Enumerative Encoding of TMTR Codes for Optical Recording Channel

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Research Article
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Abstract

We propose a new time-varying maximum transition run (TMTR) code for DVD recording systems, which has a rate Open image in new window higher than the EFMPlus code and a lower power spectral density (PSD) at low frequencies. An enumeration method for constructing the new TMTR code is presented. Computer simulations indicate that the proposed TMTR code outperforms the EFMPlus code in error performance when applied to partial response optical recording channels.

Keywords

Code Rate Modulation Code User Density Pseudorandom Binary Sequence High Code Rate 

1. Introduction

In data storage systems, a modulation code is known as Open image in new window -constrained code, where Open image in new window and Open image in new window represent the maximal and minimal number of zeros between two consecutive ones. The main function of a Open image in new window modulation code is to improve the recording density and increase the storage capacity. The timing information could also be controlled using a Open image in new window modulation code. For example, magnetic tape and disk systems often adopt Open image in new window or Open image in new window codes, while optical systems such as CD and DVD usually employ Open image in new window EFM (Eight-to-Fourteen Modulation) or Open image in new window EFMPlus [1] modulation codes.

Recent research on the Open image in new window modulation code has focused on the time-varying maximum transition run (TMTR) code [2, 3, 4, 5, 6, 7, 8, 9], which can be treated as a Open image in new window modulation code. The TMTR code matched to the partial response channel can delete some dominant error events and enhance the Euclidian distance of the partial response channel to the matched filter bound. As a result, a coding gain over the conventional scheme can be obtained when the time-varying Viterbi detector is applied to the TMTR-coded partial response channel. In a previous work [10], we proposed a new time-varying maximum transition run (TMTR) code with Open image in new window constraint for DVD recording systems, which has rate Open image in new window higher than the EFMPlus code and a lower power spectral density (PSD) at low frequencies. The Open image in new window TMTR code was realized with a look-up table, and the k-constraint was not considered during construction. In this paper, instead of a look-up table we present an enumeration method for constructing the Open image in new window codes. Based on this construction, a rate Open image in new window code with Open image in new window is found. The proposed code can achieve better timing recovery performance. We show that 387 surviving words exist with length 11 from the construction technique. This new method needs one bit of memory for encoding, but no memory is required for decoding. An enumerating algorithm is used for encoding/decoding, and a look-up table is not required.

The rest of this paper is organized as follows. In Section 2, we briefly describe the Open image in new window TMTR codes for partial response (PR) optical recording channels. In Section 3, an outline of the design methodology for constructing a high-rate TMTR code is presented. We illustrate concatenation problem between codewords and provide a solution. Section 4 introduces an enumerative coding method for TMTR codes. In Section 5, the power spectral density (PSD) of the rate Open image in new window code is evaluated and compared with the EFMPlus code. An error performance comparison between uncoded, TMTR-coded, and EFMPlus-coded EPRII optical recording channel is presented in Section 6. The conclusion is provided in Section 7.

2. TMTR Codes for Partial Response Optical Recording Channels

The maximum transition run (MTR) method is a coding method, which limits the number of consecutive potential variations being not greater than Open image in new window . The time-varying maximum transition run (TMTR) is a further modification of the MTR, which sets different constraints for the number of consecutive variations depending upon whether it starts at an odd or even position. For example, Open image in new window TMTR constraints mean that the number of consecutive 1s starting at an even position is not greater than Open image in new window and the number of consecutive 1s starting at an odd position is not greater than Open image in new window . The method can increase the minimum distance of the encoded system to an upper matched filter bound (MFB); therefore, it has the distance enhancing property. The TMTR constraint can be described using a finite state transition diagram (FSTD), given in Figure 1. The vertices at the top of the diagram represent even positions, and the number of 1s starting at the even positions can be Open image in new window only, satisfying the constraint of Open image in new window . The vertex at the bottom of the diagram represents odd positions, and the number of 1s starting at the odd positions can be Open image in new window or Open image in new window , satisfying the constraint of Open image in new window . Figure 2 shows a simplified FSTD with the Open image in new window TMTR constraints.
Figure 1

FSTD of TMTR constraint.

Figure 2

Simplified FSTD with TMTR constraint.

