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Airborne Radar Ground Clutter
Suppression Using Multitaper Spectrum
EstimationComparison with Traditional Method
Linus Ekvall
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&
Comparison with Traditional Method
Linus C. Ekvall
Lulea University of Technology
Dept. of Computer Science, Electrical and Space EngineeringDiv. Signals and Systems
20th September 2018
ABSTRACT
During processing of data received by an airborne radar one of the issues is that the typical
signal echo from the ground produces a large perturbation. Due to this perturbation it can
be difficult to detect targets with low velocity or a low signal-to-noise ratio. Therefore, a
filtering process is needed to separate the large perturbation from the target signal. The
traditional method include a tapered Fourier transform that operates in parallel with a
MTI filter to suppress the main spectral peak in order to produce a smoother spectral
output. The difference between a typical signal echo produced from an object in the
environment and the signal echo from the ground can be of a magnitude corresponding
to more than a 60 dB difference. This thesis presents research of how the multitaper
approach can be utilized in concurrence with the minimum variance estimation technique,
to produce a spectral estimation that strives for a more effective clutter suppression. A
simulation model of the ground clutter was constructed and also a number of simulations
for the multitaper, minimum variance estimation technique was made.
Compared to the traditional method defined in this thesis, there was a slight improve-
ment of the improvement factor when using the multitaper approach. An analysis of how
variations of the multitaper parameters influence the results with respect to minimum
detectable velocity and improvement factor have been carried out. The analysis showed
that a large number of time samples, a large number of tapers and a narrow bandwidth
provided the best result. The analysis is based on a full factorial simulation that provides
insight of how to choose the DPSS parameters if the method is to be implemented in a
real radar system.
Keywords: Ground clutter, tapering, multitaper, discrete prolate spheroidal sequences,
minumum variance estimation, signal processing, radar, airborne radar.
iii
PREFACE
This work concludes my MSc in Engineering Physics and Electrical Engineering at Lulea
University of Technology between the years 2013-2018. The work has been conducted in
Kalleback at Saab Surveillance who supplies solutions including security, surveillance, de-
cision support and solutions for detecting and protecting against different types of threats.
The work was done in collaboration with Carl-Henrik Hanquist who has been focusing
on analysis of DPSS parameters using full factorial design [1]. The focus of this thesis
has been to provide a simulation environment for the full factorial simulation and a
comparison with a traditional method.
I would like to thank my co-worker Carl-Henrik Hanquist for ideas and insightful dis-
cussions. Also the team at Saab Surveillance, foremost my external supervisor Bjorn
Hallberg. A sincere thanks goes to my supervisor, Professor Johan Carlson at Lulea
University of Technology.
I would also like to express my gratitude for my parents Hans Ekvall and Monica Ekvall
for supporting me through the years.
Linus C. Ekvall
Goteborg, Sweden 2018
v
ABBREVIATIONS
CNR Clutter-to-noise ratio
DFT Discrete Fourier transform
DPSS Discrete prolate spheroidal sequences
FFT Fast Fourier transform
IF Improvement factor
LST Linear subspace transform
MDV Minimum detectable velocity
MTI Moving target indicator
PRF Pulse repetition frequency
Radar Radio detection and ranging
RCS Radar cross section
SNR Signal-to-noise ratio
ULA Uniform linear array
vii
CONTENTS
Chapter 1 – Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Related work and literature review . . . . . . . . . . . . . . . . . . . . . 3
1.4 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Scientific, societal and ethical aspects . . . . . . . . . . . . . . . . . . . . 3
Chapter 2 – Theory 5
2.1 Radar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Radar cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Radar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.5 Radar equation and clutter . . . . . . . . . . . . . . . . . . . . . 9
2.2 Spectral analysis and estimation . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Autocorrelation matrix . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Tapering and window functions . . . . . . . . . . . . . . . . . . . 11
2.2.4 Moving target indicator & Minimum detectable velocity . . . . . 13
2.2.5 Parameters and ratios . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.6 Discrete prolate spheroidal sequences . . . . . . . . . . . . . . . . 14
2.2.7 Multitaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Linear subspace transform & Minimum variance estimation . . . . 18
2.3.2 Full factorial simulation theory . . . . . . . . . . . . . . . . . . . 18
Chapter 3 – Method 19
3.1 Antenna model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Clutter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Minimum variance estimation & Multitaper . . . . . . . . . . . . . . . . 22
3.3 Traditional method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Simulation environment for full factorial simulation . . . . . . . . . . . . 23
3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 Signal processing simulation environment . . . . . . . . . . . . . . 24
Chapter 4 – Results 27
4.1 Clutter generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Simulated results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Parameter analysis by full factorial simulation . . . . . . . . . . . . . . . 32
4.3.1 Full factorial simulation . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Comparison with traditional method . . . . . . . . . . . . . . . . . . . . 40
4.4.1 Computation time . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter 5 – Discussion 45
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 6 – Conclusion 47
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Appendix A – 173 factor simulation 49
ix
CHAPTER 1
Introduction
1.1 Background
The concept of radar has its origin in the 19th century where James Clerk Maxwell
predicted the existence of radio waves in his theory of electromagnetism in 1864. The
theory was verified by the experiments of the German physicist Heinrich Hertz in 1886-87
and provided insight in the reflective behaviour of electromagnetic waves. In the early
20th century these types of systems became widely available and started to be utilized
for range measurements. The first system was implemented by the German inventor
Christian Hulsmeyer who constructed a ship detection device with the intention to avoid
collisions in fog, patented in 1904 [2].
Since then the technology has taken a vast leap and the research conducted in the area to-
day is cloaked in advanced mathematical concepts which are constantly evolving. Radar
engineers often refer to the ”target” which can be an airplane, a ship, a vehicle etc.
However, this can be generalized to any object in the surrounding environment that pro-
duces a desired radar echo to show its position. Another common term is the ”clutter”
which is the radar echoes produced from unwanted objects in the propagation path such
as birds, insects, rain, sea or the ground. In some cases the clutter can cause severe
performance issues for the radar system and thus a reliable filtering process is often nec-
essary to disregard these effects. The applications of today’s radars have gone from range
and angle measurements to applications including determining target velocity, amplitude
measurement, recognition of targets based on characteristics and weather prediction to
name a few. To make these applications both effective and efficient the data received by
the radar is processed digitally to extract desired information from the received signal.
Based on the desired outcome, the data is processed to unveil information that can be
hard to extract without the digital signal processing tool [3], [4], [5].
As an electromagnetic wave propagates through space, gets reflected by an object and
is received by the radar not only the distance to the object can be extracted but also its
velocity. Since the propagating wave has a predetermined frequency upon transmission
1
2 Introduction
it is possible to compare the frequency shift caused by the Doppler effect. To determine
this frequency shift a common method is to use the Fourier transform that transforms
the wave from time to frequency domain according to
F (ω) =
∫ ∞−∞
f(t)w(t)exp(−iωt) (1.1)
in continuous time. This is a fundamental concept in the field of spectral analysis and
is crucial in extracting information such as target velocity. With slight modifications it
can be applied in the discrete time case in order to be processed digitally, which will be
discussed in this thesis.
The standard Fourier transform is weighted with a uniform window function w(t), as
seen in (1.1). From a digital signal processing standpoint this corresponds to an equal
gain for every sample in time. The uniform weighting can limit the information one wishes
to extract from the spectrum such as targets with low signal-to-noise ratio (SNR). By
applying different gain on each time sample, targets that may be misinterpreted as noise
in the uniform weighted spectrum may appear. This method of gain calibration for the
different time samples is called tapering and in digital signal processing this function is
referred to as a window. Some common windows will be discussed such as Taylor or
Gaussian. To apply the method in a broader sense it is possible to use several windows
to extract information and combine them to get a desired outcome, which is referred to
as multitapering [4], [6].
