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Airborne Radar Ground Clutter
Suppression Using Multitaper Spectrum
Estimation
&
Choosing DPSS Parameters
Carl-Henrik Hanquist
Engineering Physics and Electrical Engineering, master's level
2018
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
Airborne Radar Ground ClutterSuppression Using Multitaper
Spectrum Estimation&
Choosing DPSS Parameters
Carl-Henrik D. Hanquist
Examiner: Johan Carlson
Thesis supervisor: Bjorn Hallberg
SAMMANFATTNING
En av de storsta utmaningarna i ett flygburet radarsystem ar att urskilja ett mal fran
markekot. Problem uppstar eftersom markekot, kallat markklotter, kan vara upp emot
en miljon ganger starkare an svaret fran malet i fraga. I dagslaget anvands flera olika
filtreringmetoder i flyburna radarsystem for att urskilja malet fran markklottret, alla
har sina fordelar och nackdelar. I en ideal varld skulle det optimala filtret filtrera ut
markklottret fullstandigt och endast bevara malsignalen. Eftersom dessa filter inte exis-
terar i verkligheten efterstravas istallet ett filter med laga sidlober och minimal forlust
i signal-till-interferens ration. En typ av filter som uppvisar detta beteende ar diskreta
prolata sfaroid sekvenser (DPSS).
Denna uppsats undersoker ifall DPSS kan anvandas som viktfunktioner i multitaper
spektralestimering for att filtrera ut markklotter i en radarsignal. En enkel klottermodell
utvecklades for generering av simulerat markklotter som sedan filtrerades ut med multita-
per metoden och en traditionell metod. Resultatet visade att det var mojligt att anvanda
DPSS i multitaper spektralestimering och att metodens prestanda overstiger den tradi-
tionella meteoden, sa lange parametrar som bandbredd och antal anvanda sekvenser valjs
korrekt. Prestandaforbattringen mot den traditionella metoden uppstar mot en kostnad
i berakningstid som okar med varje DPSS ordning som anvands.
Ett full factorial experiment utfordes ocksa for att undersoka vilka parametrar som hade
storst paverkan for att maximera forbattringsfaktorn och minsta detekterbara hastighet.
Resultated visade att lag bandbredd vid generering av DPSS var att foredra, samt att ett
stort antal anvanda DPSS och tidssamples okade prestandan. Resultaten visade ocksa
att for ett okat antal tisdssamples sa maste bandbredd och antal sekvenser som anvands
justeras for att bibehalla samma niva av forbattringsfaktorn.
Slutligen rekommenderades det att framtida arbete borde fokusera pa validering med
mer avancerade klottermodeller och MTI filter i simuleringar, samt validering mot verk-
lig radar data. Om detta visar sig framgangsrikt bor optimering av berakningstid och
implementation av ett adaptivt val av DPSS bandbredd goras fore implementering i ett
radarsystem.
iii
ABSTRACT
One of the biggest challenges in any airborne radar is to distinguish a target from a strong
ground echo. The main problem is that the ground echo, called ground clutter, can be up
to a million times stronger than the response from the target in question. Today many
different filtering methods are used in airborne radar systems to separate the target signal
from the ground clutter. All of them with their own advantages and shortcomings. In
an ideal world the optimum filter would completely filter out the unwanted ground echo.
But as ideal filters don’t exist in reality a filter with low sidelobes and minimum loss in
signal-to-interference ratio is sought after. A type of filter which exhibit this behaviour
are discrete prolate spheroidal sequences (DPSS).
This thesis investigated if DPSS could be used as weight functions in multitaper spec-
trum estimation to filter out ground clutter in the radar signal. A simple clutter model
was developed for generating simulated ground clutter which was then filtered out by
multitaper and a traditional method. Results showed that it is possible to use DPSS in
multitaper spectrum estimation and that it outperforms a basic traditional method in
clutter filtration as long as parameters such as bandwidth and the number of sequences
used are chosen properly. The increase in performance against the traditional method
comes at a cost of increased computational load with each additional DPSS order used.
A full factorial experiment was also performed to investigate which parameters were
important for maximising improvement factor and minimum detectable velocity. The
results from these showed that a low bandwidth in the generation of the DPSS was
preferable and that a high number of time samples and DPSS used improved perfor-
mance. They also showed that for an increase in number of time samples the bandwidth
and number of sequences used need to be adjusted to maintain the same level of the
improvement factor.
It was concluded that future work should focus on validation with more advanced
clutter models and MTI filters in simulations as well as validation against real radar
data. If proved successful, optimisation of calculation speeds as well as implementation
of adaptive choice of DPSS bandwidth would be beneficial before being implemented in
a radar system.
v
PREFACE
This thesis was part of a project to investigate ground clutter suppression with discrete
prolate spheroidal sequence multitapering at SAAB in Gothenburg. The project was
divided into two separate thesis reports. The investigation to see if the method was pos-
sible was performed together, while specialisation into separate fields were done in the
latter part of the project. This thesis focuses on the design of a full factorial experiment
to investigate which parameters are most importance when performing the filtering. The
implementation of a simulation environment to perform the experiment as well as com-
pare the result to traditional signal processing was performed by Linus Ekvall. As such,
some parts of this thesis references Linus thesis titled: Airborne Radar Ground Clut-
ter Suppression using Multitaper Spectrum Estimation & Comparison with Traditional
Method.
Some illustrations in this report use the TikZ package ’aircraftshapes’ [1] c©2015 Stanford
Intelligent Systems Laboratory, under the MIT License.
Special thanks goes out to
Linus Ekvall, in which much of the work was conducted together with in the project.
The project would not have been possible to finish alone.
Johan Carlson, for help and support during the project, as well as providing literature
and references. He was also kind enough to provide some MATLAB code which helped
in understanding full factorial design- and model matrices.
Bjorn Hallberg, for an excellent introduction into the world of radar signal processing
and helpful advice along the way.
Jacob Nilsson, for taking the time to be my opponent and providing constructive feeback.
My fiancee, for always providing loving support and constructive feedback while at the
same time putting up with my rants and weird ideas.
Carl-Henrik Hanquist - [email protected]
vii
LIST OF ACRONYMS
CNR Clutter to Noise Ratio.
DPSS Discrete Prolate Spheroid Sequence.
DPSW Discrete Prolate Spheroid Window.
ESA Electronically Scanned Array.
FIR Finite Impulse Response.
IF Improvement Factor.
LOS Line of Sight.
MDV Minimum Detectable Velocity.
MLC MainLobe Clutter.
MTI Moving Target Indication.
PRF Pulse Repetition Frequency.
Radar RAdio Detection And Ranging.
RCS Radar Cross Section.
SINR Signal Interference plus Noise Ratio.
SLC SideLobe Clutter.
STAP Space-Time Adaptive Processing.
ULA Uniform Linear Array.
ix
MATHEMATICAL NOTATION
α Estimated model parameters
c Clutter vector
D Full factorial design matrix
Dcode Full factorial design matrix with coded values
Dexp Full factorial design matrix with experimental values
n Noise vector
Q Interference autocorrelation matrix
q Interference vector
S Steering vector
s Desired signal vector
T Linear subspace transformation matrix
X Full factorial model matrix
x Data vector
Y Full factorial performance measure matrix
β Grazing angle
l Doppler total distance rate of change
R Doppler radial distance rate of change
ν [n](k) (N,W ) Slepian function
ν [n](k) Slepian function with omitted dependency
ε Depression angle
λ Wavelength of radar signal
θ Azimuth angle
θml Azimuth angle of mainlobe
c Speed of light in a vacuum
d Element separation distance in array
di,j Design matrix element
fprf Pulse repetition frequency
fD Doppler frequency
L Number of DPSS orders
xi
N Number of time samples used
NW DPSS time half bandwidth product
R Radial distance to target
Rk Taper weights on antenna receiver
Tk Taper weights on antenna transmitter
vpl Platform velocity
vt Target velocity
W DPSS Bandwidth
xii
CONTENTS
Chapter 1 – Introduction 1
1.1 The road to airborne radar . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Introduction to radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Airborne radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Electronically scanned arrays . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Radar lobes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Related research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2 – Theory 9
2.1 Radar basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Antenna weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Doppler shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The radar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Clutter filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Clutter modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Target modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Minimum Variance Estimator . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Multitapering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Improvement factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Discrete prolate spheroidal sequences . . . . . . . . . . . . . . . . . . . . 18
2.10 Discrete prolate spheroidal window . . . . . . . . . . . . . . . . . . . . . 19
2.11 Factorial experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.11.1 Factorial design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.11.2 Statistics of an experiment . . . . . . . . . . . . . . . . . . . . . . 22
2.11.3 Estimating effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.11.4 Design and Model matrices . . . . . . . . . . . . . . . . . . . . . 24
2.11.5 Analysing the effect of terms . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3 – Method 27
3.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Comparison against a traditional method . . . . . . . . . . . . . . 27
3.1.2 Modelling the clutter . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.3 Modelling the targets . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.4 Multitapering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Full factorial experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 The performance measures . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Identifying and choosing parameters . . . . . . . . . . . . . . . . 30
3.2.3 Choosing levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.5 Simulation environment . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter 4 – Results 37
4.1 Multitaper simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Multitaper simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Comparison with the traditional method . . . . . . . . . . . . . . . . . . 38
4.4 Factorial experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter 5 – Discussion 43
5.1 Multitaper spectrum estimation . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Traditional method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 The factorial experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1 Impact on IF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.2 Impact on MDV . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.3 Parameter importance . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 Choice of DPSS parameters . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.1 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.2 Number of DPSS orders . . . . . . . . . . . . . . . . . . . . . . . 46
5.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.6 Science and ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter 6 – Conclusion 49
xiv
CHAPTER 1
Introduction
1.1 The road to airborne radar
The term radar is an acronym for Radio Detection And Ranging [2]. The principle
of radar builds upon transmitting electromagnetic waves, also called radar signals, in
a certain direction. The waves are reflected by objects in the path of the wave, these
reflections are then detected at a receiving antenna. The transmitter and receiver can
be separate antennas or a single antenna set up for both tasks. The distance to the
object can be determined by measuring the travel time of the wave that is received. If
the object is moving it also introduces a Doppler shift in the received signal, which can
be measured to calculate the objects radial velocity.
That electromagnetic waves reflect of metal objects was proven experimentally by Hein-
rich Hertz in the late 19th century and using reflected radio waves to locate objects was
suggested by Nikola Tesla in 1900. It was not until the invention of the magnetron just
before the second world war that these ideas became reality however [3]. The technol-
ogy was developed in great secrecy in several countries simultaneously before and during
the war. The cold war that followed stoked the flames of development even further and
today radar has became an essential part of many modern technologies. Radar is used
in meteorology [4], air traffic control [5], astronomy [6], flight control systems [7] and
geological observations [8] just to name a few applications.
Early airborne radar was developed as a means to intercept aircraft at night and in poor
weather during world war II. Bulky antennas were mounted to aircraft by most major air
forces, with varying success. The limitations of power output and size were among the
biggest problems in the early days [9]. As the technology improved and the antennas got
smaller their performance was increased and they could be fitted into shielding radomes.
As the development in airborne radar continued so did the development of ground
based radars, where linear array radars had been developed to increase performance.
The idea of mounting a linear array on aircraft was though of in 1950. Initially it was
proposed to be mounted on the leading edge of the wing for offensive purposes. Electrical
1
2 Introduction
and mechanical limitations at the time forced the array to be mounted along the aircraft
fuselage instead. This compromise made the array unable to fulfil the offensive purposes
initially considered, as a result side looking radars were proposed as a reconnaissance
and mapping tool instead [10].
