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Decoder Evaluation by EXIT-Chart method

The Extrinsic Information Transfer (EXIT) Chart is a method used to evaluate performance of any iterative decoder and giving a feedback on required number of iterations to reach successful decoding. It can be used in any type of iterative decoding including LDPC codes. The chart is plotted as the a priori mutual information $I_{A}$ before message is decoded and the extrinsic mutual information $I_{E}$ after decoding. Since two decoders are used and the extrinsic of the first decoder is the a priori of the second and from the second iteration the extrinsic of the second decoder is the a priori of the first. The two plots are plotted on the same chart, but with swapped axis for the second decoder. The decoding iterative behavior is given by a staircase function in the tunnel between the two curves. The staircase function starts at point (0,0) (no mutual information) and a successful decoding is achieved when the point (1,1) is reached.

In LDPC codes the first decoder is given as the variable nodes and the second decoder is given as the check nodes. Since the variable nodes are directly connected to the channel, the a priori mutual information $I_{A}$ of variable nodes decoder in the first iteration is considered the mutual information of the channel. However, the check nodes decoder (second decoder) is totally independent of the channel performance unlike the variable nodes decoder. From stated fact we can clearly show that variable nodes decoder EXIT curves get higher and closer to point (1,1) as the channel performance is better. On the other hand, check nodes EXIT curves are independent from the channel and so they stay constant. This will lead to a wider tunnel between two curves (variable and check nodes curves) which means better and faster iterative decoding performance. However, if the two curves (variable and check nodes curves) intersect at any point other than (1,1) no successful decoding is achievable, since no complete tunnel from point (0,0) and (1,1) is observed. In the next slides the performance of iterative decoding using EXIT Charts will be tested for two different channel models:

Binary Erasure Channel (BEC)

Additive White Gaussian Noise (AWGN) Channel