This paper presents a new family of convolutional codes, nicknamed turbo-codes, built from a particular concatenation of two recursive systematic codes, linked together by nonuniform interleaving. Decoding calls on iterative processing in which each component decoder takes advantage of the work of the other at the previous step, with the aid of the original concept of extrinsic information. For sufficiently large interleaving sizes, the correcting performance of turbo-codes, investigated by simulation, appears to be close to the theoretical limit predicted by Shannon.

/pdf/near-optimum-error-correcting-coding-and-decoding-turbo-1rr3xsfa1x.pdf

Near optimum error correcting coding and decoding: turbo-codes

Information Theory and Reliable Communication

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

This document provides updates to IEEE Std 802.16's MIB for the MAC, PHY and asso- ciated management procedures in order to accommodate recent extensions to the standard.

IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems Draft Amendment: Management Information Base Extensions

We estimated the world’s technological capacity to store, communicate, and compute information, tracking 60 analog and digital technologies during the period from 1986 to 2007. In 2007, humankind was able to store 2.9 × 10 20 optimally compressed bytes, communicate almost 2 × 10 21 bytes, and carry out 6.4 × 10 18 instructions per second on general-purpose computers. General-purpose computing capacity grew at an annual rate of 58%. The world’s capacity for bidirectional telecommunication grew at 28% per year, closely followed by the increase in globally stored information (23%). Humankind’s capacity for unidirectional information diffusion through broadcasting channels has experienced comparatively modest annual growth (6%). Telecommunication has been dominated by digital technologies since 1990 (99.9% in digital format in 2007), and the majority of our technological memory has been in digital format since the early 2000s (94% digital in 2007).

/pdf/the-world-s-technological-capacity-to-store-communicate-and-39dybc0mey.pdf

The World’s Technological Capacity to Store, Communicate, and Compute Information

From the Publisher:
Convolutional codes, among the main error control codes, are routinely used in applications for mobile telephony, satellite communications, and voice-band modems. Written by two leading authorities in coding and information theory, this book brings you a clear and comprehensive discussion of the basic principles underlying convolutional coding. This book can be used as a textbook for graduate-level electrical engineering students. It will be of key interest to researchers and engineers of wireless and mobile communication, satellite communication, and data communication.

Fundamentals of Convolutional Coding

A fast algorithm for searching a tree (FAST) is presented for computing the distance spectrum of convolutional codes. The distance profile of a code is used to limit substantially the error patterns that have to be searched. The algorithm can easily be modified to determine the number of nonzero information bits of an incorrect path as well as the length of an error event. For testing systematic codes, a faster version of the algorithm is given. FAST is much faster than the standard bidirectional search. On a microVAX, d/sub infinity /=27 was verified for a rate R=1/2, memory M=25 code in 37 s of CPU time. Extensive tables of rate R=1/2 encoders are given. Several of the listed encoders have distance spectra superior to those of any previously known codes of the same rate and memory. A conjecture than an R=1/2 systematic convolutional code of memory 2M will perform as well as a nonsystematic convolutional code of memory M is given strong support. >

A fast algorithm for computing distance spectrum of convolutional codes

/pdf/a-fast-algorithm-for-computing-distance-spectrum-of-3szeg1yn8d.pdf

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-one matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.

/pdf/searching-for-voltage-graph-based-ldpc-tailbiting-codes-with-4hd8nq4chl.pdf

Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth

The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra. >

Rolf Johannesson

Papers

Fundamentals of Convolutional Coding

A fast algorithm for computing distance spectrum of convolutional codes

A fast algorithm for computing distance spectrum of convolutional codes

Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth

A linear algebra approach to minimal convolutional encoders