IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 25, NO. 1, JANUARY 1, 2013
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Modified Reed–Muller Coding Scheme Made from the Bent Function for Dimmable Visible Light Communications Sunghwan Kim and Sung-Yoon Jung, Member, IEEE Abstract— In this letter, we propose a new forward error correction (FEC) coding method using a coset constructed from a bent function and a Reed–Muller (RM) code for providing accurate dimming control in on-off keying-based visible light communication (VLC) systems. By partitioning the coset made from bent function and the first-order RM codes, the new FEC scheme can be made to be suitable for dimming close to 0% or 100%. Therefore, it can contain minimal compensation symbols in supporting multiple dimming target values, which causes improved coding gain when the total amount of frame data is fixed. Simulation results for various dimming target values show that the proposed scheme leads to about 1-dB signal-to-noise ratio gain in 10−2 bit error rate compared with other three FEC codes. Index Terms— Bent function, dimming, light emitting diode (LED), Reed–Muller (RM) codes, visible light communication (VLC).
I. I NTRODUCTION
D
IMMABLE capability of visible light communication (VLC) is very important issue because it has to provide both illumination lighting and communication functions together. If we consider the hardware constraint for practical usage, OOK sending binary data by on/off pulses is adequate for VLC due to its simplicity. In VLC based on OOK, dimming is not commonly realized by amplitude control, but the ratio of ONs to OFFs due to the non-linear light current characteristic of LED. Therefore, many researchers have been developing the useful dimming support function for VLC [1]–[4]. Inverse source coding (ISC) concept is provided to show the information theoretic approach and one of its practical implementation based on inverse Huffman coding to maximize the achievable data rate under the support of dimming control [1]. Also, the use of multiple pulse position modulation (MPPM) is proposed to offer both functions of modulating data-stream and dimming at the same time [2]. In the case of VLC standard [3], [4], they are providing multiple dimming schemes:
Manuscript received April 4, 2012; revised October 6, 2012; accepted October 19, 2012. Date of publication October 23, 2012; date of current version December 14, 2012. This work was supported in part by the Excellence Program in the School of Electrical Engineering at the University of Ulsan and in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology under Grant 2011-0026560. S. Kim is with the School of Electrical Engineering, University of Ulsan, Ulsan 680-749, Korea (e-mail:
[email protected]). S.-Y. Jung is with the Department of Electronic Engineering, Yeungnam University, Gyeongsan-si 712-749, Korea (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2012.2226210
1) dimming methods combined with modulation schemes, and 2) dimming methods for flicker-free communication such as idle pattern dimming and visibility pattern dimming. Even though dimming supportability for VLC is variously announced, efficient dimming scheme considering error control codes, which is indispensable for enhancing communication performance, is rarely considered in the literature. Though forward error correction (FEC) codes such as RS (Reed–Solomon) codes or concatenated codes with RS codes and convolutional codes are supported in IEEE VLC standard, it just uses conventional FEC code for communication [3], [4], which does not consider a performance variation affected by dimming support such as compensation symbol (CS) insertion. Even in the case of ISC schemes (Inverse Huffman Coding and MPPM schemes) for VLC, there are no proposed FECs which can maintain the data correction capability with dimming support [1], [2]. Recently, efficient FEC scheme considering dimming support for VLC is proposed [5]. It can increase more coding gain in supporting the exact dimming support based on modified RM code and minimal use of CSs. In this letter, we propose an enhanced FEC coding method using a coset made from a bent function and a Reed–Muller (RM) code for dimmable VLC system. By partitioning the coset made from bent function and the first-order RM codes, the new FEC scheme can be made to provide enhanced BER performance thanks to the minimal usage of CSs. Simulation results for various dimming target values show that the proposed scheme leads to about 1dB signal-to-noise ratio (SNR) gain in 10−2 bit error rate (BER) than other conventional FEC codes (the first-order RM, Reed–Solomon (RS) [3] and modified RM codes [5]). II. S YSTEM M ODEL A. VLC System Description In Fig. 1, the transmitter of our proposed system is shown. The codeword c is generated after passing through the information vector i with k bits to the proposed FEC block. Then, the dimming compensator is used to satisfy the total number of 0’s and 1’s using CS following the given target dimming level. After the interleaving process, the OOK-modulated signal s(t) is emitted as the light signal x(t) through T LED, which has the average optical power Pt (Pt = T1 0 x(t)dt), where T denotes light signal duration. After passing through the VLC channel h(t), x(t) is received by a photodiode (PD). Then, the received signal r (t) is given as r (t) = R · x(t) h(t)+n(t) [6], where denotes
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 25, NO. 1, JANUARY 1, 2013
Dimming value
i
Channel c Dimming Encoder Compensator
Fig. 1.
