1544 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO....

1544 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 2013

NLOS UV CommunicationsUsing M -ary Spectral-Amplitude-CodingMohammad Noshad, Student Member, IEEE, Maıte Brandt-Pearce, Senior Member, IEEE,

and Stephen G. Wilson, Senior Member, IEEE

Abstract—We present an M -ary spectral amplitude code (SAC)modulation technique to improve the performance of free-spaceoptical (FSO) communication systems. Although this approachcan be used in any dispersive FSO system, in this paper we focuson non-line of sight (NLOS) ultraviolet (UV) systems relying onatmospheric scattering. Spectral amplitude encoding is appliedon a broadband UV source using the same code families forthe M -ary alphabet as used previously in SAC optical codedivision multiple access (OCDMA) systems. A differential struc-ture using two photomultiplier tubes is utilized in conjunctionwith various demodulation algorithms to decode the receivedsignal. Intersymbol interference (ISI), received beam divergenceand shot noise are considered as the main factors limiting thesystem performance. An upper bound on the bit error probabilityis presented and compared with simulation results for variousgeometries and for different code parameters. The maximum bitrate for a fixed bit error probability is calculated in terms of thelink length, and results for different alphabet sizes are shown. Bysacrificing spectral efficiency without becoming more susceptibleto ISI, the proposed system can support higher rates and longerdistances for the same performance compared with on-off keyingsystems.

Index Terms—M -ary communication, ultraviolet communi-cations, non-line of sight communication, spectral amplitudecoding.

I. INTRODUCTION

THE application of free space optics (FSO) for non-lineof sight (NLOS) communications has attracted increasing

interest because of its relative insensitivity to pointing errorsand robustness against shadowing [1]. Atmospheric scatteringin the UV band is higher than in other optical bands, and,consequently, in NLOS systems the receiver can receive morepower from the transmitter than if it used another portion ofthe optical spectrum [2], making the UV band an interestingchoice for NLOS communications [1]. Furthermore, back-ground irradiance severely limits the performance of NLOSinfrared and visible light optical communications, while theultraviolet (UV) band (200-280 nm) has low background lightbecause of sunlight filtering by the upper atmosphere [1]. Yetthe strong scattering in the NLOS UV channel imposes atemporal dispersion on the transmitted optical pulses. Thus,both intersymbol interference (ISI) and low received powercan be factors limiting the data rate in NLOS UV links [3].

Manuscript received May 30, 2012; revised October 12, 2012. The associateeditor coordinating the review of this paper and approving it for publicationwas W. Kwong.

The authors are with the Charles L. Brown Department of Electrical andComputer Engineering, University of Virginia Charlottesville, VA 22904 (e-mail: {mn2ne, mb-p, sgw}@virginia.edu).

Digital Object Identifier 10.1109/TCOMM.2013.020813.120371

In this paper we propose single-user M -ary modulation anddemodulation techniques to combat the channel degradationand increase the distance-rate product of NLOS UV systems.The distance-rate product is an important measure for evalu-ating the performance of optical communication systems. It isdefined as the product of the transmitter-receiver distance ofthe optical channel and its maximum achievable data rate fora predefined maximum acceptable bit error probability. Thisproduct is usually limited by either intersymbol interference(ISI) or low signal to noise ratio (SNR). Error correctingcodes, equalization and M -ary transmission are the mostcommon techniques for increasing the distance-rate product inboth radio frequency (RF) and optical communications. Themaximum data rate versus the distance between the transmitterand receiver has been presented in previous work on NLOSUV communications for on-off keying (OOK) and differentsystem geometries [3], [4].

In [5] we propose a novel structure using spectral amplitudecoding (SAC) for improving the performance of the single-user NLOS UV communication system. We utilize fixed cross-correlation codewords, shown to have optimal properties in[6], as modulation symbols in order to increase the bit rate ofthe NLOS UV communication system. In [5], an avalanchephotodiode (APD) array is used for extracting the receivedsignal in each wavelength. However, because of the lowgain of the APD arrays in the UV range and the smallaperture size of each element of the array, the system isnot able to transmit data for distances longer than 100 m.Photomultiplier tubes (PMT) are a better choice for detectingweak signals. Due to the high gain of the PMTs, shot noisedominates and other noise sources, such as thermal noise, canbe neglected. An experimental demonstration of a NLOS UVcommunication system using OOK and PMTs is reported in[7], but communications is still limited to 500 Kb/s and 200m range.

In this paper we consider a different transmitter/receiverstructure than in [5]. A broadband optical source is requiredfor SAC, thus, since high-power broadband UV sources arescarce, widely-available broadband spectrum visible lasers areproposed as transmitter light sources. The encoding is done inthe visible/infrared (IR) region, and then the encoded signalis frequency-doubled (or tripled) to the UV range. We usesymmetric balanced incomplete block designs (BIBD) [8] toconstruct spectrally encoded symbols. At the receiver side,one or two PMTs can be used for decoding the receivedsignal. The symbol time period, Ts, is divided into smaller

0090-6778/13$31.00 c© 2013 IEEE


Fig. 2. Geometry of the NLOS UV link.

III. CHANNEL MODEL

Analyzing the performance of NLOS UV communicationsystems requires a model of the path loss and the impulseresponse of the channel. Analytical [15], simulation [16], [17]and experimental [7] approaches have been used to obtain aprecise estimate of the impulse response and to calculate thelink loss. In this work we use the channel model presentedin [17]. In this model, the common volume between thetransmitter and receiver is divided into small cubic volumes,and then numerical integration is used to calculate the impulseresponse of the NLOS UV link.

