need some help

[ZehaiJack]

Sorry to bother you, but I am a master's student in communications, recently working on research related to HQAM. I found your H-QAM code on GitHub and read the mentioned literature. I noticed that many functionalities presented in the literature, such as the detection methods and SEP calculation methods, etc., are not reflected in the code. If possible, I hope you could share the relevant code with me. I would really appreciate it.

[Nick-Liou]

We have implemented a detection scheme, but looking at the code in the repo it seems we forgot to update it. I'll upload the rest of the files that contain a lot more with the detection. Feel free to ask questions !

[catherinepapad]

Hello @ZehaiJack All missing scripts were added. We also included a technical report we had composed. A few notes:

• We do not implement the detector mentioned in the bibliography. Instead, we proposed a different detector using a geometric transformation. This detector has complexity O(1) and hence is very fast and efficient. You can find more details about that in the report.
• Regarding the SEP, we have a script that calculates SEP and BER. Later on, we found out that the Communications Toolbox of matlab has these functions already implemented.

[ZehaiJack]

Thank you for your answers @Nick-Liou @catherinepapad ，I have a few questions about the technical report and reference 1 that I don’t understand.

• Could you explain how the time complexity of the method, the space complexity and the complexity of the idea mentioned in the technical report is calculated; and the last paragraph of the Introduction of Reference 1(On the error analysis of H-QAM constellations) mentions the average complexity O(logM) and O(M^1/2), I read along the references mentioned in the paper, but I still don’t understand how this kind of complexity is calculated. Is it convenient for you to provide me with some references or give me some explanations?
• Do you know how d_min is derived from Equation 2 in Ref. 1?
• Could you provide more details on the Gp, even closed form mentioned on page 10 of the technical report? And how to get Gp,odd in the same way?
• One of the methods("Our first thought was to utilize our knowledge of linear algebra to find the coordinates c1 and c2 which is quite easy to do, after offsetting the received symbol by v3 and multiplying with the inverse transformation matrix.") mentioned on page 11 of the technical report does not work in some cases. Could you give more details? For example, under what circumstances can it not work?
• I still don’t quite understand some of the specific implementation details of the detection method in the technical report.Is the detection method used the Cube coordinates and Axial coordinates mentioned on the Hexagonal grids website? Have you compared it with the MLD method like reference 1? Is there any relevant literature that provides more details on this detection method?
• Are this detection method and SEP calculation method applicable to other more complex channel environments(optical fiber)?

Thank you again for sharing. If you have the time to respond to the questions I mentioned above, I really appreciate it!

[catherinepapad]

Hello @ZehaiJack I will address some of your questions.

I still don’t quite understand some of the specific implementation details of the detection method in the technical report.Is the detection method used the Cube coordinates and Axial coordinates mentioned on the Hexagonal grids website? Have you compared it with the MLD method like reference 1? Is there any relevant literature that provides more details on this detection method?

Our detection method was indeed based on reference 6. This is a classic resource for game developers when dealing with hexagonal boards. From there, we got the idea for utilizing cube and axial coordinates, as well as mapping the hexagons to a 2D matrix using these coordinates. The only extra part we had to implement was the rounding. The transformations from Cartesian to axial coordinates mentioned on this site, as well as other relevant sources, are for the centers of the hexagons, so we had to take account of all points inside a hexagon. We have to clarify that the main objective of this project was to propose a detection algorithm for HQAM constellations that was faster and more efficient than the current implementation of MLD (Maximum Likelihood Detector) which had a time complexity of O(log(M)). It appears that the problem was more or less solved already by game developers, but was never used by telecom engineers. We believe we were the first to actually use this method for telecommunication purposes.

Are this detection method and SEP calculation method applicable to other more complex channel environments(optical fiber)?

This detection algorithm can be used wherever MLD can be used. The 2 main prerequisites are that: All symbols are equally likely, and The channel is only affected by AWGN. Regarding optical fibers, I don't think either of us can answer with certainty. I believe QAM modulation is used in optical communications, but I don't know the specifics of the demodulation process.

We will come back with answers for the rest of your questions later on. It's been a while since we last worked on this project, so we need some time to remember everything.

Thank you for showing interest in our work!

[Nick-Liou]

• Do you know how d_min is derived from Equation 2 in Ref. 1?

No, we tried to figure it out in the past but didn't get far. It states that it can be found in Table 6 of this paper, but the table does not clearly have values for "a". I would guess that you could solve Equation 2 with respect to "a" and use the $\frac{E_s}{d^2}$ column from the table to substitute, but I am not sure if this is what they had in mind. Note that the entry for M=8 the value of $\frac{E_s}{d^2}$ which was 6 does not agree with our results, we found 4.5. You can see the values we found for $\frac{E_s}{d^2}$ in our report page 2, section 1.1.5 Table 1

• Could you provide more details on the Gp, even closed form mentioned on page 10 of the technical report? And how to get Gp,odd in the same way?

Given that we couldn't find a closed form for $G{p,even}$ we derived it from scratch. We don't have a formal proof, but I can walk you through the logic. Basically, we used the definition for $G{p}$ and split the symbols into groups based on their value of $G_{p,si}$. Then we multiplied the number of symbols in each group with their $G{p,s_i}$ and added them together.

