea is the delta function. Nonetheless, as we will see, it arises naturally, and the Laplace transform will allow us to work easily with it. At first meeting this operation may seem a bit strange. The first is the convolution product of two functions. In the course of this unit, two important ideas will be introduced. We prove an extension of the Superposition Principle by Ambrosio-Gigli-Savaré in the context of a control problem. Then, once we solve for X(s) we can recover x(t). The Laplace transform converts a DE for the function x(t) into an algebraic equation for its Laplace transform X(s). If we think of ƒ(t) as an input signal, then the key fact is that its Laplace transform F(s) represents the same signal viewed in a different way. For example, the Laplace transform of ƒ(t) = cos(3t) is F( s) = s / (s² + 9). This operation transforms a given function to a new function in a different independent variable. Next we will study the Laplace transform. Now using Fourier series and the superposition principle we will be able to solve these equations with any periodic input. In Unit Two we learned how to solve a constant coefficient linear ODE with sinusoidal input. The following theorem says that any linear combination y Cy1(t) + Dy2(t) is also a solution to this differential equation for any constants C and D. We start with Fourier series, which are a way to write periodic functions as sums of sinusoids. Suppose that we have a linear homogenous second order differential equation d2y dt2 + p(t)dy dt + q(t)y 0 and that y y1(t) and y y2(t) are both solutions. In this unit we will learn two new ways to represent certain types of functions, and these will help us solve linear time invariant (LTI) DE’s with these functions as inputs. Modeled on the MIT mathlet Convolution Accumulation. Unit III: Fourier Series and Laplace TransformĬonvolution as a superposition of impulse responses.
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