Ajay Dholakia

Reduced-complexity decoding of LDPC codes

Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.

Table based decoding of rate one-half convolutional codes

Table based error correction and decoding of rate one-half convolutional codes is described. A new class of fast-decodeable locally invertible convolutional codes based on a one-to-one mapping between information and encoded blocks of equal lengths is defined. The syndrome is used as an address to access a correction table which stores pre-computed correction information. The correction table generation process is described and a specific table based correction algorithm is given. Performance of this scheme is analyzed and simulation results are presented. >

Table-Driven Decoding of Binary One-Half Rate Nonsystematic Convolutional Codes

Table look-up based decoding schemes for convolutionally encoded data have been designed for both block coding [5], and convolutional coding [6, 21. However, in the latter case most of the work was done for systematic codes. Nonsystematic codes offer better error correcting capability than systematic codes if more than one constraint length of received blocks are considered [7J. Our approach to table look-up based decoding of nonsystematic convolutional codes was introduced in [l, 31. A l/2-rate convolutional encoder is characterized by two generator sequences g(j) = (gf),g;j), ..., gp)), j = 1,2, where U is the constraint length of the code, i.e., the number of memory elements in a minimal realization of the convolutional code [4]. The output conatmint length i s defined as = Z(u + 1) and is equal to the number of encoded bits affected by a single input information bit [TI. An input information sequence U is encoded into two encoded output sequences v(j), j = 1,2, using v = uG, where v = ( u p ) , up), up), up), ...) is the composite encoded sequence, also called a codeword, obtained by multiplexing the two encoded sequences, and G is the semi-infinite code generator matrix [?I. The encoded bits in l/a-rate coding are generated at twice the input information rate. However, it is possible to find an encoding operation that relates blocks of input information bits to equal length blocks of encoded bits [I, 31. Proposition: For I/%-rate convolutional coding with constraint length v, there exists a correspondence between equal length blocks of input information bits and the encoded bits. The length of these corresponding blocks is 2v bits. (For proof, sec [l, 31.) We can formalises this relationship as follows. Let [ul;,i+zu-l=(ui, 2;i.j.,, .q+a.-x{ be a Zv-bit block of the input information sequence, . ,a,+g-l= uzi, uai+t, ... , uai+aV-l) be the 2u-bit block of the corresponding encoded sequence. Given the generator sequences &), j = 1,2, for a l/Z-rate convolutional code with constraint length U, we define the reduced encoding matrix as

On locally invertible rate-1/n convolutional encoders

A locally invertible convolutional encoder has a local inverse defined as a full rank w/spl times/w matrix that specifies a one-to-one mapping between equal-length blocks of information and encoded bits. In this correspondence, it is shown that a rate-1/n convolutional encoder is nondegenerate and noncatastrophic if and only if it is locally invertible. Local invertibility is used to obtain upper and lower bounds on the number of consecutive zero-weight branches in a convolutional codeword. Further, existence of a local inverse can be used as an alternate test for noncatastrophicity instead of the usual approach involving computation of the greatest common divisor of n polynomials.

A lost packet recovery technique using convolutional coding in high speed networks

Packet loss due to buffer overflow during congested traffic conditions is expected to be a major source of performance degradation in future high speed networks. Forward error correction can be employed in these networks to improve the end-to-end system performance by reconstructing lost packets at the receiver, thereby avoiding retransmissions. The authors study the feasibility of using convolutional coding for recovering lost packets. They propose a high-speed convolutional decoding technique and demonstrate its applicability in ATM networks at the ATM Adaptation Layer to recover lost packets.<<ETX>>

Error Recovery in High-Speed Networks

The performance of high-speed networks of today and the future will be primarily limited by packet loss due to buffer overflows during congested traffic conditions. A message or aprotocol data unit(PDU) is made up of many smaller packets, and any one of the packets being lost results in the entire PDU being retransmitted. The performance of end-to-end protocols degrades rapidly if error recovery is based on packet loss detection and retransmission of PDUs, i.e., on some form of ARQ. This phenomenon is more pronounced in high-speed networks since the ratio of packet transmission time to propagation delay is very small. Hence, efficient protocols must be designed to provide reliable communication using error control techniques to recover lost packets. Using FEC to reduce the retransmission frequency is an attractive solution for error recovery in high-speed networks.

High-speed table-driven correction and decoding in convolutionally encoded type-I hybrid-ARQ protocols

The authors describe the use of novel table-driven error correction and decoding in convolutionally encoded type-I hybrid-ARQ error control protocols. The table-driven correction and decoding are fast and simple, and can be used at very high data rates. The proposed type-I hybrid-ARQ protocol has simplicity comparable to that of the majority-logic decoding based hybrid-ARQ schemes. But since the retransmission requests can be finely tuned, the throughput losses are minimized and the error correction performance is comparable to that of Viterbi decoding based type-I hybrid-ARQ schemes.<<ETX>>

A variable-redundancy hybrid ARQ scheme using invertible convolutional codes

Nonstationary channels (e.g., digital mobile communication channels) require adaptive error control schemes for reliable communication. On these channels, a particular transmission may encounter no errors. Hence, it is desirable to split an encoded sequence into subsequences that are sent in successive transmissions such that each subsequence contains all the information necessary to recover the original message in case of no errors. When errors are present, the subsequences are combined to perform error correction. We show that invertible convolutional codes have this property, and can be used to provide incremental redundancy in a variable-redundancy hybrid ARQ (VR-HARQ) scheme. Invertible convolutional codes provide an alternative to polynomial division otherwise required to extract the original message in convolutional VR-HARQ schemes.<<ETX>>

Determination of inverses and syndrome formers for 2-D convolutional encoders

A representation of a 2-D polynomial convolutional encoder, analogous to the semi-infinite generator matrix representation of a 1-D encoder is presented. An algorithm to find its inverses and syndrome formers using this representation is described.

Minimal, minimal-basic, and locally invertible convolutional encoders

Rate-k/n locally invertible convolutional encoders are defined. It is shown that a basic locally invertible encoder is minimal-basic. Local invertibility is used to re-derive Forneys (1973) upper and lower bounds on the maximum number of consecutive all zero branches in a convolutional codeword. A time-domain test for minimality of an encoder is given.