# lect8 - Error Correcting Codes Combinatorics Algorithms and...

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(Fall 2007) Lecture 8: Shannon’s Noise Models September 14, 2007 Lecturer: Atri Rudra Scribe: Sandipan Kundu Atri Rudra Till now we have concentrated on the worst case noise model pioneered by Hamming. In today’s lecture, we will study some stochastic noise models (most of) which were first studied by Shannon. 1 Overview of Shannon’s Result Shannon introduced the notion of reliable communication over noisy channels. Broadly, there are two types of channels that were studied by Shannon: Noisy Channel : This type of channel introduces error in during transmission, which results in incorrect reception of the transmitted signal by the receiver. Redundancy is added at the transmitter to increase reliability of the transmitted data. The redundancy is taken o f at the receiver. This process is termed as Channel Coding . Noise-free Channel : As the name suggests, this channel does not introduce any type of error in transmission. Redundancy in source data is being used for of the compression of the source data in the transmitter. The data is decompressed at the receiver. The process is popularly known as Source Coding . The figure below presents a generic model of a communication system, which combines the two concepts we discussed above: Message Source Encoder Channel Encoder Channel Channel Channel Decoder Source Decoder Decoded Message In the figure above, source coding and channel coding are coupled. However, Shannon’s source coding theorem allows us to decouple both these parts of the communication and study each of these parts separately. Intuitively, this makes sense: if one can have reliable communication over the channel using channel coding, then for the source coding the channel e f ectively has no noise. For source coding, Shannon proved a theorem that precisely calculated the amount by which the message can be compressed: this amount is related to the “entropy” of the message. We will however, not talk about source coding in any detail in this course. From now on, we will exclusively focus on the channel coding part of the communication setup. Note that one aspect of channel coding is how we model the channel noise. We have seen Hamming’s worst case noise model in some detail. Next, we will study some specific stochastic channels. 1

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lect8 - Error Correcting Codes Combinatorics Algorithms and...

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