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As part of my undergraduate degree at the University of Aberdeen I completed an undergraduate thesis. In my thesis I considered the concept of a multi static passive radar system capable of tracking commercial civil aircraft using transmitters of opportunity.

An error is the can be defined as the difference between computed, estimated or measured value and the true, specified, or theoretically correct value. Any measurement made or position estimated by a radar system there is an associated error. Here I aim to identify the errors associated with a multi-static radar system’s geometry, where they originate and how to separate them from other system error to create a model.

Geometric Dilution of Precision

The geometric dilution of precision in a multi-static TDOA radar system is comparable to the GPS GDOP metric as described in the paper GPS GDOP metric (R. Yarlagadda, 2000). Adapting the metric as described in GPS GDOP metric (R. Yarlagadda 2000), a metric can be obtained representing the geometric error of the system separately from the error in the TDOA measurements. The configuration of transmitters and receivers relative to the target affects the precision with which the target can be located. The geometric error relates to the difference between the direct path between the transmitter and receiver and the reflected path from the transmitter to the target to the receiver. The closer the reflected path is to the direct path, the greater the error in the position. The geometric error relating to a specific transmitter and specific axis , can be calculated as in Equation 1.

Equation 1

.

The geometric error described in Equation 1 has a maximum value of one when the target lies directly on the normal between the transmitter and receiver.

Equation 2

Equation 2 shows the relationship between the error position , geometric error and the error in TDOA measurement , where is the axis and is the transmitter. Using Equation 1 and the calculus defined in GPS GDOP metric (R. Yarlagadda 2000), a GDOP metric can be defined for a multi-static radar system.

Visibility Matrix

Equation 3

Equation 3 represents the 3D visibility matrix for a multi-static radar system consisting of four transmitters and a single receiver. The dilution of precision parameters , can be obtained as shown in Equation 4.

Equation 4

DOP Equations

From the dilution of precision parameters, the typical DOP figures can be obtained as described in Equation 5, Equation 6 and Equation 7.

Equation 5

Equation 6

Equation 7

Using Equation 1 a 2D visibility matrix (Equation 8) can be generated for a multi-static system consisting of three transmitters and a receiver.

Equation 8

The dilution of precision parameters , can be obtained as shown in Equation 9.

Equation 9

From the dilution of precision parameters, the typical DOP figures can be obtained as described in Equation 10.

Equation 10

GDOP Model

Developing a model for the GDOP error for a system the distribution can be mapped relative to the transmitters and receivers. This would be partially useful for ensuring that an approach to an airport gets the best possible level of service and that there are no GDOP chimneys where the error is effectively infinite.

Using the visibility matrix as defied in Equation 4 a map can be generated representing the GDOP error for a system consisting of three transmitters and a single receiver. Each point on the map represents the GDOP error at that point due to the relative positions of the transmitters and receiver.

Diagram Representing GDOP

Figure 1

Figure 1 shows a representation of a single target, a single transmitter and a receiver in 3D. It can be seen here that for a fixed error in the TDOA measurement for any target position, the greatest position error occurs when difference between the multi-path signal and direct-path signal is at it’s lowest. The lowest difference occurs when the target is directly between the transmitter and receiver.

Figure 2

Figure 2 is a plot of the HDOP of a 2D multi-static radar system consisting of three transmitters and a single receiver. The higher HDOP values are red and the lower are in blue. Higher HDOP metric values correspond to higher geometric dilution of precision.

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As part of my undergraduate degree at the University of Aberdeen I completed an undergraduate thesis. In my thesis I considered the concept of a multi static passive radar system capable of tracking commercial civil aircraft using transmitters of opportunity.

One possible obstacle for a multi-static TDOA radar system is solving a system containing multiple targets. Solving the system quickly enough to enable a quick refresh rate, and real time operation is envisioned as a major obstacle as it could be prohibitive in terms of cost. In this entry I will discuss system complexity while tracking multiple targets, discuss the difference between real and ghost targets and approaches to reduce overall system complexity.

