2500 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 10, MAY 15, 2016

Vibration Influence on Hit Probability DuringBeaconless Spatial Acquisition

Lothar Friederichs, Uwe Sterr, and Daniel Dallmann

Abstract—Starting November 2007 and continuing through thepresent, Tesat-Spacecom has been conducting successfully opticalintersatellite communication links utilizing laser communicationpayloads built by Tesat-Spacecom under DLR funding. A total offour communication payloads are in space at the time of writing.Also optical space-to-ground communication links have been suc-cessfully conducted using a compatible ground terminal built byTesat-Spacecom and the ESA Optical Ground Station on Tenerife.All terminals are using a beaconless spatial acquisition algorithmwhich was developed by Tesat-Spacecom for intersatellite com-munication links. We present an analytical investigation on theinfluence of vibrations deferring the laser beam search pattern onthe receiving probability of the counter terminal. A Gaussian dis-tribution of the vibrations is assumed. Results are presented hereand compared to computer simulations to verify the analytical re-sults. The influence of link margin and slave terminal position inthe scan field on hit probability is addressed.

Index Terms—Beaconless spatial acquisition, EDRS, ISL, lasercommunication terminal, SGL, space-to-ground links, TESAT.

I. INTRODUCTION

THE beaconless spatial acquisition algorithm investigatedin this paper allows designing a laser communication ter-

minal without a dedicated optical beacon. Spatial acquisitionis instead performed by the communication laser before it isin communication mode. Eliminating a dedicated beacon of-fers a substantial reduction of hardware requirements. How-ever, the influence of vibrations on the search pattern might be aconcern for both, duration and reliability, of the spatial acqui-sition. Both are key requirements for commercial laser com-munication systems such as the European Data Relay System(EDRS) in which laser communication terminals of TESAT areused for data transmission between low earth orbiting (LEO)and geostationary earth orbiting (GEO) satellites.

Results of hundreds of spatial acquisitions in the last eightyears between LEO-to-LEO, LEO-to-Ground, LEO-to-GEOand GEO-to-Ground have proven that the beaconless spatialacquisition is both, fast and reliable. Details of the beaconlessspatial acquisition algorithm can be found in [1].

Manuscript received October 9, 2015; revised December 18, 2015 and Febru-ary 18, 2016; accepted March 11, 2016. Date of publication March 15, 2016;date of current version April 11, 2016.

L. Friederichs was an independent consultant in Satellite Communica-tions based at Backnang 71522, Germany. He is now with the SatelliteTelecommunications Consulting, Esslingen 73733, Germany (e-mail: Lothar.Friederichs@web.de).

U. Sterr is with the Satellite Telecommunications Consulting, Esslingen73733, Germany (e-mail: uwe.sterr@st2c.de).

D. Dallmann is with the Tesat Spacecom, Backnang 71522, Germany (e-mail:Daniel.Dallmann@tesat.de).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2016.2542918

To optimize the beaconless spatial acquisition algorithm andits parameters a thorough investigation on the influence of vi-brations on the hit probability is necessary.

Rather than on the hit probability earlier work focused onmodeling the complete beaconless spatial acquisition [5] andthe duration thereof [6]. Also the optimum divergence of thelaser beam was investigated [7].

The in depth derivation of the analytical solution is given aswell as a short description of the computer simulation. The re-sults of the analytical method are presented to give insight intothe influence of vibration and link budget on hit probability,whereas the results of the computer simulation is used to verifythe two methods mutually. This verification was a fundamen-tal step to develop a tool that calculates the spatial acquisitionprobability for Teast-Spacecom’s laser communications termi-nals. The tool takes into account characteristics of both, termi-nals and satellites, and has been used to simulate LEO-to-GEOconstellations.

Gaussian distributed vibrations of unlimited bandwidth arenot physically plausible for a mechanical system. However, itis a useful model to explore the worst-case impact of vibrationson hit probability. Furthermore, very high vibration levels arenot realistic, but are presented for completeness.

The receive part is not considered. The only criterion of a hitbeing achieved is a defined intensity level being directed intothe direction of the counter terminal at a given instance in time.Outside the beam width the signal intensity is assumed not tobe sufficient for producing a hit. Electrical characteristics likenoise or bandwidth of the receiver are not considered.

II. METHODS

A. Analytical

The analysis considers the statistics of the angular distancebetween the true line of sight (LOS), which may be assumed any-where within the maximum scan radius around the propagator-derived expected LOS, and the direction of the actual scan angleat a certain time instant according to the position on the spiralwith the superimposed vibration angle.