Modern advances in electronics and computers made it possible to fit powerful radar
on aircraft and perform increasingly complicated signal processing of the radar signals.
Examples are the APS-145 mounted on the iconic E-2C Hawkeye [11], [12], the APY-2
radar on the Boeing E-3 AWACS [13],[14] and the joint STARS radar on the Boeing E-8C
[15],[16].
1.2 Problem definitionWhile the basic concepts of radar are rather simple to explain, having the radar airborne
introduces more complexity. To start with the size and power of the radar is limited since
the platform is an aircraft. In addition the radar is no longer stationary, which means
that when detecting the velocity of a target one needs to consider the platforms velocity
as well. A secondary effect of having a moving platform is that the ground will introduce
a Doppler shift relative to the platforms velocity.
Ground clutter is a problem since the signal strength of the ground return can be up
to a million times (60 dB) stronger than an airborne target. Therefore trying to detect
a target gets harder the closer1 the target signal is to ground clutter. The immense
difference in signal strength can be compared to the difference in luminosity between a
candle and the flash from a lightning strike [17]. Methods to counter this include beam
forming of the transmitted signal as well as window weighting, also called tapering.
In the end the problem can be broken down into spectrum estimation as well as Doppler
filtering with low-sidelobe filters to identify targets in the frequency domain. Well known
spectrum estimation techniques often utilise different weights in the Fourier transforms,
often called tapers, to amplify the frequencies of interest. Which spectrum estimation
method and taper to use depends on the application and the data set on which the
estimation is to be performed.
The question arises, are there existing tapers which are better than the ones used tra-
ditionally? An ideal taper would act as a filter that only amplifies a single frequency
range of interest and completely cancel out all other frequencies, such filters do not exist.
In reality tapering functions suffer from a trade-off between sidelobe level and mainlobe
gain. That is, as the frequency of interest is amplified by the mainlobe, so are other unde-
sirable frequencies by the sidelobes, although to a lesser degree. The balance between the
amplification of the frequency of interest compared to other frequencies is important for
a good performance. A good taper function would have a low Signal to Interference plus
Noise Ratio (SINR) loss and low sidelobes. The Discrete Prolate Spheroidal Sequences
1Closer in this context refers to proximity of signals in the Doppler shift domain, or proximity in
azimuth and/or elevation angle between the target and clutter.
1.3. Introduction to radar 3
(DPSS) are a family of tapering functions that belong to this category. It is however
unclear if DPSS tapering can live up to standard radar requirements.
1.3 Introduction to radar
1.3.1 Airborne radar
As mentioned in Section 1.2 having a radar mounted on an airborne platform introduces
new complications and challenges which would not be a problem for a stationary radar
on the ground. The reason that airborne radar exist despite this is that the advantages
of an airborne surveillance radar outweigh the challenges.
A stationary ground based radar system is limited the terrain around it, for example
tall buildings and hills will severely limit the visibility of the radar. Even if the area
around the ground radar would be completely flat the maximum range is limited by the
curvature of the earth. To increase their range, ground based radars are often located
at the top of hills or mountains. For the same reason shipborne radar are mounted at
the highest point possible aboard the ship. This emphasises the main advantage of an
airborne radar, the altitude of the platform. By increasing the altitude a few thousand
meters the range of the airborne radar is increased immensely when compared to ground
based radar, as illustrated in Figure 1.1.
Ground radar limit
Airborne radar limit
Figure 1.1: Illustration of the increased range covered by an airborne radar when com-
pared to a ground radar. The target aircraft will not be detected by the ground radar
due to shadowing by the obstacle.
While mountains and other obstacles still introduce radar dead zones, also called radar
shadows, the impact of the obstacles on performance is much lower in an airborne system
when compared to a ground based system.
4 Introduction
1.3.2 Electronically scanned arrays
The simplest array configuration for an electronically scanned array (ESA) is a single
row of antenna elements spaced a distance d < λ/2 apart [18], where λ is the wavelength.
This single row ESA is a so called uniform linear array (ULA). The principle behind a
ULA, and in extension an ESA, is that each antenna element is phase shifted in relation
to the previous element. The result is that the wavefront from each individual element
interact with each other, creating a combined wavefront at a certain angle. An example
of this is shown in Figure 1.2. In a similar fashion the direction of arrival of an incoming
1 2 3 4 5 6 7
d
φ
Figure 1.2: A ULA of with antenna elements separated by a distance d. The transmitted
wave from each element is phase shifted in relation to the previous element to create a
combined wavefront at angle φ.
planar wave can be calculated as the receivers register a phase shift with respect to each
other. The received signal from a number of received wave fronts can be summed up and
post processed to get a Doppler spectrum of the signal.
1.3.3 Radar lobes
When discussing radar and signal processing tapers there exists a common term which
can give rise to some confusion, namely lobes. Although the name is the same they
represent different phenomena, both equally complex in their own right. The following
brief and heavily simplified explanation of lobes should suffice for the context of this
thesis.
In the case of radar lobes they can be though of as physical lobes originating from
an antenna array. These lobes can for example represent the distribution of the energy
that a radar system transmits and receives. The majority of the energy is in a direction
1.3. Introduction to radar 5
of interest, but at the same time some energy is transmitted to, or received from other
directions. An example of this is illustrated in Figure 1.3a, where most of the energy is
directed towards 0◦, in what is called the mainlobe. However, some energy is also concen-
trated in other seemingly random directions, in whats called sidelobes. The sidelobes are
weaker the further away from the mainlobe, with a minimum at 180◦. While the lobes
are illustrated here in a simple 2D plane, in reality they are more complex and exist in
3D space, with lobes in all directions.
0°
45°
90°
135°
180°
-135°
-90°
-45°
-60
-50
-40
-30
-20
-10
0
Example of 2D radiation pattern
(a) Antenna lobes.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Po
we
r [d
B]
Spectrum of a taylorwindow
(b) Taper lobes.
Figure 1.3: Illustration of (a) antenna lobe pattern and (b) taper lobes from a Fourier
transform with Taylor window weights. Both the radial axis in (a) and the y-axis in (b)
reads the power in dB normalised with respect to the mainlobe peak.
When speaking of lobes in signal processing they too represent an energy concentration,
but with respect to an often normalised frequency. In Figure 1.3b the power spectral
density of a taper consisting of Taylor window weights is shown. The mainlobe is situated
at 0 and the sidelobes get weaker as they get further from the mainlobe. This taper could
for example be used as a filter to amplify a target signal by placing the mainlobe at the
target frequency. The sidelobes will however add some amplification at their frequencies
as well, this is called spectral leakage and is unwanted.
6 Introduction
1.4 Clutter
Radar clutter is a term used for unwanted radar echoes from for example the ground,
sea, rain, chaff2, birds, insects and wind shear to name a few sources [2]. Ground clutter
is among the strongest and most common of these unwanted returns and filtering it out
is a core problem in radar technology.
Ground clutter can be divided into parts relating to the the cause of the clutter. Com-
mon terms for the ground clutter are altitude return, mainlobe clutter (MLC) and side-
lobe clutter (SLC). The Doppler frequency spectrum of typical ground clutter is presented
in Figure 1.4a.
MLC is caused by the mainlobe partly illuminating the ground. As the mainlobe
contains the highest concentration of energy of the transmitted radar signal, the ground
return from the mainlobe is by far the strongest. The Doppler frequency of the MLC will
depend on the direction the mainlobe. The MLC in Figure 1.4a has a positive Doppler
frequency, which means the mainlobe is angled towards the platform travel direction.
f0
MLC
SLC
Altitude return
(a) Clutter Doppler spectrum. (b) Clutter origin example.
Figure 1.4: Illustration of the (a) clutter Doppler spectrum and (b) an example of ground
clutter (black) getting mixed in with the target signal (red).
SLC is caused by the radar signal being reflected from the ground by sidelobes illu-
minating the ground. As sidelobes can not be avoided completely this will always be
present. In Doppler frequency domain the sidelobe clutter manifests itself as a almost
steady clutter floor as sidelobes in different directions get Doppler shifted by different
amounts.
Altitude return is caused by radar signals directed directly down towards the ground,
the ground then acts as a specular reflector, returning most of the energy. If in addi-
tion the ground is relatively close to the platform there is minimal dissipation of signal
2Strips of lightweight metal or metalized material dispensed in large quantities to simulate false
targets or create large clutter signals to mask actual targets [2].
1.5. Related research 7
strength. As a result the altitude return can be a significant source of clutter even if only
a small fraction of the transmitted energy is directed downward. In general it is a greater
problem for low flying fighter aircraft than for high altitude side looking surveillance
aircraft. As the name implies the altitude of the platform will have an impact on the
strength of the clutter from altitude return. A greater altitude results in a lower strength
of the altitude return.
An example on how ground clutter can become a problem is shown in Figure 1.4b.
The energy returned from the low flying target is much lower than the return from the
ground. Further more if the MLC and target have the same Doppler shift, the target
signal will be drowned in the MLC.
1.5 Related researchThe use of DPSS as taper functions in multitaper spectral estimation theory is attributed
to David J. Thomson [19]. Since then the method and different modifications [20], of
this estimation have seen use in many different fields.
DPSS tapering and Thomsons method have been considered for spectrum estimation
problems in geology where it may prove useful in signal processing of seismic activity
[21], [22], [23], [24].
Thomsons multitaper method is also used in bioengineering [25] and neuroscience [26],
[27]. It has also been suggested for use with electroencephalography (EEG) analysis of
anaesthesia and sleep [28].
Multitapering with DPSS functions is also used in passive sonar problems [29], [30],
where Thomsons mothod is used as it was designed to work with low sample support,
which make them ideal for analysing nonstationary or transient signals. This work is
relevant since sonar and radar have many similarities between them and since it can be
applied to sonar problems it should be viable for radar problems as well.
No publicly published research regarding the implementation of multitaper spectrum
estimation in airborne radar was found during the literature study of the project.
8 Introduction
1.6 ScopeThis thesis sets out to answer a set of questions outlined below. Much of the work
depended on a collaborative effort with Linus Ekvall. As such even the individual parts
relied upon each others work and the results merge together to form a complete picture.
What follows below is a description of the work division for the thesis questions.
The following thesis question was worked on by the author of this thesis
together with Linus Ekvall.
• Can Discrete Prolate Spheroid Sequences be used as multitaper functions to filter
out ground clutter in airborne radar?
The following thesis questions were investigated by Linus Ekvall and are cov-
ered with greater depth in his thesis titled: Airborne Radar Ground Clutter
Suppression using Multitaper Spectrum Estimation & Comparison with Tra-
ditional Method.
• How would the performance compare to traditional frequency estimation tech-
niques?
• How would the computational load compare to traditional frequency estimation
techniques?
The following thesis questions were investigated by the author of this thesis.
• What parameters affecting the spectrum estimation performance exists?
• How should the parameters be chosen?
• Does a optimum choice of these parameters exist?
CHAPTER 2
Theory
2.1 Radar basics
As briefly mentioned in section 1.1 the principle behind radar is that electromagnetic
waves are transmitted, reflected and received again by an antenna. The time it takes for
the transmitted wave to return again can then be used to calculate the distance from the
antenna to the object.
As an electromagnetic wave propagates the power per area unit is inversely proportional
to the square of the distance travelled from the source, this is called the inverse-square
law. Since radar needs the wave to be reflected and travel back to the receiver, the
power is decreased further as the fraction of the returned power has to travel the same
distance a second time. However the power of the reflected wave depends on numerous
parameters such as reflectivity of the target, antenna parameters and losses in energy
along the propagation path, to name a few. How the power in the received signal is
influenced by different parameters is expanded upon in section 2.2.