System Complexity

Given the real time requirements of the system and that computing resources available to any system are finite, it is important to have an idea how expensive in terms of computing power the system might be. The system will be required to compute the location of all the targets in range and again for each refresh to maintain a real time view of the targets of interest. Using an indiscriminating brute force approach would be inefficient in terms of computing power. Solving the system in this way has a complexity of approximately

Equation 1

,

for a 2D system. A 3D system has the approximate complexity of

Equation 2

.

Where is the number of targets and is the number of unique transmitter-receiver pairs in the system. Using an alternative approach the approximate complexity of the system could be reduced to

Equation 3

.

Target Solutions for Multiple Targets

Locating multiple targets is difficult, as multiple returns will be detected from each transmitter. For a system using a 2D position algorithm and 3 transmitters as described in previously, the number of possible target solutions can be shown using Equitation 4.

Equation 4

,

where n is the number of returns, excluding the direct path signal from each transmitter. For a system using a 3D positing algorithm using 4 transmitters as described previously, the number of possible target solutions can be shown using

Equation 5

,

where n is the number of returns, excluding the direct path signal from each transmitter. In this thesis, two possible methods are presented to reduce the number of possible target solutions. The first method is to analyze the behaviour of the possible solutions. The second is the use of additional transmitters to test the possible targets.

Tracking of Multiple Targets in 2D

To analyze system performance when tracking multiple targets, a simulation was devised tracking two simulated targets. The parameterized targets vectors are shown in Equation 6 for target 1 and Equation 7 for target 2. The target vectors are chosen to demonstrate the tracking two aircraft with different bearings and velocities.

Equation 6

Equation 7

The simulation runs from to.

There are two returns from each transmitter not including the direct path signal, as there are two targets as shown in Table 1.

Table 1

Transmitter	Returns
1	T1_1, T1_2
2	T2_1, T2_2
3	T3_1, T3_2

Initially the system does not know which return corresponds with which target so every combination is plotted. As per Equation 4 there is 8 possible solutions for a system tracking 2 targets using 3 transmitters. The 8 possible solutions are listed in Table 2.

Table 2

Trace	Positing Arguments			Label
1	T1_1	T2_1	T3_1	Target 1
2	T1_1	T2_1	T3_2	Ghost 1
3	T1_1	T2_2	T3_1	Ghost 2
4	T1_1	T2_2	T3_2	Ghost 3
5	T1_2	T2_1	T3_1	Ghost 4
6	T1_2	T2_1	T3_2	Ghost 5
7	T1_2	T2_2	T3_1	Ghost 6
8	T1_2	T2_2	T3_2	Target 2

The traces are plotted in Figure 1. The traces representing genuine targets are coloured blue Target 1 and green Target 2. In Figure 1 the traces that consist of components of target 1 and target 2 are shown to have wild trajectories. These are clearly out with the performance of any aircraft. This is further analyzed in Figure 2.

Figure 1 (click to enlarge)

The magnitude of the possible targets as a multiple of the constant speed of target 1 are plotted in Figure 2. The Ghost traces can be clearly distinguished from the genuine targets as their speed is magnitudes higher that that of the fastest genuine target.

Figure 2 (click to enlarge)

Analyzing Possible Target Behavior

This solution exploits the limited range of behavior that genuine targets show. False target can be identified when they show behaviour that is not within the range expected from a genuine target. Components such as velocity, position, acceleration and change in direction can be used to distinguish targets.

Analyzing Possible Targets with Additional Transmitter References

Additional transmitters can be used to corroborate possible target solutions. Using a 2D positioning algorithm to track two targets, the result is eight possible solutions as described in Equation 4. For each of these possible targets, the theoretical bi-static range is calculated for the receiver and additional transmitter giving the TDOA for each target. The synthesized TDOA is then compared to the actual returns from the additional transmitter. The error is defined as the difference between the TDOA of the synthesized return and the actual returns from the additional transmitter. As there are two targets, there are two returns received from the additional transmitter. This means that each synthesized TDOA has to be compared with two returns from the additional transmitter. Genuine targets will be distinguished from false targets, as they will have a zero difference from their corresponding return from the additional transmitter.

Corroboration with a Fourth Transmitter

An additional transmitter can be used to corroborate the possible solutions. All the possible solutions for returns describe in Table 3 are calculated as in Table 4.

