Connexions

You are here: Home » Content » Repetition Codes
Content Actions
Lenses

What is a lens?

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

  • This module is included inLens: Rice University Disability Support Services's Lens
    By: Rice University Disability Support ServicesAs a part of collection:"Fundamentals of Electrical Engineering I"

    Comments:

    "Electrical Engineering Digital Processing Systems in Braille."

    Click the "Rice DSS - Braille" link to see all content affiliated with them.

    Rice DSS - Braille

  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Fundamentals of Electrical Engineering I"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Also in these lenses

  • This module is included inLens: Connexions Books Available for Print on Demand
    By: ConnexionsAs a part of collection:"Fundamentals of Electrical Engineering I"

    Comments:

    "This book was assembled for print in July 07. A braille version of this book is being produced also."

    Click the "Printable Books" link to see all content selected in this lens.

    Printable Books
Tags

(?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Repetition Codes

Module by: Don Johnson

Summary: Explains the repetition code for error correction.

Perhaps the simplest error correcting code is the repetition code.
Repetition Code
sys32.png
Figure 1: The upper portion depicts the result of directly modulating the bit stream bn b n into a transmitted signal xt x t using a baseband BPSK signal set. R ' R ' is the datarate produced by the source coder. If that bit stream passes through a (3,1) channel coder to yield the bit stream cl c l , the resulting transmitted signal requires a bit interval TT three times smaller than the uncoded version. This reduction in the bit interval means that the transmitted energy/bit decreases by a factor of three, which results in an increased error probability in the receiver.
Here, the transmitter sends the data bit several times, an odd number of times in fact. Because the error probability p e p e is always less than 12 1 2 , we know that more of the bits should be correct rather than in error. Simple majority voting of the received bits (hence the reason for the odd number) determines the transmitted bit more accurately than sending it alone. For example, let's consider the three-fold repetition code: for every bit bn b n emerging from the source coder, the channel coder produces three. Thus, the bit stream emerging from the channel coder cl c l has a data rate three times higher than that of the original bit stream bn b n . The coding table illustrates when errors can be corrected and when they can't by the majority-vote decoder.
Coding Table
Code Probability Bit
000 1- p e 3 1 p e 3 0
001 p e 1- p e 2 p e 1 p e 2 0
010 p e 1- p e 2 p e 1 p e 2 0
011 p e 21- p e p e 2 1 p e 1
100 p e 1- p e 2 p e 1 p e 2 0
101 p e 21- p e p e 2 1 p e 1
110 p e 21- p e p e 2 1 p e 1
111 p e 3 p e 3 1
Figure 2: In this example, the transmitter encodes 0 0 as 000 000. The channel creates an error (changing a 0 0 into a 1 1) that with probability p e p e . The first column lists all possible received datawords and the second the probability of each dataword being received. The last column shows the results of the majority-vote decoder. When the decoder produces 00, it successfully corrected the errors introduced by the channel (if there were any; the top row corresponds to the case in which no errors occurred). The error probability of the decoders is the sum of the probabilities when the decoder produces 1 1.
Thus, if one bit of the three bits is received in error, the receiver can correct the error; if more than one error occurs, the channel decoder announces the bit is 1 instead of transmitted value of 0. Using this repetition code, the probability of b^ n≠0 b^ n 0 equals 3 p e 21- p e + p e 3 3 p e 2 1 p e p e 3 . This probability of a decoding error is always less than p e p e , the uncoded value, so long as p e <12 p e 1 2 .
Problem 1
Demonstrate mathematically that this claim is indeed true. Is 3 p e 21- p e + p e 3≤ p e 3 p e 2 1 p e p e 3 p e ?
[ Click for Solution 1 ]
Solution 1
This question is equivalent to 3 p e 1- p e + p e 2≤1 3 p e 1 p e p e 2 1 or 2 p e 2+-3 p e +1≥0 2 p e 2 3 p e 1 0 . Because this is an upward-going parabola, we need only check where its roots are. Using the quadratic formula, we find that they are located at 12 1 2 and 11. Consequently in the range 0≤ p e ≤12 0 p e 1 2 the error rate produced by coding is smaller.
[ Hide Solution 1 ]

Comments, questions, feedback, criticisms?

Send feedback