#### Lecture25

Fort Lewis, CSIS 190
Excerpt: ... CSIS 190 Lecture 25 Announcements Exam Friday. Review Wednesday. My notes are on the course web site. Homework due Wednesday. Monday & Wednesday next week meeting in Reed Library computer lab 2D. Extra Credit come to my office MWF this week to help build the robots. Error Correcting Codes Last time we looked at error detecting codes when we read the data, we could use the parity bit or the check character to determine if the data was correct or not. But, if it was incorrect, we could not tell what the error was. Look at example code from book: Symbol A B C D E F G H Code 000000 001111 010011 011100 100110 101001 110101 111010 Define the Hamming distance as the number of bit positions where two codes differ. In this case, the minimum Hamming distance is 3. This is an important attribute of this set of codes. Suppose we receive the code 111100 what symbol does it represent? Well, we can calculate it's Hamming distance from the real codes: Symbol A B C D E F G H Code 000000 001111 010011 011100 100110 1 ...

#### Viterbi2

UMass (Amherst), ECE 559
Excerpt: ... ded sequences (basically a FSM transitioning with time) ECE 559 VLSI Design Project Viterbi Decoder Block Diagram Input Hamming Distance Compute Metric Compare Select Path Memory Output Path Select ECE 559 VLSI Design Project Hamming Distance Bit-wise XOR comparison of received channel symbol pair and possible channel symbol pairs. I.e. the hamming distance between 01 and 11 would be 1, and the hamming distance between 01 and 10 would be 2. ECE 559 VLSI Design Project Hamming Distance Distance with 2 Input 2 Hamming Distance Module 2 2 2 00 01 10 11 ECE 559 VLSI Design Project Compute Metric Add the previous accumulated error metric to the current calculated hamming distance . ECE 559 VLSI Design Project Compute Metric From Compare Select 3 3 3 3 2 From Hamming Distance 4 Compute Metric Module 2 2 2 . . . 4 8 Error Metrics ECE 559 VLSI Design Project Compare Select Determines the smallest accumulated error metric entering each state. Reduces magn ...

#### quiz5sol

Sveriges lantbruksuniversitet, CS 371
Excerpt: ... What a code (block code)? Give a step by step explanation of how a code used to detect and correct transmission errors. In your explanation define and use the concept of Hamming distance . A code is a representation of a k bit data sequence using n bits. Each of the 2k possible sequences of k bits of data is represented by a valid code word. A code word is one of 2n possible sequences of n bits. A valid code word is a code word selected from the possible code words to represent a particular sequence of k bits of data. The Hamming distance between two code words is the number of bits that differ between the two code words. For example the Hamming distance between the code words 1000111 and 1010101 is 2, because the 3rd and 6th bits are different in each word. Valid code words are chosen from the complete set of 2n code words to dothe following: maximize the minimum Hamming distance between any pair of valid code words. Provide a unique valid code word that is a minimum Hamming distance from ...

#### Lecture01

UNC, COMP 411
Excerpt: ... xx1 1xx1 x0x1 xx01 xxxx x00x 9 1010 1x1x 1x1x x01x xxxx xx10 x0x0 10xx 10 1100 11xx 11xx xxxx x10x x1x0 xx00 1x0x 1xx0 11 1111110 1111xx0 11111xx xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 12 1111011 11110xx 1111xx1 xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 1111x1x This difference is called the " Hamming distance " "A Hamming distance of 1 is needed to uniquely identify an encoding" Comp411 Fall 2006 8/23/2006 L01 - Introduction 26 A Short Detour It is illuminating to consider Hamming distance s geometrically. 10 00 01 11 Given 2 bits the largest Hamming distance that we can achieve between 2 encodings is 2. This allows us to detect 1-bit errors if we encode 1 bit of information using 2 bits. With 3 bits we can find 4 encodings with a Hamming distance of 2, allowing the detection of 1-bit errors when 3 bits are used to encode 2 bits of information. 110 010 011 111 100 000 001 101 We can also identify 2 encodings with Hamming distance 3. This extra distance allows us to detect 2-bit errors. However, we could use t ...

