in these slides you will learn about turbo codes, Exit chat, encoding process, decoding process, trellis diagram.

1. Salim Syed
FA12-BTN-032
Shahrukh Ali Khan
FA12-BTN-020
Department of Computer Science
COMSATS Institute of IT, Abbottabad
Supervised By:
Dr. Abbas Khalid
Reliable Multimedia Transmission under Noisy
conditions

2. • TURBO codes (are a class of high performance FEC ) are the channel
coding scheme used in wireless cellular networks as they are able
to reach the Shannon limit (maximum information transfer rate of
the channel , for a particular noise level ) . In our project we
introduce turbo codes, and implement a turbo encoder/decoder.
The decoding scheme employs the BCJR algorithm, the maximum
aposteriori algoithm (MAP). Also, one of the most important parts
is the analysis of the code; is the implementation of the EXIT chart
and BER analysis. On the basis of these implementations, we were
able to study the iterative behavior of turbo codes.
Introduction

3. • A random signal will be encoded via turbo encoder.
• The encoded data will be modulated and transmitted on the channel.
• The errors caused by the channel will be detected and corrected by the
decoder.
• The decoder will be implemented using maximum a-posteriori probability
(MAP) algorithms.
• To optimize the performance of the decoder, Exit chart and BER analysis
will be made.
• The practical demonstration will be shown by transferring random bits or
a multimedia file.
Scope

4. • Turbo codes are a class of high
performance forward error correction(FEC)
codes.
• Turbo codes on source side consist of two
convolutional code separated by an inter-leaver.
• On receiver side Turbo codes consist of Two
decoders, that work cooperatively in order to
refine and improve the estimate of the original
bits.
4
Turbo Codes

5. 5
Source Encoder Modulator
DemodulatorDecoder
Channel
Destinat
ion
Convolutional code BPSK Modulation
MAP Decoding
System Block Diagram

6. 6
Convolutional code
Encoder 2
Encoder 1
Turbo Encoding (K=1/3)

7. Exit chart example
• Received information bits: : 12.6774 1.9846 2.1132 -0.0145 8.4528
19.0188 -4.2264
parity bits: 0.3422 0.2138 1.8467 -2.5767 -1.3762 2.1274 0.8901
interleave parity bits: 1.0205 2.7671 0.2240 0.6878 -1.2168 -1.2189 -1.2493

8. Exit chart example
• In the process to get the exit chart curve, the information bits are separated
into positive and negative values. Positive values are the values which are
greater then 0.5 and the negative values are lesser then 0.5.
• Positive values: 12.6774 2.1132 8.4528 -0.0145
• 1 2 3 4 5 6 7
• 0 1 0 1 1 0 1
• 0 12.6774 0 2.1132 8.4528 0 -0.0145
• The positive values are 1 and the negative values are 0
• Negative values : 1.9846 19.0188 -4.2264
• 1 2 3 4 5 6 7
• 0 1 0 1 1 0 1
1.9846 0 19.0188 0 0 -4.2264 0

9. Exit chart example
• To make a histogram we take the positive values from minimum to maximum
by giving the movement of 1
• -4.2264 -3.2264 -2.2264 -1.2264 -0.2264 0.7736 1.7736
• 2.7736 3.7736 4.7736 5.7736 6.7736 7.7736 8.773
9.7736 10.7736 11.7736 12.7736 13.7736 14.7736 15.7736
• 16.7736 17.7736 18.7736
• Here the one’s are the point in the histogram where the positive values lies
• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
• 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0
•
• 18 19 20 21 22 23 24
• 1 0 0 0 0 0 0

10. Exit chart example
• These point which points the positive values on the histogram is divided
by it sum ( sum = 4 )
• 0 0 0 0 1/4 0 1/4 0 0 0 0 0 0 1/4 0 0 0 1/4 0 0 0
0 0
• Divide
• 0 0 0 0 0.2500 0 0.2500 0 0 0 0 0 0 0.2500 0 0
0 0.2500 0 0 0 0 0 0

11. Exit chart implementation
now to make a histogram for negative values we take the negative values from
mininum to maximum by giving the movement of 1
-4.2264 -3.2264 -2.2264 -1.2264 -0.2264 0.7736 1.7736
2.7736 3.7736 4.7736 5.7736 6.7736 7.7736 8.7736
9.7736 10.7736 11.7736 12.7736 13.7736 14.7736 15.7736
16.7736 17.7736 18.7736
These point ,points the negative values on the histogram is divided by it sum (
sum = 3 ). Here 2 show that 2 points are lies at the same location
1 2 3 4 5 6 7 8 9 10 11
2 0 1 0 0 0 0 0 0 0 0
12 13 14 15 16 17 18 19 20 21 22 23 24
0 0 0 0 0 0 0 0 0 0 0 0 0

12. Exit chart implementation
Divide :
0.6667 0 0.3333 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
•
• PAp(find(pAp>0)= 0.2500 0.2500 0.2500
• 2*pAp(find(pAp>0)= 0.5 0.5 0.5
pAp(find(pAp>0))+pAm(find(pAp>0)))= 0.2500 0.2500 0.2500 + 0 0 0
• pAm(find(pAm>0)= 0.6667 0.3333
2*pAm(find(pAm>0)= 1.3334 0.6666
• pAp(find(pAm>0))+pAm(find(pAm>0)))= 0 0 0 +0.6777 0.3333
•
• to solve this whole equation the IA1(J) value is [0 0.7126]
•

13. ENCODING PROCESS

14. +
+
+
Time Input Present Future ϓk
k Uk Sk Sk+1 Uk, Pk
0 0 00 00 0,0
1 1 00 10 1,1
2 0 01 10 0,0
3 1 01 00 1,1
4 0 10 11 0,1
5 1 10 01 1,0
6 0 11 01 0,1
7 1 11 11 1,0
00 00
01 01
10 10
1111
U=0 U=1
State Table
Encoder
Trellis States

15. +
+
+
0 1 0 1 0 1 1
0 0
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111

16. +
+
+
0 1 0 1 0 1 1 0
0
0 0
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=0
0
0
0

17. +
+
+
0 1 0 1 0 1 1 0 1
0 1
0 0
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=1
1
1
0

18. +
+
+
0 1 0 1 0 1 1 0 1 0
0 1 1
1 0
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=2
0
1
1

19. +
+
+
0 1 0 1 0 1 1 0 1 0 1
0 1 1 0
1 1
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=3
1
1
0

20. +
+
+
0 1 0 1 0 1 1 0 1 0 1 0
0 1 1 0 1
1 1
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=4
0
0
0