Fig. 2 shows the geometry of the NLOS UV link that weuse in this paper. The transmitter and receiver are located atdistance D from each other. We denote the transmitter beamfull-width divergence by φ1, the receiver field of view (FOV)by φ2, the transmitter elevation angle by θ1 and the receiverelevation angle by θ2. The transmitter beam axis and receiverFOV axis are assumed coplanar for notational simplicity, butthis assumption is not required for our approach.

In NLOS UV systems, the transmitted pulse is spread in thetime domain due to the large scattering volume and the strongatmospheric scattering of the UV band. The received signalat wavelength f , p

(f)R (t), can be obtained by convolving the

channel intensity impulse response in wavelength f , hf(t),with the transmitted signal in that wavelength, p

(f)T (t). So for

the received signal intensity we have

p(f)R (t) =

∞∑k=−∞

skf

[Ff(t − kTs) ∗ hf (t)], f = 1, 2, . . . , F.

(3)

IV. RECEIVER

The optimum receiver for SAC UV systems requires FPMTs (one PMT for each wavelength), which, considering thesize and price of PMTs, is impractical and not cost effective.Here, we propose a structure using one or two PMTs, whichis sub-optimal but practical.

Our structure for the UV receiver using two PMTs (e.g.R1584; Hamamatsu1) is shown in Fig. 3. We use a UVfilter (e.g. Edmund Optics; 254 nm center wavelength, 40nm FWHM bandwidth2) to limit the background noise on thereceiver. Consequently, and considering that the background

1http://sales.hamamatsu.com/en/products/electron-tube-division/detectors/photomultiplier-tubes/part-r1584.php

2http://www.edmundoptics.com/products/displayproduct.cfm?productID=1903&dc&PageNum=2

Fig. 3. Receiver structure for the UV system using two PMTs.

radiation is extremely low in the UV range wavelengths of250 nm - 280 nm, we neglect the effect of the backgroundlight in this paper.

A. Spatial Broadening Due to the Wide Receiver FOV

In the receiver, a UV diffraction grating (e.g., 3600grooves/mm; 0.25 nm/mr dispersion at 250 nm; LaserCom-ponents3) and lens are used for decomposing the spectrum ofthe received beam. A decoder mask, composed of F elements,is placed after the diffraction grating to decode the receivedsignal. As shown in Fig. 3, we assign the z-axis perpendicularto the grooves of the grating.

According to the receiver structure in Fig. 3, the diffractiongrating reflects each wavelength with a specific angle. Thus,divergence of the received beam in the z direction causesthe signal of one wavelength to be reflected at a widerangle, causing interference between adjacent wavelengths. Asa result, each point in the decoder mask plane receives powerfrom more than one wavelength. This can be interpreted asa transformation of the divergence from the received beamangle into the wavelength domain after the diffraction grating.We denote this effect as spatial broadening, which looks likespectral broadening and causes interference between adjacentspectral bins, as illustrated in Fig. 4.

We assume that the normalized optical power distributionon the decoder mask plane at point z from wavelength f isdenoted by g(z−zf), where zf is the center of the bin f , andz − zf is the distance from the center of the beam, assumedcentered on the f th mask element. Furthermore, we assumethat g(z − zf ) is the same for all wavelengths. To describethis effect, we define the spatial broadening coefficients as

ai =1∫∞

−∞ g(z)dz

∫ 2i+12 ΔL

2i−12 ΔL

g(z)dz, (4)

where ΔL is the width of one element of the decoder mask.These spatial broadening coefficients capture the interferencethat the optical signal in one wavelength bin introduces on itsneighboring bins. We define the extent of spatial broadeningby the integer Z such that

∑Zi=−Z ai = 1 and ai ≈ 0 for

∀|i| > Z .

3http://www.lasercomponents.com/fileadmin/user upload/home/Datasheets/optometr/gratings.pdf

NOSHAD et al.: NLOS UV COMMUNICATIONS USING M-ARY SPECTRAL-AMPLITUDE-CODING 1547

Fig. 4. Effect of spatial broadening, due to the received beam divergence,on the interference between wavelengths after the diffraction grating.

Fig. 5. The variation of the spectral decoder mask in one symbol period.

Because of this spatial broadening, the optical power re-flected from the diffraction grating in the angle that correspondto the wavelength f can be written as

p(f)R (t) =

Z∑i=−Z

aip(f−i)R (t). (5)

According to (4), the ai’s are functions of ΔL = L/F ,where L is the length of the decoder mask and is assumedto be fixed. As the code length, F , increases, ΔL decreasesand, consequently, Z increases. Thus, by increasing F theinterference between the spectral bins worsens.

B. Decision Variables

In order to capture as much of the received light as possibleto decode the spectrally encoded signal, the decoder mask isa switch that modifies the beam’s direction of propagation ineach element, unlike the amplitude (on-off) encoder mask usedat the transmitter. The elements can either be micro-electro-mechanical-system (MEMS)-based mirrors with UV coating,or electro-optic crystals, such as BBO or LBO, which aretransparent in the UV range. Each element can be in one oftwo different states, named u1 and u2, and its state can bechanged between u1 and u2. By changing the state of eachelement, the beam of the corresponding wavelength can bedirected to one of the two PMTs. The states of the elementsof the decoder mask are determined by the codeword that thedecoder controller sends to the mask.

As illustrated in Fig. 5, each symbol period with durationTs is divided into M equal time slots and the decoder maskis changed in each time slot. Without loss of generality, weconsider that the decoder mask matches the mth codewordin the mth time slot. The state of element j of the decoder

mask is u1 in the mth time slot if cmj = 1 and is u2

otherwise. Because of the non-rectangular temporal shape ofthe received pulse [7], the received energies in the time slotsof one symbol period are not equal. Therefore, an interleaveris used to reorder the codes in the decoder mask, so that theaverage error probability becomes equal for all symbols.