#### errorCorrectingCodes

University of Illinois, Urbana Champaign, CS 231
Excerpt: ... or(s) detected need not be confined to the word A but an error in the parity bit will also be detected.1 Error Correction 1 In the above scheme the total number of 1s in the vector {a 0,a 1, . . . , an 1} is even; therefore, it is called an even parity scheme. If following equation is used to compute p then it is called an odd parity scheme. p = 1 a 0 a 1 . . . an 1 Definition: For two binary vectors A = {a 0,a 1, . . . ,an 1} and B = {b 0,b 1, . . . ,bn 1} the Hamming distance H(A,B) is i =0 (a i bi) i =n 1 Example For A = (0 1 0 0), B = (0 0 1 0) H(A,B) = 2 Another way to look at the Hamming distance s is that if A and B are nodes of a binary ncube then the Hamming distance between them is the minimum number of links traversed to get from node A to B (or from node B to A). It is also evident that two adjacent nodes on the binary n-cube have a Hamming distance of one. In order to introduce redundancy, we add extra bits to a data word and the result is called a code ...

#### Lecture3

University of Illinois, Urbana Champaign, ECE 556
Excerpt: ... us the generator matrix for the dual code C . Observe that G H T = 0. DEFINITION 2.3 (Systematic form). A matrix is said to be in systematic form if it contains I in some column order. So, if a code has generator matrix of the form G = [IP] then the dual code has the generator matrix H = [PT I]. Here, I stands for the identity matrix of appropriate dimension(k or n k). LECTURE 3 3 3. Hamming distance of a code We will now try to nd out how we can identify errors and correct them in linear codes where bits might be ipped in the communication channel. DEFINITION 3.1 ( Hamming distance of a code). Let C be a [n, k] linear code. The minimum Hamming distance of C is the smallest Hamming distance between any two code words of C . If a code has Hamming distance d then it means that there exists two code words which are distance d away from each other. Therefore if there are at most t = d1 errors, 2 we can correct it. We see that just specifying that a code has parameters [n, k] code is not suf ...

#### ps1

Duke, CPS 6096
Excerpt: ... . Here we explore the probabilistic properties of a simple version of random projections. 4.1 Probability of a perfect match Given two sequences v and w of the same length, let the Hamming distance between them, dH (v, w) be the number of positions in which they dier. This is a very useful notion of distance for sequences. Let the Hamming distance between two strings of length d is zero, then if we take k random positions (with replacement), what is the probability that the two strings agree at these position? What does this imply about the sensitivity of the approach? What is the probability that the two strings agree at these position if the Hamming distance is one? 4.2 Probability of a match given the Hamming distance In general, if the Hamming distance between two sequences v, w is dH (v, w), what is the probability that k random positions agree? 4.3 Collision plots A collision is an event in which two sequences have the same random projection. Plot the probability of collision of two sequence ...

#### ps1

Duke, CPS 6096
Excerpt: ... 4.1 Probability of a perfect match Given two sequences v and w of the same length, let the Hamming distance between them, dH (v, w) be the number of positions in which they differ. This is a very useful notion of distance for sequences. Let the Hamming distance between two strings of length d is zero, then if we take k random positions (with replacement), what is the probability that the two strings agree at these position? What does this imply about the sensitivity of the approach? What is the probability that the two strings agree at these position if the Hamming distance is one? 4.2 Probability of a match given the Hamming distance In general, if the Hamming distance between two sequences v, w is dH (v, w), what is the probability that k random positions agree? 4.3 Collision plots A collision is an event in which two sequences have the same random projection. Plot the probability of collision of two sequences against their Hamming distance for d = 10 and k = 3. How does it change as k increases i. ...

#### Lecture01

UNC, COMP 120
Excerpt: ... 0 + 0 = 2 11-1111110 = 1 + 1 + 1 + 1 + 1 + 1 + 0 = 6 12-1111011 = 1 + 1 + 1 + 1 + 0 + 1 + 1 = 6 1/12/2006 Lecture Property 2: Separation Page Each encoding differs from all others by at least two bits in their overlapping parts 3 4 0011 xx11 xx11 5 0101 x1x1 x1x1 0xx1 6 0110 x11x x11x 0x1x 01xx 7 0000 xxxx xxxx 00xx 0x0x 0xx0 8 1001 1xx1 1xx1 x0x1 xx01 xxxx x00x 9 1010 1x1x 1x1x x01x xxxx xx10 x0x0 10xx 10 1100 11xx 11xx xxxx x10x x1x0 xx00 1x0x 1xx0 11 1111110 1111xx0 11111xx xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 12 1111011 11110xx 1111xx1 xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 1111x1x How much information is in the last bit? 1111101 2 3 4 5 6 7 8 9 10 11 1111000 1111101 0011 0101 0110 0000 1001 1010 1100 1111110 1111x0x This difference is called the Hamming distance A Hamming distance of 1 is needed to uniquely identify an encoding Comp120 Spring 2006 1/12/2006 L01 - Introduction 25 Comp120 Spring 2006 1/12/2006 L01 - Introduction 26 A Short Detour It is illuminating to consider H ...

#### project1

UMBC, CS 313
Excerpt: ... CMSC 313, Computer Organization & Assembly Language Programming Fall 2003 Project 2: Hamming Distance Due: Tue 09/23/03, Section 0101 (Chang) & Section 0301 (Macneil) Wed 09/24/03, Section 0201 (Patel & Bourner) Objective The objective of this programming project is to practice designing your own loops and branching code in assembly language and to gain greater familiarity with the i386 instructions set. Assignment Write an assembly language program that prompts the user for two input strings and computes the Hamming distance between the two strings. The Hamming distance is the number of bit positions where the two strings differ. For example, the ASCII representations of the strings "foo" and "bar" in binary are: "foo" = 0110 0110 0110 1111 0110 1111 "bar" = 0110 0010 0110 0001 0111 0010 So, the Hamming distance between "foo" and "bar" is 8. Some details: Your program must return the Hamming distance of the two strings as the exit status of the program. This is the value stored in the EBX register just ...

#### lecture27

Sveriges lantbruksuniversitet, CS 471
Excerpt: ... CMPT 371 Data Communications and Networking Error Correction 1 Error Correction We have already looked at ways of identifying errors if they exist. Once we know we have an error what can we do? Correct the error (forward error correction, FEC) Retransmit the corrupted data (error control) To understand how Forward error correction is done we first need to understand the concepts of Hamming Distance and block coding Error control has been considered along with flow controw Janice Regan 2005 2 1 Codes Consider a incoming data stream from which we are accepting blocks of k bits. There are 2k possible different sequences of bits Each possible sequence of bits is to be represented by a codeword of length n>k bits Not all 2n possible sequences of n bits will be used to represent the 2k possible input data sequences. A valid codeword is one of the 2n possible sequences that is used to represent a possible input data sequence (to be discussed with encoding) How does one decide which will be the valid code wo ...

#### paper-37-notes

Wisconsin, ECE 902
Excerpt: ... x number calculations : This involves more complex calculations. With one MAC maybe 10 instructions would be needed.With more MACs memory would be a bottleneck. (note there are no register files here). Distance calculations take the most number of bits. There are two types of distance calculations (1) Square distance (2) Hamming distance . Hamming distance does not need MAC and only MCA unit is used. It counts how many 1s are there is a particular pattern. (Note: counting the number of 1s in software is very difficult). The ACS is specialized FIFO queue and decides which way the branch will go. Experimental Results They claim that MCA area overhead is small. They report results in MIPS but it is not definite if clock cycle time is included. Also this does not include the number of cycles of an instruction. Also in the comparison with TMS320C54x they do not address cycle time. Also they do not show comparison results for values of r>1. Power and area have not been mentioned. They do not ...

#### L01-Introduction

UNC, COMP 411
Excerpt: ... x11 xx11 5 0101 x1x1 x1x1 0xx1 6 0110 x11x x11x 0x1x 01xx 7 0000 xxxx xxxx 00xx 0x0x 0xx0 8 1001 1xx1 1xx1 x0x1 xx01 xxxx x00x 9 1010 1x1x 1x1x x01x xxxx xx10 x0x0 10xx 10 1100 11xx 11xx xxxx x10x x1x0 xx00 1x0x 1xx0 11 1111110 1111xx0 11111xx xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 12 1111011 11110xx 1111xx1 xx11 x1x1 x11x xxxx 1xx1 1x1x 11xx 1111x1x This difference is called the " Hamming distance " "A Ham ing distanceof 1 m is ne de to unique e d ly ide ntify an e ncoding" Comp411 Fall 2006 8/23/2006 L01 - I ntroduction 26 A Short Detour It is illuminating to consider Hamming distance s geometrically. 10 00 01 11 is 2. if we Given 2 bits the largest Hamming distance that we can achieve between 2 encodings This allows us to detect 1-bit errors encode 1 bit of information using 2 bits. 110 010 With 3 bits we can find 4 encodings with a Hamming distance of 2, allowing the detection of 1-bit errors when 111 3 bits are used to encode 2 bits of information. 110 011 010 011 111 100 000 001 101 Encodings ...

#### CS231Lecture6

Vanderbilt, CS 231
Excerpt: ... ror Detection For our \$17 million scenario Transmit 100012 and parity bit Chosen to make number of 1 bits even (or odd) Is there a difference between even and odd parity? For even parity codeword: 1000102 Received codeword is 0000102 Incorrect parity means error in received data Parity detects any single-bit error Actually detects any odd number of errors 13 14 Terminology Hamming Distance Number of bit positions in which two codewords differ Parity Check For even or odd parity dmin equals 2 Detects error patterns (dmin - 1) Generates an error flag Minimum Distance Minimum Hamming distance between all distinct pairs of codewords Uses of parity check Memory Disk storage A code with minimum distance dmin can detect all error patterns of weight less than or equal to (dmin 1) 15 16 Where Do Errors Come From? Voltage spikes from power supply Coupling with nearby signals Noise on transmission line Radiation Manufacturing defect Etc. One Step Further ...

#### lecture01

Toledo, ECE 1501
Excerpt: ... ng a third error! In fact, as the reader may verify, the decoder will always introduce a third error when presented with a codeword that has been corrupted in two distinct positions. This tells us that error-correction does not always help in reducing the error rate of the channel: if the channel error rate is too high, the decoder may become confused and produce an output that is even noisier than the output of the channel! 5 Hamming Distance Recall that a metric (or distance function) dened on a set X is a function d : X X [0, ) that satises for all x, y, z X 1. non-negativity: d(x, y) 0, with d(x, y) = 0 if and only if x = y; 2. symmetry: d(x, y) = d(y, x); 3. the triangle inequality: d(x, y) d(x, z) + d(z, y). In this course, the so-called Hamming distance , dened on n-tuples over alphabets, plays a pivotal role. Lecture Notes, Winter 2007 7 University of Toronto ECE 1501Error Control Codes Denition: The Hamming distance between two n-tuples x and y o ...

#### disc14

University of Illinois, Urbana Champaign, CS 232
Excerpt: ... not be conned to the word A but an error in the parity bit will also be detected. ERROR CORRECTION Denition For two binary vectors A = {a0 , a1 , ., an1 } and B = {b0 , b1 , ., bn1 }, the Hamming distance H(A, B) is the number of bits of A we need to ip to get B. In other words, mathematically, H(A, B) = i=n1 (ai bi ) i=0 Example: For A = (0100), B = (0010) H(A, B) = 2 Another way to look at the Hamming distance s is that if A and B are nodes of a binary n-cube then the Hamming distance between them is the minimum number of links traversed to get from node A to B (or from node B to A). It is also evident that two adjacent nodes on the binary n-cube have a Hamming distance of one. In order to introduce redundancy, we add extra bits to a data word and the result is called a codeword. The extra bits are selected in such a way that not only the Hamming distance between any pair of code words is large but the minimum of all the Hamming distance s 1 CS232 Spring 2006 Discussion 14: EC ...

#### CNI5e Ch 08

CSU San Marcos, CS 436
Excerpt: ... Computer Networks and Internets, 5e By Douglas E. Comer Lecture PowerPoints By Lami Kaya, LKaya@ieee.org 2009 Pearson Education Inc., Upper Saddle River, NJ. All rights reserved. 1 Chapter 8 Reliability and Channel Coding 2009 Pearson Education Inc., Upper Saddle River, NJ. All rights reserved. 2 Topics Covered 8.1 Introduction 8.2 The Three Main Sources of Transmission Errors 8.3 Effect of Transmission Errors on Data 8.4 Two Strategies for Handling Channel Errors 8.5 Block and Convolutional Error Codes 8.6 An Example Block Error Code: Single Parity Checking 8.7 The Mathematics of Block Error Codes and (n, k) Notation 8.8 Hamming Distance : A Measure of a Code's Strength 8.9 The Hamming Distance Among Strings in a Codebook 8.10 The Tradeoff Between Error Detection and Overhead 8.11 Error Correction with Row and Column (RAC) Parity 8.12 The 16-Bit Checksum Used in the Internet 8.13 Cyclic Redundancy Codes (CRCs) 8.14 An Efficient Hardware Implementation of CRC 8.15 Autom ...

#### lecture8

Loyola Maryland, CS 111
Excerpt: ... CS111 Lecture 8 Tuesday, Sept. 29th Administrivia 1. No quiz today. Really. 2. Reading: For next Tuesday read pages 99 to 136 in Murray 3. Online reading: Check out the Technology Services Technology for Undergraduate Students Go to Microsoft E-Learning Library (MELL) and try to run one Excel Course and one E-Book, each of your choice. Note they require a PC with MS XP. 4. Online powerpoint: the presentation from today is on the web site Assignment Due next Thursday (Oct. 6th) From Chapter 6 Page 253 Numbers 1, 2, 4, 5, 7. Page 255 Numbers 1, 4 Page 258 Numbers 3, 5, 6 Page 265 Numbers 1, 2, 4, 5, 6 Lecture Data codes vs. utility codes The codes here take already encoded data and modify it for a particular purpose security, data integrity, reducing storage requirements. I. Error detection and correction a. Parity bits b. Checksums c. Rectangular codes d. Hamming distance II. Encryption a. Caeser ciphers b. One time pads c. Psuedo-random d. Public key encryption (check out ...

#### ex2guide

SUNY Fredonia, CSIT 413
Excerpt: ... EXAM-II IS IN CLASS ON TUESDAY NOVEMBER 15 Max Points=100 (Weight 15% of final grade) Time = 80 minutes General Structure of Questions=5 to 7 Short Questions expecting short answers, design, problem solving Topics Included (Ch 4,5,6,7,9,12) 1. Cache principles, effective access time with cache, cache mapping 2. 3. 4. 5. options, replacement and write policy, DRAM and its variations, SRAM, EPROM, EEPROM definitions, Error detection and correction, Hamming distance , parity schemes, Hamming algorithm for error correction Secondary memory (Hard disk structure, tracks, sectors and cylinders), RAID various levels and their highlights, I/O modules, programmed I/O, interrupt driven I/O, Intel 82c59A and 82c55 controllers, types of DMA and its working mechanism, 8237 DMA controller 2s complement number representation, 2s complement addition and subtraction, unsigned multiplication, Booths algorithm for 2s complement multiplication, 2s complement division CPU Register block, instruction pipelining, p ...

#### disc11

University of Illinois, Urbana Champaign, CS 11
Excerpt: ... bit itself! Error correction The Hamming distance between binary vectors A = a0 , a1 , . . . , an1 and B = b0 , b1 , . . . , bn1 is the number of bits of A we need to ip to get B. Mathematically, i=n1 H(A, B) = i=0 (ai bi ) . Example: For A = (0100) and B = (0010), H(A, B) = 2 Another way to look at the Hamming distance s is that if A and B are nodes of a binary n-cube, then the Hamming distance between them is the minimum number of links traversed to get from node A to B (or from node B to A). It is also evident that two adjacent nodes on the binary n-cube have a Hamming distance of one. In order to introduce redundancy, we add extra bits to a data word the result is called a codeword. The extra bits are selected in such a way that the minimum Hamming distance s between all pairs of code words is maximized. 1 CS232 Spring 2007 Discussion 11: ECC Figure 1. The 3-cube Consider the case of appending a parity bit to each data word. Suppose every code word has an even numbers of 1s i ...

