2.6 Properties of fourier transform

Properties of the fourier transform

The properties of the Fourier transform are important in applying it to signal analysis and to interpreting it. The main properties are given here using thenotation that the FT of a real valued function $x(t)$ over all time $t$ is given by $ℱ\{x\}=X(\omega )$ .

1. Linear: $ℱ\{x+y\}=ℱ\{x\}+ℱ\{y\}$
2. Even and Oddness: if $x(t)=u(t)+jv(t)$ and $X(\omega )=A(\omega )+jB(\omega )$ then $u$ $v$ $A$ $B$ $|X|$ $\theta$ even 0even 0even 0odd 00 oddeven 00 even0 eveneven $\pi /2$ 0 oddodd 0even $\pi /2$
3. Convolution: If continuous convolution is definedby: $y(t)=h(t)*x(t)={\int }_{-\infty }^{\infty }h(t-\tau )x(\tau )d\tau ={\int }_{-\infty }^{\infty }h(\lambda )x(t-\lambda )d\lambda$ then $ℱ\{h(t)*x(t)\}=ℱ\{h(t)\}ℱ\{x(t)\}$
4. Multiplication: $ℱ\{h(t)x(t)\}=\frac{1}{2\pi }ℱ\{h(t)\}*ℱ\{x(t)\}$
5. Parseval: ${\int }_{-\infty }^{\infty }{|x(t)|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{|X(\omega )|}^{2}d\omega$
6. Shift: $ℱ\{x(t-T)\}=X(\omega )e^{-j\omega T}$
7. Modulate: $ℱ\{x(t)e^{j2\pi Kt}\}=X(\omega -2\pi K)$
8. Derivative: $ℱ\{\frac{dx}{dt}\}=j\omega X(\omega )$
9. Stretch: $ℱ\{x(at)\}=\frac{1}{|a|}X(\omega /a)$
10. Orthogonality: ${\int }_{-\infty }^{\infty }{e}^{-j{\omega }_{1}t}{e}^{j{\omega }_{2}t}=2\pi \delta ({\omega }_1-{\omega }_2)$