2.1.1 Antenna weights
It is common practice in radar technology to weight the antennas in an ESA with for
example a Gaussian or Taylor window at the receiver side, while the transmitter side has
uniform weights [31]. The reason for the uniform weights at the transmitter is to get as
much signal energy as possible transmitted. The Gaussian and Taylor windows are used
as they have been identified as good windows for use in radar due to their good trade off
between mainlobe widening, sidelobe levels and mainlobe directivity.
2.1.2 Orientation
When interpreting incoming signals what is known is the angle of the incoming signal as
well as at what range the signal was returned from. It is logical to describe the orientation
9
10 Theory
of the radar lobe and targets in angles with respect to the platforms travel direction. A
simplified illustration of this is presented in Figure 2.1. While many other orientation
angles exist in airborne radar, the ones presented in this section are sufficient in the
context of the thesis. Two frequent angles are the azimuth θ and the depression angle
ε, which are used to describe the direction of the main beam in relation as well as other
directions such as the line of sight (LOS) to a target. An additional angle β is named
the grazing angle and is dependant on the curvature of the earth. The assumption ε = β
is often used for low altitude radar systems.
x
y
vpl
θ
(a) Top.
x
z
vpl
ε
Ground
β
(b) Side.
Figure 2.1: Illustration of the orientation with the platform as a reference point, as seen
from (a) the top and (b) the side. vpl is the platform velocity.
2.1.3 Doppler shift
When a travelling wave gets reflected of an object it’s frequency is shifted slightly due
to the objects velocity. This frequency modulation is called a Doppler shift and is pro-
portional to the objects velocity. If the object is travelling away from the source of the
wave the Doppler shift is negative and if it is travelling towards the source the shift is
positive. The Doppler equation
fD =−lλ
(2.1)
2.1. Radar basics 11
states that the the Doppler frequency fD is equal to the rate of change in distance l
in wavelengths, l is the total difference travelled by the wave. The negative sign is
implemented to represent that closing distances (negative l) result in positive Doppler
shift. Since the transmitted radar waves have to return to the antenna the distance
travelled becomes l = 2R. R is the radial distance to a target and is inserted into Eq. 2.1
which results in
fD =−2R
λ(2.2)
where R is the radial rate of change to a radar target.
The effect is illustrated in Figure 2.2a, where the negative radial rate of change, caused
by the target moving towards the platform, compresses the wavefronts as the frequency of
the wave is increased after the reflection off the target. Similarly the positive radial rate
of change, caused by the target moving away from the platform, increases the distance
between the wavefronts as the frequency is decreased after the reflection off the target.
As both the platform and the target are moving the Doppler shift gets affected by the
rate of change in separation distance relative each other. This range rate is denoted R
and is generally equal to the sum of the magnitudes of the projections of the platform
velocity and target velocity on to the LOS, illustrated in Figure 2.2b. When applied to
Eq. 2.2 it becomes
fD =2(v1 + v2)
λ(2.3)
where v1 and v2 are the velocity projections on to the LOS for the platform and target
respectively.
As mentioned in section 1.4, the ground return also introduce a Doppler shift in the
signal proportional to the velocity of the platform. But since the radar can look in
different directions R becomes equal to a projection of the platform velocity on to the
LOS of the patch of ground being illuminated by the radar. The Doppler shift from an
illuminated section of the ground becomes
fD = 2vplcos(θ)cos(ε)
λ(2.4)
It is from Eq. 2.4 implied that the clutter is constrained to the frequency span
−2vpl
λ< fD <
2vpl
λ(2.5)
making the platform velocity an important factor. It is preferable to keep it as low as
possible to minimise the spread of the SLC in the Doppler spectrum.
2.1.4 Clutter
As stated in Eq. 2.4 a small patch of the ground being illuminated by a radar gets a
specific Doppler frequency. The next small patch next to the first one gets a different
12 Theory
vpl vt
vpl vt
(a) Doppler shift example.
vpl
vt
v1
v2
(b) Range rate example.
Figure 2.2: Illustrations of (a) the Doppler shifted reflection (red) of a transmitted wave
(black) depending on R and (b) R as a sum of the projections of the platform and target
velocity on the LOS.
Doppler frequency and so on. A consequence of this is that even the narrow mainlobe
beam illuminating a section of the ground gets spread out slightly in Doppler frequency.
The same effect also gets introduced when sidelobes illuminate the ground. Assuming
that the sidelobes are relatively evenly spread and illuminating ground in all directions
gives rise to a sidelobe clutter spectrum which is spread over the span specified in Eq. 2.5.
This spread was hinted at but not yet explained fully in Figure 1.4a. A more complete
picture of the clutter spectrum can be seen in Figure 2.3.
v
f
fnorm0−0.5 −2vpl/λPRF 2vpl/λPRF 0.5
−PRF/2 −2vpl/λ 2vpl/λ PRF/2
−vprf/2 −vplvpl vprf/2
Thermal noise level
MLC
SLC
Figure 2.3: Illustration of a clutter spectrum with thermal noise plotted against Doppler
velocity, Doppler frequency and normalised Doppler frequency.
2.2. The radar equation 13
In it the limits of the clutter spectrum are specified in velocity, frequency and nor-
malised frequency with respect to the pulse repetition frequency (PRF). Which represent
how many pulses are transmitted per second from the antenna1. In Figure 2.3 a thermal
noise level is also present, with approximately the same power at all frequencies. The
thermal noise floor is the minimum level of noise present in any electronic system and
can never be avoided completely.
2.2 The radar equation
The radar equation [32] is widely used as a tool to estimate the maximum detection range
of a radar. A simple form of the radar equation is often written as
Pr =
(PtGt
4πR2
)×( σ
4πR2
)× Ae =
PtGtσAe(4π)2R4
(2.6)
where Pr is the received power at the receiver. The first factor correspond to the power
density at a distance R, where Pt is the transmitted power, Gt is the gain of the transmit-
ter antenna. The second term correspond to the reflection and spreading on the return
path, where σ is the radar cross section (RCS) of a target. Lastly Ae is the effective area
of the antenna which receives the reflected energy.
The radar equation can be rewritten in several ways, depending on how many param-
eters needs to be considered. A useful version is when the received signal is set equal to
the minimum detectable signal of the radar Smin
R4max =
PtGtAeσ
(4π)2 Smin(2.7)
From Eq. 2.7 it can be understood that improving the range of the radar is a challenge.
For example, an improvement by a factor of two in range by only increasing the trans-
mitted power means that the transmitted power needs to be 16 times greater. Therefore
improvement in or reduction of loss in any way possible is important, but as many hard-
ware parameters are often limited in some way more and more attention has been given
to signal processing to improve the signal.
2.3 Clutter filtering
Clutter filtering in stationary ground radars is rather straight forward, at least in Doppler
domain. Since the clutter will have relative velocity of zero or close to zero it will be
centred at 0 Doppler frequency and can easily be filtered out with a finite impulse response
1In the context of this thesis the PRF only affects the Doppler frequency span of the spectrum and
clutter. It will therefore not be elaborated upon further.
14 Theory
(FIR) filter that filters out DC components. An example of such a filter is
hmti = [−1, 2,−1] (2.8)
This is called a moving-target indication (MTI) filter and is used with varying degrees of
complexity and adaptability, for the purpose of this thesis it is sufficient to think of it as a
simple DC rejection filter. The drawback of using this type of filter in a airborne system
is that any target with a Doppler frequency close to the clutter will also be affected by
the crude DC filter.
As mentioned the clutter spectrum is very narrow in stationary ground system, so only a
narrow MTI filter is needed to remove the clutter. In airborne radars however the clutter
is spread out over several frequencies with a dominant mainlobe at some frequency, as
seen in Figures 1.4a and 2.3. To filter out the clutter the MTI filter is placed at the same
frequency as the centre of the MLC. This removes the MLC and a windowed Fourier
transform can be applied to detect any targets without the interference of the MLC. The
drawback is that the SLC is still present in the signal and any target that had a Doppler
frequency close to the MLC has also been eliminated by the MTI filter. As such it is
not a very efficient method and many applications have turned to Space Time Adaptive
Processing (STAP) for more advanced ways of filtering out the clutter [33].
2.4 Clutter modellingSince clutter in general is complex and varies with geographical location and various
radar parameters simplifications need to be made for this thesis.
Altitude return is assumed to be small enough to be ignored, leaving only the SLC
and MLC to be accounted for. The SLC would be spread between the minimum and
maximum possible Doppler shifts caused by the platforms velocity. The MLC would be
steerable within the SLC interval in frequency domain, which relate to radar mainlobe
azimuth angle.
The clutter would be modelled to have a complex Gaussian distribution c ∈ CN (0, σ2c ).
The motivation behind this choice was that the side looking radar would have a clutter
cell which was limited by range resolution and azimuth angle. This would result in re-
turns from the ground from approximately the same range. As the radar beam would be
expected to encounter similar objects and surfaces with similar reflectivity, the distribu-
tion from each of them would be similar. With this assumption it would be fair to say
that the summed up radar return would be approximately Gaussian in its distribution
according to the central limit theorem in probability theory.
Complex valued additive white Gaussian noise would then be added across the full
Doppler spectrum to simulate the thermal noise n ∈ CN (0, σ2t ) present in the signal.
Since the signals were to be complex valued, both the clutter and thermal noise had
real and imaginary parts, both of which need to be generated separately to make them
independent of each other.
2.4. Clutter modelling 15
With the statistics decided upon the signal can be constructed in a way representing
reality. The signal was to be measured in terms of the signal strength registered in each
azimuth angle θ. The antenna would transmit and receive the signal with a specific field
pattern. The field pattern depended on the weights placed on the antenna elements and
which azimuth angle the mainlobe of the radar was directed in. The transmitted field
pattern can be described as
Et(θ) =K−1∑k=0
Tkexp
[j2πdk
λ(sin(θ)− sin(θml))
](2.9)
where θml is the azimuth angle of the mainlobe and Tk are rectangular window weights.
The transmitted signal would interact with the environment and get coloured by the
clutter, to lastly be returned and weighted with a standard antenna receiver taper such
as a Taylor window. The received field pattern becomes
Er(θ) =K−1∑k=0
Rkexp
[j2πdk
λ(sin(θ)− sin(θml))
](2.10)
where Rk are Taylor window weights. Both Tk and Rk had a Gaussian distributed phase
noise coded into them to represent the difference between the phase shifters in the antenna
array when transmitting and receiving. The total signal can then be expressed as
Etr(θ) = Et(θ)c(θ)Er(θ) (2.11)
where c(θ) denotes the complex valued clutter colouring of the signal at each azimuth
angle. The signal was then assigned the correct Doppler frequency according to the cor-
responding azimuth angle according to Eq. 2.4 with (ε = 0) and Eq. 2.5. By then adding
thermal noise the field pattern gets transformed into a Doppler frequency dependent
signal
Etr+n(f) = Etr(f) + n(f) (2.12)
The received power of the cluttered and noisy signal would then create the clutter spec-
trum
|Etr+n(f)|2 = |Etr(f) + n(f)|2 (2.13)
which can divided with the PRF to get the normalised signal power spectrum. In addition
to this the power of the noise can be adjusted until a power ratio between the clutter and
the noise that is suitable can be achieved. This ratio is called the clutter to noise ratio
(CNR) and is basically calculated by dividing the clutter power by the noise power. A
CNR of roughly 60 dB is not uncommon in radar applications.
