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[[Category:ECE301 S11 Exam 3 more practice]] | [[Category:ECE301 S11 Exam 3 more practice]] | ||
| − | = | + | = Problem = |
| − | + | <math>\begin{align} x(t) &= u(t) - u(t-1) \\ y(t) &= u(t+2) - u(t-2) \\ z(t) &= x(t) * y(t) \end{align}</math> | |
| − | + | ||
| − | + | ||
| − | + | ||
| + | = Solution = | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | z(t) &= y(t) * x(t) \\ | ||
| + | &= \int_{-\infty}^{\infty}x(\tau) y(t-\tau)\mathrm{d}\tau \\ | ||
| + | &= \int_{-\infty}^{\infty}\left( u(\tau) - u(\tau-1) \right) \left( u(t-\tau+2) - u(t-\tau-2) \right)\mathrm{d}\tau \\ | ||
| + | &= \int_{0}^{1}\left( u(t-\tau+2) - u(t-\tau-2) \right)\mathrm{d}\tau \\ | ||
| + | &= \left[ (t-\tau+2)u(t-\tau+2)\right]\big|_0^1 - \left[ (t-\tau-2)u(t-\tau-2)\right]\big|_0^1 \\ | ||
| + | &= \left[ (t+1)u(t+1) - (t+2)u(t+2)\right] + \left[ -(t-3)u(t-3) + (t-2)u(t-2)\right] | ||
| + | \end{align} | ||
| + | </math> | ||
[[ ECE301 S11 Exam 3 more practice|Back to ECE301 S11 Exam 3 more practice]] | [[ ECE301 S11 Exam 3 more practice|Back to ECE301 S11 Exam 3 more practice]] | ||
Revision as of 07:17, 29 April 2011
Problem
$ \begin{align} x(t) &= u(t) - u(t-1) \\ y(t) &= u(t+2) - u(t-2) \\ z(t) &= x(t) * y(t) \end{align} $
Solution
$ \begin{align} z(t) &= y(t) * x(t) \\ &= \int_{-\infty}^{\infty}x(\tau) y(t-\tau)\mathrm{d}\tau \\ &= \int_{-\infty}^{\infty}\left( u(\tau) - u(\tau-1) \right) \left( u(t-\tau+2) - u(t-\tau-2) \right)\mathrm{d}\tau \\ &= \int_{0}^{1}\left( u(t-\tau+2) - u(t-\tau-2) \right)\mathrm{d}\tau \\ &= \left[ (t-\tau+2)u(t-\tau+2)\right]\big|_0^1 - \left[ (t-\tau-2)u(t-\tau-2)\right]\big|_0^1 \\ &= \left[ (t+1)u(t+1) - (t+2)u(t+2)\right] + \left[ -(t-3)u(t-3) + (t-2)u(t-2)\right] \end{align} $
