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Unformatted text preview: ECE 5670 : Digital Communications Lecture 5: Rate Efficient Reliable Communication 1 2/10/2011 Instructor: Salman Avestimehr Introduction We now move to rate efficient reliable communication (energy efficiency tends to come for free in this scenario). In this lecture we see that there are block communication schemes smarter than the naive repetition coding seen earlier that promise arbitrarily reliable communication while still having a non-zero data rate. We begin by setting the stage for studying rate efficient reliable communication by carefully dividing the transmitter strategy of mapping the information bits to transmit voltages into two distinct parts: 1. map information bits into coded bits by adding more redundancy: the number of coded bits is larger than the number of information bits. The ratio of information bits to coded bits is called the coding rate . This process is generally called coding . 2. map coded bits directly into transmit voltages. This is done sequentially : for in- stance, if only two transmit voltages are allowed ( ± √ E ) then every coded bit is se- quentially mapped into one transmit voltage. If four possible transmit voltages are allowed ( ± √ E, ± √ E 3 ), then every two consecutive coded bits are mapped into a single transmit voltage sequentially. This mapping is typically called modulation and can be viewed as a labeling of the discrete transmit voltages with a binary sequence. The receiver could also be potentially broken down into two similar steps, but in this lecture we will continue to focus on the ML receiver which maps the received voltages directly into information bits. Focusing on a simple binary modulation scheme and the ML receiver, we see in this lecture that there are plenty of good coding schemes: in fact, we will see that most coding schemes promise arbitrarily reliable communication provided they are decoded using the corresponding ML receiver! Transmitter Design: Coding and Modulation We are working with an energy constraint of E , so the transmit voltage is restricted to be within ± √ E at each time instant. For simplicity let us restrict that only two transmit voltages are possible: + √ E and- √ E . 2 If we are using T time instants to communicate, this means that the number of coded bits is T , one per each time instant. With a coding rate of R , the number of information bits (the size of the data packet) is B = RT . Surely, R ≤ 1 in this case. The scenario of R = 1 exactly corresponds to the sequential communication scheme studied in Lecture 3. As 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 2 We will explore the implications of this restriction in a couple of lectures from now. we saw there, the reliability level approaches zero for large packet sizes. The point is that even though we have spaced the transmit voltages far enough apart (the spacing is 2 √ E in this case), the chance that at least one of the bits is decoded incorrectly approaches unity...
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