Achievable Rate Region of the Zero-Forcing Precoder in a 2 X 2 MU-MISO Broadcast VLC Channel with Per-LED Peak Power Constraint and Dimming Control

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by  Amit Agarwal

1
Achievable Rate Region of the Zero-Forcing
Precoder in a 2 × 2 MU-MISO Broadcast VLC
Channel with Per-LED Peak Power Constraint and
Dimming Control
Amit Agarwal and Saif Khan Mohammed
7 Abstract—In this paper, we consider the 2 × 2 multi-user for the recovery of the information signal [1]. Contrary to RF
1 multiple-input-single-output (MU-MISO) broadcast visible light systems, in VLC systems the modulation symbols must be
0 communication (VLC) channel with two light emitting diodes
non-negative and real valued as information is communicated
2 (LEDs) at the transmitter and a single photo diode (PD) at
each of the two users. We propose an achievable rate region by modulating the power/intensity of the light emitted by
n
oftheZero-Forcing(ZF)precoderinthis2×2MU-MISOVLC the optical source (LED). The modulation symbols are also
a
J channel under a per-LED peak and average power constraint, constrained to be less than a pre-determined value as the
where the average optical power emitted from each LED is intensity of the light emitted by the LED is peak constrained
2
fixed for constant lighting, but is controllable (referred to as
1 due to safety regulations and also due to the limited linear
dimming control in IEEE 802.15.7 standard on VLC). We
rangeofthetransferfunctionofLEDs[1],[3].Moreover,due
analyticallycharacterizetheproposedrateregionboundaryand
T] show that it is Pareto-optimal. Further analysis reveals that to constant lighting the mean value of the modulation symbol
the largest rate region is achieved when the fixed per-LED is also fixed (i.e., non-time varying) and can be adjusted
I
. average optical power is half of the allowed per-LED peak according to the users’ requirement (dimming target) [4], [5].
s
optical power. We also propose a novel transceiver architecture
c Duetotheseconstraints,analysisperformedforRFsystems
[ where the channel encoder and dimming control are separated is not directly applicable to VLC systems. For example, the
whichgreatlysimplifiesthecomplexityofthetransceiver.Acase

1 study of an indoor VLC channel with the proposed transceiver capacity of the RF single-input-single-output (SISO) additive
v reveals that the achievable information rates are sensitive to the white Gaussian noise (AWGN) channel is well known and it
3 placement of the LEDs and the PDs. An interesting observation hasbeenshownthattheGaussianinputdistributioniscapacity
5 is that for a given placement of LEDs in a 5 m × 5 m ×
achieving. For the case of the optical wireless AWGN SISO
2 3 m room, even with a substantial displacement of the users
channel with IM/DD transceiver, closed form expression for
3 fromtheiroptimumplacement,reductionintheachievablerates
0 is not significant. This observation could therefore be used to the capacity is still not known, though several inner and outer
. define “coverage zones” within a room where the reduction in bounds have been proposed [6]–[8]. However, it has been
1
the information rates to the two users is within an acceptable shown that the capacity achieving input distribution for the
0
tolerance limit.
7 IM/DD SISO AWGN optical wireless channel is discrete [9],
1 Index Terms—Visible light communication, rate region, zero- and has been computed numerically in [10]. Similarly, for
v: forcing, multi-user, multiple-input-multiple-output. the case of dimmable VLC IM/DD SISO channel with peak
i constraint,thereisnoclosedfromexpressionforthecapacity.
X I. INTRODUCTION However following a similar approach as in [6], an upper and
r Visible light communication (VLC) is a form of optical lower bound is presented in [11].
a
wirelesscommunication(OWC)technologywhichcanprovide Recently, there has been a lot of interest in multi-
high speed indoor wireless data transmission using existing user multiple-input multiple-output/single-output (MU-
infrastructure for lighting. One distinctive advantage of VLC MIMO/MISO) VLC systems, where multiple LEDs are used
technology is that it utilizes the unused visible band of the for information transmission to multiple non-cooperative PDs
electromagnetic spectrum and does not interfere with the (users) [12], [1]. Such systems have been shown to enhance
existing radio frequency (RF) communication in the UHF the system sum rate when compared to SISO VLC systems
(Ultra High Frequency) band [1], [2]. [13], [14] .
In VLC systems, it is common to use intensity modulation In [13], the information sum rate of MU-MIMO VLC
(IM)vialightemittingdiode(LEDs)fortransmissionofinfor- broadcast systems has been studied under the non-negativity
mationsignalanddirectdetection(DD)viaphotodiodes(PDs) constraint on the signal transmitted from each LED, and
also a per-LED average transmitted power constraint with
The authors are with the Department of Electrical Engineering, Indian
no dimming control. The block diagonalization precoder in
InstituteofTechnologyDelhi(IITD),Newdelhi,India.SaifKhanMohammed
isalsoassociatedwithBhartiSchoolofTelecommunicationTechnologyand [13] is used to suppress the multi-user interference and the
Management(BSTTM),IITDelhi.Email:[email protected] numerically computed achievable sum rate is shown to be
work is supported by the Visvesvaraya Young Faculty Research Fellowship
sensitive to the placement of the users and the rotation of the
(YFRF)oftheMinistryofElectronicsandInformationTechnology,Govt.of
India. PDs. However, they do not consider peak power constraints
2
which is important due to eye safety regulations and also dimming control.
due to the requirement of limited interference to other VLC 2) In Section III, We also mathematically define the pro-
systems. posed rate region of the ZF precoder for a fix dimming
Per-LED peak and average power constraint has been con- target.
sidered in [14], where the sum-rate of the zero forcing (ZF) 3) In Section IV, we analytically characterize the bound-
precoder is maximized in a IM/DD based MU-MIMO/MISO ary of the proposed rate region by deriving explicit
VLC systems. However, in many practical scenarios fairness expressions for the largest possible length along the
is required and therefore maximizing the sum rate might not u -axis of some rectangle inside R whose midpoint
2 //
alwaysbethedesiredoperatingregime.Forexamplewewould coincides with the fixed point D on the diagonal of R
//
liketofindthemaximumpossibleratesuchthateachusergets and whose length along the u -axis is given. Through
1
the same rate. Such operating points can only be obtained analysis we also show that the rate region boundary is
from the rate region characterization of the MU-MIMO VLC Pareto-optimal.
systems.In[15],authorshaveproposedinnerandouterbounds 4) We also analyze the variation in the rate region with
on the capacity region of a two user IM/DD broadcast VLC change in the dimming level. In depth analysis reveals
system where the transmitter has a single LED and each user that the largest rate region is achieved when the fixed
hasasinglePD.Per-LEDaverageandpeakpowerconstraints point D lies at the midpoint of the diagonal of R , i.e.,
//
are considered. The authors have extended their work to more when the fixed per-LED average optical transmit power
than two users in [16]. However, in both [15] and [16], the is half of the per-LED peak optical power.
