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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LPT.2016.2572300, IEEE Photonics Technology Letters ZHANG, SI-MA, WANG, ZHANG AND ZHANG, RECEIVERS AND CONSTELLATION DESIGNS FOR SPAD VLC SYSTEMS

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Low-Complexity Receivers and Energy-Efficient Constellations for SPAD VLC Systems Jian Zhang, Ling-Han Si-Ma, Bin-Qiang Wang, Jian-Kang Zhang and Yan-Yu Zhang

Abstract— In radio frequency (RF) wireless communications with additive white Gaussian noise (AWGN), the commonly used bipolar pulse amplitude modulation (PAM) is considered to be the most energy-efficient one-dimensional constellation and admits a fast maximum likelihood (ML) receiver. For visible light communication (VLC) over AWGN channels, the unipolar PAM constellations are generated by adding proper direct currents to the bipolar PAM and have the same properties. However, for VLC systems with single-photon avalanche diode (SPAD VLC), the channel has additive Poisson noise (APN) and as a result, modified PAM over APN channels is not energy-efficient and its ML receiver has exponentially increasing complexity against average bit rate per symbol. In this letter, we first propose a low-complexity Anscombe root receiver by utilizing Anscombe root (AR) transformation to approximately transform the APN channels into AWGN channels. Then, for SPAD VLC with this proposed AR receiver, an energy-efficient constellation is designed by minimizing the average transmitted optical power for a fixed minimum Euclidean distance and shown to be the squared version of unipolar PAM constellations. Furthermore, for this constellation, equally spaced threshold (EST) receiver is developed. Extensive simulations indicate that 1) the proposed receiver and the ML receiver have almost the same error performance for PAM and our optimally designed constellation, respectively; 2) our designed constellations significantly outperform the unipolar PAM constellation for ML and our proposed receivers. Index Terms— Intensity modulation with direct detection (IM/DD), visible light communications (VLC), single-photon avalanche diode (SPAD), Poisson channels, Anscombe root transformation, squared pulse amplitude modulation (SPAM).

I. I NTRODUCTION Recently, visible light communication (VLC), as a promising alternative to radio frequency (RF) system, has received extensive attention [1]. A typical VLC system consists of light emitting diodes (LEDs) and photodiodes (PDs), which are used as transmitters and receivers respectively. However, the incoherence of the LED’s output causes the VLC system to adopt simple intensity modulation and direct detection (IM/DD) [2]. Typical detectors used in VLC systems is positive-intrinsic-negative (PIN) diodes and avalanche photon diodes (APDs). These detectors are not efficient for low-power and long-distance transmission because of the existence of the transimpedance amplifier (TIA) [3]. This work was supported by NHTRDP (863 Program) of China (Grant No.2013AA013603) Jian Zhang, Ling-Han Si-Ma, Bin-Qiang Wang and YanYu Zhang are with National Digital Switching System Engineering and Technological Research Center, Henan Province (450000), China. Emails: [email protected], [email protected], [email protected] and [email protected]. Jian-Kang Zhang is with the Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, Ontario, Canada. Email: [email protected]. The work of J.-K. Zhang was partially funded by NSERC.

