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.
General Derivation: $ FT=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt $
- If z(t) = {a*x(t)+b*y(t)}, then the FT 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:
