The Gaussian ``bell curve'' is possibly the only smooth, nonzero function, known in closed form, that transforms to itself.4.15
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(4.55) |
It also achieves the minimum time-bandwidth product
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(4.56) |
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(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.