(New page: Linearity property 4.3.1 Aperiodic signal <===> Fourier transform <math>\mathcal{F}</math>{ax(t)+ by(t)} <=> aX(jw) + bY(jw) An example of this property can be used: x(t) = ...) |
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x(t) = 3u(t) + 2u(t-5) | x(t) = 3u(t) + 2u(t-5) | ||
| − | <math>\mathcal{F}</math>{x(t)} = <math>\mathcal{F}</math>{ | + | <math>\mathcal{F}</math>{x(t)} = <math>\mathcal{F}</math>{3u(t) + 2u(t-5)} |
= 3<math>\mathcal{F}</math>{u(t)} + 2<math>\mathcal{F}</math>{u(t-5)} | = 3<math>\mathcal{F}</math>{u(t)} + 2<math>\mathcal{F}</math>{u(t-5)} | ||
As you see the constants in front of those step functions are then placed in front of <math>\mathcal{F}</math> and this reflect on the linearity property so it hold true. | As you see the constants in front of those step functions are then placed in front of <math>\mathcal{F}</math> and this reflect on the linearity property so it hold true. | ||
Latest revision as of 12:57, 8 July 2009
Linearity property 4.3.1
Aperiodic signal <===> Fourier transform
$ \mathcal{F} ${ax(t)+ by(t)} <=> aX(jw) + bY(jw)
An example of this property can be used: x(t) = 3u(t) + 2u(t-5)
$ \mathcal{F} ${x(t)} = $ \mathcal{F} ${3u(t) + 2u(t-5)} = 3$ \mathcal{F} ${u(t)} + 2$ \mathcal{F} ${u(t-5)}
As you see the constants in front of those step functions are then placed in front of $ \mathcal{F} $ and this reflect on the linearity property so it hold true.
