Master Thesis
Electrical Engineering with emphasis on Signal Processing
October 2013
Evaluation of measurement data for an optical free-space aircraft to ground communication link
system
VEVEK SELVARAJ
German Aerospace Center Institute of Communication and Navigation
82234 Oberpfaffenhofen Germany
School of Engineering Blekinge Institute of Technology 37179 Karlskrona Sweden
External Advisor(s) Joachim Howarth
German Aerospace Center
Institute of Communication and Navigation 82234 Oberpfaffenhofen
Germany
Phone: +49 8153 28-1832 University Examiner:
Sven Johansson
Department of Applied Signal processing
School of Engineering
Blekinge Institute of Technology 371 79 KARLSKRONA SWEDEN
Internet: www.bth.se/com Phone: +46 455 385000 SWEDEN
Abstract
Optical free-space communication is an effective approach that can provide tap-proof, high bandwidth data communications. In this thesis there is an Optical link established between aircraft and the ground station. The link created between aircraft and ground station will go through the wavefront distortions due to the atmospheric turbulence,hence the signal quality is degraded. In order to optimise system performance the impact of atmo- spheric effects have to be evaluated for different flight conditions and link distances. To compensate the wavefront distortions Adaptive Optics can be used. In this thesis the main aim is to process the measured data from different optical sensors in order to extract amplitude and phase informa- tion of the received field. The downlink data from the aircraft to the optical ground station that are obtained from Adaptive Optics as in this case Shack- Hartmann wavefront Sensors in the ground station is analysed. The other part in the thesis is the analysis of the power vectors received during the data transmission from aircraft to the ground station. Finite State Space Markov Model is created for the analysis of the power vectors that are re- ceived with receivers of different aperture size.
Keywords: Optical Free-space communication, Power Vectors, Shack-Hartmann wavefront Sensors and Finite State Space Markov Model.
i
This thesis is the final project work for the Master of Science in Electrical Engineering with emphasis on Signal Processing at the Department of Elec- trical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden.
This thesis work has been performed at Deutsches Zentrum fr Luft- und Raumfahrt e.V., Oberpfaffenhofen, Germany, under supervision of Joachim Horwath and Ramon Mata Calvo. Without their support and guidance, this thesis would have not been possible. They guided me all through the way and made me explore lot of intricacies in the project. They gave lot of directions to make me propose a very good method. Also, i thank each and everyone in the Optical Communication group at DLR for all their friendly approach during any circumstances.
I am especially grateful to all the friends I met during my thesis in Germany, who made my staying enjoyful and my friends in Optical Com- munication group who stood by my side. Last, I would like to give my special thanks to my parents and my sister for their unconditional support.I would not have been successful professional without their love and driving force.
ii
Contents
Abstract i
Acknowledgment ii
Contents iii
List of Figures v
List of Tables vii
Abbreviations viii
Thesis Outline ix
Statement of Work x
Introduction 1
1 Introduction 1
1.1 Free-Space Optical Communication . . . 1 1.2 Wavefront Sensors . . . 4 1.3 Finite State Markov Model . . . 8
Theoretical Background 9
2 Theoretical Background 10
2.1 Shack-Hartmann Wavefront Sensor . . . 10
iii
Implementation 18
3 Implementation 18
3.1 SHS Wavefront Reconstruction . . . 18 3.2 Finite State Markov Model . . . 24
Results 27
4 Results 27
4.1 SHS Wavefront Reconstruction . . . 27 4.2 Finite State Markov Model . . . 31
Conclusions 37
5 Conclusions 38
Future Work 39
6 Future Work 39
Bibliography 40
iv
List of Figures
1 Free Space Laser Communication Architectures . . . 2
2 Optical Ground Station . . . 2
3 Atmospheric Transmission Monitor . . . 4
4 Shack-Hartmann Sensor . . . 5
5 Shack-Hartmann Raw Image in pixels . . . 5
6 Setup of the Shack-Hartmann Sensor optical system . . . 6
7 Aircraft Downlink Terminal . . . 7
8 GEC model . . . 9
9 Finite State Markov Chain . . . 9
10 Reconstruction Model . . . 11
11 Zernike Modes . . . 13
12 Residual Errors . . . 15
13 Wavefront Reconstruction Flowchart . . . 19
14 Shack-Hartmann Raw Image . . . 19
15 Shack-Hartmann Dark Image of Integration time 100s . . . . 19
16 GUI for Wavefront Reconstruction . . . 20
17 Shack-Hartmann Image after offset removal . . . 21
18 Gaussian filtered Shack-Hartmann Image . . . 21
19 Create Mask . . . 21
20 Batch Reconstruction GUI . . . 22
21 Phase of the reconstructed image . . . 23
22 Reconstructed parameters of the SH image . . . 23
23 Channel Measurement segmentation for K=8 states . . . 24
24 Power Vector for K=8 states . . . 25
25 PDF Estimate . . . 26
26 Normalized Autocovariance of channel measurements . . . 26
27 Phase of the reconstructed image . . . 27
28 Reconstructed parameters of the SH image . . . 28
29 Focus camera image analysis flowchart . . . 29
30 Focus Camera raw image . . . 29
31 Focus Camera final image . . . 30
v
36 PDF for Tangential 60Km 40cm PIN diode . . . 33
37 PDF for Tangential 60Km 5cm Steuerbord diode . . . 34
38 PDF for Tangential 60Km 5cm Backbord diode . . . 34
39 PDF Estimate for Radial Approach . . . 35
40 PDF Estimate for Tangential Approach . . . 35
41 Normalized Autocovariance for Radial Approach . . . 36
42 Normalized Autocovariance for Tangential Approach . . . 36
vi
List of Tables
1 Shack Hartmann system specifications . . . 7
2 Zernike Modes and Aberrations . . . 13
3 Residual Mean Square Error . . . 15
4 State Transition Probability Matrix . . . 25
5 Radial Approach . . . 31
6 Tangential Approach . . . 31
vii
FSO F ree Space Optics
SHWS Shack Hartmann wavef ront Sensor
FSC F inite State Channel
CCD Charge Coupled Device
FSMC F inite State M arkov Channel
OGS Optical Ground Station
OP Oberpf af f enhof en
SGL Space− to − Ground Link
OCG Optical Communication Group
DLR DeutschesZentrumf rLuf t− undRaumfahrte.V.