A schematic of the receiver is shown in Fig. 6. According to(5) and the decoder function, the incident signal in wavelengthf and time slot j upon PMT1 is cjf p

(f)R (t) and upon PMT2

is (1− cjf )p(f)R (t).

We denote the current generated by PMT1 and PMT2 asI(1)(t) and I(2)(t), and the additive noise at PMT1 and PMT2as ν(1)(t) and ν(2)(t), respectively. Without loss of generalitywe consider the k = 0 symbol time. Let I

(n)j (t) and ν

(n)j (t)

respectively represent I(n)(t) and ν(n)(t) in time slot j ofsymbol period k = 0, for n = 1, 2. Then according to [14],I(n)j (t) is given by

I(n)j (t) = GR

F∑f=1

((n − 1)− (−1)ncjf

)p(f)R (t)

+ ν(n)j (t),

(j − 1)Ts

M≤ t <

jTs

M, (6)

where n = 1, 2, j = 1, 2, . . . , M , G is the gain and R is theresponsivity of the PMTs.

An integrate-and-dump filter and a sampler are placed aftereach PMT, and a shift register of length M stores the outputsamples of all time slots. We represent the stored variables byvector r(n) = (r

(n)1 , r

(n)2 , . . . , r

(n)M ), n = 1, 2, where r

(n)j is

r(n)j =

∫ jM Ts

j−1M Ts

I(n)j (t)dt. (7)

Using (2), (5) and (6), r(n)j becomes

r(n)j =

[Z∑

i=−Z

GRai

F+ i−|i|2∑

f=1+ i+|i|2

((n − 1)− (−1)ncjf

)×

( ∞∑m=−∞

sm(f−i)

∫ jM Ts

j−1M Ts

[Ff (t − mTs) ∗ hf (t)]dt

)]

+

∫ jM Ts

j−1M Ts

[ν(n)j (t)

]dt. (8)

As mentioned above, an interleaver is used to change theorder of the codewords of the decoder mask in each symbolperiod; therefore, for mathematical simplicity, we replace the

term∫ j

M Ts

j−1M Ts

[Ff(t − mTs) ∗ hf (t)]dt in (8) with its mean,

1M

∫ Ts

0

[Ff(t−mTs)∗hf (t)]dt. Here,

∫ jM Ts

j−1M Ts

[Ff(t−mTs)∗hf (t)

]dt is the part of the optical energy transmitted in

wavelength f of symbol time m and received during timeslot j of the symbol at time k = 0.

Again for mathematical simplicity, we assume that thereceived pulse shape, i.e., Ff (t) ∗ hf (t), is the same for all


Fig. 6. Schematic illustration of the receiver, showing the decoder mask, PMTs and decision circuit for symbol k = 0. Light paths are indicated by dottedlines and electrical paths are indicated by solid lines.

wavelengths. We define the ISI coefficients, γm, as

γm =

∫ Ts

0

[Ff (t − mTs) ∗ hf (t)]dt, −K ≤ m ≤ K,

(9)

where 2K is the number of adjacent symbols influencing thedesired symbol, i.e., γi ≈ 0 for ∀|i| > K . Using (9), (8)becomes

r(n)j = GR

Z∑i=−Z

ai

( K∑m=−K

(w(n − 1)

− (−1)nS(sm, i)cTj

)γm

M

)+

∫ jM Ts

j−1M Ts

ν(n)j (t)dt, (10)

where S(sm, i) is the ith right shift of vector sm. Onelast approximation we make for analytical tractability is tooverestimate the spatial interference on sm by using cyclicright shifts of sm instead of linear right shifts. For theremainder of this paper, S(y, i) denotes the ith cyclic shiftof vector y.

The mean of r(1)j , neglecting ISI, is given by

μ(1)j = GR

γ0

M

Z∑i=−Z

aiS(s0, i)cTj . (11)

Let mk and mk be the transmitted and estimated symbolsin the kth symbol period, respectively. We define μ

(1)j given

m0 = � as μ(1|�)j = GR γ0

M

Z∑i=−Z

ai

(S(c�, i)cTj

). Since the

code is cyclic, we have µ(1|�) = S(µ(1|1), �−1), for ∀�, wherethe vector µ(1|�) = [μ

(1|�)1 , μ

(1|�)2 , . . . , μ

(1|�)M ], � = 1, 2, . . . , F .

From the receiver structure, letting μ(2|�)j be the mean of r

(2)j

given m0 = �, we know μ(1|�)j +μ

(2|�)j = GR γ0

M w; henceforth

we use GR γ0

M w − μ(1|�)j instead of μ

(2|�)j .

Because of the high gain of the PMTs, shot noise dominatesthe thermal noise and, thus, the variance of r

(n)j in (10) given

m0 = �, neglecting other noise sources, can be approximatedas

Var(r(n)j |�) = 2eΔfGμ

(n|�)j , (12)

where e is the electron charge (e = 1.6×10−19 C) and Δf isthe receiver equivalent noise bandwidth. Since in the receiver

structure the symbol period is divided into M equal time slots,Δf is proportional to M and increases as M increases. In thisstudy we assume the shot noise can be modeled as Gaussiandistributed [18].

