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− | '''Linearity''' | + | '''Linearity of CTFT''' |
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− | + | F{ax(t) + by(t)} = aX(jw) + bY(jw) | |
+ | |||
+ | ----- | ||
+ | |||
+ | By definition of Fourier Transform: | ||
+ | |||
+ | F{ax(t) + by(t)} = <math>\int\limits_{-\infty}^{\infty}(ax(t)+by(t))e^{(-\jmath wt)}dt</math> | ||
+ | |||
+ | F{ax(t) + by(t)} = <math>a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt + b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt</math> | ||
+ | |||
+ | F{ax(t) + by(t)} = aX(jw) + bY(jw) |
Latest revision as of 18:49, 8 July 2009
Linearity of CTFT
F{ax(t) + by(t)} = aX(jw) + bY(jw)
By definition of Fourier Transform:
F{ax(t) + by(t)} = $ \int\limits_{-\infty}^{\infty}(ax(t)+by(t))e^{(-\jmath wt)}dt $
F{ax(t) + by(t)} = $ a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt + b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt $
F{ax(t) + by(t)} = aX(jw) + bY(jw)