Contribution of Multidimensional Trellis Coding in VDSL Systems

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SETIT2005 20Log10(|H(f)| 0 −10 −20 −30 −40 −50 −60 −70 1300m Channel Attenuation 500m 1000m 270-1 v2.0.6 part1, 2002), we will deal with the 32QAM constellation shown in Figure 4 we used in our simulations. This constellation is invariant to 90° rotations. Therefore, to make the system transparent to 90° phase offsets, when mapping bits into constellation points: 1. 3 bits (Q3n, Q4n and Q5n) are assigned to points within a quadrant so that a 90° rotation leaves them unchanged, as shown in the constellation of Figure 4. 2. The two first bits (Q1n and Q2n) are differentially encoded to specify the quadrant, i.e., bits Q1n and Q2n will determine the change in quadrant from symbol to symbol using the rules listed in table 1. −80 0 0.5 1 1.5 2 2.5 Freq (Hz) 3 3.5 4 4.5 x 10 6 xx111 xx011 xx110 xx010 Figure 2. Signal attenuation for a 500m, 1000m and 1300m wire length 2. Crosstalk: is noise caused by electromagnetic radiation of other telephone lines physically located in close proximity in the same cable binder. Such coupling increases with frequency, so it is very harmful for VDSL, which uses bandwidth up to 12 MHz. Practically, we can distinguish two types of crosstalk: Near-End crosstalk (NEXT) caused by signals traveling in opposite directions in the same cable binder, and Far-End crosstalk (FEXT) caused by signals traveling in the same direction as shown in Figure 3. Line 1 Line i NEXT Line 1 Line i Figure 3. NEXT and FEXT in cable binder FEXT 3. Thermal or background noise: a convention in standardization committees is to model background noise as additive white Gaussian noise (AWGN) with a fixed Power Spectral Density (PSD) level equal to - 140 dBm/Hz as defined in (ETSI TS 101 270-1 v2.0.6 part1, 2002)(ETSI TS 101 270-2 v2.0.3 part2, 2002). qi Serial to Parallel Converter xx010 xx110 xx011 xx101 xx100 xx001 xx111 xx101 xx100 Q5n Q4n Q3n Q2n Q1n xx010 xx001 xx100 xx101 xx111 xx000 xx000 xx001 xx011 xx000 xx000 xx100 xx110 xx110 xx001 xx101 xx010 xx011 Table 2.1 xx111 Y2n−1 D Y1n−1 D Differential Encoder Figure 4. Symbol Mapper Y5n Y4n Y3n Y2n Y1n Complex Symbol Mapper Inputs Previous Outputs Current Outputs Q1n Q2n Y1n-1 Y2n-1 Y1n Y2n 1 0 0 0 0 1 1 0 0 1 1 0 Cn 2 Trellis Coding in AWGN Passband Channels Trellis Coded Modulation using four-dimensional constellations have a better performance in terms of complexity and coding gain over the usual twodimensional schemes (Lee-Fang Wei, 1987). Actual SCM-VDSL systems use the twodimensional Differential Quadrature Amplitude Modulation (DQAM) scheme (ETSI TS 101 270-1 v2.0.6 part1, 2002). In this section, we demonstrate that using the 4-dimensional Trellis Coded Modulation as a function of the truncation length of the Viterbi decoder can further increase the performance of this DQAM system, from a coding gain and cable range viewpoints. 2.1 DQAM Principe To fully understand the DQAM (ETSI TS 101 1 0 1 0 1 1 1 0 1 1 0 0 Table 1. Differential QAM Coding table 2.2 Trellis Coded Modulation: A modified WEI 16- State 4D Code An inherent cost of the coded schemes is that the size of the 2D constellation is doubled over uncoded schemes. This is due to the fact that a redundant bit is added every signaling interval. Otherwise, the coding gain of those coded schemes would be 3 dB greater. Using multidimensional (>2) constellations with a trellis code of rate m/m+1 (Lee-Fang Wei, 1987) can reduce that cost because fewer redundant bits are added for each 2D signaling interval. Furthermore, multidimensional encoding provides more flexibility than 2D encoding in that it can use a fractional number of bits per symbol.
