2.5 Proposed interleaver design method
3.1.1 Analysis of the reference precoding structure
For simplicity, only periodic and regular precoding patterns are considered in this analysis. For a precoding ratio ρ, 0≤ ρ≤1, the precoding pattern contains 100·ρ
% ones and 100(1−ρ) % zeros. Precoding patterns are similar to puncturing masks, except that a 1 represents a precoded data symbol and a 0, a non-precoded data symbol.
Figure 3.1: Considered PTC encoder structure.
The precoder and the TC can be seen as the outer and the inner codes of a serially concatenated code, as shown in Fig. 3.1. Therefore, the convergence behavior of the PTC can be analyzed by evaluating the EXIT curves of the corresponding outer and inner codes. Ashikhmin et al. showed in [86] that the design of capacity-approaching serially concatenated codes (among others) reduces to a curve-fitting problem. Thus, in [23], a suitable value for ρ was proposed to be identified as the one allowing the EXIT curve of the outer code to match that of the inner code as much as possible.
Fig. 3.2 shows the EXIT chart for different values ofρover the AWGN channel, evaluated at a SNR value for which the outer curve matches the one of the inner code.
Because the precoding ratio only affects the outer code (i.e., precoder) extrinsic information processing, the EXIT curve of the inner code (i.e., TC) is the same for all values ofρ. According to the EXIT chart results, 0.16 is identified as the value of
3.1. SELECTION OF A SUITABLE PRECODING STRUCTURE 67
Figure 3.2: PTC EXIT chart evaluated at Eb/N0=2 dB for different values of ρ, K= 1504, andR= 4/5 over the AWGN channel.
ρwhich makes the inner and outer curves fit best. For values of ρhigher than 0.16, the outer curve crosses the inner curve, which prevents the iterative decoding process from converging. Therefore, the PTC convergence SNR is expected to increase as the value ofρ increases to larger values than 0.16.
In order to analyze the actual impact of ρ on the PTC error rate performance, we plotted the error rate curves with uniform interleavers. Fig. 3.3 shows the corresponding curves with values of ρ ranging from 0.06 to 0.3. Sixteen decoding iterations of the decoding algorithm presented in Section 1.1.2.2 were used. As predicted in the EXIT chart analysis, an increase in the convergence threshold of the PTC is observed for values ofρ higher than 0.16. However, in spite of what was expected, the best PTC convergence performance is achieved for ρ= 0.11 not for ρ= 0.16. Even for ρ= 0.06 a better convergence performance than for ρ= 0.16 is obtained, especially in the BER curves.
These observations show that the curve-fitting approach using conventional EXIT charts is not reliable for the considered precoding structure. Then, a three-dimensio-nal (3D) EXIT chart for the PTC is proposed and studied. Because the CRSC constituent codes of the TC are symmetric, only the EXIT curve of the first con-stituent decoder of the TC, SISO 1, needs to be evaluated. Fig. 3.4 shows the 3D EXIT chart measurement scheme. At each PTC decoding iteration, the postdecoder updates thea priori information La(0,1) on d2 (the sequence at input of CRSC 1).
The EXIT chart for SISO 1, as introduced in Section 1.1.1.4, is then measured for each PTC decoding iteration. Thus, the PTC decoding trajectory can be plotted from the TC EXIT charts generated for the different PTC decoding iterations.
68 CHAPTER 3. PRECODING TECHNIQUES FOR TURBO CODES
2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 10
Figure 3.3: PTC error rate performance for different values of ρ over the AWGN channel with BPSK modulation for K= 1504 and R= 4/5.
SISO 0
Figure 3.4: 3D EXIT chart measurement for SISO 1 decoder in the PTC decoding structure.
Figs. 3.5 and 3.6 show the 3D EXIT chart evaluated at 2.43 dB, which is a value close to the decoding SNR threshold of the classical TC (see Fig. 3.7).
It can be observed that the decoding trajectory for ρ = 0.11 reaches the point (1, 1) whilst for ρ= 0.16 it is blocked. It means that a better PTC convergence performance is obtained forρ= 0.11, which is in accordance with the obtained error rate performance of Fig. 3.3. However, at 2.38 dB, the decoding trajectory for ρ= 0.06 reaches the point (1, 1), whereas it is not the case forρ= 0.11, which is not in line with the simulated error rate performance shown in Fig. 3.3.
3.1. SELECTION OF A SUITABLE PRECODING STRUCTURE 69 Consequently, the introduced EXIT chart tools (i.e., EXIT charts for serially concatenated codes and 3D EXIT charts) are not appropriate for the accurate de-termination of the best precoding ratio in terms of PTC convergence.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 3.5: 3D EXIT chart and decoding trajectory for the PTC evaluated at Eb/N0=2.43 dB forρ= 0.11,K= 1504, andR= 4/5 over the AWGN channel.
Figure 3.6: 3D EXIT chart and decoding trajectory for the PTC evaluated at Eb/N0=2.43 dB forρ= 0.16,K= 1504, andR= 4/5 over the AWGN channel.
70 CHAPTER 3. PRECODING TECHNIQUES FOR TURBO CODES
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
TC
IE1,IA2
IA1, IE2
Figure 3.7: TC EXIT chart and decoding trajectory evaluated at Eb/N0=2.43 dB over the AWGN channel.
3.1.1.1 Proposed design tool based on extrinsic information exchange
In this section, we propose a modified EXIT chart scheme that is appropriate for the selection of the precoding ratio value.
Let us first identify the shortcomings of classical EXIT charts. One important parameter for EXIT charts is the distribution of the extrinsic mutual information (MI) at the input. Traditionally in the case of TCs where one SISO decoder receives a priori information from only one other SISO decoder, a Gaussian assumption is made. In our precoded case,a priori information on some of the bits (precoded ones) is available from the SISO decoder of the accumulator (SISO 0)and from the SISO decoder of CRSC 2 (SISO 2). Therefore, the distribution of the extrinsic mutual information at the input of one component SISO decoder is now different from the TC case. It can be even questioned if it follows a Gaussian distribution at all. In order to verify the validity of our reasoning, we have compared the distribution of the extrinsic information provided at the input of the SISO 1 decoder with the one generated via a Gaussian process, both having the same measured MI value.
Figs. 3.8 (a) and (b) show a comparison of the histograms of extrinsic information measured at the input of the SISO 1 decoder and the one generated from the same Gaussian process, for ρ = 0.06 and for ρ = 0.11, respectively. As observed, the corresponding histograms are different, even for a low value of ρ (0.06), i.e., closer to a conventional TC since only 6 % of the bits are precoded. It proves that the extrinsic information samples at the input of SISO 1 cannot be generated from the same Gaussian process as in classical EXIT chart tools.
3.1. SELECTION OF A SUITABLE PRECODING STRUCTURE 71
Generated from a Gaussian process
(a)
Generated from a Gaussian process
(b)
Figure 3.8: Histograms of extrinsic informationE at the input of SISO 1 generated in SISO 0 and SISO 2 without a priori information, for ρ= 0.06 (a) and forρ= 0.11 (b), evaluated at Eb/N0 = 2 dB over the AWGN channel.
As a solution, we propose to measure the real exchange of extrinsic information between the constituent SISO decoders of the TC. In this modified PTC EXIT chart, the best PTC parameters in terms of convergence performance are identified as those allowing a crossing point (IA, IE) as close as possible to (1, 1).
The application of this modified EXIT chart to the most promising precoding ratios of Fig. 3.2 is presented in Fig. 3.9. Corresponding results validate the application of the proposed tool since the convergence threshold hierarchy is in line with simulated error rate curves of Fig. 3.3.