transmitter has only one LED. Furthermore, dimming control 5) Forpracticalscenarioswithfairnessconstraints,through
is not considered in [13]–[16]. analysis we show that the largest achievable rate pair
Thecapacity/achievablerateregionofaIM/DDbasedVLC (R1,R2) such that R2 = αR1 is given by the unique
broadcast channel where the transmitter has N > 1 LEDs intersection of the proposed rate region boundary with
and M > 1 users having one PD each, is still an open and the straight line R2 =αR1.
challengingproblem,primarilyduetothenon-negativity,peak 6) In Section V, from the point of view of practical
and average constraints on the electrical signal input to each implementation we also propose a novel transceiver
LED. architecturewherethesamechannelencodercanbeused
Inthispaper,weconsiderthesmallestinstanceofthisopen irrespective of the level of dimming control.
problem along with dimming control, i.e., with N =2 LEDs 7) Analytical results have been supported with numeri-
at the transmitter and M = 2 users (each having one PD). cal simulations in Section VI. It is observed that for
Dimming control is required in indoor VLC systems since a fixed placement of the two LEDs, the achievable
the illumination should not vary with time on its own and information rates are a function of the placement of
should be controllable by the users. Therefore, in this paper, the two PDs/users. Specifically, we observe that for
in addition to the peak and non-negativity constraints, we a given placement of the two LEDs, there exists an
constrain the average optical power radiated by each LED to optimal placement of the two users which maximizes
befixed,i.e.,non-timevarying.Subsequentlyinthispaperwe the symmetric rate. Another interesting observation is
refer to this system as the 2×2 MU-MISO VLC broadcast that in a 5 m × 5 m × 3 m (height) room with the two
system. LEDs attached to the ceiling and the two PDs placed in
The major contributions of this paper are as follows: thehorizontalplaneataheightof50cmabovethefloor,
even a user displacement of 60 cm from the optimal
1) In Section III, we propose an achievable rate region for
placementresultsinonlyapprox.a10percentreduction
the 2×2 MU-MISO VLC broadcast system with the
in the symmetric rate when compared to the symmetric
ZF precoder. In this section through analysis we show
rate with the optimal placement of PDs2. This allows
that the per-LED non-negativity and peak constraint
for substantial mobility of the user terminals around
restrictstheinformationsymbolvectorforthetwousers
their optimal placement which is specially desirable
(i.e., (u ,u )) to lie within a parallelogram R . Each
1 2 //
whentheuserterminalsaremobile/portable.Apractical
achievable rate pair (R ,R ) then corresponds to a
1 2
application of the results derived in this paper could
rectangle which lies within R . The rate R ,i = 1,2
// i
be in defining coverage zones for the PDs/users, i.e.,
to the ith user depends on the length of the rectangle
the maximum allowable displacement of the users for a
along the u -axis. Due to the same average optical
i
fixed desired upper limit on the percentage loss in the
power constraint at each LED, these rectangles should
achievable information rates.
also have their midpoint (i.e., point of intersection of
the diagonals of the rectangle) at a fixed point on the
II. SYSTEMMODEL
diagonal of R denoted by D.1 This fixed point D on
//
the diagonal of R is non-time varying, but can be We consider a 2 × 2 IM/DD MU-MISO VLC broadcast
//
controlled by the user depending upon the illumination system. The transmitter of the MISO system is equipped
requirement.Thisfeatureoftheproposedsystemenables with two LEDs and each user has a single photo-diode
1OutofthetwodiagonalsofR//,werefertotheonewhichhasoneend 2Forthisstudythedimmingcontrolissuchthattheaverageopticalpower
pointattheorigin(u1,u2)=(0,0). radiatedfromeachLEDis30percentofthepeakallowedopticalpower
3
Fig. 1: 2×2 IM/DD MU-MISO VLC broadcast system. Fig. 2: The information vector u is constrained to lie within
the parallelogram R (H) whose non-parallel sides are h
// 1
and h . The rectangular region U ×U whose length along
2 1 2
the u axis is L and that along the u axis is L and whose
1 1 2 2
(PD)(seeFig.1).3 TheLEDconvertstheinformationcarrying center lies at D(H,ξ) is denoted by Rect(L1,L2,D(H,ξ)).
electrical signal to an intensity modulated optical signal and
the PD at each user converts the received optical signal to
electricalsignal.Thetransmitterperformsbeamformingofthe normalized received signal vector is given by4
information symbols towards the two non-cooperative users.
(cid:20)y (cid:21) (cid:20)h h (cid:21)(cid:20)x(cid:48)(cid:21) (cid:20)n (cid:21)
Letu1 ∈U1 andu2 ∈U2 betheinformationsymbolsintended y1 = h11 h12 x1(cid:48) + n1 ,
for the first and second user respectively, where U1 and U 2 21 22 2 2
2 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124) (cid:123)(cid:122) (cid:125)(cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125)
are the information symbol alphabets for user 1 and user 2 (cid:44)y (cid:44)H (cid:44)x(cid:48) (cid:44)n
respectively. Let xi be the optical power transmitted from the s.t. 0≤x(cid:48) ≤1, E[x(cid:48)]=ξ, i=1,2. (5)
ithLED(i=1,2).Atanytimeinstance,thetransmittedoptical i i
power vector x(cid:44)[x1 x2]T is given by where H (cid:44)[hki]2×2 is the channel gain matrix. The channel
gain coefficients between the ith LED and the kth user is
x=Au, (1) denoted by h ,i = 1,2,k = 1,2.5 We further define h (cid:44)
ki 1
where u(cid:44)[u1 u2]T and A∈R2×2 is the beamforming ma- [h11 h21]T and h2 (cid:44) [h12 h22]T to be the channel vectors
trix.Inthispaper,weconsiderthefollowingpowerconstraints from LED 1 and LED 2 respectively. Further, n1 and n2 are
for our dimmable VLC system. the sum of the thermal noise and ambient light-induced shot
noise at the respective users6 and are independent of x(cid:48) and
TheinstantaneouspowertransmittedfromeachLEDisnon 1
x(cid:48) [12]. The noise signals are i.i.d. zero mean real AWGN
negative and is less than some maximum limit P due to 2
0 with variance σ2/P2, where σ2 is the variance of the noise
skin and eye safety regulations [3]. Further, such a maximum 0
before the scaling down of the received signal by P , i.e.,
0
limit on the transmitted power is required also due to limited
n∼N(0,(σ/P )2).
0
interference requirement to the neighboring VLC systems, i.e.
