Solutions Solutions

Solutions to Chapte 7: Channel Coding(Bernard Sklar)

Note: State diagram, tree diagram and trellis diagram for K=3 are sameonly changes will occur in the output that depends upon the connectionvector.

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Solutions Solutions

7.1 Draw the state diagram, tree diagram and trellis diagram for the K=3, rate=1/3 code generated by

g1(X ) = X + X 2, g2(X ) = 1 + X + X 2, g3(X ) = 1 + X

Output

Input U

1

2

3

States represent possible contents of therightmost K-1 register content.

For this example there are only two transitionsfrom each state corresponding to two possibleinput bits.

Solid line denotes for input bit zero, and dashedline denotes for input bit one.

Tuple State00 a10 b01 c11 d

Table: State transition table (rate=1/2, K=3)

Input State at State at Outputbit time ti time ti + 10 00 00 0001 00 10 0110 01 00 1101 01 10 1010 10 01 1111 10 11 1000 11 01 0011 11 11 010

b=10 c=01

d=11

0(000)

a=000(110)1(011)

1(101)

0(111)

1(010)

1(100)

0(001)

Input bit

Output bit

Figure: State diagram for rate=1/3 and K=3

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Solutions Solutions

1

0

000

011

100

000000

111

011a

b011

a

010

111110

001

101c

d100

b

000a

100

100

11000

111

11a

b101c

010

001110

001

101c

d010d

011b

000

111

011

101

100

000000

111

011a

b011

a

010

111110

001

101c

d100

b

111c

010

100

110000

111

011a

b101c

010

001110

001

101c

d010d

100d

110

001

a

b

a

t1 t2 t3 t4 t5

Figure: Tree diagram for rate=1/3 andK=3

Steady State

a=00000

d=11

c=01

b=10

000 000000000

011 011

111

011011 011

010

100

111111

100100100

010010

001 001001

110110110101101101

t1 t5t4t3t2 t6

Figure: Trellis diagram for rate=1/3 and K=3

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Solutions Solutions

7.2 Given a K=3 and rate 1/2 binary convolutional code with the partially completed state diagram shown in Figure 7.1 find thecomplete state diagram, and sketch a diagram for encoder

10 01

11

001(11)

0(10)

1(00)

Input bit

Output bit

Figure: State diagram for rate=1/2, K=3

States represent possible contents of therightmost K-1 register content.

For this example there are only two transitionsfrom each state corresponding to two possibleinput bits.

OutputInput U

1

2

Figure: Encoder diagram

Table: State transition table (rate=1/2, K=3)

Input State at State at Outputbit time ti time ti + 10 00 00 001 00 10 110 01 00 011 01 10 100 10 01 101 10 11 010 11 01 111 11 11 00

10 01

11

0(00)

000(01)1(11)

1(10)

0(10)

1(00)

1(01)

0(11)

Input bit

Output bit

Figure: Modified State diagram

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Solutions

7.3 Draw the state diagram, tree diagram and trellis diagram for the convolutional encoder characterized by the block diagramin Figure p7.2

Output

Input U

1

2

States represent possible contents of therightmost K-1 register content.

For this example there are only two transitionsfrom each state corresponding to two possibleinput bits.

Solid line denotes for input bit zero, and dashedline denotes for input bit one.

Tuple State00 a10 b01 c11 d

Table: State transition table (rate=1/2, K=3)

Input State at State at Outputbit time ti time ti + 10 00 00 001 00 10 100 01 00 111 01 10 010 10 01 111 10 11 010 11 01 001 11 11 10

10 01

11

0(00)

a=000(11)1(10)

1(01)

0(11)

1(10)

1(01)

0(00)

Input bit

Output bit

Figure: State diagram for rate=1/3 and K=3Manjunatha. P (JNNCE) Coding Techniques July 5, 2013 5 / 11

Solutions

1

0

00

10

01

0000

11

10a

b10

a

10

1111

00

01c

d01

b

00a

01

01

1100

11

11a

b01c

10

0011

00

01c

d10d

10b

00

11

10

01

01

0000

11

10a

b10

a

10

1111

00

01c

d01

b

11c

10

01

1100

11

10a

b01c

10

0011

00

01c

d10d

01d

11

00

a

b

a

t1 t2 t3 t4 t5

Figure: Tree diagram for rate=1/3 andK=3

Steady State

a=0000

d=11

c=01

b=10

00 000000

10 10

11

1010 10

10

01

1111

010101

1010

00 0000

111111010101

t1 t5t4t3t2 t6

Figure: Trellis diagram for rate=1/3 and K=3

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Solutions Solutions

7.5 Consider the convolutional encoder shown in Fiugre a)Write the connection vectors and polynomials for this encoder.b)Draw the state diagram, tree diagram and trellis diagram

Solution: g1 = [1 0 1], g2 = [0 1 1]

g1(X ) = 1 + X 2, g2(X ) = X + X 2

Output

Input U

1

2

States represent possible contents of therightmost K-1 register content.

For this example there are only two transitionsfrom each state corresponding to two possibleinput bits.

Tuple State00 a10 b01 c11 d

Table: State transition table (rate=1/2, K=3)

Input State at State at Outputbit time ti time ti + 10 00 00 001 00 10 100 01 00 111 01 10 010 10 01 011 10 11 110 11 01 101 11 11 00

10 01

11

0(00)

000(11)1(10)

1(01)

0(01)

1(00)

1(11)

0(10)

Input bit

Output bit

Figure: State diagram for rate=1/3 and K=3Manjunatha. P (JNNCE) Coding Techniques July 5, 2013 7 / 11

Solutions Solutions

1

0

00

10

11

0000

01

10a

b10

a

00

0111

10

01c

d11

b

00a

11

11

1100

01

11a

b01c

00

1011

10

01c

d00d

10b

00

01

10

01

11

0000

01

10a

b10

a

00

0111

10

01c

d11

b

01c

00

11

1100

01

10a

b01c

00

1011

10

01c

d00d

11d

11

10

a

b

a

t1 t2 t3 t4 t5

Figure: Tree diagram for rate=1/3 andK=3

Steady State

a=0000

d=11

c=01

b=10

00 000000

10 10

01

1010 10

00

11

0101

111111

0000

10 1010

111111010101

t1 t5t4t3t2 t6

Figure: Trellis diagram for rate=1/3 and K=3

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Solutions Solutions

7.6 An encoder diagram is shown in Figure. Find the encoder output for an input sequence 1 0 0 1 0 1 0

Solution: g1 = [1 0 1], g2 = [1 1 1]

g1(X ) = 1 + X 2, g2(X ) = 1 + X + X 2

Output

Input U

1

2

Table: State transition table (rate=1/2, K=3)

Input Register State at State at Outputbit Contents time ti time ti + 1- 000 00 00 001 100 00 10 110 010 10 01 010 001 01 00 111 100 00 10 110 010 10 01 011 101 01 10 000 010 10 01 01

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Solutions Solutions

7.6 Figure shows an encoder for a (3,2) convolutional code. Find the transfer function T (D) and minimum free distance for thiscode. Also, draw the state diagram for the code.

Output

Input U1

2

2

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References

S. Lin and D. J. C. Jr., Error Control Coding, 2nd ed. Pearson / Prentice Hall, 2004.

R. Blahut, Theory and Practice of Error Control Codes, 2nd ed. Addison Wesley, 1984.

J. G. Proakis, Digital communications, 4th ed. Prentice Hall, 2001.

J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed. Prentice Hall,2002.

S. Haykin, Digital communications, 2nd ed. Wiley, 1988.

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