SER Performance of CC-DH-PIM

The comparative studies of upper error bounds (given by (4.8) and (4.10) for a general convolutional encoder and (4.13) for CC-DH-PIMΓ) can be a good measure to find the

effectiveness of convolutional code for DH-PIMΓ. Figure 4.8 shows the predicted SER

performance against the SNR for (4.8), (4.10) and (4.13) for 16-DH-PIM2. For an SER

≤ 10-6

, the upper bound for CC-DH-PIMΓ and the general upper bound almost overlap,

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less than or equal to the upper bound for its counterparts, it is expected that it should give similar or even improved SER performance compared to the other convolutional coded modulation schemes such as OOK for the same value of uncoded SER. This is because CC-DH-PIMΓ has fixed header patterns that limit the number of possible paths

in the Trellis diagram. As a result, the expansion coefficient βd of (4.7) is different for

the CC-DH- PIMΓ. The coefficients of 3rd and 4th terms of (4.10) and (4.13) are (12, 32)

and (9, 20), respectively. Although the first two terms are the dominating terms in determining the error bound at lower SER values, the contribution of other terms cannot be neglected for higher SER values. Hence, the error bounds (4.10) and (4.13) will have noticeable differences at higher uncoded SER values. In other words, as the number of possible paths decreases, the probability of error decreases. This is because there is only a finite set of paths to choose from, thus making the decoding process much simpler.

For DH-PIMΓ scheme, a (3, 1, 2) convolutional encoder and the Viterbi algorithm with

the ‗hard‘ decoding is utilized. Assuming an ideal channel with single-sided PSD and Ib = 200 μA, the proposed CC-DH-PIMΓ system is simulated using

Matlab. The simulation parameters used and the flow chart for determining the error rates for the CC-DH-PIM are given in Table 4.2 and Figure 4.9, respectively.

Figure 4.8: The SER against the SNR for different error bounds for CC-DH-PIM2.

-2 -1 0 1 2 3 4 5 6 7 10-10 10-8 10-6 10-4 10-2 100 SNR (dB) P se -C C Error bound (4.10) Error bound (4.12) Error bound (4.15)

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Start

Select parameters (M, Rb , Γ, Ib , SNR)

Generate 105 M-bit binary number

Encode into DH-PIM sequence

Apply convolutional coding

Add noise

Matched filter sampled at Ts

Viterbri algorithm

Count number of slot errors

Calculate SER

End Threshold detector

Figure 4.9: A flow chart diagram for determining the SER of the CC-DH-PIM system.

Table 4.2: The simulation parameters for CC-DH-PIMΓ.

Parameters Values

M 3 and 4

Γ 1 & 2

Detector responsivity R 1 A/W Background noise current Ib 200 µA

Threshold level k 0.5

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Figure 4.10 provides the predicted and simulated results for SER performance for the standard DH-PIM1&2 and CC-DH-PIM1&2 schemes for M = 3 and 4. For Pse < 10-2 the

predicted and simulated curves for the CC-DH-PIMΓ match each other reasonably well.

Since it is somewhat difficult to ascertain the exact Hamming distance for the convolutional code in the theoretical analysis, only the upper bound is considered. The simulated SER performance is very close to the upper bound, but is always less than or equal to it. The code gain decreases with the SNR and at very low values of SNR (or SER > 0.1) the code gain is negative. At higher values of SER, introducing coding contributes to more correlated errors from the uncorrelated random errors [210], thus giving rise to additional errors and hence degrading the error performance. A code gain of more than 4 dB is observed for Pse of 10-4 compared to the standard DH-PIM2 scheme

for M = 3 and 4, respectively. To achieve a certain SER, the SNR required decreases as

M increases. This is expected since the SER of uncoded DH-PIM scheme decreases

with M [112]. A difference of almost ~3 dB in the SNR is observed at the Pse of 10-4 for M of 3 and 4. Similarly in the case of DH-PIM1 scheme, a code gain of more than ~ 4

dB is observed at a SER of 10-4.

Detailed comparison of the SER performance of the standard DH-PIMΓ with other

modulation schemes is given in [112]. Here the comparison of the CC-DH-PIMΓ (upper

bound) to the standard DH-PIMΓ and PPM schemes is made, see Figure 4.11. CC-DH-

PIM1 offers the best performance compared to the uncoded PPM and DH-PIM1, as

expected. However, the Trellis coded PPM scheme does outperform all other coded modulation schemes including the CC-DH-PIM1. It is expected that the coded PPM

offers the least optical power requirement compared to the uncoded PPM. The performance of 16-CC-DH-PIM2 is very close to that of the standard 16-DH-PIM1. The

SER of the 16-CC-DH-PIM2 runs almost parallel to the standard 16-DH-PIM1, 16-

CC-DH-PIM2 requiring just more than 0.5 dB of SNR to achieve the same SER

performance. To achieve a SER of 10-6, 16-CC-DH-PIM1 requires ~5 dB lower SNR

compared to the standard DH-PIM1, while 16-DH-PIM2 requires ~ 4 dB more SNR

compared to the 16-CC-DH-PIM2. The reduction in SNR means a reduction in the

transmitted power to achieve the same error probability. Note that a reduction of 3 dB in SNR is equivalent to half the transmitter power. The gain in SNR could be used to increase the link length or trade for performance.

80 (a)

(b)

Figure 4.10: The theoretical and simulation SER versus the SNR for: (a) standard 8 & 16-DH-PIM2 and 8 & 16-CC-DH-PIM2, and (b) standard 8 & 16-DH-PIM1 and 8 & 16-

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Figure 4.11: The theoretical SER against the SNR for 16-PPM and coded and uncoded DH-PIM1&2.

The performance of the CC-DH-PIMΓ system can further be enhanced by increasing the

constraint length, which results in increased code gain but at the cost of increased system complexity. Thus, there exists a trade-off between complexity and performance. A ½ rate convolutional encoder with a constraint length of 7, which are readily available, is used to assess the performance of different CC-DH-PIMΓ systems.

Simulated Pse performance of 16-CC-DH-PIMΓ with a generator of (133, 171) in octal

number is depicted in Figure 4.12. Also shown for comparison is Pse of 16-DH-PIMΓ

and 16-CC-DH-PIMΓ. The best SER performance is observed for CC-DH-PIMΓ with a

constraint length K of 7 for Pse ≤ 10-2. At Pse = 0.005, Pse curves of CC-DH-PIMΓ with K of 3 and 7 intersect, indicating no code gain with higher values of constraint length.

However, at Pse ≤ 10-3,there is a marked improvement in the performance when using a

longer constraint length. An additional ~ 2 dB code gain can be achieved at an SER of 10-5 using an encoder with K = 7 compared to the encoder with K =3. This additional code gain can be used to increase the link range in indoor OWC system.

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Figure 4.12: The SER against SNR for 16-CC-DH-PIMΓ with constraint lengths of 3

and 7.