Cideciyan et al. [11] suggested an advanced signal processing technique, the partial response and maximal likelihood (PRML) channel, to further increase the recording densities and reliability over that achieved by the conventional peak detector. The signal processing technique employing the PRML channel has become a standard widely used in most of today's data storage systems. The most popular partial response system for optical recording has the form Open image in new window , where Open image in new window is a nonnegative integer. The PR systems with Open image in new window and Open image in new window are referred to as the PRII and EPRII systems, respectively. Karabed and Siegel [12] proposed a class of modulation codes that take advantage of the well-defined spectral nulls presented in partial response channels. The time-varying maximum transition run (TMTR) code [2, 3, 4, 5, 6, 7, 8, 9], which can be treated as a Open image in new window modulation code, has recently been studied for partial response channels. The TMTR code matched to the partial response channel can delete some dominant error events and enhance the Euclidian distance of the partial response channel.

Vannucci and Foschini [13] described a powerful algorithm to search for the minimum Euclidean distance Open image in new window for Open image in new window partial response channels. They found that the shortest error event achieving Open image in new window has the type of Open image in new window for most of Open image in new window partial response channels. As a matter of fact they found that those error events of the form Open image in new window always have a distance less than the matched filter bound Open image in new window which is defined as the distance corresponding to the one-bit error event.

If the error event Open image in new window can be forbidden to occur in coded sequences for Open image in new window partial response channels, the minimum distance of the channels can be increased to Open image in new window resulting in a coding gain of Open image in new window dB. With NRZI modulation there are four pairs of binary coded sequences, which could generate the error event Open image in new window shown as follows:

         

The TMTR modulation code with constraint Open image in new window can be used to forbid the occurrence of sequences 111 and 011, and as a result error event Open image in new window would not occur in the detection of Open image in new window channels, and a coding gain of 3 dB can be obtained. The channel capacity of the Open image in new window TMTR code is equal to Open image in new window , which indicates that a codeword with length 11 bits at least is required to encode or represent a byte (8-bit) message.

3. Construction for Open image in new windowTMTR Codes

A TMTR code is specified as Open image in new window constraint, where k is the maximum number of consecutive zeros, Open image in new window and Open image in new window constraints represent the maximum numbers of consecutive ones starting from an even position and an odd position, respectively. This construction is based upon Open image in new window constraint. Both Open image in new window and Open image in new window represent the maximum number of ones after the last zero and the maximum number of zeros after the last one, respectively. In similar, parameters Open image in new window and Open image in new window represent the maximum number of ones before the first zero and the maximum number of zeros before the first one. Any two sequences satisfying Open image in new window constraint can be freely concatenated without violating the Open image in new window constraint. In order to reduce the consecutive zero length in the sequences after concatenation, the following substitution rule is applied: assume that a sequence Open image in new window is followed by a sequence Open image in new window , then

if Open image in new window has more than Open image in new window zeros before the first one, and the last bit of Open image in new window is a zero, then flip the first 2 bits of Open image in new window into two ones, for example, Open image in new window ;

otherwise, use Open image in new window as the encoder output.

In the case of Open image in new window , a long sequence of consecutive zeros is spread into two parts by "11" in the beginning of the 2nd sequence. Because Open image in new window , a sequence beginning with "11" is not an original code.

Rates of some constructed codes using this method are listed in Table 1. As displayed, a rate Open image in new window code with Open image in new window and Open image in new window constraints is found, and this code has the Open image in new window constraint. Sequences of even lengths satisfying the Open image in new window constraint can be freely concatenated without violating the Open image in new window constraint. Odd lengths sequences, however, cannot be freely concatenated without violating the Open image in new window constraint. To solve this problem, assuming that there is a modulo- Open image in new window counter synchronized to the data, the two transitions in arrow can end at times Open image in new window and Open image in new window relative to counter. The even and odd positions in a codeword of 11 bits are given as (e o e o e o e o e o e). For example, a sequence of 3 codewords will be

where the 1st line expresses the positions of the code bits. The 2nd line expresses the even/odd code bit positions. The 3rd line expresses the maximum number of consecutive "1" starting at the position. There is no two consecutive "2" in the 3rd line. It means that no dominant error event Open image in new window will occur. To obtain the coding gain of this encoder, a time-varying Viterbi detector is required. The trellis diagrams of the Viterbi detector for even and odd times are shown in Figure 7(c). The Viterbi detector for code bit stream positions will be the same as shifting the 2nd line to right by 1 position. The result is shown in the 4th line. The even or odd Viterbi detector properties must match the bit position shown in the 4th line.