More recently, a new type of detector, single photon avalanche diode (SPAD), is proposed and studied in [4]–[6]. The SPAD detector without TIA is highly sensitive and thus, can detect a single photon. For this reason, the SPAD detector is suitable in some special applications where the received signal is very weak [7]–[9]. Unlike traditional PDs and APDs, the detection output of SPAD is the number of photons, which, thus can be modelled as a single photon counter. The photon counting process of an ideal photon counter can be modelled using Poisson statistics, and the noise generated by the process is called shot noise. Due to the significant difference of additive Poisson noise (APN) from additive white Gaussian noise (AWGN), the traditional constellation and detection method designed for VLC over AWGN channels can not be directly generalized to the APN channel [10]–[14]. For VLC over AWGN channels, pulse amplitude modulation (PAM) is appealing due to its high bandwidth efficiency and fast maximum likelihood (ML) detection. However, this is no longer true for APN channels, for which the ML receiver of PAM has higher complexity [15]. The aforementioned factors indeed motivate us to propose a low-complexity receiver and design an energy-efficient constellation for this receiver. Our contribution in this letter can be summarized as follows: 1) A low-complexity receiver is proposed by using the Anscombe root transformation; 2) Specifically for this receiver, the most energy-efficient constellation is designed by minimizing the average optical power for a fixed distance metric; and 3) For the proposed constellation, a receiver with equally spaced thresholds is developed. II. C HANNEL M ODEL AND ML R ECEIVER In this section, we first introduce channel model of SPAD VLC and then, will discuss the complexity of ML receiver for SPAD VLC systems. A. Channel Model Let us consider an SPAD VLC system with a transmitter LED and a receiver SPAD array. For SPAD VLC systems, the SPAD detection rate is usually constrained by the dead time, during which individual SPAD in an array can only detect one photon. For this reason, if the number of the incoming photons is larger than the maximum number of counted photons of this array, the received signals will suffer from clipping distortion. In this letter, we follow the assumption in [3], and assume that the time for the SPAD array to count photons is much longer than the dead time. For such system, the transmitted signal, defined by s, is nonnegative to satisfy the unipolarity requirement of intensity modulation and belongs to a given

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LPT.2016.2572300, IEEE Photonics Technology Letters 2

ZHANG, SI-MA, WANG, ZHANG AND ZHANG, RECEIVERS AND CONSTELLATION DESIGNS FOR SPAD VLC SYSTEMS

K

−1 constellation S = {si }i=2 , where K is a positive integer. i=0 Then, the output of the SPAD, z, is given by [3]

z = (CT /E)s + N T + p,

(1)

and thus: s

r 1 3 1 E(ˆ z ) = 2 E(z) + − ≈ 2 E(z) + 8 48E 2 (z) 8

(6)

where z is the photon number counted by an SPAD array, p is an additive Poisson noise scalar, C denotes the photon detection efficiency of the SPAD, N means the dark count rate of an SPAD, and E is defined by the single photon energy. In addition, T is the time interval between two transmissions. B. ML Receiver

telling us that we can use the number of counted photons to approximate the signal expectations in the transformation domain. Thus, the original APN channel is approximately transformed into well-established AWGN channels

For presentation simplicity, let α and β be defined by α = CT /E and β = N T . According to [3], z follows Poisson distribution with mean given by

where zˆ is pthe output of the transformation, sˆ can be expressed as sˆ = 2 αs + β + 3/8 and n is an additive Gaussian noise with mean 0 and variance approximating 1. 2) Proposed Anscombe Root Receiver: Following the ML receiver of AWGN channels, the ML threshold for the transformed signal, zˆ, in (7) is given by, p p (1) (8) ti = λi + 3/8 + λi+1 + 3/8,

λ = αs + β = E(z),

(2)

Then, the respective probability density function of z and p can be expressed as: Pr(z = n) = e−λ

λ−(n+λ) λ−n , Pr(p = n) = e−λ . n! (n + λ)!

Now, we consider the ML receiver for on-off keying (OOK), Pr(z=n|s=0) i.e, s ∈ {0, 1}, over APN channels. If Pr(z=n|s=1) ≥ 1, then, the output of ML receiver is given by “0”. Otherwise, the ML estimate of s is “1”. It should be noticed that for the reason that the significant difference of the likelihood function from that of Gaussian channels, there is no closed-form ML receiver even for this binary modulation. Therefore, for the commonly K −1 used PAM (s ∈ {i}i=2 ) constellation, the ML receiver i=0 for APN channels has the same complexity as an exhaustive search [15], which exponentially grows against increasing bit number per symbol, say, K. In this letter, our main task is to propose a low-complexity receiver, for which an energy-efficient constellation will be designed. III. P ROPOSED R ECEIVERS AND C ONSTELLATIONS Now, we formally state our main results in this letter. A. Proposed Anscombe Root Receiver Our main idea is to use Anscobe transformation at the receiver side to approximate the transformed APN channels by square-root AWGN channels. 1) Anscombe Root Transformation: From the results in [16], [17], given a random variable z obeying Poisson distribution, p the Anscombe root transformation of z is expressed as 2 z + 3/8, which can be well approximated by a Gaussian random variable with unitary variance. Now, we use the Anscombe transformation at the receiver side of APN channels as p zˆ = 2 z + 3/8, (3) According to [16], [17], the approximation of the variance of zˆ, D(ˆ z ), is proved to be, D(ˆ z ) = 1 + 1/(16E 2 (z)),

