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lecture7

# lecture7 - ECE 5670 Digital Communications Lecture 7...

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ECE 5670 : Digital Communications Lecture 7: Capacity of the AWGN Channel 1 2/17/2011 Instructor: Salman Avestimehr Introduction In the last two lectures we have seen that it is possible to communicate rate eﬃciently and reliably. In this lecture we will see what the fundamental limit to the largest rate of such a reliable communication strategy is. This fundamental limit is called the capacity . Examples We can see what the fundamental limits to reliable communication are in the context of the scenarios in the last two lectures: 1. AWGN channel : With binary modulation and random linear coding at the transmit- ter and ML decoding at the receiver, we have seen that the largest rate of reliable communication is R = 1 log 2 1 + e SNR 2 . (1) This is the capacity of the AWGN channel when the transmitter is restricted to do linear coding and binary modulation. 2. Erasure channel : We developed this model in Lecture 6 in the context of simplifying the receiver structure. But it is a very useful abstract model on its own right and widely used to model large packet networks (such as the Internet). The basic model is the following: the transmitter transmits one bit at a time (you could replace the word “bit” by ”packet”). The receiver either receives the bit correctly or it is told that the bit got erased . There is only a single parameter in this channel and that is the rate of erasures (the chance that any single transmit bit will get erased before reaching the receiver): p . What is the largest data rate at which we can hope to communicate reliably? Well, since only a single bit is sent at any time, the data rate cannot be more than 1 bit per unit time. This is rather trivial and we can tighten our argument as follows: the receiver receives only a fraction 1 p of the total bits sent (the remaining p fraction of the total bits sent got erased). So, the data rate for reliable communication could not have been any more than the fraction of bits that the receiver got without erasures.

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lecture7 - ECE 5670 Digital Communications Lecture 7...

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