HW 3.3 - Ryne Rayburn ECE301 Summer2009 - Rhea

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) $

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)}]} $

Back to Homework 3