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# book_8 - Chapter 8 Detection co ding and deco ding 8.1...

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Chapter 8 Detection, coding, and decoding 8.1 Introduction The previous chapter showed how to characterize noise as a random process and this chapter uses that characterization to retrieve the signal from the noise corrupted received waveform. As one might guess, this is not possible without occasional errors when the noise is unusually large. The objective then, is to retrieve the data while minimizing the eﬀect of these errors. This process of retrieving data from a noise corrupted version is known as detection . Detection, decision making, hypothesis testing, and decoding are synonyms. The word detection refers to the eﬀort to detect whether some phenomenon is present or not on the basis of obser- vations. For example, a radar system uses the observations to detect whether or not a target is present; a quality control system attempts to detect whether a unit is defective; a medical test detects whether a given disease is present. The meaning of detection has been extended in the digital communication ﬁeld from a yes/no decision to a decision at the receiver from a ﬁnite set of possible transmitted signals. Such a decision from a set of possible transmitted signals is also called decoding , but here the possible set is usually regarded as the codewords in a code rather than the signals in a signal set. 1 Decision making is, again, the process of deciding between a number of mutually exclusive alternatives. Hypothesis testing is the same, and here the mutually exclusive alternatives are called hypotheses. We use the word hypotheses for the possible choices in what follows, since the word conjures up the appropriate intuitive image of making a choice between a set of alternatives, where only one alternative is correct and there is a possibility of erroneous choice. These problems will be studied initially in a purely probabilistic setting. That is, there is a probability model within which each hypothesis is an event. These events are mutually exclusive and collectively exhaustive, i.e., the sample outcome of the experiment lies in one and only one of these events, which means that in each performance of the experiment, one and only one hypothesis is correct. Assume there are M hypotheses 2 , labeled a 0 ,... ,a M 1 . The sample outcome of the experiment will lie in one of these M events. This deﬁnes a random symbol U 1 As explained more fully later, there is no fundamental diﬀerence between a code and a signal set. 2 The principles here apply essentially without change for a countably inﬁnite set of hypotheses; for an uncount- ably inﬁnite set of hypotheses, the process of choosing an hypothesis from an observation is called estimation . Typically, the probability of choosing correctly in this case is 0 and the emphasis is on making an estimate that is close in some sense to the correct hypothesis.

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book_8 - Chapter 8 Detection co ding and deco ding 8.1...

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