PIN P hoto Intrinsic
AO Adaptive Optics
SHS Shack Hartmann Sensor
FSC F inite− state channel
GEC Gilbert− Elliott channel
SNR Signal− Noise Ratio
GUI Graphical user interf ace
COG Centre of Gravity
PDF P robability Distribution F unction ALTAO Alternative Adaptive Optics
viii
Thesis Outline
Adaptive optics is the technology for correcting random optical wavefront distortions in real time. An adaptive optics system measures the distortion with a wavefront sensor and adapts a wavefront corrector to reduce the phase distortion and retrieve the original signal.
The research aim of this thesis work is divided into two parts. The first part shows the Wavefront reconstruction using SHWS of the ALTAO measurement campaign that took place in October 2012 and June 2013 with different specifications. The final outcome of this part shows the effective reconstruction of wavefront with increased lenslets per array. The other part is the development of Finite State Space Markov model, where the received power vectors from the ground station are modeled according to FSMC model.
ix
Chapter 1: The first chapter is the introduction of the thesis and this part gives brief introduction about Optical Free Space communication along with the adaptive optics system explaining in detail working of the Shack- Hartmann Wavefront Sensor. The chapter also gives brief introduction about the Finite State Space Markov Model.
Chapter 2: In the second chapter, the theoretical background of the wave- front reconstruction of the Shack-Hartmann Wavefront Sensor is explained.
The chapter also introduces the theoretical background involved in the de- velopment of the Finite State Space Markov Model with properties like Probability distribution function and autocovariance function.
Chapter 3: In the third chapter, the implementation principle involved in the wavefront reconstruction and Finite State Space Markov Model is ex- plained in detail. The simulation procedure for the wavefront reconstruction and Markov Model is also discussed.
Chapter 4: In the fourth chapter, the results achieved based on the imple- mentation principle explained in the Chapter 3 are shown. The results from the simulation is analysed and compared.
Chapter 5: In the last chapter, the conclusions of the thesis is presented by summarizing the general results of the simulation.
x
Chapter 1
Introduction
1.1 Free-Space Optical Communication
Free-space optical communication is considered to be one of the key tech- nologies for realizing very high speed multi gigabit per second large capacity aerospace communication [1]. Optical free-space Communication is a rela- tively new technology which uses high data transmission rates and provides high bandwidth, wireless and communication between remote sites. For the past thirty years several advancement has been made in electro-optics and opto-electronics, which uses latest developments in the fiber communications [1]. Free-space optics extends the communications to remote sites involving mobile partners like satellites or aircraft’s. Ground-to ground can be used for short-ranges where no fiber infrastructure is available, but the goal are satellites and aircraft’s. Several systems have been developed based on Op- tical free space communication wherein it has been developed for ground- to-ground, ground-to-aircraft, ground-to-satellite and satellite-to-satellite.
Figure 1. shows the Optical free space laser architectures for above men- tioned scenarios. [1].
1
Intersatellite link
Aircraft-Ground link Satellite-
Ground link
Figure 1: Free Space Laser Communication Architectures
Figure 2: Optical Ground Station
The main advantages of using Optical Communication systems are [1],
· No spectrum regulations
· High data rates
CHAPTER 1. INTRODUCTION 3
· Tap-proof
· Low power consumption
· Interference insensitivity
· Portable and quick deployment
The Optical Communication Group in The German Aerospace Cen- ter(DLR) was started in late 1980’s with focus on the study of optical free space communication. The goal is to develop new technologies for free-space optical communications. The role that plays the atmosphere is one of the study topics in the group, since the signal is distorted when the signal propa- gates through the atmosphere. The OCG at DLR, investigates new concepts and technologies in order to improve the performance of the free space opti- cal links between ground stations, satellites and aircrafts.In the Institute of Communication and Navigation at DLR, on the top of the building is the OGS which is used for the reception of the signals from the aircraft. The OGS situated in the building is shown in the Figure 2 [3]. Under ALTAO a measurement campaign was organized in October 2012 and June 2013 with the scope of atmospherical turbulence characterization, in the frame of de- veloping new Adaptive Optics technologies for communication applications.
