1. Diagram the encoders for a. A systematic (3,2,3) convolutional code b. A systematic (4,3,1) convolutional code Label the input and output rates and the current input state and state at arbitrary time. Answer: Output rate 3/2 r b Input rate r b M j-6 M j-4 M j-2 M j M j-7 M j-5 M j-3 M j-1 + Any combination of bits in registers, but must include m j-6 and/or m j-7 State Input (a)
Solution 2 DC
Sep 15, 2015
Solution 2
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1. Diagram the encoders fora. A systematic (3,2,3) convolutional codeb. A systematic (4,3,1) convolutional code
Label the input and output rates and the current input state and state at arbitrary time.Answer:
Outputrate 3/2 rb
Inputrate rb
Mj-6 Mj-4 Mj-2 Mj
Mj-7 Mj-5 Mj-3 Mj-1
+Any combination of bits in registers, but must include mj-6 and/or mj-7
State Input(a)
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(b)
+
Mj-3 Mj
Mj-5 Mj-2
Mj-4 Mj-1
State Input
Any combination of bits in register, but must include mj-5, and mj-4, and/or mj-3.
Input raterb
Output rate4/3 rb
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2. A (3,1,2) achieves maximum free distance when
a. Construct the code trellis and state diagramb. Find the state and output sequence produced by the input sequence
1011001111.
'2j j jx m m= '' ''' '2j j j jx x m x= =
Answer:
'' ''' '2j j j jx x m x= = ' 2j j jx m m=
'' '''2 1j j j j jx x m m m = =
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a=00
b=01
c=10
d=11
a
b
c
d
000
111
111000011
100
100011
a
b
d
c
000
111
111
000011
100
100
011
Input: 1 0 1 1 0 0 1 1 1 1State: a b c b d c a b d d dOutput: 111 011 000 100 100 111 111 100 011 011
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3.
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Answer:In the answer, binary digits are shown while using long division. If you are easier with polynomial notation, then just write the respective polynomials for all the vectors used in calculation.
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Trellis
01 11 01 01 11 01 10 11
00
11
00
11
0110
10
01
11
00
00
11
10
01
10
01
000
001
010
011
100
101
110
111
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01 11 01 01 11 01 10 111
1
3
3
1
1
4
4
2
2
1
3
5
3
5 2
5 2
5 4
5 4
2 7
4 5
4 3
2 5
-
01 11 01 01 11 01 10 111
1
3
3
1
1
4
4
2
2
1
3
5
3
2
2
4
4
2
4
3
2 5 3
5 3
5 4
5 4
4 4
2 6
2 4
4 2
-
01 11 01 01 11 01 10 111
1
3
3
1
1
4
4
2
2
1
3
5
3
2
2
4
4
2
4
3
2 3
3
4
4
4 4
2
2
2 3 5
3 5
2 5
6 3
-
01 11 01 01 11 01 10 111
1
3
3
1
1
4
4
2
2
1
3
5
3
2
2
4
4
2
4
3
2 3
3
4
4
4 4
2
2
2 3
3
2
3
4 3
5 3
5 3
-
01 11 01 01 11 01 10 111
1
3
3
1
1
4
4
2
2
1
3
5
3
2
2
4
4
2
4
3
2 3
3
4
4
4 4
2
2
2 3
3
2
3
3
3
3
Maximum like hood Decoded : 00 11 00 01 10 01 10 11Decoded sequence: 0 1 1 1 1 0 0 0
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