Performance of Convolutional Coded Dual Header

Performance of Convolutional Coded Dual Header Pulse Interval Modulation in Infrared Links S. Rajbhandari, Z. Ghassemlooy, and N. M. Aldibbiat Optical Communications Research Group, NCRLab, School of Computing, Engineering & Information Sciences, Northumbria University, Ellison Building, Newcastle upon Tyne, NE1 8ST, UK [email protected], [email protected], and [email protected]

Abstract — The paper presents analysis and simulation for the convolutional coded DH-PIM using ½ convolutional code with the constraint length of 3. Decoding is implemented using the Viterbi algorithm. The proposed scheme is simulated for a constraint length of 7 and the results are compared with a number of different modulation techniques. DH-PIM with convolutional coding requires about 4-5 dB lower SNR for a given slot error rate compared with the un-coded DH-PIM.

with convolutional coding, and PPM. The rest of the paper is organised as follows. System description is outlined in Section II, whereas convolutional coded DH-PIM is introduced in section III. Results and discussion are given in Section IV, and finally concluding remarks are presented in Section V. II. SYSTEM

I.

INTRODUCTION

The use of infrared (IR) indoor wireless communications has attracted interest from researchers worldwide because of the advantages IR links offer over radio links such as a huge unregulated bandwidth, high data rates, immunity to electromagnetic interference, relative security since it does not pass through walls, and the ability to reuse the same wavelength within the same environment [1-4]. However, IR links have limitations such as multipath induced distortion for non-line-of-sight links, ambient artificial light interference and power restrictions due to the eye safety [1-4]. To overcome some or all of the limitations it is important to select a modulation technique offering power and bandwidth efficiencies. Different modulation techniques have been suggested and investigated thoroughly for IR links such as on-off keying (OOK), pulse position modulation (PPM), digital pulse interval modulation (DPIM), and dual header pulse interval modulation (DH-PIM) [5]. OOK is the simplest among these techniques, offering efficient bandwidth requirements; however, it suffers severely from high power requirement. PPM offers power efficiency at the cost of huge bandwidth requirements [5]. DPIM requires no symbol synchronisation, and offers an improvement in bandwidth efficiency compared with PPM and power efficiency compared with OOK and PPM [5]. DHPIM with built-in symbol and slot synchronisation capabilities offers the best transmission capacity and requires lower transmission bandwidth compared with PPM and DPIM [6-7]. Detailed analysis of standard DH-PIM is available in the literature [6], but no work on coded DH-PIM has been reported yet. In this paper, convolutional coding is applied to DH-PIM to improve the error performance. We compare the performances of standard DH-PIM, DH-PIM

The block diagram for the convolutional coded DHPIM (CC-DH-PIM) is given in Fig. 1. DH-PIM is encoded before being transmitter over an optical channel. The channel is assumed to be ideal with no loss or mutipath dispersion. The noise added is white Gaussian representing the contribution from the ambient light sources, which is the dominant noise source. The receiver front end employs a match filter, a sampler and the threshold detector to regenerate the coded DH-PIM before passing it to the Viterbi decoder to recover the original DH-PIM sequence. Since for every input bit, a ½ encoder produces two output bits, the bandwidth requirement for convolved DH-PIM is approximately double that of the standard DH-PIM, provided that the DH-PIM sequence is long enough so that when adding the last two zeros to clear the memory will not affect the bandwidth requirement significantly. Since this modulation scheme has a non-fixed symbol structure therefore an error in one symbol will affect the preceding symbols. Therefore it would not be possible to assess the error performance in terms of bit error rate, instead slot error rate and packet error rate are used. The probability of slot error rate (SER) for DH-PIM for line-of-sight configuration and under certain limitations is given by [6]:

Fig. 1: Block diagram of DH-PIM system employing convolutional coding

⎛ μkR P ⎞ ⎛ ⎞⎤ 1 ⎡ ⎟ + 3αQ⎜ μ (1 − k ) R P ⎟⎥ ⎢ 4 L − 3α Q⎜ ⎜ ηR ⎟ ⎜ 4 L ⎢⎣ ηRb ⎟⎠⎥⎦ b ⎠ ⎝ ⎝

(

Pse =

)

where μ = 32M L / 9α

2

(1)

, Q (v ) = (1 / 2π ) exp(− 0.5 x 2 )dx , k ∫

is the threshold level, L is average symbol, Rb is the bit rate, η is noise power spectral density, α > 0 is an integer and M is the input bit resolution. Here we use the symbol of L-DHPIMα where L = 2M. α = 2 for the mathematical derivative and simulation. The probability of packet error rate (PER) Ppe is given by [6]:

Ppe =

⎛ μkR P ⎞ ⎛ ⎞⎤ N Pkt ⎡ ⎟ + 3αQ⎜ μ (1 − k ) R P ⎟⎥ ⎢ 4 L − 3α Q⎜ ⎜ ηR ⎟ ⎜ 4M ⎢ ηRb ⎟⎠⎥⎦ b ⎠ ⎝ ⎝ ⎣

(

)

(2) where Npkt is the packet length in bits. Assuming that the assumption given in [6] still holds we have used (1) in our mathematical analysis. III.