21. +
+
+
0 1 0 1 0 1 1 0 1 0 1 0 1
0 1 1 0 1 1
0 1
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=5
1
0
1

22. +
+
+
0 1 0 1 0 1 1 0 1 0 1 0 1 1
0 1 1 0 1 1 1
0 0
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
00 00
01 01
10 10
1111
K=6
1
0
1

23. +
+
u
P11
P12
Resetting the encoder

24. 1 1
+
+
00 00
01 01
10 10
1111
00
01
10
11
01
Resetting the encoder (0)

25. 0 1
+
+
00 00
01 01
10 10
1111
00
01
10
11
01
Resetting the encoder (1)

26. 0 0
+
+
00 00
01 01
10 10
1111
01
10
11
01
00
Resetting the encoder (2)

27. Turbo Decoding

28. DECODING PROCESS

29. • Information bits (uk):
0, 1, 0, 1, 0, 1, 1
• ˆParity bits (pk):
0, 1, 1, 0, 1, 1, 1
• Flushing bits:
1, 1, 0, 1
Noise
Channel (σ): 0.972849
Actual Data

30. • Information bits (Xk):
−3.24327, 2.73651,−2.13742, 0.729417, 0.200324, 2.83389, 2.74642
• ˆParity bits (pk):
−0.765662, 1.44776, −0.654661, −0.562306, 0.524247, 1.12587, 1.10221
• Flushing bits:
0.861011, 0.102445,−0.808592,−0.445921
Noise
Channel (σ): 0.972849
Received Data

31. Branch Matric()
ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
ˆj = 0, 1, 2, 3 (= # of states in the trellis).
ˆk = 0, 1, 2, ..n (= # of information bits).
ˆ𝑢 𝑘 = 𝑘 𝑡ℎ transmitted information bit
ˆ𝑝 𝑘 = 𝑘 𝑡ℎ transmitted parity bit
ˆ𝑥 𝑘 = 𝑘 𝑡ℎ received information bit
ˆ𝑦 𝑘
= 𝑘 𝑡ℎ
received parity bit
𝐴 𝑘
𝑖
= a priori probability (i=0 if 𝑢 𝑘
is 0 and i=1 otherwise).

32. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 0)

33. Branch Metric (k = 0)
00 00
01 01
10 10
1111
0.0340244
3.59164e-005
k: 0
Systematic Bit: -3.24327
Parity Bit: -0.765662
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

34. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 1)

35. Branch Metric (k = 1)
00 00
01 01
10 10
1111
1.32133e-005
0.000428981
k: 1
Systematic Bit: 2.73651
Parity Bit: 1.44776
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

36. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 2)

37. Branch Metric (k = 2)
00 00
01 01
10 10
1111
0.237016
0.44258919
k: 2
Systematic Bit: -2.13742
Parity Bit: -0.654661
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

38. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 3)

39. Branch Metric (k = 3)
00 00
01 01
10 10
1111
0.0930673
0.434728
k: 3
Systematic Bit: 0.729417
Parity Bit: -0.562306
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

40. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 4)

41. Branch Metric (k = 4)
00 00
01 01
10 10
1111
0.0684456
0.104518
k: 4
Systematic Bit: 0.200324
Parity Bit: 0.524247
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

42. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 5)

43. Branch Metric (k = 5)
00 00
01 01
10 10
1111
1.94825e-005
0.00777034
k: 5
Systematic Bit: 2.83389
Parity Bit: 1.12587
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

44. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 6)

45. Branch Metric (k = 6)
00 00
01 01
10 10
1111
2.91557e-005
0.00966591
k: 6
Systematic Bit: 2.74642
Parity Bit: 1.10221
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

46. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 7)

47. Branch Metric (k = 7)
00 00
01 01
10 10
1111
0.0422176
0.260426
k: 7
Flushing Bit 1: 0.861011
Flushing Bit 2: 0.102445
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

48. ϓ 𝑘
𝑗
= 𝐴 𝑘
𝑖
. exp(−
𝑥 𝑘−𝑢 𝑘 .2
2𝛿2 +
𝑦 𝑘−𝑝 𝑘
2𝛿2 )
Branch Metric (k = 8)

49. Branch Metric (k = 8)
00 00
01 01
10 10
1111
0.41691
0.0755168
k: 8
Flushing Bit 1: --0.808592
Flushing Bit 2: --0.445921
a Priori: 0
Probability of ‘0’: 0.5
Probability of ‘1’: 0.5

50. Forward Recursion (α)
01
ϓ 𝑎
⍺3 = ⍺1 ϓ 𝑎+ ⍺2 ϓ 𝑏

51. Forward Recursion (α)
00 00
01 01
10 10
1111
⍺0
00
⍺1
00
⍺0
01
⍺0
10
⍺0
11
⍺1
01
⍺1
10
⍺1
11
ϓ0
ϓ3
ϓ1
ϓ2
ϓ5
ϓ4
ϓ6
ϓ7
⍺1
00
= ⍺0
00
ϓ0+⍺0
01
ϓ3
⍺1
01
= ⍺0
10
ϓ5+⍺0
11
ϓ6
⍺1
10
= ⍺0
00
ϓ1+⍺0
01
ϓ2
⍺1
11
= ⍺0
10
ϓ4+⍺0
11
ϓ7
⍺1 = ⍺1
00
+ ⍺1
01
+ ⍺1
10
+ ⍺1
11
⍺1
00
= ⍺1
00
/⍺1
⍺1
01
= ⍺1
01
/⍺1
⍺1
10
= ⍺1
10
/⍺1
⍺1
11
= ⍺1
11
/⍺1
These parameters can be normalized as:

52. Forward Recursion (α)
00 00
01 01
10 10
1111
⍺ = 1 ⍺ = 0
⍺ = 0 ⍺ = 0
⍺ = 0
⍺ = 0
⍺ = 0
⍺ = 0
0

53. Forward Recursion (k=0)
⍺1
00
= (1)(0.0340244) + (0)(7.12211e−006)= 0.0340244
⍺1
01
= (0)(0.00674693) + (0)(3.59164e-005) = 0
⍺1
01
= (0)(0.0340244) + (1)(7.12211e-006) = 7.12211e-006
⍺1
01
= (0)(0.00674693) + (0)(3.59164e-005) = 0
⍺ 𝟏 =0.0340244 + 0 + 7.12211e-006 + 0 = 0.0340315
⍺1
00
=
⍺1
00
⍺1
= 𝟎. 𝟗𝟗𝟗𝟕𝟗𝟏
Normalization:
00 00
01 01
10 10
1111
0.0340244
3.59164e-005
⍺ = 0.99979⍺ = 1
⍺ = 0 ⍺ = 0
⍺ = 0 ⍺ = 0.0002
⍺ = 0 ⍺ = 0
⍺1
01
=
⍺1
01
⍺1
= 𝟎
⍺1
10
=
⍺1
10
⍺1
= 𝟎. 𝟎𝟎𝟎𝟐𝟎𝟗𝟐𝟖
⍺1
11
=
⍺1
11
⍺1
= 𝟎
1