1.2 Problem statement
One of the challenges in airborne radar systems is that the typical signal echo received
from the ground can be more than a million times larger than the signal echo from a
target, such as another airborne unit or a moving vehicle on the ground. In order to
suppress the magnitude of the unwanted ground clutter and increase the echo from the
target there are different techniques associated to this pursuit, one which is digital signal
processing, which will be studied in this thesis.
The purpose of this thesis is to investigate the possibilities of using multitapering to
suppress the ground clutter. The focus will be on time filtering regarding the difference in
Doppler frequency of the stationary ground and moving targets. This is a relatively com-
prehensive subject and has been divided into two subcategories in order to be manageable
within a masters thesis project. In this thesis I will provide a simulation environment for
performance measurements and comparison with traditional signal processing methods,
whereas my co-worker Carl-Henrik Hanquist will provide an analysis of discrete prolate
spheroidal sequence (DPSS) parameters using full factorial design [1].
1.3. Related work and literature review 3
1.3 Related work and literature reviewRelevant theory in the area includes [7] in which some common ”amplitude-weighting”
or tapering functions are compared in terms of integration loss, main lobe broadening
and sidelobe structure. Also the integration loss factor is defined, which characterizes
the decrease in SNR resulting from the inclusion of tapering. It is also explained that the
suppression of the sidelobes by amplitude weighting is proportional to the SNR. One of
many tapering functions called DPSS, also know as Slepian functions, are investigated.
Due to the orthogonal nature of these functions they can be utilized as combinations of
each other and thus be used in multitaper research. The theory is explained in [8] and
also in great detail in [9] where the discrete case is presented. Additional papers on the
DPSSs includes [10], [11] and [12].
The multitaper method is an approach for spectral density estimation, developed by
David J. Thomson. He chose to use the the DPSSs for multitapering [13]. In [6], cal-
culation methods suited for software implementation are presented and were analyzed
for theoretical insight. [5] provides necessary theoretical information regarding spectral
resolution, spectral masking, the modified periodogram and windowing to name a few
concepts. In [14], the different versions of the radar equation and radar cross section are
presented and [3] presents radar theory.
1.4 GoalThe main objective is to compare different kinds of multitaper for evaluation of the clutter
suppression measured by the improvement factor and minimum detectable velocity and
how they compare to traditional methods. In addition to this, a simulated estimate of
the ground clutter in order to be able to suppress it. To achieve this purpose simulations
in MATLAB were constructed for different methods and this is also where the majority
of the work was focused.
1.5 Scientific, societal and ethical aspectsThe applications of the methods presented in this thesis are primarily designed to be used
for airborne radars. From a scientific standpoint the objective is to improve the accuracy
of detecting targets before a noisy background and investigate any improvements and
limitations that the multitaper method provides. However, this can also be viewed from
the perspective of defence and security where the information provided by the radar can
be used to warn authority of different types of threats or illegal activity at a relatively
early stage, in order to provide decision support. The author would like to remind the
reader that the concepts presented in this thesis should be used with caution and that
any implementation pursuits will be for the benefit of society, defence or security.
CHAPTER 2
Theory
2.1 Radar theoryCompared to ground based radar, an airborne radar has an extended range due to its
elevated position and allows a longer visible range before the horizon creates shadows,
which are hidden areas on the ground the radar can not reach. Due to the mobility of the
platform it is also possible to extract high resolution images via the synthetic aperture
radar technology. The airborne radar clearly provides an advantage when it comes to
detecting targets early at a far distance but includes an extended clutter area depending
on the altitude of the platform [4].
A radar placed on a moving platform receives Doppler shifted echoes which are induced
by the velocity of the platform and the velocity of the object reflecting the wave. Specif-
ically for the ground clutter case, the high-powered signal echo from the ground right
underneath the platform has zero Doppler shift and is responsible for a high powered
spectral peak [3].
This spectral peak shows the maximum energy return but is typically not bounded to
zero Doppler but rather extends to a broader band of frequencies. The spread is due to
small ground movement variations such as moving trees or waves and is also dependent
on the angle of incidence of the electromagnetic wave [4].
The IEEE standard letter designations for radar frequency bands [15] are used to be
able to distinguish the electromagnetic spectral bands from each other and are presented
in Table 2.1.
5
6 Theory
Table 2.1: IEEE standard letter designation for radar-frequency bands.
Band designation Frequency range Abbrevation
HF 3-30 MHz High frequency
VHF 30-300 MHz Very High frequency
UHF 300-1000 MHz Ultra High Frequency
L 1-2 GHz Long wave
S 2-4 GHz Short wave
C 4-8 GHz Compromise between S and X
X 8-12 GHz X for cross (as in crosshair)
Ku 12-18 GHz Kurz-under
K 18-27 GHz Kurz (German for ”short”)
Ka 27-40 GHz Kurz-above
V 40-75 GHz
W 75-110 GHz W follows V in the alphabet
mm or G 110-300 GHz
2.1.1 Wave propagation
A radar operates by transmitting electromagnetic energy into the environment and re-
ceives information based on the energy that is reflected back from objects in its path.
One of the most basic concepts is to use this to measure distance. Assuming free-space
propagation it is possible to determine the range to an object in its path according to
R =ct
2(2.1)
where R is the one-way distance to the object, c is the speed of the electromagnetic
wave i.e the speed of light and t is the time delay between transmitting and receiving the
pulse. Since the electromagnetic wave travels twice the distance it is divided by two [3].
2.1.2 Radar cross section
Radar cross section (RCS) is a unit measured in [m2] which determines the detectability
of a target. As an electromagnetic wave hits an object the energy is scattered in all
directions and only a limited portion of the energy is backscattered in the direction of
the radar. The formal definition of the RCS is
σ = limR→∞
4πR2 |Es|2
|E0|2(2.2)
where E0 is the electric-field strength of the incident planar wave colliding with an
2.1. Radar theory 7
object in the far field. Es is the electric-field strength of the scattered wave at a distance
R from the object [14].
2.1.3 Radar equation
There are numerous variations of the radar equation in which simplifications for a de-
sired approximation can be chosen. In this thesis two variations will be explained where
the first is a classic example of the received and transmitted power ratio with the as-
sumption that the transmitter and receiver have different gains due to the nature of
antenna tapering. The other can be used to calculate the clutter-to-noise ratio (CNR)
and will be explained in Section 2.1.5. As the electromagnetic wave propagates through
the atmosphere its behaviour can be described by the ratio of received-signal power to
transmitted-signal power as given by
PrPt
=GTGRλ
2F 2TF
2R
(4π)3R4σ (2.3)
Table 2.2: Parameter description for (2.3).
PR Received-signal power
PT Transmitted-signal power
λ Wavelength
GR(θ, φ) Antenna gain on receive
θ Azimuth
φ Elevation
GT Antenna gain on transmit
FT Pattern propagation factor for transmitting-antenna-to-target path
FR Pattern propagation factor for target-to-reveiving-antenna path
σ Radar target cross section
R One way propagation distance
The parameters that need further explaining are FT and FR that account for a design
where the target is not in the beam maxima but can vary within the beam. More
specifically the factors FT and FR represents the ratio of the field strength E at the
targets position to the corresponding field strength E0 if the target would have been
located at the same distance R but in the maximum gain direction assuming free-space
propagation [14]. Apart from the power ratio the radar equation (2.3) also provides a
method for calculating the maximum detection range which can be derived with some
algebraic manipulation.
