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Discrete Memoryless Channel

DISCRETE MEMORYLESS CHANNEL IN DIGITAL COMMUNICATION


The channel is the propagation medium or electromagnetic route that connects the transmitter and receiver. By modeling a channel as a statistical model, the features of the channel may be investigated. One of these channel models is the Discrete Memoryless Channel (DMC).


As depicted in Figure 1, a discrete memoryless channel is a statistical model having an input X, and an output Y. Y is a noisy variant of X as an output. Both X and Y are randomly generated variables.


The channel receives an input symbol X from an alphabet X for each unit of time (signaling interval). As a result, it produces the output symbol Y from the alphabet Y. When both the alphabets X and Y have finite sizes, the channel is said to be "discrete." When the current output symbol depends solely on the current input symbol and none of the prior inputs, the channel is said to be "memoryless".

                                               Figure1: Discrete Memoryless Channel


An input alphabet x = {x0, x1, x2, …. xj-1} and an output alphabet y = {y0, y1, y2, …. yk-1} are used to represent the channel. The Probability of  transition is given by


(𝑦𝑘/𝑥𝑗) = 𝑃(𝑌=𝑦𝑘/𝑋=𝑥𝑗) for all j and k………………………………… (1)

We have, 0 ≤ (𝑦𝑘/𝑥𝑗) ≤ 1 for all j and k.


• The conditional probability that the channel output Y = yk, given that the channel input X = xj, is called the transition probability (yk/xj). There is a chance that mistakes will occur throughout the information transmission process over the channel.


• When k = j, the transition probability (yk/xj) provides the conditional probability of receiving the signal properly.


• A conditional probability of error is indicated when k ≠ j.


Channel Matrix


• The discrete memoryless channel may be characterized by its whole set of transition probabilities, which are organized in the form of a matrix as shown below. This matrix is termed a channel matrix.

Probability of error


• The average probability of symbol error, Pe, is defined as the probability that the output random variable Yk is different from the input random variable Xj, averaged over all k ≠ j.


• The difference (1 – Pe) is the average probability of receiving the signal correctly.


Binary Symmetric Channel


The Binary Symmetric Channel (BSC) is a discrete memoryless channel with j = k = 2 that is a special case of the discrete memoryless channel. Figure 2 shows the transition probability diagram for this channel. 

Figure 2: Binary Symmetric Channel


Two input symbols (x0 = 0, x1 = 1) and two output symbols (y0 = 0, y1 = 1) make this channel. Because the probability of receiving a 1 if a 0 is transmitted is the same as the probability of receiving a 0 if a 1 is sent, the channel is symmetric. Its channel matrix is as follows:

where p denotes the probability of a common transition or the conditional probability of error.



This post first appeared on Electronics And Communications, please read the originial post: here

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Discrete Memoryless Channel

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