C. Symbol Detectors

At the end of the symbol period, the stored variables in theshift registers are fed into a symbol-by-symbol demodulator,which then estimates the symbol mk for the kth time instant(ignoring ISI). Three different decision rules are considered:

1) Optimum Single-PMT Detector: In this case the decisionis made based on a maximum likelihood (ML) rule using onlythe output of either PMT1 or PMT2. This is the simplest andleast expensive option. The decision criterion for this detectorcan be written as

m0 = arg max1≤�≤M

Pr(r(n)|m0 = �), (13)

where n = 1 or 2, m0 is the symbol transmitted in the k =0 symbol period and Pr(r(n)|m0 = �) is the probability ofreceiving vector r(n), given m0 = �.

2) Optimum Dual-PMT Detector: Given equally-likelytransmitted symbols, the optimum detector can be imple-mented based on a ML rule using the outputs of both PMT1and PMT2. For this detector, since the shot noise componentsin different time-slots have unequal powers, the decision rulein its most simple form is quadratic. Since the noise terms ofPMT1 and PMT2 are independent, the decision criterion forthis case can be written


Pr(r(1)|m0 = �)Pr(r(2)|m0 = �). (14)

3) Optimum Linear Dual-PMT Detector: In linear de-tectors, the decision statistic, written as a vector x =[x1, x2, . . . , xM ], is generated from r(1) and r(2) as

x = r(1)Q+ r(2)Q′, (15)

where Q = [Q1 Q2 . . . QM ] and Q′ = [Q′1 Q′

2 . . . Q′M ]

are M × M matrices, formed by columns Q� =

[q1�

q2�

. . . qM�

]T

and Q′� = [q′

1�q′2�

. . . q′M�

]T

. Then thedecision is made based on the following rule:


x�. (16)


For the optimal linear detector, Q and Q′ are chosen tominimize the symbol error probability.

Note that if a0 = 1 and ai = 0 for i �= 0, the three detectorsdescribed (the optimum dual-PMT detector, optimum lineardual-PMT detector and optimum single-PMT detector usingPMT2) all reduce to

m0 = arg min1≤�≤M

r(2)� . (17)

Since the performance of these detectors is not a functionof the Hamming distances between the symbols, the fixedweight and fixed cross-correlation code families proposedhere, further described in [6], may not be the optimum codes,i.e., those that have the best performance. Finding the optimumcodes for each one of these detectors is beyond the scope ofthis paper.

V. ANALYTICAL BIT ERROR PROBABILITY CALCULATION

In this section, we calculate the optimum Q and Q′ of thelinear dual-PMT detector, and obtain expressions for the biterror probability of this receiver by considering the shot noise,temporal dispersion and spatial broadening effects. As shownbelow in Section VI, the symbol error rate of the linear dual-PMT detector is close to the error rate of the quadratic optimaldual-PMT detector, and acts as a bound on its performance.Therefore, and for mathematical simplicity, we analyze the biterror probability of only the linear dual-PMT detector in thissection.

A. Optimal Linear Dual-PMT Detector for ISI-Free System

Generally, the symbol error probability can be written as

Ps =

M∑�=1

Pr(m0 = �)(1− Pr(�|�)). (18)

where we define Pr(�′|�) .= Pr(m0 = �′|m0 = �). As-

suming the decision criterion to be as (16), an error arisesif for at least one �′ (�′ �= �) we have x�′ > x�, andtherefore, Pr(�′|�) = Pr(x�′ > x�|m0 = �). The outputsamples in Fig. 6 at time slot j are modeled as Gaussiandistributed, and denoted as r

(n)j ∼ N (μ(n|�)

j , Var(r(n)j |�)).

Thus, x� − x�′ is also a Gaussian random variable withmean μ(�−�′) =

∑Mj=1 μ

(1|�)j (qj� − qj�′ ) + μ

(2|�)j (q′j� − q′j�′)

and variance Var(�−�′) =∑M

j=1 Var(r(1)j |�)(qj� − qj�′ )2 +

Var(r(2)j |�)(q′j� − q′j�′)2. Then

Pr(�′|�) = 1

2erfc( μ(�−�′)√

2Var(�−�′)

). (19)

Let Q∗ and Q′∗ be the solutions of the optimum detectorin the ISI-free case. Since the symbols are cyclic shifts, Q∗

and Q′∗ are circulant matrices, so Q∗� = S(Q∗

1, � − 1) andQ′∗

� = S(Q′∗1, � − 1), for ∀�. Therefore, Pr(�′|�), using (12),

becomes

Pr(�′|�) =

1

2erfc

(√1

4eΔfG

µ(1|�)(Q∗

� −Q∗�′)− µ(1|�)

(Q′∗

� −Q′∗�′)

√µ(1|�)Q�,�′ +

(GRw γ0

M− µ(1|�)

)Q

′�,�′

),

(20)

where Q′�,�′ =

[(q∗j� − q∗j�′ )

2]M

j=1and Q�,�′ =

[(q′∗j� −

q′∗j�′)2]M

j=1are M × 1 matrices. According to (19), adding a

constant value to all qj�’s or q′j�’s does not change Pr(�′|�). Onthe other hand, for the optimum linear detector q∗i� = q∗j� and

q′∗i� = q′∗j� if μ(n|�)i = μ

(n|�)j . So, without loss of generality,

for time slots j such that μ(n|�)j = GR γ0

M λ, q∗j� and q′∗j� arechosen to be zero.