8 Theory
2.1.4 The Doppler effect
As previously mentioned, a radar can be used to determine the velocity of an object by
utilizing the fact that a moving target will produce a Doppler shifted echo corresponding
to the radial velocity of the target with respect to the platform, which is the projection
of the target velocity in the line of sight to the platform.
In the one dimensional case the target is traveling in the same or opposite direction
of the platform, tail-on or nose-on respectively. For a target approaching in this manner
the radial velocity is equivalent to the velocity of the target.
The Doppler frequency can be derived as
fd = −2R
λ(2.4)
where λ is the wavelength of the electromagnetic wave and R is the time derivative of
R.
Vpl VT
Figure 2.1: Illustration of a target approaching nose-on. Vpl is the velocity of the platform and
VT is the velocity of the target.
If a target appears at an angle α relative to the platform it is important to note that
the assumption of R = VT is no longer valid. The Doppler shift of the echo in this two
dimensional case corresponds to the radial velocity of the target according to
fd =2vpλcos(α) (2.5)
where
vp = Vpl + VT (2.6)
2.1. Radar theory 9
αV1
Vpl
V2
VT
Figure 2.2: Illustration of a target approaching at an angle α. Vpl is the velocity of the platform.
VT is the velocity of the target. V1 is the radial velocity of the platform and V2 is the radial
velocity of the target.
In airborne radar systems, the Doppler effect can be used to identify ground clutter [4]
which will be discussed in more detail in Section 2.2.4.
2.1.5 Radar equation and clutter
As previously explained, the clutter corresponds to signal echoes generated from reflec-
tions by undesired objects. To make an estimate of the clutter it can be put in relation
to noise as the CNR. The noise often corresponds to perturbations in the electronics
such as thermal noise in the receiver. Assuming no external losses in the propagation
path or elsewhere, the CNR can be described by modifying the radar equation (2.3) and
including the noise spectral density kBTs so that
C
N=PT τ GRGT λ
2
(4π)3R4 kB Tsσc (2.7)
10 Theory
Table 2.3: Parameter description for (2.7).
PT Transmitted power
τ Pulse length
GR Antenna gain on receive
GT Antenna gain on transmit
λ Wavelength
σc Radar cross section of the clutter resolution cell
R One way propagation distance
kB Boltzmans constant
TS System noise temperature
2.2 Spectral analysis and estimation
The applications of spectral analysis are typically used in interference spectrometry;
Wiener filter design for signal recovery and image restoration; during channel equalization
design in communication systems, to name a few [6]. In this section a summary of the
theoretical concepts regarding radar applications are presented.
2.2.1 Fourier transform
The Fourier transform is used to convert a signal from time to frequency domain and is
a fundamental concept in spectral analysis. The continuous time Fourier transform was
mentioned in (1.1), but since this is a function of a continuous variable ω, it is not suited
for digital processing. For a finite length sequence x(n) with N samples the discrete
Fourier transform (DFT), which is a function of an integer variable k, can be applied
according
X(k) =N−1∑n=0
x(n)w(n)exp(−j2πkn/N) (2.8)
The DFT can be computed very efficiently by using an algorithm called the fast Fourier
transform (FFT) which is based on the DFT concept [5].
2.2.2 Autocorrelation matrix
The autocorrelation is a second-order statistical characterization of a discrete-time ran-
dom process that can be represented in matrix form. If
x = [x(0), x(1), ..., x(p)]T (2.9)
2.2. Spectral analysis and estimation 11
is a vector of p+ 1 values of the process x(n), the outer product
xxH =
x(0)x∗(0) x(0)x∗(1) ... x(0)x∗(p)
x(1)x∗(0) x(1)x∗(1) ... x(1)x∗(p)...
......
x(p)x∗(0) x(p)x∗(1) ... x(p)x∗(p)
(2.10)
is a (p+ 1) × (p+ 1) matrix. If x(n) is wide sense stationary the autocorrelation can be
derived by taking the expected value E{xxH} and using the Hermitian symmetry of the
autocorrelation rx(k) = r∗x(−k), the autocorrelation matrix can be derived according to
Q =
rx(0) r∗x(1) r∗x(2) · · · r∗x(p)
rx(1) rx(0) r∗x(1) · · · r∗x(p− 1)...
......
...
rx(p) r∗x(p− 1) r∗x(p− 2) · · · rx(0)
(2.11)
which is a Hermitian and Toeplitz matrix [5] that will be used for clutter filtering.
2.2.3 Tapering and window functions
The window functions are used to weight the signal amplitude in order to balance the
tradeoff between spectral resolution, leakage and mismatch loss. The spectral resolution
is the ability to know how the signal energy is distributed in frequency space and is related
to the main lobe width of the windowed spectrum. With ideal resolution it is possible
to detect two different signals no matter how close they are in frequency. The amount
of leakage is a consequence of using a finite signal, where every frequency component
is responsible for the energy distribution in the frequency span. The leakage measured
corresponds to the ability to detect a weak signal in the presence of a neighbouring strong
signal [5].
The most common window function is the rectangular window which is evenly weighted
over all discrete inputs, sometimes referred to as ”uniform weighting” and is the default
window when performing a Fourier transform, excluding a window function. The function
has unity in magnitude for all elements of the time vector and zero otherwise. Thus
creating a rectangular shape. Although the window function can be used arbitrarily
and with some creativity to get a desired result, there are some commonly established
windows used in the field.
Apart from the rectangular window there are many others, often composed of some
variations of a sinusoid or exponential function. For example, the Gaussian window and
the Taylor window as seen in Figure 3.1. The Gaussian window function corresponds to
a sequence described as
wG(n) =
{e−n
2/2σ2, −(N − 1)/2 ≤ n ≤ (N − 1)/2
0, otherwise
12 Theory
where N is the number of samples and σ is the Gaussian standard deviation which is
a parameter that can be varied for desired spectral resolution.
The Taylor window [16] corresponds to
wT (n) =
{1 +
∑n−1m=1 Fmcos
(2πmnN−1
), |n| ≤ N−1
2
0, otherwise
where Fm = F (m, n, η) are Taylor coefficients of the mth order. η and n show ratio of
mainlobe over sidelobe level and number of sidelobes at equal level, respectively.
A graphical example is provided in Figure 3.1 where the functions are defined in the
time domain.
0 50 100 150 200 250
Time [Samples]
0
0.2
0.4
0.6
0.8
1
Am
plit
ude
Window functions
Taylor window
Gaussian window
Rectangular window
Figure 2.3: Example of three commonly used windows for signal manipulation, all with N = 256
samples.
In general, most window functions have a parameters such as σ, n or η responsible for
the tradeoff between spectral resolution and leakage.
The window functions are responsible for the modified periodograms. Where the dis-
crete modified periodogram is the Fourier transform of the windowed signal such as
Pper =1
N
∣∣∣XN(ejω)∣∣∣2 =
1
N
∣∣∣ ∞∑n=−∞
x(n)ω(n)e−jω∣∣∣2 (2.12)
2.2. Spectral analysis and estimation 13
where N is the number of samples, x(n) is the input, ω(n) is the window function and
XN is the Fourier transform of the windowed signal [5].
2.2.4 Moving target indicator & Minimum detectable velocity
The moving target indicator (MTI) has the capacity of detecting moving targets before
an interfering background. In this concept the difference between target and clutter
velocity is exploited for target detection. The pulse Doppler radar transmits coherent
pulses and measures the phase of the backscattered echoes, where the phase shift is
directly proportional to the radial velocity of the object in the propagation path. This
principle can also be used in ground clutter suppression where the central peak of the
clutter dominates at the zero Doppler frequency. By using a MTI filter that acts on the
received signal as a notch at the zero Doppler frequency, the clutter peak can be reduced.
