We will find the Fourier transform
of the Gaussian bell-shaped function
where and are positive constants.
We get first
Completing the square in
and substituting , we may write
does not depend on at all. In fact, we have
Hence we may evaluate as
(see the area under Gaussian curve). Putting this value to (3) yields the result
Thus, we have gotten another Gaussian bell-shaped function (4) corresponding to the given Gaussian bell-shaped function (2).
Interpretation. One can take for the breadth of the bell the portion of the abscissa axis, outside which the ordinate drops under the maximum value divided by , for example. Then, for the bell (2) one writes
whence giving, by evenness (http://planetmath.org/EvenFunction) of the function, the breadth . Similarly, the breadth of the bell (4) is . We see that the product
If is the time and is the on a system of oscillators with their natural frequencies, then in the formula
of the inverse Fourier transform, means the amplitude of the oscillator with angular frequency . We can infer from (5) that the more localised ( small) the external force is in time, the more spread ( great) is its spectrum of frequencies, i.e. the greater is the amount of the oscillators the force has excited with roughly the same amplitude. If one, conversely, wants to better the selectivity, i.e. to compress the spectrum narrower, then one has to spread out the external action in time. The impossibility to simultaneously localise the action in time and enhance the selectivity of the action is one of the manifestations of the quantum-mechanical uncertainty principle, which has a fundamental role in modern physics.
- 1 Я. Б. Зельдович & А. Д. Мышкис: Элементы прикладной математики. Издательство ‘‘Наука’’. Москва (1976).
- 2 Ya. B. Zel’dovich and A. D. Myshkis: ‘‘Elements of applied mathematics’’. Nauka (Science) Publishers, Moscow (1976).
|Date of creation||2013-03-22 18:38:25|
|Last modified on||2013-03-22 18:38:25|
|Last modified by||pahio (2872)|