0≤x ≤P , i=1,2. (2) III.ANACHIEVABLERATEREGIONOFTHECHANNELIN(5)
i 0
Since our VLC system is dimmable we further impose a In this section, we derive an achievable rate region for the
per-LED average power constraint of the type channelin(5)usingtheZFprecoder.Forthe2×2MU-MISO
system discussed in section II, the ZF precoding matrix is
E[x ]=ξP , i=1,2, (3)
i 0 uniquely givenby A=P H−1, i.e., x(cid:48) =x/P =Au/P =
0 0 0
where 0 ≤ ξ ≤ 1 is the dimming target [5]. For the sake of P0H−1u/P0 = H−1u. Thus the received signal vector is
analysis,wedefinex(cid:48) (cid:44) xi, i=1,2asthenormalizedpower given by
i P0
transmitted from each LED. Consequently, the normalized y =Hx(cid:48)+n=HH−1u+n=u+n. (6)
optical power transmitted from each LED must satisfy the
following constraints given by i.e., there is no multi-user interference (MUI). Since
0≤x(cid:48) ≤1 & E[x(cid:48)]=ξ, i=1,2. (4) u=Hx(cid:48) =[h h ][x(cid:48) x(cid:48)]T, (7)
i i 1 2 1 2
Assuming y , k = 1,2 to be the normalized received elec-
k
trical signal at the kth user (after scaling down by P0), the 4Insubsequentdiscussions,by“receivedelectricalsignal”,werefertothe
“normalizedreceivedelectricalsignal”.
5Notethathki’sarenonnegativeandmodeltheoverallgainsoftheline
ofsight(LOS)opticalpathbetweentheith LEDandthekth userandalso
theresponsivityofthePDofthekth user[3].
3Since each of the two users has a single PD we will be interchangeably 6Notethattheabovenoiseimpairmentsofthereceivedsignalarethemain
usinguserandPDinsubsequentdiscussions. impairmentsthatarecommonlyassumedinVLCsystems[3].
4
and 0 ≤ x(cid:48) ≤ 1,i = 1,2 (see (4)) it follows that, the correspondtoanachievableratepairforthebroadcastchannel
i
information signal vector u must be limited to the region in (5). Since a rectangle in the u − u plane corresponds
1 2
to a unique U and U and vice versa, it follows that any
R (H)(cid:44){u | u=Hx(cid:48), 0≤x(cid:48) ≤1,0≤x(cid:48) ≤1}. (8) 1 2
// 1 2 rectangle lying inside the parallelogram R (H) and having
//
The region R (H) is a parallelogram with its two non its midpoint at D(H,ξ) will correspond to an achievable
//
parallelsidesash andh (seeR (H)inFig.2).Inaddition rate pair. In this paper, for the broadcast channel in (5), we
1 2 //
tothis,thediagonaloftheparallelogramR (H)isthevector therefore propose an achievable rate region which consists of
//
h +h as shown in Fig. 2. ratepairscorrespondingtosuchrectangles(onesuchrectangle
1 2
Let E[u] (cid:44) [E[u ] E[u ]]T be the mean information is shown in Fig. 2). We define our proposed rate region more
1 2
symbolvector.From(7)and(4),themeaninformationsymbol precisely in the following. Towards this end, we first formally
vector is given by define the achievable rate of a SISO AWGN optical channel,
wherethetransmittedinformationsymbolisconstrainedtolie
E[u]=HE[x(cid:48)]=[h h ]E[x(cid:48) x(cid:48)]T (=a)ξ(h +h ), (9) in an interval.
1 2 1 2 1 2
wherestep(a)followsfrom(5).Therefore,themeaninforma- Result 1. [From [6], [10]] The achievable information rate
tion symbol pair (E[u ], E[u ]) is a point corresponding to of a SISO channel y = u+n (where u ∼ Unif[a,b] and
1 2
the tip of the vector E[u]=ξ(h +h ). From (8) it is clear n∼N(0,(σ/P )2)dependsontheinterval[a,b]onlythrough
1 2 0
that the vector (h +h ) is a diagonal of R (H) (see Fig. its length L=|b−a|, and is given by the function
1 2 //
2). For a given 0 ≤ ξ ≤ 1, the tip of the mean information
C(L=|b−a|,P /σ)(cid:44)I(u;y), (13)
symbol vector ξ(h +h ) is therefore a fixed point on the 0
1 2
diagonal (h1+h2). We denote this point by hereUnif[a,b]denotetheuniformdistributionintheinterval
[a,b] and I(u;y) is the mutual information between u and y.
D(H,ξ)=(E[u ],E[u ])
1 2
=(ξ(h11+h12), ξ(h21+h22)). (10) Result 2. [From [6], [10]] The function C(L,P0/σ) is
continuouswithrespecttoLandincreasesmonotonicallywith
With the ZF precoder, the broadcast channel in (5) is reduced increasing L for a fixed P /σ.
0
totwoparallelSISO(single-inputsingle-output)opticalchan-
nelsbetweenthetransmitterandthetwousers(see(6)).Since Let Rect(L1,L2,D(H,ξ)) denote the unique rectangle
u ∈ U and u ∈ U are independent and originate from having its midpoint as D(H,ξ) and whose length along the
1 1 2 2
different codebooks, it follows that (u1,u2)∈U1×U2. From u1 axis is L1 and that along the u2 axis is L2 (see Fig. 2).
(8), we know that (u1,u2) must belong to the parallelogram Any such rectangle Rect(L1,L2,D(H,ξ)) ⊂ R//(H) will
R (H) and therefore correspond to an achievable rate pair given by
//
U1×U2 ⊂R//(H). (11) (R1,R2)(cid:44)(cid:0)C(L1,P0/σ),C(L2,P0/σ)(cid:1) (14)
For a given (H,P /σ,ξ) the proposed achievable rate region
In general we choose U and U to be intervals of the type 0
1 2
for the ZF precoder is given by
[a,b] [9]. Let the length of the intervals U and U be L and
1 2 1
L respectively, i.e. |U | = L , |U | = L . With U and U
2 1 1 2 2 1 2
asintervals,itisclearthatU ×U mustbearectanglewhose R (H,P /σ,ξ)(cid:44) ∪{C(L ,P /σ),C(L ,P /σ)}, (15)
1 2 ZF 0 1 0 2 0
lengthalongtheu1 axisisL1 andthatalongtheu2 axisisL2. (L1,L2)∈(L1,L2)∈S
Inthispaperweassumeu1 andu2,tobeuniformlydistributed where S (cid:44) {(L1 ≥ 0,L2 ≥ 0)| ∃ Rect(L1,L2,D(H,ξ))
in the interval U1 and U2 respectively.7 Therefore, for a given ⊂R//(H)}.
