Fourier Transform

Signals and Spectra
- sinc function and rectangular pulse
Fourier Transform
- \(F(w) = \int f(x) e^{- 2 \pi w i x} dx\)
- \(f(x) = \int F(w) e^{2 \pi w i x} dw\)
Properties of Fourier Transform
- Linearity
- Scaling
- Time Shifting: \(f(x-x_0) \leftrightarrow e^{-iwx_0} F(w)\)
- Frequency Shifting: \(e^{-iw_0 x} f(x) \leftrightarrow F(w-w_0)\)
- Duality: \(f(x) \leftrightarrow F(w) \Leftrightarrow F(x) \leftrightarrow f(-w)\)
Convolution
- \(y = h*x \Leftrightarrow Y=HX\)

Reference