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Project Your final project is to create and analyze a plot showing the bit error performance afforded by various channel coding schemes for the binary symmetric channel = c X*Y x = Y as a function of c € [0, .11. All of the coding schemes are targeting of a rabe of R = 1/2 information bits per channel use. The plota should be all on the same axes, with the horizontal axis being the probability e that the channel leaves the bit unchanged and the vertical axis being the bit error rate P € (0,21. 1. Fundamental Limits: (a) Error-free Shannon Limit: a vertical line at the channel probability c above which arbitrarily reliable communication at a rate 3 is possible. (b) Shannon Limit Allowing Errors: a curve indicating the minimum required channel probability c to communicate at a rate of } message bits per channel use while achieving a message bit error rate no greater than p. 2. Practical Coding Schemes: Solve any fao of the following, for eztra credit, solve more. (a) Repelition Code: Plot the bit error performance, as a function of c, of the simple code that repeats each message bit three times, sends each of the three copies of the message bit over the channel, and decodes the three bits based on wether the majority of the three received bits were 1 or 0. (b) Reed Solomon Code Decoded with Berlekamp Masseg: Plot a monte carlo estimate of the bit error performance of a t-error correcting Reed-Solomon code over GF(q) with t = L ST with q = 2m and m of your choosing and a Berlekamp Massey decoding algorithm, as a function of c. (c) Convolutional Code: A convolutional code of your selection with rate $ and decoded with a soft decoder. (d) Turbo Code: A turbo code of your selection with rabe } decoded with the standard turbo decoder. (e) LDPC Code: A LDPC code of your selection with rabe } with a belief propagation decoder. 3. Analysis: (a) At a target bit error rate of 10-5, how much gap in channel correct transmission probability c is there between the best scheme you examined and the fundamental limit? 1

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import numpy as np
from itertools import product

from math import log2
from scipy.optimize import fsolve
from matplotlib import pyplot as pp
from numpy import linspace

# R = Cap(c) / (1-H2(p_b))
# where R=1/3, Cap(c)=1-H2(c) is channel capacity and p_b is target bit error rate
# so the equation is H2(p_b) = 1-Cap(c)/R = 1-3Cap(c) = 1-3(1-H2(c))

def H2(p):
    return -p*log2(p) - (1-p)*log2(1-p)...

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