Fourier Transform Properties
1. Linearity
$$ ax(t)+bv(t) \longleftrightarrow aX(\omega)+bV(\omega) $$
2. Time Shift
$$ x(t-c) \longleftrightarrow e^{-j\omega c} X(\omega ) $$
3. Time Scaling
$$ x(at) \longleftrightarrow \frac 1a X \left( \frac {\omega}{a} \right), \ a \neq 0 $$
4. Time Reversal
$$ x(-t) \longleftrightarrow X(-\omega ) $$
5. Multiply by tn
$$ t^nx(t) \longleftrightarrow j^n \frac {d^n}{d\omega^n} X(\omega), \ n=1,2,3,\cdots $$
6. Multiply by Complex Exponential
$$ e^{j\omega_ot}x(t) \longleftrightarrow X(\omega-\omega_o), \ \omega_o \ real $$
7. Multiply by Sine
$$ \sin (\omega_ot)x(t) \longleftrightarrow \frac j2 \left[ X(\omega + \omega_o) – X(\omega – \omega_o) \right] $$
8. Multiply by Cosine
$$ \cos (\omega_ot)x(t) \longleftrightarrow \frac 12 \left[ X(\omega + \omega_o) + X(\omega – \omega_o) \right] $$
9. Time Differentiation
$$ \frac {d^n}{dt^n} x(t) \longleftrightarrow (j\omega)^n X(\omega), \ n=1,2,3,\cdots $$
10. Time Integration
$$ \int\limits_{-\infty}^t x(\lambda)d\lambda \longleftrightarrow \frac {1}{j\omega}X(\omega)+\pi X(0) \delta(\omega) $$
11. Convolution in Time
$$ x(t)\cdot h(t) \longleftrightarrow X(\omega)\cdot H(\omega) $$
12. Multiplication in Time
$$ x(t) \cdot w(t) \longleftrightarrow \frac {1}{2\pi} X(\omega)\cdot W(\omega) $$
13. Parseval’s Theorem (General)
$$ \int\limits_{-\infty}^{\infty} x(t) \overline {v(t)} dt = \frac {1}{2\pi} \int\limits_{-\infty}^{\infty} X(\omega)\overline {V(\omega)} d\omega $$
14. Parseval’s Theorem (Energy)
$$ \int\limits_{-\infty}^{\infty} x^2(t)dt = \frac {1}{2\pi} \int\limits_{-\infty}^{\infty} |X(\omega)|^2 d\omega $$
15. Duality: If x(t) ↔ X(ω)
$$ x(t) \longleftrightarrow 2\pi x(-\omega) $$