I was unable to solve a convolution question.Question is attached herewith.
I don't know how to initiate for solving the problem and what is the final expression that prove the convolution of a top-hat function with itself is the triangle function.
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$\begingroup$ Hi! you may initiate by writing down the convolution integral, and the piecewise defintion of your "top-hat" function.. $\endgroup$ – Fat32 Jun 10 '17 at 10:15
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When you slide a rectangular signal over another rectangular signal and take the area under the curve, i.e., perform convolution, at first there is no overlap. Then, the overlap gradually starts increasing in a linear fashion. When the two rectangular signals are exactly on top of each other, that is the point of maximum overlap. After than, there is a gradual decrease until there is no overlap. The resulting convolution signal is then a triangular signal.
Update: A rectangular signal is given by \begin{equation*} \prod (t) = \begin{cases} 1 & 0 \le t \le 1 \\ 0 & otherwise \end{cases} \end{equation*} Its convolution is given by $$z(t) = \int _{\tau=-\infty}^{\infty} \prod(\tau) * \prod(t-\tau) d\tau$$ which turns out to be \begin{equation} z(t) = \begin{cases} 0 & -\infty \le t \le 0 \\ t & 0 \le t \le 1 \\ 2 - t & 1 \le t \le 2 \\ 0 & 2 \le t \le \infty \end{cases} = \Lambda(t) \end{equation}
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I also don't know what is top hat function. But, I know convolving rectangular function with itself will result in triangular function.
The below image illustrates more, what hidden meaning in the equation you need to prove.