(4)

Combining the equality, D(z) = E(z 2 ) − E 2 (z), with (3) and (4), we obtain E(4z+3/2)−E 2 (ˆ z ) = D(ˆ z ) = 1+1/(16E 2 (z)),

(5)

zˆ = sˆ + n

(7)

where λi = αsi + β is defined by (2). Then, we propose the Anscombe root receiver as follows: Anscombe Root (AR) Receiver: Let s ∈ S = K −1 {si }i=2 . Given the received signal z defined by (1), the i=0 output of the proposed AR receiver is determined in the following two steps: p 1) AR transformation: zˆ = 2 z + 3/8. K (1) (1) −2 2) Decision threshold: let T (1) = {ti }i=2 , where ti i=0 is defined in (8). Then, the output of AR receiver, sˆ as the estimate of s, is determined by (1) a) If zˆ < t0 , then, sˆ = s0 ; (1) (1) b) If ti−1 < zˆ < ti for i = 1, · · · , 2K − 2, then, sˆ = si ; (1) c) If zˆ > t2K −2 , then, sˆ = s2K −1 . To make our presentation as clear as possible, we specifically use our proposed receiver to recover the transmitted symbol from the received signal specifically for PAM. Example 1: The constellation of 4-PAM is S = {0, 1, 2, 3}. Using (8) and letting β˜ = β +3/8, the decision threshold set is expressed as (1)

T (1) = {tk : k = 0, 1, 2},

(9)

(1)

where tk is defined by (8). Then, sˆ can be attained by 1) 2) 3) 4)

(1)

If zˆ ≤ t0 , then, sˆ = 0; (1) (1) If t0 < zˆ ≤ t1 , then, sˆ = 1; (1) (1) If t1 < zˆ ≤ t2 , then, sˆ = 2; Otherwise, sˆ = 3.

B. Constellation Designs To simplify the design problem, when we design the constellation, the influence of the dark count is not considered, i.e., letting β = 0. This assumption is reasonable due to the fact that the dark count photon is relatively smaller than the photon caused by signal, especially when the signal is large. For this reason, we ignore the fractional term 38 for the signal design. For our proposed AR receiver, if β˜ = β+3/8 is treated as zero, then, the performance is dominated by the minimum


This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LPT.2016.2572300, IEEE Photonics Technology Letters

of √ the √distance among all the distinct transmitted signal, say, | s− sˆ|. Therefore, the design problem is formulated below: Problem 1: For any given positive integer K, find a constellation S of size 2K and with all the 2K entriesPbeing nonnegative such that the average optical power 21K s∈S s is minimized under constraint that the minimum Euclidean distance between the square roots of any two √ signals, √ distinct s, sˆ ∈ S, is normalized, i.e., mins,ˆs∈S,s6=sˆ | s − sˆ| = 1. whose solution is stated in the following theorem. Theorem 1: The optimal solution to Problem 1 is given by K −1 S = {i2 }i=2 . i=0 Proof: Let us consider any given constellation S˜ = K −1 {si }i=2 satisfying that √ all the√elements of S˜ are noni=0 negative and mins,ˆs∈S,s6 | s − sˆ| = 1. Without loss of ˜ =sˆ generality, we assume that the elements of S˜ satisfy√that s K > · · · > s0 . Then, the assumption mins,ˆs∈S,s6 ˜ =sˆ | s− √2 −1 √ √ sˆ| = 1 tells us that si+1 − si ≥ 1. In other words, for √ √ √ any 1 ≤ i ≤ 2K − 1, si ≥ 1+ si−1 ≥ i+ s0 ≥ i. Therefore, X s∈Sˆ