Interleaver
s(t) OOK Modulator
LED
x(t)
Block diagram of the transmitter of the proposed VLC system.
convolution; R is the PD conversion efficiency (A/W); and n(t) is additive white Gaussian noise (AWGN) which contains the shot and thermal noise. Because we use OOK modulation for data transmission, the received SNR (SNRrx ) is expressed as [7] 2 R · PrSignal (1) SNRrx = 2 2 2 σshot + σthermal + R · PrISI T where PrSignal (= ∞0 x(t) h(t)dt) is the desired signal power, PrISI = T x(t) h(t)dt is the received power because of inter-symbol interference. A variance of the 2 = 2qy PrSignal + PrISI B + shot noise is given as σshot 2q Ibg I2 B [7]; where q is the electronic charge, B is equivalent noise bandwidth, and Ibg and I2 are background current and the noise bandwidth factor, respectively. In addition, 2 the thermal noise variance is also expressed as σthermal = 8πkTk 16π 2 kTk 2 2 2 3 η AI B + η A I B [7]; where the first two 2 3 G gm terms represent feedback-resistor noise and FET channel noise, respectively. K is Boltzmann’s constant, Tk is absolute temperature, G is the open-loop voltage gain, η is the fixed capacitance of PD/unit area, is the FET channel noise factor, gm is the FET transconductance, and I3 = 0.0868. Using a matched filter out of an OOK demodulator, soft values, which are log-likelihood ratio (LLR) values for interleaved codeword bits and compensation symbols, are generated. Then, the deinterleaver is used to rearrange the order of LLR values for decoding and eliminate signals for dimming compensation. Finally, ML estimator estimates the transmitted information using metrics made from summation of LLR values corresponding to codeword bits for each candidate. B. Proposed Coding Scheme Let us assume that the length of transmitted packet is n = n c +n d , where n c and n d denote the number of codeword bit and the additive compensation symbols, respectively. The dimming target value is considered as 25% (or 75%) and 12.5% (or 87.5%). The channel codes for two dimming target value (25% and 75%) are similarly constructed but the n d compensation symbols are all zeros for 25% dimming target value and all ones for 75% dimming target value, respectively. Similarly, the difference between dimming target value 12.5% and 87.5% is shown in the compensation symbols. Therefore, we only consider two dimming target values (25% and 12.5%). In this letter, a modified FEC code made from a coset of RM codes is discussed. Among various RM codes, the first-order RM code, RM(1, m), is considered for easy implementation. The properties such as weight distribution of RM(1, m) had been studied in [8]. The basis of the conventional RM(1, m) are composed of a all-one vector v0 with length 2m and m vectors with length
2m which are expressed as v1 , v2 , …, vm . For example, when m is 4, the basis vectors of the conventional RM(1, 4) are given as v0 v1 v2 v3 v4
= [1 = [0 = [0 = [0 = [0
1 1 0 0 0
1 0 1 0 0
1 1 1 0 0
1 0 0 1 0
1 1 0 1 0
1 0 1 1 0
1 1 1 1 0
1 0 0 0 1
1 1 0 0 1
1 0 1 0 1
1 1 1 0 1
1 0 0 1 1
1 1 0 1 1
1 0 1 1 1
1] 1] 1] 1] 1].
