II.%20Linear%20Block%20Codes

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II. Linear Block Codes
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Digital Communication Systems
Source of Information
User of Information
Source Encoder
Source Decoder
Channel Encoder
Channel Decoder
Modulator
De-Modulator
Channel
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Motivation for Channel Coding
B
B

• PrB?Bp
• For a relatively noisy channel, p (i.e.,
  probability of error) may have a value of 10-2
• For many applications, this is not acceptable
• Examples
• Speech Requirement PrB?Blt10-3
• Data Requirement PrB?Blt10-6
• Channel coding can help to achieve such a high
  level of performance

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Channel Coding
W1W2.. Wn
B1B2.. Bk
B1B2.. Bk
W1W2.. Wn
Channel Decoder
Channel Encoder
Physical Channel

• Channel Encoder Mapping of k information bits in
  to an n-bit code word
• Channel Decoder Inverse mapping of n received
  code bits back to k information bits
• Code Rate rk/n
• rlt1

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What are Linear Block Codes?
Linear Block Codes

• Information sequence is segmented into message
  blocks of fixed length.
• Each k-bit information message is encoded into an
  n-bit codeword (ngtk)

Binary Block Encoder
2k n-bit DISTINCT codewords
2k k-bit Messages
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What are Linear Block Codes?
Linear Block Codes

• Modulo-2 sum of any two codewords is
• also a codeword
• Each codeword v that belongs to a block code C is
  a linear combination of k linearly independent
  codewords in C, i.e.,

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Linear Independence

• A set of vectors g0, g1,, gk-1 are linearly
  independent if there exists no scalars u0, u1,,
  uk-1 that satisfy

Unless u0u1 uk-10

• Examples
• 0 1 0 , 1 0 1, 1 1 1 are
• Linearly Dependent
• 0 1 0 , 1 0 1, 0 0 1 are
• Linearly Independent

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Why Linear?

• Encoding Process
• Store and Index 2k codewords of length n
• Complexity
• Huge storage requirements for large k
• Extensive search processing for large k
• Linear Block Codes
• Stores k linearly independent codewords
• Encoding process through linear combination of
  codewords g0, g1,, gk-1 based on input message
  uu0, u1,, uk-1

Generator Matrix
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Example
Message Codeword
0000 0000000
0001 1010001
0010 1110010
0011 0100011
0100 0110100
0101 1100101
0110 1000110
0111 0010111
1000 1101000
1001 0111001
1010 0011010
1011 1001011
1100 1011100
1101 0001101
1110 0101110
1111 1111111
g3
g2
g1
u 0 1 1 0
g0
Linear Block Encoder (vu.G)
v g1g2 v 1 0 0 0 1 1 0
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Example
Linearly Dependent
u 1 0 0 1
u 0 1 1 1
Block Encoder (vu.G)
Block Encoder (vu.G)
v g1g2g3 v 0 1 1 1 0 0 1
v g0g3 v 0 1 1 1 0 0 1
NOT DISTINCT
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Linear Systematic Block Codes
n-k bits
k bits
Redundant Checking Part
Message Part
p-matrix
kxk- identity matrix
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The Parity Check Matrix

• For any k x n matrix G with k linearly
  independent rows, there exists an (n-k) x n
  matrix H (Parity Check Matrix), such that
• G.HT0

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Example
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Encoding Circuit
Input u
Message Register
u0
u1
u2
u3
To channel
Output v



v1
v2
v0
Parity Register
v0 v1 v2 u0 u1 u2 u3
u0 u1 u2 u3
Encoder Circuit
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Syndrome

• Characteristic of parity check matrix (H)

v
r
Channel
Syndrome
v
rve

e
Error Pattern
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Error Detection
r is NOT a codeword
An Error is Detected What Options do we have?

• Ask for Retransmission of Block
• Automatic Repeat Request (ARQ)
• Attempt the Correction of Block
• Forward Error Correction

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Undetectable Error Patterns

• Can we be sure that rv ??
• NO! WHY?

• How many undetectable error patterns exist?
• 2k-1 Nonzero codeword means
• 2k-1 undetectable error patterns

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Syndrome Circuit
r1
r2
r3
r4
r5
r6
r0



s0
s1
s2