In this subsection, we will remark on the simulation results presented in the previous sub-section.
As having been pointed out, we can observe from Figures 4.1 and 4.3 that the minimum pairwise non-coherent distance does not seem to be a decisive factor for codes’ performances.
On the one hand, Figure 4.1 hints that the code with the largest minimum pairwise non-coherent distance, i.e., “5-5”, gives the worst performance. On the other hand, Figure 4.3 shows the opposite as the code with larger minimum pairwise non-coherent distances usually perform better. A better criterion that can well-connect the pairwise non-coherent distances and the codes’ performances should be adopted. So we turn to the so-called cumulative distance function, based on which the method to select non-coherent convolutional codes is proposed (see (3.2)).
We then apply the convolutional code selection method defined in (3.2) for different codeword lengths, varying from N = 6 to N = 12. The selection results, together with the simulated optimal designs, are listed in Table 4.1. Notably, it is sometimes hard to identify the optimal code design from simulations because their performance curves are close to each other, and one code may outperform the other code in one region but worse in the other region. If such occurs, we will list two or even three as the optimal code designs. There however exist certain code designs that exhibit “strange” CDF patterns like “5-5” and “7-7”, of which the performances are very bad. Particularly, these two codes have many sudden increases in their CDFs.
Next, from Figures 4.5, 4.6 and 4.7, we observed that type 1 code always perform the worst among the tested three types, which seems to hint that our design favors a shorter code.
However, there is no apparent winner between type 2 and type 3 codes. To be specific, type
Table 4.1: Convolutional codes identified by the proposed method in (3.2) and the optimal convolutional code designs obtained from simulations. The codeword lengths vary from N = 6 to N = 12. The code rate is 1/2.
type codeword length code selected optimal code
3 N = 6 7-5 5-6 or 7-5
2 N = 8 5-6 4-7 or 5-6 or 7-5
1 N = 10 7-4 7-4
0 N = 12 7-4 5-7 or 7-4 or 7-5
Table 4.2: Convolutional codes identified by the proposed method in (3.2). The codeword lengths vary from N = 6 to N = 12. The code rate is 2/3. Since there are many choices for rate 2/3 codes, to identify the optimal code structures via simulations turns out to be infeasible. Hence, we did not show the optimal code design in this table.
type codeword length code selected
3 N = 6 7-6; 7-5
2 N = 8 5-6; 7-5
1 N = 10 5-6; 7-5
0 N = 12 7-5; 7-6
2 code performs better than type 3 code when they are concatenated with the 3GPP/LTE specified turbo code; however, the winner changes when they are concatenated with the (37, 21) turbo code and the turbo code in [11].
Although we infer that the concatenated coding system that we propose favors a shorter non-coherent convolutional code, the results on type 2 and 3 codes do not support this inference. Hence, we conduct another set of simulations by adding type 0 code, of which the codeword length is 12.
As such, in this series of simulations, four types of codeword lengths are used. They are type 0, type 1, type 2 and type 3, which respectively correspond to codeword lengths 12, 10, 8 and 6. The results are summarized in Figures 4.11, 4.12 and 4.13. These simulations then clearly show that the longest code, i.e., type 0, has the worst performance. However, by
following this trend, type 3 code should always have the best performance but it sometimes performs worse than the type 2 code. An possible cause could be that the the code selected by our method may not be the one with the optimal performance.
In order to verity this interpretation, we additionally perform the following simulations.
We replace the non-coherent convolutional codes selected by our method with the optimal non-coherent convolutional codes obtained from simulations. The simulation results are illustrated in Figures 4.14, 4.15 and 4.16. Here, we only focus on codes of lengths 6 and 8. These figures clearly indicate that the shorter type 3 codes are always superior than the longer type 2 codes, thereby confirming our interpretation.
Finally, we remark from Figures 4.8, 4.9 and 4.10 that the performances using our non-coherent convolutional code together with the turbo code are seemingly better than using the LS estimators with the same turbo code. This implies that when the estimation window size of the LS estimator is short such as 6 and 8, it may be advantageous to consider replacing the conventional LS estimator with our non-coherent convolutional code.
