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Uncertainty Principle

Module by: Richard Baraniuk

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
fig1.png
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=1∥f∥2∫-∞∞t2|ft|2dt σ f 2 1 f 2 t t 2 f t 2 (1)

Frequency Width

σ F 2=12π∥f∥2∫-∞∞ω2|Fⅈω|2dω σ F 2 1 2 f 2 ω ω 2 F ω 2 (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):

ft= A 1 ⅇ-αt2 f t A 1 α t 2 (4)
Fⅈω= A 2 ⅇ-ω22α F ω A 2 ω 2 2 α (5)
Figure 3
fig3.png
Figure 4
fig4.png

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