Table 3

Transmitter	Returns
1	T1_1, T1_2
2	T2_1, T2_2
3	T3_1, T3_2
4	T4_1, T4_2

Table 4

Target Solution	Positing Arguments			Label
$1	T1_1	T2_1	T3_1	Target 1
$2	T1_1	T2_1	T3_2	Ghost 1
$3	T1_1	T2_2	T3_1	Ghost 2
$4	T1_1	T2_2	T3_2	Ghost 3
$5	T1_2	T2_1	T3_1	Ghost 4
$6	T1_2	T2_1	T3_2	Ghost 5
$7	T1_2	T2_2	T3_1	Ghost 6
$8	T1_2	T2_2	T3_2	Target 2

Table 5

Target Solution			Label
	T4_1	T4_2
$1			Target 1
$2			Ghost 1
$3			Ghost 2
$4			Ghost 3
$5			Ghost 4
$6			Ghost 5
$7			Ghost 6
$8			Target 2

Using the known locations of the additional transmitter and receiver, the bi-static ranges are calculated for each possible target solution. Using the bi-static ranges, the equivalent TDOA timings are calculated. These are then compared to the actual returns received from the additional transmitter T4_1 and T4_2 as shown in Table 5. All solutions that have an equivalent TDOA that is not close to equal to the measured bi-static ranges T4_1 and T4_2 can be excluded. The remaining two are the genuine target solutions.

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The following is a list of resources that I have used in the past to search for technical papers.

INFOMINE. Not to be confused with InfoMine, a resource for mining and mineral exploration technology. INFOMINE is an virtual library of internet resources. Typically the results from the search are good but you have to ensure that you specify the resource access as free on the search page and not to include UC subscription eJournals and eBooks on the results page.

Intute. Intute is a free online service. All material returned by Inture is evaluated and selected by a network of subject specialists. Intute does not make it clear whether this involves peer review of the material. Regardless of that technicality, the seems to work as the quality of the results are good.

The University of Glasgow, Enlighten. Enlighten offers access to the University’s Institutional Repository. Despite this being only a fraction of the university research output, this is still a valuable resource. In the future the university aims make available where possible, the full text, for all peer-reviewed, published research outputs produced by university staff.

CiteSeerx. CiteSeerx is a scientific literature digital library and search engine that focuses primarily on the literature in computer and information science. CiteSeerx is loosely based on the previous CiteSeer search engine.

Directory of Open Access Journals. The Directory of Open Access Journals lists open access journals, that is, scientific and scholarly journals that meet high quality standards by exercising peer review or editorial quality control and are free to all from the time of publication.

Open J-Gate. Open J-Gate is another directory of open access journals. Open J-Gate claims to be the largest list of open access journals available but it is important to note that unlike The Directory of  Open Access Journals the majority are not peer reviewed or exposed to editorial quality control.

arXiv. arXiv is an directory service in the fields of physics, mathematics, non-linear science, computer science, quantitative biology and statistics. The contents of arXiv conform to Cornell University academic standards.

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In this entry I will describe the geometry of multi-static passive radar and how it obtains a target solution. The multi-static passive TDOA radar example in this entry consists of a single receiving station that monitors the direct signals from the transmitters and indirect, multi-path signals from potential targets. Other configurations are possible but the 2D operation described requires 3 transmitter – receiver combinations. To obtain a target the system uses the time difference of arrival (TDOA) of the multi-path signal compared to the direct path signal from the same transmitter. Using these timings, solutions for possible target locations can be generated. I will cover the basic bi-static geometry on which this system relies and how it is incorporated into multi-static system. A 2D solution will also be derived from scratch for a multi-static system.

Bi-Static Radar Geometry

Figure 1

Figure 1 illustrates a simple bi-static radar system consisting of a transmitter , a target and a receiver . A distance separates the transmitter and receiver . The sum of the distances between the target and the transmitter and the distance between the target and receiver is the range of the bi-static radar. By measuring the time difference of arrival of the reflected path via the target relative to the direct path signal from the transmitter the relationship between the direct path and the bi-static range can be determined as demonstrated in Equation 1

Equation 1

,

where is the speed of propagation of the signal through air.