54. Forward Recursion (k=1)
⍺2
00
=(0.99979) (1.32133e−005) + (0)(0.0914347)
=1.32105e−005
⍺2
01
= (0)(0.000281634) + (0.0002)(0.00428981)
= 8.57962e-007
⍺2
01
= (0) (1.32133e-005) + (0.99979) (0.0914347)
= 0.0914155
⍺2
01
= (0.0002) (0.000281634) + (0) (0.00428981)
= 5.63268e-008
⍺ 𝟐 = 1.32105e-005 + 8.57962e-007 + 0.0914155
+ 5.63268e-008 = 0.0914296
⍺2
00
=
⍺2
00
⍺2
= 𝟎. 𝟎𝟎𝟎𝟏𝟒𝟒𝟒𝟖𝟖
Normalization:
00 00
01 01
10 10
1111
1.3233e005
0.00428981
⍺ = 0.0001⍺ = 0.99979
⍺ = 0 ⍺ = 9.82𝑒 − 006
⍺ = 0.0002 ⍺ = 0.999
⍺ = 0 ⍺ = 6.16𝑒 − 007
⍺2
01
=
⍺2
01
⍺2
= 𝟗. 𝟑𝟖𝟑𝟖𝟓𝐞 − 𝟎𝟎𝟔
⍺2
10
=
⍺2
10
⍺2
= 𝟎. 𝟗𝟗𝟗𝟖𝟒𝟔
⍺2
11
=
⍺2
11
⍺2
= 𝟔. 𝟏𝟔𝟎𝟔𝟕𝐞 − 𝟎𝟎𝟕
2

55. Forward Recursion (k=2)
⍺3
00
=(0.0001)(0.237016) + (9.81924e006)(0.000649159)
= 3.42526e-005
⍺3
01
= (6.4465e-007)(0.0594244) +(0.999845) (0.00258919)
= 0.00258883
⍺3
01
= (9.81924e-006)(0.237016) + (0.0001)(0.000649159)
= 2.42111e-006
⍺3
01
= (0.9998)(0.059424) + (6.44653e-007) (0.00258919)
= 0.0594152
⍺ 𝟑 = 3.42526e-005 + 0.00258883 + 2.42111e-006
+ 0.0594152 = 0.0620407
⍺3
00
=
⍺3
00
⍺3
= 𝟎. 𝟎𝟎𝟎𝟓𝟓𝟐𝟎𝟗𝟗
Normalization:
00 00
01 01
10 10
1111
0.237016
0.00258919
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576
⍺3
01
=
⍺3
01
⍺3
= 𝟎. 𝟒𝟏𝟕𝟐𝟕𝟗
⍺3
10
=
⍺3
10
⍺3
= 𝟑. 𝟗𝟎𝟐𝟒𝟔𝐞 − 𝟎𝟎𝟓
⍺3
11
=
⍺3
11
⍺3
= 𝟎. 𝟗𝟓𝟕𝟔𝟖𝟏
3
⍺ = 0.0001
⍺ = 9.82𝑒 − 006
⍺ = 0.999
⍺ = 6.16𝑒 − 007

56. Forward Recursion (k=3)
⍺4
00
= (0.0005)(0.0930673) + (0.0417279)(0.132484)
= 0.00557966
⍺4
01
= (0.957681)(0.0283623) + (3.9e-005)(0.434728)
= 0.027179
⍺4
01
= (0.0417279)(0.0930673) + (0.0005)(0.132484)
= 0.00395665
⍺4
01
= (3.9e-005)(0.0283623) + (0.957681)(0.434728)
= 0.416332
⍺ 𝟒 = 0.00557966 + 0.027179 + 0.00395665 +
0.416332 = 0.453047
⍺4
00
=
⍺4
00
⍺4
= 𝟎. 𝟎𝟏𝟐𝟑𝟏𝟓𝟗
Normalization:
00 00
01 01
10 10
1111
0.0930673
0.434728
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.0087
⍺ = 0.9189
⍺4
01
=
⍺4
01
⍺4
= 𝟎. 𝟎𝟓𝟗𝟗𝟗𝟏𝟓
⍺4
10
=
⍺4
10
⍺4
= 𝟎. 𝟎𝟎𝟖𝟕𝟑𝟑𝟒𝟏
⍺4
11
=
⍺4
11
⍺4
= 𝟎. 𝟗𝟏𝟖𝟗𝟓𝟗
4
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576

57. Forward Recursion (k=4)
⍺5
00
= (0.0123158)(0.0684456) + (0.0599917)(0.31646)
= 0.0198279
⍺5
01
= (0.918959)(0.207239) + (0.00873342)(0.104518)
= 0.191357
⍺5
01
= (0.0599917)(0.0684456) + (0.0123158)(0.31646)
= 0.00800363
⍺5
01
= (0.00873342)(0.207239) + (0.918959)(0.104518)
= 0.0978577
⍺ 𝟓 = 0.0198279 + 0.191357 + 0.00800363 +
0.0978577 = 0.317046
⍺5
00
=
⍺5
00
⍺5
= 𝟎. 𝟎𝟔𝟐𝟓𝟒
Normalization:
00 00
01 01
10 10
1111
0.0684456
0.104518
⍺ = 0.06254
⍺ = 0.6035
⍺ = 0.0252
⍺ = 0.3086
⍺5
01
=
⍺5
01
⍺5
= 𝟎. 𝟔𝟎𝟑𝟓𝟔𝟐
⍺5
10
=
⍺5
10
⍺5
= 𝟎. 𝟎𝟐𝟓𝟐𝟒𝟒𝟒
⍺5
11
=
⍺5
11
⍺5
= 𝟎. 𝟑𝟎𝟖𝟔𝟓𝟒
5
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.0087
⍺ = 0.9189

