By Jørn Justesen, Søren Forchhammer

This entire advent to two-dimensional (2-D) info conception and coding presents the main ideas for modeling info and estimating their details content material. all through, detailed emphasis is put on functions to transmission, garage, compression, and mistake defense of photo details. The e-book starts off with a self-contained advent to info concept, together with techniques of entropy and channel means, which calls for minimum mathematical heritage wisdom. It then introduces error-correcting codes, relatively Reed-Solomon codes, the fundamental tools for error-correction, and codes appropriate to facts prepared in 2-D arrays. universal innovations for information compression, together with compression of 2-D facts in line with software of the fundamental resource coding, also are coated, including a complicated bankruptcy devoted to 2-D restricted coding for garage functions. quite a few labored examples illustrate the speculation, while end-of-chapter workouts try the reader's figuring out, making this an incredible ebook for graduate scholars and likewise for practitioners within the telecommunications and information garage industries.

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**Example text**

5) is satisfied. 6) where H is the binary entropy function. By using this approximation in the expression for the weight distribution and taking logarithms we get a bound for long codes, namely the Gilbert bound (asymptotic version): there exist long binary (n, k, d) codes when d/n and the rate R = k/n satisfy H (d/n) < 1 − R. 7) It is not known whether there exist long binary codes with fixed rate 0 < R = k/n < 1 and larger minimum distances. Also there is no known construction (with complexity polynomial in n) for constructing long codes that reach this bound.

The Markov chain is given by P(1|1) = 1/2 and P(0|0) = 7/8. (a) Calculate the stationary distribution and the entropy of the Markov chain. (b) The black and white runs will alternate. Determine two Huffman codes, one for coding white runs and one for coding black runs. (For the white runs you introduce an escape character for runs longer than, say, 5. ) (c) Calculate the average code length for the run-length coding in (b). 3 A Markov chain with alphabet {a, b, c} has transition probabilities 1/2 1/2 M = 0 1/2 1/2 0 0 1/2.

2). If the symbol distribution is known, we can consider a more limited maximization. Consider a finite-state source with given 0/1 adjacency matrix, T. Let the states and the output be identified as in Markov chains, and let the stationary probability distribution, p∗ , on the symbols be given. We can then find the Markov chain with maximum entropy which is consistent with T and has a given stationary distribution, p∗ . First note that, if the columns of the transition matrix, P, for a Markov chain are multiplied by the corresponding stationary probability, the rows also sum to the stationary distribution.