s≥

K 2X −1

i=0

si ≥

K 2X −1

i=1

i2 =

2K (2K − 1)(2K+1 − 1) 6

K −1 where the equality holds if Sˆ = {i2 }i=2 . Therefore, the i=0 K −1 optimal solution to Problem 1 is given by S = {i2 }i=2 , i=0 completing the proof of Theorem 1. It can be seen that our proposed design is the squared version of common used PAM constellation. For this reason, we name this design as squared PAM (SPAM).

C. Modified Receiver With Equally Spaced Thresholds For our proposed SPAM, if the thresholds of the Anscombe receiver approximately distribute with an equal interval, then, the AR receiver’s complexity can be further reduced. Thus, low complexity decision method can be obtained via modification of the thresholds. Specific analysis and method of revision will be given below. Taking the 4-SPAM as an example, the constellation is S = {0, 1,√ 4, 9}. The thresholds can be calculated by (8) as T ideal = { αP k : k = 1, 3, 5} with dark count and fractional term 3/8 ignored, which means that, in the ideal situation, the proposed constellation can make the thresholds of Anscombe receiver distributing with an equal interval. However the dark count causes a bias leading to the practical q on the constellation q (1) 2 ˜ ˜ i=2 without ones T = { i α + β + (i + 1)2 αP + β} i=0 equal interval. Thus, we propose to revise the T ideal to approximate T (1) and meanwhile, maintain the former with an equal interval. (2) Therefore, the modified qthreshold = q set is T K (2) i=2 −2 (2) ˜ ˜ {ti }i=0 , where t0 = β + αP + β and for i = 1, · · · , 2K − 2, q √ 2 (2) ti = αP (2i + 1) + αP (2K−1 − 1) + β˜ q √ 2 + αP (2K−1 ) + β˜ − αP 2K − 1 To illustrate the approximation reasonability, the asymptotical behaviour of the revised thresholds for 4-SPAM is shown in

Threshold Values

3

150 20

40

10

20

100 0

50 0 −120

−110−109.5 −109−108.5 −108

−115

−110

−94

−92

−105 −100 −95 −90 Optical Irradiance (dBm)

−90

−85

−80

Fig. 1. Comparison of decision threshold values of the optimal T (1) (circle), ideal T ideal (solid) and revised T (2) (cross) ones for 4-SPAM.

the Fig. 1. It can be seen that our modification is reasonable when the transmitted optical power is sufficiently high. The (1) t main reason is that lim i(2) = 1. Then, the corresponding P →∞ ti receiver is presented below. Equally Spaced Threshold (EST) Receiver: Given the output of AR transformation, zˆ, the output of EST receiver sˆ is given by (2)