(2)
For even m, the binary vector f , which is called a bent function, is defined as follows. f = v 1 v 2 + v 3 v 4 + · · · + v m−1 v m . The multiplication a b and addition a+b of the vector a and b are component-wise AND and XOR operations, respectively [8]. When m is 4, the bent function is given as f = v 1v 2 + v 3v 4 = [0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]. The proposed codeword can be determined as a coset made from f . The weight of the coset is 6 or 10 when m is 4. Let generator matrix G for the first-order RM code, RM(1, m), be defined as tr tr tr tr (3) G = v tr 0 v1 v2 · · · vm where tr means the transpose of matrix. From information vector i with length m + 1 for the RM(1, m), the codeword c with length 2m is given as c = i G. The coset C made from f and c is represented as follows.
C = c|c = c + f, ∀c ∈ RM(1, m) . Let A j (B) be the number of codewords with weight j in the set B. Then, the weight distribution of C, WD(C) is given as [8] A j (C) = 2m , for j = 2m−1 ± 2m/2−1 WD(C) = (4) 0, otherwise. For two weights, two subsets, C1 and C2 can be defined as
C1 = c|w(c) = 2m−1 − 2m/2−1 , ∀c ∈ C
∀c ∈ C C2 = c|w(c) = 2m−1 + 2m/2−1 , where w(c) means the weight of codeword c. Let length of information vector i be m. Then there is a bijection between any information and the subset. Let ϕ1 or ϕ2 be the bijective functions between i and C1 or between i and C2 , respectively. One example of ϕ is the mapping based on the ascending sort of information i and subset if the vectors of information or codeword can be regarded as binary number. The subset C1 or C2 can be used for dimming target value less or more than 50%, respectively. If the coset C1 is used without compensation symbols, the dimming value (dim) is =(2m−1 − 2m/2−1 ). To keep the dimming target value as 25%, compensation symbol with zero bits should be included in the
KIM AND JUNG: MODIFIED RM CODING SCHEME MADE FROM THE BENT FUNCTION
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0
TABLE I
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PARAMETER C OMPARISON OF P ROPOSED S CHEME [5] AND C ONVENTIONAL RM C ODES FOR F IXED I NFORMATION L ENGTH
25, 75
12.5, 87.5
Parameter nc nd k nc nd k
Proposed Scheme 2m m−1 2 −2m/2−1 m 2m 3×2m −2m/2+2 m
Reference in [5] 2m 2m m 2m 3×2m m
Conventional RM Codes 2m−1 3×2m−1 m 2m−1 7×2m−1 m
-1
10
BER
Dimming (%)
-2
10
Dim=25%, Proposed Dim=25%, Ref.[5] Dim=25%, RS Dim=25%, RM Dim=12.5%, Proposed Dim=12.5%, Ref.[5] Dim=12.5%, RS Dim=12.5%, RM
-3
10
TABLE II A CHIEVABLE D ATA R ATES D AND A CTUAL C ODE R ATE Ra A CCORDING THE D IMMING VALUES W HEN O PTICAL R ATE I S 15 MH Z AND m I S 4
-4
Dimming (%) 25, 75 12.5, 87.5
Proposed Scheme
Reference in [5] Conventional RM Codes
D (Mb/s) 2.727
Ra
D (Mb/s)
Ra
0.182
D (Mb/s) 1.875
0.125
1.875
0.125
1.250
0.083
0.9375
0.0625
0.9375
0.0625
Ra
packet. Let the minimum compensation symbols with zero bits be n d . Then, n d should be satisfied as following equation. 2m−1 − 2m/2−1 = 0.25, 2m + n d
n d = 2m − 2m/2+1 .