SETIT2005 We will focus, now, on the case where the number Q of information bits transmitted per signaling interval is equal to 5. These five information bits arriving in the current signaling interval n are denoted as I1n, I2n… and I5n. A 2/3 rate, 16-state code with a 4D rectangular constellation of 211 points and a Minimum Square Euclidean Distance (MSED) d 2 0 is shown in Figure 5. The 4D constellation is constructed from a 48-point 2D constellation partitioned into eight subsets with enlarged MSED equal to 4d 2 0 , as explained in (Lee- Fang Wei, 1987). I5_n+1 I4_n+1 I3_n+1 I2_n+1 I1_n+1 I5_n I4_n I3_n I2_n I1_n DIFFERENTIAL ENCODING I3_n’ I2_n’ W2n 2T W3n 2T Trellis ENCODER W1n 2T W4n 2T I3_n’ I2_n’ I1_n Y0_n BIT CONVERTOR T 2 T Z10_n Z9_n Z8_n Z7_n Z6_n Z5_n Z4_n Z1_n+1 Z0_n+1 Z1_n Z0_n Exlusive OR Signaling Interval Delay Element Figure 5. 16-State code with 4D Rectangular constellation If we denote the current and next states of the trellis encoder as W1pW2pW3pW4p, p=n and n+2, the corresponding 16-state trellis diagram is shown in Figure 6. CURRENT NEXT STATE STATE 10log 2 ⎛ 4d0 ⎜ ⎜ 31.33d ⎜ 2 d0 ⎜ 2 ⎝ 20d0 ⎞ ⎟ 2 0 ⎟ 10 = ⎟ ⎟ ⎠ 4.0713 dB Where 31.33 d 2 0 is the average power of the 4D constellation, and 20d 2 0 is the average power of the 32QAM. 2.3 Simulation Results Bit Error Rates (BER) for the two different systems have been simulated, the uncoded system (32 DQAM), and the 4D 16-state code TCM system with a Viterbi decoder using a truncation length equal to K. In this stage of simulation, the system is simulated only with the AWGN disturbance. The results of the BER simulations, for 105 information bits sent, are shown in Figure 7. This figure shows the BER values for different Signal-to- Noise Ratios (SNR) and for different values of the truncation length K. Both systems have the same information rate (5 information bits/symbol period). Table 2 shows that the BER decreases with K. However, the BER values become very similar for the values of K that exceed 125. BER 10 −1 10 −2 10 −3 32QAM non codée MCT:Tronc 10 MCT:Tronc 40 MCT:Tronc 125 MCT:Tronc 500 MCT:Tronc 5000 MCT:No Tronc 4D SUBSET 0 2 1 3 4 6 5 7 2 0 3 1 W1n W2n W3n W4n W1n+2 W2n+2 W3n+2 W4n+2 0 0 0 0 0 0 0 0 0 2 0 0 0 1 3 1 0 0 0 1 0 0 1 0 0 0 1 0 6 4 7 5 1 3 0 2 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 10 0 SNR (dB) 10 −4 10 −5 0 5 10 15 20 25 5 7 4 6 3 1 2 0 7 5 6 4 2 0 3 1 6 4 7 5 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 Figure 7. 4D TCM performance in AWGN channel As shown in Figure 7, asymptotically, the system employing 4D TCM code gains approximately 4 dB of SNR over the uncoded system. 0 2 1 3 1 0 1 0 1 0 1 0 4 6 5 7 3 1 2 0 7 5 6 4 1 3 0 2 5 7 4 6 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 Figure 6. Trellis Diagram of 16-State code of Figure 5 The coding gain of the trellis coded modulation over the uncoded 32QAM therefore is: SNR = 18 K 10 20 40 125 BER 0.0171 0.0096 0.0065 0.004 K 500 5000 No Trunc BER 0.0051 0.0031 0.0039 SNR = 19 K 10 20 40 125 BER 0.0054 0.0026 0.0018 5.6e-4 K 500 5000 No Trunc BER 3e-4 2e-4 2.7e-4
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Contribution of Multidimensional Trellis Coding in VDSL Systems

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