The goal of the measurement campaign was to obtain an overall picture of the atmospheric turbulence. For that purpose, several instruments where synchronized. Each instrument took images or recorded samples of the in- coming signal at the same time. The instruments were: a focus camera, a pupil camera, a Shack-Hartmann sensor and three power sensors, one using the telescope aperture of 40cm and two using two collocated aperture of 5cm.The overall setup of the OGS consists of Focus Camera, Pupil Cam- era, Shack-Hartmann Wavefront Sensor, PIN Power sensors and Differential Image motion monitor. The overall setup is shown in the Figure 3 [3].
Figure 3: Atmospheric Transmission Monitor
1.2 Wavefront Sensors
Adaptive Optics is a technology based on a principle to correct the wave front distortions or aberrations which is induced by atmospheric turbu- lence. When transmitting the optical signal through the atmosphere, the signal is distorted by the atmospheric turbulence. Atmospheric turbulence comes from index-of-refraction fluctuations. Small changes in the index-of- refraction distorts the signal phase. Phase distortions produces interference (constructive and destructive) of the wave with itself which is intensity fluc- tuations, i.e. scintillation. Speckles can be seen, some areas are bright and some are dark. All the scenarios is before the telescope, also when signal arrive at the receiver. Now if we want to collect all the light, put the light into a fiber and also we use a lens to focus the light into a fiber. If we use a small aperture, we will get a lot of signal fluctuations and sometimes we will see dark areas and some times we will see bright areas. Therefore we will use a big aperture, taking a lot of speckles both dark and bright area and make sure that we get always same power. In that way we can reduce our power scintillations and this is aperture averaging effect.But now if we focus the light we will not get one spot, but a lot of speckles at the focus, dancing and changing which will be impossible to couple into a Single Mode optical fiber since it has too much losses. The speckles at the focus are due
CHAPTER 1. INTRODUCTION 5
to the phase distortions of the incoming wave so we need to correct these distortions. AO uses the technique to correct these distortions.
Adaptive Optics system consists of three basic components. They are Wavefront sensors, Deformable mirrors and Simulation Software. The oper- ation methodology of the system is, the wavefront sensors collect information from the received wavefront and send it to the system which in turn creates the signal to control the deformable mirrors. Based on the signal received the deformable mirror wavefront phase to correct the aberrations. The sys- tem software reconstructs the wavefront phase based on the wavefront sensor measurements.
Figure 4: Shack-Hartmann Sensor
Figure 5: Shack-Hartmann Raw Image in pixels
optics system to measure the atmospheric turbulence or aberrations is the Shack-Hartmann sensors. The schematic representation of Shack-Hartmann sensor is given in Figure 4 [4]. It consists of an array of lenslets of same focal length and CCD camera. The incoming wavefront is focused into multiple spots by an array of lenses. The incoming wavefront create spots on a CCD camera. If the wavefront is plane we get an image with centered focus spot and if it is distorted the focus spots gets displaced from the center.
Measurements of the displacement of the centroid of each spot from its expected position are directly proportional to the wavefront gradient. The tilt of the wavefront for each lenses is calculated from the position of the focal spot on the sensor. The slopes of the spots in the sub aperture is calculated and with these the wavefront is reconstructed. Figure 5, shows the raw image captured using SHWS. Each tiny spot of the image represents the spot created by each lenslet array.
Figure 6: Setup of the Shack-Hartmann Sensor optical system
CHAPTER 1. INTRODUCTION 7
Figure 7: Aircraft Downlink Terminal
The SHWS of the OGS-OP was designed to measure the phase of the incoming beam with high resolution. Each focus spot covers an area of 12x12mm in the entrance pupil of the telescope.
Table 1: Shack Hartmann system specifications SHS Specifications
Camera Type Xenics Xeva XS
Lenslet-Diameter/Pitch[mm] 0.150000
FocusLength[mm] 3.600000
Wavelength[nm] 1550.000000
TelescopeApertureDiameter[m] 0.400000
Magnification 80.000000
PixelSize[μm] 30.000000
ExpectedSpotRadius[Pixel] 1.53430
ExposureTime[μs] 100.000000
Gain 140
TargetFrameRate 1000
Pixel/Lenslet 5.000000
ROILef t 0
ROIT op 0
ROIW idth 320
ROIHeight 256
Digital Shift 2
The intensity distribution can be either read from the SHS with high temporal resolution (20fps) or with high spatial resolution from the Turbu-
the number of lenslets, 33x33. It makes only sense to crosscheck with the pupil camera, but the intensity is measured by the pupil camera. The Table 1 lists the Shack Hartmann Sensor system specifications. Figure 6 shows the optical setup of the SHS and the SHS camera, a Xenics Xeva XS and uses InGaAs technology, also for 1550nm. Two relay lenses form a new conjugate plane for the lenslet array location. The lenslet array consists of 33x33 lenslets with a diameter of 150μm and a focal length of 36mm.
Effective focal length including the telescope with a magnification of 80 is fef f=1.4mm.
The aircraft downlink terminal of DLR attached in DO-228 aircraft is shown in Figure 7 [3].
