C OMMUNICATIONS IN I NFORMATION AND S YSTEMS
Vol. 7, No. 2, pp. 177-194, 2007 c
2007 International Press
005
ACTIVE POINTING CONTROL FOR SHORT RANGE FREE-SPACE
OPTICAL COMMUNICATION ∗
ARASH KOMAEE † , P. S. KRISHNAPRASAD † , AND PRAKASH NARAYAN †
Abstract.
Maintaining optical alignment between stations of a free-space optical link requires an active pointing mechanism to persistently aim an optical beam toward the receiving station with an acceptable accuracy. This mechanism ensures delivery of maximum optical power to the receiving station in spite of the relative motion of the stations. In the active pointing scheme proposed in the present paper, the receiving station estimates the center of the incident optical beam based on the output of a position-sensitive photodetector. The transmitting station receives this estimate via an independent communication link and uses it to accurately aim at the receiving station. The overall mechanism which implements this scheme can be described in terms of a diffusion process which modulates the rate of a doubly stochastic space-time Poisson process. At the receiving station, observation of the space-time process over a subset of R 2 is provided in order to control the diffusion process. Our goal is to determine a control law, measurable with respect to the history of the space-time process, which minimizes a quadratic cost functional.
1. Introduction.
Free-space optical communication is increasingly regarded as a high-bandwidth power-efficient means for point-to-point communication. The range of applications include fixed-location terrestrial communication [1], communication between mobile robots [2], airborne communication [3], and intersatellite communication [4].
In free-space optical communication using (narrow) laser beams, it is necessary to maintain the alignment of the transmitter and the receiver in spite of their relative motion. This relative motion might be caused by the mobile nature of the stations, mechanical vibration, or accidental shocks. Maintaining alignment is achieved through two operations: spatial tracking and active pointing. Spatial tracking properly adjusts the orientation of the receiver to hold the transmitter within its field of view [5].
Pointing serves to aim the transmitted beam toward the receiver within an acceptable accuracy [5]. While coarse pointing is essential for initiating a free-space optical link, active pointing—a persistent fine pointing operation—is required during data transmission in order to compensate for pointing error caused by relative motion.
The last operation attempts to maximize the received optical power during the course
∗ This research was supported by the Army Research Office under ODDR&E MURI01 Program Grant No. DAAD19-01-1-0465 to the Center for Communicating Networked Control Systems
(through Boston University), by the National Science Foundation under Grant No. ECS 0636613, and by the Army Research Office under ARO Grant No. W911NF0610325.
† The authors are with the Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD 20742, USA, E-mail: { akomaee, krishna, prakash } @umd.edu
177

178 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN of communication. In this paper, our focus is on active pointing for short range applications.
The one-way optical link under consideration comprises an optical transmitter and an optical receiver which are subject to relative motion. The optical transmitter is equipped with a servo-driven pointing assembly which can control the azimuth and elevation of a transmitting laser source. The optical beam emitted by the laser source has a nonuniform intensity profile which is assumed to be Gaussian [5]. Normally, the aperture of the receiver is smaller than the received optical beam, so that the receiver can collect only a fraction of the optical beam. In order to enlarge this captured fraction, the goal of active pointing is to hold the center of the optical beam at the center of the receiving aperture. The receiver employs a position-sensitive photodetector to measure the intensity profile of the optical beam that strikes its aperture. The output of the photodetector is used to estimate the center of the received optical beam, which is then conveyed to the transmitter through an optical link or a low-bandwidth
RF channel. The pointing assembly then adjusts the orientation of the transmitter based on this estimate.
The performance of the proposed active pointing scheme depends significantly on the accuracy of the estimate of the beam center. In order to achieve a good estimate of the beam center, it is necessary that the size of the receiving aperture be comparable with the size of the beam. This requirement limits the application of our method to short distance links.
The rest of this paper is organized as follows. In the next two sections we develop a stochastic model for the overall scheme and state the control problem associated with the model. Based on this model, in Section 4, we formulate and solve the problem of estimating the center of the beam. The results of this section will be used later in
Section 5 to discuss the optimal control problem.
2. The Model.
The structure of our model follows that introduced in [6] for spatial tracking systems. Although the model in [6] describes a spatial tracking system rather than an active pointing one, since the two systems share similar components, the models for pointing assembly, relative motion, and the photodetector are adopted from [6]. We refer the reader to [6] for a detailed description and justification of the models.
Let the two-dimensional vector θ t denote the azimuth and elevation angles of the transmitter axis with respect to some fixed coordinate system, where the subscript t indicates time dependence. Similarly, α t denotes the azimuth and elevation angles of the line-of-sight of the stations (passing through the center of receiving aperture) with respect to the same coordinate system. The pointing error is ϕ t
= θ t
− α t
. We assume that the receiving aperture is held perpendicular to the line-of-sight by means

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 179 of a spatial tracking system. Then, for a small pointing error, the displacement of the center of optical beam with respect to the center of receiving aperture is given by y t
= lϕ t
, where l is the distance between the stations which is assumed to be a constant.
The pointing assembly is an electro-mechanical system with input vector u t
R
2 elevation angles. The associated stochastic state space model is
∈ and output vector θ t
∈ R
2 , which correspond, respectively, to the azimuth and
(1) dx p t
= A p t x p t dt + B t p u t dt + D p t dw t p
θ t
= C t p x p t where x p t
∈ R n p is the state vector, { w p t
, t > 0 } is a m p
-dimensional standard Wiener process, and A p t
, B p t
, D p t
, and C t p are known uniformly bounded matrices of appropriate dimensions. The use of a linear model for the pointing assembly is justified by the fact that the system operates over small angles during the active pointing regime.
We model α t by a Gauss-Markov stochastic process [6] described by the state space equations
(2) dx d t
= A d t x d t dt + D d t dw d t
α t
= C d t x d t with state vector x d t
∈ R n d
, m d
-dimensional standard Wiener process { w d t
, t > 0 } , and known uniformly bounded matrices A d t
, D d t
, and C d t
The displacement vector y t
= lϕ t of proper dimensions.
is a linear function of x p t and x d t
, so that (1) and (2) can be combined in a compact form:
(3) dx t
= A t x t dt + B t u t dt + D t dw t y t
= C t x t with state vector x t
∈ R n and m -dimensional standard Wiener process { w t
, t > 0 } , where n = n p
+ n d and m = m p
+ m d
. The initial state x
0 is assumed to be Gaussian x
0
Let Φ k
: R k × R k × R k × k → R
+
Σ
0
, and independent of { w t be a Gaussian map defined as
, t > 0 } .
Φ k
( z ; ¯ Θ) = (2 π )
− k/ 2
(det Θ)
− 1 / 2 exp −
1
2
( z − ¯ )
T
Θ
− 1
( z − ¯ ) .
Let r denote the position vector of an arbitrary point on the plane of the receiving aperture with respect to a coordinate system centered at the center of the aperture.
Then, for a Gaussian beam centered at y t
= C t x t
, the optical intensity I t
( r ) over the plane of the aperture is proportional to
I t
( r ) ∝ Φ
2
( r ; C t x t
, R t
)

180 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN where R t
= R T t is a 2 × 2 positive definite matrix describing the shape of the beam.
For a circular symmetric beam with constant radius ̺ > 0 we have R t
= ̺
2
I
2
1
.
Let A denote the area of the receiving aperture. In a practical system, the optical field over the receiving aperture is focused on a photodetector of small surface area by means of a focusing lens. The photodetector measures the intensity profile of the imaged optical field, which is a scaled-down version of the optical intensity over the receiving aperture. Therefore, we consider the combination of the lens and the photodetector as a virtual photodetector of area A , i.e., we assume that the virtual photodetector provides the observation of the optical intensity over an area A .
The position-sensitive photodetector is a photoelectron converter whose surface is partitioned into small regions. The output of each region counts the number of converted electrons regardless of their location within the region. The photoelectron conversion rate depends linearly on the optical power absorbed by the region.
Generally, a photoelectron converter is modeled by a Poisson process with a rate proportional to the impinging optical power [5, 7]. In the present case, where the optical power is a stochastic field, the output of each region is modeled by a doubly stochastic Poisson process. We assume that the receiver employs a high spatial resolution photodetector. Following [6], we use an infinite resolution model for such a sensing device. This idealized model, which is characterized by a doubly stochastic space-time Poisson process, provides a reasonable approximation for a high spatial resolution photodetector.
The rate of the space-time process which models the output of the photodetector is proportional to the optical intensity I t
( r ). Thus, introducing a proportionality constant µ t
> 0, we express the rate as
λ t
( r, x t
) = µ t
Φ
2
( r ; C t x t
, R t
) .
In a general situation, µ t is a nonnegative stochastic process representing the random optical fade caused by atmospheric turbulence and the information-bearing signal modulating the optical beam. However, here we simplify the model by assuming that µ t is deterministic and nonnegative.
The space-time Poisson process, defined over [0 , ∞ ) × A , characterizes the occurrence of discrete events (e.g., release of a single electron) with a temporal component t ∈ [0 , ∞ ) and a spatial component r ∈ A . For T and S Borel sets in [0 , ∞ ) and A , respectively, let N ( T × S ) denote the number of points occurring in T × S . Define the random variable
ρ ( T × S ) =
Z
T ×S
λ t
( r, x t
) dtdr.
1
I k is the k × k identity matrix

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 181
Then N ( T × S ) is a conditionally Poisson random variable with conditional probability distribution
Pr { N ( T × S ) = n | ρ ( T × S ) } = e − ρ ( T ×S ) ρ n n !
( T × S )
.
Moreover, for disjoint T
1
× S
1 and T
2
× S
2
, the random variables N ( T
1
× S
1
) and
N ( T
2
× S
2
), conditioned on ρ ( T
1
× S
1
) and ρ ( T
2
× S
2
) are (conditionally) independent.
With (Ω , F , P ) as the underlying probability space for the stochastic model above, define B t as the σ -algebra generated by the space-time process over [0 , t ).
We define the counting process N t as the number of points that occur during [0 , t ) over the entire surface of the photodetector regardless of their location, i.e., N t
=
N ([0 , t ) × A ).
3. Problem Statement.
The central objective of an active pointing system is to maintain the centroid of the optical beam as close as possible to the center of the photodetector. This control task can be interpreted as one of minimizing y t with respect to some appropriate norm. We adopt the quadratic form
(4) J = E
"
Z
T
0 x
T t
Q t x t
+ u
T t
P t u t dt + x
T
T
Sx
T
# with P t
= P t
T > 0, Q t
= Q T t
> 0, and S = S T > 0, as the cost functional. For purpose of active pointing, a reasonable choice is Q t
= C
T t
C t
, P t
= I
2
, and S = 0.
We say u t is an admissible control if u t is B t
-measurable and the solution to (3) is well-defined. Based on the cost functional (4), the control problem can be defined as:
Subject to state space equation (3) , find the admissible control u t that minimizes the cost functional (4) .
An intermediate step for solving the control problem is to obtain the posterior density p x t
( x | B t
). In the next section we discuss this problem and develop an approximation for this posterior density. Employing this approximation, we propose a solution for the optimal control problem.
4. Estimation Problem.
Let ( t k − 1
, t k
] be the interval between the ( k − 1) th and k th occurrences of the space-time process, and let r k be the location of k th occurring point. For a function b t
( r, ξ t
) that is continuous in r and left-continuous in t and ξ t
, the stochastic differential equation dξ t
=
Z
A b t
( r, ξ t
) N ( dt × dr ) is defined such that dξ t
= 0 during ( t k − 1
, t k
] and ξ t encounters a jump of b t k
( r k
, ξ t k
) at t = t k
.

182 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
For the model of Section 2, Rhodes and Snyder [8] derived a stochastic partial differential equation describing the time-evolution of the posterior density p x t
( x | B t
).
This equation is expressed by
(5) dp x t
( x | B t
) = L { p x t
( x | B t
) } dt + p x t
( x | B t
)
Z
λ t
( r, x ) ˆ − 1 t
( r ) − 1 N ( dt × dr )
A
− p x t
( x | B t
)
Z
A
λ t
( r, x ) − λ t
( r ) drdt
λ t
( r ) = E [ λ t
( r, x t
) | B t
] and L{·} is the forward Kolmogorov operator associated with (3) and defined as
L{·} = − n
X
∂ [( A t x + B t u t
) ( · )] i i =1
/∂x i
+
1
2 n
X n
X
∂
2 i =1 j =1
D t
D
T t
( · ) ij
/∂x i
∂x j
.
Also, for special case A = R
2 , it was shown in [8] that the solution of (5) is Gaussian x t and conditional covariance Σ t which satisfy the stochastic differential equations
(6)
(7) d ˆ t d Σ t
= A t
ˆ t dt + B t u t dt +
Z
R
2
M t
( r − C t x t
) N ( dt × dr )
= A t
Σ t dt + Σ t
A
T t dt + D t
D
T t dt − M t
C t
Σ t dN t with initial states ˆ
0
= ¯
0 and Σ
0
= ¯
0
. In these equations, we have M t
= Γ t
(Σ t
) where Γ t
( · ) is defined as
Γ t
(Σ) = Σ C T t
C t
Σ C T t
+ R t
− 1
.
The aperture of a practical receiver has finite area so that the ideal condition
A = R
2 is not feasible. In practice, where A 6 = R
2 , the filtering problem associated with (5) is infinite-dimensional. However, when A is large enough compared with the size of the optical beam, a finite-dimensional approximation is reasonable. The fact that p x t
( x | B t
) is Gaussian for A = R
2 motivates us to consider a Gaussian approximation for p x t
( x | B t
) when A 6 = R
2 . In the reminder of this section, we develop a method to determine the mean and covariance matrix of such a Gaussian approximation. The cumulant generating function associated with p x t
( x | B t
) plays a central role in this development.
The conditional cumulant generating function of x t given B t is defined as
ψ t
( s ) = ln E exp jω
T x t
| B t jω = s
, and can be expanded in terms of conditional cumulants κ i
1 t i
2
··· i j [9] as
(8) ψ t
( s ) =
∞
X X j =1 I n j
1 j !
κ i
1 t i
2
··· i j s i
1 s i
2
· · · s i j

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 183 where I j n = { 1 , 2 , . . . , n } j and s = ( s
1
, s
2
, . . . , s n x t and Σ t are represented in terms of the first and second order cumulants as ˆ t
= κ
1 t
, κ
2 t
, . . . , κ n t and Σ t
= [ κ ij t
]. The time-evolution of ψ t
( · ) is described by a partial differential integral equation derived from (5) and is stated next.
Theorem 4.1.
Let ψ t
( · ) be the conditional cumulant generating function of x t given B t where x t is the solution of (3) and B t is defined in Section 2. Then, the time-evolution of ψ t
( · ) is described by
(9) dψ t
( s ) = s
T
+
Z
A
A t
∂ψ t
( s )
∂s
+ B t u t dt +
1
2 s
T
D t
D
T t s dt ln β t
( r, s ) − ln β t
( r, 0) N ( dt × dr ) −
Z
A
β t
( r, s ) − β t
( r, 0) drdt where β t
( · , · ) is defined as
(10) β t
( r, s ) = exp {− ψ t
( jω ) } E exp jω
T x t
λ t
( r, x t
) | B t jω = s
.
Moreover, if the Fourier transform of λ t
( r, · ) ,
Λ t
( r, jω ) =
Z
R n
λ t
( r, x ) exp − jω
T x dx exists, β t
( · , · ) can be expressed as
(11) β t
( r, s ) =
1
(2 π ) n
Z
R n
Λ t
( r, jν ) exp ψ t
( jν + s ) − ψ t
( s ) dν.
Proof . See Appendix A.1.
The time-evolution of the cumulants is described by a (generally infinite) set of nonlinear stochastic differential equations driven by the space-time point process
N ( T × S ). This set of equations can be derived from (9) by matching the coefficients of corresponding s i
1 s i
2
· · · s i j on the two sides of (9). We can usually suppose that the first few cumulants approximate p x t
( x | B t
) with an acceptable precision. This means that the infinite set of equations can be approximated by a finite-dimensional one.
Regarding this approach, two issues should be addressed. First, we need to compute β t
( · , · ) in terms of the cumulants via equations (10) or (11) and expansion (8), which is not straightforward for an arbitrary number of cumulants. Second, when we truncate (8) to a limited number of terms, the corresponding approximation for p x t
( x | B t
) might not be a valid probability density function, i.e., it might be negative for some x . When we limit the expansion (8) to the first and second order terms
(Gaussian approximation), these difficulties are avoided. In this case, β t
( · , · ) can be easily calculated and the truncated expansion leads to a valid probability density.
In Appendix A.2, we have used the method above to approximate p x t
( x | B t
) with x t and covariance