(5)
Similarly, compensation symbols for dimming 75% are a vector of one bits with length 2m − 2m/2+1 . To examine the properties of the proposed scheme, it is compared with the scheme in [5] and conventional first-order RM codes with compensation symbols. Let k be a length of information bits. For proper comparison, it is considered that the information length is fixed as m and n d is different according to dimming target values. Parameters for different dimming target values are listed in Table I. As shown in Table I, the proposed scheme brings the minimum CSs and it results in the improved coding gain when the total amount of data in a frame is fixed. In table II, the achievable date rates D (Mbps) and actual code rate Ra (= k/(n c +n d )) according to dimming value are listed when m is 4. The optical rate (same meaning with LED bandwidth) is assumed as 15MHz, which is one of the candidates in VLC standard [3]. III. S IMULATION R ESULTS We consider the case information size m is fixed as 4 and dimming target values are 25% (or 75%), or 12.5% (or 87.5%). For VLC channel, we use the channel environment and parameters for determining noise variances in [7]. In order to investigate the performance in the operating SNRrx range of 0–20 dB (the average SNRrx under the VLC environment in [7] is 15.4 dB), we change the distance between LED and PD from 1.68 to 5.34 meters. Fig. 2 shows the BER performances of the proposed scheme (‘Proposed’), reference one in [5] (‘Ref. [5]’), and conventional first order RM codes (‘RM’) according to the received Eb /N0 ((Eb/N0 )rx = SNRrx /Rc ) for different dimming target values. For fair comparison among
10
0
2
4
6
8
10 12 Eb/N0 (dB)
14
16
18
20
Fig. 2. BER performances of the proposed [5] conventional first-order RM and the RS (15, 2) codes for different diming target values when m is 4.
the BER performances of FEC schemes, (Eb /N0 )rx is considered. In addition, the performance of RS (Reed–Solomon) (15), (2) codes (‘RS’) adopted in VLC standard [3], whose code rate is similar with our scheme, is additionally inserted in Fig. 2 to show the advantage of our proposed scheme for VLC. For 25% dimming (‘Dim = 25%’), the (Eb /N0 )rx gain of the proposed scheme is nearly 1dB than others when BER is 10−2 . The gain can be obtained since the codeword length of the proposed scheme is longer than that of the compared ones. IV. C ONCLUSION To increase the coding performance guaranteeing the accurate dimming target values, new coding methods are proposed by using the coset made from a bent function and corresponding compensation symbols. From simulation results, the proposed scheme shows a better BER performance than the conventional ones guaranteeing the exact dimming support. R EFERENCES [1] J. K. Kwon, “Inverse source coding for dimming in visible light communications using NRZ-OOK on reliable links,” IEEE Photon. Technol. Lett., vol. 22, no. 19, pp. 1455–1457, Oct. 1, 2010. [2] K. Lee and H. Park, “Modulations for visible light communications with dimming control,” IEEE Photon. Technol. Lett., vol. 23, no. 16, pp. 1136–1138, Aug. 15, 2011. [3] IEEE Standard for Local and Metropolitan Area Networks-Part 15.7: Short-Range Wireless Optical Communication Using Visible Light, IEEE Standard 802.15.7, 2011. [4] S. Rajagopal, R. D. Roberts, and S.-K. Lim, “IEEE 802.15.7 visible light communication: Modulation schemes and dimming support,” IEEE Commun. Mag., vol. 50, no. 3, pp. 72–82, Mar. 2012. [5] S. Kim and S.-Y. Jung, “Novel FEC coding scheme for dimmable visible light communication based on the modified Reed–Muller codes,” IEEE Photon. Technol. Lett., vol. 23, no. 20, pp. 1514–1516, Oct. 15, 2011. [6] T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 2892–2900, Jun. 2009. [7] T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron., vol. 50, no. 1, pp. 100–107, Feb. 2004. [8] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. New York: North-Holland, 1997.
