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n18 - CS 70-2 Spring 2009 Discrete Mathematics and...

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CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Lecture 18 Inference Example 3: Communication over Physical Channels Question: I have one bit of information that I want to communicate over a physical channel. I map the value of the bit to a voltage signal with amplitude + a or - a to transmit over the channel, which corrupts it by additive Gaussian 1 noise. The receiver needs to guess the transmitted bit from the received signal. How much improvement in reliability do I get by repeating my transmission n times? Comment: This is similar to the communication problem considered in Lecture Note 16. There, both the transmitted symbols and the received symbols are discrete (0 or 1) and the channel randomly flips the transmitted symbols. (This channel is called the binary symmetric channel.) In the present problem, the transmitted signal is still discrete (either + a or - a is transmitted) but the received signals are continuous- valued due to the continuous-valued noise. Most physical channels are analog and so this is a natural model. In fact, the discrete received symbols considered in the earlier lecture can be viewed as a quantization of the received analog signal through a 1-bit analog-to-digital convertor. We are now directly modeling the underlying analog physical channel. The parameter a is the amplitude of the transmitted signal, and is dictated by the energy or device limitation of the transmitter. Modeling The situation is shown in Figure 1. Let X ( = + a or - a ) be the value of the voltage signal I want to transmit (to represent the value of the information bit). Assume that X is equally likely to be + a or - a . The received signal Y i on the i th repetition of X is Y i = X + Z i , i = 1 , 2 ,..., n with Z i being Gaussian distributed with mean zero and variance σ 2 . (This is often abbreviated as Z i N ( 0 , σ 2 ) .) The addition is now real addition. The Z i ’s are assumed to be mutually independent across different repetitions of X and also independent of X . The Gaussian distribution is found to be a reasonable model for the noise in many physical channels. One reason is that the noise is often the aggregation of many small noise sources, and the Central Limit Theorem, to be discussed in the next lecture, implies that the aggregate should look Gaussian. Single Transmission Let’s first solve the special case when the signal X is transmitted only once. The problem is: given the received signal Y = X + Z 1 Recall that ”Gaussian” and ”normal” are two equivalent names for the same distribution. In this note, in keeping with standard practice in the communications community, we will use the term ”Gaussian.” We shall use the term ”normal” in other contexts. CS 70-2, Spring 2009, Lecture 18 1
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Figure 1: The system diagram for the communication problem.
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