58. Forward Recursion (k=5)
⍺6
00
= (0.0625394)(1.94825e-005) + (0.603561)(0.083888)
= 0.0506332
⍺6
01
= (0.308655)(0.000210334) + (0.0252443)(0.00777034)
= 0.000261077
⍺6
01
= (0.603561)(1.94825e-005) + (0.0625394)(0.0838888)
= 0.00525811
⍺6
01
= (0.0252443)(0.000210334) + (0.308655)(0.0077034)
= 0.00240366
⍺ 𝟔 = 0.0506332 + 0.000261077 + 0.00525811 +
0.00240366 = 0.0585561
⍺6
00
=
⍺6
00
⍺6
= 𝟎. 𝟖𝟔𝟒𝟔𝟗𝟔
Normalization:
00 00
01 01
10 10
1111
1.94825e-005
0.0077034
⍺ = 0.8647
⍺ = 0.0044
⍺ = 0.0898
⍺ = 0.0410
⍺6
01
=
⍺6
01
⍺6
= 𝟎. 𝟎𝟎𝟒𝟒𝟓𝟖𝟓𝟗
⍺6
10
=
⍺6
10
⍺6
= 𝟎. 𝟎𝟖𝟗𝟕𝟗𝟔𝟐
⍺6
11
=
⍺6
11
⍺6
= 𝟎. 𝟎𝟒𝟏𝟎𝟒𝟖𝟔
6
⍺ = 0.06254
⍺ = 0.6035
⍺ = 0.0252
⍺ = 0.3086

59. Forward Recursion (k=6)
⍺7
00
= (0.864696)(2.91557e-005) + (0.004458)(0.0992647)
= 0.000467791
⍺7
01
= (0.0410488)(0.00029941)+(0.089796)(0.0096659)
= 0.000880253
⍺7
01
= (0.0044585)(2.91557e-005) + (0.86469)(0.0992647)
= 0.0858339
⍺7
01
= (0.0897962)(0.00029941) + (0.04104)(0.0096659)
= 0.00042366
⍺ 𝟕 = 0.000467791 + 0.000880253 + 0.0858339
+ 0.00042366 = 0.0876056
⍺7
00
=
⍺7
00
⍺7
= 𝟎. 𝟎𝟎𝟓𝟑𝟑𝟗𝟕𝟒
Normalization:
00 00
01 01
10 10
1111
⍺ = 0.0053
⍺ = 0.010
⍺ = 0.9797
⍺ = 0.0048
⍺7
01
=
⍺7
01
⍺7
= 𝟎. 𝟎𝟏𝟎𝟎𝟒𝟕𝟗
⍺7
10
=
⍺7
10
⍺7
= 𝟎. 𝟗𝟕𝟗𝟕𝟕𝟔
⍺7
11
=
⍺7
11
⍺7
= 𝟎. 𝟎𝟎𝟒𝟖𝟑𝟔
7
⍺ = 0.8647
⍺ = 0.0044
⍺ = 0.0898
⍺ = 0.0410
2.91557e-005
0.00966591

60. Forward Recursion (k=7)
⍺8
00
= (0.864696)(2.91557e-005) + (0.004458)(0.0992647)
= 0.000467791
⍺8
01
= (0.0410488)(0.00029941)+(0.089796)(0.0096659)
= 0.000880253
⍺8
01
= (0.0044585)(2.91557e-005) + (0.86469)(0.0992647)
= 0.0858339
⍺8
01
= (0.0897962)(0.00029941) + (0.04104)(0.0096659)
= 0.00042366
⍺ 𝟖 = 0.00345747 + 0.255391 + 0.00213605 +
0.0526077= 0.313592
⍺8
00
=
⍺8
00
⍺8
= 𝟎. 𝟎𝟏𝟏𝟎𝟐𝟓𝟒
Normalization:
00 00
01 01
10 10
1111
⍺ = 0.0110
⍺ = 0.8143
⍺ = 0.0068
⍺ = 0.1677
⍺8
01
=
⍺8
01
⍺8
= 𝟎. 𝟖𝟏𝟒𝟒𝟎𝟓
⍺8
10
=
⍺8
10
⍺8
= 𝟎. 𝟎𝟎𝟔𝟖𝟏𝟏𝟓𝟓
⍺8
11
=
⍺7
11
⍺8
= 𝟎. 𝟏𝟔𝟕𝟕𝟓𝟖
8
⍺ = 0.0053
⍺ = 0.010
⍺ = 0.9797
⍺ = 0.0048
0.0422176
0.260426

61. Forward Recursion (k=8)
⍺9
00
= (0.0110)(0.416991) + (0.8143)(0.0294307)
=0.0285523
⍺9
01
= (0.1677)(0.162511) + (0.0068)(0.0755168)
= 0.0277666
⍺9
01
= (0.8143)(0.416991) + (0.0110)(0.0294307)
= 0.33988
⍺9
01
= (0.0068)(0.162511) + (0.1677)(0.0755168)
= 0.0137692
⍺ 𝟗 = 0.00345747 + 0.255391 + 0.00213605 +
0.0526077= 0.409968
⍺9
00
=
⍺9
00
⍺9
= 𝟎. 𝟎𝟔𝟗𝟔𝟒𝟓𝟑
Normalization:
00 00
01 01
10 10
1111
⍺ = 0.0697
⍺ = 0.0677
⍺ = 0.8289
⍺ = 0.0336
⍺9
01
=
⍺9
01
⍺9
= 𝟎. 𝟎𝟔𝟕𝟕𝟐𝟖𝟖
⍺9
10
=
⍺9
10
⍺9
= 𝟎. 𝟖𝟐𝟗𝟎𝟒
⍺9
11
=
⍺7
11
⍺9
= 𝟎. 𝟎𝟑𝟑𝟓𝟖𝟔𝟐
9
⍺ = 0.0110
⍺ = 0.8143
⍺ = 0.0068
⍺ = 0.1677
0.41691
0.0755168

62. Forward Recursion
00 00
01 01
10 10
1111
⍺ = 0.99979⍺ = 1
⍺ = 0 ⍺ = 0
⍺ = 0 ⍺ = 0.0002
⍺ = 0 ⍺ = 0
00
01
10
11
⍺ = 0.0001
⍺ = 0.999
⍺ = 6.16𝑒 − 007
00 00
01 01
10 10
1111
00
01
10
11
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.0087
⍺ = 0.9189
⍺ = 0.06254
⍺ = 0.6035
⍺ = 0.0252
⍺ = 0.3086
⍺ = 9.82𝑒 − 006

