ISIT2000, Sorrento, Italy, June 25{30, 2000
An Interleaver Design Algorithm based on a Cost Matrix for Turbo Codes
Didier Le Ruyet
Hong Sun
Han Vu Thien
Laboratoire Signaux et Systemes Departement Electronic and Laboratoire Signaux et Systemes Conservatoire National des Arts et Information Engineering Conservatoire National des Arts et Metiers Huazhong University of Science Metiers 75141 Paris Cedex 03, France and Technology 75141 Paris Cedex 03, France 430074 Wuhan, China e-mail:
[email protected] e-mail:
[email protected] e-mail:
[email protected]
Abstract | In this paper the design of interleavers with C = f ( ) = l0 + l1 + l2 + + lw,1 g u D
for Turbo Codes is considered. The proposed algorithm is based on a Hamming weight cost matrix. It optimizes both the minimal distance of Turbo Codes and the passing of extrinsic information. Simulation results show that for short lengths these interleavers improve the error performances at high SNR.
It is admitted that the interleaver is the key element of Turbo Codes [1] [2]. In order to optimize the distance spectrum and the minimal distance of Turbo Codes, the interleaver should map input sequences ( ) which generate low weight output sequences 1 ( ) with interleaved sequences ( ) which generate high weight output sequences 2 ( ), and vice versa. Due to the iterative structure of the turbo decoder, the interleaver should also guarantee a good passing of extrinsic information from one decoder to the other. The proposed interleaver optimizes both these two criteria. In order to increase the minimal distance, a Hamming weight cost matrix is used for the construction. The second goal is achieved since the proposed interleaver belongs to the family of cycle optimized interleavers [3]. The interleaver is built element by element using a tree search method. Let u = [ 0 1 N ,1 ] and v = [ 0 1 N ,1 ] respectively be the input and output sequences of the interleaver. We have the relation P : v = uP where = f ij gN N with ij 2 f0 1g and j ij = i ij = 1. We can also de ne the interleaver with the permutation vector = [ (0) (1) (2) ( , 1)] where ( ) = , ij = 1 u D
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-choice of ( ) : let E 1 = f j nj MAX g (3) E 2 = f j j , j + j ( ) , ( )j = ,1 ,2 , , 2g (4) ( ) 2 E = E 1 \ E 2. ( ) is chosen randomly in E . Equations (1) and (2) reduce the set C of input sequences ( ) to test for each . Equation (3) corresponds to the minimal distance constraint. Equation (4) corresponds to the cycle optimized constraint which imposes that two bits separated by bits ( , 2) in the input sequence ( ) should be separated by at least , 2 , bits in the sequence ( ). E is the set of all the new positions satisfying both constraints. This method allows us to build an interleaver with minimal distance MAX and minimum cycle p . From [3], it is possible to build an interleaver with + 2. If the tree search fails (E = ), the procedure should be started again. To obtain an interleaver with the greatest minimal distance, the procedure must be repeated by increasing the value MAX until it is no longer possible to build the interleaver. e n
I. Introduction
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