Fabrizio Pollara

Turbo codes for PCS applications

Turbo codes are the most exciting and potentially important development in coding theory in many years. They were introduced in 1993 by Berrou, Glavieux and Thitimajshima, and claimed to achieve near Shannon-limit error correction performance with relatively simple component codes and large interleavers. A required E/sub b//N/sub o/ of 0.7 dB was reported for BER of 10/sup -5/ and code rate of 1/2. However, some important details that are necessary to reproduce these results were omitted. This paper confirms the accuracy of these claims, and presents a complete description of an encoder/decoder pair that could be suitable for PCS applications. We describe a new simple method for trellis termination, we analyze the effect of interleaver choice on the weight distribution of the code, and we introduce the use of unequal rate component codes which yields better performance. Turbo codes are extended to encoders with multiple codes and a suitable decoder structure is developed, which is substantially different from the decoder for two-code based encoders.

Soft-Input Soft-Output Modules for the Construction and Distributed Iterative Decoding of Code Networks

Soft-input soft-output building blocks (modules) are presented to construct and iteratively decode in a distributed fashion code networks, a new concept that includes, and generalizes, various forms of concatenated coding schemes. Among the modules, a central role is played by the SISO module (and the underlying algorithm): it consists of a four-port device performing a processing of the sequences of two input probability distributions by constraining them to the code trellis structure. The SISO and other soft-input soft-output modules are employed to construct and decode a variety of code networks, including “turbo codes” and serially concatenated codes with interleavers.

Multiple turbo codes

We introduce multiple turbo codes and a suitable decoder structure derived from an approximation to the maximum a posteriori probability (MAP) decision rule, which is substantially different from the decoder for two-code-based encoders. We developed new rate 1/3 and 2/3 constituent codes to be used in the turbo encoder structure. These codes, for 2 to 32 states, are designed by using primitive polynomials. The resulting turbo codes have rates b/n, b=1, 2 and n=3, 4, and include random interleavers for better asymptotic performance. A rate 2/4 code with 16QAM modulation was used to realize a turbo trellis coded modulation (TTCM) scheme at 2 bit/sec/Hz throughput, whose performance is within 1 dB from the Shannon limit at a BER=10/sup -5/.

Serial concatenated trellis coded modulation with rate-1 inner code

We develop new, low complexity turbo codes suitable for bandwidth and power limited systems, for very low bit and word error rate requirements. Motivated by the structure of previously discovered low complexity codes such as repeat-accumulate (RA) codes with low density parity check matrix, we extend the structure to high-level modulation such as 8PSK, and 16QAM. The structure consists of a simple 4-state convolutional or short block code as an outer code, and a rate-1, 2 or 4-state inner code. Two design criteria are proposed: the maximum likelihood design criterion, for short to moderate block sizes, and an iterative decoding design criterion for very long block sizes.

Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes

We propose new decoding algorithms to be embedded in the iterative decoding strategy of parallel concatenated convolutional codes. They are derived from the optimum maximum-a-posteriori algorithm and permit a continuous decoding of the coded sequence without requiring trellis termination of the constituent codes. Two basic versions of the continuous algorithm and their suboptimum simplifications are described. Simulation results refer to the applications of the new algorithms to a highly efficient rate 1/3 concatenated code; they show performance only 0.6 dB worse than the Shannon limit.

Serial concatenated trellis coded modulation with iterative decoding

We propose a design approach for serial concatenation of an outer convolutional code and an inner trellis code with multilevel amplitude/phase modulations using a bit-by-bit iterative decoding scheme. An example is given for throughput of 2 bits/sec/Hz with 2/spl times/8PSK modulation to clarify the approach. In this example, an 8-state outer code with rate 4/5 and a 2-state inner trellis code with 5 inputs and 2/spl times/8PSK outputs per trellis branch were used. The performance of this code with input block of 16384 bits is within 1.1 dB from the Shannon limit for 8PSK at a bit error probability of 5/spl times/10/sup -8/ for 2 bits/sec/Hz with 10 iterations,.

A VLSI decomposition of the deBruijn graph

The deBruijn graph <italic>B<supscrpt>n</supscrpt></italic> is the state diagram for an <italic>n</italic>-stage binary shift register. It has 2<italic><supscrpt>n</supscrpt></italic> vertices and 2<supscrpt><italic>n</italic> + 1</supscrpt> edges. In this papers, it is shown that <italic>B<subscrpt>n</subscrpt></italic> can be built by appropriately “wiring together“ (i.e., connecting together with extra edges) many isomorphic copies of a fixed graph, which is called a <italic>building block</italic> for <italic>B<subscrpt>n</subscrpt></italic>. The efficiency of such a building block is refined as the fraction of the edges of <italic>B<supscrpt>n</supscrpt></italic> which are present in the copies of the building block. It is then shown, among other things, that for any α < 1, there exists a graph <italic>G</italic> which is a building block for <italic>B<supscrpt>n</supscrpt></italic> of efficiency > α for all sufficiently large <italic>n</italic>. These results are illustrated by describing how a special hierarchical family of building blocks has been used to construct a very large Viterbi decoder (whose floorplan is the graph <italic>B<supscrpt>13</supscrpt></italic>) which will be used on NASAs <italic>Galileo</italic> mission.

Rate-distortion efficiency of subband coding with integer coefficient filters

Discusses efficient image compression algorithms that are suitable for low-complexity implementation in deep space probes. Subband coding or equivalent wavelet transforms provide some advantages over traditional block transforms in terms of rate distortion performance under the mean square-error criterion and according to subjective visual evaluation. Furthermore, the possibility of choosing a desired resolution, independently in the time or frequency domain, is important to preserve the specific features that are most relevant for substantiating scientific findings. A low-complexity version of a JPEG-like algorithm based on the integer cosine transform (ICT), is being implemented in software on the Galileo spacecraft. The most promising candidate for improving the current ICT-based algorithm is a subband coding method that uses quadrature mirror filters (QMF) with lattice structure and integer coefficients. Lattice QMF filters are paraunitary perfect reconstruction (PR) filters. This provides a simple method to obtain PR filters with integer coefficients, without the need for any additional constraint to be satisfied for guaranteeing the PR property. Similarly, orthonormal wavelet transforms can be easily implemented by de signing a two-channel paraunitary QMF bank and then using a tree structure. This means that the wavelet orthonormality properties can be retained even when the coefficients are quantized or subject to certain constraints.<<ETX>>

Serial turbo trellis coded modulation with rate-1 inner code

We develop new, low-complexity turbo codes suitable for bandwidth and power limited systems. These codes are constructed as an extension of repeat-accumulate codes to high-level-modulations. Two design criteria are proposed, based on maximum-likelihood decoding and on Gaussian density evolution in iterative decoding.

Finite-state codes

A class of codes called finite-state (FS) codes is defined and investigated. The codes, which generalize both block and convolutional codes, are defined by their encoders, which are finite-state machines with parallel inputs and outputs. A family of upper bounds on the free distance of a given FS code is derived. A general construction for FS codes is given, and it is shown that in many cases the FS codes constructed in this way have a free distance that is the largest possible. Catastrophic error propagation (CEP) for FS codes is also discussed. It is found that to avoid CEP one must solve the graph-theoretic problem of finding a uniquely decodable edge labeling of the state diagram. >