lecture2 - ECE 5670 : Digital Communications Lecture 2:...

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ECE 5670 : Digital Communications Lecture 2: Optimum Receiver 1 1/27/2011 Instructor: Salman Avestimehr 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 speciFc interest will be on the Gaussian statistics, it helps (for later lectures) to study the more general situation. ±or a Fxed transmission strategy, we will derive the optimum receiver in terms of minimizing the unreliability of communication. Towards doing this, we formally deFne what unreliability means by carefully looking at the di²erent sources of randomness and what statistical assumptions we make about them. We conclude with a fundamental relation between the variance of the noise σ 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 don’t 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. ±or 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. ±igure 1 illustrates the action taken at the receiver. 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 1
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Channel - y (information about signal and channel) Receiver - ˆ b =0or1 Figure 1: The basic block diagram of a receiver. Formal Defnition o± Reliable Communication Consider a single bit to be communicated reliably. Figure 2 diagrammatically illustrates the familiar bits-to-voltages mapping at the transmitter. - r E 0 r E 1 Figure 2: Mapping for sending 1 bit across an AGN channel.
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell.

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lecture2 - ECE 5670 : Digital Communications Lecture 2:...

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