Other than OOK systems, the MD decoding using the proposed distance functions is also applicable to TM systems with minor modifications. Under the assumption that the receiver can distinguish different types of molecules perfectly once they are caught, a TM system can be regarded as a superposition of multiple OOK sub-systems that operate in-dependently with each sub-system consisting of only a single type of molecules. With this, we propose to calculate the distance values of this “compound system” based on the distance values of the OOK sub-systems. Suppose that there are M types of molecules in a type-based modulation system, and we want to define the distance value between the codeword x and the received pattern ˆy (Notice that ˆy is a compound pattern, not a code-word.) By transforming the codeword x into x1, . . . , xM, with xi being the transmission pattern corresponding to type-i molecules, and decomposing ˆy into y1, . . . , yM with yi be-ing a codeword in the OOK sub-system correspondbe-ing to type-i molecules, the distance between x and ˆy for the compound system is given by
dcomp(x, ˆy) =
∑M i=1
cid(xi, yi), (5.1)
where d is the distance function (either Hamming, probability-based or pattern-based) of the sub-system and ci is the weighting for type-i molecules and∑Mi=1ci = 1. The weight-ing cican be chosen according to the diffusion properties of each molecule due to, for ex-ample, different radius and different diffusion coefficients. In case that the molecules have similar diffusion behaviors, we can assign equal coefficients ci = 1/M , i = 1, 2, . . . , M for simplicity.
We use Fig. 5.1 to demonstrate the minimum distance decoding process for the type-based modulation systems. Assume that there are three different types of molecules A, B,
Figure 5.1: Demonstration of the minimum distance decoding process for the type-based modulation systems. The TM system is decomposed into three OOK sub-systems. Molecule types A, B, and C represent data bits ‘01’, ‘10’, and ‘11’, re-spectively, and a silence represents ‘00’. We use the [4, 2] block code with codebook {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1)} for example.
and C. We use the [4, 2] block code with codebook{(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1)}. If the data bits are ‘00’, for instance, the coded pattern is ‘(0, 0, 0, 1)’, the transmitting pattern will be ‘01’ for type-A molecules and ‘00’ for both type-B and type-C molecules. Equal coefficients ci = 13are chosen, and the pattern-based distance is used for demonstration. During decoding, if the receiver detects ‘01’ for type-A, ‘00’ for type-B, and ‘10’ for type-C molecules, the distance between the codeword (0, 0, 0, 1) and the de-tected results is calculated as13{dpatt((0, 1), (0, 1))+dpatt((0, 0), (0, 0))+dpatt((0, 0), (1, 0))} =
1
3(0 + 0 + 1) = 13. Similarly, the distance is 13(2 + 1 + 1) = 43 between the codeword (0, 0, 1, 0) and the detected results, 13(1 + 0 + 2) = 1 between the codeword (0, 1, 1, 1) and the detected results, and 13(2 + 2 + 2) = 2 between the codeword (1, 0, 1, 1) and the detected results. It can be seen that the distance is minimized by decoding the codeword
as (0, 0, 0, 1), which represents data bits ‘00’.
5.3 Numerical Results
In this section we numerically compare the SERs of the coded and uncoded scenar-ios under different decoding criteria for TM scenario of both active and passive trans-port systems. The simulation parameters are set to be the same as the OOK systems for molecules type A, B, and C: the diffusion coefficient D = 10−6cm2/s, the distance be-tween the transmitter and the receiver d = 20 µm, and the drift velocity v = 2 µm/s in active transport. The channel codes used for simulations are also [4, 2] block codes, which means that there are(244)= 1820 possible codebook selections. The joint coding-modulation we mentioned in Section 2.4 is applied, and the actual coded transmission pattern for each symbol has 2 bits rather than 4 bits as the OOK system because each molecule of type A, B, and C represents two coded bits. There are 464 decoding rules for each codebook selection, where the 4 means 4 codewords for a codebook, and the 64 means (22)3 for 3 types of molecule and the 22for 4 possible decomposed codeword pat-terns{(0, 0), (0, 1), (1, 0), (1, 1)}. When calculating the probability-based distance, we use N = 2. In order to compare the uncoded and coded scenario fairly, we use the same average bit duration T , which is the average period to send one information bit.