Knowing the bi-static range allows the target to be positioned somewhere on an ellipse. The sum of the distances from the points of the ellipse to the transmitter and receiver is equal to the bi-static range .

Figure 2

This relationship can also be demonstrated as in Equation 2

Equation 2

.

The location of the transmitter is at . The location of the receiver is at and the target is located . Expanding Equation 2 gives

Equation 3

.

Multi-Static Radar Geometry

Figure 3

Figure 3 shows a multi-static system consisting of four transmitters, a target and a receiver. There is a direct path signal, from every transmitter to the receiver. A multi path signal from each of the transmitter is reflected by the target and is intercepted by the receiver. The path the reflected signal follows from the target to the receiver is common for all transmitters .

2D Operation

A two dimensional solution for the target can be found using measurements of the TDOA of the reflected signal compared to the direct signal from each transmitter. The 2D solution requires three TDOA measurements from three transmitters , and . Using the TDOA values and the known locations of the transmitters Equation 6 can be rewritten for each transmitter

Equation 4

Equation 5

Equation 6

.

Equation 4 and Equation 5 can be rearranged into

Equation 7

and

Equation 8

.

Squaring both sides of Equation 7 and Equation 8 gives

Equation 9

Equation 10

.

Rearranging Equation 9 and Equation 10 in terms of

Equation 11

Equation 12

.

Equating Equation 11 and Equation 12 gives

Equation 13

Rearranging Equation 0‑10 in the form of

Equation 14

gives

Equation 15

Equation 16

Equation 17

Using Equation 4 and Equation 6 rearranging into the form

Equation 18

gives

Equation 19

Equation 20

Equation 21

Equating Equation 14 and Equation 18

Equation 22

and rearranging for gives

Equation 23

.

Subtitlinginto Equation 14 for gives

Equation 24

The next post will expand the 2D derivation into a solution in 3D. An extra dimension will be added using an additional transmitter. The concept of a geometric error will also be introduced.

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The following is an excerpt from a group project completed in my finial year at the University of Aberdeen. Dr. Kaliyaperumal Nakkeeran supervised the project. The group project title was Simulation of Pulse Propagation in Optical Fiber Transmission Systems. The aim of the project was to simulate the pulse propagation in dispersion managed (DM) optical fiber transmission systems. Effects like optical losses, group-velocity dispersion and self-phase modulation are analyzed in this project. Part of my contribution to the project was the production of a MATLAB simulation of the pulses’ propagation through the wave-guide.

Introduction

The aim of this group project was to simulate the pulse propagation in dispersion managed (DM) optical fibre transmission systems. Effects like optical losses, group-velocity dispersion and self-phase modulation are analyzed in this project.

Nonlinear pulse propagation in a fiber can be described by the generalized nonlinear Schrödinger equation (NLSE), which has the form,

,

where ψ(z, t) is the slowly varying envelope amplitude of the electric field measured in units of square root of Watts at position z, at time t. Subscript z or t denotes the partial derivative with respect to that letter. β₂(z) and β₃(z) represent the second and third order dispersion coefficients respectively. γ(z) represents the strength of the nonlinearities.

Split Step Fourier Method

Description

The non-linear Schrödinger equation describes the propagation of the wave through an optical wave-guide. The equation is described in Equation 2‑1. The equation represents the non-linear response of the dielectric to the electric field of the propagating light wave.

Equation 2‑1

The non-linear Schrödinger equation consists of two main components, the linear component as described in Equation 2‑2 and the non-linear component as described in Equation 2‑3. The non-linear and the linear components of the equation is solvable separately however non-linear Schrödinger equation does not have a simple solution.

Equation 2‑2

Equation 2‑3

If the only a step h is taken along the waveguide z, then the linear and non-linear components can be treated separately resulting in only a small numerical error.  These steps are illustrated in Figure 2‑1. 

In each step in the splits-step method there are three operations, involving the non-linear and linear components. The non-linear transfer function is applied from  to , then from  to . The linear transfer function is then applied from  to . This is illustrated in Figure 2‑3.

Derivation

Linear Component

The linear component of the operation can be obtained from the non-linear Schrödinger’s equation (Equation 2‑4).