This can be applied in the time domain by convolution according to
y(n) = c(n) ∗ hMTI(n) (2.13)
where c(n) is the clutter and hMTI(n) is the MTI filter, and thus suppressing the domi-
nating spectral peak.
When receiving ground clutter of large amplitude the detection of targets with low
radial velocity is limited by the minimum detectable velocity (MDV). This factor is
dependent on the spectral estimation and the frequency spread of the mainlobe clutter.
As the radial velocity of the target approaches zero it will fall into the clutter region and
eventually enter a blind zone where it will be left undetected or interpreted as clutter [4].
2.2.5 Parameters and ratios
To make a measurement of how the SNR of the input relates to the SNR of the output,
the improvement factor (IF) is defined in [4] as
IF =P outs
P outn
/P ins
P outn
(2.14)
When analyzing window properties the IF can be used to define the SNR improvement
post Fourier transform. Assuming that any noise from the receiver is white Gaussian
noise, the IF of the specific window under analysis is equivalent to
IFw =|∑∞
k=−∞wA(k)|2∑∞k=−∞ |wR(k)|2
(2.15)
where ωA corresponds to the analyzed window and ωR is a rectangular window to act
as a reference. This ratio can be left as is, in linear format or expressed in dB.
14 Theory
2.2.6 Discrete prolate spheroidal sequences
Before proceeding with multitaper theory a commonly used term in the field, called
discrete prolate spheroidal sequences (DPSS) must be explained. These sequences, also
referred to as Slepian (David Slepian) sequences, corresponds to an approach to solve
the spectral concentration problem, which is a sought time sequence that maximizes the
spectral concentration within a chosen frequency interval. These sequences have some
specific characteristics which are desirable when applying multiple window functions for
a given signal. Each sequence have orthogonal properties to one another which is a useful
quality when trying to focus the energy within a narrow frequency band.
There are some different ways to derive the DPSSs where the methods vary in accuracy
and computation time. In [6], 4 ways are explained, these are,
• Calculating DPSSs from the defining equation
• Calculating DPSSs from numerical integration
• The tridiagonal formulation
• Substitutes for the DPSSs
In this section the DPSSs from the defining equation will be explained where the accuracy
is traded for a slightly longer computation time. For a faster but less accurate derivation,
the tridiagonal formulation could be used.
According to [6], the sequence of length N with the highest concentration of energy
in the frequency span [-W,W] is the eigenvector v0(N,W ) which is derived from the
eigenvalue λ0(N,W ) in the equation
Avk(N,W ) = λk(N,W )vk(N,W ) (2.16)
where vk(N,W ) is an N × 1 vector and W can be chosen between 0 and 1/2.
A is a N ×N matrix with element (t, t′) equivalent to
A =sin(2πW (t′ − t))
π(t′ − t)(2.17)
In order to avoid numerical errors in the calculation, the values corresponding to the
elements where t = t′ are set to 2W since
lim(t−t′)→0
sin(2πW (t′ − t))π(t′ − t)
= 2W (2.18)
When deriving the eigenvalues and eigenvectors from (2.16) for higher orders than
zero, there exists a finite set of K eigenvectors that have corresponding eigenvalues close
to one, which implies large energy concentration within [-W,W]. The number can be
approximated to be less than the Shannon number 2NW∆t, where NW is refered to as
2.2. Spectral analysis and estimation 15
the time half bandwidth. This implies that for sequences where ∆t = 1, the number can
be approximated to
K = 2NW − 1 (2.19)
and is used to set a threshold for the maximum number of sequences to be used for
multitapering. This is because the eigenvalues corresponding to higher order DPSSs
decrease rapidly after this threshold and therefore the energy is no longer concentrated
optimally within [-W,W] [6].
By solving (2.16) with A defined as in (2.17) the DPSSs can be derived. Note that
if vk(N,W ) is an eigenvector of (2.16) then so is cvk(N,W ), where c is any non zero
constant which implies that scaling and polarity convention is necessary. The energy is
normalized over all taper elements such as
N−1∑t=0
vt,k(N,W ) = 1 (2.20)
for even, symmetric tapers (k=0,2,4...) the average of the taper elements are made
positive so that
N−1∑t=0
vt,k(N,W ) > 0 (2.21)
and for odd tapers (k=1,3,5,...) the tapers are made to start with a positive first lobe
such that
N−1∑t=0
(N − 1− 2t)vt,k(N,W ) > 0 (2.22)
By applying these corrections to the derived sequences it is possible to see the resulting
sequences before and after the scaling and polarity correction in Figure 2.4.
16 Theory
0 50 100 150 200 250
Time [Samples]
-0.15
-0.1
-0.05
0
0.05
0.1
Am
plit
ud
e g
ain
Before correction DPSS
Zeroth
First
Second
Third
Fourth
(a)
0 50 100 150 200 250 300
Time [Samples]
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Am
plit
ud
e g
ain
After correction DPSS
Zeroth
First
Second
Third
Fourth
(b)
Figure 2.4: (2.4a) shows the DPSSs before corrections of lobe characteristics, with parameters
N = 256, NW = 6 and W = NW/N . (2.4b) shows the DPSSs after the corrections of lobe
characteristics with equivalent parameters.
This method provides a good discrete approximation but might need some modifica-
tions for large W.
2.2.7 Multitaper
The multitaper method is used in spectral density estimation where several window
functions can be used to get a better approximation of the spectral density function.
The method was introduced by David J. Thomson in [13], which involves the use of
multiple orthogonal tapers (DPSSs). This approach is based on taking the arithmetic
average of the first K DPSSs. In this section a discrete derivation will be provided based
on the derivations in [6] and [13].
Assume that the time series of realizations X1, X2, ..., XN of a stationary process Xt
has zero mean, variance σ2 and spectral density function S(·). Also, assume a sampling
interval ∆t between each realization so that the Nyquist frequency is f(N) ≡ 1/(2∆t)
with sample size N . By averaging the first K DPSSs, where K is limited by the Shannon
number, the estimated spectral density function can be derived according to
S(mt)(f) =1
K
K−1∑k=0
S(mt)k k = 1, 2, ..., K (2.23)
2.2. Spectral analysis and estimation 17
where
Sk(f) ≡ ∆t
∣∣∣∣∣N∑t=1
vt,kXte−i2πft∆t
∣∣∣∣∣2
(2.24)
is the spectrum of Xt with the kth order DPSS denoted vt,k, where vt,k is assumed to be
normalized according to
N∑t=1
v2t,k = 1 (2.25)
Each DPSS taper corresponds to a spectral window such as
Hk(f) ≡ ∆t
∣∣∣∣∣N∑t=1
vt,ke−i2πft∆t
∣∣∣∣∣2
(2.26)
The average of the first K = 4 DPSSs tapers corresponds to a spectral shape shown in
Figure 2.5.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency (units of )
-20
-10
0
10
20
30
40
50
Am
plit
ude [dB
]
Average of the first 4 DPSSs
Figure 2.5: Spectral shape corresponding to the average of the first 4 DPSSs i.e K = 4 and a
time half bandwidth NW=2.5.
18 Theory
2.3 Other
2.3.1 Linear subspace transform & Minimum variance estimation
The linear subspace transform (LST) explained in this section has traditionally been
used to cancel interfering sources on a sensor array. In order to avoid a cumbersome
procedure that involves an inversion of the covariance matrix or autocorrelation matrix
of large sample size, the idea of the LST is to reduce the vector space given by the
array size and perform clutter rejection at the subspace level [4]. This is implemented
by defining the LST matrix T. The variables directly affected by the LST is the steering
vector s, the signal plus clutter vector x and the autocorrelation matrix Q according to
sT = T∗s; xT = T∗x; QT = T∗QT (2.27)
where ∗ denotes the complex conjugate transpose. The linear processor in the subspace
domain is derived by the minimum variance estimation according to
wT = Q−1T sT (2.28)
The efficiency of the linear processor wT can be characterized by the improvement
factor which is calculated according to
IF =w∗T sT s
∗TwT tr(Q)
w∗TQTwT s∗s(2.29)
where tr(Q) denotes the trace of Q.