63. Backward Recursion
ϓ 𝑎
𝞫3 𝞫1
𝞫2
𝞫3 = 𝞫1 ϓ 𝑎+ 𝞫2 ϓ 𝑏

64. 00 00
01 01
10 10
1111
β0
00
β1
00
β0
01
β0
10
β0
11
β1
01
β1
10
β1
11
ϓ0
ϓ3
ϓ1
ϓ2
ϓ5
ϓ4
ϓ6
ϓ7
β0
01
= β1
10
ϓ2+β1
00
ϓ3
β0
10
= β1
11
ϓ4+β1
01
ϓ5
β0
11
= β1
01
ϓ6+β1
11
ϓ7
⍺1 = ⍺1
00
+ ⍺1
01
+ ⍺1
10
+ ⍺1
11
β0
00
= β0
00
/⍺1
β0
01
= β1
01
/⍺1
β0
10
= β0
10
/⍺1
β0
11
= β0
11
/⍺1
These parameters can be normalized as:
Backward Recursion (β)

65. Backward Recursion (β)
00 00
01 01
10 10
1111
β = 1 β = 0
β = 0 β = 0
β = 0
β = 0
β = 0
β = 0 0

66. Backward Recursion (k=8)
β8
00
= (1)(0.416991) + (0)(0.0294307) =0.416991
β8
01
= (0)(0.416991) + (1)(0.0294307) = 0.0294307
β8
10
= (0)(0.162511) + (0)(0.0755168) = 0
β8
11
= (0)(0.162511) + (0)(0.0755168)= 0
⍺ 𝟗 = 0.409968
β8
00
=
β8
00
⍺9
= 𝟏. 𝟏𝟎𝟏𝟔𝟗𝟕
Normalization:
00 00
01 01
10 10
1111
⍺ = 0.0697
⍺ = 0.0677
⍺ = 0.8289
⍺ = 0.0336
β8
01
=
β8
01
⍺9
= 𝟎. 𝟎𝟕𝟏𝟕𝟕𝟔𝟕
β8
10
=
β8
10
⍺9
= 𝟎
β8
11
=
β8
11
⍺9
= 𝟎
1
⍺ = 0.0110
⍺ = 0.8143
⍺ = 0.0068
⍺ = 0.1677
0.41691
0.0755168
β = 1.101697 β = 1
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟕𝟔𝟕 𝛃 = 𝟎
𝛃 = 𝟎 𝛃 = 𝟎
𝛃 = 𝟎 𝛃 = 𝟎

67. Backward Recursion (k=7)
β7
00
= (1.01697)(0.0422176) + (0)(0.323372) = 0.042934
β7
01
= (0)(0.0422176) + (1.01697)(0.323372) = 0.32886
β7
10
= (0)(0.0524218) + (0.0717767)(0.260426) = 0.0186925
β7
11
= (0.0717767)(0.0524218) + (0)(0.260426)= 0.0037626
⍺ 𝟖 = 0.313592
β7
00
=
β7
00
⍺8
= 𝟎. 𝟏𝟑𝟔𝟗𝟏
Normalization:
00 00
01 01
10 10
1111
β7
01
=
β7
01
⍺8
= 𝟏. 𝟎𝟒𝟖𝟔𝟗
β7
10
=
β7
10
⍺8
= 𝟎. 𝟎𝟓𝟗𝟔𝟎
β7
11
=
β7
11
⍺8
= 𝟎. 𝟎𝟏𝟏𝟗𝟗𝟖𝟔
2
⍺ = 0.0053
⍺ = 0.010
⍺ = 0.9797
⍺ = 0.0048
⍺ = 0.0110
⍺ = 0.8143
⍺ = 0.0068
⍺ = 0.1677
0.0422176
0.260426
β = 1.101697
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟕𝟔𝟕
𝛃 = 𝟎
𝛃 = 𝟎
β = 0.136882
𝛃 = 𝟏. 𝟎𝟒𝟖𝟒𝟕
𝛃 = 𝟎. 𝟎𝟓𝟗𝟔𝟎
𝛃 = 𝟎. 𝟎𝟏𝟏𝟗𝟗𝟗𝟔

68. Backward Recursion (k=6)
β6
00
= (0.13682)(2.91557e − 005) + (0.05959)(0.0992647)
= 0.00591967
β6
01
= (0.059595)(2.91557e − 005)+(0.136882)(0.0992647)
= 0.0135893
β6
10
= (0.011996)(0.000299416)+(1.04847)(0.00966591)
= 0.010138
β6
11
= (1.04847)(0.000299416) +(0.011996)(0.00966591)
= 0.000429881
⍺ 𝟕 = 0.087606
β6
00
=
β6
00
⍺7
= 𝟎. 𝟎𝟔𝟕𝟓
Normalization:
β6
01
=
β6
01
⍺7
= 𝟎. 𝟏𝟓𝟓𝟏𝟏𝟗
β6
10
=
β6
10
⍺7
= 𝟎. 𝟏𝟏𝟓𝟕𝟐𝟑
β6
11
=
β6
11
⍺7
= 𝟎. 𝟎𝟎𝟒𝟗𝟎𝟕
β = 0.06757
𝛃 = 𝟎. 𝟏𝟓𝟓𝟏𝟏𝟖
𝛃 = 𝟎. 𝟏𝟏𝟓𝟕𝟐𝟑
𝛃 = 𝟎. 𝟎𝟎𝟒𝟗𝟎𝟔𝟗
00 00
01 01
10 10
1111
⍺ = 0.0053
⍺ = 0.010
⍺ = 0.0048
3
⍺ = 0.8647
2.91557e-005
0.00966591
⍺ = 0.9797⍺ = 0.0898
⍺ = 0.0410
⍺ = 0.0044
β = 0.136882
𝛃 = 𝟏. 𝟎𝟒𝟖𝟒𝟕
𝛃 = 𝟎. 𝟎𝟓𝟗𝟔𝟎
𝛃 = 𝟎. 𝟎𝟏𝟏𝟗𝟗𝟗𝟔

69. Backward Recursion (k=4)
β4
00
= (0.165809)(0.0684456) + (0.0206017)(0.31646)
= 0.0178685
β4
01
= (0.0206017)(0.0684456) + (0.165809)(0.31646)
= 0.053882
β4
10
= (0.00120834)(0.207239) + (0.0968433)(0.104518)
= 0.0103723
β4
11
= (0.096843)(0.20723)+(0.0012083)(0.104518)
= 0.020196
⍺ 𝟓 = 0.317046
β4
00
=
β4
00
⍺5
= 𝟎. 𝟎𝟓𝟔𝟑𝟓𝟗𝟒
Normalization:
β4
01
=
β4
01
⍺5
= 𝟎. 𝟏𝟔𝟗𝟗𝟓
β4
10
=
β4
10
⍺5
= 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
β4
11
=
β4
11
⍺5
= 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
β = 0.0563593
𝛃 = 𝟎. 𝟏𝟔𝟗𝟗𝟓
𝛃 = 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
𝛃 = 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
00 00
01 01
10 10
1111
5
β = 0.165809
𝛃 = 𝟎. 𝟎𝟗68433
𝛃 = 𝟎. 𝟎𝟐𝟎𝟔𝟎𝟏𝟕
𝛃 = 𝟎. 𝟎𝟎𝟏𝟐𝟎𝟖𝟑
0.0684456
0.104518
⍺ = 0.06254
⍺ = 0.6035
⍺ = 0.0252
⍺ = 0.3086
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.0087
⍺ = 0.9189