More precisely:

• The symbols in the center of the constellation (green) that have 6 neighbors (4 with 1 bit difference and 2 with 2 bits difference, since we use Gray Coding) they have $G_{p,si} = \frac{1+1+1+1+2+2}{6} = 4/3$. The number of symbols that are in this group are $(a{even}-2)^2$ (they form a square). Which they give the term $\frac{4}{3}(a_{even}-2)^2$
• The symbols at the top and bottom side of the constellation (purple) have 4 neighbors (3 with 1 bit difference and 1 with 2 bits difference) thus $G_{p,si} = \frac{1+1+1+2}{4} = 5/4$. The number of symbols that are in this group are $2(a{even}-2)$ (one row for the top and one row for the bottom). So they give the term $\frac{5}{4} * 2(a_{even}-2)$
• Then there are the symbols on the left and the right (orange and light blue), which are split into two categories
  • The outer symbols (orange) with 3 neighbors, which differ by only 1 bit thus $G_{p,si} = \frac{1+1+1}{3} = 1$ and there are in total $a{even}-2$ symbols in this group. So the term for this group is $a_{even}-2$
  • The inner symbols (light blue) with 5 neighbors, (3 with 1 bit difference and 2 with 2 bits difference) thus $G_{p,si} = \frac{1+1+1+2+2}{5} = 7/5$. There are $a{even}-2$ symbols in this group. The term of this group is $\frac{7}{5}(a_{even}-2)$
• Finally, there are the 4 symbols (dark red) in the corners that have 2 or 3 neighbors which their total sum of the different bits gives 14/3

Adding all the above terms together, we have:

$$ \sum{i=0}^{M-1} G{p,si} = \frac{4}{3}(a{even}-2)^2 + \frac{5}{4} * 2(a{even}-2) + (a{even}-2) + \frac{7}{5}(a_{even}-2) + 14/3 $$

$$ \sum{i=0}^{M-1} G{p,si} = \frac{4}{3}(a{even}-2)^2 + \frac{49}{10} (a_{even}-2) + 14/3 $$

after dividing with M it simplifies to the expression in our report.

As for the odd case, we had started working on it, but it was significantly more complex than the above (due to the way the Gray Code mapping is generated for even order constellations) and thus we didn't complete it. If you end up solving it, please let me know because it has been bothering me ever since.

[Nick-Liou]

• One of the methods("Our first thought was to utilize our knowledge of linear algebra to find the coordinates c1 and c2 which is quite easy to do, after offsetting the received symbol by v3 and multiplying with the inverse transformation matrix.") mentioned on page 11 of the technical report does not work in some cases. Could you give more details? For example, under what circumstances can it not work?

To be honest, we don't quite remember. We had tried, but we had faced some problem. You can give it a shot to see where it falls apart, and who knows you might come up with a clever solution for it.

• Could you explain how the time complexity of the method, the space complexity and the complexity of the idea mentioned in the technical report is calculated; and the last paragraph of the Introduction of Reference 1(On the error analysis of H-QAM constellations) mentions the average complexity O(logM) and O(M^1/2), I read along the references mentioned in the paper, but I still don’t understand how this kind of complexity is calculated. Is it convenient for you to provide me with some references or give me some explanations?

The time and space complexity is a metric that can be used to describe an algorithm. They are terms from computer science and there are plenty of online resources where you can learn the basics. For example here and here and for even more detail here. As for the complexity of the idea, it isn't something that can be measured, we just stated that it is more complex that simply checking against all the symbols to find the closest one (which you will understand that it has a time complexity of O(M)).

We hope we answered your questions. Let us know if something is unclear.

[catherinepapad]

Just out of curiosity, what is your research about? Is it your thesis or a class project?

[ZehaiJack]

Thank you very much for your help, I am a master's student at Shanghai University, and my master's research direction is the research of symbol detection algorithms for hexagonal constellation modulation.

Beside，if you have the opportunity to come to Shanghai, China, please don’t hesitate to contact me and I will show you culture、delicious food、landscape about the city of Shanghai. Thank you again for your help, and hope we keep in touch with each other in the process of later research.

[Nick-Liou]

You're very welcome! We're glad we could assist you. Your master's research direction sounds interesting, we would love to take a look at your thesis when the time comes. Are you planning to write it in English?

We appreciate your kind invitation to visit Shanghai. We don't have any plans to visit Shanghai in the foreseeable future, but we will definitely reach out if we have the opportunity to come to Shanghai in the future.

Let's indeed keep in touch as you progress with your research. Feel free to share updates or reach out if you have any questions or insights to discuss.

[ZehaiJack]

The thesis may be written in Chinese., but it will have an English abstract. But don't worry about it, let's keep in touch with each other! Besides, Shanghai is a very nice city, welcome to visit!

[Nick-Liou]

Hello @ZehaiJack, how is your thesis progress going?

After our last talk, I spend some extra time on the problem and I found the closed formula for the Gp_odd. You can find it in the newly uploaded file.

Does your University have specific guidelines for referencing and acknowledging contributions? If so, please let us know what information you need.

[ZehaiJack]

Oh that's so great for the Gp_odd's closed formula. In the past months, we have been on summer vacation in China, so there has been no progress in scientific research. If i need your information in the future, i will contact you in time.