4. Enumerative Encoding Open image in new window TMTR Codes

Let us lexicographically order the binary sequences of length Open image in new window by

An enumerating encoder maps a set of consecutive integers onto a lexicographically ordered set of sequences. In order to describe the enumerating encoder/decoder, some notations will be defined as follows.

  1. (D.1)

    Open image in new window is the lexicographically ordered set of Open image in new window sequences of length Open image in new window .

     
  2. (D.2)

    R( Open image in new window ) is the number of sequences Open image in new window such that Open image in new window .

     
  3. (D.3)
     
  4. (D.4)

    res( Open image in new window ) is the sequence obtained by modifying the first nonzero bit of Open image in new window to zero.

     
  5. (D.5)

    Open image in new window is the minimum sequence among sequences in Open image in new window and having the first symbol one at position Open image in new window .

     
  6. (D.6)

    Open image in new window is the maximum sequence among sequences in Open image in new window and having the first symbol one at position Open image in new window .

     
  7. (D.7)
     
  8. (D.8)
     
By definitions (D.5) and (D.6), it is easy to see that
The Open image in new window 's and Open image in new window 's can be obtained by the following recursive relation with initial values Open image in new window :
For illustration, consider Open image in new window sequences with length Open image in new window ; one has
Similarly, when using a codeword of length Open image in new window and code rate of Open image in new window , the enumerating encoding method is as follows:
Let Open image in new window be a subset of Open image in new window consisting of sequences with no more than Open image in new window leading zeros. Then the number of elements of Open image in new window is given as
We can then encode an integer Open image in new window to a sequence Open image in new window using the enumerative algorithm given in Figure 3 (assuming the previous block is Open image in new window ). The decoding is simply done by
For illustration, the mapping relationship between data and codewords of a rate Open image in new window code satisfying Open image in new window constraint is given in Table 2. The data to codeword mapping for a rate Open image in new window code satisfying Open image in new window constraint is listed in Table 3.
Table 2

Codebook of rate Open image in new window code.

Data

Codeword

Data

Codeword

Open image in new window

Open image in new window

Open image in new window

Open image in new window

000

0001or 1101

100

0110

001

0010

101

1000

010

0100

110

1001

011

0101

111

1010

Table 3

Data to codeword mapping for the rate Open image in new window code.

 

Data

Codeword

 

Data

Codeword

 

Data

Codeword

 

Data

Codeword

 

Open image in new window

Open image in new window

 

Open image in new window

Open image in new window

 

Open image in new window

Open image in new window

 

Open image in new window

Open image in new window

0

00000000

00000001000

42

00101010

00001100100

94

01011110

00100100100

178

10110010

01001000000

  

or 11000001000

  

or 11001100100

95

01011111

00100100101

179

10110011

01001000001

1

00000001

00000001001

43

00101011

00001100101

96

01100000

00100100110

180

10110100

01001000010

  

or 11000001001

  

or 11001100101

97

01100001

00100101000

181

10110101

01001000100

2

00000010

000000010

44

00101100

00001100110

98

01100010

00100101001

182

10110110

01001000101

  

or 11000001010

  

or 11001100110

99

01100011

00100101010

183

10110111

01001000110

3

00000011

00000010000

45

00101101

00001101000

100

01100100

00101000000

184

10111000

01001001000

  

or 11000010000

  

or 11001101000

101

01100101

00101000001

185

10111001

01001001001

4

00000100

00000010001

46

00101110

00001101001

102

01100110

00101000010

186

10111010

01001001010

  

or 11000010001

  

or 11001101001

103

01100111

00101000100

187

10111011

01001010000

5

00000101

00000010010

47

00101111

00001101010

104

01101000

00101000101

188

10111100

01001010001

  

or 11000010010

  

or 11001101010

105

01101001

00101000110

189

10111101

01001010010

6

00000110

00000010100

48

00110000

00010000001

106

01101010

00101001000

190

10111110

01001010100

  

or 11000010100

  

or 11010000001

107

01101011

00101001001

191

10111111

01001010101

7

00000111

00000010101

49

00110001

00010000010

108

01101100

00101001010

192

11000000

01001010110

  

or 11000010101

  

or 11010000010

109

01101101

00101010000

193

11000001

01001011000

8

00001000

00000010110

50

00110010

00010000100

110

01101110

00101010001

194

11000010

01001011001

  

or 11000010110

  

or 11010000100

111

01101111

00101010010

195

11000011

01001011010

9

00001001

00000011000

51

00110011

00010000101

112

01110000

00101010100

196

11000100

01001100000

  