16 Theory
2.5 Target modellingTarget modelling in the clutter and noise spectrum is fairly straight forward. As a target
in general is small and rather far away it means that its Doppler spectrum is narrow
and its power is often much lower than the clutter itself. The easiest way to implement
a target in the spectrum was to add a single harmonic with a specific frequency in
the return signal. This manifested as a single spike at the specified frequency with a
adjustable amplitude. The amplitude of the targets can then adjusted by trial and error
until a target.
2.6 Minimum Variance EstimatorTo construct an optimum processor with the goal of improving the SINR of the signal
an optimum linear weighting can be constructed
wopt = γQ−1s (2.14)
where γ is a scalar constant, which in the case of SINR maximisation has no influence
and can be set to unity [33]. The matrix Q is the interference autocorrelation matrix
built up as
Q = E[qq∗] =
Rq(0) R∗
q(1) R∗q(2) . . . R∗
q(N − 1)
Rq(1) Rq(0) R∗q(1) . . . R∗
q(N − 2)...
......
. . ....
Rq(N − 1) Rq((N − 2)) Rq((N − 3)) . . . Rq(0)
(2.15)
where ∗ denotes the complex transpose and Rq(lag) denotes a certain element from the
autocorrelation function Rq. For example the autocorrelation matrix for analytical white
gaussian noise would result in the identity matrix, as the noise only correlates with itself
at zero lags. The output of this optimum processor is the scalar
y = x∗wopt (2.16)
The reason for it being scalar is that the optimum weighting is constructed to search for
signal s with a certain frequency. Which means that a new wopt needs to be constructed
for each frequency searched for. In the case of 128 discrete frequencies it means the
calculation of 128 different wopt where each new scalar y is the signal estimate at that
certain frequency the weights were optimised for.
2.7 MultitaperingMultitapering in spectrum estimation is equivalent to using a linear subspace transfor-
mation and performing the clutter or interference rejection in that subspace. This means
2.7. Multitapering 17
less calculations are needed, which is important when performing calculations in real-
time, as is done in radar systems. In this thesis the nomenclature outlined in [33] was
followed to present multitapering in subspace form. Consider a data vector
x = s+ q = s+ c+ n (2.17)
constructed from the column vectors
x =
x1
x2...
xn
; s =
s1
s2...
sn
; q =
q1
q2...
qn
; c =
c1
c2...
cn
; n =
n1
n2...
nn
(2.18)
where s is the desired signal, q is the interference constructed from the clutter c and
noise n. Multitapering performs the minimum variance estimation calculations but with
a set of orthogonal tapers. This means that if there are N discrete frequencies and L
tapers used, the complexity increases and the number of calculations needed increase.
To combat this the linear transform matrix T of size N × L is introduced
T =
w1,1S1 w1,2S1 . . . w1,LS1
w2,1S2 w2,2S2 . . . w2,LS2...
.... . .
...
wN,1SN wN,2SN . . . wN,LSN
, S =
exp [j2πfst1]
exp [j2πfst2]...
exp [j2πfstN ]
(2.19)
Each column of T consists of the weights for each discrete frequency belonging to each
taper used. These tapers are then steered out to a single discrete frequency fst by
the steering vector S, note that S 6= s. The signal vector s gets introduced into the
multitapering as a part of x from Eq. 2.17, while S is responsible for steering which
discrete frequency to investigate. Setting fst to a certain frequency is equivalent to
investigating the frequency response of each taper at that specific frequency.
The matrix T is used to perform the following transformations
xt = T ∗x St = T ∗S
qt = T ∗q Qt = T ∗QT (2.20)
where T ∗ denotes complex transpose of T . The data, steering and interference vectors
have been transformed by Eq. 2.20 to column vectors of L elements, similarly the size
of the noise covariance matrix has been reduced down from N × N to L × L. This is
the main benefit of the subspace transformation, as the smaller Qt is easier to estimate
than the large Q. Since L << N in most cases, this transform significantly reduces the
computation time of the spectrum estimation.
The optimum weights for the subspace transformation becomes
wt = γQ−1t St (2.21)
18 Theory
and similarly the signal estimate becomes
yt = x∗twt (2.22)
where the signal has been estimated for a single discrete frequency, but with optimum
contributions from all L tapers and their weights. To construct a complete spectrum
estimation in this way the calculations of Eq. 2.19, 2.20, 2.21 and 2.22 have to be repeated
for all discrete frequencies in the spectrum.
2.8 Improvement factorThe improvement factor (IF) is a commonly used factor as a performance measure of the
radar. It is defined as the SINR of the output of the clutter filter divided by the SINR
of the input to the clutter filter [2]. It is calculated as
IF =SINRout
SINRin=
P outs
P outq
P ins
P inq
=
w∗ss∗ww∗Qws∗s
tr(Q)
(2.23)
where the subscripts s and q stands for signal and interference respectively, while tr()
denotes the trace of a matrix. In the case of the subspace transform vectors the IF
become
IF =w∗
t StS∗t wttr(Q)
w∗t QtwtS∗S
(2.24)
as shown in [33]. This IF can then be compared to the IF of an optimum processor,
such as the minimum variance estimator. The optimum IF for the minimum variance
estimator becomes
IFopt =S∗Q−1Str(Q)
S∗S(2.25)
2.9 Discrete prolate spheroidal sequencesDPSS are also called Slepian sequences, named after David S. Slepian for work on the
subject, starting with [34].
The origin of the DPSS is the problem of finding a bandlimited signal that is maximally
concentrated within a certain time interval [35]. That is, which time-limited sequence
x(t) maximises the ratio
µ =
∫ B−B |X(f)|2df∫ 12
− 12
|X(f)|2df(2.26)
2.10. Discrete prolate spheroidal window 19
where X(f) is the Fourier transform of x(t) and Eq. 2.26 is a measure of energy concen-
tration within the bandwidth [−B,B]. The limits on the integral in the denominator rep-
resent the entire frequency band [−12, 1
2] normalised with respect to sampling frequency.
In extension the same concentration problem is extended into a index-limited discrete
problem where the sequence x[n] that maximise the concentration ratio is searched for.
Maximising µ results in the sequence x[n], which can be shown to be a scalar multiple of
the zero-order DPSS [36]. How to find these sequences is outlined in section 2.10.
DPSS are defined by their sequence length, N as well as the frequency bandwidth
(−W,W ) in which they are maximised [36]. The DPSS is therefore written as
ν [n](k) (N,W ) (2.27)
where k is the Slepian order and n is the time index. Throughout this thesis however
the dependency on N and W is omitted and the Slepians are instead written as
ν [n](k) (2.28)
for simplicity. The DPSS of order k is defined as the real solution to the system of
equations
N−1∑m=0
sin((2πW (n−m)))
π(n−m)ν [n](k) = λ(k)ν [n](k) , n = 0,±1,±2, ... (2.29)
where λ(k) are the eigenvalues of the system with their dependency on N and W omitted.
The problem can be reduced to solving for the eigenvectors in a matrix. The eigenvalues
will however be clustered close to 0 or 1, this result in the eigenvectors being numerically
ill conditioned. This means that in practice DPSS are not calculated by solving Eq. 2.29,
instead DPSS are calculated as described in the following section.
2.10 Discrete prolate spheroidal window
In [37] DPPS were presented in an easier to calculate form, which meant that a Discrete
Prolate Spheroidal Window (DPSW) could easily be calculated. Slepian showed that if
the DPSS were index limited to n ∈ (0, N − 1) they would also satisfy the equation
1
2n(N − n)ν [n− 1](k) +
[cos(2πW )
(N − 1
2− n
)2
− θk
]ν [n](k) +
+1
2(n+ 1)(N − 1− n)ν [n+ 1](k) = 0 (2.30)
where θk are the eigenvalues of Eq. 2.30, the eigenvalues dependency on W and N has
been omitted. The main benefit of the reformulation from Eq. 2.29 to Eq. 2.30 was
20 Theory
that the eigenvalues become well spaced and as a result the eigenvectors become well
conditioned numerically. The difference equation in Eq. 2.30 can also be reformulated
and solved by finding the eigenvectors of a tridiagonal matrix A. A summary of this can
be found in [36], the resulting matrix become
{A(N,W )}i,j =
12i(N − i) , j = i− 1
cos(2πW )(N−1
2− i)2
, j = i12(i+ 1)(N − 1− i) , j = i+ 1
0 , otherwise
(2.31)
with eigenvalues denoted
0 < λ(k) = θk < 1 (2.32)
the reason for the eigenvalues in the two problems are the same is outlined in section VII
in [36]. The eigenvalues can be calculated by a suitable eigendecomposition algorithm
such as the one outlined for use with MATLAB in Appendix 1 of [36]. The algorithm
approximates the dominant eigenvalue, which is coupled to a corresponding sequence
{xk}, which approaches the eigenvector. The calculated sequence is equal to the approx-
imated DPSW, which is a scalar multiple of the zero-order DPSS. Finding the second
most dominant eigenvalue will correspond to the DPSW which is a scalar multiple of the
first-order DPSS, and so on.
As mentioned there are several algorithms that can be used, one of which is the power
method presented in [36]. However, MATLAB has a built in function dpss() in the signal
processing toolbox which is useful for quickly constructing any DPSS. A comparison
between a DPSS of the 0th order generated by the algorithm in [36] and the built in
function in MATLAB can be seen in Figure 2.4. These DPSS orders were generated
with a sequence length of N = 31 and a half time bandwidth of NW = 3.1. The Half
time bandwidth is the product of the sequence length and half bandwidth in which the
energy is concentrated. In this case W = 3.131
= 0.1 which means that the main peak
in the spectrum of the window should be within that limit, which it is when looking at
Figure 2.4a.
Comparing the two algorithms show that the difference is less than 10−11, see Fig-
ure 2.4b. But as the error will vary with error thresholds set in the power method
algorithm it is safe to assume that the built in function in MATLAB should be the
preferred method for DPSS generation.
An example of the first three orders of a DPSS generated by the function can be seen in
Figure 2.5. These functions were generated with a half time bandwidth of NW = 2.5 and
128 samples, resulting in an energy concentration within the frequency W = 2.5128≈ 0.02,
as seen in Figure 2.5b.
As can be hinted at slightly in Figure 2.5b the energy concentration of higher order
DPSS start to approach the specified limit W . With orders high enough the energy
2.10. Discrete prolate spheroidal window 21
0 10 20 30
Time samples
0
0.05
0.1
0.15
0.2
0.25
0.3
Am
plit
ud
e
DPSS
0 0.1 0.2 0.3 0.4
Normalised Frequency
-140
-120
-100
-80
-60
-40
-20
0
Po
wer
[dB
]
Power Spectral Density
Power method
MATLAB function
(a) Algorithm comparison.
0 5 10 15 20 25 30
Time sample
0
0.5
1
1.5
2
2.5
3
3.5
4
Diffe
ren
ce
10-12 Difference between methods
(b) Algorithm difference.
Figure 2.4: DPSS generated from two different algorithms showing the sequences in (a)
time and frequency as well as (b) a comparison of the error between the algorithms.
0 20 40 60 80 100 120
n
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Am
plit
ud
e
Slepian sequences: N = 128, NW = 2.5
0th
1st
2nd
(a) Time domain.
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Normalised freq.
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Pow
er
[dB
]
Spectrum of slepian sequences: N = 128, NW = 2.5
0th
1st
2nd
(b) Frequency domain.