For an optimum processor wT the improvement factor is
IFopt = s∗Q−1str(Q)
s∗s(2.30)
which represents the theoretical limit of this method [4].
2.3.2 Full factorial simulation theory
The idea of the full factorial simulation is to provide an analysis of how the DPSS
parameters, and combinations of parameters affect the output. This is done by full
factorial experimental design where the theory is explained in [17] and the simulation
method is designed by my co-worker Carl-Henrik Hanquist which is available in [1].
CHAPTER 3
Method
3.1 Antenna model
The antenna model used for clutter generation is a uniform linear array (ULA). The
ULA model provides signal information that can determine the received energy direction
angle in azimuth but not in elevation. The received electromagnetic wave is assumed to
reach the antenna as a plane wave and the object or clutter to be in the far field distance
according to the Fraunhofer distance equation Rf > 2D2/λ. Where D is the length of the
antenna and λ is the wavelength. The space between the antenna elements is assumed
to be d = 0.45λ.
Figure 3.1: Antenna model
This representation of the electromagnetic wave lets the antenna receive input data
that will be phase shifted replicas at each antenna element. The far field equation can
19
20 Method
be defined as a function of the mainlobe angle sin(θml) and the angle of incidence sin(θ).
To get a uniform sampling u = sin(θ) and uml = sin(θml) are used according to
E(u, uml) =1
M
M−1∑m=0
√GTGRexp
(−i2π
λmd(u− uml)
)(3.1)
where GT and GR are the taper gain on transmit and receive respectively, m correspond-
ing to the antenna elements from 0 to M − 1 and d is the distance between the antenna
elements.
3.1.1 Clutter model
The spectral spread of the ground clutter is limited within a frequency span that corre-
sponds to the velocity of the platform, that is the interval [−Vpl/2,Vpl/2]. Within this
interval, two-dimensional colored Gaussian noise is assumed to be present according to
nc =σ2c√2N (0, 1) + i
σ2c√2N (0, 1) (3.2)
due to variations in the ground clutter. In (3.2) σ2c is the noise variance and N (0, 1) is
the normal distribution with zero-mean and unit-variance. Frequencies extending these
bounds are limited by the pulse repetition frequency (PRF) according to
VPRF =λPRF
2(3.3)
in the interval [−VPRF/2,VPRF/2], where white noise from the receiver is assumed to
be present due to variations in the electronics such as thermal noise etc. This noise is
modeled as a uniform noise floor with spectral amplitude nt.
3.1. Antenna model 21
Figure 3.2: Illustration of the ground clutter spectrum with a positive mainlobe angle θml. The
clutter response is limited by the platform velocity in the interval [−Vpl,Vpl]. The spectra is
limited by the PRF in the interval [−VPRF /2,VPRF /2]. The spectral peak at Vplsin(θml) is
determined by the mainlobe angle.
The power of the spectrum generated from (3.1) is derived according to
Pnc = |E(u, uml)|2 (3.4)
The data generated from the clutter spectrum is used to calculate the autocorrelation
matrix. It is estimated by using the first N − 1 values of the power spectrum of the
clutter according to
rx(n) =N−1∑k=0
Pnc(n)exp(−2πkn/N) (3.5)
and the autocorrelation matrix can be derived according to (2.11) as
Q =
rx(0) r∗x(1) r∗x(2) · · · r∗x(p)
rx(1) rx(0) r∗x(1) · · · r∗x(p− 1)...
......
...
rx(p) r∗x(p− 1) r∗x(p− 2) · · · rx(0)
(3.6)
which will be used for clutter suppression.
22 Method
3.2 Minimum variance estimation & Multitaper
The multitaper method is used in accordance with the minimum variance estimation
where it is assumed that the radar operates in the S-band region with a medium PRF.
Target signals are modeled as
s(n) = Aei2πfn (3.7)
where A is the amplitude of the signal and f is the desired Doppler frequency.
The received signal consists of the target signal, the ground clutter and thermal noise
according to
x(n) = s(n) + c(n) + nt(n) (3.8)
where c(n) is the clutter and nt(n) is the thermal noise. In order to avoid calculations
that includes the inversion of a large sample autocorrelation matrix the LST method is
utilized as explained in Section 2.3.1. Here we define T to be
T =
s(1)v1,1 s(1)v1,2 · · · s(1)v1,K
s(2)v2,1 s(2)v2,2 · · · s(2)v2,K...
.... . .
...
s(N)vN,1 s(N)vN,2 · · · s(N)vN,K
(3.9)
where s(n) is the steering vector and vn,k is the kth order DPSS at time sample n.
By defining T the vector space is reduced according to (2.27) such as
sT = T∗s; xT = T∗x; QT = T∗QT (3.10)
and the linear processor in the subspace domain is derived from the minimum variance
estimation
wT = Q−1T sT (3.11)
By iteration, the steering vector is updated for every Doppler channel in order to
produce a result for leakage and to give a spectral estimation output according to
y = x∗TwT (3.12)
For every Doppler channel the improvement factor is calculated according to (2.29)
and as a reference with theoretically optimum value for wT the optimum IF is calculated
according to (2.30).
3.3 Traditional method
To evaluate the multitaper method it is compared to a traditional method defined in
this section. The traditional method presented consists of a single tapper Taylor window
wtrad(n) with a common MTI filter defined as
3.4. Simulation environment for full factorial simulation 23
hMTI = [1,−2, 1] (3.13)
To implement the method a replacement of the LST matrix T in (3.9) is necessary
since only one taper is used. For this case T is modified to be a single column matrix
defined by the the Hadamard product of the traditional taper and the steering vector
according to
wtrad(n) = wtrad(n) ◦ s(n) (3.14)
By applying the MTI filter by convolution the LST matrix is defined according to
Ttrad = wtrad ∗ hMTI (3.15)
and thus only using a single tapper but including the MTI filter.
Apart from this substitution of T the derivation of the improvement factor follows
the equivalent procedure as in Section 3.2. To determine how the multitaper approach
compares to the traditional method the MDV and the IF in the thermal noise band is
compared. The derivation of these values are explained in Section 3.4.2.
3.4 Simulation environment for full factorial simulationThe characteristics of the DPSSs are derived from parameters including the number of
time samples N , the number of DPSSs used L and the bandwidth W in which one
wishes to concentrate the energy. Also the cross term of N and W called the time
half bandwidth, denoted NW is investigated. In order to analyze how these parameters
depend on each other and how they influence the MDV, IF and the spectral output, a
full factorial simulation is performed. The advantage of this approach is that it avoids
varying one variable at a time which can cause performance results at a local minimum.
The method of this simulation have been divided into two parts. In [1] the method for
the full factorial design matrix, model matrix and visualization design of the results are
presented and also a discussion regarding parameter choices. In this thesis the simulation
environment of the signal processing is explained and the derivation of MDV and IFnoiseis defined.
3.4.1 Overview
The simulation environment is made in MATLAB and the overview of the system is
presented in Figure 3.3. The blue parts are treated in this thesis and the pink parts,
including the full factorial method, are explained in [1]. The simulation is based on the
methods described in previous sections where the parameter values of the DPSSs N , L,
and W are updated for every iteration until the number of test parameters defined by
the experimental space of the design matrix reaches its endpoint.