70. Backward Recursion (k=3)
β3
00
= (0.0563593)(0.0930673) + (0.0327154)(0.132484)
= 0.00957947
β3
01
= (0.0327154)(0.0930673) + 0.0563593)(0.132484)
= 0.0105114
β3
10
= (0.0637005)(0.0283623) + (0.16995)(0.434728)
= 0.0756887
β3
11
= (0.16995)(0.0283623) + (0.0637005)(0.434728)
= 0.0325126
⍺ 𝟒 = 0.453047
β3
00
=
β3
00
⍺4
= 𝟎. 𝟎𝟐𝟏𝟏𝟒𝟒𝟓
Normalization:
β3
01
=
β3
01
⍺4
= 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟕
β3
10
=
β3
10
⍺4
= 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
β3
11
=
β3
11
⍺4
= 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
β = 0.0211445
𝛃 = 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟔
𝛃 = 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
00 00
01 01
10 10
1111
6
β = 0.0563593
𝛃 = 𝟎. 𝟏𝟔𝟗𝟗𝟓
𝛃 = 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
𝛃 = 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
0.0930673
0.434728
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.0087
⍺ = 0.9189
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576

71. Backward Recursion (k=2)
β2
00
= (0.0211445)(0.237016)+(0.167066)(0.000649159)
= 0.00512004
β2
01
= (0.167066)(0.237016) + (0.0211445)(0.000649159)
= 0.039611
β2
10
= (0.0717642)(0.0594244) + (0.0232016)(0.00258919)
= 0.00432462
β2
11
= (0.0232016)(0.0594244) + (0.0717642)(0.00258919)
= 0.00156455
⍺ 𝟑 = 0.0620407
β2
00
=
β2
00
⍺3
= 𝟎. 𝟎𝟖𝟐𝟓𝟐𝟕𝟏
Normalization:
β2
01
=
β2
01
⍺3
= 𝟎. 𝟔𝟑𝟖𝟒𝟔𝟗
β2
10
=
β2
10
⍺3
= 𝟎. 𝟎𝟔𝟗𝟕𝟎𝟔𝟏
β2
11
=
β2
11
⍺3
= 𝟎. 𝟎𝟐𝟓𝟐𝟏𝟖𝟐
β = 0.082573
𝛃 = 𝟎. 𝟔𝟑𝟖𝟒𝟔𝟗
𝛃 = 𝟎. 𝟎𝟔𝟗𝟕𝟎𝟔𝟏
𝛃 = 𝟎. 𝟎𝟐𝟓𝟐𝟏𝟖𝟐
00 00
01 01
10 10
1111
7
β = 0.0211445
𝛃 = 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟔
𝛃 = 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
0.237016
0.00258919
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576
⍺ = 0.0001
⍺ = 9.82𝑒 − 006
⍺ = 0.999
⍺ = 6.16𝑒 − 007

72. Backward Recursion (k=1)
β1
00
= (0.08252)(1.32133e − 005) + (0.069706)(0.091434)
= 0.00637465
β1
01
= (0.06970)(1.32133e − 005) + (0.082527)(0.0914347)
= 0.00754678
β1
10
= (0.0252182)(0.000281634)+(0.638469)(0.00428981)
= 0.00274601
β1
11
= (0.638469)(0.000281634)+(0.0252182)(0.00428981)
= 0.000287996
⍺ 𝟐 = 0.0914296
β1
00
=
β1
00
⍺2
= 𝟎. 𝟔𝟗𝟕𝟐𝟏𝟗
Normalization:
β1
01
=
β1
01
⍺2
= 𝟎. 𝟎𝟖𝟐𝟓𝟒𝟐
β1
10
=
β1
10
⍺2
= 𝟎. 𝟎𝟑𝟎𝟎𝟑𝟒𝟐
β1
11
=
β1
11
⍺2
= 𝟎. 𝟎𝟎𝟑𝟏𝟒𝟗𝟗𝟐
β = 0.0697721
𝛃 = 𝟎. 𝟎𝟖𝟐𝟓𝟒𝟐
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟑𝟒𝟐
𝛃 = 𝟎. 𝟎𝟎𝟑𝟏𝟒𝟗𝟑
00 00
01 01
10 10
1111
8
β = 0.082573
𝛃 = 𝟎. 𝟔𝟑𝟖𝟒𝟔𝟗
𝛃 = 𝟎. 𝟎𝟔𝟗𝟕𝟎𝟔𝟏
𝛃 = 𝟎. 𝟎𝟐𝟓𝟐𝟏𝟖𝟐
1.3233e005
0.00428981
⍺ = 0.0001⍺ = 0.99979
⍺ = 0 ⍺ = 9.82𝑒 − 006
⍺ = 0.0002 ⍺ = 0.999
⍺ = 0 ⍺ = 6.16𝑒 − 007

73. Backward Recursion (k=0)
β0
00
= (0.06973)(0.03403) + (0.0300341)(7.12211e − 006)
= 0.00237246
β0
01
= (0.030035)(0.034024)+(0.06972)(7.12211e − 006)
= 0.00102239
β0
10
= (0.003149)(0.0067469)+(0.0825419)(3.59164e − 005)
= 2.42169e-005
β0
11
= (0.082541)(0.0067469)+(0.0031492)(3.59164e − 005)
= 0.000557018
⍺ 𝟏 = 0.0340315
β0
00
=
β0
00
⍺2
= 𝟎. 𝟎𝟔𝟗𝟕𝟏𝟑𝟔
Normalization:
β0
01
=
β0
01
⍺2
= 𝟎. 𝟎𝟑𝟎𝟎𝟒𝟐𝟒
β0
10
=
β0
10
⍺2
= 𝟎. 𝟎𝟎𝟎𝟕𝟏𝟐
β0
11
=
β0
11
⍺2
= 𝟎. 𝟎𝟏𝟔𝟑𝟔𝟕𝟕
β = 0.0697136
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟒𝟐𝟒
𝛃 = 𝟎. 𝟎𝟎𝟎𝟕𝟏𝟏𝟕
𝛃 = 𝟎. 𝟎𝟏𝟔𝟑𝟕
00 00
01 01
10 10
1111
9
β = 0.0697721
𝛃 = 𝟎. 𝟎𝟖𝟐𝟓𝟒𝟐
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟑𝟒𝟐
𝛃 = 𝟎. 𝟎𝟎𝟑𝟏𝟒𝟗𝟑
0.0340244
3.59164e-005
⍺ = 0.99979⍺ = 1
⍺ = 0 ⍺ = 0
⍺ = 0 ⍺ = 0.0002
⍺ = 0 ⍺ = 0

