By Solomon W. Golomb

Basic options in details concept and Coding is an outgrowth of a one­ semester introductory path that has been taught on the collage of Southern California because the mid-1960s. Lecture notes from that path have advanced according to pupil response, new technological and theoretical increase­ ments, and the insights of college individuals who've taught the direction (in­ cluding the 3 of us). In proposing this fabric, we've made it obtainable to a extensive viewers through restricting necessities to uncomplicated calculus and the ele­ mentary thoughts of discrete chance conception. to maintain the cloth compatible for a one-semester path, we now have restricted its scope to discrete details thought and a basic dialogue of coding thought with no precise remedy of algorithms for encoding and deciphering for numerous particular code periods. Readers will locate that this ebook deals an surprisingly thorough remedy of noiseless self-synchronizing codes, in addition to the good thing about challenge sections which have been honed through reactions and interactions of a number of gen­ erations of brilliant scholars, whereas Agent 00111 presents a context for the dialogue of summary concepts.

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Chapter 1 40 Thus we see that for an irreducible homogeneous Markov source of period I, the entropy of the source is given by H(M IMOO) = lim -kl H(MI X ... X M k ) k-oo fit lim P(k) = k-oo (77) where fi is defined in Equation 67. As a word of caution, remember that the discussion here has assumed a finite number of states in the source. Generalizing to infinite state sources may lead to difficulties. A final result for homogeneous irreducible Markov sources involves the relationship between long sample sequences generated by the information source and the statistical description of the source.

M(k)] M(lj, ... ,M(k)EM (81) As k increases, this mean value approaches the entropy of the source. lim E{Id k-'>oo = H(M IMOO) (82) If the value of h for a particular sequence M(l), ... , t of its (83) then M(1), ... , M(k) is said to be t-typical. We now prove the following relations for homogeneous irreducible Markov sources. 2a. Asymptotic Equipartition Property (Shannon-McMillan Theorem) The sequence of random variables h, k = 1,2, ... , converges in probability to the source entropy H(M IMOO); that is, for any t > 0, lim Pr{ Ih - H(MIMoo)1 ~ t} = k-'>oo ° Proof: The random variable h can be rewritten in state notation for the nih-order Markov source as 48 Chapter 1 h == == k1 log Pr[s(n), sen + 1), ...

Using Chebychev's inequality from probability theory, develop your own proof of the asymptotic equipartition property for memoryless information sources having finite symbol alphabets. 3. Consider the following functions + kb e ak + e bk j;(k) = e ak fz(k) = Jj(k) = ea(k+b) ft(k) = kbe ak J5(k) = bke ak where a and b are constants. What are the asymptotic exponential rates of growth of these functions as k increases? Introduction 53 4. Clarify the role of the choice of information units in the asymptotic equipartition property and the bounds on a typical sequence count by indicating what changes would have to be made if the entropy H(M IMOO) were specified in bits.

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