The source of the vibrations are mainly from the transmittingsatellite platform, but can also originate from the communica-tions terminal itself. Fig. 1 illustrates the scan geometry withmaster terminal (M) and slave terminal (SL) (also designatedas “target”). The figure represents the scan geometry by itsparallel projection onto the (x,y)-plane orthogonal to the ex-pected LOS which is derived from and remains permanentlytracked according to the propagator data. Since the angles builtup in the scan constellation of M and SL are very small, the

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FRIEDERICHS et al.: VIBRATION INFLUENCE ON HIT PROBABILITY DURING BEACONLESS SPATIAL ACQUISITION 2501

Fig. 1. Scan geometry, showing a situation leading to a hit.

two-dimensional (2-D) representation in the (x,y)-plane are ac-curate enough to allow the x and y coordinates being angles andnot physical distances, i.e., x and y are measured in units of rador μrad instead of m or km. In this sense, “angle vector” �z fromM to SL is determined by its modulus |�z| = z, the offset anglefrom the expected LOS, and its elevation angle ζ with respectto the positive x axis. Similarly, vector �r from M to point SPon a spiral arm has the elevation angle ϕ. In order to identifythat SP is meant to be on the κth spiral arm between κ · 2π and(κ + 1) · 2π, the actual elevation angle is denoted as ϕ − κ · 2π.

The platform carrying the transmitting terminal M as wellas the platform carrying the receiving terminal SL are subjectto small attitude vibrations. The vibrations of the M platformare comparable in size to the laser scan beam angle, whereassimilar vibrations of the SL platform are much smaller thanthe field of view of the receive sensor of the SL terminal. Thislimits the scenario to be considered to the impact of vibrationsof the master Terminal M only. In Fig. 1, the angular distancebetween the true slave position (SL) and the end point of vector�r nominally moving along an Archimedes’ spiral with super-imposed random vibration vector �n.

The x and y components of �w can be considered as ran-dom variables shifted from the origin by scan position specific

offsets. When �w is expressed by its inline (i) and quadrature (q)components instead of the x and y components, the mathematicssimplify.

With |�w|, the modulus of �w, the criterion used in this anal-ysis to discriminate between hit and no-hit events is that a hitrequires |�w| to be less or equal to half the scanning beam widthBs , whereas a no-hit is associated to |�w| > Bs/2. Thus, noconsideration is given to the actual beam characteristic withinthe beam width Bs . The signal strength at scan beam edge isthus established as a reference for the definition of the spatialacquisition link margin.

The analytical approach includes the following steps, start-ing from independent Gaussian probability density functions(PDFs) of the x and y vibration noise components:

1) Determine the PDFs of the inline and quadrature vibrationnoise components.

2) Derive the PDFs of the squared noise components. Asexpected, these PDFs are χ2 PDFs.

3) Derive the PDF of |�w|2 , the squared length of vector �w,as the sum of the squared i and q components by meansof convolution in analytical form.

4) Derive the PDF of |�w|, the absolute value of vector �w.5) Integrate the PDF obtained for |�w| over the range from

0 to Bs , to obtain the probability of scan hit for a singlesampling instant.

6) Combine the hit probabilities for all sampling instants todetermine the total hit probability.

In performing the steps above, the methods given in [2] areapplied to compute the PDF of a random variable defined as afunction of another random variable.

1) Mathematical Formulation: The movement of a point onthe spiral is described by the time-dependent position vector inthe (x,y) coordinate system

�r(t) =B

2π· ϕ(t) ·

(cos ϕ(t)sin ϕ(t)

)(1)

where B is the radial distance between two adjacent arms of thespiral. The SL position with respect to master is indicated bythe vector

�z = z ·(

cos ζsin ζ

). (2)

The vibration noise vector is

�n =(

nx

ny

). (3)

Its components nx and ny are independent Gaussian variableswith zero means, i.e., E {nx} = μx = 0, E {ny} = μy = 0 andequal variances, i.e., E

{n2

x

}= σ2

x = σ20 and E

{n2

y

}= σ2

y =σ2

0 .The PDFs of the noise components are

px(nx) =1√

2π · σ0exp

(− n2

x

2σ20

)(4)

py (ny ) =1√

2π · σ0exp

(−

n2y

2σ20

)(5)

2502 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 10, MAY 15, 2016

For ease of mathematical formulation, the (x, y) co-ordinate system is now transformed into the local (i, q)coordinate system with the origin at the SL and the i-axis point-ing through the considered point on the scan spiral (SP). How-ever, the distance a = |�a| = (a2

x + a2y )1/2 between SL and SP

is invariant to the transformation and can therefore still be ex-pressed in (x, y) coordinates:

ax = z cos ζ − B

2πϕ · cos ϕ (6)

ay = z sin ς − B

2πϕ · sin ϕ. (7)

