Math Problems and solutions
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11.3.1 Convolution of functions
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11.3.2 Convolution of functions
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11.3.3 Convolution of functions
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11.3.4 Convolution of functions
11.3.5 Convolution of functions
11.3.6 Convolution of functions
11.3.7 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 \[ \begin{array}{l} A(-2,0), B(-1,2), C(1,2), D(2,0), \quad \psi: A B C D \text { (broken - line), } \\ \psi(x)=0 \text { when } x \notin[-2,2] . \end{array} \]](/storage/uploads/task_mls/445/en/11.3.1.png){kind=link}
![Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes a value, equal to zero, when \( x \notin[-1,4] \), and when \( x \in[-1,4] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A(-1,0), B(1,2), C(2,-2), D(4,-2), \\ g(x)=\left\{\begin{array}{cc} 0, & x<0 \\ 1, & 0 \leq x<1 \\ 0, & x \geq 1 \end{array}\right. \end{array} \]](/storage/uploads/task_mls/446/en/11.3.2.png){kind=link}
![Problem: For piecewise constant functions \( f(x) \) and \( g(x) \) of the form \[ \begin{array}{c} f(x)=\left\{\begin{array}{cc} 0, & x<-1 \\ 1, & -1 \leq x<0 \\ -2, & 0 \leq x<2 \\ 0, & x \geq 2 \end{array}\right. \\ g(x)=\left\{\begin{array}{cc} 0, & x<0 \\ 1, & 0 \leq x<1 \\ -2, & 1 \leq x<3 \\ 0, & x \geq 3 \end{array}\right. \end{array} \] find the cross-covariance and cross-correlation functions.](/storage/uploads/task_mls/447/en/11.3.3.png){kind=link}
![Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes a value, equal to zero, when \( x \notin\left[x_{1} ; x_{4}\right] \), and when \( x \in\left[x_{1} ; x_{4}\right] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A\left(x_{1} ; 0\right), \quad B\left(x_{2} ; a\right), \quad C\left(x_{3} ; b\right), \quad D\left(x_{4} ; b\right) . \\ x_{1}=-2, \quad x_{2}=1, \quad x_{3}=3, \quad x_{4}=4, \quad a=2, \quad b=-1 . \end{array} \] The function \( g(x) \) has the form \[ g(x)=\left\{\begin{array}{ll} 0, & x<0 \\ 1, & 0 \leq x<1 \\ 0, & x \geq 0 \end{array}\right. \]](/storage/uploads/task_mls/926/en/11.3.4.png){kind=link}
![Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes the value, equal to zero, when \( x \notin\left[x_{1} ; x_{4}\right] \), and when \( x \in\left[x_{1} ; x_{4}\right] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A\left(x_{1} ; a\right), \quad B\left(x_{2} ; a\right), \quad C\left(x_{3} ; b\right), \quad D\left(x_{4} ; 0\right) . \\ x_{1}=-1, \quad x_{2}=1, \quad x_{3}=4, \quad x_{4}=6, \quad a=-1, \quad b=2 . \end{array} \] The function \( g(x) \) has the form \[ g(x)=\left\{\begin{array}{ll} 0, & x<0 ; \\ 1, & 0 \leq x<1 ; \\ 0, & x \geq 0 . \end{array}\right. \]](/storage/uploads/task_mls/928/en/11.3.6.png){kind=link}
