By Stefan M. Moser
This easy-to-read consultant presents a concise creation to the engineering heritage of recent verbal exchange platforms, from cell phones to information compression and garage. heritage arithmetic and particular engineering innovations are saved to a minimal in order that just a uncomplicated wisdom of high-school arithmetic is required to appreciate the fabric coated. The authors start with many useful purposes in coding, together with the repetition code, the Hamming code and the Huffman code. They then clarify the corresponding info concept, from entropy and mutual details to channel capability and the knowledge transmission theorem. eventually, they supply insights into the connections among coding conception and different fields. Many labored examples are given during the publication, utilizing useful functions to demonstrate theoretical definitions. workouts also are incorporated, permitting readers to double-check what they've got realized and achieve glimpses into extra complicated themes, making this excellent for a person who wishes a brief creation to the topic
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Extra resources for A Student’s Guide to Coding and Information Theory
Why do we need this assumption? Would the situation be different without this assumption? Take the case of Pr[s = 0] = 0 and Pr[s = 1] = 1 as an example. 9) = Pr(Writing in (0 0 0)) · Pr(0 → 0) · Pr(0 → 0) · Pr(0 → 1) 1 = · (1 − p) · (1 − p) · p 2 (1 − p)2 p = . 2 Similarly, we have Pr(Writing in (1 1 1) and reading out (0 0 1)) = (1 − p)p2 . 13) These together show that Pr(Reading out (0 0 1)) = Pr(Writing in (0 0 0) and reading out (0 0 1)) + Pr(Writing in (1 1 1) and reading out (0 0 1)) (1 − p)p .
38) The last logic deduction relies on the following two facts: (1) we assume there is only one bit in error, and (2) all columns of H are distinct. With the above arguments, we get the following result. 14 lowing: The (7, 4) Hamming code can be classified as one of the fol- (1) a single-error-correcting code, (2) a double-error-detecting code. Proof The proof of the Hamming code being able to correct all one-bit errors follows from the same logic deduction given above. To establish the second claim, simply note that when two errors occur, there are two 1s in the error pattern e, for example e = (1 0 1 0 0 0 0).
Hence, Pr( n=1 A ) is the event that all events A occur at the same time. The notation ∏ is a shorta1 · a2 · · · an . hand for multiplication: ∏n=1 a 16 Error-detecting codes The probability of a single error in the message is given by np(1 − p)n−1 . 14) errors is given by the ( + 1)th term in the binomial exn 1 = (1 − p) + p n n 2 n p(1 − p)n−1 + p (1 − p)n−2 = (1 − p)n + 1 2 0 n n p +···+ n n(n − 1) 2 p (1 − p)n−2 + · · · + pn . 17) n(n − 1) 2 p (1 − p)n−2 . 18) 2 We can obtain the probability of an even number of errors (0, 2, 4, .
A Student’s Guide to Coding and Information Theory by Stefan M. Moser