5.3.1 Active Transport via Brownian Motion with Drift
In Fig. 5.2, we show the SER distribution over all codebooks under different decoding rules for TM systems when T = 100 s. We compare the minimum distance decoding rules of using the probability-based distance, the pattern-based distance, and the Hamming distance, along with using the optimal decoding rule. The codebooks are then indexed by the order of descending SER using the optimal decoding rule. Similar to the case of OOK systems, we see that most of the codebooks do not lead to better performance than the uncoded system, and there are only 60, 38 and 37 codebooks that result in better performance than the uncoded system when using the optimal decoding rule, MD decoding
with the probability-based distance and the pattern-based distance, respectively. The SER of the uncoded scenario is 0.24%. The SERs of the best codebook are 0.11% using the optimal decoding rule, 0.11% using the MD decoding with the probability-based distance, 0.11% using the MD decoding with the pattern-based distance, and 0.25% using the MD decoding with the Hamming distance, respectively. When the Hamming distance is used as the MD decoding rule, all codebooks result in worse performance than the uncoded system.
0 200 400 600 800 1000 1200 1400 1600 1800
0
Coded, decoding by Hamming distance Coded, decoding by pattern−based distance Coded, decoding by probability−based distance Coded, optimal decoding rule
Uncoded
Figure 5.2: SER distribution under different decoding rules for TM systems with T = 100 s. The codebooks are indexed by the order of descending SER using the optimal decoding rule.
The SER performances of the TM system under various T using the optimal decod-ing rule, the MD decoddecod-ing with the probability-based distance, the pattern-based distance, and the Hamming distance are shown in Fig. 5.3. The results of the uncoded scenario are also plotted for reference. We can see that the joint coding-modulation scheme can effec-tively lower the SER. Both the probability-based distance and the pattern-based distance is overlapped with the optimal decoding rule, showing the effectiveness of the proposed distances. Compared to the OOK system (Fig. 4.3), the SERs for the TM system are at
least an order lower, meaning that using multiple types of molecules (if they are available) can greatly improve the system performance.
0 100 200 300 400
Coded, decoding by Hamming distance Uncoded
Coded, decoding by pattern−based distance Coded, decoding by probability−based distance Coded, optimal decoding rule
Figure 5.3: Comparison of SER performances for the uncoded TM system and the coded TM systems employing optimal decoding rule, MD decoding with probability-based tance, MD decoding with pattern-based distance, and MD decoding with Hamming dis-tance.
To understand more about the proposed distances, and the Hamming distance, we com-pare their performances with the optimal decoding rule under different T . In Fig. 5.4, we show the amount of codebooks decoding by MD with the three distances performing ex-actly as the optimal decoding rule. We can see that the amount of codebooks grow from about 100 to 700 when T grows from 20 s to 400 s for both the probability-based distance and the pattern-based distance. When T is larger than 300 s, the pattern-based distance even has slightly more codebooks with the same performance as the optimal decoding rule than the probability-based distance, showing that in the case of the TM active transport system, the method of summing crossover levels used in the pattern-based distance for approximating the actual channel coding distance between codewords is more optimal.
The cases of decoding with the Hamming distance are much worse and thus not suitable for decoding in active transport, and there are only about 50 codebooks performing as the
optimal decoding rule.
20 100 200 300 400
0 100 200 300 400 500 600 700 800
T (s)
Amount of Codebook
Probability−based distance Pattern−based distance Hamming distance
Figure 5.4: The amount of codebooks decoding by MD with the probability-based, the pattern-based, and the Hamming distance having the same performance as the optimal decoding rule under different T for TM systems.
As shown in Fig. 5.5, we compare the performance between the probability-based, the pattern-based, and the Hamming distance, and shows the amount of codebook about decoding by which distance would have the best SER performance among the three dis-tances. We can see that the probability-based distance outperforms other distances with having approximately 1400 codebooks performing the best among all cases of T . The pattern-based distance catches up with growing from 700 codebooks, and reaching to 1500 codebooks when T = 400 s. The Hamming distance has only about 100 codebooks hav-ing the lowest SER. We conclude that the probability-based distance leads to better results in most practical cases, and when T is larger than 300 s, the pattern-based distance per-forms as good. Therefore, we suggest decoding with the probability-based distance when T ≤ 300 s, and switch to the pattern-based distance when T is larger.
20 100 200 300 400
Figure 5.5: The amount of codebook performing the best among MD decoding with the probability-based, the pattern-based, and the Hamming distance under different T for TM systems.
5.3.2 Passive Transport via Brownian Motion
In Fig. 5.6, we show the SER distribution over all codebooks under different decoding rules for TM systems when T = 100 s. We compare the minimum distance decoding rules of using the probability-based distance, the pattern-based distance, and the Ham-ming distance, along with using the optimal decoding rule. The codebooks are indexed by the order of descending SER using the optimal decoding rule. There are 951 codebooks decoding with the optimal decoding rule performing better than the uncoded system. The best SERs are 4.25% using the optimal decoding rule, 4.68% using the MD decoding with the Hamming distance, 4.33% using the MD decoding with the probability-based distance, and 4.64% using the MD decoding with the pattern-based distance, respectively;
while the SER is 8.96% for the uncoded scenario. We can see from Fig. 5.6 that the per-formance of the SER decoding with Hamming distance is worse than the case with the probability-based for passive transport. The comparison between the Hamming distance and the pattern-based distance can’t be clearly observed in Fig. 5.6, and we will further
discuss the comparison between them later in this subsection.
0 200 400 600 800 1000 1200 1400 1600 1800
0.04
Coded, decoding by pattern−based distance Coded, decoding by Hamming distance Coded, decoding by probability−based distance Coded, optimal decoding rule
Uncoded
Figure 5.6: SER distribution under different decoding rules for TM systems with T = 100 s. The codebooks are indexed by the order of descending SER using the optimal decoding rule.
The SER performances of the type-based modulation system under various T using the MD decoding with the probability-based distance, the pattern-based distance, the Ham-ming distance are shown in Fig. 5.7. The results of the uncoded scenario and the results of using the optimal decoding rule are also plotted for reference. We can see that the joint coding-modulation scheme can effectively lower the SER. The SER results of the Hamming distance is slightly worse than the probability-based distance and the pattern-based distance. However, the Hamming distance has the advantage of low computational complexity; therefore, choosing the Hamming distance or the probability-based and the pattern-based distance can be considered a tradeoff between SER performance and com-putational complexity. Compared to the OOK system (Fig. 4.7), the SERs for the TM system are lower for about half an order, meaning that using multiple types of molecules (if they are available) can greatly improve the system performance.
Parallel to the OOK system, we compare the performances of the probability-based
0 100 200 300 400 500 10−2
10−1
T (s)
SER
Uncoded
Coded, decoding by Hamming distance Coded, decoding by pattern−based distance Coded, decoding by probability−based distance Coded, optimal decoding rule
Figure 5.7: Comparison of SER performances for the uncoded TM system and the coded TM systems employing optimal decoding rule, MD decoding with probability-based tance, MD decoding with pattern-based distance, and MD decoding with Hamming dis-tance.
distance, the pattern-based distance, and the Hamming distance with the optimal decod-ing rule under different bit duration T to understand more about the performance compar-ison between the distances. In Fig. 5.8, we show the number of codebooks decoding by MD with the probability-based, the pattern-based, and the Hamming distance performing having the same performance as the optimal decoding rule.
There are about 70, 50 and 15 codebooks having the same performance as the opti-mal decoding in the case of decoding by the probability-based distance, the pattern-based distance and the Hamming distance, respectively. We can see that more codebooks have the same performance as the optimal decoding in the case of decoding by the probability-based distance. Although the codebook amount of having the same performance as the optimal decoding rule is lower than the OOK system, we can see from Fig. 5.7 that the SER results of the distances are still close to the SER of the optimal decoding rule.
Similarly, we compare the performance between the probability-based, the pattern-based, and the Hamming distance, and shows the amount of codebook about decoding
20 100 200 300 400 500
Figure 5.8: The amount of codebooks decoding by MD with the probability-based, the pattern-based, and the Hamming distance performing having the same performance as the optimal decoding rule under different T for TM systems.
by which distance would have the best SER performance among the three distances in Fig. 5.9. We can see that the probability-based distance as well outperforms other dis-tances with having more than 1200 codebooks performing the best among all cases of T . About 450 codebooks perform better with decoding by the pattern-based distance than the Hamming distance (about 300 codebooks.) Like the scenario of OOK system, the probability-based distance leads to better results in most practical cases. Note that, however, the pattern-based distance and the Hamming distance have the advantage of low computational complexity. Therefore, choosing the probability-based distance, the pattern-based distance, or the Hamming distance can be considered a tradeoff between SER performance and computational complexity.
20 100 200 300 400 500 0
200 400 600 800 1000 1200 1400 1600
T (s)
Amount of Codebook
Probability−based distance Pattern−based distance Hamming distance
Figure 5.9: The amount of codebook performing the best among MD decoding with the probability-based, the pattern-based, and the Hamming distance under different T for TM systems.
Chapter 6
Conclusions and Future Research
In this thesis, we have examined the idea of adopting channel coding for diffusion-based molecular communications. We have proposed two categories of distance functions, the probability-based distance and the pattern-based distance, for diffusion-based molec-ular communication systems. Numerical results have shown that the proposed distance functions are more suitable as channel code design metrics for diffusion channels and the proposed distance functions lead to much better SER performance when used as minimum distance decoding rule than the traditionally used Hamming distance.
In both on-off keying (OOK) and synchronous type-based modulation (TM) for active transport, the SER results decoding with our proposed distances for minimum distance decoding is nearly identical to using the optimal decoding rule, and significantly outper-forms the Hamming distance for channel decoding. None of the cases decoding with the Hamming distance performs better than the uncoded scenario, showing that the traditional Hamming distance is no longer a good distance function. Since the probability-based dis-tance results in better performance than the pattern-based disdis-tance in most practical cases while the pattern-based distance has the advantage of lower computational complexity, choosing the probability-based distance or the pattern-based distance can be considered a tradeoff between SER performance and computational complexity.
Channel coding in passive transport can improve the SER performance more than in the case of active transport. In OOK systems of the passive transport, the SERs of the best performance codebook for the probability-based, the pattern-based and the Hamming
dis-tance is similar, and there are more codebooks decoding by the probability-based disdis-tance perform as well as the optimal decoding. The SER results decoding with the Hamming distance performs slightly worse than than the cases with the proposed distances in TM systems for passive transport. Channel coding can indeed improve the symbol error rate (SER) results when encoding with proper codebooks and decoding with suitable distance functions.
The effort in our work serves as a beginning study of a brand new channel coding con-cept for diffusion-based molecular communications. The concon-cept mentioned in this thesis can be applied to any diffusion-based channel if the delay distribution is known. The work can also be extended for other cases of level-based and type-based modulation schemes in diffusion-based molecular communications. More scenarios can be considered, such as applying different channel coding methods or taking the constraint of molecular amount into consideration. Using the two proposed distance functions to design suitable code-books will also be discussed as a future work.
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