Subspace Based Error and Erasure Correction with DFT Codes for ...

1

Subspace Based Error and Erasure Correction with DFT Codes for Wireless Channels Gagan Rath and Christine Guillemot Abstract—This paper deals with the decoding of low-pass DFT codes in presence of both errors and erasures. We propose a subspace based approach for the error localization which is similar to the subspace approaches followed in the array signal processing for direction of arrival (DOA) estimation. The basic idea is to divide a vector space into two orthogonal subspaces of which one is spanned by the error locator vectors. The locations of the errors are estimated from the spanning eigenvectors of the complement subspace. However, unlike the subspace approach in DOA estimation, which is similar to estimating the subspaces from the syndrome covariance matrix after a projection, in the proposed approach the subspaces are estimated from the modified syndrome covariance matrix after a whitening transform. Simulation results with a Gauss-Markov source reveal that the proposed algorithm is more efficient than the coding theoretic approach on impulsive channels, and than the subspace approach with projection on lossy channels.

I. I NTRODUCTION All-IP based multimedia communication on wireless and mobile networks at the same, or at least comparable, ”carrier-class” and bandwidth efficiency as circuit-switched subsystems of 2G and 3G remains a challenging issue. QoS provisioning becomes all the most challenging in a global mobility context with highly varying channel characteristics (bandwidth, throughput, error rates, fading and erasure characteristics...). Traditionally, error control codes are implemented in the link layer and are often complemented by an error detection mechanism in the transport layer in order to cope with residual errors. For example, the UDP error detection mechanism, applied on both headers and payloads, leads to discard all packets containing bit errors at the expense of bandwidth. The equivalent channel is often referred to as an erasure channel. Such erasures are often coped with by retransmission mechanisms and/or by forward error correction at a packet level at the expense of additional delays. However, QoS requirements of multimedia applications make them highly delay-sensitive even though some corruptions of bits may be tolerated. It is becoming a common understanding that different service classes built upon link layers making use of different transmission modes (e.g., with different channel codes, modulation and access methods), as already present in 3G systems [1], need to be defined to best meet heterogeneous sources requirements. Concurrently, recent initiatives within the IETF and 3GPP (e.g. the UDP-lite and ROHC initiatives) acknowledge that transmission modes, offering guarantees in delay however offering no guarantee in error rates, are required to achieve high utilization of scarce wireless resources. In the above framework, the end-to-end channel, as seen by the application layer, will be characterized by both erasures and errors. The design of joint source-channel coding systems, guided by an optimum trade-off between compression efficiency and error and/or erasure resilience depending on the link characteristics, becomes a key issue. In this context, this paper addresses the subject of channel error correction with DFT codes [2–7] in presence of erasures. DFT codes are being considered for join-

t source-channel coding in order to provide robustness to data loss or corruption in communication channels [8–10]. The basic idea consists in incorporating redundancy in a message by transforming a -sample message vector to an -sample codevector with . The code is characterized by the property that the discrete Fourier transform (DFT) of every codevector vanishes for a fixed set of frequencies called the parity frequencies or the syndrome frequencies. The DFT coefficients of a received vector over the syndrome frequencies, called the syndrome, indicate the presence of sample errors if they are nonzero, and are used to localize and correct those errors.







It is known that DFT codes are cyclic codes in the complex field [2]. However, the code properties do not hold once the codevectors are quantized. The error correction problem becomes analogous to the problem of estimation of directions and amplitudes of plane waves (DOAs) incident on a uniform linear array [10–13]. The erasure of code samples becomes analogous to the knowledge of some of the directions. However, unlike in the DOA estimation, where there are many sets of observations (snapshots) available over some given time interval, in the error correction problem, there is only one set of syndrome coefficients for each received vector. Further, the occurrence of errors and erasures is associated with discrete frequencies. Therefore the DOA estimation techniques need to be adapted to the error localization problem. In this paper, we first extend the error localization algorithm for binary BCH codes in presence of erasures to the real field [3]. The algorithm uses the concept of an erasure locator polynomial to generate modified syndrome coefficients and then applies the ”errors only” localization procedure over those coefficients. Then we present a subspace based algorithm to localize the channel errors which is derived along the lines of the subspace based approaches to DOA estimation. This approach first projects the syndrome vectors of appropriate length onto the orthogonal complement of the erasure subspace (defined in section IIIB) and then applies the ”errors only” subspace approach on the projected vectors. Because of its higher complexity, we present another algorithm which applies the ”errors only” subspace approach on the modified syndrome coefficients. Then we consider the quantization of codevectors and modify all algorithms in order to accommodate the localization errors due to the quantization noise. With quantization noise, the presented subspace approaches become similar to the popular MUSIC algorithm for DOA estimation [12, 13] but have limitations due to the finite number of syndrome coefficients. We establish relationships among the three localization algorithms and compare their performances by deriving a performance measure. The relationships give us insight into the limitations of the coding theoretic approach relative to the subspace approaches. Finally we present some simulation results in order to test and validate the