or 11000011000

  

or 11010000101

113

01110001

00101010101

197

11000101

01001100001

10

00001010

00000011001

52

00110100

00010000110

114

01110010

00101010110

198

11000110

01001100010

  

or 11000011001

  

or 11010000110

115

01110011

00101011000

199

11000111

01001100100

11

00001011

00000011010

53

00110101

00010001000

116

01110100

00101011001

200

11001000

01001100101

  

or 11000011010

  

or 11010001000

117

01110101

00101011010

201

11001001

01001100110

12

00001100

00000100000

54

00110110

00010001001

118

01110110

00101100000

202

11001010

01001101000

  

or 11000100000

  

or 11010001001

119

01110111

00101100001

203

11001011

01001101001

13

00001101

00000100001

55

00110111

00010001010

120

01111000

00101100010

204

11001100

01001101010

  

or 11000100001

  

or 11010001010

121

01111001

00101100100

205

11001101

01010000001

14

00001110

00000100010

56

00111000

00010010000

122

01111010

00101100101

206

11001110

01010000010

  

or 11000100010

  

or 11010010000

123

01111011

00101100110

207

11001111

01010000100

15

00001111

00000100100

57

00111001

00010010001

124

01111100

00101101000

208

11010000

01010000101

  

or 11000100100

  

or 11010010001

125

01111101

00101101001

209

11010001

01010000110

16

00010000

00000100101

58

00111010

00010010010

126

01111110

00101101010

210

11010010

01010001000

  

or 11000100101

  

or 11010010010

127

01111111

00110000001

211

11010011

01010001001

17

00010001

00000100110

59

00111011

00010010100

128

10000000

00110000010

212

11010100

01010001010

  

or 11000100110

  

or 11010010100

129

10000001

00110000100

213

11010101

01010010000

18

00010010

00000101000

60

00111100

00010010101

130

10000010

00110000101

214

11010110

01010010001

  

or 11000101000

  

or 11010010101

131

10000011

00110000110

215

11010111

01010010010

19

00010011

00000101001

61

00111101

00010010110

132

10000100

00110001000

216

11011000

01010010100

  

or 11000101001

  

or 11010010110

133

10000101

00110001001

217

11011001

01010010101

20

00010100

00000101010

62

00111110

00010011000

134

10000110

00110001010

218

11011010

01010010110

  

or 11000101010

  

or 11010011000

135

10000111

00110010000

219

11011011

01010011000

21

00010101

00001000000

63

00111111

00010011001

136

10001000

00110010001

220

11011100

01010011001

  

or 11001000000

  

or 11010011001

137

10001001

00110010010

221

11011101

01010011010

22

00010110

00001000001

64

01000000

00010011010

138

10001010

00110010100

222

11011110

01010100000

  

or 1100100000

  

or 11010011010

139

10001011

00110010101

223

11011111

01010100001

23

00010111

00001000010

65

01000001

00010100000

140

10001100

00110010110

224

11100000

01010100010

  

or 11001000010

  

or 11010100000

141

10001101

00110011000

225

11100001

01010100100

24

00011000

00001000100

66

01000010

00010100001

142

10001110

00110011001

226

11100010

01010100101

  

or 11001000100

  

or 11010100001

143

10001111

00110011010

227

11100011

01010100110

25

00011001

00001000101

67

01000011

00010100010

144

10010000

00110100000

228

11100100

01010101000

  

or 11001000101

  

or 11010100010

145

10010001

00110100001

229

11100101

01010101001

26

00011010

00001000110

68

01000100

00010100100

146

10010010

00110100010

230

11100110

01010101010

  

or 11001000110

  

or 11010100100

147

10010011

00110100100

231

11100111

10000000100

27

00011011

00001001000

69

01000101

00010100101

148

10010100

00110100101

232

11101000

10000000101

  

or 11001001000

  

or 11010100101

149

10010101

00110100110

233

11101001

10000000110

28

00011100

00001001001

70

01000110

00010100110

150

10010110

00110101000

234

11101010

10000001000

  

or 11001001001

  

or 11010100110

151

10010111

00110101001

235

11101011

10000001001

29

00011101

00001001010

71

01000111

00010101000

152

10011000

00110101010

236

11101100

10000001010

  

or 11001001010

  