Similarly, the variance of the noise components is invariantto the coordinate transformation, i.e., σ2

i = σ2q = σ2

0 . Thus, thecomposite vibration noise, denoted as σ2

n , is twice the varianceof a single component, in whichever coordinate system it isconsidered

σ2n = 2σ2

0 . (8)

The difference vector �w between the target �z and the vibrationnoise loaded scan vector �r + �n is �w = �r + �n − �z, written in (i,q) as

�w = �a + �n =(

a0

)+(−ni

nq

). (9)

Normalizing to the spiral arm distance B, the square of itsabsolute value is

|�w|2

B2 =( a

B− ni

B

)2+(nq

B

)2or s = (A − Ni)2 + N 2

q (10)

where

s =|�w|2

B2 (11)

A =a

B(12)

Ni =ni

Band Nq =

nq

B. (13)

The criterion of scan success can now be expressed by

s =|�w|B2

2

≤(

12

Bs

B

)2

or w =√

s =|�w|B

≤ Bs

2B. (14)

The normalized random variables Ni and Nq have the PDF

pi/q (N) =1√

2π · σexp

(− N 2

2σ2

)(15)

with σ = σ0B . N stands for Ni or Nq .

The normalized standard deviation of the composite noise is

σN =√

2σ. (16)

The quantities

X = (A − Ni)2 , Y = N 2q , and s = X + Y (17)

represent new random variables for which the PDFs are deter-mined here below from those of Ni and Nq .

Fig. 2. Transformation Ni to X.

a) PDFs of Variables X and Y: The sample space of vari-able Ni is transformed by a parabolic function (12) into that ofvariable X, whereby X is confined to non-negative values.

As Fig. 2 illustrates, the X sample points of the interval[0 ≤ X ≤ ξ] correspond to Ni sample points in the interval[A −

√ξ ≤ Ni ≤ A +

√ξ].

Therefore, the probability of X being within[0 ≤ X ≤ ξ] equals the probability of Ni being within[A −

√ξ ≤ Ni ≤ A +

√ξ], i.e.,

P (X ≤ ξ) = P (A −√

ξ ≤ Ni ≤ A +√

ξ)

= P (Ni ≤ A +√

ξ) − P (Ni ≤ A −√

ξ).

(18)

Then

pX (ξ) =dPdξ

=d

dξ

∫ A+√

ξ

−∞pi(Ni)dNi

− d

dξ

∫ A−√

ξ

−∞pi(Ni)dNi. (19)

Thus, using the differentiation formula [3]

d

dy

β (y )∫α

f [x]dx = β′(y) · f [β(y)] (20)

the PDF of X, related to that of Ni , is

pX (ξ) =1

2√

ξ

[pi(A +

√ξ) + pi(A −

√ξ)]

=1

2√

ξ

⎡⎢⎢⎣

1√2πσ

exp(− (A +

√ξ)2

2σ2

)

+1√2πσ

exp(− (A −

√ξ)2

2σ2

)⎤⎥⎥⎦ (21)

pX (ξ) =1√2πσ

1√ξ· exp

(−A2 + ξ

2σ2

)

· cosh(

A√

ξ

σ2

). (22)

FRIEDERICHS et al.: VIBRATION INFLUENCE ON HIT PROBABILITY DURING BEACONLESS SPATIAL ACQUISITION 2503

The PDF of Y has, in analogy, the same structure as equation(22), but with A = 0, i.e.,

pY (η) =1√2πσ

1√

η· exp

[− η

2σ2

]. (23)

It is worth noting that pX (ξ) and pY (η) represent PDFs ofindependent χ2-distributions with one degree of freedom, non-central for pX (ξ) and central for pY (η).

b) PDF of Variable s: According to equation (17) the vari-able s is defined as the sum of the two independent random vari-ables X and Y. Its PDF is therefore obtained from the convolutionof the X and Y PDFs. Thus,

ps(s) =∫ s

0pX (X = ξ) · pY (Y = s − ξ)dξ. (24)

Inserting pX and pY according to (22) and (23) yields

ps(s) =1

2πσ2 · exp(− A2

2σ2

)· exp

(− s

2σ2

)

·∫ s

0cosh

{A√

ξ

σ2

}1√

ξ√

s − ξdξ. (25)

The substitution ξ/s = cos t makes the term

1√ξ/s

√1 − ξ/s

dξ

s

to become −2 dt and transforms the integration limits to t(ξ =0) = π/2 and t(ξ = s) = 0.