24 Method
Design matrix
Organize input values
Clutter generation
Autocorrelation
matrix generation
Generation of DPSSs
Minimum vari-
ance estimation
IF calculation
MDV and IFnoisecalculation
Save results
Model matrix
Visualization
Update parameters
Figure 3.3: Flowchart of full factorial simulation, the blue areas corresponds to the signal
processing environment which is treated in this thesis. The pink parts are explained in [1].
3.4.2 Signal processing simulation environment
In this section a detailed explanation of the simulation environment is provided.
Organize input values The design matrix specifies the number of experiments needed
in order to satisfy the chosen number of parameter values to be investigated. Three dif-
ferent experimental spaces are defined in order to compare the influence of the stochastic
processes and eliminate any random fluctuations that might cause misinterpretations of
the results. The simulations are for 33, 93 and 173 experimental spaces. The parameters
are organized in order to be adaptable for variations in the experimental space and to be
3.4. Simulation environment for full factorial simulation 25
updated according to the design matrix. The values are chosen according to [1] and can
be seen in Table 3.1.
Ex.Space/Parameter 33 93 173
L {3,6,9} {1,3,5,...,17} {1,2,...,17}N {200,350,500} {200,300,...,1000} {200,...,1000}W {0.025,0.065,0.1} {0.02,0.04,...,0.1} {0.02,...,0.1}
Table 3.1: Parameter values for the different experiments.
A limit for the number of DPSSs to be used in the multitaper approach are set to
be less than 20 in order to restrict the computational time. In this section the defined
parameter values of L, N and W are chosen within a predefined interval [1] and then
sampled with equal space within this interval for desired precision.
In order to avoid combinations of parameters which forces the number of DPSS orders
to exceed the Shannon number rule in (2.19), a restriction is set to ignore these iterations.
This is clearly visualized in the image representation plots in Section 4.3.1 and discussed
in [1].
Clutter & Autocorrelation matrix generation The clutter generation is indepen-
dent of the DPSS parameters apart from the integration time N . In order to generate
clutter with matching sampling time and length, the spectrum is therefore regenerated for
each iteration. The same reasoning also applies to the generation of the autocorrelation
matrix.
Generation of DPSS The generation of DPSSs are updated for every iteration since
all parameter values change.
Minimum variance estimation To reduce the vector space and thus decrease com-
putational time the LST is applied in accordance with the minimum variance estimation
and multitaper approach.
IF calculation The IF is calculated for each iteration as a vector, in which each value
corresponds to a discrete frequency. In order to have a resulting measure of how the
parameters affect the result, the optimum improvement factor IFopt is calculated, to act
as a reference according to (2.30).
MDV & IFnoise calculation The resulting improvement factor is divided into two
categories. The first is the MDV, which is calculated based on the width of the IF peak
26 Method
at the -20 dB amplitude. The second is the improvement factor in the noise band IFnoise,
which is calculated based on the average of the values in this region, since it may have
some slight deviations.
Save results As the iteration is completed the parameters are updated according to
the values defined in the design matrix, the saved results are the corresponding values of
MDV and IFnoise for every iteration.
Post processing The post processing that includes the model matrix and visualiza-
tions are explained in [1] where the results are organized.
CHAPTER 4
Results
4.1 Clutter generation
The parameters that must be set for the clutter generation are the number of time
samples N , the size of the Nt × Nt autocorrelation matrix Q, platform velocity vpl,
PRF , the wavelength of the transmitted electromagnetic wave λ, mainlobe direction
uml, CNR, taper on receive winR, phase noise variance on receive pn, and thermal noise
floor amplitude nt.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Am
pitu
de
[d
B]
Clutter spectrum
(a)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Am
pitu
de
[d
B]
Clutter spectrum
(b)
Figure 4.1: Spectral response of the clutter model with 60 dB CNR and noise floor amplitude
nt = 0.02. (a) shows a response for a mainlobe angle of 0 rad and (b) show the response for a
mainlobe angle corresponding to a spectral peak at 0.1π rad.
27
28 Results
N Nt vpl PRF λ uml CNR winR pn256 256 100 m/s 10 kHz 0.1 m 0/0.1π rad 60 dB Taylor 3◦ degrees
Table 4.1: Parameters used for clutter generation.
In Figure 4.1 the spectrum is derived assuming the values in Table 4.1. The influence of
the mainlobe angle is clearly seen as the spectral peak is shifted with the corresponding
angle.
The autocorrelation matrix derived from the first N − 1 values of the clutter spectrum
corresponding to the derivation with uml = 0 rad can be seen in Figure 4.2.
Amplitude
50 100 150 200 250
50
100
150
200
250
0.75
0.8
0.85
0.9
0.95
1
(a) Amplitude
Phase
50 100 150 200 250
50
100
150
200
250 -0.15
-0.1
-0.05
0
0.05
0.1
0.15
(b) Phase
Figure 4.2: Autocorrelation matrix characteristics where (a) shows the amplitude and (b) shows
the phase in radians for the clutter spectrum with spectral peak at 0 rad.
4.2 Simulated results
By comparing the performance results of IFnoise and the MDV with CNRs of 70,60 and
50 dB a general sense of how the performance responds to increased CNR is achieved.
The main issue as the CNR increases is that the clutter region dominates the spectra to
the point where the definition of the MDV becomes unreliable and might misinterpret
parts of the clutter variations as the main peak width, if it reaches the -20dB level. This
issue is avoided for simulations not exceeding a CNR of 70 dB. In Figures 4.3-4.5 the
behaviour of the results are presented. The figures show the theoretically optimal results
4.2. Simulated results 29
IFopt vs the results achieved when using specified DPSS parameters IFnorm. Both IFoptand IFnorm are normalized in order to provide an accurate comparison.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
(a) Normalized improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
(b) Normalized improvement factor in [dB]
Figure 4.3: Improvement factor of the multitaper method with N = 256, NW = 10, L = 19 and
CNR = 50dB. (a) shows the normalized improvement factor and (b) shows the improvement
factor in dB.
30 Results
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
(a) Normalized improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
(b) Normalized improvement factor in [dB]
Figure 4.4: Improvement factor of the multitaper method with N = 256, NW = 10, L = 19 and
CNR = 60dB. (a) shows the normalized improvement factor and (b) shows the improvement
factor in dB.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
(a) Normalized improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
(b) Normalized improvement factor in [dB]
Figure 4.5: Improvement factor of the multitaper method with N = 256, NW = 10, L = 19 and
CNR = 70dB. (a) shows the normalized improvement factor and (b) shows the improvement
factor in dB.
4.2. Simulated results 31
The performance results for variations in the CNR are presented in Table 4.2. IFnoiseshows a relatively small or no deviation based on different CNR level changes. The
MDV however shows a broadened behaviour with increased CRN. This behaviour could
be explained intuitively. With an increased CNR the magnitude of the spectral peak is
increased but also its frequency width, which implies a decreased detectability of targets
approaching the zero Doppler frequency.
CNR 50 dB 60 dB 70 dB
MDV 0.0118 0.0275 0.0627
IFnoise 0.9444 0.9441 0.9441
Table 4.2: Performance results for different CNR.
The resulting spectral estimations derived from (3.12) are shown in Figure 4.6, where
three targets are included in the received clutter signal. The targets are located in the
normalized frequency spectra at positions corresponding to, a target in the clutter span,
close to the main clutter peak and in the noise band. That is at −0.1, 0.03 and 0.4
respectively. With a CNR of 60 dB all three targets can be recognized whereas with a
CNR of 70 dB the target closest to the main clutter peak falls in to the clutter region
and is no longer detectable.
