lecture-1.md (1736B)
1 +++ 2 title = "Lecture 1" 3 template = "page-math.html" 4 +++ 5 6 Some definitions: 7 - digit: 0, 1 8 - word: sequence of digits 9 - length: digits in word (|word|) 10 - binary code: set C of words 11 12 assumptions about transmission channel: 13 - length of sent == length of received 14 - easy to find beginning of first word 15 - noise scattered randomly (not in bursts) 16 17 properties of binary channel: 18 - symmetric if 0 and 1 transmitted with same accuracy 19 - reliability: probability that digit sent == digit received 20 - we assume ½ ≤ p < 1 21 22 information rate of code (length n) = $\frac{1}{n} \log_{2} |C|$ 23 24 ## Most likely codeword 25 Let $\phi_{p} (v,w)$ be probability that if word v sent over BSC with reliability p, word w is received. 26 27 $\phi_{p} (v, w) = p^{n-d} (1-p)^d$ if v and w disagree in d positions. 28 29 if v₁ and w disagree in d₁, and v₂ and w in d₂, then $\phi_{p} (v_{1}, w) \leq \phi_{p} (v_{2}, w)$ iff d₁ ≥ d₂. 30 - English: the most likely word disagrees in least digits 31 32 ## Weight and distance 33 34 K = {0, 1}, $K^{n}$ = set of all binary words of length n 35 36 (Hamming) weight: number of ones 37 38 (Hamming) distance: number of differing digits between words 39 40 ## Max likelihood decoding (MLD) 41 Complete: 42 - if one word min distance, decode to that 43 - else, arbitrarily select one of closest 44 45 Incomplete: 46 - if one word min distance, decode to that 47 - else, ask for retransmission 48 - look for smallest weight in error patterns with C, e.g. 0+w and 1+w 49 - retransmit if same weight 50 51 Reliability: probability that if v sent over BSC of prob b, then IMLD concludes that v was sent 52 53 $\theta_{p} (c, v) = \sum_{w \in L(v)} \phi_{p} (v, w)$ where $L(v) = \lbrace words \in K^{n} \enspace \| \enspace \text{IMLD correctly concludes v sent} \rbrace$ 54 55