| Line 3: | Line 3: | ||
----- | ----- | ||
| − | ax(t) + by(t) | + | 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)