Figure 2.5: The first three DPSS orders generated with the built in dpss() function in
MATLAB shown in (a) time domain and (b) frequency domain.
eventually leaks outside this limit, reducing the usefulness of those orders. The choice of
bandwidth with regard to the number of samples is a complex problem and the choice
will have an direct impact on the usefulness of the sequences in spectrum estimation
problems. For a given half time bandwidth there exists a maximum of
Lmax = 2NW − 1 (2.33)
useful orders with eigenvalues sufficiently close to unity for them to be used, normally
22 Theory
fewer than the maximum useful orders are used. The choice of the number of orders to
be used is dependent on the application and processing power.
The same goes for the choice of bandwidth, depending on the application the choice of
bandwidth and the resulting choice of half time bandwidth can have a great impact on
performance in spectrum estimation. Typically the default value used in many applica-
tions is a half time bandwidth value of NW = 4 for different reasons. It is in no way an
optimal value as shown in [38], but few methods seem to exist to aid in the choice of the
bandwidth.
2.11 Factorial experiments
When performing experiments or simulations it is sometimes important to get results
from as few experiments as possible. Examples of this are if the experiments require a
lot of resources, money or simulation time to complete. Factorial experimental design is
a way to realise this by restricting the number of parameters and their levels to get the
most out of few experiments.
2.11.1 Factorial design
There are several different designs for a factorial experiment which offer different advan-
tages, but the basic 2p and 3p designs are sufficient to introduce the theory. The 22 and
23 designs can be seen in Figure 2.6, illustrating the high/low levels of each parameter
xk coded to ±1. In Figure 2.6 only two levels are used, but two levels are only enough
to estimate linear and cross effects. To estimate second order effects a minimum of three
levels are needed, and so on. The benefit of more levels is that it enables estimation of
quadratic terms, but to only estimate linear and cross term effects two levels are sufficient
[39].
2.11.2 Statistics of an experiment
The statistics of an experiment need to be evaluated, this is mostly done with variance
analysis, sometimes called ANOVA [40], [41]. The time limit for this project did not allow
for the use of ANOVA methods, but they are a critical tool in deciding which results are
statistically significant. The reason being that any estimation of a parameter from a data
set becomes less accurate if the data set is decreased. A small data set which has a large
variance in the parameters impact on a performance measure will result in a incorrect
estimate of the parameters impact. Thus ANOVA becomes increasingly important with
a decreasing number experiments performed.
Additionally the variance will be affected by how many levels are used in each parameter
that should be estimated. Having 30 levels of parameter 1, while only having 3 levels of
parameter 2 will result in an uneven estimation in terms of variance. This is the reason
2.11. Factorial experiments 23
x1
x2
(−1,−1) (+1,−1)
(−1,+1) (+1,+1)
(a) 22 design.
x1
x3
x2
(+1,−1,−1)(−1,−1,−1)
(−1,−1,+1) (+1,−1,+1)
(+1,+1,+1)(−1,+1,+1)
(−1,+1,−1) (+1,+1,−1)
(b) 23 design.
Figure 2.6: Example of the basic (a) 22 and (b) 23 factorial designs.
that factorial designs are symmetrical.
2.11.3 Estimating effects
To estimate the effect of the different parameters a model first needs to be chosen. Con-
sider a system output y which is assumed to be able to be approximated by a polynomial
with cross terms and second order terms. It is dependent on three parameters x1, x2, x3
each able to take on three different levels. This results in 33 experiments and the ability
to approximate the system output of each experiment as
y(x; α) = α0 + α1x1 + α2x2 + α3x3 + α12x1x2 + α13x1x3 + α23x2x3+
+ α123x1x2x3 + α11x21 + α22x
22 + α33x
23 (2.34)
In matrix notation all experiments can be expressed as
y1
y2...
y27
=
1 x1,1 x1,2 x1,3 x1,1x1,2 x1,1x1,3 x1,2x1,3 x1,1x1,2x1,3 x2
1,1 x21,2 x2
1,3
1 x2,1 x2,2 x2,3 x2,1x2,2 x2,1x2,3 x2,2x2,3 x2,1x2,2x2,3 x22,1 x2
2,2 x22,3
......
......
......
......
......
...
1 x27,1 x27,2 x27,3 x27,1x27,2 x27,1x27,3 x27,2x27,3 x27,1x27,2x27,3 x227,1 x2
27,2 x227,3
α0
α1
α2
α3
α12
α13
α23
α123
α11
α22
α33
(2.35)
24 Theory
where the first subscript denotes experiment number and the second denotes the param-
eter. Each new experiment contains a unique combination of parameter levels according
to a chosen factorial design. In compressed notation Eq. 2.35 can be written as
y = Xα (2.36)
where the parameters α can be estimated with least-squares estimation according to
α = (XT X)−1XT y (2.37)
2.11.4 Design and Model matrices
The matrix X is called the model matrix and as mentioned above each row represent an
equation for a unique choice of levels in the parameters. As such X contain the complete
set of equations for all possible combinations of the parameters for the given factorial
design. In addition, all possible combinations of levels can be summarised in a design
matrix D, which can be used to construct the model matrix X depending on how many
orders and cross terms which are to be estimated.
For example with a 23 factorial design the possible combinations result in a design
matrix
D =
−1 −1 −1
−1 −1 +1
−1 +1 −1
−1 +1 +1
+1 −1 −1
+1 −1 +1
+1 +1 −1
+1 +1 +1
(2.38)
where each column is a parameter and each row is a unique combinations of the coded
levels for the parameters. A design matrix for a 33 factorial design would thus have a
size of 27× 3.
Constructing a model matrix from the design matrix is rather straight forward as the
model matrix is just a combination of the elements already present in the design matrix.
Taking the design matrix in Eq. 2.38 and producing a model matrix that estimates first
order and cross order terms would result in
X =
1 d1,1 d1,2 d1,3 d1,1d1,2 d1,1d1,3 d1,2d1,3
1 d2,1 d2,2 d2,3 d2,1d2,2 d2,1d2,3 d2,2d2,3
1 d3,1 d3,2 d3,3 d3,1d3,2 d3,1d3,3 d3,2d3,3
1 d4,1 d4,2 d4,3 d4,1d4,2 d4,1d4,3 d4,2d4,3
1 d5,1 d5,2 d5,3 d5,1d5,2 d5,1d5,3 d5,2d5,3
1 d6,1 d6,2 d6,3 d6,1d6,2 d6,1d6,3 d6,2d6,3
1 d7,1 d7,2 d7,3 d7,1d7,2 d7,1d7,3 d7,2d7,3
1 d8,1 d8,2 d8,3 d8,1d8,2 d8,1d8,3 d8,2d8,3
=
+1 −1 −1 −1 +1 +1 +1
+1 −1 −1 +1 +1 −1 −1
+1 −1 +1 −1 −1 +1 −1
+1 −1 +1 +1 −1 −1 +1
+1 +1 −1 −1 −1 −1 +1
+1 +1 −1 +1 −1 +1 −1
+1 +1 +1 −1 +1 −1 −1
+1 +1 +1 +1 +1 +1 +1
(2.39)
2.11. Factorial experiments 25
which is simple and quick to construct. The elements di,j are simply the elements from
Eq. 2.38 and the same method can be used for any number of levels and orders to be
estimated. The method also lends itself well for automated generation in a script or
function.
2.11.5 Analysing the effect of terms
When the effect of the terms are examined a change is made to Eq. 2.37 which becomes
A
B
C
AB
AC
BC
ABC
AA
BB
CC
= 2
α1
α2
α3
α12
α13
α23
α123
α11
α22
α33
= 2(XT X)−1XT y (2.40)
where the coefficient corresponding to α0 has been omitted as it is a constant effect which
is of less interest when investigating the relationships between the parameters. The factor
two is added since the terms estimated by the least-squares fit, model the effect on [0,+1],
but the values are actually changed over [−1,+1] according to the fractal design chosen
[39].
The Pareto plot
After having estimated the effect of each term they can be visualised in a Pareto plot. It
is a bar graph where each bar is equal to the double magnitude of each term coefficient as
specified in Eq. 2.40. The terms are sorted in descending order according to how big their
impact is on the performance measure when the corresponding term is increased. A blue
bar correspond to a positive effect on the performance measure as the term is increased.
Similarly the red bar correspond to a negative effect on the performance measure as the
term is increased. The scale on the x-axis is of no great importance as it will be affected
by the values in the parameters, it is instead the size of the bars relative to each other
that is of interest. Figure 2.7 shows an example of such a plot.
The analysis of some terms are straight forward, however interpretation of the cross
terms and the higher order terms become less obvious. When looking at Figure 2.7 it can
be seen from C and A that increasing x3 and x1 has a positive effect on the performance
measure, where x3 is dominant. Increasing x2 on the other hand, has a negative effect
on the performance measure, as seen from B. A faulty conclusion is thus to increase x1
and x3 while keeping x2 as low as possible and that should maximise the performance.
26 Theory
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Double magnitude of effect
ABC
CC
AA
BB
B
BC
AB
A
AC
C
Term
Example Pareto plot
(a) Pareto plot example.
0 0.5 1 1.5 2 2.5 3
x
-6
-4
-2
0
2
4
6
y
y=+A*x + AA*x2
y=+A*x - AA*x2
y=-A*x + AA*x2
y=-A*x - AA*x2
(b) Term interpretation aid.
Figure 2.7: An example of a Pareto plot and an aiding graph in how to interpret the size
and sign of the effects in relation to each other.
But in reality increasing both x1 and x3 has a large negative effect as visible from the
AC bar in Figure 2.7, while at the same time AB and BC are positive. So there exist
a trade-off between the parameters which is large enough that it should not be ignored.
The three parameter cross term ABC is on the other hand rather small and does not
contribute much to the performance. The fact that it is present signifies that all three
parameters have a complicated releationship.
The second order terms can be interpreted as a curvature of the performance measure
surface as the parameters increase. For example the negative AA bar can be interpreted
as the positive effect of increasing x1 is decreasing as it grows larger. A comparison
can be made to the red line in Figure 2.7b as the effects are coefficients in front of an
estimated polynomial, see Eq. 2.34 and Eq. 2.40. Similarly the positive CC bar means
that the positive effect of increasing x3 increases even more as it gets larger, as in the
blue line in Figure 2.7b. Finally the negative BB bar means that the harmful effect on
performance from increasing x2 increases even further as it grows, as with the purple line
in Figure 2.7b.
CHAPTER 3
Method
3.1 Simulation
The simulation part of the work was performed in the software MATLAB. Initially the
traditional methods were investigated to get an understanding of how traditional radar
Doppler filtering operates. The modelling of clutter was implemented as a function
for use with multitapering techniques later on. The introduction of targets in the the
background clutter spectrum was implemented to have something to filter out. Finally
the clutter was filtered out with a multitaper spectrum estimation function.
3.1.1 Comparison against a traditional method
The multitaper spectrum estimation was compared against a traditional MTI filter by
comparing the IF of the two methods. Additionally a simulation was performed to
estimate the average computation time and compare it against the multitaper spectrum
estimation. The method for the comparison against the traditional method is outlined
in greater detail in the partner thesis written by Linus Ekvall, titled: Airborne Radar
Ground Clutter Suppression using Multitaper Spectrum Estimation & Comparison with
Traditional Method.
3.1.2 Modelling the clutter
An example of a clutter spectrum generated from Eq. 2.13 is presented in Figure 3.1,
the CNR in the simulation was set to 70 dB. The other parameters were the same values
as in Table 3.1, with exceptions for W and L as the clutter spectrum generation was
unaffected by these parameters. The spectrum has been normalised with respect to the
maximum peak corresponding to the mainlobe clutter return.