8 8.5 9 9.5 10 10.5 11 11.5 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
(37,21) turbo code with type 0, 1, 2 and 3 convolutional codes
type 0 type 1 type 2 type 3
Figure 4.11: Performances of the concatenated system that employs the (37, 21) turbo code [7] concatenated with the non-coherent convolutional codes of type 0, 1, 2, 3. Here, the codeword lengths at type 0, 1, 2, 3 codes are respectively 12, 10, 8 and 6. The inner convolutional code rate is 2/3 and the outer turbo code rate is 1/3
8 8.5 9 9.5 10 10.5 11 11.5 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
3GPP/LTE specified turbo code with type 0, 1, 2 and 3 convolutopnal codes
type 0 type 1 type 2 type 3
Figure 4.12: Performances of the concatenated system that employs the 3GPP/LTE specified turbo code [12] concatenated with the non-coherent convolutional codes of type 0, 1, 2, 3.
Here, the codeword lengths at type 0, 1, 2, 3 codes are respectively 12, 10, 8 and 6. The inner convolutional code rate is 2/3 and the outer turbo code rate is 1/3
8 8.5 9 9.5 10 10.5 11 11.5 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
Shu Lin’s turbo code with type 0, 1, 2 and 3 convolutional codes
type 0 type 1 type 2 type 3
Figure 4.13: Performances of the concatenated system that employs the turbo code from [11]
concatenated with the non-coherent convolutional codes of type 0, 1, 2, 3. Here, the codeword lengths at type 0, 1, 2, 3 codes are respectively 12, 10, 8 and 6. The inner convolutional code rate is 2/3 and the outer turbo code rate is 1/3
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
(37,21) turbo code with the best (2,1) convolutional codes in type 2 and 3
type 2 type 3
Figure 4.14: Performances of the concatenated system that employs the (37, 21) turbo code [7] concatenated with the optimal non-coherent convolutional codes of type 2 and 3. The inner convolutional code rate is 1/2 and the outer turbo code rate is 1/3
6 6.5 7 7.5 8 8.5 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
3GPP/LTE specified turbo code with the best (2,1) convolutional codes in type 2 and 3
type 2 type 3
Figure 4.15: Performances of the concatenated system that employs the 3GPP/LTE specified turbo code [12] concatenated with the optimal non-coherent convolutional codes of type 2 and 3. The inner convolutional code rate is 1/2 and the outer turbo code rate is 1/3
6 6.5 7 7.5 8 8.5 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
Shu Lin’s turbo code with the best (2,1) convolutional codes in type 2 and 3
type 2 type 3
Figure 4.16: Performances of the concatenated system that employs the turbo code from [11]
concatenated with the optimal non-coherent convolutional codes of type 2 and 3. The inner convolutional code rate is 1/2 and the outer turbo code rate is 1/3
Chapter 5
Conclusion and Future Work
In this thesis, we proposed a simple non-coherent-distance-based method to select a good convolutional code to co-work with the outer turbo code in an unknown frequency-selective fading environment. By using the selected inner non-coherent convolutional code and the corresponding GLRT decoding criterion, a better performance than the conventional LS estimator can be obtained when the codeword length/estimation window size is small. When the code rate is half, the codes that our method locates are the ones with the simulated optimal performance. When the code rate increases to 2/3, however, our method may locate the second-best convolutional code structure in the sense of simulated performance.
A side observation from our simulations that the proposed concatenated system prefers using a shorter inner convolutional code. This could be due to that in a dynamic fading environment, it may be better to perform frequent channel estimation based on a smaller window and resort the error protection task to the long outer turbo code.
Our experiments hint that when code rates increase, our method may not select the best code in performance. So some modification may be needed when the code rates further increase. This shall be a useful future research direction. In addition, we restrict the modu-lation schemes to BPSK in our design and simumodu-lations. It shall be more practical to extend our design to a higher order modulations like QPSK and 16-QAM.
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