Introductory concepts - Practical Biomedical Signal Analysis Using MATLAB - page 15

FIGURE 1.5: Illustration of the uncertainty principle. As the observation time
shrinks, the time localization of the observation improves, but the estimate of the
frequency deteriorates.

It turns out that the product of the time and frequency span is bounded [Folland
and Sitaram, 1997]:

1
16π 2

σ t σ f ≥

(1.41)

The equality is reached by Gabor functions (Gaussian envelope modulated by co-
sine). It is important to realize this property, especially when working with time-
frequency representations of a signal. Many different methods of time-frequency
representations make various trade-offs between time and frequency span but in
each of them the inequality (1.41) holds.

1.5 Hypotheses testing
1.5.1 The null and alternative hypothesis
The key to success in statistical hypothesis testing is the correct formulation of
the problem. We need to specify it as two options: the null hypothesis H 0 and the
alternative one H 1 . The two hypotheses must be disjoint and complementary. We
usually try to put the option which we would like to reject as the null hypothesis,
since we can control the probability of erroneous rejection of the null hypothesis.