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Proof

We omit the details of the proof of the rise time form of the uncertainty principle which are found in PVI and mention instead a few insights found along with the proof. The ``slack'' in the uncertainty principle relates to comparing the duration of a signal $\Lambda T$ to the duration of its autocorrelation $\Lambda T_{\rm auto}$.Generally speaking, it is easy to find a long signal that has short autocorrelation. Just take an arbitrary short signal and convolve it using a lengthy all-pass filter. Conversely, we cannot get a long autocorrelation function from a short signal. A good example is the autocorrelation of a rectangle function, which is a triangle. The triangle appears to be twice as long, but considering that the triangle tapers down, it is reasonable to assert that the $\Lambda T$'s are the same. Thus, we conclude that  
 \begin{displaymath}
\Lambda T_{\rm auto}
\quad \leq \quad
\Lambda T\end{displaymath} (4)
From this inequality may be proven the inequality in the uncertainty relation  
 \begin{displaymath}
\Lambda T \ \Lambda F \quad \geq \quad 1\end{displaymath} (5)

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Stanford Exploration Project
3/1/2001