2

developed theory. To our knowledge, there is no prior work on the application of DFT codes for both error and erasure correction with quantization. II. DFT C :A  An 
 DFT code is a linear block code whose generator matrix consists of any columns from the inverse DFT matrix of order
 [2]. Let  denote the DFT matrix of order
  whose   element is equal to   !#" . Let   denote the conjugate transpose of  . An 
 DFT code is  defined by the generator matrix $ given as $&%' )( , where ( is an
+* binary matrix such that its ,-/. th element  is equal to 0 iff the . th column of $ has column index , in  . A matrix, 1 , of the code consists of the remaining 2 parity %unitary 3
5property 4checkcolumns of the inverse DFT matrix. Due to the  of  , 1 $7698;:= . This implies that, for  any codevector 2 @ , 1 @A6B8C:=<  . Therefore every DFT codevector has DFT coefficients equal to zero. These coefficients ODES

REVIEW

$

samples and recovering the erased samples. The pseudoinverse of the generator matrix applied on the decoded codevector completes the decoding process by producing the corresponding message vector [15]. In the following, we consider the case when the codevectors are transmitted without quantization. The more realistic situation with quantized codevectors is treated in section IV. Let denote the received vector when the transmitted codevector is corrupted by the error vector and the erasure vector . The vectors and are assumed to have components each. The error vector has nonzero components only at the erroneous sample locations. Similarly the erasure vector has nonzero components only at the erased sample locations. Since the errors are assumed to occur only at the non-erased samples, the indices of the nonzero elements of are mutually exclusive with those of the nonzero elements of . We assume that the components of corresponding to the erased samples are set to zero. The received vector can be expressed as

f

d@

e

d

e ef




f

ef

d)6g@ G e G fh^ (1) Therefore the syndrome of the received vector is given as D i 6j1  d)6j1  @ G e G f  6g1  Ee G f   (2) D
L 2 2   o  n  where i %9k lm0 -lm `^_^`^Wlm is a column vector of length . The superscript p denotes the matrix transposition operation. Let there2 be q sample erasures in the received vector where  F 0 s r qtr . We assume that there is at least one erasure, othE
F erwise it becomes a ”errors only” case. With q errors. erasures, any 2 = J v L M  H 2 G BCH DFT code can correct up to  I  u 4 q 0 2  J=LvLetM . Letthe 2I KJLNM 2 received samples contain w errors where w rxI 4yq ,, z| {,  `, ^_z|^`{ ^a`,^_z ^`^adenote the indices of the erroneous samples and ` z ~ { }   , denote the indices
G q . of the erased samples.  Let €)%   ‚b Y„ƒ "  , …†6]0`^_^`^a-w A. Error localization – coding theoretic approach   DFT code defined by the generator matrix $O%QP > R( S>T The coding theoretic approach is an extension of the error and where the non-zero elements of the
U* rectangular ma- erasure localization algorithm for binary BCH codes to the real trix ( are given as 6X0 , for field [3, 17]. First the contribution of erasures to the syndrome V 6X0 ,is(!assumed YZY 6 ( to  beY[ > odd. Y Low-pass ,\6]0_^`^`^ab 4c0  J=L (!. VWHere are termed as syndrome coefficients. DFT codes are cyclic codes in the complex field [2]. Within the class of DFT codes, there exist BCH codes in the real field and the complex field. If the parity frequencies are spaced by , where is relatively prime to , then the DFT code is a BCH code in the complex field [2]. If, in addition, the parity frequencies are such that the complex conjugate of every column of the generator matrix also belongs to it, then, through elementary column operations, the BCH code can be made real [2]. A BCH DFT code is an MDS code. An DFT code which is an MDS code in the complex field or the real field has minimum Hamming distance [3]. Therefore it can correct up to sample errors and recover up to sample erasures. For the details about the existence of real BCH codes for different combinations of and , the reader is referred to the exposition by Marshall [2]. In this paper, due to the advantage of implementation, we will consider only the low-pass real BCH

real DFT codes do not exist when is even [2]. The theory that follows applies to all BCH DFT codes and is not restricted to low-pass DFT codes only. III. R EAL E RROR

AND

E RASURE C ORRECTION C ODES

WITH

DFT

All DFT codevectors are transmitted to a receiver over some communication media which might add random noises to their samples. Further, in the context of a packet network, all samples of a codevector may not reach the receiver due to packet losses giving rise to erasures. In this framework, the error correction is the problem of estimating the corresponding message vector from the received samples of a codevector. Since the locations of the erased samples are known to the receiver, the procedure consists of two steps: localizing the erroneous samples in the received samples, and finding the values of the error and the erasure vectors. Once the error and the erasure vectors are decoded, the codevector can be decoded by modifying the erroneous

is eliminated by using an erasure locator polynomial defined as

‡  ˆ  % ‰ } 0Œ4 z|{ Y ˆ    6 ‡ V G ‡ ˆ   G ^`^_^ G ‡ } ˆ  }  (3)  ‹‡ Y Š  ‡  where V 690 .  ˆ has roots at  z|{ `^_^`^~ z`{~} , which cor respond locations. The inverse DFT of Ž% ‡k V  ‡  `^_to^`^~the ‡ } erasure n  loca-8