CHAPTER 6
REVIEW AND ANALYSIS
OF OPTIMUM CODE RATE
OF REED-SOLOMON CODES
IN AWGN CHANNEL
Introduction
Simulation Model
Simulation Results and Analysis
Conclusion
REVIEW AND ANALYSIS OF OPTIMUM CODE RATE OF
REED-SOLOMON CODES IN AWGN CHANNEL
- Introduction
Reed-Solomon codes are nonbinary cyclic codes with symbols made up of m-bit sequences, where m is any positive integer having a value greater than 2. R-S (N, K) codes on m-bit symbols exist for all N and K for which
0 < K < N < 2m + 2 (1)
where, K is the number of data symbols being encoded, and N is the total number of code symbols in the encoded block. For the most conventional R-S (N, K) code [27],
(N, K) = (2m - 1, 2m - 1 - 2t)
where, t is the symbol-error correcting capability of the code and n - k = 2t is the number of parity symbols.
Reed-Solomon codes achieve the largest possible code minimum distance for any linear code with the same encoder input and output block lengths. For nonbinary codes, the distance between two codewords is defined (analogous to Hamming distance) as the number of symbols in which the sequences differ. For Reed- Solomon codes, the code minimum distance is given by [27]
dmin = N - K + 1
The code is capable of correcting any combination of t or fewer errors, where t can
be expressed as [27]
t = 
In review paper[1], it is stated that for Reed-Solomon(R-S) codes, the optimum code rate that minimize the required Eb/No is about 0.6 to 0.7 for a Gaussian channel, 0.5 for a Rician-fading Channel(for K=7dB), and 0.3 for a Rayleigh-fading channel. However, the performance curves are plotted for BPSK modulation and an RS (31, K) code for various channel types. This statement is not generalize to all conventional RS(N,K) codes e.g. it is observed that for N=3 only one (N,K) tuple exist i.e. RS(3,1) for which the code rate is 0.3333.
- Simulation Model
Fig. 6.1 Simulated Model of Reed-Solomon Coding with M-QAM in AWGN Channel
- Simulation Results
Fig. 6.2 Coding Gain for various RS(7, k) codes with 8-QAM and AWGN channel
Fig. 6.3 Coding Gain for various RS(15, k) codes with 16-QAM and AWGN channel
Fig. 6.4 Coding Gain for various RS(31, k) codes with 32-QAM and AWGN channel
Fig. 6.5 Various RS(31, k) codes with 32-QAM and AWGN channel
Fig. 6.6 Coding Gain for various RS(63, k) codes with 64-QAM and AWGN channel
Fig. 6.7 Various RS(63, k) codes with 64-QAM and AWGN channel
Fig. 6.8 Coding Gain for various RS(127, k) codes with 128-QAM and AWGN channel
Fig. 6.9 Various RS(127, k) codes with 128-QAM and AWGN channel
Fig. 6.10 Coding Gain for various RS(255, k) codes with 256-QAM and AWGN channel
Fig. 6.11 Various RS(255, k) codes with 256-QAM and AWGN channel
Table 6.1 RS (N, K) codes
m
|
q = 2m
|
N = q-1
|
K
|
dmin
|
R = K/N
|
Wt. of code gain
at fixed BER
|
2
|
4
|
3
|
1
|
3
|
0.333333
|
Perfect
|
3
|
8
|
7
|
5
|
3
|
0.714286
|
Perfect
|
3
|
5
|
0.428571
|
Fair
| |||
1
|
7
|
0.142857
|
Fair
| |||
4
|
16
|
15
|
13
|
3
|
0.866667
|
Fair
|
9
|
7
|
0.600000
|
Perfect
| |||
5
|
11
|
0.333333
|
Fair
| |||
3
|
13
|
0.200000
|
Fair
| |||
5
|
32
|
31
|
23
|
9
|
0.741935
|
Fair
|
21
|
11
|
0.677419
|
Fair
| |||
19
|
13
|
0.612903
|
Perfect
| |||
17
|
15
|
0.548387
|
Good
| |||
15
|
17
|
0.483871
|
Fair
| |||
6
|
64
|
63
|
57
|
7
|
