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Unformatted text preview: 6.450 Introduction to Digital Communication November 13, 2002 MIT, Fall 2002 Lecture 17-18: Detection 1 Introduction We have studied modulation and demodulation, and studied how noise processes can cor- rupt the received waveform, and thus corrupt the demodulated waveform. We now must study how the transmitted data can be retrieved from this noisy demodulated waveform. As one might guess, it is not possible to do this 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 decide whether some phenomenon is present or not on the basis of some observations. For example, a radar system uses the data 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 has been extended in the communication field to detect which one, among a set of mutually exclusive alternatives, is correct. 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. Decoding is the process of mapping the received signal into one of the possible set of code words or transmitted symbols. We use the word hypotheses for the possible choices in what follows, since the word conjures up the appropriate intuitive image. 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 mu- tually 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, numbered 0 , 1 ,... ,m- 1, and let H be a random variable whose sample value is the correct hypothesis j , 0 j m- 1 for that particular sample point. The probability of hypothesis j , p H ( j ), is denoted p j and is usually referred to as the a priori probability of j . There is also a random variable (rv) Y , called the observation. This is the data on which the decision must be based. We observe a sample value y of Y , and on the basis of that observation, we want to make a decision between the possible hypotheses. The observation could equally well be a complex random variable, a random vector, a random procss, or a chance variable and these generalizations are discussed as we proceed....
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