184 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
Σ t of this Gaussian approximation are solution to the stochastic differential equations
(12)
(13) d ˜ t
= A t x t dt + B t u t dt +
Z
A
M t
( r − C t x t
) N ( dt × dr ) − µ t h t d
˜ t
= A t
Σ t dt + ˜ t
A
T t dt + D t
D
T t dt − t
C t
˜ t dN t
+ µ t
H t x t
, x
Σ t t
, Σ t dt dt with initial state ˜
0
= ¯
0 and ˜
0
Σ
0
. Here, ˜ t
= Γ t
R n and H t
( · , · ) : R n × R n × n → R n × n are defined as
˜ t
, and h t
( · , · ) : R n
× R n × n
→
(14)
(15) h t
( x, Σ) =
H t
( x, Σ) =
Z
Γ t
(Σ) ( r − C t x ) Φ
2 r ; C t x, C t
Σ C
T t
Z
A
A
Γ t
(Σ) C t
Σ C
T t
+ R t
− ( r − C t x ) ( r
+
−
R
C t t x dr
)
T
Γ
T t
(Σ)
· Φ
2 r ; C t x, C t
Σ C
T t
+ R t dr.
x t
Σ t x t and Σ t
, not their exact values.
Remark 4.1.
Equations (14) and (15) imply that as A → R
2 , h ( · , · ) → 0 and
H ( · , · ) → 0 , then as a consequence, the approximate estimator (12) , (13) tends to exact estimator (6) , (7) . In this sense, we can say that (12) , (13) is an asymptotically optimal estimator.
5. Control Problem.
We exploit the results of the previous section in solving the control problem as Theorem 5.1 below. Before stating this result, we fix some notation. Let Σ = [ σ ij
] denote a symmetric n × n matrix and f (Σ) be a scalar function of Σ. Assume that the partial derivatives of f (Σ) with respect to the elements of Σ exist. We denote by ∂f (Σ) /∂ Σ a n × n symmetric matrix F (Σ) = [ F ij
(Σ)] such that
F ii
= ∂f /∂σ ii and F ij
= (1 / 2) ∂f /∂σ ij for i = j . Let g t
( x, Σ) be a scalar function of x ∈ R n and n × n symmetric matrix Σ. Assume that the partial derivatives of g t
( x, Σ) with respect to x and Σ exist. Define the linear operator L t
{·} as
(16)
L t
{ g t
( x, Σ) } =
Z
A g t x + Γ t
(Σ) ( r − C t x ) , Σ − Γ t
(Σ) C t
Σ − g t
( x, Σ)
· Φ
2 r ; C t x, C t
Σ C
T t
+ R t dr
− ( ∂g t
( x, Σ) /∂x )
T h t
( x, Σ) + tr { ( ∂g t
( x, Σ) /∂ Σ) H t
( x, Σ) } .
Finally, we use k · k 2
P t to denote ( · )
T
P t
( · ).

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 185
Theorem 5.1.
Let x ∈ R n and Σ be a n × n symmetric matrix. Suppose that g t
( x, Σ) is the backward solution of the partial differential equation
(17)
−
∂
∂t g t
( x, Σ) =
∂
∂x g t
( x, Σ)
+ tr
T
A t x −
1
4
∂
∂x g t
( x, Σ)
T
∂
∂ Σ g t
( x, Σ) A t
Σ + Σ A
T t
+ D t
D
T t
B t
P t
− 1
B
T t
+ Q t
Σ
+ x
T
Q t x + µ t
L t
{ g t
( x, Σ) }
∂
∂x g t
( x, Σ) with boundary condition g
T
( x, Σ) = x T Sx . Then the cost functional (4) can be expressed as
(18)
J = g
0
˜
0
,
˜
0
+ E
"
Z
T
0
δ t dt
#
+ E
"
Z
T
0 u t
+
1
2
P t
− 1
B
T t
∂
∂ ˜ t g t
˜ t
,
˜ t
2
P t dt
#
, where
δ t
=
Z
R n
(19) x
T
Q t x +
Z
A g t
˜ t
+ ˜ t
( r − C t
˜ t
) ,
˜ t
−
˜ t
C t
˜ t
− g t x t
,
˜ t
λ t
( r, x ) dr
· p x t
( x | B t
) − ˜ x t
( x | B t
) dx is the error term resulting from replacing the posterior density p x t
( x | B t
) by its
Gaussian approximation ˜ x t
( x | B t
) .
Proof . See Appendix A.3.
The first term in the right side of (18) clearly does not depend on u t and so is not involved in the minimization. While the hard-to-compute error term δ t in (18) depends on u t
, it is supposed to be small. Therefore, in minimizing (18), we ignore δ t and only minimize the third term. We note that the minimum of the third term is 0 and is achieved when u t is given by
(20) u
∗ t
= −
1
2
P t
− 1
B
T t
Then the cost associated with u ∗ t will be
∂
∂ ˜ t g t
˜ t
,
˜ t
.
J ∗ = g
0 x
0
, Σ
0
+
Z
T
0
E δ t
| u t
= u ∗ t dt.
When A = R
2 , the solution to (17) can be simplified. This is stated as the following theorem which confirms that the optimal control in (20) is consistent with that obtained for A = R
2 by Rhodes and Snyder [8]. This shows that the approximation tends to the exact solution as A tends to R
2 .

186 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
Theorem 5.2.
When A = R
2
, the backward solution of the partial differential equation (17) with boundary condition g
T
( x, Σ) = x
T
Sx can be expressed as
(21) g t
( x, Σ) = x
T
K t x + f t
(Σ) where K t is the solution to the Riccati equation
(22) K t
= − K t
A t
− A
T t
K t
+ K t
B
T
P t
− 1
B
T t
K t
− Q t with K
T
= S , and f t
(Σ) is the backward solution to the partial differential equation
−
∂
∂t f t
(Σ) = tr
∂
∂ Σ f t
(Σ) A t
Σ + Σ A T t
+ D t
D T t
+ Q t
Σ
(23) + µ t f t
(Σ − Γ t
(Σ) C t
Σ) − f t
(Σ) + µ t tr { Γ t
(Σ) C t
Σ K t
} with boundary condition f
T
(Σ) = 0 .
Proof . See Appendix A.4.
We see from (20) and (21) that when A = R
2 , the optimal control is given by
(24) u ∗ t
= − P t
− 1
B
T t
K t x t with associated optimal cost
(25) J
∗ T
0
K
0
¯
0
+ f
0
¯
0
.
While the optimal control (24) has been obtained by Rhodes and Snyder [8], the value of the corresponding optimal cost (25) is newly obtained here.
6. Conclusion.
Optimal control and estimation problems associated with a diffusion process modulating the rate of a doubly stochastic space-time Poisson process are discussed. It is assumed that the observation of the space-time process over a subset of R
2 is used for the purpose of estimation and control. Our prime motivation for examining this optimal control problem is its application to active pointing in freespace optical communication. The main contributions of the paper are a suboptimal estimator for state of the diffusion process and a suboptimal control associated with a quadratic cost functional. Further, it is shown that when the observation is available over all of R
2 , our results tend toward the results of Rhodes and Snyder [8] for the latter case.
Appendix A. Proof of the Theorems.
(26)
A.1. Proof of Theorem 4.1.
The Fourier transform of (5) is given by [8] dφ t
( jω ) = E exp jω
T x t jω
T
( A t x t
+ B t u t
) −
1
2
ω
T
D t
D
T t
ω B t
+
Z
A
E h exp jω
T x t
λ t
( r, x t
) ˆ
− 1 t
( r ) − 1 B t i
N ( dt × dr )
Z
− E h exp jω
T x t
λ t
( r, x t
) −
ˆ t
( r ) B t i drdt.
A dt

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 187
Let t
1
< t
2
< t
3
< · · · be the occurrence times of the space-time process N ( T × S ).
During the interval ( t k
, t k +1
) , k = 0 , 1 , 2 , . . .
, the first integral in the right side of (26) is zero, thus we can write (26) as d exp { ψ t
(27)
( jω ) } = E exp jω
T x t jω
T
( A t x t
+ B t u t
) −
1
2
ω
T
D t
D
T t
ω B t
−
Z
A
E h exp jω
T x t
λ t
( r, x t
) −
ˆ t
( r ) B t i drdt.
Continuing, we write dt exp
(28)
{ ψ t
( jω ) } dψ t
( jω ) = exp { ψ t
( jω ) } jω
T
−
Z
A
E h exp jω T x t
A t
∂ψ t
( jω )
∂jω
+ B t u t
−
1
2
ω
T
D t
D
T t
ω dt
λ t
( r, x t
) − λ t
( r ) B t using the identity
E x t exp jω
T x t
| B t
=
=
∂
∂jω
E exp jω
T x t
| B t
∂ψ t
( jω )
∂jω exp { ψ t
( jω ) } .
i drdt,
Multiplying both sides of (28) by exp {− ψ t
( jω ) } and substituting β t
( r, jω ) from (10) into the resulting equation, we obtain
(29) dψ t
( jω ) = jω
T
A t
∂ψ t
( jω )
∂jω
+ B t u t dt −
1
2
ω
T
D t
D
T t
ω dt
−
Z
A
β t
( r, jω ) − β t
( r, 0) drdt.
The discontinuity at t = t k is treated as follows. Let r k be the spatial component of the event occurring at t k
. Then, from (26) we find
φ t
+ k
( jω ) − φ t − k
( jω ) = E which can be simplified as h exp jω
T x t − k
λ t − k r k
, x t − k
λ − 1 t − k
( r k
) − 1 B t − k i
,
φ t
+ k
( jω ) = E h exp jω
T x t − k
λ t − k r k
, x t − k
B t − k i
λ
− 1 t − k
( r k
) .
Multiplying both sides of this equation by exp − ψ t − k obtain
( jω ) and taking logarithms, we
ψ t
+ k
( jω ) − ψ t − k
( jω ) = ln exp − ψ t − k
( jω ) E h exp jω
T x t − k
λ t − k r k
, x t − k
− ln ˆ t − k
( r k
)
= ln β t − k
( r k
, jω ) − ln β t − k
( r k
, 0) .
B t − k i
Combining this with (29) and replacing jω by s , we obtain (9).

188 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
From definition of β t
( r, jω ) (10), we have
(30) β t
( r, jω ) = exp {− ψ t
( jω ) }
Z
R n p x t
( x | B t
) exp jω
T x λ t
( r, x ) dx.
Note that p x t
( x | B t
) is the Fourier transform of exp { ψ t
( jω ) } , and so we can write p x t
( x | B t
) =
1
(2 π ) n
Z
R n exp { ψ t
( jν ) } exp − jν
T x dν.
Upon substituting this into (30) and interchanging the order of integration 2 , we obtain
β t
( r, jω ) =
1
(2 π ) n
Z
R n exp ψ t
( jν ) − ψ t
( jω )
Z
R n
λ t
( r, x ) exp n
− j ( ν − ω )
T x o dxdν.
Replacing the second integral above by Λ t
( r, jν − jω ) and changing the variable of integration ν with ν + ω , we get
β t
( r, jω ) =
1
(2 π ) n
Z
R n
Λ t
( r, jν ) exp ψ t
( jν + jω ) − ψ t
( jω ) dν.
Finally we obtain (11) upon replacing jω with s .
A.2. Derivation of (12), (13).
We first state a technical lemma from [10] which will be used later in deriving (12) and (13). For sake of completeness, we repeat below the proof from [10].
Lemma A.1.
Let z k
, ¯ k
∈ R k , z l
∈ R l , and Θ k and Θ l be respectively k × k and l × l positive definite matrices. Assume that G is any l × k matrix. Then we have
(31)
Z
R k
Φ k
( z k z k
, Θ k
) Φ l
( z l
; Gz k
, Θ l
) dz k
= Φ l z l
; G ¯ k
, Θ l
+ G Θ k
G
T
.
Proof . Denoting the Fourier transform of the left side of (31) by F l
( ω l
), we can write
F l
( ω l
) =
Z
R k
Φ k
( z k
; ¯ k
, Θ k
) exp jω l
T
Gz k
−
1
2
ω T l
Θ l
ω l dz k
= exp jω
T l
G ¯ k
−
1
2
ω l
T G Θ k
G T ω l exp −
1
2
ω T l
Θ l
ω l
= exp jω T l
G ¯ k
− 1
2
ω l
T Θ l
+ G Θ k
G T ω l
Taking inverse Fourier transform of the expression above, we get the right side of (31).
s T x t
ψ t
( s ) =
+ 1
2 s T Σ t s is Gaussian with mean ˜ t
Σ t
. With this approximate probability density function and with λ t
( r, x t
) = µ t
Φ
2
( r ; C t x t
, R t
), the approximation of β t
( · , · ) is given by
˜ t
( r, s ) = exp n
−
˜ t
( s ) o
Z
R n
Φ n x ; ˜ t
,
˜ t exp s
T x µ t
Φ
2
( r ; C t x, R t
) dx.
2 This interchange is permissible since for any fixed t , the integrand is continuous in x and ν .

