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Unformatted text preview: ECE 562: Advanced Digital Communication Lecture 2: Histogram to Optimum Receiver Introduction In this lecture we focus our study on how to use the detailed statistical knowledge available in the histogram of the noise in doing reliable communication at a desired level of reliability. Though our specific interest will be on the Gaussian statistics, it helps (for later lectures) to study the more general situation. For a fixed transmission strategy, we will derive the optimum receiver in terms of minimizing the unreliability of communication. Towards doing this, we formally define what unreliability means by carefully looking at the different sources of randomness and what statistical assumptions we make about them. We conclude with a fundamental relation between the variance of the noise 2 , the transmit energy constraint E , and the reliability of communication. Sources of Randomness There are two sources of randomness from the perspective of the receiver: one intrinsic (the information bits itself is unknown) and the other extrinsic (the additive noise introduced by the channel). The receiver typically knows some statistical information about these sources of knowledge. Statistics of the bit : this is the fraction of bits that are 0. If there is some prior information on how likely the transmitted information bit is say, 1, then that could factor in the decision rule. In the extreme instance, if we somehow knew before the communication process that the information bit is 1 for sure, then we dont need to worry about the received voltage. We just decide at the receiver that the information bit is 1. Many a time, no such prior knowledge is available. In this case, we suppose that the information bit is equally likely to be 1 or 0. Noise Statistics : knowing whether the noise is more likely to be small or large will help the receiver make the decision. For instance, if the noise is more likely to be near zero than large, the receiver would likely pick the nearer of the two possible transmit voltages as compared to the received voltage (the so-called nearest-neighbor rule). One of the main conclusions at the end of Lecture 2 is that additive noise in the physical world is (far) more likely to be near its mean than away from it. Figure 1 illustrates the action taken at the receiver. Formal Definition of Reliable Communication Consider a single bit to be communicated reliably. Figure 2 diagrammatically illustrates the familiar bits-to-voltages mapping at the transmitter. 1 Channel- y (information about signal and channel) Receiver- b = 0 or 1 Figure 1: The basic block diagram of a receiver.- r- E r E 1 Figure 2: Mapping for sending 1 bit across an AGN channel....
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This note was uploaded on 01/30/2009 for the course ECE 562 taught by Professor Pramodviswanath during the Fall '08 term at University of Illinois at Urbana–Champaign.
- Fall '08