To use the clutter and noise in the processing the frequency dependent signal in Eq. 2.12
27
28 Method
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Pow
er
[dB
]
Clutter spectrum
Figure 3.1: Simulated clutter with CNR = 70 dB.
was Fourier transformed back to the time domain to receive a time signal
q(t) = F−1{Etr+n(f)} (3.1)
containing clutter and noise. This was then used to form the autocorrelation matrix Q
to later filter out the clutter.
3.1.3 Modelling the targets
With the clutter and noise in the time domain from Eq. 3.1, the targets were inserted
into the signal by simply adding single harmonics with specified normalised frequencies
x(t) = q(t) + s1(t) + s2(t) + s3(t) (3.2)
where si(t) were single harmonics with a specific frequency. The amplitudes of the targets
were adjusted by trial and error until they were hardly visible inside the clutter.
3.1.4 Multitapering
The multitaper spectrum estimation was implemented as a MATLAB script which for
set parameter values generated clutter, a spectrum estimation and a plot of the IF of the
spectrum.
It worked as follows: To start off the target signals, clutter, interference autocorrelation
matrix and slepians to be used were generated. The multitapering was then performed
for each discrete frequency in the spectrum according to the following algorithm.
1. Form steer vector S
3.2. Full factorial experiment 29
2. Calculate linear subspace transform T with Eq. 2.19
3. Transform vectors and matrices with Eq. 2.20
4. Calculate optimum weights with Eq. 2.21
5. Calculate spectrum estimate with Eq. 2.22
6. Calculate optimum IF and multitaper IF
7. Repeat until the spectrum estimation and IF have been calculated for all discrete
frequencies
When the algorithm finishes all the data to produce plots for the clutter, spectrum
estimation and IF would be readily available.
A simulation with the implemented algorithm was performed to confirm if the multi-
taper spectrum estimation could be used to filter out the clutter. The filter was used on
a signal containing clutter and noise with a CNR of 70 dB which also contained three
inserted targets to be filtered out. The MLC was steered to the normalised frequency
of 0, equivalent to the radar mainlobe directed straight out of the side of the aircraft.
The normalised frequencies of the targets were ft = {−0.09, 0.03, 0.4} where the first two
were positioned close to the MLC and the 0.4 target was a control target placed in the
noise region. A successful experiment would filter out the clutter and make the targets
visible as spikes well above the clutter and noise in the resulting Doppler spectrum.
3.2 Full factorial experiment
3.2.1 The performance measures
The performance measure chosen was the improvement factor in the noise region IFnoise.
Also, since full factorial experiments support more than one performance measure for
each model estimation a second performance measure was included. Namely something
called minimum detectable velocity (MDV). The performance measures are illustrated in
Figure 3.2.
The IFnoise measure was taken as an average of the calculated IF in the noise region.
The IF in the noise region could be regarded as wholly uninteresting when the focus of
the thesis is on clutter suppression. The reason this performance measure was chosen
was first of all for the simplicity of calculating it, but also that a change in the IF in the
noise region also follows the change of IF as a whole in the complete spectrum.
The MDV measure was defined as the width in frequency where the IF has dropped 20
dB below the optimum IF. The frequency width represents the Doppler shift bandwidth
within which a target lower than 20 dB SINR would be undetected. This typically hap-
pens at a bandwidth centred around the same Doppler frequency as the MLC. Assuming
30 Method
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [dB
]
Performance measures
IF
IFopt
Figure 3.2: Example of the performance measures in a IF plot over the normalised
spectrum. The rectangle indicates the noise region of the spectrum where an average is
taken. The arrow indicates an example bandwidth where a target will not be detected
due to the MLC, equivalent to a 10 dB MDV limit.
that the target signal strength would remain at the same level, the target would have to
move outside the Doppler bandwidth in which it is invisible. To do this it would have
to move at a radial velocity relative the platform greater than half the MDV bandwidth.
For example if the MDV bandwidth was 32 Hz, the targets relative radial velocity would
have to be large enough to generate > 16 Hz in Doppler frequency.
3.2.2 Identifying and choosing parameters
First all variables though to have an impact in some way on either performance or
simulation time were listed, see Table 3.1. All of variables were changed one at a time in
the multitaper spectrum estimation simulation to see if they increased performance and
how much they affected the simulation time.
The impact on simulation time was graded on a scale from 1 (none/minimal) to 5
(extreme). The variables CNR, fprf, vpl, λ, d, θml and nφ simply affected the clutter
spectrum in some way. Since the clutter only had to be generated a few times, changing
these parameters did not impact simulation time in a major way.
The tapering weights for the transmitted and received signals in the generation of the
clutter did increase simulation time minimally when increasing N , but that is not why
they have been assigned the value 5 on the simulation time. As the other parameters,
they only need to be called once every time the clutter is generated. However, if they were
to be changed to some other taper weights, a completely new set of simulations would
have to be performed. For each new set of taper weights the simulation time would be
doubled. In addition many of the parameters affecting the generation of the simulation
time would have to be manually adjusted until the clutter generated from the new taper
3.2. Full factorial experiment 31
Table 3.1: All parameters thought to have an impact on the multitaper performance and
simulation time. The simulation time impact was graded between 1 and 5.
Parameter SymbolSimulation
timelevels Values
DPSS bandwidth W 1 17 {0.02, 0.025, . . . , 0.1} [Hz]
Time samples N 4 17 {200, 250, . . . , 1000} [-]
Slepian orders L 3 17 {1, 2, . . . , 17} [-]
Clutter to noise ratio CNR 1 1 60 [dB]
PRF fprf 1 1 10000 [Hz]
Platform velocity vpl 1 1 100 [m/s]
Wavelength λ 1 1 0.1 [m]
Element separation d 1 1 0.45λ [m]
Tapering weights out Tk 5 1 Uniform [-]
Tapering weights in Rk 5 1 taylorwin(N,4,-40) [-]
Mainlobe angle θml 1 1 0 [rad]
Element phase noise nφ 1 1 3 [-]
would produce a well behaved clutter spectrum in accordance with theory. That manual
adjustment time has been taken into account in the simulation time impact assigned to
these parameters.
The DPSS bandwidth did not impact simulation time much either as it was only used
in the generation of the slepians. As with the generation of the clutter this was not time
consuming and was only performed a few times per simulation.
The number of time samples used in the simulation had a big impact on the simulation
time. Mainly since it increased the size of the vectors and matrices which were operated
upon in the simulation. The simulation time grew significantly as N increased if the
number of slepian orders used were kept to a constant value. As it increased it also
allows for more usable slepians, in line with Eq. 2.33. If all available orders were used
the simulation time increased even further. The reason was that the number of slepians
used directly increased the size of the subspace vectors in Eq. 2.20. Since these need to
be calculated for each Doppler channel they are a time consuming calculation. Similarly
increasing W while keeping N constant will also unlock more usable slepian orders. This
is not as heavy an impact on the simulation time as increasing N . This made both N
and L among the dominant parameters affecting simulation time.
With the investigation of the time impact performance was at the same time observed.
From the simulations it was clear that the focus should lie on the parameters W , N and
L as they impacted the performance the most. While the other parameters also impacted
performance they did this by modifying the clutter in various degrees. Since the goal
was not to investigate the clutter model but rather the performance of DPSS multitaper
spectrum the choice was made to keep the clutter parameters constant. Motivating a full
32 Method
factorial model where bandwidth, time samples and DPSS orders were investigated.
3.2.3 Choosing levels
First off the variables which affected the clutter were considered. Here the most important
choice was the tapering weights in the clutter modelling. A set of weights that represented
a real spectrum well enough were already identified and were thus kept constant.
The CNR was simply a scaling factor between the energy in the noise and the clutter.
It was set to 60 dB as this level made the SLC visible in the spectrum without interfering
with the MDV measure. The PRF was set to 10000 Hz and the wavelength was set to
0.1 m as this was considered common levels for a airborne surveillance radar system [42].
Platform velocity was set to 100 m/s as a low velocity would be preferred for airborne
surveillance radar systems, since it limits the spread of the SLC according to Eq. 2.5.
Element separation was set to 0.45λ m to satisfy the distance separation of d < λ/2
between elements for a ULA. The mainlobe angle was set to a constant value of 0 rad
for simplicity, since it did not impact the performance measures.
The element phase noise was set to a constant scalar that adjusted the variance of
a N (0, 1) distribution. The scalar was chosen by trial and error. The final value pro-
vided the clutter spectrum with close to constant SLC level next to the mainlobe, as to
approximate the theoretical clutter spectrum.
The three variable parameters were to be chosen uniformly between a minimum and a
maximum value. The bandwidth was set to be
0.01 ≤ W ≤ 0.1 (3.3)
where the maximum was decided first. The whole idea of spectrum estimation with
first tapering the sequence to be estimated is that the taper has its energy concentrated
within a limited bandwidth. Too wide of a bandwidth and the resolution of the spectrum
estimation decreases. A DPSS bandwidth of 0.1 meant that the tapers energy would be
concentrated within the frequency normalised span (−0.1, 0.1), which was 20 % of the
total normalised spectrum. Investigating bandwidths above this did not seem reasonable.
A lowest W level of 0.01 was decided upon as lower bandwidths would force high values
of N according to Eq. 2.33 and high filter sidelobes.
The time samples were chosen to be in the range
50 ≤ N ≤ 1000 (3.4)
The whole problem of spectrum estimation in radar can be summed up as having too few
samples to get a good estimation. Generalised low values could therefore be somewhere
in the range of 50-200 samples. The highest value was set to 1000 time samples which
is very unusual as it would represent something along the lines of focusing on a single
stationary target.
3.2. Full factorial experiment 33
As mentioned above, the levels are subject to Eq. 2.33, but the lowest level was set to 1
order, representing a single window tapering with a 0th order slepian. The highest order
would be limited, but assuming that N and W are big enough there still exist a limit to
how many windows can be used. This limit is governed by hardware and computational
power mainly. It is not reasonable to compute multitaper calculations with thousands
of tapers in a real-time system. In reality only a few tapers can be used, so 20 windows
were chosen to be the upper limit.
1 ≤ L ≤ 20 (3.5)
The levels in Eq. 3.3, Eq. 3.4 and Eq. 3.5 differ from the ones specified in Table 3.1.
The cause for this was that the parameters were chosen according to the number of levels
of the full factorial experiment and to conform with Eq. 2.33, detailed further in the
following section.
3.2.4 Experimental setup
After having decided upon the important parameters and how many levels to use, two
design matrices would be generated. The first one holding the coded values would be
used to estimate a model, called Dcode. The second one would hold the actual values and
be used as an input for the simulation environment to produce the performance measures
Y , this second design matrix was called Dexp. Generating these design matrices would
be rather straight forward normally but for this case the process had to be tweaked a bit.
The reason for this was that the maximum orders of usable DPSS orders Lmax according
to Eq. 2.33 limits how large L can become for a given combination of N and W . This
resulted in the experimental space being restricted, an example of this can be seen in
Figure 3.3. Restricting the experimental space makes some of the combinations in the
basic design matrices invalid. The solution to this was to simply delete the rows in the
design matrices which correspond to the invalid experiments.
Initially smaller factorial designs were experimented with to examine how computa-
tionally heavy the simulations would be. It quickly became clear that the computational
load was very low, as such the number of levels in the design was increased up to a
point where the computational time became inconvenient. In the end 17 levels for each
parameter was chosen, resulting in 4913 different simulations. Some of these simulations
were restricted by Eq. 2.33, after removing those experiments 4849 valid experiments
remained. This restriciton left the experimental space uneven which would affect the
variance of each parameters estimation. But with the large number of simulations the
impact was thought to be kept to a minimum.
