Basic Stochastic Process Concepts
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Knowledge of stochastic processes is required for navigation system design and analysis, yet many students nd the topic di cult This is in part due to the lack of familiarity with the role and utility of stochastic processes in daily life After introducing a few basic concepts, this section presents a few motivational examples that should be understandable to readers with an engineering background Random variables are functions that map random experimental outcomes to real numbers The idea of a random variable can be more clearly motivated after a few key terms are de ned A random experiment in105
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CHAPTER 4 STOCHASTIC PROCESSES
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volves a procedure and an observation of an experimental outcome There is some degree of uncertainty in either the procedure or the observation, which necessitates the experiment The uncertainty often arises from the extreme complexity of modeling all aspects of the experiments procedure and observation with su cient detail to determine the output Consider the example of a coin toss The procedure involves balancing the coin on the thumbnail, icking the thumb, allowing the coin to land and settle to a stationary position on a at surface The observation involves determining which surface of the coin is on top Most people have performed this experiment numerous times Almost nobody would seriously consider trying to determine the initial forces and torques applied by the thumb to the coin, the aerodynamics during ight, or the impact forces and torques upon landing that would be required to predict the result of any given toss The complexity of such a deterministic prediction motivates the use of a (fair) coin toss as an unbiased tool in many decision making situations The utility of probability theory and stochastic processes is that it provides the analyst with the ability to make quantitative statements about events or processes that are too complicated to allow either exact replication of experiments or su ciently detailed deterministic analysis For example, even though an analyst might be incapable of predicting the outcome of a single coin toss, by analyzing a su ciently large number of experiments, the analyst might be able to predict which surface of an unfair coin has a higher likelihood of landing upwards To be useful, the theory for this analysis should allow quantitative statements concerning the number of observations required to achieve any desired level of con dence in such a prediction In fact, such analysis is quite commonplace The e cacy of random analysis has motivated a mathematically sophisticated theory for probability and probabilistic reasoning This theory is based on measure theory Using probability theory, we analyze models of experiments instead of the experiments themselves Therefore, the analyst must be careful to consider whether the probabilistic model is a su ciently accurate model of the experiment
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The topic of stochastic processes is very important to the study of navigation system design as it is the foundation on which we build our ability to quantitatively analyze the propagation of uncertainty through the navigation system as a function of time A thorough presentation of the related topics of statistics, probability theory, random variables, and stochastic processes is well beyond the scope of this text Several excellent texts are mentioned at the conclusion of this chapter Instead, this chapter focuses on topics selected for their relevance to the design and analysis of navigation systems The following examples are intended to motivate the topics
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41 BASIC STOCHASTIC PROCESS CONCEPTS
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Figure 41: Oscilloscope image of a sinusoidal input discussed in the balance of this chapter Example 41 Most engineers have used an oscilloscope at some point in their careers An example image from an oscilloscope screen is shown in Figure 41 This image is created by directly connecting the output of a sinusoidal source to the oscilloscope input The image clearly shows a noisy version of a sinusoidal signal This gure is included to clearly exemplify the concept of measurement noise The noise is the result of several complex physical processes, eg, thermal noise in the probes and electronic parts Quite often it is accurate to describe a measurement in the form y(t) = Sx(t) + (t), where y represents the measurement, x represents the actual value of some physical property, S is a scale factor, and represents measurement noise Typically, the manufacturer of the instrument is careful to design the instrument such that the scale factor S is constant and accurately known and that the measurement noise is not related to the signal x In addition, the manufacturer will often provide the purchaser with the quantitative description of suitable for further analysis To be able to communicate with sensor manufacturers and other navigation engineers and to be able to perform quantitative analysis, the analyst must be familiar with stochastic process de nitions and methods
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