Then the integral in (25) becomes

−∫ 0

π/22 · cosh

{A√

s

σ2 cos t

}dt

=∫ π/2

0

[exp

(A√

s

σ2 cos t

)

+ exp(−A

√s

σ2 cos t

)]dt. (26)

The exponential terms are periodic in t and can therefore beexpressed as series of trigonometric functions with modifiedBessel functions of first kind as coefficients [4]

exp(z cos θ) =∞∑

n=0

εnIn (z) · cos(nθ) (27)

with εn ={

1 for n = 02 for n > 0 .

Then

[exp(z cos θ) + exp(−z cos θ)]

=∞∑

n=0

εn [In (z) + In (−z)] · cos(nθ). (28)

Using the symmetry relation

In (z) + In (−z) = [1 + (−1)n ] · In (z)

={

2I2n (z) for even n0 for odd n

(29)

integral in (26) can now be written as

∞∑n=0

εn2I2n

(A√

s

σ2

)·∫ π/2

0cos(2nt)dt

= π ·∞∑

n=0

εn · 2I2n

(A√

s

σ2

)(30)

because of∫ π/2

0cos(2nt) dt =

{π/2 for n = 00 for n > 0 . (31)

Replacing the integral in (25) by the right-hand expression of(30), the PDF of variable s is obtained as

ps(s) = exp(− A2

2σ2

)· 12σ2 exp

(− s

2σ2

)· I0

(A√

s

σ2

).

(32)The standard deviation σ refers to the vibration process in one

coordinate axis, the composite noise has the standard deviationσN =

√2 σ [see. (6)]. Equation (18) can thus be written as

ps(s) = exp(−A2

σ2N

)· 1σ2

N

exp(− s

σ2N

)· I0

(2A

√s

σ2N

).

(33)According to (14), the criterion of a scan hit is

s =|�w|B2

2

≤(

Bs

2B

)2

or w =√

s =|�w|B

≤ Bs

2B. (34)

The probability for s ≤ S is obtained from integrating thePDF of (33) over the interval 0 ≤ s ≤ S, where the parameterA is specific to the sampling point on the SP determined by thescan angle φ(tk ) = φk at time tk and is therefore called Ak .This is equivalent to integrating the PDF of variable w =

√s,

designated as pw (w), over the interval 0 ≤ w ≤ W . The latteris used here, since it provides also a convenient way to checkthe plausibility of the result.

c) PDF of Variable w: The PDF of variable w =√

s isobtained from the PDF of ps(s) by the following step:

P (w ≤ W ) = P (s ≤ S) =∫ S=W 2

0ps(s)ds. (35)

The derivative with respect to the upper bound is

pw (W ) =d

dW

{∫ W 2

0ps(s)ds

}=

d

dW

{W 2} · ps(W 2)

= 2W · ps(W 2). (36)

Let W take any possible value of w, then pw (w) is the PDFof the variable w.

Inserting (33) into (36), pw (w) is now obtained as

pw (w) =2w

σ2N

exp(−w2 + A2

σ2N

)· I0

(2Aw

σ2N

). (37)

The form of (37) corresponds to the Rice probability densityfunction known from signal transmission theory.

2504 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 10, MAY 15, 2016

With reference to Fig. 1, the meaning of symbols in (37) is:

A =[A2

x + A2y

]1/2(38)

Ax =ax

B=

z

Bcos ζ − 1

2πϕ cos ϕ (39)

Ay =ay

B=

z

Bsin ζ − 1

2πϕ sin ϕ (40)

σN = σ/B (41)

with

σ =√

σ2x + σ2

y =√

2σx =√

2σy (42)

I0(.) represents the modified Bessel function of first kind andorder zero.

2) Probability of Scan Hit: The probability that w is withinthe interval 0 ≤ w ≤ Bs/2B at sampling time tk with the in-herent constant A(tk ) is computed from the integral

Pk = P (w ≤ Bs/2B | tk ) = Pk =∫ Bs /2B

0pw (w |tk )dw.

(43)This integral cannot be solved analytically in closed form,

but is well suited for numerical integration due to the regularbehaviour of function pw (w |tk ).

