Fourier Transform Properties – Fourier series formulas

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)

$$x(t) \longleftrightarrow 2\pi x(-\omega)$$