ENHANCING ERROR LOCALIZATION OF DFT CODES BY WEIGHTED ℓ 1 -NORM MINIMIZATION Anil Kumar A and Anamitra Makur School of Electrical and Electronic Engineering Division of Information Engineering Nanyang Technological University Singapore 639798 Email: anil.kumar@pmail.ntu.edu.sg and eamakur@ntu.edu.sg ABSTRACT We consider the problem of decoding of real BCH dis- crete Fourier transform codes (RDFT) which are con- sidered for joint source channel codes to provide robust- ness against errors in communication channels. In this paper, we propose to combine the subspace based al- gorithm like MUSIC algorithm with ℓ 1 -norm minimiza- tion algorithm, which is promoted as a sparsity solution functional, to enhance the error localization of RDFT codes. Simulation results show that the combined al- gorithm performs better over the performances of these individual algorithms. 1. INTRODUCTION Error correction coding over real and complex fields, as opposed to finite-fields was introduced by Marshall [1]. Within the class of codes defined for error correction over real fields, the discrete Fourier transform (DFT) codes forms an important class. DFT codes are being considered for joint source channel coding for providing robustness to errors over communication channel [2, 3] and in particular in hiperlan2 [4]. An (N,K) DFT code is a linear block code whose generator matrix consists of any K columns of an IDFT matrix of order N [1]. The parity check matrix consists of the remaining (N − K) columns. Within the class of DFT codes, real number Bose-Chaudhuri-Hocquengem (BCH) DFT codes (RDFT) are possible if the spacing of parity frequencies are relatively prime to N and complex conjugate columns are selected for the generator matrix. The generator matrix of such a code can be defined as G =  (N/K)W h N ΣW K , and the parity check matrix as H = W h N×d , where W h N denotes inverse DFT matrix of order N , Σ denotes an N × K binary matrix whose nonzeros elements are Σ 00 =1, Σ ii =Σ N-i,k-i = 1 for i =1, ..., (K − 1)/2, and d =(N − K) which denotes the indexes (K + 1)/2, ..., N − (K + 1)/2 of an IDFT matrix of order N [5]. A RDFT code is a maximum distance separable code. The minimum distance of this code is d + 1 and hence it can correct up to ⌊d/2⌋ errors. Several algorithms have been proposed for decoding of DFT codes [3] - [8]. Rioul [6] modified the Peter- son - Gorenstein - Zeiler (PGZ) algorithm which is used for decoding RS codes over finite fields, for decoding real BCH codes in the presence of background noise. The authors in [3], [7] showed that the problem of error correction of RDFT codes is analogous to the problem of complex sinusoidal estimation. Subspace based ap- proaches like MUSIC, ESPRIT were applied in [5] for es- timating these complex frequencies. All these proposed algorithms decode perfectly under no background noise. But in the presence of background noise like the quan- tization noise which affects all locations, the decoding is not exact and depends on the algorithm. In general, subspace based algorithms performs better compared to other algorithms in the presence of noise. The problem of sampling and recovering sparse sig- nals [9, 10] is analogous to the problem of decoding RDFT codes. The techniques that have been pro- posed by using the annhilating filter in [10] and by ℓ 1 - norm minimization in [11] can also be used for decoding RDFT codes. However in the presence of the noise these techniques perform worse than the subspace based tech- niques. Recently it is shown in [12] that using weighted ℓ 1 - norm minimization, which is achieved by the application of an appropriate weighting matrix, the performance of ℓ 1 -norm minimization can be enhanced, and the en- hancement is significant. The weighting should reflect the solution, for which an a priori information is re- quired. In this paper we propose to use a two step algo- rithm. In the first step a subspace based algorithm like MUSIC will be applied. An application of a posteriori test followed by MUSIC algorithm will confirm whether the decoding is proper. If the a posteriori test fails then in the second step a sub matrix of the parity check ma- trix will be formed, which includes those columns which may correspond to the potential error locations. This sub matrix will be formed with the help of the a priori output obtained by the MUSIC algorithm. A weighted ℓ 1 -norm minimization algorithm will be applied upon this sub matrix which will be of a smaller dimension than the original parity check matrix. Simulation results in Section 4 shows that the proposed two step algorithm performs much better than the performance obtained by the application of each of these individual algorithms. The organization of the paper is as follows: In Sec- tion 2 we give a brief overview of the subspace algorithm for error localization, Section 3 describes the proposed two step algorithm, simulation results are provided in Section 4 and finally, Section 5 concludes the paper. 16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP