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ANSWER-1
INTRODUCTION
Polar Codes ,introduced by Arıkan, are known for the capacity approaching with low hardware complexity. We simulated a Polar Encode
Decoder system on MATLAB in order to evaluate Polar Codes.Tu
o Codes employs the iterative BCJR algorithm for reliable and efficient decoding, Arıkan suggested a new encoding/decoding scheme, Polar Codes [4], that proven accomplishes the limit of Binary Discrete Memoryless Channel (BDMC)with low multifaceted nature.
BDMCs parameters
In this section we define two parameters of BDMCs, namely shared data and the Bhattacharyya parameters .
Definition:- Mutual information I(W) of a BDMC with input alphabets x={0,1}, output alphabet y,and the transmission W(y|x) is defined as :
The capacity of symmetric BDMC is equal to the shared data among information and yield of the channel expecting uniform dispersion on the sources of info. I(W) is a proportion of rate in a channel, and as it is notable, solid co
espondence is conceivable over a symmetric BDMC at any rate up to I(W).
The Bhattacharyya parameter of given channel is defined as:
The Bhattacharyya parameter is a measure of reliability of a channel , since Z(W) is the upper bound on the probability of maximum-likelihood (ML) decision e
or when W is used once to transmit a 0 or 1.
ANSWER NO -2
Polar Codes
Polar Codes are linear codes, proven to achieve the capacity of a symmetric BDMC. The basic idea is to create N different channels , from N independent copies of a BDMC W, 1 ≤ i ≤ N , through a linear transformation. As N increases these channels are polarized. The term “polarized” means that transformed channels are either very good, and their mutual information is close to 1, or very bad while their mutual information is close to 0, asymptotically.
As an example of BDMC, one binary erasure channel (BEC), where the input bit is taken as X and the output values chosen as 0 or 1. If the erasure probability of this BEC is 0.5, the co
esponding capacity is I(W) = 1 − 0.5 = 0.5. joining two autonomous uses of this BEC, a compound channel (W,W) is gotten which has two input bits [x1,x2] and two output bits [y1,y2]. The associated capacity is 2I(W). By using a modulo-2 operation between two independent BECs, is obtained the equivalent compound, which has two inputs bit [u1,u2] and two output bits [y1,y2], while the capacity of this channel is still 2I(W).Now chain rule is applied of shared (mutual ) information, after that decomposition of this channel into two synthesized channels denoted as W+ and W−. After combining and splitting, two independent BECs with the same reliability, are changed into two polarised channels and the limit of two channels is unchangeable
     I(W+) + I(W−) = 2I(W).
Arıkan derived the mutual information of these two channels as :
    .1)
     I(W+) = 2I(W) − I(W)2
    (1.2)
     I(W−) = I(W)2 .
    (1.3)
It was proved by Arıkan that the bad channel W− which has smaller capacity than given BEC W, On other side W+ has a larger capacity that is good channel, that is, I(W−) ≤ I(W) ≤ I(W+).
Considering (1.2) and (1.6) the polarizing transformation can be continuously applied over N = 2n independent uses of given BEC W and the capacities of all the polarized channels can be recursively evaluated. After the polarization effect on given BEC W, only good channels W+ can transmit, since their capacity is very large. On the other hand bad channels W− are considered as frozen and both transmitter and receiver use a default value, for a sake of simplicity 0, for these channels.
    Polar Codes Construction
Polar Code Construction (PCC) is any algorithm that selects K best among N possible polar bit-channels at the SNR (Signal-to-Noise-Ratio) in terms of BER (Bit e
or ratio). Optimal Polar Code construction is hard and therefore many sub-optimal Polar Code construction have been proposed [5] at different computation complexities. The earliest Polar Code Construction is from Arıkan using Bhattacharyya parameters. Arıkan proved that a pair of upper bounds on the Bhattacharyya parameters of BEC involves as {z,z} → {2z − z2,z2} at each polarizing transform. In Section 2.3 we assume that the initial value (γ) for the polarizing transformation is γ = 0.5, co
esponding to the worst BER. This is the Bhattacharyya parameter of the BEC, therefore it may be replaced by exp(RE
No). So the original γ = 0.5 can be obtained at dB. The
PCC algorithm uses (1.2) and (1.3) in a recursive manner, and then sorts final values. K among these with larger numeric value are assigned to a new vector, Frozen vector, with the value of 1, while the remaining N − K values are assigned to the same vector with value of 0. Information bits are assigned at the same position, where ones are located in Frozen vector.
Polar Encode
There are mainly three scheme of polar encoding: systematic, concatenated coding and non-systematic [3]. Here code length given as N = 2n and K information bits, the encoder’s input consists of K information bits and N −K Frozen bits. Codeword x can be obtained as follows
    x = uGN ,    (1.4)
where is the bit-reversal permutation matrix and is the n-th
Kronecker power,
        (1.5)
    
    
    
    
As an example, the generator matrix for N = 8...