The Gaussian ``bell curve'' is possibly the only smooth, nonzero function, known in closed form, that transforms to itself.^{4.15}
(4.55) |
It also achieves the minimum time-bandwidth product
(4.56) |
(4.57) |
Since the true Gaussian function has infinite duration, in practice we must window it with some usual finite window, or truncate it.
Depalle [58] suggests using a triangular window raised to some power for this purpose, which preserves the absence of side lobes for sufficiently large . It also preserves non-negativity of the transform.