The union bound on the symbol error probability is givenby

Ps,UB =M∑

�=1

M∑�′=1�′ �=�

1

MPr(�′|�). (21)

Since the symbols and the channel action (after interleaving)are symmetric, (21) can be simplified to

Ps,UB =M∑

�′=2

Pr(�′|1). (22)

For high SNRs the largest Pr(�′|1) becomes the dominantterm. So, minimizing the maximum Pr(l′|1), over 2 ≤ l′ ≤ M ,asymptotically minimizes the symbol error probability. Conse-quently, the solution to the following minimax problem givesus the Q∗ and Q′∗ of the asymptotically optimum lineardetector:

(Q∗,Q′∗) = arg minQ,Q′

max2≤�′≤M

Pr(�′|1). (23)

In general, (23) is not an easy problem to solve. Here, werestrict ourselves to a specific case, which is valid for mostpractical systems. We assume that the spatial broadening issmall enough to neglect ai for |i| ≥ 2, i.e., the only nonzeroai’s are a0, a−1 and a1. We also assume that the spatialbroadening is symmetric, i.e., a1 = a−1, and therefore, wehave a0 + 2a1 = 1. In this case, since μ

(1|1)j = GR γ0

M λ,for j = 3, 4, . . . , M − 1, q∗j1 = q′∗j1 = 0 for j =

3, 4, . . . , M − 1. We also have μ(1|1)1 = GR γ0

M (a0w + 2a1λ)

and μ(1|1)2 = μ

(1|1)M = GR γ0

M (a1w + (a0 + a1)λ), and thus,q∗21 = q∗M1 and q′∗21 = q′∗M1. Hence, (20) becomes as (24)at the top of the next page, where ζ1 = (a0 − a1)(w − λ),ζ2 = a1(w−λ), ζ3 = a0(w−λ), ξ1 = (1−a1)w+(1+a1)λ,ξ2 = a1w + (2− a1)λ, and ξ3 = a0w + (2− a0)λ.

From Appendix A, the minimum of Pr(2|1) = Pr(M |1)over all Q and Q′ is

1

2erfc

(√wRγ0

2eΔf

√ζ21

ξ1(2w − ξ1)+

ζ22ξ2(2w − ξ2)

), (25)

which is obtained for q∗11 = ζ1

ξ1+ ζ2

ξ2and q∗21 = ζ2

ξ2. Substituting

these values in (24) we get Pr(2|1) ≥ Pr(�′|1) for 3 ≤ �′ ≤M −1. For any other q11, q21, q′11 and q′21, max

2≤�′≤MPr(�′|1) is

larger than (25) and thus, (25) is the solution of the minimaxproblem in (23).

B. Symbol Error Probability

To calculate the bit error probability in the presence of ISI,first we obtain an upper bound on the symbol error probabilityand then we calculate a bound on the bit error probability.


Pr(�′|1) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

12erfc

(√Rγ0

4eΔfM(q∗

11−q∗21)ζ1+q∗

21ζ2−(q′∗11−q′∗

21)ζ1−q′∗21ζ2√

(q∗11−q∗

21)2ξ1+q∗

212ξ2+(q′∗

11−q′∗21)

2(2w−ξ1)+q′∗21

2(2w−ξ2)

), �′ = 2, M ;

12erfc

(√Rγ0

4eΔfMq∗11ζ3+q∗

21ζ2−q′∗11ζ3−q′∗

21ζ2√q∗12ξ3+q∗

212ξ2+q′∗

112(2w−ξ3)+q′∗

212(2w−ξ2)

), �′ = 3, M − 1;

12erfc

(√Rγ0

4eΔfMq∗11ζ3+2q∗

21ζ2−q′∗11ζ3−2q′∗

21ζ2√q∗211ξ3+2q∗2

21ξ2+q′∗211(2w−ξ3)+2q′∗2

21(2w−ξ2)

), otherwise.

(24)

Pr(�′|1) = 1

2

(1

M

)2K M∑m−K=1

· · ·M∑

m−1=1

M∑m1=1

· · ·M∑

mK=1

erfc

(√1

4eΔfG

(K∑

i=−K

γi

γ0µ(1|mi−1)

)[(Q∗

1 −Q∗�′)− (Q′∗

1 −Q′∗�′)]

√(K∑

i=−K

γi

γ0µ(1|mi−1)

)(Q1,�′ − Q

′1,�′)+

(GR w

M

K∑i=−K

γi

)∥∥∥Q′∗1 −Q′∗

�′

∥∥∥2)

. (26)

We assume the detector used is the optimal linear detectorignoring ISI derived above. For the upper bound, we use theunion bound in (22), where Pr(�′|1), considering ISI, becomesas (26), given at the top of the page.

C. Bit Error Probability

Having bounded the symbol error probability, we still needthe bit error probability in order to be able to comparethe system performance with OOK systems. Assume b�, alog2 M digit binary number, is the binary sequence assignedto symbol �. When m0 = � is transmitted and m0 = �′

is received, d(b�, b�′) bits are decoded incorrectly, whered(b�, b�′) denotes the Hamming distance between the binaryvectors b� and b�′ . According to [19], for equally likelysymbols we have the following inequality for the bit errorprobability, Pb,

Pb ≤ M

2(M − 1)Ps,UB

.= Pb,UB , (27)

where Pb,UB is an upper bound on Pb.

VI. NUMERICAL RESULTS

In this section, numerical results are presented to comparethe performance of the proposed M -ary transmission fordifferent code-lengths with previously proposed OOK modu-lation. In this work, the main limiting factor on the bit-rate isinter-symbol interference, and other pulsed modulations, suchas PPM, perform much worse than OOK, because of theirshort pulse duration. Hence, in our numerical results we onlycompare the performance with OOK.

We use Paley, projective geometry (PG) and twin primepower (TPP) difference sets [8] as code families in theseresults. For these code families F = 2w +1 and w = 2λ+1.For the bit-symbol mapping, since the code-lengths are in gen-eral not powers of two, we concatenate multiple symbols andassign longer binary sequences to them [20]. The transmittedpower is assumed to be 1 W. The gain G of the PMTs is107 and their responsivity R is 0.3 A/W. Three geometriesare considered, described in Table I.

