Homework 1: Noissy channel simulation

This week on information theory class we discuss the channel transmission systems topic way to connect a transmitter to a receiver in some information transferring. The homework for this week is to build a binary symmetric channel simulation, them make data transmission by the channel and then make some analysis of the simulation output.

Binary symmetric channel simulation
The simulation works this way:
The program receives as parameters the 0 symbol frequency in a word, and a matrix describing the probability of a Symbol to become in another on the transmission, our symbols are just "0" and "1" because we are working with a binary alphabet.

First we simulate a message, for this propose this is done using the "generate_word" method that receives as input the word length and using Bernoulli trials with the 0 frequency obtained as parameter, simulate symbol by
symbol of the word.

Next thing to do is to send the message, this is done with the method "word_transmission" walking symbol by symbol in the word and applying a Bernoulli trial to each symbol, using the matrix of change in a transmission received as a parameter.


https://github.com/MaxHernandez/teoria_informacion

Simulation output analysis
For the analysis part i build a bash script that execute the last python script
and make graphs with the output value, on the graph the Y axis shows the percent of success sending and the X shows the length of the word sent.

https://github.com/MaxHernandez/teoria_informacion

With 90% of frequency occurrence of "0" symbols:
X axis shows the length of the word sent
Y axis shows the percent of success sending a message

With 50% of frequency occurrence of "0" symbols:
X axis shows the length of the word sent
Y axis shows the percent of success sending a message

With 70% of frequency occurrence of "0" symbols:
X axis shows the length of the word sent
Y axis shows the percent of success sending a message

Estimating the asymptotic probability
I could not estimate the accurate asymptotic probability of the output values, but i arrive to the conclusion using the last graphs that the output behaves as an exponential function with negative input $exp(-xC)$ and increasing the probabilityity of success of no change of symbols in transmissions must increase the function constant input $C$.

This is the last graph, but adding the $exp(-x)$ function, as we look the output behaves as and exponential function but i couldn't accurate the function.

References:
http://en.wikipedia.org/wiki/Binary_symmetric_channel
http://elisa.dyndns-web.com/~elisa/teaching/comp/info/channel.pdf