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# Fourier Transform Properties ## Linearity The Fourier transform is a linear operation. $a \cdot f(t) + b \cdot g(t) \stackrel{\mathcal{F}}{\longleftrightarrow} a \cdot F(f) + b \cdot G(f)$ ## Time Scaling $f(at) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{|a|}F(\frac{f}{a})$ ## Time...

# Fourier Transform Properties ## Linearity The Fourier transform is a linear operation. $a \cdot f(t) + b \cdot g(t) \stackrel{\mathcal{F}}{\longleftrightarrow} a \cdot F(f) + b \cdot G(f)$ ## Time Scaling $f(at) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{|a|}F(\frac{f}{a})$ ## Time Shifting $f(t - t_0) \stackrel{\mathcal{F}}{\longleftrightarrow} e^{-j2\pi ft_0}F(f)$ ## Frequency Shifting $e^{j2\pi f_0t}f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(f - f_0)$ ## Conjugation $f^*(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F^*(-f)$ ## Convolution $f(t) * g(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(f) \cdot G(f)$ ## Multiplication $f(t) \cdot g(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(f) * G(f)$ ## Differentiation $\frac{d}{dt}f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} j2\pi fF(f)$ ## Integration $\int_{-\infty}^{t} f(\tau) d\tau \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{j2\pi f}F(f) + \frac{1}{2}F(0)\delta(f)$ ## Duality $F(t) \stackrel{\mathcal{F}}{\longleftrightarrow} f(-f)$