3 - ECE 562: Advanced Digital Communication Lecture 3:...

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ECE 562: Advanced Digital Communication Lecture 3: Sequential and Block Communication Introduction In the previous lecture, we have studied reliable communication of a bit (or a handful of bits) at a given time instant. In practice, there tend to be several hundreds of thousands of bits that need to be communicated reliably, but over multiple time instants. A natural scheme that communicates a whole bunch of information bits over multiple time instants comes is sequential Communication : read the information bits serially, say k at a time, and transmit them sequentially at different time instants. We will see that this scheme, while simple, has limitations. In particular, as time grows the reliability level approaches zero. To ameliorate this situation, we turn to block communi- cation , where all the voltages at different times are picked jointly as a function of all the information bits. In this lecture we will see a simple block communication scheme, the so- called repetition coding strategy. We see that it promised arbitrarily reliable communication, but at the cost of arbitrarily small data rate and energy efficiency. Channel Model Corresponding to multiple transmissions, we have multiple received voltages. To figure out how to process these received voltages into a decision on the transmitted information bits, we first need a statistical model for how the additive noise varies over time . Our discussion in Lecture 1 lets us argue that the statistics of the additive noise at any time instant is Gaussian. Without much loss of generality, we can suppose the mean and the variance are unchanged over time. In practice, it is typically also a good assumption to suppose that the additive noise at one time instant has little relation statistically to that at the next time instant. In other words, the additive noises at different times are statistically independent from each other. Such a noise model is said to be white ; this is as opposed to the “colored” noise where the value at one time instant sheds some information at another. The received voltage at time m y [ m ] = x [ m ] + w [ m ] , (1) is the sum of the transmitted voltage at time m and the noise at time m . Since the noise statistics are Gaussian, we will refer to this channel as the additive white Gaussian noise channel, or simply the AWGN channel. Reliability of Sequential Communication Now, we have argued earlier that information bits can be well modeled as statistically in- dependent of one another. In sequenctial communication different bits are sent at different times. This means that the transmitted voltages at different times are also statistically independent of one another. Since the additive noises at different times are statistically 1
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independent of one another, we conclude that the received voltages are also statistically in- dependent of each other. In other words, sequential transmission over an AWGN channel naturally leads to sequential reception as well: at each time
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This note was uploaded on 01/30/2009 for the course ECE 562 taught by Professor Pramodviswanath during the Fall '08 term at University of Illinois at Urbana–Champaign.

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3 - ECE 562: Advanced Digital Communication Lecture 3:...

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