U and U , the mean information symbol pair (E[u ],E[u ])
1 2 1 2
will lie at the point of intersection of the two diagonal of the
IV. CHARACTERIZINGTHEBOUNDARYOFTHERATE
rectangle U × U . We will subsequently call this point of
intersection1as the2 “midpoint” of the rectangle U ×U and REGIONRZF(H,P0/σ,ξ)
1 2
will denote it by C(U ,U ). In this section, we completely characterize the boundary of
1 2
From(10),itfollowsthatthemeaninformationsymbolpair the rate region, RZF(H,P0/σ,ξ), for a fixed (H,P0/σ,ξ).
must exactly coincide with D(H,ξ), i.e. Towards this end, for each information rate R1 achievable by
the first user, we find the corresponding maximum possible
C(U ,U )=D(H,ξ) (12)
1 2 information rate R achievable by the second user. Each pair
2
ofR anditscorrespondingmaximumpossibleR istherefore
The ZF precoder transforms the broadcast channel into two 1 2
a point on the boundary of the proposed rate region. By
parallel SISO channels y = u +n ,u ∈ U ,i = 1,2. Let
i i i i i
increasing R from 0 to its maximum possible value, all such
R and R denote the information rates achieved on these 1
1 2
(R ,R ) pairs characterize the boundary of the rate region.
SISO channels with u distributed uniformly in U . Any given 1 2
i i
From (15), we know that any achievable rate pair (R ,R )
U and U satisfying the conditions in (11) and (12) would 1 2
1 2
in the proposed rate region R (H,P /σ,ξ) corresponds
satisfytheopticalpowerconstraintsin(4)andwouldtherefore ZF 0
to some rectangle Rect(L ,L ,D(H,ξ)). The rate to the
1 2
ith user, i.e. R = C(L ,P /σ),i = 1,2 depends only
7AthighSNR(P0/σ>>1),uniformlydistributedinformationsymbolis i i 0
nearcapacityachieving[10]. on the length of this rectangle along the ui-axis. Since the
5
C(L,P /σ) function is monotonic and continuous in its first Rect(L ,L ,D(H,ξ))(L ≥0)whichliescompletelyinside
0 1 2 2
argument, each value of R corresponds to a unique L the parallelogram R (H), is given by
i i //
and vice versa. Therefore, towards characterizing the bound-
Lmax(ξ)(cid:44) max L
ary of R (H,P /σ,ξ), we note that for a given R ,i.e., 1 1
ZF 0 1 L1≥0,L2≥0
for a given length L1 along the u1-axis, we would like Rect(L1,L2,D(H,ξ))⊂R//(H)

to find the largest possible R2,i.e., the largest possible L2 −2ξdet(H) , 0≤ξ ≤1/2
such that the rectangle Rect(L1,L2,D(H,ξ)) lies entirely = max(h21,h22) (21)
inside R (H). Hence, we can characterize the boundary −2(1−ξ)det(H), 1/2≤ξ ≤1.
// max(h21,h22)
of R (H,P /σ,ξ) simply by varying L = x from 0 to
ZF 0 1
its maximum possible value (denoted by Lm1ax(ξ)), and for Proof: See Appendix A.
each value of L = x ∈ [0,Lmax(ξ)] we find the largest
1 1
possibleL2 =Lξ2(x)whichgivesusacorrespondingratepair It is clear from (21) in Proposition 1 that Lm1ax(ξ) is a
(R1,R2) = (C(L1 = x,P0/σ),C(L2 = Lξ2(x),P0/σ)) on continuous function of ξ and Lm1ax(ξ)=Lm1ax(1−ξ).
the boundary of the rate region R (H,P /σ,ξ).
ZF 0
For a given (L = x,L = Lξ(x)) the corresponding Remark 1. The function Lmax(ξ) is a continuous function of
1 2 2 1
information rate pair lies on the boundary of the proposed ξ and is symmetric about ξ =1/2, i.e.
rate region R (H,P /σ,ξ). We denote this information rate
ZF 0 Lmax(ξ)=Lmax(1−ξ), 0≤ξ ≤1 (22)
pair by (RBd(x,P /σ,ξ),RBd(x,P /σ,ξ)). From (14), this 1 1
1 0 2 0
information rate pair is given by
From (21) it is clear that since det(H) < 0 (see (20))
R1Bd(x,P0/σ,ξ)(cid:44)C(L=x,P0/σ). (16) Lm1ax(x)islinearlyincreasingfor0≤ξ ≤1/2andislinearly
decreasing for 1/2 ≤ ξ ≤ 1. Hence Lmax(ξ) has a unique
1
RBd(x,P /σ,ξ)(cid:44)C(L=Lξ(x),P /σ). (17) maximum at ξ =1/2.
2 0 2 0
This then completely characterizes the boundary of the rate
region R (H,P /σ,ξ), which is given by8 Remark 2. The function Lmax(ξ) has its unique maximum at
ZF 0 1
ξ =1/2, i.e.
(cid:16) (cid:17)
RZBFd(H,P0/σ,ξ0)≤(cid:44)x≤L∪m1ax(ξ)R1Bd(x,P0/σ,ξ),R2Bd(x,P0/σ,ξ) a0rg≤ξm≤a1x Lm1ax(ξ)=1/2 (23)
= ∪ (cid:0)C(x,P /σ),C(Lξ(x),P /σ)(cid:1) (18)
0 2 0
0≤x≤Lm1ax(ξ) Proposition 2. For a given L1 = x ∈ [0,Lm1ax(ξ)], the
It is noted that the analysis done in this paper is applicable largest possible L ≥ 0 such that there exists a rectangle
2
toanyplacementoftheusersandtheLEDs.Subsequently,we Rect(x,L ,D(H,ξ))⊂R (H), is given by
2 //
follow the following convention that, by LED 1 we shall refer
Lξ(x)(cid:44) max L
to the LED whose channel vector has a higher inclination 2 2
L2≥0
angle (from the u1 axis) than the inclination angle of the Rect(x,L2,D(H,ξ))⊂R//(H)
channel vector of the other LED. =2min(Lup,ξ(x),Ldown,ξ(x)), (24)
2 2
Lettheinclinationofthevectorh andh fromtheu axis
1 2 1 where Lup,ξ(x) is given by
be θ and θ respectively (see Fig. 2). From our definition of 2
1 2
LED 1 and LED 2 (see the above paragraph), it follows that Case I: 0≤ξ ≤ h11
h11+h12
θ1 >θ2. Therefore it follows that tanθ1 >tanθ2. Since Lup,ξ(x)=(cid:110)−ξdet(H)−x2h21, 0≤x≤Lmax(ξ) (25)
tanθ =h /h , tanθ =h /h . (19) 2 h11 1
1 21 11 2 22 12
Case II: h11 ≤ξ ≤1
Hence, tanθ1 >tanθ2 implies that h11+h12
h21/h11−h22/h12 >0, Lup,ξ(x)=(cid:40)−(1−ξ)deht1(2H)−x2h22, 0≤x≤η3(ξ)
h11h22−h12h21 <0, i.e. 2 −ξdet(hH11)−x2h21, η3(ξ)≤x≤Lm1ax(ξ)
det(H)<0 (20) (26)
In the following proposition, we first compute the maximum where η3(ξ)(cid:44)2ξh12−2(1−ξ)h11.
