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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.

Figure 1
Figure 1 (fig1.png)
As TT increases, f⁢t f t widens and F⁢i⁢ω F ω narrows. In fact CTFT applied to f⁢a⁢t f a t yields 1a⁢F⁢i⁢ωa 1 a F ω a (and vice versa). This means that f⁢t f t and F⁢i⁢ω 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=1∥f∥2⁢∫−∞∞t2⁢|f⁢t|2d t σ f 2 1 f 2 t t 2 f t 2
(1)

Frequency Width

σ F 2=12⁢π⁢∥f∥2⁢∫−∞∞ω2⁢|F⁢i⁢ω|2d ω σ F 2 1 2 f 2 ω ω 2 F ω 2
(2)
Figure 2.
Figure 2
Figure 2 (fig2.png)

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):

f⁢t= A 1 ⁢e−(α⁢t2) f t A 1 α t 2
(4)
F⁢i⁢ω= A 2 ⁢e−ω22⁢α F ω A 2 ω 2 2 α
(5)
Figure 3
Figure 3 (fig3.png)
Figure 4
Figure 4 (fig4.png)

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