For simplicity, the transmitted signal is assumed to havea rectangular pulse shape in time, and the received pulse is

TABLE IPARAMETERS OF THREE DIFFERENT GEOMETRIES

Case A Case B Case Cθ1 75◦ 60◦ 40◦

θ2 45◦ 60◦ 40◦

φ1 10◦ 2◦ 5◦

φ2 50◦ 50◦ 50◦

D 400 m 400 m 400 m

obtained by convolving the transmitted signal with the im-pulse response of the channel, where the impulse response iscomputed using the numerical integration approach, presentedin [17]. In addition, the spatial distribution of the transmittedbeam is assumed to be a Gaussian function.

Fig. 7 compares the symbol error probability of differentdetectors using simulation results. These results are for caseA in Table I and for F = 23. For these system parameters,a0 = 0.78 and a1 = a−1 = 0.11. The simulation result forthe optimum detector neglecting spatial broadening, in (17),is also plotted. According to these results, for this code theoptimum single detector using PMT2 has a better performancecompared to the same detector using PMT1. This means thatfor the optimum single detector, the lowest error probabilityis achieved when the decoder mask matches the complementsof the codewords. OOK using a single-PMT results in anerror rate between the two. The difference between the errorrates of the optimum single-PMT2 detector and the optimumdual-PMT detector shows the performance price that is paidfor having one PMT instead of two. The performance ofthe optimum linear dual-PMT detector is close to that ofthe optimum dual-PMT detector, and since it has a lowercomplexity, we prefer to use this detector in the receiver.Henceforth all results assume this detector, and assume thatsimilar conclusions hold if another detector is chosen.

Fig. 8 shows the Pb versus the code length, F , for thethree physical geometries in Table I. The analytical resultsof the upper bounds are compared with simulation results.The bit-rate is assumed to be fixed for all code lengths(Rb = 1 Mb/s). For small F ’s, ISI is the main limitingfactor on system performance. For a fixed bit rate, however,by increasing the number of symbols (F ), the symbol ratedecreases, and, consequently, the ISI effect is reduced. On


Fig. 7. Simulated symbol error probability, Ps, versus the received power forOOK and M -ary SAC using an optimum dual PMT detector, optimum singlePMT1 detector, optimum single PMT2 detector, optimum linear detector andoptimum detector neglecting spatial broadening for case A and F=23.

Fig. 8. Pb of the optimum linear dual-PMT detector versus the code length,F , for cases A, B and C defined in Table I.

the other hand, for large F ’s the number of time slots isgreater, and, therefore, the optical energy in each time slotis reduced, thus, SNR limits the Pb. In addition, accordingto Section IV-A, the spatial broadening effect increases byincreasing F and becomes another source of degradation forlonger code lengths. Therefore, there is an optimum value forF considering these three factors.

For low transmitter and receiver inclination angles theoptimum modulation is OOK, since the ISI is low for thesegeometries. In Fig. 8, for cases A and B, defined in Table I,the ISI limits the Pb and the optimum values of F are 15 and19, respectively. In case C, since the elevation angle of boththe receiver and the transmitter is small, the impulse responseis shorter than the former cases and the ISI effect is smaller.Consequently, the optimum value of F is 7. Also, according tothese results the union bound is notably close to the simulationresults and can be used as a good approximation for the Pb.

The maximum achievable bit rate to sustain a Pb of at most3× 10−5 is plotted in Fig. 9 versus the distance between thetransmitter and receiver. The parameters of case B in TableI are assumed to obtain these results. For this Pb and short

Fig. 9. Maximum attainable bit-rate versus distance for F=7 and comparisonto the ISI and SNR limited bit-rates, assuming case B for a Pb = 3× 10−5.

distances the bit rate is limited by ISI for every constellationsize. By increasing the distance, since the loss increases, theperformance becomes limited by the SNR. For small distances,larger F s are able to achieve higher data rates compared tosmaller F ’s. But, as the distance increases, the SNR affects thePb, and the optimum F decreases. So for long ranges, smallF ’s or OOK become better choices for data transmission.

The tradeoff between SNR and ISI is also illustrated inFig. 9, where we use two bounds to approximate the maximumachievable bit rate: the ISI-limited bit rate, and the SNR-limited bit rate. At short ranges the ISI is the main limitingfactor for the maximum bit rate, and the bit rate matches theISI-limited bound. By increasing the range, the SNR limitbecomes smaller than the ISI limit and bounds the maximumbit rate. The ISI limit can be obtained by assuming the ISIis the only limiting factor on the bit rate. In this case the Pb

depends only on the ISI coefficients; to have a fixed Pb itis essential to fix these coefficients. As seen in (9), the ISIcoefficients depend on the impulse response and symbol time,Ts. By increasing the distance, D, the time scale of the impulseresponse is widened proportionally. Hence, Ts is increasedproportionally in order to keep the ISI coefficients constantas well. So the ISI-limited symbol rate and, accordingly, theISI-limited bit rate decreases inversely proportional to D.

In the SNR limit, Pb depends only on the received energyper symbol. According to [4] the received energy per symbolin NLOS UV links can be approximated as

Es(D) =αTs

De−βD, (28)

where α and β depend on the system and geometric pa-rameters. Therefore, for longer ranges, Ts needs to increaseexponentially as D increases for a fixed Es(D).

