Skip to content Skip to navigation

Connexions

You are here: Home » Content » Noisy Channel Coding Theorem

Navigation

Lenses

What is a lens?

Definition of 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.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

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.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Fundamentals of Electrical Engineering I"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice DSS - Braille display tagshide tags

    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.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection:"Fundamentals of Electrical Engineering I"

    Comments:

    "The course focuses on the creation, manipulation, transmission, and reception of information by electronic means. It covers elementary signal theory, time- and frequency-domain analysis, the […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

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

Noisy Channel Coding Theorem

Module by: Don Johnson. E-mail the author

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: Describes the Noisy Channel Coding Theorem.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

As the block length becomes larger, more error correction will be needed. Do codes exist that can correct all errors? Perhaps the crowning achievement of Claude Shannon's creation of information theory answers this question. His result comes in two complementary forms: the Noisy Channel Coding Theorem and its converse.

Noisy Channel Coding Theorem

Let EE denote the efficiency of an error-correcting code: the ratio of the number of data bits to the total number of bits used to represent them. If the efficiency is less than the capacity of the digital channel, an error-correcting code exists that has the property that as the length of the code increases, the probability of an error occurring in the decoded block approaches zero.

∀E,E<C:limN→∞Prblock error=0 E E C N block error 0 (1)

Converse to the Noisy Channel Coding Theorem

If E>C E C , the probability of an error in a decoded block must approach one regardless of the code that might be chosen.

limN→∞Prblock error=1 N block error 1 (2)
These results mean that it is possible to transmit digital information over a noisy channel (one that introduces errors) and receive the information without error if the code is sufficiently inefficient compared to the channel's characteristics. Generally, a channel's capacity changes with the signal-to-noise ratio: As one increases or decreases, so does the other. The capacity measures the overall error characteristics of a channel—the smaller the capacity the more frequently errors occur—and an overly efficient error-correcting code will not build in enough error correction capability to counteract channel errors.

This result astounded communication engineers when Shannon published it in 1948. Analog communication always yields a noisy version of the transmitted signal; in digital communication, error correction can be powerful enough to correct all errors as the block length increases. The key for this capability to exist is that the code's efficiency be less than the channel's capacity. For a binary symmetric channel, the capacity is given by

C=1+ p e log2 p e +1− p e log21− p e bits/transmission C 1 p e 2 p e 1 p e 2 1 p e bits/transmission (3)
Figure 1 shows how capacity varies with error probability. For example, our (7,4) Hamming code has an efficiency of 0.570.57, and codes having the same efficiency but longer block sizes can be used on additive noise channels where the signal-to-noise ratio exceeds 0dB 0 dB .

Figure 1: The capacity per transmission through a binary symmetric channel is plotted as a function of the digital channel's error probability (upper) and as a function of the signal-to-noise ratio for a BPSK signal set (lower).
capacity of a channel
capacity of a channel (capacity1.png)

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of 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.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks