Beneficios y resultados de la aplicación del procedimiento para la

5.5.1 Introduction of MECs

As stated at the start of this chapter, it is essential to develop energy efficient communication techniques for MC. Choosing codewords for each source outcome such that the mean codeword energy is less than any other choice of codeword mappings is called minimum energy coding. A novel, minimum energy coding scheme, which is proved to be energy efficient, is provided in [212] for a THz wireless nanosensor network. Unlike conventional studies, MEC maintains the desired code distance to provide reliability while minimizing energy. In theory, by using OOK modulation, MEC with Hamming distance constraints can reduce energy consumption by minimising the average weight of codewords [212]. In this section, the MEC proposed in [212], which is considered reliable and suitable for nanoscale communications, is used as the channel code to improve the system performance in the proposed diffusion-based MC system, especially the energy performance. With OOK modulation, the average codeword energy is the symbol energy times average codeword weight, which means that average energy can be minimized by minimizing the average code weight. In other words, codewords with a lower weight result in reduced energy consumption since the transmission of a ‘0’ symbol requires less energy than the transmission of a ‘1’ symbol. Therefore, codeword weights and original message-codeword mappings are chosen such that the expected code weight is minimized at the cost of increased codeword length, hence increased delay. The idea of using low weight channel codes together with OOK modulation to reduce energy consumption is first introduced in [213] for sensor networks, in which

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codewords are sorted in increasing code weight order and original symbols are assigned in decreasing probability order, such that the most probable source symbol is mapped to the codeword with the smallest weight, resulting to the optimum average code weight. Development of reliable MECs has been an open issue, however it is the first attempt in MC, which has met with some success [214].

The source message, which is of length , can be encoded into a codeword which is of length in the following way. For a given set of source symbols, which have a specific source distribution, and a given set of codewords, codewords are sorted in increasing code weight order and source symbols are assigned in decreasing probability, which has been explained above. For example, the least probable source symbol is mapped to the largest weight codeword. In the work of this thesis, it is assumed that each codeword has the same probability of occurrence as the source outcomes to which they are mapped, since no source coding mechanism is applied. For OOK modulation, transmitting bit ‘0’ requires no energy. Thus, as has been stated earlier, the minimisation of energy consumption is equivalent to the minimisation of the average codeword weight. The weight enumerator of a code ℂ is the polynomial _ℂ( ) = ∑ Å , where Å is the number of codewords with weight and is a symbol which is called an indeterminate that does not represent any value.

Assuming that is the number of codewords, is the minimum Hamming distance and is the maximum probability in the source probability distribution, the weight enumerators of MEC codewords are given by [212]:

ℂ( ) =

+ ( − 1) > 0.5

+ ( − 1) ≤ 0.5 (5.15)

134 ( ) = ⎩ ⎪ ⎨ ⎪ ⎧ (1 − ) , > 0.5 2 , < 0.5, 2 − , < 0.5, (5.16)

The MEC only provides the limitation of the length of codeword and the minimum weight, rather than the actual codeword. Thus, different codebooks can exist for a single Hamming distance. For MECs, the decoding method is minimum distance decoding which means that the received -tuple is mapped to the closest codeword in terms of Hamming distance. More errors can be corrected when the minimum Hamming distance increases with the codeword length but this leads to a larger number of error patterns, which will decrease the reliability of the MEC [212]. The minimum codeword length is given by [212]:

= + ( − 2)

2 (5.17)

Here, is the minimum codeword length required to satisfy the MEC weight enumerator for given values of and . It is important since it indicates the minimum delay due to transmission of codewords.

Using equations (5.15) and (5.17), sample codebooks for < 0.5 and > 0.5 with = 3 and = 3 can be generated as:

1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 ; 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 (5.18)

Generally, with the increase of the minimum Hamming distance between the codewords, more errors can be corrected. However, the length of the codewords will also increase with the code distance, leading to a larger number of error patterns. Thus, increasing code distance does not necessarily increase the channel reliability. Hence, it is required to analyse the error correction capability of MEC. Codes with

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distance can correct errors, and the channel reliability increases with code distance, since more error patterns can be corrected. Thus after minimum distance decoding of a MEC, the probability that transmitted codeword is correctly decoded is given by:

= (1 − ) (5.19)

where is the average bit error probability.

To calculate the energy consumption of MEC, method given in Section 5.4 can be applied with results given in the subsection below.

5.5.2 Analytical results

In this section, the error correction capability and energy efficiency of MECs are investigated via numerical evaluations of analytical parameters in MATLAB. Specifically, MECs are compared with Hamming (7, 4) and (15, 11) codes in terms of BER and energy consumption. MECs satisfy the minimum Hamming distance required by Hamming codes so here this is set to three. An ( , ) code maps 2 source words into length channel codewords. Thus, for better comparison, the corresponding MECs are thus = 2 and = 2 . Also, the bacterial population in the receiver node is 100. The error correction and energy performances of MECs and Hamming codes over a 4μm transmission distance are illustrated in Figure 5.7 and Figure 5.8.

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Figure 5.7 BER comparison between MECs and Hamming codes.

Figure 5.8 Average energy per bit comparison between MECs and Hamming codes.

Figure 5.7 shows that both Hamming Codes and MECs improve system performance for a transmission distance of 4 . The coding gain can be determined as above by

101 102 103 10-10 10-8 10-6 10-4 10-2 100

Molecules per bit

B E R Uncoded (7,4) Hamming (15,11) Hamming MEC (M=16,distance-3) MEC (M=2048,distance-3) 0 50 100 150 200 250 300 350 400 450 500 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Molecules per bit

E n e rg y p e r b it , J (7,4) Hamming (15,11) Hamming MEC (M=16,distance-3) MEC (M=2048,distance-3)

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taking the ratio of the number of molecules for a given BER in the uncoded and coded cases. When the receiver node contains = 100 bacteria, the number of molecules per bit is 280 for (15, 11) Hamming code, 330 for (7, 4) Hamming code, 140 for = 2 MEC code, 50 for = 2 MEC code and 410 for uncoded information symbols for a BER of 10 which gives a relatively effective error free operation. Thus, it can be derived that at 10 BER level, the coding gains for the Hamming codes are 0.94 dB and 1.65 dB for the (7, 4) and (15, 11) codes respectively, and for the MECs, the figures are 4.67 dB and 9.14 dB for = 2 and = 2 respectively. In general, MECs have a better performance than Hamming codes with a larger coding gain. Also, the system performance is better with a lengthy codeword. However, for MECs, since increasing the number of codewords means increasing the amount of information to be transmitted, more reliable channels are required to transmit the codewords, which is intuitively expected. Figure 5.8 shows that MECs exhibit superior average energy per bit values. The extra energy needed to deal with unreliable decoding for small numbers of molecules per bit becomes negligible as the number of molecules per bit increases.