1.3 Finite State Markov Model
The study of finite state communication channels with memory was started by Shannon[5] in 1957. Shannon proved that channel measurements could be calculated from the receiver, not necessarily from the receiver. In 1960, Gilbert introduced a new type of FSC model where the channel output de- pends on channel states and inputs. The channel output was the probabilis- tic function of current channel input and current channel state. Gilbert’s model[6] was the first channel model with memory. Elliott [7] improvises Gilbert model with evaluation and comparing the error rate performance of error correcting and error detecting codes. The model developed is Gilbert- Elliott model. The basic GEC model is shown in Figure 8, where G is good state and B is bad state. The channel states are statistically independent of channel input symbols and unknown to the transmitter or receiver. In 1968, McCullough [8] introduced more channel states in the model. During the same year, Gallager [9] further developed FSMC with advanced information theory. The model developed has properties where the channel state tran- sition is driven and controlled by the channel input. The channel state is statistically independent of channel input. From then on several researches had been proposed but the work done by Wang and Moayeri FSMC model in 1995 [10] for fading channels in mobile radio communications is being widely used. In the work, there is an explicit relation between the fading channel and FSMC states. Also, each FSMC state represents the range of SNR which represents error probability in the state. Based on this, they provided ex- pressions for states, state transition probabilities and error probability in each state.
CHAPTER 1. INTRODUCTION 9
Figure 8: GEC model
In this thesis, FSMC model was considered for K=8 and K=64 states with different vector lengths. The channel measurements, as in this case the power vectors received with different aperture size is segmented into 8 power levels based on the power in three diodes. Once after the segmentation is done, the state transition matrix for K=8 or K=64 is calculated. A Markov chain is a mathematical model for stochastic systems whose states, discrete or continuous, are governed by state transition matrix.
Figure 9: Finite State Markov Chain
Figure 9 shows the model of Finite state space markov chain, where s0, s1, .., sK−1 are the individual states and their respective probabilities is pi,j with K states.
Theoretical Background
2.1 Shack-Hartmann Wavefront Sensor
The principle operation of Shack-Hartmann gradient Sensor is simple as it is based on the gradient measurement of the spot in each lenslet. The theoretical steps that are involved for the wavefront reconstruction is,
· Gradient Estimation
· Geometry matrix calculation
2.1.1 Gradient Estimation
The distortion caused by the atmospheric turbulence makes the input wave- front produce a local gradient α(x, y) over each lenslet. The lenslet ar- ray converts wavefront gradients into measurable spot displacements. The diffracted spots are usually calculated by finding the centroids along x-axis and y-axis. For this, a rectangular grid of apertures is created matching exactly to the rectangular grid of photo-detectors in CCD camera. The rectangular grid has small cells based on the number of lenslet array used in the CCD camera. From the displacements of the spot in each lenslet grid, the center of gravity is calculated.
x =
imax i=imin
jmax
j=jminI(i, j).i
imax
i=imin
jmax
j=jminI(i, j) .s (2.1) where x is the centroid in x direction, I(i, j) is the intensity measured in ith row and jth column and s is spacing of pixels along x or y axes. The threshold for intensity is set for the calculation of the center of gravity, this is mainly used to minimize the effects of noise due to camera. Similarly the centroid in y direction is also calculated using the formula below,
10
CHAPTER 2. THEORETICAL BACKGROUND 11
y =
imax i=imin
jmax
j=jminI(i, j).j
imax
i=imin
jmax
j=jminI(i, j) .s (2.2) The gradient of the wavefront is calculated from the motion of diffracted spot from the reference spot position. The change of position of the diffracted spot is Δx. The position of the center in the grid is found and with the spac- ing or distance between aperture array and detector array, the gradient of the wavefront is calculated.
The reconstruction of the SHS is based on the principle that gradient g is proportional to difference between two neighbouring phase points ϕ1 and ϕ2.
g1 ∝ ϕ2− ϕ1 (2.3)
Figure 10: Reconstruction Model
In Figure 10 the nodes at which the wavefront reconstruction is repre- sented as wi,j where i = 1, 2, .., N and j = 1, 2, .., N are the indexes in x and y direction [11]. Similarly the respective slopes are represented as sxi,j and syi,j in x and y direction. In the square array, there are N (N − 1) x slopes and y slopes. From Figure 10, the relation between wavefront slopes and nodes is given by the relation,
sx11= w21− w11
sx21= w31− w21 sx31= w41− w31
(2.4)
sy11= w12− w11
sy12= w13− w12 sx13= w14− w13
(2.5)
From the linear equations in (2.6) and (2.7), they can be represented in a matrix form. Similarly for the whole array of the gradient measurements is given by a relation,
−
→gi = Ageo−→ϕi+ n (2.6) where Ageo is the geometry matrix and n is noise added in the measure- ment process. From equation (2.6), if Ageo and −→gi is given, it is required to find wavefront vector −→ϕi. The noise n is assumed to be random, uncorre- lated and equal on all slope measurements. The solution can be found by multiplying each side of the equation (2.6) by transpose of Ageo and then inverting the resulting square matrix which gives,
−
→ϕi = (ATgeoAgeo)−1ATgeo−→gi (2.7) Since the geometry matrix is not a square matrix, it can be inverted using least square method or better by using Singular Value Decomposition, where Ageo is given by,
Ageo = UgeoSgeoVgeo (2.8) From equation (2.8), Ageo is the geometry matrix of C columns and R rows, i.e. a matrix with dimension RxC, then Ugeo is a RxC column- orthogonal matrix, Sgeo is a diagonal matrix with dimension CxC, and Vgeo is a CxC orthogonal matrix. For the wavefront reconstruction the dimen- sions are given by C = N2 and R = 2N (N− 1).