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION
A simple calculation yields that exp n
− ψ t
( s ) o
Φ n x ; ˜ t
, Σ t exp s
T x = Φ n x ; ˜ t
Σ t s, Σ t
.
189
Then, using Lemma A.1, we get
˜ t
( r, s ) = µ t
Φ
2 r ; C t
˜ t
+ C t
˜ t s, C t
˜ t
C
T t
+ R t
, which leads to
(32)
− 1 ln ˜ t
( r, s ) − ln ˜ t
( r, 0) = s
T ˜ t
C
T t
C
−
1
2 s T Σ t
C T t t
˜ t
C
C t
T t
Σ t
+ R t
C T t
+ R t
( r − C t
˜ t
)
− 1
C t
Σ t s and
(33)
β t
( r, s ) − β t
( r, 0) = µ t
Φ
2
· s
T ˜ r ; C t x t
, C t
Σ t
C
T t
+ R t t
C
T t
C t
˜ t
C
T t
+ R t
− 1
( r − C t
˜ t
)
+
1
2 s T Σ t
C t
T
−
1
2 s
T
Σ t
C
T t
C t
Σ t
C T t
+ R
C t
Σ t
C
T t
+ R t t
− 1
( r − C t x t
)
2
− 1
C t
Σ t s + O k s k
3
.
We combine (32), (33), and (9), and match the coefficients of s T ( · ) and s T ( · ) s from both sides to obtain (12), (13).
A.3. Proof of Theorem 5.1.
Our proof consists of the following four steps.
Step I: Using properties of conditional expectation, it is easy to show that
E x
T t
Q t x t
= E h
T t
Q t
ˆ t
+ tr { Q t
Σ t
} i
.
Then the cost functional (4) can be expressed as
(34) J = E
"
Z
T
0
T t
Q t x t
+ tr n
Q t
˜ t o
+ u T t
P t u t
+ δ 1 t dt + x T
T
Sx
T
# where δ 1 t is defined as
δ
1 t
= tr n
Q t x t
T t
− ˜ t
˜
T t
+ Σ t
− Σ t o
.
Step II: For t > 0 and for any positive ǫ , ∆ N t
, N t + ǫ
− N t is a conditionally Poisson random variable with stochastic rate ǫ t
=
Z t + ǫ
Z t A
λ
τ
( r, x
τ
) drdτ.

190 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
Thus, using the law of total probability, we can write
(35)
Pr ∆ N t
= 1 B t
= E Pr ∆ N t
= 1 |
¯ ǫ t
B t
= E ǫ t exp − ¯ ǫ t
B t
= ǫq t
+ O ǫ
2 where q t is defined as
(36) q t
=
Z
A
E [ λ t
( r, x t
) | B t
] dr.
In a similar manner, we can show that
(37)
Pr ∆ N t
= 0 B t
= 1 − ǫq t
+ O ǫ
2
Pr ∆ N t
> 2 B t
= O ǫ
2
.
Let the random vector R ∈ R
2 ing [ t, t + ǫ ). We show that denote the location of a single event occurring dur-
(38) p
R
( r | ∆ N t
= 1 , B t
) =
E [ λ t
( r, x t
) | B t
] q t
I
A
( r ) + O ( ǫ ) where I
A
( r ) = 1 if r ∈ A and I
A
( r ) = 0 otherwise. For this purpose, let D ( r ) ⊂ A denote a square with side length ∆ r and centered at r ∈ A . Defining T = [ t, t + ǫ ) and using Bayes’ rule, we can write p
R
( r | ∆ N t
= 1 , B t
) = lim
∆ r → 0
∆ r
− 2
Pr { R ∈ D ( r ) | ∆ N t
= 1 , B t
}
(39)
= lim
∆ r → 0
∆ r
− 2
Pr { N ( T × D ( r )) = 1 | ∆ N t
= 1 , B t
}
= lim
∆ r → 0
1
∆ r 2
·
Pr { N ( T × D ( r )) = 1 , N ( T × A ) = 1 | B
Pr { ∆ N t
= 1 | B t
} t
}
.
Note that the event of N ( T × D ( r )) = 1 and N ( T × A ) = 1 is equivalent to the event of N ( T × D ( r )) = 1 and N ( T × ( A − D ( r ))) = 0. Therefore, defining
X t
= { x
τ
| τ ∈ T } and using the law of total probability and properties of a space-time
Poisson process, we get
(40)
Pr { N ( T × D ( r )) = 1 , N ( T × A ) = 1 | B t
}
= E h
Pr { N ( T × D ( r )) = 1 , N ( T × ( A − D ( r ))) = 0 | X t
, B t
} B t i
= E h
Pr { N ( T × D ( r )) = 1 | X t
} Pr { N ( T × ( A − D ( r ))) = 0 | X t
} B t i
= E
"
Z
T ×D ( r )
λ
τ
( s, x
τ
) dτ ds 1 − O ǫ ∆ r
2
= ǫ ∆ r 2 E [ λ t
( r, x t
) | B t
] + O ǫ ∆ r 3 + O ǫ 2 ∆ r
B
2 t
.
#

