Understanding Convolution Integral in Time Domain Analysis of First-Order DEs, Lecture notes of Differential and Integral Calculus

Download Understanding Convolution Integral in Time Domain Analysis of First-Order DEs and more Lecture notes Differential and Integral Calculus in PDF only on Docsity! 2.1 The Convolution Integral So now we have examined several simple properties that the differential equation satisfies linearity and time-invariance. We have also seen that the complex exponential has the special property that it passes through changed only by a complex numer the differential equation. Also, we have discussed the roll of tansforms, as representing arbitrary inputs via the superpositions of complex exponentials. This discussion is often called a ”frequency domain analysis”. Frequency domain analysis studyies the outputs of linear and time-invariant systems via their response to complex exponentials. Now we turn our focus to a pure time domain analysis, understanding the response of the differential equation directly in terms of its time domain inputs. For this we explore the ”convolution integral”. We do this by solving the first-order differential equation directly using integrating factors. For this, examine the differential equation and introduce the integrating factor f(t) which has the property that it makes one side of the equation into a total differential. Define
 
 
    
 which implies

 
 
 This implies the integrating factor is
  , and using the boundary condition y(−∞) = 0 the total differential is solved giving        We have almost arrived at our convolution formula. For this introduce the unit step function, and the definition of the convolution formulation. The unit-step function is zero to the left of the origin, and 1 elsewhere:   1,   0 0,   0 Definition 2.2. Given time signals f(t), g(t), then their convolution is defined as
    
      Proposition 2.1. The output of this first order differential equation with input x(t) is given according to       To see this, simply use the property of the unit step to rewrite the solution of Eqn. 13 according to