Using the Matlab function svds the singular vectors Ugeo , Sgeo and Vgeo are calculated and it is given by,
[geoU, geoS, geoV ] = svds(sA, r) (2.9) The wavefront nodes can be calculated by using the formula,
−
→ϕi = Vgeo.((1./Sgeo‘ )).UgeoT .A‘geo (2.10)
2.1.3 Modal Analysis
Zernike polynomials are used to represent a wavefront over a circular aper- ture [12]. The polynomials found are used to represent the aberrations of the optical system. The lower order polynomials like tilt, defocus, astigmatism
CHAPTER 2. THEORETICAL BACKGROUND 13
and coma give rise to aberrations. Figure 11 shows the Zernike modes and Table 2 shows the corresponding aberrations.
Figure 11: Zernike Modes
Table 2: Zernike Modes and Aberrations
Zernike Mode Polar Coordinate Form Aberration
Z2 2rcosθ X-axis tilt
Z3 2rsinθ Y-axis tilt
Z4 √
3(2r2− 1) Defocus
Z5 √
6r2sin2θ Astigmatism
Z6 √
6r2cos2θ Astigmatism
Z7 √
8(3r3− 2R)sinθ Y-axis Coma
Z8 √
8(3r3− 2R)cosθ X-axis Coma
Z9 √
8r3sin3θ Y-axis Trefoil
Z10 √
8r3cos3θ X-axis Trefoil
Z11 √
5(6r4− 6r2+ 1) Spherical
Z12 √
10(10r4− 3r2)cos2θ Astigmatism 5th order
Z13 √
10(10r4− 3r2)sin2θ Astigmatism 5th order
Z14 √
10r4cos4θ Ashtray
Z15 √
10r4sin4θ Ashtray
The zernike polynomials Zj(ρ, ψ) can be represented in terms of wave- front phase φ in polar coordinates as [13],
φ(ρ, ψ) =
∞ j=1
ajZj(ρ, ψ) (2.11)
The Zernike coefficients aj is defined by,
aj =
1
0
2Π
0
φ(ρ, ψ)Zj(ρ, ψ)ρ.dρ.dψ (2.12)
If the aberrations in the wavefront are corrected, the interesting thing to note is how much wavefront distortions remains and it is given by Zernike polynomials [13]. Let J be the number of modes corrected and the corrected phase is given by,
φc(ρ, ψ) =
J j=1
ajZj(ρ, ψ) (2.13)
The mean square residual error is given by,
Δ =
[φ(ρ, ψ)− φc(ρ, ψ)]2Zj(ρ, ψ)ρ.dρ.dψ (2.14)
CHAPTER 2. THEORETICAL BACKGROUND 15
By substituting Eq. (2.11) to Eq. (2.12), the approximation formula for the maen suare residual error is given as,
Δ = 0.2944J−√3/2(D/r0)5/3 (2.15) Table 3 gives the residual error for the first 8 modes and Figure 12 illustrates the dependence of the residual error on the ratio D/r0 .
Table 3: Residual Mean Square Error Residual Error Corrected Terms σ12= 1.0299.(D/r0)5/3 Piston
σ22= 0.582.(D/r0)5/3 Piston, Tilt σ32= 0.134.(D/r0)5/3 Piston, Tilt σ42= 0.111.(D/r0)5/3 Piston, Tilt, Defocus σ52= 0.0880.(D/r0)5/3 Piston, Tilt, Defocus, Astigmatism σ62= 0.0648.(D/r0)5/3 Piston, Tilt, Defocus, Astigmatism σ72= 0.0587.(D/r0)5/3 Piston, Tilt, Astigmatism, Coma σ82= 0.0525.(D/r0)5/3 Piston, Tilt, Astigmatism, Coma σJ2 = 0.2944J−√3/2(D/r0)5/3 For larger value of J
Figure 12: Residual Errors
A finite-state Markov chain can be directly applied to model the time- varying behaviour of discrete fading communication channels [10] [14]. The main working aspect of the algorithm is to divide the channel measurements as in this case, the power vectors received by three diodes namely 40cm PIN diode, 5cm Steuerbord diode and 5cm backbord diode into finite number of discrete levels which are mapped as states. The transition matrix is devel- oped based on the transitions between the respective states.
Let Sn={s0, s1, s2, ..., sK−1} represents finite number of K states, where {Sn} denotes state of Markov process at time step n.
Let h be the measured channel measurements and K is the number of states. The partitioning interval Δ is given by,
Δ = hmax− hmin
K (2.16)
where hmax and hmin are maximum and minimum power levels of the downlink data. A Markov model is simple and computationally efficient since it is completely defined by the state transition matrix T = [tj,k]K×K.
[tj,k] = P r(Sn+1 = sk|Sn= sj) = Nj,k
K−1
l=0 Nl,j = Nj,k
Nj , j, k∈ 0, 1, 2, ..., K − 1 (2.17) where Nj,k is the number of observed transitions from state sj to state sk and Nj is the total number of times the channel measurement was at state sj. The state transition probability has the property that the sum of elements in each row is equal to 1. In order to ensure the well-defined probabilities, the following conditions must satisfy,
K−1 l=0
tk,l = 1,∀k ∈ 0, 1, 2, ..., K − 1 (2.18)
0≤ pi,j ≤ 1, i, j = 1, 2, .., n (2.19)
n j=1
pi,j = 1, i = 1, 2, .., n (2.20)
The partition used to define the state transition probability matrix is Uniform partition. In general, channel measurements are divided with
CHAPTER 2. THEORETICAL BACKGROUND 17
thresholds that are uniformly spaced. The methodology behind the par- tition is defining the desired thresholds and deriving the state probabilities.