32 Results
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-70
-60
-50
-40
-30
-20
-10
0
Am
plit
ude [
dB
]
Multitaper
(a) 60 dB CNR
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-70
-60
-50
-40
-30
-20
-10
0
Am
plit
ude [
dB
]
Multitaper
(b) 70 dB CNR
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Am
pitud
e [d
B]
Clutter spectrum
(c) 60 dB CNR
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Am
pitud
e [d
B]
Clutter spectrum
(d) 70 dB CNR
Figure 4.6: Spectral estimation of clutter and three targets with N = 256, L=19 and NW=10.
(a) shows a spectral estimation with a 60 dB CNR and (b) shows the spectral estimation with
70 dB CNR. The spectra of the underlying ground clutter is seen in (c) and (d).
4.3 Parameter analysis by full factorial simulation
The performance of the multitaper method in the simulation environment is based on two
factors, the MDV and IFnoise. The MDV is defined to be the main peak frequency span
at -20 dB and IFnoise is the improvement factor in the frequency span with only thermal
noise present. Due to a small variation of the IF in this frequency span an average of
the amplitude in this region is calculated and defined as noise level improvement factor
denoted IFnoise. The values are compared to the optimal improvement factor defined in
(2.30). The parameter values investigated in this section are
4.3. Parameter analysis by full factorial simulation 33
• N (Number of time samples)
• W (Defines the bandwidth for energy concentration)
• L (Number of DPSSs used for multitaper)
Since the full factorial simulation is a measure of how the parameter values affect the
results on average, this section is dedicated to show the results but also to interpret and
analyze the simulation results.
4.3.1 Full factorial simulation
To be able to interpret the figures that are presented in this section one must keep in
mind that the results are an on average representation of the impact each parameter has
for the performance of the system. The results presented in this section represents a
93 factor simulation. The resolution of the simulation is smaller for the 93 case but the
same conclusions can be drawn from the result of the larger 173 simulation. Thus, it is
excluded in this section but added in Appendix A. The 33 simulations is also excluded
in this thesis but can be found in [1] for interested readers.
The experimental space is shown in Figure 4.7, where the right corner has a steep
descent due to the limitation of DPSSs orders defined by the Shannon number according
to (2.19).
0
0.1
0.08
200
W
0.06 400
N
600
10
0.04
L
800
Experimental space
0.02 1000
20
Figure 4.7: Experimental space for the 93 factor simulation.
34 Results
The results of the simulations are presented in Pareto plots which show the average
impact the increased parameters have on the performance that is the MDV and IFnoise.
This visualization also provides a measure of how the combination of increased parameter
values affect the performance which can be seen in Figure 4.8, where the full factorial
parameter A corresponds to W , B to N and C to L. The color code of the bars represent
positive impact (blue) and negative impact (red) on the performance with increased
parameter values. For methodology and theory see [1].
0 0.5 1 1.5 2 2.5 3
Effect
AA
AB
ABC
BC
A
AC
C
BB
B
CC
Te
rm
MDV
(a) MDV
0 1 2 3 4 5
Effect 107
BB
ABC
CC
AC
AB
BC
AA
A
B
C
Te
rm
IFnoise
(b) IFnoise
Figure 4.8: Pareto plots of the 93 factor model where (a) shows the average parameter impact
on MDV performance. (b) shows the average parameter impact on improvement factor in the
noise band.
Figure 4.8 shows four different types of of values. That is the factors A,B and C. The
interaction terms AB, AC and BC. The quadratic terms AA, BB, CC and the triple
interaction term ABC.
The parameter impact shows that out of the three variables under inspection N has
the greatest positive impact on the MDV when increased, followed by L. The bandwidth
W has on average a negative impact on the MDV.
The quadratic term AA (blue) represents a decrease of the negative impact of W with
increased parameter value. BB (red) represents a decrease of the positive impact with
increased parameter value of N . CC (red) represents a decreased positive impact by
increasing L.
The interaction term AC (blue) represents a positive impact of the MDV if both W and
4.3. Parameter analysis by full factorial simulation 35
N are increased. AB (blue) represents a positive impact if both W and N are increased.
BC (red) represents a negative impact by increasing both N and L.
The corresponding results for the improvement factor in the noise band are presented
in Figure 4.8b. With the purpose of improving IFnoise the parameter that has the largest
positive impact on the result when increased is the number of DPSSs L, followed by N .
An increased bandwidth W has a negative impact on the performance of IFnoise.
The quadratic term AA (blue) represents a decreasing negative impact with an in-
creased W . BB (red) represents a decreasing positive effect with increased N . CC (red)
represents a decreasing positive effect with increased L.
The interaction term AC (red) represents a negative impact on IFnoise if both W and
L are increased. BC (blue) represents a positive impact when increasing both N and L.
AB (red) represents a negative impact if both W and N are increased.
As a summary of the results; both MDV and IFnoise benefits from a large number of
DPSS orders L and a large number of time samples N . A small bandwidth W is more
important for optimization of IFnoise than for the MDV but does have a negative impact
with increased value for both. Overall, the variation of DPSS parameters affect IFnoisemore than the MDV.
In Figure 4.9 the results of IFnoise are presented in three sub figures where the header
indicates a constant value of a parameter, the x-axis is the variation of the second and
the y-axis is the variation of the third. The dark blue areas indicate ignored simulations
as previously stated. Figure 4.9a shows that, given a parameter value L, the best result
for all cases studied is to use the highest value of N and the lowest value of W . Figure
4.9b shows that for a given N , the best result is achieved from using a large L and a
narrow W . Figure 4.9c shows that for a given W , the best result is derived when using
the largest N and the largest L. It is however clear that the lowest parameter value of
W provides the best result, as seen by the colored scale.
36 Results
(a) L (b) N
(c) W
Figure 4.9: Image representation of how parameter values affect the result of IFnoise.
The corresponding case study for the MVD is shown in Figure 4.10. Since the optimal
result for the MDV is a low value, the blue values indicate improved results. In this
case the results are not as easily interpreted as for the IFnoise case. Figure 4.10a shows
that, for a given L, a higher value of N provides better results. For the lowest value of
N = 200 the bandwidth W = 0.07 provides the worst result. Note that there is an outlier
for L = 1, which indicates that the largest N and W should be avoided. Figure 4.10b
shows that for a given N the best result is derived from using a high L and a narrow W .
Figure 4.10c shows that, for a given W the best result is derived when using a large N
and a large L.
4.3. Parameter analysis by full factorial simulation 37
(a) L (b) N
(c) W
Figure 4.10: Image representation of how the parameter values affect the result of MDV.
Figure 4.11 provides a representation of where IFnoise have constant value, represented
in a contour figure. Figure 4.11a shows a smooth representation of the previously stated
effect that for a given L, the performance benefits from large N and a small W . Figure
4.11b shows that for a given N , IFnoise benefits from a large L and a small W . Figure
4.11c shows that for a given W IFnoise benefits from a large L and a large N . Overall,
the behavior is different in a sense where some parameter show almost a linear increase
with increased parameter value, while some show more of an exponential or quadratic
behaviour when responding to an increased parameter value.
38 Results
(a) L (b) N
(c) W
Figure 4.11: The lines of the contour plots shows the constant value of IFnoise and also its
behaviour due to parameter changes.
The corresponding contour figures for the performance of the MDV is represented in
Figure 4.12. Figure 4.12a shows that for a given L the worst result is when N = 200
and W = 0.07. The performance of the MDV increases with an increasing N . W shows
fluctuating results but appears not to have great impact on the performance. Figure
4.12b shows that for a given N , the best performance of the MDV is derived using a
narrow W and a large L. It can be seen from the figure that increasing L has a larger
impact for a wide W than for a narrow W . Figure 4.12c shows that, for a given W ,
the best result is given by using a large L and a large N . The result of this figure shows
more fluctuations than the previous.