The scan process is organized such that the signal receivedby the sensor on-board the slave terminal is sampled at a rate of25 kHz, i.e., in time steps of 40 μs which correspond to 4 μraddistance of scan points along the spiral track, i.e., less than B/2.Fig. 1 illustrates the situation just for one sampling instant tk .In the result, this yields a probability of success, or scan hit, Pk

for each sampling instant tk . Since the vibration noise vectors attimes tk and tiwith i �= k are independent, all hit probabilitiesPk are independent. The hit probability resulting for the totalscan process thus is

Phit = 1 −K∏

k=1

(1 − Pk ) (44)

where K is the total number of sampling instants.a) Special Case: Vibration Level Approaching Zero When

Slave is Located in the Middle Between Two Spiral Arms: It isworth to consider the probability of hit for the vibration levelapproaching zero in cases where the spiral does not pass throughthe slave terminal position. Then the constants Ak will not van-ish. Now the case σN → 0 lets the argument of the modifiedBessel function and thereby the Bessel function itself takes verylarge values. An approximation is therefore used [4]

I0(z) ≈ exp(z)√2πz

for |z| 1. (45)

Equation (37) then becomes

pw (w) ≈ 1σN

√w

πA· exp

[−(

w − A

σN

)2]

for σN → 0.

(46)In (46), the constant A is a function of tk , the sampling time

A = A(tk ) = Ak .

Replacing 1/σN by ρ, the integral to be considered becomes

Pk =∫ Bs /2B

0

√w

Ak· ρ√

πexp

[−ρ2(w − Ak )2] dw (47)

of which the limit for σN → 0 or ρ → ∞ is wanted

limρ→∞

Pk =∫ Bs /2B

0

√w

Ak

· limρ→∞

{ρ√π

exp[−ρ2(w − Ak )2]} dw. (48)

It is now useful to observe that the limit expression in theintegrand is the continuous representation of the unit impulse orδ-function [2]

limρ→∞

{ρ√π

exp[−ρ2(w − Ak )2]} = δ(w − Ak ). (49)

Therefore, making use of the sifting property of the δ-function, i.e., ∫ +∞

−∞f(x) · δ(x − x0)dx = f(x0) (50)

the limit for Pk is now

limρ→∞

Pk =∫ Bs /2B

0

√w

Ak· δ(w − Ak )dw

=

{√Ak

Ak= 1 if Ak ≤ Bs

2B

0 if Ak > Bs

2B

. (51)

In the sense of the model used as the basis of this analysis,(51) means that all sampling points inside the circle with radiusBs /2 around the SL will lead to limρ→∞ Pk = 1, whereas allsampling points outside this circle imply limρ→∞ Pk = 0.

Reversely, if circles of radius Bs /2 are drawn around eachsampling point on the spiral, the areas outside these circles are“white patches” positioned along a virtual spiral track half-waybetween the SP tracks. These white patches are not covered bythe scan process, but only in the limiting case of σN → 0. Inthis context, the ratio of scan beam width to spiral arm distanceis essential, as is illustrated in Fig. 3 below.

In Fig. 3(a) with scan beam width equal to the spiral arm dis-tance, numerous small white patches can be observed betweenthe circles. Their number is considerably reduced when Bs isslightly enlarged by 6.5% [see Fig. 3(b)]. Some small whitepatches are marked by ellipses. Fig. 3(c) with 10% increase ofscan beam width demonstrates that no white patches will occuranymore.

b) Behaviour of Hit Probability for Non-Vanishing Vibra-tion Level: A transition range exists when σvibr increases fromzero to moderate levels of about 1 μrad in those cases wherethe slave terminal is located at or near the middle between twospiral arcs. In the end, however, the scan hit probability buildsup for further increasing vibration levels and approaches thevalues obtained for the cases where the slave terminal is wellaway from the center location between two spirals.

There is no analytical way to predict the hit probability be-haviour in this transitional range of very small to moderate

FRIEDERICHS et al.: VIBRATION INFLUENCE ON HIT PROBABILITY DURING BEACONLESS SPATIAL ACQUISITION 2505

Fig. 3. Examples of scan coverage in case of no vibration for scan beamwidths of 1.0 B, 1.065 B and 1.1 B.

TABLE ICOMPARISON OF ANALYTICAL AND ESTIMATED RESULTS

FOR HIGH VIBRATIONS

Bs /B Ph i t , analysis (%) Ph i t , estimated (%)

1.0 81.57 81.521.065 85.31 85.271.1 87.07 87.041.2 91.24 91.211.3 94.26 94.241.4 96.37 96.351.5 97.77 97.76

vibration levels. The only way to obtain insight is by numericalevaluation or by simulation.