The power vectors that are received during the experiments are sampled at 20k samples per second. The sampled data is then used for the model.
A parameter auto-correlation function is also considered for FSMC model.
The model with 8 and 64 states is validated by auto-correlation function.
The analytical expression for the function is,
R[m] = E[X(x0)X(xm)]
=
n i=1
n j=1
X(i)X(j)P r[xm = j, x0 = i]
R[m] =
n j=1
X(j)πj
n i=1
X(i)pmi,j (2.21)
where x(·) is the output of the chain corresponding to the state. The matrix form is represented as,
R[m] = xTΦPmx (2.22)
with Φ is the diagonal matrix and P is the transition matrix.
Implementation
3.1 SHS Wavefront Reconstruction
The GUI is created using Matlab for the reconstruction of the SHS wave- front. The main GUI consists of following steps,
· Load SHS, Dark Images and AER file
· Create Mask
· Generate Geometric Matrix
· Wavefront Reconstruction
The flowchart for the above process is given in the Figure 13. First in the process the images received from the downlink data that will be analysed by SHWS is loaded. Once the image is loaded, the dark image with same integration time is also loaded. The dark image contains the offset values of the downlink data images. For every downlink data image, the dark image varies corresponding to their respective integration time.
The resulting image after the offset removal, that is after subtracting the raw image with the dark image is used for further processing. Figures 14 and 15 show the SHS raw image and the respective dark image with the corresponding integration time.
18
CHAPTER 3. IMPLEMENTATION 19
Figure 13: Wavefront Reconstruction Flowchart
The GUI created for this process is shown in Figure 16. The GUI is coded with three main parts,
· Loading Images
· Creating a reference mask which will be used for all Images
· Reconstructing the Wavefront based on the Geometric matrix
Figure 14: Shack-Hartmann Raw Im- age
Figure 15: Shack-Hartmann Dark Im- age of Integration time 100s
Figure 16: GUI for Wavefront Reconstruction
The image after removing the offset is filtered using the Gaussian filter of kernel size 4x4 for smoothing of the image[15]. Figure 17 and 18 show the SH image after offset and noise removal. The grid is created over the resulting image. The grid corresponds to the number of lens in the lenslet array. By placing the grid over the spot of the SH image, the mask has to be created on the SH image. For creating mask the following steps plays a major role,
· Grid position in x and y direction
· Finding COG at each sub grid
· Marking the valid spots
CHAPTER 3. IMPLEMENTATION 21
Figure 17: Shack-Hartmann Image af- ter offset removal
Figure 18: Gaussian filtered Shack- Hartmann Image
The mask is created in a such a way that, the edge spots are marked in red color circle and valid spots are marked as green color circles. Figure 19 shows the mask that is placed over SH image. Finally the mask created is saved as mask.mat, which serves as the reference for all the SH images.
Figure 19: Create Mask
In the reconstruction, we extract three main parameters from the SH images. The parameters that are extracted from the SH images are,
· Phase of each spots of SH image
· Intensity of each spots of SH image
Figure 20 shows the Reconstruction GUI where in each section shows the different properties of the image. One with the SH image with the grid over it, Focus spot of the image, Phase of the image in radians and Intensity of the image.
Figure 20: Batch Reconstruction GUI
The intensity of the image is found based on the criteria where it will check for the maximum value in each sub grid. Figure 21 and Figure 22 show the phase of the reconstructed image and the total reconstructed parameters of the SH image.
CHAPTER 3. IMPLEMENTATION 23
Figure 21: Phase of the reconstructed image
Figure 22: Reconstructed parameters of the SH image
The main aspect in Markov modelling is to divide the channel measurements into finite number of states. As in this case, the channel measurements are divided into K = 8 and K = 64 states. The channel measurements received from PIN 40cm, 5cm backbord and 5cm steuerbord diode are analysed. The steps for creating Markov model is,
· Segmenting the channel measurements based on power levels
· Compute State Transition Matrix
· Generate Markov Chain
The channel measurements are divided into finite states for K = 8 states.
These 8 states are shown in Figure 23. The segmentation of the channel measurements is actually segmenting the power vector with equally spaced states based on their power levels.
Figure 23: Channel Measurement segmentation for K=8 states Once the channel measurements are segmented for finite states, the state transition matrix is calculated. Using the state transition matrix the Markov chain is generated. The markov chain for the respective power vector is shown in the Figure 24.
CHAPTER 3. IMPLEMENTATION 25
Figure 24: Power Vector for K=8 states
The sampling rate of the measurements is 20k samples/s and the mea- surement data received during the night flight on June. 6, 2013 at 8:06 PM during clear weather. The samples range from minimum 8.32mV to 10.55mV is divided into K = 8 states. The corresponding state transition matrix for K = 8 is shown in the Table 4. The power vector obtained from the Markov model is then subjected for K = 8 and K = 64 states to cal- culate the PDF and it is compared with the original power vector. Also the autocovariance of the channel measurements are calculated for original vector and it is compared with the autocovariance for K = 8, K = 64 , seen in Figure 25 and Figure 26.
