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# Uncertainty Principle

Module by: Richard Baraniuk. E-mail the author

Summary: Introduction to the Uncertainty Principle.

The width of a signal is proportional to the reciprocal of the width of its CTFT.

Recall Figure 1.

As TT increases, ft f t widens and Fiω F ω narrows. In fact CTFT applied to fat f a t yields 1aFiωa 1 a F ω a (and vice versa). This means that ft f t and Fiω F ω cannot be narrow simultaneously.

Let's make this mathematical:

Definition 1: width
Define the width of a function as its normalized standard deviation.

## Time Width

σ f 2=1f2t2|ft|2d t σ f 2 1 f 2 t t 2 f t 2
(1)

## Frequency Width

σ F 2=12πf2ω2|Fiω|2d ω σ F 2 1 2 f 2 ω ω 2 F ω 2
(2)
Figure 2.

## Uncertainty Principle

σ f σ F 12 σ f σ F 1 2
(3)
In other words, the time width times the frequency width is greater than or equal to cst.

Proof: use C.S.T.

The signal that has equality in bound (which means that it is the most concentrated signal in both time and frequency under this measure) is the Gaussian (Figure 3 and Figure 4):

ft= A 1 e(αt2) f t A 1 α t 2
(4)
Fiω= A 2 eω22α F ω A 2 ω 2 2 α
(5)

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