1) If zˆ ≤ t0 , then, sˆ = 0; (2) (2) 2) If ti−1 < zˆ ≤ ti for 1 ≤ i ≤ 2K − 2, then, sˆ = i2 ; 3) Otherwise, sˆ = (2K − 1)2 . It should be noted that the closed-form decision thresholds of our proposed receivers enable us to efficiently recover the transmitted signals. The complexity of our proposed EST receiver is equivalent to that of ML detection for PAM over AWGN channels and thus, very low for our energy-efficient SPAM. From Fig. 1, we can see that the proposed decision thresholds approach to those numerical values for the ML detection. In other words, the approximation impact of our proposed receivers on the error performance is expected to be unnoticeable in high power regimes, which will be verified in the ensuing simulation section. IV. S IMULATION R ESULTS In this section, we carry out computer simulations to investigate the performance of our proposed receivers and signal designs. The simulation parameter setup follows that of [3] (i.e. α = 4.52 × 1014 s/J and β = 7.27). More detail of simulations is provided below: 1) ML and AR Receivers: The performance of AR receiver is compared with ML receiver in Figs. 2(a) and 2(b). To make the comparison as comprehensive as possible, we simulate for different modulation order and constellations. The result indicates that the proposed receiver has performance approaching the ML receiver for diverse modulation orders when PAM or our proposed constellation is adopted. 2) PAM and SPAM: Then, we compare the error performance of our proposed constellation with PAM using the same receiver. Simulation results for ML receiver and the proposed receiver are respectively shown in Figs. 2(c) and 2(d), which indicate that the SPAM constellation has a better performance than PAM. In addition, the power gains are about 2dB, 3dB and 4dB respectively when the modulation order is 4, 8 and 16, which is expected to become larger against increasing modulation order.


This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LPT.2016.2572300, IEEE Photonics Technology Letters 4

ZHANG, SI-MA, WANG, ZHANG AND ZHANG, RECEIVERS AND CONSTELLATION DESIGNS FOR SPAD VLC SYSTEMS 0

10

0

−2

10

Bit Error Rate

Bit Error Rate

10

16−PAM, ML 16−PAM, AR 8−PAM, ML 8−PAM, AR

−4

10 −120

4−PAM, ML 4−PAM, AR

−115

−110

16−SPAM, ML −2

10

−85

−80

8−SPAM, ML 8−SPAM, EST 4−SPAM, ML

−4

−105 −100 −95 −90 Optical Irradiance (dBm)

16−SPAM, EST

10 −120

4−SPAM, EST

−115

−110

−105 −100 −95 Optical Irradiance (dBm)

−90

−85

(a) AR and ML receivers for PAM Fig. 3. Error rate comparison of ML and EST receivers for SPAM with T = 1ms, and K = 2, 3, 4.

0

Bit error rate

10

16−SPAM, ML −2

10

proposed constellation has been shown to outperform PAM constellations with both receivers. In future, we will consider the low-complexity detection and energy-efficient design of signal waveform for time-continuous SPAD VLC systems.

16−SPAM, AR 8−SPAM, ML 8−SPAM, AR 4−SPAM, ML

−4

10 −120

4−SPAM, AR

−115

−110


−90

−85

(b) AR and ML receivers for SPAM 0

Bit Error Rate

10

16−PAM, ML −2

10

16−SPAM, ML 8−PAM, ML 8−SPAM, ML 4−PAM, ML

−4

10 −120

4−SPAM, ML

−115

−110


−90

−85

−80

−85

−80

(c) SPAM and PAM with the ML receiver. 0

Bit Error Rate

10

16−PAM, AR −2

10

16−SPAM, AR 8−PAM, AR 8−SPAM, AR 4−PAM, AR

−4

10 −120

4−SPAM, AR

−115

−110


−90

(d) SPAM and PAM with the AR receiver. Fig. 2.

Error rate comparison with T = 1ms and K = 2, 3, 4.

3) ML and EST: In Fig. 3, EST receiver is compared with ML receiver, and only the performance of SPAM scheme is shown in the figure, because the revisional receiver is specially designed for this constellation. Fig. 3 indicates that a low complexity method is obtained by revising the threshold to equal interval at the cost of slight degradation of its performance. Furthermore, the curves of EST receiver asymptotically approach those of the ML receiver. This phenomenon agrees with the comparison result in Fig. 1. V. C ONCLUSION In this letter, we have investigated the signal detection and designs of SPAD-VLC systems. For such system over APN channels, we have proposed a low-complexity receiver based on Anscombe root transformation of the received signals. The closed-expression of this proposed receiver has enabled us to attain a tractable and useful design criterion and design onedimensional energy-efficient constellations. Simulation results have shown that our proposed low-complexity receiver has almost the same error performance as the ML receiver for PAM and our proposed SPAM, respectively. In addition, our

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