Equation 2‑4

The second derivative of the amplitude can be substituted with the Fourier transform as described in Equation 2‑5,

Equation 2‑5

.

Rearranging for  gives Equation 2‑6,

Equation 2‑6

.

Equation 2‑7

Integrating  with respect to  between  and , and with respect to  over o to h as in Equation 2‑8,

Equation 2‑8

,

giving Equation 2‑9.

Equation 2‑9

Equation 2‑10

Re-arranging into a transfer function between the current step and the next step values as in Equation 2‑11,

Equation 2‑11

.

When implemented in MATLAB Equation 2‑11 takes the form:

i = sqrt(-1);

A = A.*exp(-alpha*(h/2)+i*beta/2*w.^2*(h/2));

Non-Linear Component

The non-linear component of the operation can be obtained from the non-linear Schrödinger’s equation (Equation 2‑12).

Equation 2‑12

The equation is then rearranged in terms of , as in Equation 2‑13

Equation 2‑13

.

Integrating  with respect to  between  and , and with respect to  over 0 to h as in Equation 2‑14,

Equation 2‑14

,

giving Equation 2‑15

Equation 2‑15

.

Equation 2‑16

Equation 2‑17

Re arranging into a transfer function between the current step and the next step values as in Equation 2‑18,

Equation 2‑18

.

When implemented in MATLAB Equation 2‑18 takes the form:

i = sqrt(-1);

A = A.*exp(i*gamma*(abs(A).^2)*h);

Demonstration of Algorithm

The MATLAB program is demonstrated in Figure 2‑2. In this graph a generic pulse is shown evolving as it propagates through a glass wave-guide.

Figure 2‑3

Flow Diagram of Procedure

Figure 2‑4

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This entry will cover some of the basic principals of a passive radar system. Using an example system bellow the basic principles of a multi-static TDOA passive radar system will be explained. The basic outline provided in this post will be built upon in the future.

The above diagram details the proposed operation of a multi-static TDOA radar system. This system is passive as it uses transmitters of opportunity opposed to an active which transmitts its own signals. This example multi-static system consists of four transmitters, a target and a receiver. A single receiving station monitors the direct signals from the transmitters and indirect, multi-path signals from potential targets.  The direct path signals propagate directly from every transmitter to the receiver. Multi path signals from each of the transmitters are reflected by the target and intercepted by the receiver. The path the reflected signal follows from the target to the receiver and is common for all transmitters .

The signals are modulated in such a way that signals from different transmitters are easy for the receiver distinguish. The time difference of arrival (TDOA) is the delay between the arrival of a multi-path signal and the arrival of the same direct path signal from the same transmitter. The timing is recovered from the signals through the correlation. Using these timings, solutions for possible target locations can be generated.

A basic break down of a multi-static TDOA radar’s sub-systems is presented bellow.

In the above diagram the system is broken down in to seven main components; Antenna array, reference antenna, antenna array signal conditioning, reference antenna signal conditioning, signal de-modulation, adaptive correlation and target plotting.

The antenna array intercepts the scattered signals from potential targets. This sub system is represented as an array as is it expected to employ some form beamforming. The beamforming will allow the antenna to be tuned to be more sensitive in a particular direction. This will allow the directional gain of the antenna to be optimized for multi-path signals from potential targets. Information on potential targets can be extracted from the antenna array by steering it towards the signal of a target and extracting the angle. The angle can be used in the plotting of potential targets. The  Signal conditioning for the antenna array will optimize the intercepted signals for demodulation.

The reference antenna receives direct path signals from the transmitters. The direct path signals intercepted are used to aid de-modulation of the scattered signals from potential targets. TDOA information is extracted from the  de-modulated scattered signals by cross-correlating them with the de-modulated direct-path signals. The target plotting sub-system takes TDOA information from multiple scattered signals from multiple transmitters to plot potential targets.

The next post will discuss target plotting using TDOA information and distinguishing clutter from targets of interest.

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As part of the finial year of my degree at the University of Aberdeen I completed an undergraduate thesis. In my thesis I considered the concept of a multistatic passive radar system capable of tracking civil aircraft using transmitters of opportunity.