All results obtained from the evaluation of (44) for the totalhit probability as function of the vibration standard deviationfor a given Bs /B ratio prove to end up at the same asymptoticvalue, irrespective of the actual offset of the slave terminal fromthe propagator derived position, be it small or medium, or verylarge. Unfortunately, (44) is not suited to derive this asymptoticvalue.

There is, however, a simple approach to estimate this value:At high offsets the spiral arcs may be linearized within a rect-angular window of width k1 · B and height k2 · vtT , whereB is the spiral arm distance, vt is the tangential velocity, andT is the sampling interval. In total, this window encompassesthe area Aw = k1B · k2vtT = k · B · vtT with k = k1 · k2 , thetotal number of possible, non-overlapping circular areas with ra-dius Bs/2. If one of these circular areas is covered by the largelyexcursed scan beam (which is surely expected to happen), a hitis produced. The probability of one such hit is proportional tothe ratio of the circle area and the large window area

Pone hit = π(Bs/2)2/(k · vtT · B) (52)

Vice versa, the probability that the circle is not hit, the prob-ability is

1 − Pone hit = 1 − π(Bs/2)2/(k · vtT · B). (53)

For the total number k of circles with radius Bs/2, the non-hitprobability must be taken to power k. Consequently, the total hitprobability for all circles in the window is

Phit = 1 −[1 − π(Bs/2)2

k · vtBT

]k

= 1 −[1 − πB

k · 4 · vtT·(

Bs

B

)2]k

(54)

Now the window is extended and k approaches infinity. Thus,the wanted asymptotic value of the hit probability for σN → ∞is obtained

limk→∞

Phit = 1 − exp

(− πB

4vtT·(

Bs

B

)2)

(55)

Numeric evaluation of (55) coincides with high accuracy withthe values taken from Figs. 4–12 as shown in Table I.

2506 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 10, MAY 15, 2016

Fig. 4. Scan hit probability for small slave offset right between spiral armswith different Bs /B ratios.

Fig. 5. Scan hit probability for medium slave offset right between spiral armswith different Bs /B ratios.

B. Computer Simulation

The computer simulation randomly defines a position of thecounter terminal and calculates the angular distances to thatposition for a defined number of vibration disturbed spirals. Ahit is achieved if the angular distance is below the maximumallowed value. This maximum angular distance translates via thelaser beam model to a minimum directed intensity into directionof the counter terminal. The ratio spirals triggering a hit and thetotal number of spirals defines the hit probability.

III. RESULTS

A. Analytical

Equation (44) has been numerically evaluated for vibrationlevels in the range from 0.02 to 80 μrad with the slave locatedat a series of offset angles z ranging from close to the center

Fig. 6. Scan hit probability for large slave offset right between spiral armswith different Bs /B ratios.

Fig. 7. Scan hit probability for small slave offset at four tens of spiral armdistance with different Bs /B ratios.

of the spiral up to several hundreds of spiral arm distances, inparticular around the value of 2B, 10B, and 300B.

When considering the scan hit probability for increasing off-set from the spiral center, the first observation is that the scanhit probability does not globally depend on the offset of theslave terminal from the propagator-derived LOS. This becomesevident from comparison of Figs. 4, 5, and 6 or Figs. 7, 8, and 9or Figs. 10, 11, and 12. It is instead the fine position of the slaveterminal between the spiral arms, which effectively determinesthe hit probability.

Detailed results are presented in Figs. 4 through 12. In ad-dition to cases with close-by as well as far-out SLs, where theslave is assumed somewhere between the undistorted spiral arms(here at 0.4 B from a spiral arm), also some specific cases wereconsidered, i.e., when the slave is located exactly in the middleof two spiral arms and when it is directly on the spiral curve.

FRIEDERICHS et al.: VIBRATION INFLUENCE ON HIT PROBABILITY DURING BEACONLESS SPATIAL ACQUISITION 2507

Fig. 8. Scan hit probability for medium slave offset at four tens of spiral armdistance with different Bs /B ratios.

Fig. 9. Scan hit probability for large slave offset at four tens of spiral armdistance with different Bs /B ratios.

For all SLs different values of the scan beam width have beenconsidered, from 1.0 up to 1.5 spiral arm distances.

Case 1: Slave locations right between two spiral arms at small,medium and large offsets from predicted LOS.

The influence of sampling time variation is displayed in Fig. 5as an example. The standard sampling time is assumed to startat the origin of the SP in 40 μs intervals if not explicitly statedotherwise. In case of the two lower sets of curves in Fig. 5, avariation of time reference is introduced by the time offset t0 insteps of 10 μs. A noticeable effect on the hit probability curvesis observed only at vibration levels of less than about 2 μrad.In general, the influence of time variation on the hit probabilityhas to be investigated for each case separately. Since vibrationsbelow 2 μrad are not to be anticipated in practice, no furtherinvestigation has been conducted.

