Airborne Radar Ground Clutter Suppression Using Multitaper ...1249906/FULLTEXT01.pdf · Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with

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)


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


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)


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


-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


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


-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


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


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

dB

]


IFnorm

IFopt




factor in dB.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


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


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

dB

]


IFnorm

IFopt




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


-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


-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


-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


-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


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.


(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.


(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


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


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

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


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


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

dB

]


IFnorm

IFopt

IFtrad


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


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


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

dB

]


IFnorm

IFopt

IFtrad


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


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


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

IF [

dB

]


IFnorm

IFopt

IFtrad


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.

REFERENCES

[1] C. Hanquist, Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum

Estimation & Choosing DPSS parameters. 2018.

[2] P. Devine, Radar level measurement: the user’s guide. Vega Controls, 2000.

[3] G. W. Stimson, “Introduction to airborne radar hughes aircraft company, el,” Se-

gundo, CA, 1983.

[4] R. Klemm, Principles of space-time adaptive processing. No. 12, IET, 2002.

[5] M. H. Hayes, Statistical digital signal processing and modeling. John Wiley & Sons,

2009.

[6] D. B. Percival and A. T. Walden, Spectral analysis for physical applications. Cam-

bridge university press, 1993.

[7] C. Temes, “Sidelobe suppression in a range-channel pulse-compression radar,” IRE

Transactions on Military Electronics, vol. 1051, no. 2, pp. 162–169, 1962.

[8] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, fourier analysis

and uncertainty - I,” Bell Labs Technical Journal, vol. 40, no. 1, pp. 43–63, 1961.

[9] D. Slepian, “Prolate spheroidal wave functions, fourier analysis and uncertainty - V:

The discrete case,” Bell Labs Technical Journal, vol. 57, no. 5, pp. 1371–1430, 1978.

[10] H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, fourier analysis

and uncertainty-II,” Bell Labs Technical Journal, vol. 40, no. 1, pp. 65–84, 1961.

[11] H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, fourier analysis

and uncertainty-III: The dimension of the space of essentially time-and band-limited

signals,” Bell Labs Technical Journal, vol. 41, no. 4, pp. 1295–1336, 1962.

[12] D. Slepian, “Prolate spheroidal wave functions, fourier analysis and uncertainty-IV:

extensions to many dimensions; generalized prolate spheroidal functions,” Bell Labs

Technical Journal, vol. 43, no. 6, pp. 3009–3057, 1964.

59

60

[13] D. J. Thomson, “Spectrum estimation and harmonic analysis,” Proceedings of the

IEEE, vol. 70, no. 9, pp. 1055–1096, 1982.

[14] M. Skolnik, “Radar handbook second edition,” 1990.

[15] J. Bruder, J. Carlo, J. Gurney, and J. Gorman, “Ieee standard for letter designations

for radar-frequency bands,” IEEE Aerospace & Electronic Systems Society.

[16] R. Ghavamirad, H. Babashah, and M. A. Sebt, “Nonlinear fm waveform design to

reduction of sidelobe level in autocorrelation function,” in Electrical Engineering

(ICEE), 2017 Iranian Conference on, pp. 1973–1977, IEEE, 2017.

[17] J. Carlson, Measurement Systems Engineering. JEC Engineering and Media Pro-

duction AB, 2018. Ch. 8.

Documents

Airborne Radar Ground Clutter Suppression Using Multitaper ...1249906/FULLTEXT01.pdf · Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with