value of L1 and subsequently we derive the maximum value Ldown,ξ(x) is given by
of L for each value of L . 2
2 1
Case I: 0≤ξ ≤ h12
Proposition 1. The largest possible value of L (i.e., h11+h12
1
length of the interval U1) such that there exists a rectangle Ldown,ξ(x)=(cid:40)−ξdet(hH12)−x2h22, 0≤x≤η4(ξ)
R1B8dF(rxo,mP0/(1σ6,)ξ)aannddR(12B7d()x,itP0i/sσ,cξl)earrequthiraetsththeecoemxpaucttaticoonmopfuLtaξ2ti(oxn)foorf 2 −(1−ξ)deht1(1H)−x2h21,η4(ξ)≤x≤Lm1ax(ξ(2)7)
which we derive closed form expressions in the next section. Computation
of R1Bd(x,P0/σ,ξ) and R2Bd(x,P0/σ,ξ) also requires us to compute the where η4(ξ)(cid:44)2(1−ξ)h12−2ξh11.
C(L,P0/σ)functionwhichisdonenumerically.
6
Case II: h11h+12h12 ≤ξ ≤1 rate pair (R1,R2) such that R2 =αR1. This operating point
could make sense, if for example the average data throughput
(cid:110)
Ldown,ξ(x)= −(1−ξ)det(H)−x2h21,0≤x≤Lmax(ξ) requestedbyuser2isαtimesthatofthethroughputrequested
2 h11 1
(28) by user 1.
Moreover, for a given α > 0 and P /σ, the maxi-
Proof: See Appendix B. 0
mum achievable rate pair of the form (r,αr) is given by
Lemma 1. The function Lξ(x) (0 ≤ x ≤ Lmax(ξ)) is a (Rα (ξ),αRα (ξ)) where Rα (ξ) is defined as
2 1 max max max
monotonically decreasing and continuous function of x.
Rα (ξ)(cid:44) max r. (32)
max (cid:12)
Proof: From Proposition 2 it is clear that for a given r(cid:12)(r,αr)∈RZF(H,P0/σ,ξ)
ξ both Lup,ξ(x) and Ldown,ξ(x) are continuous and mono-
2 2 Theorem2. Rα (ξ)isuniqueand(Rα (ξ),αRα (ξ))lies
tonically decreasing function of x. From this it follows that max max max
on the boundary RBd(H,P /σ,ξ).
Lξ(x) = 2min(Lup,ξ(x),Ldown,ξ(x)) is a continuous and ZF 0
2 2 2
decreases monotonically with increasing x. Proof: See Appendix E.
Lemma 2. The proposed rate region boundary Remark 3. From the proof in Appendix E it is clear that
RBd(H,P /σ,ξ) is Pareto-optimal. That is, for any two rate Theorem2isnon-trivialasitdependsonthemonotonicityand
ZF 0
pairs (a,b) and (a(cid:48),b(cid:48)) on the boundary RBd(H,P /σ,ξ), if continuity of Lξ(x), which is shown in Lemma 1. If Lemma 1
ZF 0 2
a(cid:48) ≥ a then it must be true that b(cid:48) ≤ b and if b(cid:48) ≤ b then it were not true, Theorem 2 would not hold.
must be true that a(cid:48) ≥a.
Result 4. Using Theorem 2 and (30) of Result 3 it follows
Proof: Let (a,b) and (a(cid:48),b(cid:48)) be any two rate pairs that for a given α>0, Rα (ξ) is symmetric about ξ =1/2,
max
on the boundary RZBFd(H,P0/σ,ξ) such that a(cid:48) ≥ a. Then i.e.
from ((16) and (17)) it follows that there exists 0 ≤
x ≤ Lmax(ξ) and 0 ≤ x(cid:48) ≤ Lmax(ξ) such that a = Rmαax(ξ)=Rmαax(1−ξ), ∀ α>0,ξ ∈[0,1]. (33)
1 1
C(x,P /σ),b = C(Lξ(x),P /σ) and a(cid:48) = C(x(cid:48),P /σ),
0 2 0 0 Corollary 2.1. From the geometrical interpretation of The-
b(cid:48) = C(Lξ2(x(cid:48)),P0/σ), where the functions C(x,P0/σ) is orem 2 it follows that (Rα (ξ),αRα (ξ)) lies on the
max max
defined in (13). From Result (2), we know that for a given intersection of the straight line R = αR and the rate
2 1
P0/σ, C(x,P0/σ) is a continuous and monotonically in- region boundary RBd(H,P /σ,ξ). Further, from the Pareto-
creasing function of its first argument. Since C(x(cid:48),P0/σ) = optimalityoftheproZpFosedra0teregionboundary,itfollowsthat
a(cid:48) ≥ a = C(x,P0/σ), it follows that x(cid:48) ≥ x. From there is only a unique point of intersection between the line
fLuenmctmioan1,ofwex,knaonwd tthhaetreLfoξ2(rex)Lisξ(axm(cid:48))on≤otoLnξic(axl)l,yadnedcrehaesnincge R2 =αR1 and RZBFd(H,P0/σ,ξ).
2 2
b(cid:48) = C(Lξ(x(cid:48)),P /σ) ≤ C(Lξ(x),P /σ) = b. Similarly, it
2 0 2 0
can also be shown that, if b(cid:48) ≤ b then it must be true that A. Maximum symmetric rate Rsym(ξ)
a(cid:48) ≥a. This completes the proof.
Note that for the special case of α=1,Rα (ξ) is nothing
max
Lemma 3. For a given 0 ≤ ξ ≤ 1 and x ∈ [0,Lm1ax(ξ)], the but the maximum achievable symmetric rate which we shall
function Lξ(x) is symmetric about ξ =1/2, i.e. denote by
2
Lξ(x)=L1−ξ(x), 0≤ξ ≤1,x∈[0,Lmax(ξ)]. (29) Rsym(ξ)(cid:44)Rmα=a1x(ξ). (34)
2 2 1
From Theorem 2 it is clear that the maximum symmetric
Proof: See Appendix C.
rate is nothing but the largest rate R such that the rate
Using Lemma 3 along with the definition of the rate region
pair (R,R) lies on the boundary RBd(H,P /σ,ξ). From the
boundary in (18) we get the following result. ZF 0
characterization of the boundary points in (18), it follows that
Result 3. The proposed rate region boundary there must exist (x,Lξ(x)) for some 0 ≤ x ≤ Lmax(ξ) such
2 1
RBd(H,P /σ,ξ) is symmetric about ξ =1/2, i.e. that
ZF 0
RBd(H,P /σ,ξ)=RBd(H,P /σ,(1−ξ)), ∀ξ ∈[0,1]. R=C(x,P /σ), and R=C(Lξ(x),P /σ) (35)
ZF 0 ZF 0 0 2 0
(30)
and therefore
Thefollowingtheoremshowsthatfor0≤ξ ≤1,thelargest x=Lξ(x) (36)
2
rate region is achieved when ξ =1/2.
sincefromResult2weknowthatC(L,P /σ)isacontinuous
0
Theorem 1. For a fixed ξ ∈[0,1], andmonotonicfunction.From14itfollowsthatthereexistsa
rectangle Rect(x,Lξ(x),D(H,ξ))⊂R (H) corresponding
R (H,P /σ,ξ)⊆R (H,P /σ,1/2). (31) 2 //
ZF 0 ZF 0 to the rate pair (R,R) where x satisfies 36.
