(New page: Proving that the Continuous-Time Fourier Transform demonstrates linearity Property: F(a x(t) + b y(t)) = a X(jw) + b Y(jw) Derivation: F(a x(t) + b y(t)) = <math>\int\limits_{-\infty}...) |
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Property: | Property: | ||
| − | F(a x(t) + b y(t)) = a X(jw) + b Y(jw) | + | <math>F(a x(t) + b y(t)) = a X(jw) + b Y(jw) </math> |
Derivation: | Derivation: | ||
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| − | F(a x(t) + b y(t)) = | + | <math>F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}[a x(t)+b y(t)] e^{(-jwt)}dt </math> |
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| − | F(a x(t) + b y(t)) = a X(jw) + b Y(jw) (definition of linearity) | + | <math>F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}a x(t) e^{(jwt)}dt + \int\limits_{-\infty}^{\infty}b y(t) e^{(-jwt)}dt </math> |
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| + | <math>F(a x(t) + b y(t)) = a \int\limits_{-\infty}^{\infty}x(t) e^{(jwt)}dt + b \int\limits_{-\infty}^{\infty}y(t) e^{(-jwt)}dt </math> | ||
| + | |||
| + | |||
| + | <math>F(a x(t) + b y(t)) = a X(jw) + b Y(jw) </math> (definition of linearity) | ||
Latest revision as of 04:45, 9 July 2009
Proving that the Continuous-Time Fourier Transform demonstrates linearity
Property:
$ F(a x(t) + b y(t)) = a X(jw) + b Y(jw) $
Derivation:
$ F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}[a x(t)+b y(t)] e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}a x(t) e^{(jwt)}dt + \int\limits_{-\infty}^{\infty}b y(t) e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = a \int\limits_{-\infty}^{\infty}x(t) e^{(jwt)}dt + b \int\limits_{-\infty}^{\infty}y(t) e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = a X(jw) + b Y(jw) $ (definition of linearity)
