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How can I solve the convolution for this example in steps?

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How can I solve the convolution for this example in steps?

Guest Jun 17, 2017
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#1
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Any convolution is defined as
$$(f*g)(t)=\displaystyle\int^{\infty}_{-\infty}f(x)g(t-x)dx=\int^{\infty}_{-\infty}g(x)f(t-x)dx$$

Trying to do it... Never done this type of question before.

$$(f*h)(3)=\displaystyle\int^{\infty}_{-\infty}f(x)h(3-x)dx=\int^{\infty}_{\infty}h(x)f(3-x)dx$$

But f(x) is only defined in {-1<=x<=1|x$$\in\mathbb{Z}$$}

h(x) is only defined in {1<=x<=5|x$$\in\mathbb{Z}$$}

So Idk what does it mean by integrating f(x)h(3-x) from infinity to negative infinity... How on Mars could I do that?

MaxWong  Jun 18, 2017
#2
+168
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$$\text{The definition of the discrete convolution is:}$$

$$f*h[n]=\sum\limits^\infty_{m=-\infty}f[m]g[n-m].$$

$$\text{We require the solution for the case }n=3:$$

$$f*h[3]=\sum\limits^1_{m=-1}f[m]g[3-m],$$

$$\text{since } f[m]=0, \text{for: } m\neq-1,0,1.$$

$$\text{Our solution is:}$$

\begin{align*} f*h[3]&=f[-1]g[4]+f[0]g[3]+f[1]g[2]\\ &=aI_4+bI_3+cI_2. \end{align*}

Honga  Jun 18, 2017
#3
+168
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$$\text{An intuitive approach of solving discrete convolution is sliding an inverted version of }\\f\text{ over }h. \text{ For }n=0\text{ we have:}$$

$$\begin{array}{c| c c} m&-1&0&1&2&3&4&5\\\hline h&0&0&I_1&I_2&I_3&I_4&I_5\\ f&c&b&a&0&0&0&0\\\hline f*h[0]&0&0&aI_1&0&0&0&0 \end{array}$$

$$\text{the convolution product in this case is the product of each column.}\\ \text{Now we slide }f\text{ three places:}$$

$$\begin{array}{c| c c} m&-1&0&1&2&3&4&5\\\hline h&0&0&I_1&I_2&I_3&I_4&I_5\\ f&0&0&0&c&b&a&0\\\hline f*h[3]&0&0&0&cI_2&bI_3&aI_4&0 \end{array}$$

$$\text{and we have the same solution.}$$

Honga  Jun 18, 2017
edited by Honga  Jun 18, 2017
edited by Honga  Jun 18, 2017

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