11.3.1 Convolution of functions

Problem: For the given functions \( \varphi \) and \( \psi \) : a) plot the graph of the functions \( \varphi \) and \( \psi \); b) calculate the convolution \( \varphi * \psi \) of the functions \( \varphi \) and \( \psi \); c) plot the graph of the convolution \( \varphi * \psi \). The function \( \varphi \) is given by the formula, the graph of the function \( \psi \) is broken-line, connecting the points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \) (outside the segment \( \left[x_{1}, x_{4}\right] \) the function is equal to zero). \[ \varphi(x)=\operatorname{rect} x=\eta\left(\frac{1}{2}-|x|\right)=\left\{\begin{array}{c} 1,-\frac{1}{2}\frac{1}{2} \end{array}\right. \] where \( \eta(t)=\left\{\begin{array}{l}0, t<0 \\ 1, t \geq 0\end{array}\right. \) is Heaviside step function \[ \begin{array}{l} A(-2,0), B(-1,2), C(1,2), D(2,0), \quad \psi: A B C D \text { (broken - line), } \\ \psi(x)=0 \text { when } x \notin[-2,2] . \end{array} \]