We will find the Fourier transform

F(ω)

:= ∫ −∞∞f(t)e−iωt dt

(1)

of the Gaussian bell-shaped function

f(t)

= Ce−at2 ,

(2)

where C and a are positive constants.

We first obtain

Completing the square gives

With the substitution

we may write

F(ω)	= ∫ −∞∞e−a2 e− dt
	= e− ∫ le−z2 dz,	(3)

where l is a line in the complex plane parallel to the real axis and passing through the point

Now define

We can show that I_y does not depend on y. Indeed,

	= ∫ −∞∞e−(x+iy)2 dx
	= −2i∫ −∞∞e−(x+iy)2 (x + iy) dx
	= ix=−∞x=∞
	= 0.	(4)

Hence

Substituting this result into (3) gives

F(ω)

= e− .

(5)

Thus the Fourier transform of the Gaussian in (2) is another Gaussian.

Interpretation. One can take the breadth of the bell to be the portion of the abscissa axis outside which the ordinate drops below the maximum value divided by e. For the Gaussian in (2),

so

By the evenness of the function, the breadth is

Similarly, from (5), the breadth of the transformed bell is

Therefore,

Δt Δω

= 8.

(6)

Thus, for this choice of breadth convention, the product has a constant value. More generally, other choices of pulse shape and width measure lead to analogous inverse relations between temporal width and frequency width. The two breadths are therefore inversely proportional in this sense.

If t is time and f is the action of a force on a system of oscillators with their natural frequencies, then the inverse Fourier transform is

In this representation, F(ω) describes the frequency-domain amplitude associated with angular frequency ω. Equation (6) illustrates that a more localized action in time, corresponding to a smaller Δt, requires a broader frequency spectrum, corresponding to a larger Δω. Conversely, obtaining greater frequency selectivity requires the action to be spread over a longer interval of time.

This time-frequency tradeoff is closely related to the mathematical structure underlying the quantum-mechanical uncertainty principle, also historically called a principle of indetermination.

References