Fractal Speech Processing
Including numerous applications and examples, this book offers the results of research carried out to develop novel fractal-based techniques for the analysis of speech and audio signals.
TABLE OF CONTENTS Fractal Speech Processing
Chapter 3: Statistical Analysis, Entropy and Bayesian Estimation
The processes discussed so far have not taken into account the statistical nature of a speech signal. To do this, another type of approach needs to be considered, based on Bayesian estimation. Bayesian estimation allows digital filters to be constructed whose performance is determined by various parameters that can be determined approximately from the statistics of the data.
3.1 The Probability of an Event
Suppose we toss a coin, observe whether we get heads or tails and then repeat this process a number of times. As the number of trials increases, we expect that the number of times a head or a tail occurs is half the number of trials. In other words, the probability of getting heads is 1 /2 and the probability of getting tails is also 1 /2. Similarly, if a die with six faces is thrown repeatedly then the probability that it will land on any particular face is 1 /6.
In general, if an experiment is repeated N times and an event x occurs n times, then the probability of this event P( x) is defined as
The exact probability is the relative frequency of an event as the number of trials tends to infinity. In practice, only a finite number of trials can be conducted and we therefore redefine the probability of an event x as
3.1.1 The joint probability
Suppose we have two coins, which we label A and B. We toss...
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