Fig. 10. Scan hit probability for small slave offset right on spiral arm withdifferent Bs /B ratios.

Fig. 11. Scan hit probability for medium slave offset right on spiral arm withdifferent Bs /B ratios.

Case 2: Slave locations at an arbitrary position between twospiral arms at small, medium and large offsets from predictedLOS.

Case 3: Slave locations right on spiral arm at small, mediumand large offsets from predicted LOS.

The most important conclusion from the results obtained isthat the hit probability asymptotically approaches a constantvalue for large vibration levels which is 0.816 if the scan beamwidth Bs equals the spiral arm separation B, and increases to0.9776 for Bs = 1.5B.

For low vibration levels with σ below about 5 μrad, adeformation of the curves is observed leading to develop a rela-tive minimum. The closer Bs approaches the spiral arm distanceB, the more pronounced the effect is.

Even greater influence on this effect comes from the slaveterminal position. When the slave gets close to the exact middlebetween two SP arms, i.e., at B/2 distance from either arm, a

2508 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 10, MAY 15, 2016

Fig. 12. Scan hit probability for large slave offset right on spiral arm withdifferent Bs /B ratios.

Fig. 13. Improvement of Phit and associated reduction of scan beam gain ΔGfor enlarged scan beam width.

singular behaviour occurs for the case of Bs = B. In this case,and only here, the scan hit probability curve approaches zerowhen σVibr decreases to very small values. In the extreme,when σVibr reaches zero, i.e., when there is no vibration, thescan process is no longer stochastic. The slave terminal will inthis extreme case be hit by the scan beam, if the beam of widthBs = B points exactly to the intersections of the spiral armswith the line through the origin and the slave terminal position.Only at these two points, the slave terminal is just still withinthe scan beam, equivalent to scan hit. However, the samplinginstants will in general not coincide with these points in time,which explains the zero hit probability in this condition.

When σVibr increases from zero to values in the order of a fewμrad, the scan hit probability approaches the normal behaviour.

It is important to observe, that with an increase in scan beamwidth, even if only small, the short falling effect of hit probabilitydisappears, as the curves for Bs ≥ 1.065B prove.

TABLE IICOMPARISON OF ANALYTICAL AND SIMULATION RESULTS

B Bs σv ib r hit prob. hit prob.(μrad) (μrad) (μrad) z/B analysis (%) simulated (%)

8.6 8.6 20 300 81.6 828.6 12.9 20 300 97.8 98.18.6 8.6 2 300.4 84.63 84.2

No such effects at all, however, occur when the SL fallsdirectly on the spiral track.

An important final conclusion becomes evident from the fig-ures above: When Bs is raised above 1.065 B, the minimum scanprobability which is achieved at high vibration level, increasessignificantly. It reaches values above 0.9775 for Bs/B = 1.5,as shown in Fig. 13.

The improvement of scan hit probability due to enlarged scanbeam width implies a corresponding reduction of scan beamgain which, however, can be tolerated as long as the overallacquisition link margin still is sufficient. The actual scan hitprobability performance is thus determined by the available linkmargin during the acquisition phase. In conclusion, the scan hitprobability can be expected well above the curves for Bs = B.

B. Computer Simulation and Comparison to AnalyticalResults

Computer simulated results were generated for case 3 withz/B = 300 (see Fig. 12) and different values of Bs vibrationlevel and slave terminal position as shown in Table II.

IV. CONCLUSION

The comparison between analytical and simulated resultsshows very good agreement. Therefore, the computer simulationcan be used to calculate hit probability for any kinds of vibrationsincluding vibrations described in time domain. In particular, thesimulation tool is not limited to Gaussian distributed vibrations.This proves very helpful since the time domain format is com-monly used in discussions with satellite manufacturers.

The analytical results indicate for Bs = B that:1) for high vibration levels, σvibr→�, the hit probability is

0.816.2) any (unknown) position of the slave within the scan field

will be hit with a probability of at least 0.816, as long asσvibr � 2.5 μrad.

3) for lower vibration levels, σvibr < 2.5 μrad, hit probabilityvalues below 0.816 may occur, but only when the slave islocated within 0.4 to 0.6 times the spiral arm distance.

However, higher hit probabilities will be achieved if the ratioBs /B is increased. This can be achieved by reducing B whichresults in a longer spatial acquisition duration. The other possi-bility is to improve the link budget in order to allow increase ofBs . The latter conforms perfectly to the scan beam real shapewhich is Gaussian.