I aim to use this journal as a means of exploring the ideas outlined in my thesis and expand upon them. Using this journal I will break down my thesis and re-examine the background, principals, motivations, benefits and applications of a multistatic passive radar system.

This entry will discuss the motivations behind a civil aviation application for a passive radar system.

Background

Civil aviation infrastructure is under immense stress. Passengers are demanding better quality of service, fewer delays, that there be lower prices and a guaranteed level of safety. Current civil aviation traffic management systems are based on technology that was developed in the 1950s and 1970s. These systems are currently reaching their maximum capacity amidst contending with accelerating passenger numbers, which in the UK are set to double from their 1998 high of 33.6 million passengers to a predicted 71 million passengers in 2020 (Department for Transport, 2004). This tremendous stress will take these systems to their limit. To scale and maintain the current systems to meet the predicted traffic in 2020 would cost between €14 and €18 billion per year (European Commission Directorate General for Energy and Transport, 2005). This is not sustainable.

The Single European Sky Industrial and Technological programme (SESAR) aims to ensure the safety and efficiency of air traffic Europe wide by developing new organisational methods as well as developing a new generation of air traffic management systems. The fulfilment of SERARs objectives will involve the reorganization and standardization of European airspace alongside the modernization of air traffic management systems.

The Galileo European satellite navigational system is the world’s first that is open to all. When complete the system will consist of 30 satellites and ground stations that provide navigational and positioning information. The system’s operation is technically identical to the American Global Positioning System. The American system is primarily a military system and offers no guarantee of precision or reliability for civilian users. The American system can also be switched off for all but the American military if there were any perceived threat. The European Galileo system will offer everybody everywhere guaranteed reliability. This allows the Galileo system to be used as a reliable source of navigation in many transport applications. One specific application (and one of the technical challenges detailed in the SERAR project) is the freeing of aircraft from the rigid trajectories between fixed points that current systems impose. The benefits of freeing aircraft from fixed trajectories are gained through more direct flight routes, thereby saving time, reducing congestion, reducing noise, saving fuel and reducing carbon emissions. Areas where trajectories have been opened and new routes established will not be covered by current air traffic management radar. To maintain a high level of aviation safety, the monitoring of aircraft must contain systems redundancy i.e. a minimum of two systems monitoring any aircraft at one time. An efficient solution to this problem is a low cost multi-static radar system.

Multi-static radar has the potential to provide a wide area of coverage using only receivers of modest performance. This is achieved by exploiting cheap computing power. Powerful computer systems capable of the complex digital signal processing required in a multi-static system are available at a relatively low cost. This allows cost effective systems to be built to cover wide areas that may be used in areas where traditional mono-static active radar systems would not be cost effective. A multi-static system also contains inherent redundancy.

Motovation

A passive system could be a potential competitor to current mono-static systems in terms of cost. A passive radar system has inherent advantage over a traditional mono static system in that it uses signals of opportunity which are free and numerous. A monostaic system has to transmit its own signal, which requires complex switching systems to allow it to both transmit and receive using the same antenna. The proposed passive system uses a bistatic geometric configuration, which does not require switching.

The proposed system has many potential applications and markets. Theses include:

• Civil aviation in Europe. Opening up flight paths and expanding current capacity.
• Civil aviation in developing countries. Providing low cost coverage for airports that would not be able to afford expensive monostatic systems.
• Security and military. A system could be easily being made portable. The proposed system also lends itself to military and security applications, as it is inherently difficult to jam and its use impossible to detect preventing the user from becoming a target.

New technology such as the Galileo European satellite navigational system could change the civil aviation industry and produce opportunities for a multistatic passive radar system. The Galileo European satellite navigational system is the world’s first that is open to all. One specific application (and one of the technical challenges detailed in the SERAR project) is the freeing of aircraft from the rigid trajectories between fixed points that current systems impose. Where there will be areas where the trajectories have been opened and new routes established there would be areas, which will not be covered by current air traffic management radar.  To maintain a high level of aviation safety, the monitoring of aircraft must contain systems redundancy i.e. a minimum of two systems monitoring any aircraft at one time.  Monitoring the positions of civil aircraft as a secondary system is an example of such an opportunity.

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