Table 4: State Transition Probability Matrix
k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7
j = 0 0.9967 0.0033 0 0 0 0 0 0
j = 1 0.0080 0.9605 0.0316 0 0 0 0 0
j = 2 0 0.0021 0.9360 0.0618 0 0 0 0
j = 3 0 0 0.0115 0.9569 0.0315 0 0 0
j = 4 0 0 0 0.0212 0.9501 0.0287 0 0
j = 5 0 0 0 0 0.0446 0.9466 0.0088 0
j = 6 0 0 0 0 0 0.0997 0.8986 0.0017
j = 7 0 0 0 0 0 0 0.1633 0.8367
Figure 25: PDF Estimate
Figure 26: Normalized Autocovariance of channel measurements
Chapter 4
Results
4.1 SHS Wavefront Reconstruction
The aircraft downlink data that are analysed using SHWFS are processed based on the Reconstruction GUI that have been explained in Chapter 3.
The raw image obtained is pre-processed with certain steps which makes the image valid for the reconstruction. From the image the phase and intensity are extracted. From the phase and intensity by taking Fourier transform the Focus spot is found. Figure 27 and 28 show the phase, intensity and generated focus spot.
Figure 27: Phase of the reconstructed image
27
Figure 28: Reconstructed parameters of the SH image
The image captured by the Focus camera is shown in the Figure 30.
The image received is the raw image and it is subtracted from the offset image where the background light is removed. The focus spot of the image captured by the image moves a lot, in order to make the focus spot to be in centre the following steps have to be followed. Figure 29 shows the procedure for the focus camera analysis. Figure 30 shows the raw image from the Focus camera and it is processed based on the steps that is given in the flowchart and the resulting image is shown in Figure 31.The resulting image is compared with the focus spot generated from the reconstructed SHS image. The focus spot generated is compared with the focus spot that has been captured by the Focus camera. Figure 32 shows the focus spot of Focus camera image, focus spot of SHS image, intensity of SHS image and phase of the SHS image.
CHAPTER 4. RESULTS 29
Figure 29: Focus camera image analysis flowchart
Figure 30: Focus Camera raw image
Figure 31: Focus Camera final image
Figure 32: Focus Spot Comparison
From the Figure 32, we can see the focus spot captured by the Focus camera is less confined or precise in terms of the focus spot that is recon-
CHAPTER 4. RESULTS 31
structed by the Shack Hartmann wavefront sensor. This trade off is mainly because the Focus camera use only one lens whereas in Shack Hartmann wavefront sensor it uses number of lens in a lenslet array. Even by using Shack Hartmann wavefront sensor, the received focus spot has some dis- turbances. It means that the Shack-Hartmann is not reconstructing the wavefront focus spot correctly. The reason is due to the spatial resolution or angular resolution is not enough for the effective reconstruction. This problem can be overcome by using even more number of lens in a lenslet array.From the Figure 32, it is shown that SHWFS works better when com- pared with the Focus camera and also shows good agreement over the focus spot. The focus spot generated from the Shack-Hartmann looks better than the measured focus spot.
4.2 Finite State Markov Model
The channel measurement or the power vector received during the flight trial on June. 6,2013 are analysed. The PDF for the radial approach is analysed in the steps for every 1km at 8:02 PM during clear weather conditions at 66-79km. Figure 33,34 and 35 shows the PDF for the Radial approach. Sim- ilarly for the Tangential approach the at 60km, the power vectors in the step of every 1s is analysed. The reason for choosing the power vectors during radial approach at 66-79km and tangential approach at 60 km is because during these trial period there are no offset values due to clear weather con- ditions. The main reason to choose these ranges is that it occured at 8:02 PM and external disturbances like sun and fog are not there. Figures 36, 37 and 38 show the PDF for the Tangential approach. The Tables 5 and 6 show the Radial and Tangential approach measurements for every 1s.
Table 5: Radial Approach Radial
7− 12Km 14− 19Km 19− 22Km 36− 39Km 48− 51Km 51− 59Km 62− 65Km 66− 79Km
Table 6: Tangential Approach Tangential
20Km 40Km 60Km
Figure 33: PDF for Radial 66-79Km 40cm PIN diode
Figure 34: PDF for Radial 66-79Km 5cm Steuerbord diode
CHAPTER 4. RESULTS 33
Figure 35: PDF for Radial 66-79Km 5cm Backbord diode
Figure 36: PDF for Tangential 60Km 40cm PIN diode
Figure 37: PDF for Tangential 60Km 5cm Steuerbord diode
Figure 38: PDF for Tangential 60Km 5cm Backbord diode
During the tangential approach at 60 km, the power vector is segmented for every 1 seconds and Figures 36, 37 and 38 show the probability distri- bution of the power vectors during the Tangential period for every 1 sec- onds.In FSMC model, the power vectors from both the radial and tangential
CHAPTER 4. RESULTS 35
approach are analysed for all the three diodes. The respective probability distribution of the original power vector, Markov chain generated for K = 8 and K = 64 is presented in the Figures 39 and 40.