The maximum distance-rate product is presented in Fig. 10in terms of the number of symbols, F , for Pb = 3×10−5. Theoptimum code length is larger for shorter distances and thismaximum distance-rate product increases by decreasing thelength of the link. Although for OOK and F = 7 this productis similar over a range of distances, for larger code lengthsit varies considerably. This is because for smaller symbolsizes, the distance-rate product stays close to the ISI limiteven for long ranges, while for a larger number of symbols


Fig. 10. Maximum attainable distance bit-rate product for a Pb = 3×10−5

versus the number of symbols, F , for case B and for D = 100 m, D = 400m and D = 1 km, with and without spatial dispersion.

it separates from the ISI limit at short distances and thendecreases exponentially. Using our SAC system increases thedistance-rate product by a factor varying from 3 to 5 comparedto OOK, depending on the transmission distance.

In Fig. 10 the maximum distance-rate product for theoptimum linear dual-PMT detector is also plotted neglectingthe spatial broadening. Neglecting the spatial broadening de-creases the second term inside the erfc(·) function in (25).Thus, for a fixed Pb, we need a smaller γ0, which is pro-portional to Es(D). From (28), for a fixed D, decreasingEs(D) decreases Ts, i.e., increases the data-rate. In otherwords, by neglecting the spatial broadening we can achievehigher data-rates. According to these results, neglecting thespatial broadening affects the maximum distance rate productfor F ’s larger than 30, while it has little effect for smaller F ’s.

VII. CONCLUSION

In this paper a M -ary transmission technique using spectralencoding for application in NLOS UV systems is proposed. Atransmitter structure using encoding in the visible/IR domainand wavelength conversion from visible/IR to UV, and areceiver structure using two PMTs are presented. Cyclic codefamilies are used for encoding the data to simplify transmitterand receiver structures. Several symbol detectors differing inperformance and complexity are proposed. An upper boundfor evaluating the performance of the system is derived andcompared with simulation results. The upper bound closelyapproximates the bit error probability for various geometries.The maximum achievable data rate for different distancesand symbol sizes is calculated, and the data rate is shownto be bounded by ISI and SNR limits. Larger constellationsizes reach higher data rates at short ranges, compared tosmaller code lengths. In conclusion, the proposed techniquehas a better performance compared to OOK for practicallink distances and geometries. OOK performs better only fordistances longer than 2 km, which results in data rates ofaround a few kb/s.

Performance comparisons between systems utilizing thesuggested technique and systems using an equalizer or errorcorrecting code will be the subject of future work.

VIII. ACKNOWLEDGMENT

This research was funded by the National Science Founda-tion (NSF) under grant number ECCS-0901682.

APPENDIX A

In this appendix, in order to find the coefficients of theoptimal linear detector in (23), we obtain the optimum valuesfor y1, y2, . . . , yn to maximize

Hn =

[α1y1 + α2y2 + · · ·+ αnyn

]2β1y2

1 + β2y22 + · · ·+ βny2

n

. (A.1)

Using mathematical induction, we prove that the maximum ofHn in (A.1) is given by

maxy1,...,yn

Hn =

n∑i=1

α2i

βi,

and the solution is yi =αiβ1

βiα1y1, i = 1, 2, . . . , n, where y1 can

be any value. For the base step of the induction, we verify thisstatement for n = 2. For H2 = (α1y1+α2y2)

2/(β1y21+β2y2

2),the maximum is α2

1

β1+

α22

β2, and it is obtained for y2 = α2β1

β2α1y1.

Our inductive assumption is that the statement is true forn = m − 1, and then we prove it is also true for n = m.In order to maximize Hm, the partial derivative of Hm withrespect to ym should be zero, i.e., ∂Hm

∂ym= 0. The solution of

this equation is

ym =α1

β1

(β1y21 + β2y2

2 + · · ·+ βm−1y2m−1)

(α1y1 + α2y2 + · · ·+ αm−1ym−1), (A.2)

for which Hm becomes

Hm =α2

m

βm+ Hm−1, (A.3)

where

Hm−1 =

[α1y1 + α2y2 + · · ·+ αm−1ym−1

]2β1y2

1 + β2y22 + · · ·+ βm−1y2

m−1

.

By maximizing Hm−1 we maximize Hm in (A.3). Accordingto the inductive assumption, the maximum of Hm−1 is

maxy1,...,ym−1

Hm−1 =

m−1∑i=1

α2i

βi,

which is obtained for yi =αiβ1

βiα1y1, i = 1, 2, . . . , m − 1. Sub-

stituting these values in (A.2) and (A.3), we get ym = αmβ1

βmα1y1

and

maxy1,...,ym

Hm =

m∑i=1

α2i

βi,

and this completes the inductive step. Using these results, (25)follows from (24).


REFERENCES

[1] Z. Xu and B. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag., vol. 46, no. 5, 2008.

[2] G. Shaw, A. Seigel, and J. Model, “Ultraviolet comm links for dis-tributed sensor systems,” IEEE LEOS Newsletter, vol. 19, no. 5, 2005.

[3] H. Ding, G. Chen, A. Majumdar, B. Sadler, and Z. Xu, “Modelingof non-line-of-sight ultraviolet scattering channels for communication,”IEEE J. Sel. Commun., vol. 27, no. 9, pp. 1535–1544, 2009.

[4] Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performancestudy of solar blind non-line-of-sight ultraviolet short-range communi-cation links,” Opt. Lett., vol. 33, no. 16, pp. 1860–1862, 2008.

[5] M. Noshad and M. Brandt-Pearce, “NLOS UV communication systemsusing spectral amplitude coding,” in 2011 IEEE Workshop OpticalWireless Commun., GLOBECOM.

[6] ——, “Expurgated PPM using balanced incomplete block designs,”IEEE Commun. Lett., vol. 16, no. 7, pp. 968–971, 2012.

[7] G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultra-violet pulse broadening in short-range non-line-of-sight communicationchannels,” Opt. Exp., vol. 18, no. 10, pp. 10 500–10 509, 2010.

