(New page: '''''Linearity'' - Property of Continuous Time Fourier Transform''' Linearity States that the FT of {a*x(t)+b*y(t)} will be equal to {a*X(w)+b*Y(w)} if the signal is truly linear. '''Gen...) |
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'''''Linearity'' - Property of Continuous Time Fourier Transform''' | '''''Linearity'' - Property of Continuous Time Fourier Transform''' | ||
| − | Linearity States that the | + | Linearity States that the <math>\mathcal{F}</math> of {a*x(t)+b*y(t)} will be equal to {a*X(w)+b*Y(w)} if the signal is truly linear. |
| − | '''General Derivation:''' <math> | + | '''General Derivation:''' <math>\mathcal{F}=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt</math> |
| − | * If z(t) = {a*x(t)+b*y(t)}, then the | + | * If z(t) = {a*x(t)+b*y(t)}, then the <math>\mathcal{F}</math> is <math>Z(w)=\int\limits_{-\infty}^{\infty}(a*x(t)+b*y(t))e^{(-\jmath wt)}dt</math> |
** <math>Z(w)=\int\limits_{-\infty}^{\infty}a*x(t)e^{(-\jmath wt)}dt+\int\limits_{-\infty}^{\infty}b*y(t)e^{(-\jmath wt)}dt</math> | ** <math>Z(w)=\int\limits_{-\infty}^{\infty}a*x(t)e^{(-\jmath wt)}dt+\int\limits_{-\infty}^{\infty}b*y(t)e^{(-\jmath wt)}dt</math> | ||
*** <math>Z(w)=a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt+b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt</math> | *** <math>Z(w)=a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt+b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt</math> | ||
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'''Example:''' | '''Example:''' | ||
| + | Signal <math>x(t)=1, 0<t\le1; 2, 1<t\le2; 0, else=u(t)+u(t-1)-2u(t-2)</math> | ||
| + | * <math>\mathcal{F} of x(t)=X(w)=\mathcal{F}{[u(t)+u(t-1)-2u(t-2)]}</math> | ||
| + | ** <math>X(w)=\mathcal{F}{[u(t)]}+\mathcal{F}{[u(t-1)]}-2\mathcal{F}{[u(t-2)]}</math> | ||
| + | *** <math>X(w)=\mathcal{F}{[u(t)]}+e^{(-\jmath w)}\mathcal{F}{[u(t)]}-2e^{(-\jmath 2w)}\mathcal{F}{[u(t)]}</math> by Time Shifting Property from Table 4.1, page 328. | ||
| + | **** <math>X(w)=\mathcal{F}{[u(t)]}{[1+e^{(-\jmath w)}-2e^{(-\jmath 2w)}]}</math> | ||
| + | ***** By table look-up, <math>\mathcal{F}{[u(t)]}={[\frac{1}{\jmath w}+\pi\delta{(w)}]}</math> | ||
| + | ****** Therefore, <math>X(w)={[\frac{1}{\jmath w}+\pi\delta{(w)}]}{[1+e^{(-\jmath w)}-2e^{(-\jmath 2w)}]}</math> | ||
Revision as of 08:23, 8 July 2009
Linearity - Property of Continuous Time Fourier Transform
Linearity States that the $ \mathcal{F} $ of {a*x(t)+b*y(t)} will be equal to {a*X(w)+b*Y(w)} if the signal is truly linear.
General Derivation: $ \mathcal{F}=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt $
- If z(t) = {a*x(t)+b*y(t)}, then the $ \mathcal{F} $ is $ Z(w)=\int\limits_{-\infty}^{\infty}(a*x(t)+b*y(t))e^{(-\jmath wt)}dt $
- $ Z(w)=\int\limits_{-\infty}^{\infty}a*x(t)e^{(-\jmath wt)}dt+\int\limits_{-\infty}^{\infty}b*y(t)e^{(-\jmath wt)}dt $
- $ Z(w)=a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt+b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt $
- Since $ X(w)=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt $ (Same for Y(w))
- Therefore, $ Z(w)=a*X(w)+b*Y(w) $
- Since $ X(w)=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt $ (Same for Y(w))
- $ Z(w)=a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt+b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt $
- $ Z(w)=\int\limits_{-\infty}^{\infty}a*x(t)e^{(-\jmath wt)}dt+\int\limits_{-\infty}^{\infty}b*y(t)e^{(-\jmath wt)}dt $
Example: Signal $ x(t)=1, 0<t\le1; 2, 1<t\le2; 0, else=u(t)+u(t-1)-2u(t-2) $
- $ \mathcal{F} of x(t)=X(w)=\mathcal{F}{[u(t)+u(t-1)-2u(t-2)]} $
- $ X(w)=\mathcal{F}{[u(t)]}+\mathcal{F}{[u(t-1)]}-2\mathcal{F}{[u(t-2)]} $
- $ X(w)=\mathcal{F}{[u(t)]}+e^{(-\jmath w)}\mathcal{F}{[u(t)]}-2e^{(-\jmath 2w)}\mathcal{F}{[u(t)]} $ by Time Shifting Property from Table 4.1, page 328.
- $ X(w)=\mathcal{F}{[u(t)]}{[1+e^{(-\jmath w)}-2e^{(-\jmath 2w)}]} $
- By table look-up, $ \mathcal{F}{[u(t)]}={[\frac{1}{\jmath w}+\pi\delta{(w)}]} $
- Therefore, $ X(w)={[\frac{1}{\jmath w}+\pi\delta{(w)}]}{[1+e^{(-\jmath w)}-2e^{(-\jmath 2w)}]} $
- By table look-up, $ \mathcal{F}{[u(t)]}={[\frac{1}{\jmath w}+\pi\delta{(w)}]} $
- $ X(w)=\mathcal{F}{[u(t)]}{[1+e^{(-\jmath w)}-2e^{(-\jmath 2w)}]} $
- $ X(w)=\mathcal{F}{[u(t)]}+e^{(-\jmath w)}\mathcal{F}{[u(t)]}-2e^{(-\jmath 2w)}\mathcal{F}{[u(t)]} $ by Time Shifting Property from Table 4.1, page 328.
- $ X(w)=\mathcal{F}{[u(t)]}+\mathcal{F}{[u(t-1)]}-2\mathcal{F}{[u(t-2)]} $