Figure 39: PDF Estimate for Radial Approach
Figure 40: PDF Estimate for Tangential Approach
Figure 41: Normalized Autocovariance for Radial Approach
Figure 42: Normalized Autocovariance for Tangential Approach
CHAPTER 4. RESULTS 37
The autocovariance function for the radial and tangential approaches of the power vectors for three diodes is shown in Figures 41 and 42. The re- sults obtained from the FSMC model for K=8, K=64 and experimental data received from the aircraft are compared with the properties of probability and autocovariance. Figures 39 and 40 show the probability distribution of the measured channel samples in comparison with the samples gener- ated by the Markov model. From the probability distribution plots, the actual distribution from the measured data relates same as the distribution from the Markov model for K=64 states. Even from the autocovariance distribution it can be noticed that the experimental data distribution and Markov model distribution for K=64 states shows the similar properties.
The Markov model can be modelled beyond K=64 states and it shows the good properties with the experimental data. Even the properties using more states is that it has more or less same properties compared to K=64 states.
Figures 41 and 42 show the ability of the FSMC model with the autoco- variance property, where K=64 fits well with the measured channel samples when compared with K=8 states. The comparison justifies that the samples generated by the FSMC model give good channel behaviour.
Conclusions
In this project, the downlink measurement data from Shack-Hartmann Wave- front Sensor are processed and analysed. The parameters phase, intensity and focus spot of the data are extracted from the measurement data. GUI was created for the reconstruction of the measured data that has been anal- ysed by the Shack-Hartmann Wavefront Sensor and the above mentioned parameters are extracted from the reconstructed data based on the the- oretical background. The focus spot from the reconstructed wavefront is compared with the focus spot captured by the focus camera. The result shows Shack-Hartmann Wavefront Sensor shows good agreement over the reception of the data very effectively.
A Finite state markov model was derived for the received power vector mea- surements from the aircraft and it was trained for K=8 and K=64 states.
The markov chain are generated from the markov model for both tangential and radial approach. The resulting vector from the respective states for both radial and tangential approach was compared with the measured power vec- tor with the property of probability distribution and autocovariance. The model depicts in particular for the 64-state showed good agreement towards channel measurements. It is validated by the distribution and autocorrela- tion properties. The power vector analysed in this project was during the clear weather conditions without any offset values.
38
Chapter 6
Future Work
In the future work, the number of lenslet array can be increased in the Shack-Hartmann Wavefront Sensor for the precise reception of the downlink data. Also during normal weather conditions, the power vectors has to be analysed by removing the offset values and subjecting the measurement data to the Finite state markov model.
39
[1] Arun K. Majumdar and Jennifer C. Ricklin. Free-Space Laser commu- nications, Principles and Advances. Number ISBN: 987-0-387-28652-5.
Springer, 2008.
[2] Kevin Shortt, Dirk Giggenbach, Ramon Mata-Calvo, Florian Moll, Christian Fuchs, Christopher Schmidt, Joachim Horwath, Jack Yeh, Vevek Selvaraj, and Ranjoy Banerjee. Channel characterization for air- to-ground free-space optical communication links, 2014.
[3] Dirk Giggenbach. ”Lasercomm Activities at the German Aerospace Centers Institute of Communications and Navigation”. Proc. Interna- tional Conference on Space Optical Systems and Applications (ICSOS), October 2012.
[4] Wenhan Jiang Chaohong Li, Hao Xian and Changhui Rao. Topics in Adaptive Optics. 2012-01-20.
[5] Claude E. Shannon. Certain results in coding theory for noisy channels.
Information and Control, 1(1):6 – 25, 1957.
[6] E. N. Gilbert. Capacity of a burst-noise channel. Bell System Technical Journal, 39(5):1253–1265, 1960.
[7] E. O. Elliott. Estimates of error rates for codes on burst-noise channels.
Bell System Technical Journal, 42(5):1977–1997, 1963.
[8] Richard H. McCullough. The binary regenerative channel. Bell System Technical Journal, 47(8):1713–1735, 1968.
[9] Robert G. Gallager. Information Theory and Reliable Communication.
Number ISBN: 978-0-471-29048-3. Wiley, 1969.
[10] Hong-Shen Wang and N. Moayeri. Finite-state markov channel-a useful model for radio communication channels. Vehicular Technology, IEEE Transactions on, 44(1):163–171, 1995.
[11] John W. Hardy. Adaptive Optics for Astronomical Telescopes. Number ISBN: 978-0195090192. Oxford University Press, 1998.
40
BIBLIOGRAPHY 41
[12] Guang ming Dai and Virendra N. Mahajan. Zernike annular polyno- mials and atmospheric turbulence. J. Opt. Soc. Am. A, 24(1):139–155, Jan 2007.
[13] Robert J. Noll. Zernike polynomials and atmospheric turbulence. J.
Opt. Soc. Am., 66(3):207–211, Mar 1976.
[14] Qinqing Zhang and S.A. Kassam. Finite-state markov model for rayleigh fading channels. Communications, IEEE Transactions on, 47(11):1688–1692, 1999.
[15] Pei-Yung Hsiao, Shin-Shian Chou, and Feng-Cheng Huang. Generic 2-d gaussian smoothing filter for noisy image processing. In TENCON 2007 - 2007 IEEE Region 10 Conference, pages 1–4, Oct 2007.