74. 00 00
01 01
10 10
1111
⍺0
00
β1
00
⍺0
01
⍺0
10
⍺0
11
β1
01
β1
10
β1
11
ϓ0
ϓ3
ϓ1
ϓ2
ϓ5
ϓ4
ϓ6
ϓ7
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝝀 𝑘
𝑖
𝑎𝑠
λ 𝑘
0
= ⍺ 𝑘
00
ϓ0 β 𝑘+1
00
+⍺ 𝑘
01
β 𝑘+1
10
ϓ2
+ ⍺ 𝑘
10
ϓ4 β 𝑘+1
11
+⍺ 𝑘
11
β 𝑘+1
01
ϓ6
Soft Output (𝐿 𝑘)
λ 𝑘
1
= ⍺ 𝑘
00
ϓ1 β 𝑘+1
10
+⍺ 𝑘
01
β 𝑘+1
00
ϓ3
+ ⍺ 𝑘
10
ϓ5 β 𝑘+1
01
+⍺ 𝑘
11
β 𝑘+1
11
ϓ7
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝐿 𝑘 𝑎𝑠:
𝐿 𝑘=log(λ 𝑘
1
/λ 𝑘
0
)
𝐻𝑎𝑟𝑑 𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛:
If 𝐿 𝑘> 0 then uk=1 otherwise uk=0.

75. λ0
0
= (1)(0.0340244)(0.0697219) +
(0)(0.0340244)(0.0300341) +
(0)(0.00674693)(0.00314992) +
(0)(0.00674693)(0.0825419) = 0.00237225
λ0
1
= (1)(7.12211e − 006)(0.0300341) +
(0)(7.12211e − 006)(0.0697219) +
(0)(3.59164e − 005)(0.0825419) +
(0)(3.59164e − 005)(0.00314992)
= 2.13906e-007
β = 0.0697136
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟒𝟐𝟒
𝛃 = 𝟎. 𝟎𝟎𝟎𝟕𝟏𝟏𝟕
𝛃 = 𝟎. 𝟎𝟏𝟔𝟑𝟕
00 00
01 01
10 10
1111
1
β = 0.0697721
𝛃 = 𝟎. 𝟎𝟖𝟐𝟓𝟒𝟐
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟑𝟒𝟐
𝛃 = 𝟎. 𝟎𝟎𝟑𝟏𝟒𝟗𝟑
0.0340244
3.59164e-005
⍺ = 0.99979⍺ = 1
⍺ = 0 ⍺ = 0
⍺ = 0 ⍺ = 0.0002
⍺ = 0 ⍺ = 0
Soft Output (k = 0)
𝐿0=log(2.13906e-007/0.0023722)
= -9.31381
𝑈0= 0

76. λ1
0
= (0.999791)(1.32133e − 005)(0.0825273) +
(0)(1.32133e − 005)(0.0697061) +
(0.00020928)(0.000281634)(0.0252182) +
(0)(0.000281634)(0.638469) = 1.09172e-006
λ1
1
= (0.999791)(0.0914347)(0.0697061) +
(0)(0.0914347)(0.0825273) +
(0.00020928)(0.00428981)(0.638469) +
(0)(0.00428981)(0.0252182) = 0.0063728
1
Soft Output (k = 1)
𝐿1=log(0.0063728 /1.09172e-00)
= 8.683
𝑈1= 1
β = 0.0697721
𝛃 = 𝟎. 𝟎𝟖𝟐𝟓𝟒𝟐
𝛃 = 𝟎. 𝟎𝟑𝟎𝟎𝟑𝟒𝟐
𝛃 = 𝟎. 𝟎𝟎𝟑𝟏𝟒𝟗𝟑
00 00
01 01
10 10
1111
2
β = 0.082573
𝛃 = 𝟎. 𝟔𝟑𝟖𝟒𝟔𝟗
𝛃 = 𝟎. 𝟎𝟔𝟗𝟕𝟎𝟔𝟏
𝛃 = 𝟎. 𝟎𝟐𝟓𝟐𝟏𝟖𝟐
1.3233e005
0.00428981
⍺ = 0.0001⍺ = 0.99979
⍺ = 0 ⍺ = 9.82𝑒 − 006
⍺ = 0.0002 ⍺ = 0.999
⍺ = 0 ⍺ = 6.16𝑒 − 007
0

77. λ2
0
= (0.000144489)(0.237016)(0.0211445) +
(9.81924e − 006)(0.237016)(0.167066) +
(0.999845)(0.0594244)(0.0717642) +
(6.44653e − 007)(0.0594244)(0.0232016)
= 0.004265
λ2
1
= (0.000144489)(0.000649159)(0.167066) +
(9.81924e −006)(0.000649159)(0.0211445) +
(0.999845)(0.00258919)(0.0232016) +
(6.44653e − 007)(0.00258919)(0.0717642)
= 6.008e-005
1
Soft Output (k = 2)
𝐿2=log(6.008e-005 / 0.004265)
= -4.26252
𝑈2= 0
2
0 1
β = 0.082573
𝛃 = 𝟎. 𝟔𝟑𝟖𝟒𝟔𝟗
𝛃 = 𝟎. 𝟎𝟔𝟗𝟕𝟎𝟔𝟏
𝛃 = 𝟎. 𝟎𝟐𝟓𝟐𝟏𝟖𝟐
00 00
01 01
10 10
1111
3
β = 0.0211445
𝛃 = 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟔
𝛃 = 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
0.237016
0.00258919
⍺ = 0.0005
⍺ = 0.0417
⍺ = 3.9𝑒 − 005
⍺ = 0.9576
⍺ = 0.0001
⍺ = 0.999
⍺ = 6.16𝑒 − 007