Proof: See Appendix D. Since x = Lξ(x) it follows that this rectangle is infact
2
TheproposedrateregionboundaryRBd(H,P /σ,ξ)canbe a square. Further, from the definition of Lξ(x) in (24) it
ZF 0 2
used to compute many practical operating points. Consider a follows that this is the largest sized square whose midpoint
case where we are interested in finding the largest achievable is at D(H,ξ) and has side length x.
7
and the block diagram in Fig. 3(c) depicts the receiver. The
working of this transceiver is as follows.
Consider a scenario where the rate requested by User 1
and User 2 are Rtgt bpcu and Rtgt bpcu respectively and to
1 2
satisfy the lighting requirement inside the room the required
dimming target is ξ. We call this rate pair (Rtgt,Rtgt), as
1 2
the target rate pair of the system. The Tx controller first
checks if this target rate pair lies in the proposed achievable
rate region R (H,P /σ,ξ) (see Section IV). If the target
ZF 0
rate pair lies inside the proposed achievable rate region, (i.e.,
(Rtgt,Rtgt) ∈ R (H,P /σ,ξ)) then the Tx controller flags
1 2 ZF 0
1, otherwise it flags 0 (see status output of the Tx controller
in Fig. 3(b)). If this flag is 1, then the Tx controller provides
L and L , the lengths of the intervals U and U . From (15)
1 2 1 2
we know that since (Rtgt,Rtgt) ∈ R (H,P /σ,ξ), there
(a) Transmitter(Tx)blockdiagram. 1 2 ZF 0
must exist some (L ,L ) such that Rtgt = C(L ,P /σ)
1 2 1 1 0
and Rtgt = C(L ,P /σ). From Result 2 we also know that
2 2 0
for a given P /σ, C(x,P /σ) is a monotonic function of
0 0
x, and therefore there exists a corresponding inverse func-
tion C−1(R,P /σ) such that C−1(C(L,P /σ),P /σ) = L
0 0 0
and C(C−1(R,P /σ),P /σ) = R. It then follows L =
0 0 i
C−1(Rtgt,P /σ), i=1,2 see Fig. 3(c). In the Tx controller
i 0
we also have a block which outputs the mean information
symbol vector ξ(h +h ) = [ξ(h +h ) ξ(h +h )]T
1 2 11 12 21 22
(defined in (9)).
Further,inFig.3(a)theinformationbitsforuser1anduser
2arecodedseparatelyusingindependentcodebookseachhav-
ingi.i.d.codewordsymbolswhichareunifromlydistributedin
(b) TxControllerblockdiagram. [−1/2,1/2]. The codeword symbols for user 1 and user 2 are
denoted by u(cid:48) and u(cid:48) respectively (note that u and u are
1 2 1 2
the information symbols for User 1 and User 2 respectively).
From Section III, we know that the information symbols for
the ith user must be uniformly distributed in the interval U
i
i.e.,u ∈U =[ξ(h +h )−L /2,ξ(h +h )+L /2](since
i i i1 i2 i i1 i2 i
thehorizontallengthoftherectanglecorrespondingtotherate
pair (Rtgt,Rtgt) is L , the vertical length of this rectangle is
1 2 1
(c) Receiverblockdiagramforithuser,i=1,2. L2 and its midpoint is D(H,ξ)). Therefore, starting with the
codewordsymbolu(cid:48) wecangettheinformationsymbolu by
Fig. 3: A novel transceiver architecture for the proposed 2×2 i i
MU-MISO VLC system with dimming Target of ξ and target ui =Liu(cid:48)i+ξ(hi1+hi2), i=1,2. (37)
rate pair (Rtgt,Rtgt).
1 2 ThisisalsoshowninFig.3.Theinformationvector[u u ]T is
1 2
thenprecodedwithH−1 andscaledbyP togivethetransmit
0
signalvector[x x ]T.Itisnotedthattheproposedtransmitter
1 2
Hence,themaximumachievablesymmetricratecorresponds
architecture in Fig. 3(a) allows us to use the same channel
tothelargestsizedsquarewhichiscompletelyinsideR (H)
// encoder/codebook irrespective of the dimming target ξ. This
and has its midpoint at D(H,ξ).
isbecausetheeffectofthedimmingcontrolisonlyinshifting
the mean of the information symbols (u ,u ) (see the adders
1 2
V. ANOVELTRANSCEIVERARCHITECTURE in Fig 3(a)).9
At the receiver after performing the operations shown in
In this section we propose a novel transceiver architecture Fig. 3(c), we obtain the received vector as given by (5).
for the practical implementation of the proposed 2×2 MU-
MISO VLC system to achieve any rate pair (R ,R ) ∈
1 2 VI. NUMERICALRESULTSANDDISCUSSIONS
R (H,P /σ,ξ) (see Section IV), under a per-LED peak
ZF 0
power constraint of P and a controllable dimming target. In this section, we present numerical results in support of
0
the results reported in previous sections. For all numerical
In Fig. 3, we have shown the block diagram of both the
results we consider an indoor office room environment where
transmitter and the receiver. The block diagram in Fig. 3(a)
depicts the transmitter, the block diagram in Fig. 3(b) depicts
9Note that different target rates can be achieved by the same codebook
thecontrollerforthetransmitterwhichwecallasTxcontroller throughpuncturingofthecodewords.
8
TABLE I: System Parameters used for Simulation
PD area 1 cm2
Receiver Field of Veiw (FOV) 60 [deg.]
Refractive index of a lens at the PD 1.5
Semi-angle at half power 70 [deg.]
Fig. 5: Plot between Maximum Symmetric Rate and dimming
Target, ξ.
and towards the interior of the parallelogram R (H). This
//
allows us to fit bigger rectangles and hence the rate region
expands. As ξ is increased beyond ξ = 0.5 the midpoint of
Fig. 4: Rate region boundary, RBd(H,P /σ,ξ) for different
ZF 0 the rectangles moves towards the other end of the diagonal
values of dimming target, ξ.
(h +h )andhencethesizeoftherectanglesreducesthereby
1 2
shrinking the rate region.