4.3. Parameter analysis by full factorial simulation 39
(a) L (b) N
(c) W
Figure 4.12: The lines of the contour plots shows the constant value of the MDV and also its
behaviour due to parameter changes.
If the multitaper method is to be implemented in a real radar system, the analysis
presented in this section could be used as a guide in order to find the optimal values
for the given system. The limiting factors for this implementation are the number of
time samples N and the computation time. By using a large number of tapers L the
computation time is increased which will be treated in Section 4.4.1
40 Results
4.4 Comparison with traditional methodThe comparison of the multitaper method with the traditional method is based on four
different multitaper iterations where four different parameter values of the DPSSs are
used. A limitation of the number of sequences is set to 20 in order to not overextend
the computation time. The values used corresponds to a small L, a medium L, a large
L and a value of L that produces a result comparable to the traditional method. The
bandwidth is selected to satisfy the Shannon number with N = 256. For all simulations
a CNR of 60 dB has been used. The results are presented in Table 4.3.
x MDV IFnoise Improvement IFnoise Improvement MDV
L = 19, NW = 10 0.0275 0.9441 0.8903 dB 0
L = 11, NW = 6 0.0275 0.9024 0.6938 dB 0
L = 5, NW = 3 0.0275 0.7850 0.0890 dB 0
L = 3, NW = 2 0.0353 0.6465 -0.7540 dB -0.0078
Traditional 0.0275 0.7692 x x
Table 4.3: Result of different multitapers and comparison with traditional method.
The impact of the number of orders used according to Table 4.3 show a small or no
influence on the MDV. The only case with a slight descent is when using 3 orders. But
when comparing the IFnoise values a strong correlation of a large number of orders and
a higher improvement factor is shown. The simulation also show that in order to have a
reasonably close comparison with the traditional method 5 numbers of orders would be
recommended. The greatest improvement of IFnoise is when using 19 orders which results
in an improvement of 0.8903 dB. In the clutter region between -0.2 and 0.2 normalized
frequency, the graphical illustrations shows that as the improvement factor in the noise
band is reduced, so is the improvement factor in the clutter region.
4.4. Comparison with traditional method 41
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
IFtrad
(a) Improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
IFtrad
(b) Improvement factor [dB]
Figure 4.13: Improvement factor with a large number of orders L=19 and NW=10.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
IFtrad
(a) Improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
IFtrad
(b) Improvement factor [dB]
Figure 4.14: Improvement factor with a medium number of orders L=11 and NW=6.
42 Results
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
IFtrad
(a) Improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
IFtrad
(b) Improvement factor [dB]
Figure 4.15: Improvement factor with a small number of orders L=3 and NW=2.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IF
Improvement Factor
IFnorm
IFopt
IFtrad
(a) Improvement factor
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
IF [
dB
]
Improvement factor [dB]
IFnorm
IFopt
IFtrad
(b) Improvement factor [dB]
Figure 4.16: Improvement factor with number of orders L=5 and NW=3, for comparison with
traditional method.
4.4. Comparison with traditional method 43
4.4.1 Computation time
The computation time is calculated in Matlab. The time is based on an average of
1000 simulations made for each number of tapers used with 256 time samples N and a
bandwidth W of 0.04.
In Figure 4.17, the red line represents the average computation time for the multitaper
approach, using 1 to 19 tapers. The blue line represents the average computation time
of the traditional method, using a Taylor window with a [1,-2,1] MTI filter. The dotted
line is the standard deviation of the traditional method and the gray bars represents
the standard deviation of the multitaper approach. The green circle indicates where the
performance of IFnoise for the multitaper method surpassed the traditional method. So
by using 256 time samples and a bandwidth of 0.04, at least 11 tapers had to be used in
order to increase the performance. Overall, increasing the number of tapers also increases
the computation time as can be seen in Figure 4.17.
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 4.17: Computation time of the multitaper approach compared to the traditional method.
CHAPTER 5
Discussion
5.1 Discussion
The antenna model used in this thesis is a ULA which defines the position of the antenna
elements. This representation of the antenna is a common simplification when a specific
part of the radar is studied, in this case the Doppler radar. The single ULA model
places constraints on the received energy direction to be distinguishable in azimuth but
not in elevation. This may affect the results since a real radar system typically has a
set of ULAs placed in parallel to each other which allows 3 dimensional target position
identification. Since the scope of this thesis is based on evaluation of clutter suppression
the direction of the clutter signal is assumed not to be a factor that play a significant part
in the derivation of the results. The single ULA model makes it easier to evaluate the
sought performance measure and the focus can be directed towards the clutter magnitude.
For the purpose of this thesis the model offers simulation performance that is easy to
interpret without a significant risk to produce a biased result if the method is chosen to
be implemented in a real radar system.
The clutter model is adaptable for different types of simulated noise variations, such
as phase noise on receive, ground clutter variations, and thermal noise levels that may
vary in the receiver.
The full factorial simulation provides an on average estimate of how the DPSS param-
eters impact the results and can be used as a guide for the parameter choices. Since
the same conclusion could be drawn from the 93 and 173 factor simulations a larger sim-
ulation seemed unnecessary. The reason that the 33 experiment is excluded is simply
because it provided less information about the parameter relations.
The results show that the multitaper method partly provides an improvement compared
to the traditional method used. The traditional method have not been optimized to a
larger extent. The choice of the Taylor window in accordance with the classical [1,-2,1]
MTI is a decision based on literature review and what seems to be common in the field.
But for future references a different type of window with different lobe characteristics
45
46 Discussion
could be analyzed as well as a different type of MTI filter with varied notch characteristics.
CHAPTER 6
Conclusion
6.1 ConclusionThe multitaper approach provides an improved improvement factor in the noise band
of at least 0.8903 dB compared to the traditional method. There is room for further
improvement if the number of time samples exceeds 256 and the number of used tapers
exceeds 19, which would trade increased performance for longer computation time. There
is a correlation of high improvement factor in the noise band with a large number of
DPSSs used if the Shannon number rule is used for selection of bandwidth W for energy
concentration and the number of time samples N .
The minimum detectable velocity does not seem to have a drastic improvement com-
pared to the traditional method.
6.2 Future workExtensions for this work could be to apply the method on real time radar data, where the
autocorrelation matrix is constructed from real time signal inputs that includes ground
clutter. Another extension could be to use both temporal and spatial processing in a
space-time adaptive processing environment.
47
APPENDIX A
173 factor simulation
0 0.5 1 1.5 2 2.5
Effect
AA
AB
BC
ABC
A
AC
C
BB
CC
B
Term
MDV
(a) MDV
0 1 2 3 4 5 6
Effect 107
BB
ABC
CC
AB
AC
BC
AA
A
B
C
Term
IFnoise
(b) IFnoise
Figure A.1: Pareto plots for the 173 simulation.
49
50 Appendix
Figure A.2: Experimental space for the 173 simulation.
51
(a) L (b) N
(c) W
Figure A.3: IF image figures of the 173 simulation.
52 Appendix
(a) L (b) N
(c) W
Figure A.4: MDV image figures of the 173 simulation.
53
(a) L (b) N
(c) W
Figure A.5: IF contour figures of the 173 simulation.
54 Appendix
(a) L (b) N
(c) W
Figure A.6: MDV contour figures of the 173 simulation.
55
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=W
B=N
C=L
Term
IFnoise
Figure A.7: normalized IF Pareto plot of the 173 simulation.
56 Appendix
(a) L (b) N
(c) W
Figure A.8: Normalized IF image figures of the 173 simulation.
57
(a) L (b) N
(c) W
Figure A.9: Normalized IF contour figures of the 173 simulation.
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