0.904762
|
Fair
|
51
|
13
|
0.809524
|
Fair
| |||
45
|
19
|
0.714286
|
Fair
| |||
43
|
21
|
0.68254
|
Fair
| |||
41
|
23
|
0.650794
|
Good
| |||
39
|
25
|
0.619048
|
Perfect
| |||
m
|
q = 2m
|
N = q-1
|
K
|
dmin
|
R = K/N
|
Wt. of code gain
at fixed BER
|
6
|
64
|
63
|
37
|
27
|
0.587302
|
Good
|
35
|
29
|
0.555556
|
Good
| |||
33
|
31
|
0.523810
|
Fair
| |||
31
|
33
|
0.492063
|
Fair
| |||
25
|
39
|
0.396825
|
Fair
| |||
19
|
45
|
0.301587
|
Fair
| |||
7
|
128
|
127
|
115
|
13
|
0.905512
|
Fair
|
101
|
27
|
0.795276
|
Fair
| |||
81
|
47
|
0.637795
|
Good
| |||
79
|
49
|
0.622047
|
Good
| |||
77
|
51
|
0.606299
|
Perfect
| |||
75
|
53
|
0.590551
|
Good
| |||
73
|
55
|
0.574803
|
Good
| |||
63
|
65
|
0.496063
|
Fair
| |||
51
|
77
|
0.401575
|
Fair
| |||
39
|
89
|
0.307087
|
Fair
| |||
8
|
256
|
255
|
229
|
27
|
0.898039
|
Fair
|
205
|
51
|
0.803922
|
Fair
| |||
159
|
97
|
0.623529
|
Good
| |||
157
|
99
|
0.615686
|
Good
| |||
M
|
q = 2m
|
N = q-1
|
K
|
dmin
|
R = K/N
|
Wt. of code gain
at fixed BER
|
8
|
256
|
255
|
155
|
101
|
0.607843
|
Perfect
|
153
|
103
|
0.600000
|
Good
| |||
151
|
105
|
0.592157
|
Good
| |||
103
|
153
|
0.403922
|
Fair
|
- Conclusion
From Figures 6.2-11, simulation data is summarized in Table 6.1. and thereby BER performances of various conventional RS(N,K) are weighted as Fair, Good and Perfect (N, K) tuple.
CHAPTER 7
SUMMARY AND CONCLUSION
Summary and Conclusions
Scope for Future Work
SUMMARY AND CONCLUSION
- Summary and Conclusions
The goal of first part of this thesis is to review fundamental relationships used in designing digital communication systems. First, we examined the BER performance of bandwidth-limited and power-limited systems and how such conditions influence the design. These are summarized in Figures 3.26 and 3.30
Additionally, we examined important types of block codes such as Hamming codes, Golay codes, Reed-Solomon codes. These techniques are summarized in Fig. 3.29 Further, we examined effect of convolution codes, Reed-Solomon codes, OFDM, MIMO, MIMO-TCM and MIMO-OFDM on Rayleigh channel to mitigating the effect of multipath Rayleigh fading channel. These techniques are summarized in Fig. 3.32
In second part of this thesis, we implemented Multilevel QoS system and 8-Offset PSK. BER performance of different levels of QoS system is shown in Fig. 4.2. Constellation diagram of two variants of 8-OPSK is shown in Figures 5.1 and 5.2
In third part of this thesis, we examined Reed-Solomon (R-S) codes, a powerful class of non-binary block codes, particularly useful for correcting burst errors. Because coding efficiency increases with code length, R-S codes have a special attraction. We determine the perfect RS(N,K) tuple for N:3, 7, 15, 31, 63, 127 and 255. These codes are summarized in Table 6.1
Scope for Future Work
- Future work will include a comparison of different MIMO-OFDM architectures with and without space–time coding, and developing a channel estimator for high mobility wireless communications on Multipath Rayleigh Fading Channel.
- Offset modulation (OM-OFDM) will be implemented to control the peak-to-average power ratio (PAPR) of an orthogonal frequency division multiplexing (OFDM) signal.
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