or 11010101000

153

10011001

01000000010

237

11101101

10000010000

30

00011110

00001010000

72

01001000

00010101001

154

10011010

01000000100

238

11101110

10000010001

  

or 11001010000

  

or 11010101001

155

10011011

01000000101

239

11101111

10000010010

31

00011111

00001010001

73

01001001

00010101010

156

10011100

01000000110

240

11110000

10000010100

  

or 11001010001

  

or 11010101010

157

10011101

01000001000

241

11110001

10000010101

32

00100000

00001010010

74

01001010

00100000001

158

10011110

01000001001

242

11110010

10000010110

  

or 11001010010

75

01001011

00100000010

159

10011111

01000001010

243

11110011

10000011000

33

00100001

00001010100

76

01001100

00100000100

160

10100000

01000010000

244

11110100

10000011001

  

or 11001010100

77

01001101

00100000101

161

10100001

01000010001

245

11110101

10000011010

34

00100010

00001010101

78

01001110

00100000110

162

10100010

01000010010

246

11110110

10000100000

  

or 11001010101

79

01001111

00100001000

163

10100011

01000010100

247

11110111

10000100001

35

00100011

00001010110

80

01010000

00100001001

164

10100100

01000010101

248

11111000

10000100010

  

or 11001010110

81

01010001

00100001010

165

10100101

01000010110

249

11111001

10000100100

36

00100100

00001011000

82

01010010

00100010000

166

10100110

01000011000

250

11111010

10000100101

  

or 11001011000

83

01010011

00100010001

167

10100111

01000011001

251

11111011

10000100110

37

00100101

00001011001

84

01010100

00100010010

168

10101000

01000011010

252

11111100

10000101000

  

or 11001011001

85

01010101

00100010100

169

10101001

01000100000

253

11111101

10000101001

38

00100110

00001011010

86

01010110

00100010101

170

10101010

01000100001

254

11111110

10000101010

  

or 11001011010

87

01010111

00100010110

171

10101011

01000100010

255

11111111

10001000000

39

00100111

00001100000

88

01011000

00100011000

172

10101100

01000100100

   
  

or 11001100000

89

01011001

00100011001

173

10101101

01000100101

   

40

00101000

00001100001

90

01011010

00100011010

174

10101110

01000100110

   
  

or 11001100001

91

01011011

00100100000

175

10101111

01000101000

   

41

00101001

00001100010

92

01011100

00100100001

176

10110000

01000101001

   
  

or 11001100010

93

01011101

00100100010

177

10110001

01000101010

   

5. Power Spectral Density

In DVD systems the power spectral density at low frequency, referred to as the low-frequency content, of the encoded data sequences should normally be maintained as low as possible to alleviate interference with pilot and focus servo signals. For example, in addition to satisfying the Open image in new window constraint, the Open image in new window EFMPlus code employed in the DVD system is also designed to achieve very low low-frequency content to reduce the interference between the written signal and the servo signal. Efficient low-frequency component suppression is a crucial criterion for the Open image in new window Open image in new window TMTR code rate. The low-frequency content is based upon the running digital sum (RDS) Open image in new window given by
where Open image in new window represents the writing sequences. A lower RDS results in lower low-frequency content. The power spectral density of a sequence can be expressed as
For a bounded RDS, that is, Open image in new window , C is a constant, and the DC content is then given as
that indicates a DC-free content. As shown in (11), the number of sequences with length 11 satisfying Open image in new window is Open image in new window , but it only requires Open image in new window codewords in the rate Open image in new window TMTR code. The surplus Open image in new window code sequences can be exploited for minimizing both Open image in new window -constraint and low-frequency content. The Open image in new window surplus code sequences are then used to suppress the low-frequency content by reducing the running digital sum (RDS). There are two tables (a main table and a substitute table) used in the encoder, as shown in Figure 4. The main table consists of 256 codewords (i.e., sequences Open image in new window corresponding to Open image in new window in the enumerating algorithm), and the substitute table with Open image in new window codewords (i.e., sequences Open image in new window corresponding to Open image in new window ) is used to minimize the power spectral density at low frequencies.
Figure 4

Codebook of rate Open image in new window TMTR code.

