Gaussian

The Gaussian ``bell curve'' is the only smooth function that transforms to itself:

$$\frac{1}{\sigma\sqrt{2\pi}}e^{-t^2 \left / 2\sigma^2\right.} \leftrightarrow e^{-\omega^2 \left/ 2\left(1/\sigma\right)^2\right.}$$

It also achieves the minimum time-bandwidth product

$$\sigma_t\sigma_\omega = \sigma\times (1/\sigma) = 1$$

when ``width'' of a function is defined as the square root of its second central moment. For even functions $w(t)$ ,

$$\sigma_t \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{\int_{-\infty}^\infty t^2 w(t) dt}.$$

• Since the true Gaussian function has infinite duration, in practice we must window it with some finite window.
• Philippe Depalle suggests using a triangular window raised to some power $\alpha$ for this purpose.
  • This choice preserves the absence of sidelobes for sufficiently large $\alpha$ .
  • It also preserves non-negativity of the transform