CONVOUTIONAL CODED DH-PIM

Convolutional encoding with the Viterbi decoding is a forward error correction (FEC) technique suitable for a channel corrupted mainly by additive white Gaussian noise [8-9]. A convolutional encoder is a finite state machine with K-shift register [10] with a predefined connection to n module-2 adders. A k-input bit sequence to an encoder produces an n-output bits sequence, and hence the code rate, r can be approximated as [11]: k (3) r= n

The symbol (K, n, k) denotes a convolutional encode with a constraint length of K and the code rate of k/n. The convolutional encoder can be described by its generator matrix [9]. For this simulation, we use (3, 1, 2) with the generation matrix of g1 = [111] and g2 = [101]. The state diagram of the general encoder used for the simulation is given in Fig. 2.

Fig. 2: The state diagram for general ½ convolutional encoder.

TABLE 1 The OOK, coded and un-coded DH-PIM2 for M = 3, and α =2 OOK (M=3) 000 001 010 011 100 101 110 111

DH-PIM2

CC-DH-PIM2

100 100 0 100 00 100 000 110 000 110 00 110 0 110

11 10 11 11 10 11 00 11 10 11 00 00 11 10 11 00 00 00 11 01 01 11 00 00 11 01 01 11 00 11 01 01 11 00 11 01 01

CC-DH-PIM2 preceded by 2M-1 00 10 11 00 10 11 00 00 10 11 00 00 00 10 11 00 00 00 00 01 01 11 00 00 00 01 01 11 00 00 01 01 11 00 01 01

DH-PIM2 has two header H1= [100] and H2 = [110] followed by a number of zeros depending upon the input symbol [6]. H1 and H2 will produces a new header sequences of [11 10 11] and [11 01 01] respectively provided the encoder initial state is ‘a’, which can be verified by using the state diagram in Fig. 2. The encoder will be in the state ‘a’ after each symbol since there is two zeros at the end of each DH-PIM symbol except the symbol with decimal equivalent of the 2M-1. At the end of transmission of the symbol, the encoder will be in the state ‘c’, and the current symbol will have different header pattern than previously stated. Depending on the input, the two headers when preceded by a symbol with a decimal equivalence of 2M-1 are [00 10 11] and [00 01 01]. Thus the CC-DH-PIM will have four header and limited possible paths in the Trellis diagram as shown in Fig. 3. So CC-DH-PIM will have different transfer function and error probabilities compared to other modulation scheme. For determining the transfer function, the modified state diagram of Fig. 2 is shown in Fig. 4. The exponent of D in the figure is the Hamming weight of the encoder output corresponding to the branch, the exponent of L is a counting variable to calculate the number of branches in any path and I is used to denote that the transition is due to input bit 1. The transfer function of the encoder is given by [11]: T ( D, L, I ) = D 5 L3 I + D 6 L4 I 2 (1 + L) + D 7 L5 I 3 (1 + L) 2 + ... (4)

Fig. 3: Trellis diagram of CC-DH-PIM2

IV.

Fig. 4: Modified State diagram for general convolutional encoder The minimum free distance dfree for the encoder is 5, increasing with the constraint length as tabulated in [9] for different constraint length and different code rates. Applying the Viterbi algorithm [10] to decode the coded sequence, the upper bound for the probability of bit error is given by [9]: Pb
-1 dB. The code gain seems to increase with the increase in the SNR. Theoretically the (7, 1, 2) should give much better code gain as the SER tends to zero. We did not

simulate for very high values of SNR because of long computation time. For SER of 10-5 the SNR gain of coded DH-PIM with K = 7 are ~2 dB and ~6 dB compared with the coded DH-PIM with K =3 and un-coded DH-PIM, respectively.

V.

CONCLUSIONS

In this paper we presented the mathematical analysis and simulation result for (3, 1, 2) CC-DH-PIM2 employing the Viterbi ‘hard’ decoding. A code gain of more than 3 and 4 dB was achieved for Pse of 10-4 compared with the un-coded DHPIM2 for M = 3 and 4, respectively. The (7, 1, 2) CC-DH-PIM showed improved performance than the (3, 1, 2). The improvement in the error performance of course is achieved at the cost of the reduced transmission through-put compared with the un-coded case.

Fig. 8: The SER against SNR for code and un-coded DH-PIM for constraint length of 3 and 7.

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