FRIEDERICHS et al.: VIBRATION INFLUENCE ON HIT PROBABILITY DURING BEACONLESS SPATIAL ACQUISITION 2509

The results presented show the potential of parameterizing thebeaconless spatial acquisition algorithm to meet requirementsconcerning reliability and speed.

ACKNOWLEDGMENT

The LCT (Laser Communication Terminal) development andthe in-orbit verification was sponsored by the German SpaceAgency DLR/BMWi under 54YH1344.

REFERENCES

[1] U. Sterr, M. Gregory, and F. Heine, “Beaconless acquisition for ISL andSGL, summary of 3 years operation in space and on ground,” in Proc. Int.Conf. Space Opt. Syst. Appl., May 2011, pp. 38–43.

[2] W. B. Davenport and W. L. Root, An Introduction to Random Signals andNoise. New York, NY, USA: McGraw-Hill, 1958.

[3] I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Mathematik, 3rdEded., Leipzig, Germany: B.G. Teubner, 1960.

[4] M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions,Frankfurt, Germany: Main Verlag Harry Deutsch, 1984.

[5] C. Hindman and L. Robertson, “Beaconless satellite laser acquisition—Modeling and feasibility,” in Proc. Mil. Commun. Conf., Oct./Nov. 2004,vol. 1, 2004, pp. 41–47.

[6] X. Li, S. Yu, J. Ma, and L. Tan, “Analytical expression and optimizationof spatial acquisition for intersatellite optical communications,” Opt. Exp.,vol. 19, no. 3, pp. 2381–2390, Jan. 2011.

[7] M. Toyoshima, T. Jono, K. Nakagawa, and A. Yamamoto, “Optimum di-vergence angle of a Gaussian beam wave in the presence of random jitterin free-space laser communication systems,” J. Opt. Soc. Amer. A, vol. 19,no. 3, pp. 567–571, 2002.

Lothar Friederichs was born in Wuppertal, Germany, in 1938. He receivedthe Diplom-Ingenieur degree in electrical communications technology from theTechnical University of Aachen, Aachen, Germany, in 1964. From 1964 to 1968,he was a System Engineer with the Space Communications Department of AEG-Telefunken (later ANT), Backnang, Germany. In 1969, he became the Head ofthe Satellite Systems Engineering Section. From the mid-1970s until 1994, hewas responsible for the communications payload engineering activities of ANT.Since 1995, he has been an Independent Consultant in the field of SatelliteCommunications. In this role, he was an Advisor to a number of space projectsin and outside Europe. His consulting activities include engineering supportand theoretical work of which an investigation on the phase probability densityfunction of an M-ary PSK modulated carriers with added narrowband Gaussiannoise is an example (www.audens.de/pdf_downloads/ProbDensFunctions.pdf).During recent years, he entered investigating problems relating to acquisitionof in-orbit laser links.

Uwe Sterr was born in Stuttgart, Germany, in 1966. He received the B.Sc.degree in communications technology from the University of Applied ScienceEsslingen, Esslingen am Neckar, Germany, in 1992, the M.Sc. degree in digitaldata communication systems from the University of Brunel, Uxbridge, U.K.,in 1994, and the Ph.D. degree in antenna technique from the University ofLondon, Queen Mary, and Westfield College, London, U.K., in 1998. Since1998, he has been working in the field of optical intersatellite communication.He was in charge of the in-orbit verification of Tesat-Spacecom’s LEO-LEOsystem and takes part in the in-orbit verification of Tesat-Spacecom’s GEO-LEOsystem. Since 2010, he has been the CEO of the Satellite TelecommunicationConsulting Company ST2C, Esslingen, Germany. ST2C develops simulationtool for beaconless spatial acquisition systems and supports in-orbit activitiesof laser communication systems

Daniel Dallmann was born in Heilbronn, Germany, in 1971. He received theDiploma degree in electrical engineering with a focus on telecommunicationsfrom the University of Stuttgart, Stuttgart, Germany, in 1998. He started hiscarrier with Bosch-Telecom, Grasbrunn, Germany, in the field of software en-gineering. Between 1998 and 2003, his main task was software developmentin the frame of a feasibility study for an in-orbit processing system includinga demonstration prototype of a packet switching system. Since 2003, the mainemphasis of his work has been Tesat-Spacecom’s optical intersatellite laser com-munication terminal, currently responsible as a System Engineer for softwareand algorithm aspects of pointing, acquisition, and tracking.

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