In Fig. 5, for a fixed user separation of s = 4 m, an LED
the room is 5 m × 5 m and its height is 3 m. The two LEDs separation of d = 60 cm and symmetric placement of LEDs
are attached to the ceiling and the two PDs (users) are placed and PDs, we plot the maximum achievable symmetric rate
at a height of 50 cm from the floor of the room. The two Rsym(ξ) (cid:44) Rα=1(ξ) as a function of varying ξ ∈ [0,1].
max
LEDs and the PDs lie in a plane perpendicular to the floor We numerically find this operating point by considering all
of the plane. The LEDs are placed 60 cm apart and the ratio possiblepointsintheR -R planewhichlieintheachievable
1 2
P0 is fixed to 70 dB. The channel gains are modeled for an rate region and also lie on the line R =R . Then among all
σ 2 1
indoorlineofsight(LOS)channel.Theotherparametersused these possible points we choose the one which has the largest
for simulation are given in Table I. All these parameters and component along the R axis. From the figure it is observed
1
the channel model are taken from prior work [3], [14], [17], thatthevariationinthemaximumsymmetricratewithchange
[18]. in the dimming target ξ is small when ξ is around 1/2, as
InFig.4,foraLEDseparationof0.6mandPD(user)sepa- compared to when min(ξ,1−ξ) is small. For example, when
rationof4msuchthattheplacementofboththeLEDsandthe ξ is reduced from ξ = 1/2 to ξ = 0.4 (i.e., 20% reduction),
PDsissymmetric10,weplottheproposedrateregionboundary the corresponding maximum symmetric rate drops only by
RZBFD(H,P0/σ,ξ) (see 18), for ξ =0.1,0.2,0.3,0.4,0.5,0.7. 11%. However when ξ is reduced by 20% from ξ = 0.07
For a given ξ, it is observed that the boundary is indeed to ξ = 0.056, the maximum symmetric rate decreases by
Pareto-optimal as is stated in Lemma 2. We also observe that approximately 25%. From this it appears that the maximum
asξincreasesfromξ =0.1toξ =0.5,therateregionexpands symmetric rate is lesser sensitive to variations in the dimming
and then it shrinks with further increase in ξ from ξ = 0.5 target around ξ = 1/2 as compared to variations around
onwards to ξ = 1. We have also observed that rate region smaller values of ξ. It is also observed that symmetric rate
boundary is same for both ξ = 0.3 and ξ = 1−0.3 = 0.7 Rsym(ξ) is symmetric about ξ =1/2 as is stated in Result 4
as is stated in Result 3 (see the dotted line and the solid line (symmetric rate is nothing but Rα (ξ) for α=1).
max
marked with circle in Fig. 4). It is also observed that ξ =1/2 Wenextstudythevariationinthemaximumsymmetricrate
gives us the largest rate region as is stated in Theorem 1. when the two users (PDs) are moved along a line parallel to
The expansion/shrinking of the rate region with changing ξ is the ceiling (at a height of 50 cm above the floor) while the
explained in the following. two LEDs are stationary and fixed to the ceiling with a fixed
For a given ξ, the points on the rate region boundary separation of 60 cm between them and the dimming target is
correspond to rectangles in the u1 − u2 plane having their also fixed to ξ =0.1. Further, the two LEDs and the two PDs
midpoints at D(H,ξ), i.e., on the diagonal (h1 +h2) and are co-planar. In Fig. 6, we plot the symmetric rate on the
at a distance of ξ||h1+h2|| from the origin. As ξ increases, vertical axis as a function of the displacement11 of the two
the midpoint of the rectangles move away from the origin users from the origin (origin is the point of intersection of the
10Boththelinesegmentjoiningthetwousersandthelinesegmentjoining 11Displacementisnothingbutthedistanceoftheuserfromtheorigin(see
thetwoLEDshavethesameperpendicularbisector. Fig.1.
9
(see the discussion in Section IV-A for the correspondence
betweenthelargestsquareandthemaximumsymmetricrate).
With further increase in the separation between the two users
the angular separation between the channel vectors does not
increase as sharply as before. At the same time, due to
increased path loss from the LEDs to the users, the area of
R (H) starts decreasing which results in the decrease in the
//
maximum symmetric rate. This can be seen in the figure, as
thecolourchangesbackfromwhitetogray,aswemovefrom
thedisplacementvector(−1.2,1.2)to(−2.5,2.5).Thisshows
thatthemaximumsymmetricrateisdependentonthelocation
of the users and therefore there is an optimal location12 for
both the users which results in the highest symmetric rate. In
Fig. 6 the optimum location is (−1.2,1.2), or (1.2,−1.2).
Next in Fig. 7, for a fixed LED separation of 60 cm
Fig. 6: Maximum symmetric rate vs displacement of the two
we plot the percentage loss in the maximum symmetric rate
users from the origin.
Rsym(ξ) (w.r.t. the symmetric rate at the optimum location)
with the users’ displacement from their optimum location for
two different values of ξ =0.1,0.3.
It is observed that the percentage loss increases with in-
creasing displacement of the PDs from their optimal location.
Further, the increase in the percentage loss is small when the
displacementissmallascomparedtowhenthedisplacementis
large.Forexample,withξ =0.3,thepercentagelossincreases
only by 6% as the displacement increases from 0 cm to 40
cm. However with a further increase in displacement from 40
cmto80cm,thepercentagelossincreasessharplyfrom6%to
30%.Asimilarbehaviorisalsoobservedwithξ =0.1,though
for a given displacement the loss is greater when ξ = 0.1 as
compared to when ξ = 0.3. A practical application of this
study could be in defining coverage zones for the PDs, i.e.,
themaximumallowabledisplacementforafixeddesiredupper
Fig. 7: Plot between percentage loss in Rsym(ξ) and users limitonthepercentageloss. Forexample,inthecurrentsetup
with ξ = 0.3, for a 20% upper limit on the percentage loss,
displacement from their optimum location.
the maximum allowable displacement is roughly 70 cm. It
therefore appears that indoor VLC systems allow for a lot
of flexibility in the movement of the user terminals without
perpendicular bisector of the line joining the LEDs with the
significant loss in the information rate.
line joining the two users, see Fig . 1). In Fig. 6 a positive
displacement implies that the user PD is located on the right
side of the origin and vice versa (see Fig. 1).
VII. CONCLUSION
It is observed that the maximum symmetric rate is almost
zeroifthedisplacementofboththeusersissame,i.e.,thetwo We have proposed an achievable rate region for the 2×2
users are almost co-located. In the figure this is represented MU-MISObroadcastVLCchannelunderper-LEDpeakpower
by the dark black region. This is expected since in that case constraintanddimmingcontrol.Theboundaryoftheproposed
the channels to the users is also the same and hence the rate region has been analytically characterized. We propose
performance of the ZF precoder degrades. From the figure we a novel transceiver architecture to implement such systems.
observethatstartingwithboththeusersattheorigin,asuser2 Interestingly, the design of encoder/codebook is independent
moves towards the right and User 1 moves towards left the of the dimming target, which reduces the complexity of the
maximum symmetric rate increases sharply (in the figure the transceiver. Work done in this paper reveals that, in an indoor
colourchangesfromdarkblacktolightblacktograytowhite setting, the two users have enough mobility around their
as the displacement vector moves from (0,0) to (−1.2,1.2)). optimal placement without sacrificing their information rates.
