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Lenses

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

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

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

What are tags?

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

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Inside Collection (Course):

Course by: Behnaam Aazhang. E-mail the author

Channel Coding

Module by: Behnaam Aazhang. E-mail the author

Summary: A description of channel coding, in particular linear block codes.

Channel coding is a viable method to reduce information rate through the channel and increase reliability. This goal is achieved by adding redundancy to the information symbol vector resulting in a longer coded vector of symbols that are distinguishable at the output of the channel. Another brief explanation of channel coding is offered in Channel Coding and the Repetition Code. We consider only two classes of codes, block codes and convolutional codes.

Block codes

The information sequence is divided into blocks of length k k. Each block is mapped into channel inputs of length n n. The mapping is independent from previous blocks, that is, there is no memory from one block to another.

Example 1

k=2 k 2 and n=5 n 5

0000000 00 00000
(1)
0110100 01 10100
(2)
1001111 10 01111
(3)
1111011 11 11011
(4)
information sequence ⇒ codeword (channel input)

A binary block code is completely defined by 2k 2 k binary sequences of length n n called codewords.

C= c 1 c 2 c 2 k C c 1 c 2 c 2 k
(5)
c i 01n c i 0 1 n
(6)
There are three key questions,
1. How can one find "good" codewords?
2. How can one systematically map information sequences into codewords?
3. How can one systematically find the corresponding information sequences from a codeword, i.e., how can we decode?
These can be done if we concentrate on linear codes and utilize finite field algebra.

A block code is linear if ciC c i C and cjC c j C implies cicjC c i c j C where is an elementwise modulo 2 addition.

Hamming distance is a useful measure of codeword properties

d H cicj= # of places that they are different d H c i c j # of places that they are different
(7)
Denote the codeword for information sequence e 1 =100000 e 1 1 0 0 0 0 0 by g 1 g 1 and e 2 =010000 e 2 0 1 0 0 0 0 by g 2 g 2 ,…, and e k =000001 e k 0 0 0 0 0 1 by g k g k . Then any information sequence can be expressed as
u= u 1 u k = i =1k u i e i u u 1 u k i 1 k u i e i
(8)
and the corresponding codeword could be
c= i =1k u i g i c i 1 k u i g i
(9)
Therefore
c=uG c u G
(10)
with c=01n c 0 1 n and u01k u 0 1 k where G= g 1 g 2 g k G g 1 g 2 g k , a k kxn n matrix and all operations are modulo 2.

Example 2

In Example 1 with

0000000 00 00000
(11)
0110100 01 10100
(12)
1001111 10 01111
(13)
1111011 11 11011
(14)
g 1 =01111T g 1 0 1 1 1 1 and g 2 =10100T g 2 1 0 1 0 0 and G=( 01111 10100 ) G 0 1 1 1 1 1 0 1 0 0

Additional information about coding efficiency and error are provided in Block Channel Coding.

Examples of good linear codes include Hamming codes, BCH codes, Reed-Solomon codes, and many more. The rate of these codes is defined as kn k n and these codes have different error correction and error detection properties.

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PDF | EPUB (?)

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Collection to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags?

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

Module to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags?

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