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 191
Substituting (35) and (40) into (39), we obtain (38).
Let ˜ x t
( x | B t
) be the Gaussian approximation of p x t
( x | B t
).
Then using
Lemma A.1, we can write
(41)
Z
E [ λ t
( r, x t
) | B t
] = ˜ x t
+
R n
Z
R n
( x p x t
| B
( x t
|
) λ
B t t
)
( r, x
− ˜
) x t dx
( x | B t
) λ t
( r, x ) dx
= µ t
Φ
2 r ; C t
˜ t
, C t
˜ t
C
T t
+ R t
+
Z
R n p x t
( x | B t
) − ˜ x t
( x | B t
) λ t
( r, x ) dx.
Step III: Let g t
( x, Σ) be a scalar function of x ∈ R n and n × n symmetric matrix Σ.
Assume that the partial derivatives of g t
( x, Σ) with respect to t , x and Σ exist. Using the law of total probability we can write
E h g t + ǫ x t + ǫ
, Σ t + ǫ
B t i
=
∞
X
E h g t + ǫ k =0 x t + ǫ
, Σ t + ǫ
B t
, ∆ N t
= k i
Pr { ∆ N t
= k | B t
} .
Substituting Pr { ∆ N t
= k | B t
} from (35) and (37) into the previous expression, and using the law of total probability again, we find
E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t i
= E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t
, ∆ N t
= 0 i
(1 − ǫq t
)
+ E h
E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t
, ∆ N t
= 1 , R = r i
B t
, ∆ N t
= 1 i ǫq t
+ O ǫ
2
.
Inserting (36) and (38) above and rearranging terms, we obtain
E h g t + ǫ
˜ t + ǫ
,
˜ t + ǫ
B t i
= E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t
, ∆ N t
= 0 i
+ ǫ
Z
A
E h g t + ǫ x t + ǫ
, Σ t + ǫ
B t
, ∆ N
− E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t
, ∆ N t
= 0 i t
= 1 , R = r i
(42) · E [ λ t
( r, x t
) | B t
] dr + O ǫ
2
.
Conditioned on B t and ∆ N t
= 0, (12) and (13) can be solved during [ t, t + ǫ ) to obtain
(43) x t + ǫ x t
+ ǫA t x t
+ ǫB t u t
− ǫµ t h t x t
, Σ t
+ O ǫ
2
˜ t + ǫ
Σ t
+ ǫA t
˜ t
+ ǫ
˜ t
A
T t
+ ǫD t
D
T t
+ ǫµ t
H t
˜ t
,
˜ t
+ O ǫ
2
.
Also, conditioned on B t
, ∆ N t
= 1, and R = r , we can write
(44)
˜ t + ǫ
= ˜ t
+ ˜ t
( r − C t x t
) + O ( ǫ )
˜ t + ǫ
Σ t
−
˜ t
C t
˜ t
+ O ( ǫ ) .

192 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
Inserting (43) and (44) into (42), and linearizing with respect to ǫ , we obtain
E h g t x t + ǫ
,
˜ t + ǫ
B t i
= g t
˜ t
,
˜ t
+ ǫ
∂
∂t g t x t
, Σ t
+ ǫ
∂
∂ ˜ t g t x t
, Σ t
T
A t
˜ t
+ B t u t
− µ t h t x t
+ ǫ tr
∂
∂ Σ t g t x t
, Σ t
A t
˜ t
Σ
+ ǫ
Z
A h g t
˜ t
+ ˜ t
( r − C t x t
) ,
˜ t
− t
A
T t
˜ t
C
+ t
D t
D
T t
+ µ t
˜ t
− g t x t
,
H t
˜ t x t
, Σ t i
· E [ λ t
( r, x t
) | B t
] dr + O ǫ
2
.
, Σ t t
T
We substitute E [ λ t
( r, x t
) | B t
] from (41) into the expression above and use the linear operator L t
{·} defined by (16) to obtain the simplified form
(45) E h g t x t + ǫ
, Σ t + ǫ
B t i
= g t
+ ǫ x t
, Σ t
+ ǫ
∂
∂t g t
∂
∂ ˜ t g t
˜ t
,
˜ t x t
T
, Σ
( t
A t
˜ t
+ B t u t
)
+ ǫ tr
∂
∂
˜ t g t x
+ ǫµ t
L t n g t x t
,
˜ t t
, Σ t o
+ ǫδ
2 t
A t
Σ t
+ O ǫ
2
Σ t
A T t
+ D t
D T t where the error term δ 2 t is defined as
δ
2 t
=
Z
R n
Z
A h g t
˜ t
+ ˜ t
( r − C t x t
) ,
˜ t
−
˜ t
C t
˜ t
− g t x t
,
˜ t i
· p x t
( x | B t
) − ˜ x t
( x | B t
) λ t
( r, x ) drdx.
Define the nonlinear operator K t
{·} by
K t
{ g t
( x, Σ) } =
∂
∂t
−
1
4 g t
( x, Σ) +
∂
∂x g t
( x, Σ)
∂
∂x g t
( x, Σ)
T
T
B t
P t
− 1
B T t
A t x
∂
∂x g t
( x, Σ)
+ tr
∂
∂ Σ g t
( x, Σ) A t
Σ + Σ A T t
+ D t
D T t
+ Q t
Σ
+ x
T
Q t x + µ t
L t
{ g t
( x, Σ) } .
Then, (45) can be rewritten as
E h g t + ǫ x t + ǫ
,
˜ t + ǫ
B t i
= g t
˜ t
,
˜ t
+ ǫ u t
+
1
2
P t
− 1
B
T t
∂
∂ ˜ t g t
˜ t
,
˜ t
− ǫ ˜
T t
Q t
˜ t
+ tr n
Q t
˜ t o
+ u
T t
P t u t
+ ǫ K t n g t x t
,
˜ t o
+ O ǫ
2
.
2
P t
+ ǫδ
2 t

SHORT RANGE FREE-SPACE OPTICAL COMMUNICATION 193
Now, let g t
( · , · ) be the backward solution of the partial differential equation (17) with boundary condition g
T
( x, Σ) = x
T
Sx . This implies that K t g t
˜ t
,
˜ t
= 0. Under this condition, we take expectation from the equation above to get
E h g t + ǫ x t + ǫ
, Σ t + ǫ i
= E h g t x t
, Σ t i
(46)
+ ǫ E
" u t
+
1
2
P t
− 1
B
T t
∂
∂ ˜ t g t
˜ t
,
˜ t
2 #
P t
− ǫ E h
˜
T t
Q t
˜ t
+ tr n
Q t
˜ t o
+ u
T t
P t u t i
+ O ǫ
2
+ ǫ E δ
2 t
.
Step IV: We partition the interval [0 , T ) into K subintervals [ t k
, t k +1
) , k = 0 , 1 , . . .
,
K − 1, where t
0 g t
K
˜ t
K
,
˜ t
K
= 0, t
K
= T , and t k +1
− t k
, ǫ k
> 0. Recalling that
, we approximate the cost functional (34) by the finite sum x T
T
Sx
T
=
J ≃ J
K
=
K − 1
X ǫ k
E h
˜
T t k
Q t k
˜ t k k =0
+ tr n
Q t k
˜ t k o
+ u
T t k
P t k u t k
+ δ
1 t k i
+ E h g t
K
˜ t
K
,
˜ t
K i
.
This finite sum can be rearranged as
J
K
=
K − 2
X ǫ k
E h
˜
T t k
Q t k k =0 t k
+ tr n
Q t k t k o
+ u
T t k
P t k u t k
+ δ
1 t k i
+ E h
δ
1 t
K
−
1 i
+ ǫ
K − 1
E h
˜
T t
K
−
1
Q t
K −
1
˜ t
K −
1
+ tr n
Q t
K −
1
˜ t
K −
1 o
+ u
T t
K
−
1
P t
K −
1 u t
K −
1 i
+ E h g t
K
˜ t
K
,
˜ t
K i
.
In the right side above, we replace E h g t
K t
K
, minor manipulations, and upon defining δ t
= δ 1 t t
K
+ δ 2 t i by the right side of (46). With according to (19), we find that
J
K
=
K − 2
X ǫ k
E h
˜
T t k
Q t k
˜ t k k =0
+ E h g t
K
−
1 t
K
−
1
,
+ tr n
Q t k
˜ t k o
+ u
T t k
P t k u t k
+ δ
1 t k i t
K
−
1 i
"
+ ǫ
K − 1
E u t
K
−
1
+
1
2
P t
− 1
K
−
1
B
T t
K
−
1
∂
∂ ˜ t
K
−
1 g t
K
−
1 t
K
−
1
,
+ ǫ
K − 1
E δ t
K
−
1
+ O ǫ
2
K − 1
.
t
K
−
1
2
P tK
−
1
#
Repeating this procedure for k = K − 2 , K − 3 , . . . , 1 , 0, we obtain
J
K
= E h g t
0
˜ t
0
,
˜ t
0 i
+
K − 1
X ǫ k
E [ δ t k
] k =0
+
K − 1
X ǫ k
E
" k =0 u t k
+
1
2
P t
− 1 k
B T t k
∂
∂ ˜ t k g t k t k
, t k
2
P tk
#
+
K − 1
X
O ǫ 2 k k =0
Finally, we take the limit of J
K as K → ∞ and max ǫ k
→ 0 to obtain (18).
.

194 ARASH KOMAEE, P. S. KRISHNAPRASAD, AND PRAKASH NARAYAN
A.4. Proof of Theorem 5.2.
For A = R
2 and g t
( x, Σ) given by (21), we can show that
L t
{ g t
( x, Σ) } = f t
Σ − Γ t
(Σ) C t
Σ − f t
(Σ) + tr { Γ t
(Σ) C t
Σ K t
} which is clearly not dependent on x . Therefore, (17) can be decomposed into two independent equations: the partial differential equation (23) with boundary condition f
T
(Σ) = 0 and the following equation
−
∂ x
T
K t x
∂t
=
∂ x T K t x
∂x
!
T
A t x −
1
4
∂ x T K t x
∂x
!
T
B t
P t
− 1
B T t
∂ x T K t x
∂x
!
+ x T Q t x with boundary condition x T K
T x = x T Sx . This equation holds for any arbitrary x if and only if K t satisfies (22) with terminal condition K
T
= S .
REFERENCES
[1] H. A. Willebrand and B. S. Ghuman , Fiber optics without fiber , Spectrum, IEEE, 38(2001), pp. 40–45.
[2] D. Kerr, K. Bouazza-Marouf, K. Girach, and T. C. West , Free space laser communication links for short range control of mobile robots using active pointing and tracking techniques , in: IEE Colloquium on Optical Free Space Communication Links, pp. 11/1 – 11/5, Feb.
1996.
[3] J. Maynard , System design considerations , in: Laser Satellite Communication (M. Katzman, ed.), Ch. 2, pp. 11–67, Prentice-Hall, Inc., 1987.
[4] V. W. S. Chan , Optical satellite networks , IEEE, Journal of Lightwave Technology, 21(2003), pp. 2811–2827.
[5] R. M. Gagliardi and S. Karp , Optical Communication . John Wiley & Sons, Inc., 2th ed.,
1995.
[6] D. L. Snyder , Applications of stochastic calculus for point process models arising in optical communication , in: Communication Systems and Random Process Theory (J. K. Skwirzynski, ed.), pp. 789–804, Sijthoff and Noordhoff, Alphen aan der Rijn, The Netherlands, 1978.
[7] J. W. Goodman , Statistical Optics . John Wiley & Sons, Inc., 1985.
[8] I. B. Rhodes and D. L. Snyder , Estimation and control performance for space-time pointprocess observations , IEEE Transactions on Automatic Control, AC-22(1977), pp. 338–346.
[9] J. E. Kolassa , Series approximation methods in statistics . Lecture notes in statistics, Springer-
Verlag, 1994.
[10] J. A. Gubner , Filtering results for a joint control-decoding problem in optical communications ,
Master’s thesis, University of Maryland, 1985.