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- Apr 14, 2013

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One of the techniques we are using at the digital communications to improve the reliability of a noisy communication channel, is to repeat a symbol many times.

For example, we can send each symbol $0$ or $1$ say three times. More precisely, applying the rule of majority, a $0$ (or $1$) is sent as $000$ (or $111$ respectively) and is decoded at the receiver with $0$ (or $1$) if and only if the received sequence of three symbols contains at least two $0$ (or $1$ respectively).

In this case we consider a message that is sent through a noisy communication channel and consists of $n$ symbols ($0$ or $1$). At the transmission each symbol can be altered (independent from the other ones) due to the noise. To increase the reliability of the transmission each symbol is repeated $r$ times. The probability of correct transmission of each symbol ($0$ or $1$) is $p$. Applying the rule of majority, a symbol $0$ (or $1$) is decoded correctly at the receiver if and only if the received sequence of $r$ symbols contains at least $k$ $\ 0$ (or $1$ respectively), with $k>\frac{r}{2}$.

Calculate:

(a) the probability a sent symbol to be decoded correctly.

(b) the probability the whole message to be decoded correctly.

(c) the probability to be altered not more than $x$ symbols of the message, with $x<n$.

For (a) do we have to consider that at least $k$ symbols have to be decoded correctly with probability $p$ ?