Thishappensbecauseastheusersmoveawayfromeachother, Ourworkcanalsobeappliedtoa2-Dsetting,wheretheusers
their channels become distinct i.e., the angles between the are allowed to move in a plane rather than being restricted to
vectors h1 and h2 increases and hence the area of R//(H) a line.
increases.Thisresultsinanincreaseinthelargestsizedsquare
thatcanbefitintoR (H)withcenteratD(H,ξ).Thisthen
// 12By the optimum user location we mean the displacement vector of the
impliesthatthemaximumsymmetricratewouldalsoincrease usersatwhichwegetmaximumRsym(ξ).
10
APPENDIXA
PROOFOFPROPOSITION1
Proof: Under the condition in (20), to find Lmax(ξ) we
1
need to consider three scenarios that cover all geometrically
possible parallelograms R (H): (a) (h < h and h >
// 11 12 21
h ); (b) (h ≤h ); and (c) (h ≤h and h >h ).
22 21 22 12 11 21 22
For a given dimming target, ξ, let L denote the length
3
of the longest line segment parallel to the u -axis lying
1
completely inside R (H) and whose midpoint coincides
//
with the point D(H,ξ) (D(H,ξ) is defined in (10)). For
any rectangle Rect(L ,L ,D(H,ξ)) ⊂ R (H), its side
1 2 //
along the u axis is a line segment inside R (H). From
1 //
the definition of L , it follows that L ≤ L for any rect-
3 1 3
angle Rect(L ,L ,D(H,ξ)) ⊂ R (H). Additionally, the
1 2 //
longest line segment of length L corresponds to a rectangle Fig. 8: Partition of the parallelogram OABC (cid:44)R (H) into
3 //
Rect(L ,L = 0,D(H,ξ)) ⊂ R (H). Hence, it is clear three different regions for the scenario (h <h and h >
3 2 // 11 12 21
that Lmax(ξ)=L , i.e. h ). Note that AA(cid:48) and CC(cid:48) are both parallel to the u axis.
1 3 22 1
Lmax(ξ)(cid:44) max L
1 1
L1≥0,L2≥0
Rect(L1,L2,D(H,ξ))⊂R//(H) completely inside R//(H). it follows that
= max L1 (38) Lmax(ξ)=2 min(PP ,PP ), (42)
{L1>0|Rect(L1,L2=0,D(H,ξ))⊂R//(H)} 1 1 2
Inthefollowing,wefirstlyevaluatetheexpressionforLm1ax(ξ) whereboththelinesegmentsPP1 andPP2 areparalleltothe
forscenario(a),i.e.,whenthechannelgainssatisfy(h11 <h12 u1-axis. Further, P1 lies on the line OA whereas P2 lies on
and h21 >h22). Towards this end, we partition R//(H) into the line OC as shown in Fig. 8. Next, we evaluate PP1 and
three regions, Region i,i=1,2,3, as is shown in Fig. 8. We PP2. To this end, from Fig. 8, we compute the length of the
now derive an expression for Lm1ax(ξ) depending upon the line segment PP1 as follows
region where D(H,ξ) lies. In Fig. 8 we denote D(H,ξ) by
PP =PP −P P
the point P if D(H,ξ) lies in Region 1, by the point Q if 1 3 1 3
(a)
D(H,ξ) lies in Region 2 and by the point S if D(H,ξ) lies = ξ(h +h )−OP /tanθ
11 12 3 1
in Region 3. Next, we compute Lmax(ξ) when D(H,ξ) lies
1 (b)
= ξ(h +h )−ξ(h +h )h /h
in Region 1. 11 12 21 22 11 21
Computation of Lm1ax(ξ) when P =D(H,ξ)∈Region 1 : =ξ(h12h21−h11h22)/h21 =−ξ det(H)/h21, (43)
The point D(H,ξ) belongs to Region 1 if and only if
where step (a) follows from the fact that, PP is equal to
3
the co-ordinate of the point D(H,ξ) along the u -axis and
0≤OP ≤OT, (39) 1
therefore from (10), we have PP = ξ(h +h ). In step
3 11 12
where the point T denote point of intersection of the diagonal (a) we have also used the fact that since OP P is a right
1 3
OB and CC(cid:48). Further, the line CC(cid:48) is the line parallel to the angle triangle having ∠OP P = θ . Hence, it follows that
3 1 1
u1-axis. Next, by looking at the right angle triangle OT1T P1P3 = OP3/tanθ1. Step (b) also follows from two facts.
in Fig. 8, it follows that OT = TT1/sinγ, where γ denotes Firstly, OP3 is equal to the co-ordinate of the point D(H,ξ)
inclinationofthediagonal,OB,oftheparallelogramR//(H) alongtheu2-axisandthereforefrom(10),wehavethatOP3 =
from the u1-axis. Since, from Fig. 8, TT1 =h22, and sinγ = ξ(h21+h22) and secondly, from (19), we know that tanθ1 =
(h21+h22)/OB, it follows that h21/h11.SimilarlyfromFig.8,wecalculatethelengthofPP2
h OB as follows
OT = 22 (40)
h21+h22 PP2 =P2P3−PP3
Since the point P is nothing but the point D(H,ξ), from (a)
= OP /tanθ −ξ(h +h )
3 2 11 12
(10),itfollowsthatOP =ξOB.UsingOP =ξOB and(40)
(b)
in (39) we have that D(H,ξ)∈Region 1 if and only if = ξ(h +h )h /h −ξ(h +h )
21 22 12 22 11 12
0≤ξ ≤ h22 , (41) =ξ(h12h21−h11h22)/h22 =−ξ det(H)/h22, (44)
h +h
21 22 where step (a) follows from the fact that, PP is equal to
3
Forallsuchvaluesofthedimmingtarget,ξ,satisfying(41),it the co-ordinate of the point D(H,ξ) along the u -axis and
1
follows that D(H,ξ)∈Region 1. Next, we evaluate Lmax(ξ) therefore from (10), we have that PP = ξ(h + h ).
1 3 11 12
when 0≤ξ ≤h22/(h21+h22). In step (a) we have also used the fact that OP3P2 is a
Since Lmax(ξ) is the length of the line segment parallel right angle triangle having ∠OP P = θ . Hence, it follows
1 2 3 2
to the u -axis having its midpoint at point P and lying that P P = OP /tanθ . Step (b) follows from two facts.
1 2 3 3 2