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  

$$\Lambda T_{\rm auto} \quad \leq \quad \Lambda T$$ (4)

From this inequality may be proven the inequality in the uncertainty relation  

$$\Lambda T \ \Lambda F \quad \geq \quad 1$$ (5)