78. λ3
0
= (0.000552098)(0.0930673)(0.0563593) +
(0.0417279)(0.0930673)(0.0327154) +
(3.90246e − 005)(0.0283623)(0.0637005) +
(0.957681)(0.0283623)(0.16995)
= 0.0047462
λ3
1
= (0.000552098)(0.132484)(0.0327154) +
(0.0417279)(0.132484)(0.0563593) +
(3.90246e − 005)(0.434728)(0.16995) +
(0.957681)(0.434728)(0.0637005)
= 0.0268373
1
Soft Output (k = 3)
𝐿3=log(0.0268373 / 0.0047462)
= 1.73245
𝑈3= 1
2
0 1 0
4
β = 0.0211445
𝛃 = 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟔
𝛃 = 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
00 00
01 01
10 10
1111
β = 0.0563593
𝛃 = 𝟎. 𝟏𝟔𝟗𝟗𝟓
𝛃 = 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
𝛃 = 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
0.0930673
0.434728
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.9189
⍺ = 0.0005
⍺ = 3.9𝑒 − 005
⍺ = 0.9576

79. λ4
0
= (0.0123158)(0.0684456)(0.165809) +
(0.0599917)(0.0684456)(0.0206017) +
(0.00873342)(0.207239)(0.00120834) +
(0.918959)(0.207239)(0.0968433)
= 0.0186698
λ4
1
= (0.0123158)(0.31646)(0.0206017) +
(0.0599917)(0.31646)(0.165809) +
(0.00873342)(0.104518)(0.0968433) +
(0.918959)(0.104518)(0.00120834)
= 0.00343263
1
Soft Output (k = 4)
𝐿4=log(0.0034326 / 0.0186698)
= -1.693
𝑈4= 0
2
0 1 0 1
5
β = 0.0563593
𝛃 = 𝟎. 𝟏𝟔𝟗𝟗𝟓
𝛃 = 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
𝛃 = 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
00 00
01 01
10 10
1111
β = 0.165809
𝛃 = 𝟎. 𝟎𝟗68433
𝛃 = 𝟎. 𝟎𝟐𝟎𝟔𝟎𝟏𝟕
𝛃 = 𝟎. 𝟎𝟎𝟏𝟐𝟎𝟖𝟑
0.0684456
0.104518
⍺ = 0.06254
⍺ = 0.6035
⍺ = 0.3086
⍺ = 0.0123
⍺ = 0.0087
⍺ = 0.9189

80. λ5
0
= (0.0625394)(1.94825e − 005)(0.0675718) +
(0.603561)(1.94825e − 005)(0.115723) +
(0.0252443)(0.000210334)(0.00490699) +
(0.308655)(0.000210334)(0.155118)
= 1.15395e-005
λ5
1
= (0.0625394)(0.0838888)(0.115723) +
(0.603561)(0.0838888)(0.0675718) +
(0.0252443)(0.00777034)(0.155118) +
(0.308655)(0.00777034)(0.00490699)
=0.00407062
1
Soft Output (k = 5)
𝐿5=log(0.00407062 / 1.15395e-005)
= 5.86577
𝑈5= 1
2
0 1 0 1 0
6
β = 0.0211445
𝛃 = 𝟎. 𝟎𝟐𝟑𝟐𝟎𝟏𝟔
𝛃 = 𝟎. 𝟏𝟔𝟕𝟎𝟔𝟔
𝛃 = 𝟎. 𝟎𝟕𝟏𝟕𝟔𝟒𝟐
00 00
01 01
10 10
1111
β = 0.0563593
𝛃 = 𝟎. 𝟏𝟔𝟗𝟗𝟓
𝛃 = 𝟎. 𝟎𝟑𝟐𝟕𝟏𝟓𝟒
𝛃 = 𝟎. 𝟎𝟔𝟑𝟕𝟎𝟎𝟓
0.0930673
0.434728
⍺ = 0.0123
⍺ = 0.0599
⍺ = 0.9189
⍺ = 0.0005
⍺ = 3.9𝑒 − 005
⍺ = 0.9576

81. λ6
0
= (0.864696)(2.91557e − 005)(0.136882) +
(0.00445859)(2.91557e − 005)(0.059595) +
(0.0897962)(0.000299416)(0.011996) +
(0.0410488)(0.000299416)(1.04847)
= 1.66676e-005
λ6
1
= (0.864696)(0.0992647)(0.059595) +
(0.00445859)(0.0992647)(0.136882) +
(0.0897962)(0.00966591)(1.04847) +
(0.0410488)(0.00966591)(0.011996)
=0.00609064
1
Soft Output (k = 6)
𝐿6=log(0.00609064 / 1.66676e-005)
= 5.90104
𝑈6= 1
2
0 1 0 1 0 1 1
7
β = 0.06757
𝛃 = 𝟎. 𝟏𝟓𝟓𝟏𝟏𝟖
𝛃 = 𝟎. 𝟏𝟏𝟓𝟕𝟐𝟑
𝛃 = 𝟎. 𝟎𝟎𝟒𝟗𝟎𝟔𝟗
00 00
01 01
10 10
1111
⍺ = 0.0053
⍺ = 0.010
⍺ = 0.0048
⍺ = 0.8647
2.91557e-005
0.00966591
⍺ = 0.9797
⍺ = 0.0044
β = 0.136882
𝛃 = 𝟏. 𝟎𝟒𝟖𝟒𝟕
𝛃 = 𝟎. 𝟎𝟓𝟗𝟔𝟎
𝛃 = 𝟎. 𝟎𝟏𝟏𝟗𝟗𝟗𝟔

82. • An EXIT chart (Extrinsic information transfer
chart) is a tool to aid the construction of good
iteratively-decoded error-correcting codes
such as Turbo Codes.
• Exit Chart is one of the most important part is
the analysis of the code.
• Using the EXIT charts we will study the
behavior of our turbo codes.
82
EXIT Chart

83. Use Case Diagram

84. Encoder Testing
Test Name Encoding
Test Date 08/3/2016
Application Name Reliable Multimedia Transmission under Noisy Conditions
Input Binary data of Image
Output Encoded Bits
Test Conductor Salim Syed and Sharukh Ali Khan
Verified By Dr.Abbas Khalid

85. Decoder Testing
Test Name Decoding
Test Date 26/3/2016
Application Name Reliable Multimedia Transmission under Noisy Conditions
Input Binary data of Image
Output Decoded Bits and Graph
Test Conductor Salim Syed and Sharukh Ali Khan
Verified By Dr.Abbas Khalid

86. Exit Chart Analysis
Test Name EXIT chart analysis
Test Date 15/4/2016
Application Name Reliable Multimedia Transmission under Noisy Conditions
Input System Generated Bits
Output Graph and Received Bits
Test Conductor Salim Syed and Sharukh Ali Khan
Verified By Dr.Abbas Khalid

87. BER Analysis
Test Name BER analysis
Test Date 22/4/2016
Application Name Reliable Multimedia Transmission under Noisy Conditions
Input Image file
Output Graph and Received image file
Test Conductor Salim Syed and Sharukh Ali Khan
Verified By Dr.Abbas Khalid