#### disc11

University of Illinois, Urbana Champaign, CS 232
Excerpt: ... Error correction The Hamming distance between binary vectors A = a0 , a1 , . . . , an1 and B = b0 , b1 , . . . , bn1 is the number of bits of A we need to ip to get B. Mathematically, i=n1 H(A, B) = i=0 (ai bi ) . Example: For A = (0100) and B = (0010), H(A, B) = 2 Another way to look at the Hamming distance s is that if A and B are nodes of a binary n-cube, then the Hamming distance between them is the minimum number of links traversed to get from node A to B (or from node B to A). It is also evident that two adjacent nodes on the binary n-cube have a Hamming distance of one. In order to introduce redundancy, we add extra bits to a data word the result is called a codeword. The extra bits are selected in such a way that the minimum Hamming distance s between all pairs of code words is maximized. 1 CS232 Discussion 11: ECC Figure 1. The 3-cube Consider the case of appending a parity bit to each data word. Suppose every code word has an even numbers of 1s in it. From this, we can ...

#### ErrorCD

UPenn, CIT 595
Excerpt: ... ould be in data or parity Not entirely fool proof CIT 595 16 - 5 CIT 595 16 - 6 Limitations of Parity Cannot Limitations of Parity (contd.) Parity Code (Even) 0 1 1 0 Code Word 000 011 101 110 100 101 110 111 determine which bit position has a problem If 001 is encountered, it is not a valid code-word and hence error is detected The correct code-word could either be 101 or 011 but we cannot tell Data Word 00 01 10 11 000 001 010 011 What happens if the code word is subjected to two-bit error? E.g. 011 became 000 while transmission According to parity scheme, no error is detected but error! In general, If an odd number of bits (including the parity bit) are changed from a set of bits then parity bit will be incorrect and will thus indicate that an error has occurred CIT 595 8 possible, only 4 correct code-words 16 - 7 CIT 595 16 - 8 2 Hamming Distance and Error Detection Hamming Distance = # of bit positions in which two code words differ E.g. 10001001 and 10110001 ha ...

#### cs274_lecture11

Indiana University-Purdue University Fort Wayne, CS 274
Excerpt: ... 74 Data Communications 6 Codeword nbit codeword m data bits r check bits n = m + r DQL CS 274 Data Communications 7 Hamming Distance The number of bit positions in which two codewords different two codewords 10001001 and 10110001 Hamming distance = 3 If two codewords are a Hamming distance d apart, it will require d singlebit errors to convert one into the other DQL CS 274 Data Communications 8 Example Complete Legal Codewords All possible nbit codewords: 2n All possible mbit data: 2m Out of total 2n codewords, only 2m are legal codewords (mbit data plus rbit check bits) Hamming distance for a complete codewords = the minimum Hamming distance between any two codewords in the complete code DQL CS 274 Data Communications 9 Questions a. Was a received legal codeword transmitted without errors? b. Does a received illegal codeword indicate a transmission error? DQL CS 274 Data Communications 10 Hamming Distance for Error Corre ...

#### Lecture6

Vanderbilt, CS 231
Excerpt: ... encoded 100012 and transmitted The received data is 000012 Only one bit was changed, but the impact is quite significant! 8 4 Parity Bit Simple Error Detection For our \$17 million scenario Transmit 100012 and parity bit Chosen to make number of 1 bits even (or odd) Is there a difference between even and odd parity? For even parity codeword: 1000102 Received codeword is 0000102 Incorrect parity means error in received data Parity detects any single-bit error Actually detects any odd number of errors 9 Terminology Hamming Distance Number of bit positions in which two codewords differ Minimum Distance Minimum Hamming distance between all distinct pairs of codewords A code with minimum distance dmin can detect all error patterns of weight less than or equal to (dmin 1) 10 5 Parity Check For even or odd parity dmin equals 2 Detects error patterns (dmin - 1) Generates an error flag Uses of parity check Memory ...