Choosing 17 levels for each parameter was done from within their respective spans
of interest in Eq. 3.3, 3.4 and 3.5. The bandwidth was chosen as 17 uniformly spaced
values between the the two limits detailed in Eq. 3.3. This could not be done for the
number of timesamples and DPSS orders, however as both these parameters had to be
34 Method
0.1
0.08
W
0.060
1000
0.04800
N
Experimental space
5
600
4000.02
200
10L
15
20
Figure 3.3: An example of a restricted experimental space, each blue point represents
a parameter combination, each red point represents maximum allowed order L for that
combination of N and W .
integer values. As a solution N was chosen to begin at 200, resulting in 17 uniformly
spaced values in increments of 50 timesamples up to the maximum limit of 1000 samples.
Similarly L was chosen to be the first 17 DPSS orders instead of 20. As a result the full
factorial experiment levels were chosen as detailed in Table 3.1.
An identical estimation as performed on IFnoise was also performed on the same set
of data, but normalised with respect to N . The reason for this was that increasing N
will always increase the IF due to the way the IF is defined. This is well known in radar
signal processing, which is why a normalised version is often more interesting. With the
normalisation of the IF it will directly affect the estimation of the parameter terms. The
most significant change is that N will become a negative term, as a increase in N will
lower the normalised value of the IF. The resulting Pareto plot might therefore indicate
which of W and L is of greatest importance.
3.2.5 Simulation environment
The simulation environment for implementing the full factorial experiments could be
said to consist of three parts. Pre-processing with the creation of design matrices, the
simulation itself, and post-processing to estimate the effects and visualise them. An
illustration of the simulation environment is outlined in Figure 3.4.
The Design matrix generation block constructed a coded design matrix with all possible
3.2. Full factorial experiment 35
Design matrix generation
Simulation
Model matrix generation
Effect estimation
Visualisation
Dexp
Dcode
Dcode, Dexp
X
αY
1
2 3
Figure 3.4: Illustration of the simulation environment consisting of (1) Pre-processing,
(2) Simulation and (3) Post-processing.
combinations for high and low values. An identical matrix then had experimental values
corresponding to the high and low values assigned to it. The experimental values are
checked to see if they fulfil Eq. 2.33. If they do not fulfil the DPSS order requirement,
that row, corresponding to the invalid experiment was deleted. The same row was then
deleted in the coded design matrix as they need to be identical to be able to estimate
the effects. The experimental matrix was fed to the simulation and visualisation blocks,
while the coded matrix was fed to the model matrix generation and visualisation blocks.
The simulation environment itself contained the calculation of the performance mea-
sures IF and MDV, for the set of variables along with their corresponding levels specified
in the experimental design matrix. The simulation output was a column vector contain-
ing the performance measures used in post processing. A more detailed description of the
simulation environment is outlined in the partner thesis written by Linus Ekvall, titled:
Airborne Radar Ground Clutter Suppression using Multitaper Spectrum Estimation &
Comparison with Traditional Method.
The model matrix generation block created a model matrix based on the design matrix
and which polynomial should be estimated. In the case of this thesis the design matrix
constructed estimated main effects, two parameter cross terms, three parameter cross
terms and second order terms, as in Eq. 2.34. The design matrix was then fed to the
effect estimation block.
The effect estimation block estimates the parameters with Eq. 2.40 and feeds the esti-
mation to the visualisation block. Lastly the resulting Pareto, interaction, colour coded
36 Method
images and contour plots were produced by the visualisation block.
CHAPTER 4
Results
4.1 Multitaper simulation 1
The values that differed from those in Table 3.1 for simulation 1 were W = 0.02, N = 200,
L = 7 and CNR = 70 dB. Three targets were present at the normalised frequencies
ft = {−0.09, 0.03, 0.4}. The target at -0.09. was placed at the border between the SLC
and MLC. The target at 0.03 was placed within the MLC and the target at 0.4 was placed
in the noise region for comparison. The resulting spectrum estimation of the simulation
is presented in Figure 4.1a. The achieved IF can be seen in Figure 4.1b, where it is
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
-10
0
10
20
30
40
50
60
70
80
90
Pow
er
[dB
]
Spectrum estimation
(a) Spectrum estimation.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Norm
alis
ed IF
Improvement Factor
IF
IFopt
(b) Normalised IF.
Figure 4.1: The result from a simulation with 7 DPSS sequences used as multitapers to
detect targets in the presence of clutter and noise.
normalised and compared to the optimum IF. The difference between the optimum IF
37
38 Results
and the multitaper IF in the noise region was approximately 0.72 dB for the simulation.
4.2 Multitaper simulation 2
The values that differed from those in Table 3.1 for simulation 2 were W = 0.02, N = 500,
L = 19 and CNR = 70 dB. Compared to simulation 1 only the number of time samples
and DPSS orders used was changed. Three targets were again present at the normalised
frequencies ft = {−0.09, 0.03, 0.4}. The multitaper spectrum estimation and resulting
improvement factor comparison can be seen in Figure 4.2. The difference between the
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
-20
0
20
40
60
80
100
Pow
er
[dB
]
Spectrum estimation
(a) Spectrum estimation.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalised freq.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Norm
alis
ed IF
Improvement Factor
IF
IFopt
(b) Normalised IF.
Figure 4.2: The result from a simulation with 19 DPSS sequences used as multitapers to
detect targets in the presence of clutter and noise.
optimum IF and the multitaper IF in the noise region was approximately 0.25 dB for the
simulation.
4.3 Comparison with the traditional method
The simulations which compared multitaper spectrum estimation with an optimum pro-
cessor and a traditional method are shown in Figure 4.3. They show the normalised
improvement factor for different slepian bandwidths and number of slepian orders used.
The multitaper spectrum estimation IF was called IFnorm, the optimum processor IF was
called IFopt and finally the traditional IF was called IFtrad. The number of time samples
were set to a constant value of N = 256, while W and L were varied. The remaining
parameter settings in the simulation were the same as in Table 3.1. Figure 4.4 shows
the average computation time for a multitaper spectrum estimation calculation in the
4.4. Factorial experiment 39
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1IF
Improvment Factor
IFnorm
IFopt
IFtrad
(a) W = 0.0078 & L = 3
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvment Factor
IFnorm
IFopt
IFtrad
(b) W = 0.0117 & L = 5
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvment Factor
IFnorm
IFopt
IFtrad
(c) W = 0.0234 & L = 11
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvment Factor
IFnorm
IFopt
IFtrad
(d) W = 0.0391 & L = 19
Figure 4.3: Normalised improvement factor for different bandwidths and number of
slepian orders used. All plots were simulated with N = 256 time samples used and
a CNR of 60 dB. The frequency on the x-axis has been normalised.
simulation environment. N was set to 256 and W was set to 0.04, while the number
of DPSS orders used were varied. Each dot represent the average time calculated from
1000 multitaper simulations while the blue line represent the average time for a tradi-
tional method. The grey bars represent the standard deviation of each average, while the
dotted line represent the standard deviation of the traditional method. The green circle
marks the point at which the multitaper spectrum estimations performance overcame
the traditional method.
40 Results
2 4 6 8 10 12 14 16 18
Orders used
0
0.05
0.1
0.15
0.2
0.25
Tim
e [s]
Averaged computation time: N=256, W=0.04
Trad.
Mult.
Trad.
Mult.
Figure 4.4: Computation time comparison between DPSS multitapering and a tradi-
tional MTI filter. Values calculated as an average of 1000 simulations. The green circle
represents how many orders were needed to outperform the MTI filter.
0 1 2 3 4 5 6
Effect 107
BB
ABC
CC
AB
AC
BC
AA
A
B
C
Term
IFnoise
(a) IF.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Effect
ABC
AB
CC
AC
BC
BB
AA
A
B
C
Term
IFnoise
(b) Normalised IF.
Figure 4.5: Pareto plots from the factorial experiment with (a) IFnoise and (b) IFnoisenormalised with respect to N as performance measures.
4.4 Factorial experimentThe results from the effect estimation on the IF and normalised IF performance measures
are presented in Figures 4.5a and 4.5b respectively. Colour coded images and contour
4.4. Factorial experiment 41
plots of both performance measures for a few different bandwidths can be seen in Fig-
ure 4.6. In similar fashion the result from the effect estimation on the MDV performance
measure is presented in Figure 4.7. With colour coded images and contour plots shown
in Figure 4.8.
(a) Colour scaled IF. (b) Contour plot of IF.
(c) Colour scaled normalised IF. (d) Contour plot of normalised IF.
Figure 4.6: (a) and (b) shows the IF for some different bandwidth choices as L and N
varies. (c) and (d) shows the normalised IF with respect to N .
42 Results
0 0.5 1 1.5 2 2.5
Effect
AA
AB
BC
ABC
A
AC
C
BB
CC
B
Term
MDV
Figure 4.7: Pareto plot from the factorial experiment with MDV as a performance mea-
sure.
(a) Colour scaled MDV. (b) Contour plot of MDV.
Figure 4.8: The MDV for some different bandwidth choices as L and N varies.
CHAPTER 5
Discussion
5.1 Multitaper spectrum estimation
The results presented in Figures 4.1 and 4.2 shows that multitaper spectrum estimation
with slepian sequences can indeed be used to filter out clutter. The resulting spectrum
estimation has seen a reduction in clutter as it is filtered out, while the MLC still impact
the spectrum and traces of the SLC remain.
In Figure 4.1a the target inserted in the noise interval of the spectrum is clearly visible
as it is not affected by the clutter. The target inserted at -0.09 was just barely visible as
a peak a few dB higher than any of the SLC peaks, while the target at 0.03 was to close
to the MLC to be visisble in the spectrum estimation. The reason that the 0.03 target
close to the MLC was hidden after the filtering can be seen in Figures 4.1b, where the IF
at 0.03 is very low, resulting in little to no improvement of the SINR at that frequency.
When N was increased from 200 to 500 time samples it yielded an additional 12 DPSS
orders for the given bandwidth, which could be used in the spectrum estimation. The
positive effect of this is clearly visible in Figure 4.2a where all three targets became visible
in the spectrum estimation. The target in the noise interval of the spectrum was again
unaffected by the clutter and as such was clearly visible. The target at -0.09 has become
clearly visible at almost the same power as the target in the noise interval. The target
closest to the MLC at 0.03 has in this plot become visible and has a power a few dB above
the rest of the clutter. The reason why the two targets in the clutter has become stronger
is easily spotted in Figure 4.2b. The interval in which the IF is very low, the MDV, has
shrunk, and the IF has increase at all frequencies when compared to Figure 4.1b. The IF
increase at all frequencies is the cause for the increased strength in the target at -0.09,
while the narrowing of the MDV has caused the 0.03 target to appear.
As two parameters have been changed between these two experiments nothing can be
said about which parameter caused the improvement in IF and MDV. By changing one
parameter at a time one could probably identify which parameter caused which per-
formance measure to increase, however no information about parameter interaction or
43
44 Discussion
second order effects would be gained. As the bandwidth was kept constant between
both experiments it is also not investigated properly. As such the full factorial experi-
ment where all parameters are changed at the same time would be essential to get more
information on the process.
5.2 Traditional method
The comparison between the optimum IF, the multitaper method and the traditional
method were compared in Figure 4.3 for different combinations of bandwidth and DPSS
orders, while the number of time samples was kept constant. Figures 4.3b, 4.3c, and
4.3d showed that the DPSS multitaper method can indeed outperform the traditional
methods as long as the parameters are chosen properly. If the parameters are chosen
poorly the multitaper would result in a worse result than for traditional methods, as
evident in Figure 4.3a.
While the multitaper outperformed the traditional method in Figure 4.3b, it was with
a slim margin. The small performance increase for using 5 tapers might perhaps not be
worth the increased computational load that multitapering brings.
Figure 4.4 showed a comparison in computation time in the simulation between the
traditional and multitaper method. For the given N and W the multitaper method
performed better than the traditional MTI filter when more than 10 DPSS orders were
used. The average computation time is however more than doubled due to the increased
sizes of the matrices and vectors being calculated. How many DPSS orders that have
to be used to outperform the traditional method was however dependent on the choice
of parameters. With well chosen parameters fewer DPSS orders could outperform the
traditional method. Similarly, badly chosen parameters could result in a sever increase
in computational time.
Additional work beyond the scope of this thesis would be needed to conclude where
the trade-off between calculation time and performance improvement lies for different
combinations of the parameters.
5.3 The factorial experiment
Before discussing the result of the factorial experiments in this thesis it is important to
note that the limited time of the project did not allow for proper analysis variance. For
full factorial experiments to be used correctly when searching for optimum parameters
ANOVA should be used. A small problem with the results were that Eq. 2.33 limited the
experimental space, as seen in Figure 3.3. This would directly impact the variance of the
parameters and was unwanted. However the large number of experiments was deemed
enough to minimise any negative impact from this.
The interpretation of the Pareto plots can be tricky, it is therefore valuable to keep
5.3. The factorial experiment 45
Section 2.11.5 and Figure 2.7b in mind as the results are discussed. As for the performance
measures, it is important to remember that while a high IF value increases performance,
it is the opposite for the MDV, which should be kept as low as possible. In other words
a positive effect on the IF means a increased value, while a positive effect on the MDV
means a decreased value.
5.3.1 Impact on IF
Figure 4.5a showed that the biggest factors affecting the IF were of the first order. The
biggest effect was from the number of DPSS orders used (L), followed by the number of
time samples (N) and lastly the bandwidth (W ). Increasing L and N would provide a
strong positive effect on the IF, while increasing W had a negative effect.
The AA term in Figure 4.5a indicate that the negative effect of increasing W decrease
as the bandwidth increases. Similarly the BB and CC terms indicate that the positive
effect of increasing N and L decreases as they grow. The remaining cross terms indicate
a relatively strong relationship between the parameters, which was expected.
The Pareto plot with the estimated effects of the parameters on the normalised IF can
be seen in Figure 4.5b. As predicted N does not improve the normalised IF as the IF is
normalised with respect to N . The result of which is that increasing N has a negative
effect.
More interestingly, the large positive effect from increasing L still dominates over the
negative effect from increasing W . In fact most terms related to W and L remain largely
the same when compared in size with each other between Figures 4.5a and 4.5b. While the
terms related to N has changed somewhat in size and switched sign, with the exception
of the ABC term which has become almost insignificant.
The most interesting results from the IF estimation can be seen in Figures 4.6. The
plots indicate that for a given bandwidth, an increase in N must be compensated with
additional DPSS orders in the multitapering to reach the same IF as achieved for the
lower value on N . In similar fashion for any given L, the bandwidth has to be decreased
when N is increased, if the same level of performance is to be obtained.
This analysis points toward a relationship between L and W , where they can comple-
ment each other to achieve a similar performance for different values of N . This is a
rather important result as the number of time samples used in airborne radar will vary
for various reasons. Knowing this relationship would permit changing W and L to reach
a desired IF level, regardless of the number of time samples received.
5.3.2 Impact on MDV
The impact on MDV was dominated by the positive first order effect was B in Figure 4.7.
The second most dominant first order effect was C. This indicate that the most important
parameters for improving MDV was N and to a lesser degree L. However, the CC and
BB bars indicate that this positive effect quickly diminishes as N and L are increased.
46 Discussion
This can also be seen in Figure 4.8 where increasing N initially had a positive effect of
minimising the MDV. This effect diminishes quickly however as seen in Figure 4.8b. The
same diminishing performance for L can also be seen in Figure 4.8b to an even greater
degree.
The effect of the bandwidth was rather small as evident from the A term in Figure 4.8b.
increasing W should on average result in a higher (worse) MDV. This is hinted at in
Figure 4.8, where the lowest bandwidth yield the lowest MDV values.
5.3.3 Parameter importance
The most important parameters for increasing the IF are arguably the bandwidth and
number of DPSS orders. While it is true that the number of time samples used have
an impact on the improvement factor, the number of samples that you get is something
which you sometimes have little control over. Therefore the bandwidth and number of
DPSS orders used are the most important parameters.
For improving the MDV the most important parameters would be the number of time
samples as well as the number of DPSS orders used. But as there was a rapid efficiency
decrease when increasing both parameters there exist a point at which increasing them
no longer is useful. The reason being that increasing the number of DPSS orders directly
impact the size of transformation matrix T , which governs the computational load in
the multitaper spectrum estimation. As such the use of DPSS multitaper spectrum
estimation is most likely not an efficient way to improve MDV.
5.4 Choice of DPSS parameters
5.4.1 Bandwidth
The results point toward choosing a small bandwidth to concentrate the energy within.
This makes sense in a spectrum estimation sense where the targets are single harmonics
and possibly placed close together in Doppler frequency. In the sense of the IF the impact
of increasing the bandwidth is a significant decrease in performance.
The negative impact of the bandwidth on the MDV is not as heavy. The results point
toward using a low bandwidth to increase performance of the IF and the MDV.
5.4.2 Number of DPSS orders
The results indicate that the rule for choosing the number of DPSS orders that increases
IF and MDV performances seems to be: higher is better. Overall it creates an improve-
ment and it seems proper that a method which chooses the optimum weights from a set
number of tapers will increase in effectiveness as the number of tapers to choose from
is incrased. The trade off here is against computational load, which increases with the
5.5. Future work 47
number of tapers used. A real radar system will ultimately be limited to how many
calculations it can perform by hardware, as such a maximum limit to the DPSS tapers
used is either limited by hardware or Eq. 2.33.
If the IF on the other hand would be desired to be set to a constant level instead of a
maximum level, which might be preferable in a radar system. The goal would be changed
to adjust the combination of bandwidth and DPSS orders used for any given value of
time samples available.
5.5 Future work
This thesis set out to investigate if DPSS could be used with multitaper spectrum es-
timation and perform better than traditional methods, as well as investigating some of
the parameters effect on the spectrum estimation. Along the way several simplifications
have been made and there is more work to be done to improve on the results.
Comparison against additional traditional methods
In this thesis only a single, very basic MTI filter was used as a comparison. More
advanced MTI filters exist as they have been used extensively over the years. Performance
comparison of DPSS multitaper spectral estimation against a more sophisticated MTI
filter would be necessary before replacing the traditional methods.
Validation against real data
Though originally planned, no validation against real data could be performed within
the time limit of the project. Therefore there exists significant work to be done within
validation of the method against real data.
Improved clutter model
One of the biggest simplifications in this thesis is the clutter model, as mentioned
throughout the thesis, clutter is complex and creating an accurate model of it is not a
trivial matter. The simplifications in the clutter model made in this thesis are far from
working clutter models used in the research and the aerospace industry today. As such
it only serves as a basis for showing that DPSS multitapers could be useful to filter
out the MLC and some SLC in a radar system. There may well exist forms of clutter
where the DPSS multitaper method in this thesis is inferior to other filtering methods.
It is therefore recommended that the DPSS multitaper method should be investigated
on more advanced clutter models to verify the usefulness of the method.
Optimisation
As this thesis has put little to no work into optimising the simulations or calculations,
further refinements in computational load comparison with traditional methods should be
performed. The added computational load might motivate using traditional methods in
some cases where the increased computation time makes multitaper spectrum estimation
a poor choice.
48 Discussion
Selection of bandwidth
The choice of bandwidth has a great impact on overall performance and impacts the
number of DPSS orders available for a certain number of time samples. This thesis has
found that overall a low bandwidth is preferable, but as there might arise situations
where few time samples exist and many DPSS orders are required, a higher bandwidth
might be used. It is therefore recommended that a method is developed for adaptive
choice of DPSS bandwidth for achieving the best result possible.
5.6 Science and ethicsThe scientific goal of this thesis has been focused on signal processing. To be more
specific, on how to distinguish a signal of interest in the presence of another unwanted
strong signal combined with noise. A method that has the possibility to do this has been
tested and evaluated. The method itself is applicable on many different array systems,
it could for example be used to improve radar in autonomous cars once scanned arrays
get smaller. It could also be used in ultrasound arrays to get rid of the respective clutter
in those measurements.
In the thesis the method has been applied for use in airborne surveillance radar. The
ethics of using this method in defence applications should not be forgotten. It has the
potential to be used to safeguard human life from potential harm by providing early
warnings of potential danger. However, as with many applications it has the potential
to be used to do harm as well. Anyone using the work in this thesis should therefore
consider the ethics of using it and any negative impact it could have on society and the
world as a whole.
CHAPTER 6
Conclusion
This thesis set out to answer the questions outlined in Section 1.6. The answers to these
questions are outlined below.
• Can Discrete Prolate Spheroid Sequences be used as multitaper functions to filter
out ground clutter in airborne radar?
Yes, Discrete Prolate Spheroidal Sequences can indeed be used as multitaper functions
to filter out ground clutter in airborne radar. The choice of DPSS parameters heavily
influence the performance, but once chosen correctly this thesis has demonstrated that
they are able to filter out ground clutter.
• How would the performance compare to traditional frequency estimation tech-
niques?
This thesis has shown that the performance of DPSS used as tapers in multitaper spec-
trum estimation can outperform traditional methods as long as the DPSS parameters
are chosen with care. As only a single traditional method has been investigated in this
thesis, validation against additional and more sophisticated traditional methods have
been suggested.
• How would the computational load compare to traditional frequency estimation
techniques?
A computation time simulation was presented in this simulation that showed that the
computational load for multitaper spectrum estimation are higher than in a traditional
method as the computations increase with each taper used.
• What parameters affecting the spectrum estimation performance exists?
This thesis has identified the DPSS generating parameters bandwidth W and the number
of DPSS orders L to be the parameters having the biggest effect on the spectrum estima-
tion. The number of time samples used N also affects performance, but it was assumed
49
50
that only limited control over this parameter exist. The biggest uncertainty in the effect
on performance has also been identified as the clutter, as it constantly changing. This
led to the suggestion of validation against advanced clutter models in the future.
• How should the parameters be chosen?
This thesis has identified the following:
For improving the SINR, equivalent to increasing the IF, this thesis has identified that
the number of DPSS orders used should be as high as possible, with the bandwidth being
as low as possible.
For improving the MDV the most important parameters were the number of time
samples and number of DPSS orders. Keeping both at high values had a positive effect
on the MDV. The effectiveness of increasing these parameters decreased rapidly however.
If instead keeping a constant IF as the number of time samples varies is of interest,
the number of DPSS orders and the bandwidth has been identified to have a relationship
which should be investigated further.
• Does a optimum choice of these parameters exist?
This thesis has found no true optimum for choice of parameters or universal rule that
holds for maximising the performance measures. The clutter, combination of parameter
levels and hardware limitations of a system would put limits on what in the end is
considered optimum. A low bandwidth and many DPSS orders seem to optimal for a
unrestricted increase of the IF. While the same is true for the MDV, the effectiveness
is much lower. As such using DPSS multitaper spectrum estimation as a method to
improve MDV is not recommended.
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