The PSD of both EFMPlus code and Open image in new window TMTR code can be computed using the fast Fourier transform (FFT)

where Open image in new window

Figure 5 depicts the power spectral density (PSD) versus the normalized frequency for the Open image in new window EFMPlus code, the rate Open image in new window Open image in new window TMTR code, and the Open image in new window    Open image in new window TMTR code with method of enumerative, respectively. As shown, at frequency of Open image in new window , the PSDs for both EFMPlus and TMTR codes are Open image in new window  dB, Open image in new window  dB and Open image in new window  dB, respectively. The result indicates that the Open image in new window Open image in new window TMTR code achieves a lower low-frequency content (with 5 dB lower) than the EFMPlus code. In Figure 5, we notice that the rate Open image in new window Open image in new window TMTR code has only Open image in new window  dB at frequency of Open image in new window and it is not suitable for optical recording systems, although it has a higher code rate compared to the Open image in new window Open image in new window TMTR code.
Figure 5

Power spectral density.

6. Simulation Results

The superiority of the rate Open image in new window Open image in new window TMTR code over the EFMPlus code is also demonstrated on error performance through a computer simulation on the EPRII optical channel of the form Open image in new window . Figure 6 depicts the optical recording system for simulation. An 8-state transition diagram, as depicted in Figure 7(a), can describe the EPRII optical channel. During simulation, a Gaussian optical recording channel was assumed with impulse response Open image in new window given by
where Open image in new window represents the user information density. Open image in new window is the channel bit period assumed to be one ( Open image in new window ) for uncoded systems in simulation. The optical recording channel is then corrupted with the additive white Gaussian noise (AWGN), and at receiver an ideal EPRII equalizer is employed to equalize the optical channel in the form of Open image in new window .
Figure 6

Optical recording system model.

Figure 7

Trellis diagram. (a) EPRII channel. (b) EFMPlus-coded EPRII channel. (c) TMTR-coded EPRII channel.

The Viterbi detector is then employed on the 8-state trellis diagram of the EPRII optical channel, to recover binary-recorded data from the equalized and sampled output. Note that for the EFMPlus-coded EPRII channel the 8-state trellis diagram can be reduced to a 6-state trellis diagram due to the Open image in new window constraint, as shown in Figure 7(b), while for the TMTR-coded EPRII channel the 8-state trellis diagram is still required with some branches deleted as depicted in Figure 7(c). Figure 8 demonstrates the bit error rate versus the signal to noise ratio (SNR) for TMTR-coded, EFMPlus-coded and uncoded EPRII optical recording channels at user density Open image in new window . The SNR in dB is defined in this paper as the ratio of the input complex waveform signal energy generated by a 127-bit pseudorandom binary sequence to the channel noise energy of the same duration. As shown both the TMTR-coded and EFMPlus-coded EPRII systems achieve better performance than the uncoded EPRII system. This is because both coded EPRII systems have a coding gain of 3 dB when these modulation codes are considered in the 8-state trellis diagram of the EPRII system during detection. This leads to a coding gain of 3 dB. As also can be seen in this figure, the TMTR-coded EPRII system improves the EFMPlus-coded EPRII system by approximately 1 dB in bit error rate. The coding loss of the EFMPlus-coded EPRII system is due to the bit rate loss. Figure 9 shows the signal to noise (SNR) required to achieve a bit rate error of Open image in new window , as a function of user density, at a rate of Open image in new window    Open image in new window TMTR code and a rate of Open image in new window EFMPlus code, applied to an EPRII optical channel. Figure 9 shows how the TMTR code provides little coding gain at user densities below 1.2 but increases coding gain at higher densities. At a user density of Open image in new window , the TMTR code on the EPRII optical channel provides nearly 2.7 dB of coding gain above the rate of Open image in new window EFMPlus code on the EPRII optical channel. Therefore, from the performance comparison made in Figure 9, it can be seen that even a greater improvement in coding gain could be achieved for TMTR-coded EPRII system at higher user densities.
Figure 8

Error performance of TMTR-/EFMPlus-coded and uncoded EPRII systems.

Figure 9

The SNR in dB required to achieve a 10 -5 BER versus user density.

7. Conclusion

In this paper, we present an enumeration method for constructing Open image in new window codes, and the look-up table described in a previous work is not required for the encoder/decoder. Based on the construction, a rate Open image in new window Open image in new window TMTR code is found. This code can achieve lower power spectral density at low frequencies compared to the EFMPlus code. In addition, computer simulations reveal that the rate Open image in new window TMTR code outperforms the EFMPlus code in error performance when applied to partial response optical recording channels.

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Copyright information

© Hui-Feng Tsai. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Computer Science and Information EngineeringChing Yun UniversityJhongli CityTaiwan

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