Polar Codes by Orhan Gazi

Author:Orhan Gazi
Language: eng
Format: epub, pdf
ISBN: 9789811307379
Publisher: Springer Singapore

Any discrete channel is indicated is indicated by the letter W.

3.1.1 Split Channels and Capacity of Split Channels When BECs are Employed

A discrete communication channel has a set of inputs and a set of outputs. Between every pair of output and input symbols, we have channel transition probabilities, and discrete channel is characterized by these set of transition probabilities. The data bits are coded and the code bits are transmitted through the discrete channel in a sequential manner.

Sequential transmission of code bits through the discrete channel W can also be considered as multiple use of the channel in a parallel manner as discussed in Chap. 1. Assume that there are N code bits transmitted through the identical channels in a parallel manner. The overall system between data bits and channel outputs can also be considered as a single channel having multiple inputs and multiple outputs. The decoding operation is a sequential process, and for the decoding of any data bit, we use all the available information at our hands. For the decoding of the first data bit, the available information is the all channel outputs. Then, using all the channel outputs we determine the value of the first data bit.

For the decoding of the second data bit, the available information that can be used is the channel output values, and the decision result of the first bit. Then, we can consider a channel whose input and outputs are the second data bit, and channel outputs together with the decision result of the first data bit.

In general, for the decoding of any data bit, we can define a channel whose input and outputs are the data bit to be determined and the channel outputs with the decision results of the previous bits. Such channels are called split channels.

In Fig. 3.2, a polar encoder structure for together with two DMCs is depicted.

Fig. 3.2Polar encoder structure for together with two DMCs

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