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# Fourier Transform Properties ## Linearity $a f(t) + b g(t) \leftrightarrow aF(\omega) + bG(\omega)$ ## Time Shifting $f(t - t_0) \leftrightarrow e^{-j\omega t_0}F(\omega)$ ## Frequency Shifting $e^{j\omega_0 t}f(t) \leftrightarrow F(\omega - \omega_0)$ ## Scaling $f(at) \leftrightarrow \fr...

# Fourier Transform Properties ## Linearity $a f(t) + b g(t) \leftrightarrow aF(\omega) + bG(\omega)$ ## Time Shifting $f(t - t_0) \leftrightarrow e^{-j\omega t_0}F(\omega)$ ## Frequency Shifting $e^{j\omega_0 t}f(t) \leftrightarrow F(\omega - \omega_0)$ ## Scaling $f(at) \leftrightarrow \frac{1}{|a|}F(\frac{\omega}{a})$ ## Time Reversal $f(-t) \leftrightarrow F(-\omega)$ ## Differentiation in Time $\frac{d}{dt}f(t) \leftrightarrow j\omega F(\omega)$ $\frac{d^n}{dt^n}f(t) \leftrightarrow (j\omega)^n F(\omega)$ ## Integration in Time $\int_{-\infty}^{t} f(\tau) d\tau \leftrightarrow \frac{1}{j\omega}F(\omega) + \pi F(0)\delta(\omega)$ ## Differentiation in Frequency $t f(t) \leftrightarrow j\frac{d}{d\omega}F(\omega)$ $t^n f(t) \leftrightarrow j^n \frac{d^n}{d\omega^n}F(\omega)$ ## Multiplication $f(t)g(t) \leftrightarrow \frac{1}{2\pi}F(\omega) * G(\omega)$ ## Convolution $f(t) * g(t) \leftrightarrow F(\omega)G(\omega)$ ## Parseval's Theorem $\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |F(\omega)|^2 d\omega$