[8] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs,2nd edition. Chapman and Hall-CRC, 2007.

[9] H. Kumagai, “Fine frequency tuning in sum-frequency generation ofcontinuous-wave single-frequency coherent light at 252 nm with dual-wavelength enhancement,” Optics Lett., vol. 32, no. 1, pp. 62–64, 2007.

[10] A. Kokh, T. Bekker, K. Kokh, N. Kononova, V. Vlezko, P. Villeval,S. Durst, and D. Lupinski, “Nonlinear LBO and BBO crystals for ex-treme light sources,” in Proc. 2010 Int. Conf. Advanced OptoelectronicsLasers, pp. 127–129.

[11] M. Ebrahim-Zadeh, “Efficient ultrafast frequency conversion sourcesfor the visible and ultraviolet based on BiB3O6,” IEEE J. Sel. TopicsQuantum Elect., vol. 13, no. 3, pp. 679–691, 2007.

[12] H. Wang and A. M. Weiner, “A femtosecond waveform transfer tech-nique using type II second harmonic generation,” IEEE J. QuantumElectron., vol. 40, no. 7, pp. 937–945, 2004.

[13] P. R. Prucnal, Optical Code Division Multiple Access: Fundamentalsand Applications. CRC Press, 2006.

[14] M. Noshad and K. Jamshidi, “Bounds for the BER of codes with fixedcross correlation in SAC-OCDMA systems,” J. Lightw. Tech., vol. 29,no. 13, pp. 1944–1950, July 2011.

[15] L. Wang, Z. Xu, and B. Sadler, “An approximate closed-form link lossmodel for non-line-of-sight ultraviolet communication in noncoplanargeometry,” Optics Lett., vol. 36, no. 7, pp. 1224–1226, 2011.

[16] R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channelmodeling incorporating multiple scattering interactions,” J. Opt. Soc.Am. A, vol. 28, no. 4, pp. 686–695, Apr. 2011.

[17] A. Gupta, M. Noshad, and M. Brandt-Pearce, “NLOS UV channelmodeling using numerical integration and an approximate closed-formpath loss model,” in Proc. 2012 SPIE Conf. Free-Space AtmosphericLaser Commun.

[18] G. Chen, F. Abou-Galala, Z. Xu, and B. M. Sadler, “Experimentalevaluation of LED-based solar blind NLOS communication links,” Opt.Exp., vol. 16, no. 19, pp. 15 059–15 068, 2008.

[19] D. A. Guimaraes, Digital Transmission: A Simulation-Aided Introduc-tion with VisSim/Comm. Springer, 2009.

[20] M. K. Simon and V. A. Vilnrotter, “Performance analysis and tradeoffsfor dual-pulse PPM on optical communication channels with directdetection,” IEEE Trans. Commun., vol. 52, no. 11, pp. 1969–1979, Nov.2004.

Mohammad Noshad (S’07) received the B.Sc.degree from University of Tabriz, Tabriz, Iran, in2007, and the M.Sc. degree from Sharif Universityof Technology (SUT), Tehran, Iran, both in electri-cal engineering. Since January 2011, he has beenworking towards the Ph.D. degree in the Charles L.Brown Department of Electrical and Computer En-gineering at University of Virginia, Charlottesville,VA. From May 2010 to December 2010 he was aresearcher in i2cat foundation in Barcelona, Spain.

M. Noshad received “Louis T. Rader GraduateResearch Award” from Electrical and Computer Engineering Department,University of Virginia, in 2012. He is also the recipient of the 2012 “CharlesL. Brown Fellowship for Excellence”, as well as the “TRANE GraduateFellowship” and “University of Virginia Engineering Foundation Fellowship”.He received the “Best Paper Award” at the IEEE Globecome 2012. Hisresearch interests include free-space optical communications, optical CDMAnetworks, coding and modulation, and network coding.

Maıte Brandt-Pearce (SM’99) received her B.S. inElectrical Engineering from Rice University in 1985.She completed an M.E.E. in 1989 and a Ph.D. inElectrical Engineering in 1993, both also from RiceUniversity. Dr. Brandt-Pearce is currently a profes-sor in the Charles L. Brown Department of Electricaland Computer Engineering at the University of Vir-ginia. Her research interests lie in the mathematicaland numerical description and optimization of sys-tems with multiple simultaneous components fromdifferent sources and corrupted by non-Gaussian

noise. This interest has found applications in a variety of research projectsincluding spread-spectrum multiple-access schemes, multiuser demodulationand detection, study of nonlinear effects on fiber-optic multiuser/multichannelcommunications, optical networks subject to physical layer degradations, free-space optical multiuser communications, biomedical data processing, andradar signal processing and tracking of multiple targets. Dr. Brandt-Pearcehas over 100 major journal and conference publications.

Stephen G. Wilson received degrees from IowaState University, University of Michigan, and Uni-versity of Washington, all in electrical engineering.He held a staff engineering position at Boeing(Seattle), before joining the faculty at the Universityof Virginia, where he has been since 1976. He iscurrently Professor of Electrical and Computer Engi-neering, and conducts research in digital modulationand coding, signal processing for communication,and applications of information theory. He is au-thor of Digital Modulation and Coding (Pearson

Prentice-Hall), and was associate editor for Coding Theory and Applications,IEEE TRANSACTIONS ON COMMUNICATIONS. Prof. Wilson has been recog-nized for outstanding teaching with the University of Virginia’s DistinguishedProfessor Award, and the State Council of Education in Virginia’s OutstandingFaculty Award.

1544 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO....

Documents

Transcript of 1544 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO....