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lecture6 - ECE 5670 Digital Communications Lecture 6...

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ECE 5670 : Digital Communications Lecture 6: Reliable Communication with Erasures 1 2/15/2011 Instructor: Salman Avestimehr Introduction So far, we have seen that arbitrarily reliable communication is possible at non-zero rates provided the receiver is well designed. In this lecture we will take a closer look at simplifying (in a computational sense) the complexity of the receiver design. We break up the receiver into two steps: 1. Demodulation : Map the analog received voltage to a finite number of discrete levels. To be concrete, we focus on the following situation: let the transmit voltages be binary (just as in the previous lecture). Then we map the analog voltage into one of three possible levels. Two of them correspond to the two levels of the binary modulation at the transmitter. The third, called an erasure , models the scenario when the received analog voltage is not enough to make a decision one way or the other. 2. Decoding : The second step involves taking the erasures into careful account and recov- ering the original information bits. Receiver Design in Two Steps For concreteness, the discussion in this lecture is limited to binary modulation on the AWGN channel: y [ m ] = x [ m ] + w [ m ] , m = 1 . . . T ; (1) i.e., the transmit voltage is restricted to be ± E . Further, the transmitter is assumed to be broken up into the two steps described in the previous lecture (sequential modulation and linear coding). For concreteness let us suppose that E is transmitted when the corresponding coded bit is 0 and E is transmitted when the corresponding coded bit is 1. The ML receiver (from the previous lecture) took the T received voltages and mapped them directly to the information bits. While this is the optimal design, it is also prohibitively expensive from a computational view point. Consider the following simpler two-stage design of the receiver. 1. Demodulation : At each time m the received voltage y [ m ] is mapped into one of three possible choices: Let us fix c (0 , 1). (a) If y [ m ] ≤ − c E (2) then we map into a 0. 1 Based on lecture notes of Professor Pramod Viswanath at UIUC.
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Figure 1: Demodulation Operation. (b) If y [ m ] > c E (3) then we map into a 1. (c) In the intermediate range, i.e., c E y [ m ] c E (4) we map into an “e” (standing for an erasure).
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