TAILIEUCHUNG - Digital Signal Processing Part 2

This raw curve can be converted into the complete Gaussian by adding an adjustable mean, µ, and standard deviation, F. In addition, the equation must be normalized so that the total area under the curve is equal to one, a requirement of all probability distribution functions. This results in the general form of the normal distribution, one of the most important relations in statistics and probability: EQUATION 2-8 Equation for the normal distribution, also called the Gauss distribution, or simply a Gaussian. In this relation, P(x) is the probability distribution function, µ is the mean, and σ is the standard deviation. P. | Chapter 2- Statistics Probability and Noise 27 This raw curve can be converted into the complete Gaussian by adding an adjustable mean p and standard deviation a. In addition the equation must be normalized so that the total area under the curve is equal to one a requirement of all probability distribution functions. This results in the general form of the normal distribution one of the most important relations in statistics and probability EQUATION 2-8 Equation for the normal distribution also called the Gauss distribution or simply a Gaussian. In this relation P x is the probability distribution function p is the mean and s is the standard deviation. P x 1 e - x p 2 2 2 yfiBo Figure 2-8 shows several examples of Gaussian curves with various means and standard deviations. The mean centers the curve over a particular value while the standard deviation controls the width of the bell shape. An interesting characteristic of the Gaussian is that the tails drop toward zero very rapidly much faster than with other common functions such as decaying exponentials or 1 x. For example at two four and six standard 5 -5-4-3-2-101234 x FIGURE 2-8 Examples of Gaussian curves. Figure a shows the shape of the raw curve without normalization or the addition of adjustable parameters. In b and c the complete Gaussian curve is shown for various means and standard deviations. -5 -4 -3 -2 -1 0 1 2 3 4 5 x 28 The Scientist and Engineer s Guide to Digital Signal Processing deviations from the mean the value of the Gaussian curve has dropped to about 1 19 1 7563 and 1 166 666 666 respectively. This is why normally distributed signals such as illustrated in Fig. 2-6c appear to have an approximate peak-to-peak value. In principle signals of this type can experience excursions of unlimited amplitude. In practice the sharp drop of the Gaussian pdf dictates that these extremes almost never occur. This results in the waveform having a relatively bounded appearance with an apparent peak-to-peak .

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