Anti-jamming MTI Radar using Variable Pulse-Codes*

by

Kenny Lin

B.S. Electrical EngineeringU.S. Naval Academy, 2000

Submitted to the Department of Electrical Engineering and Computer Science inpartial fulfillment of the requirements for the degree of

Master of Science in Electrical Engineering and Computer Science

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

May 2002

@ Kenny Lin, MMII. All rights reserved.

BARKERMASSACHS iij

OF TECHNOLOGY

JUL 3 12002

LIBRARIES

The author hereby grants to MIT permission to reproduce and distribute publiclypaper and electronic copies of this thesis document in whole or in part.

A u th o r ......................................................... I ..... .... .........Department of Electrical Engineering an4 Computer Science

May 24, 2002

Certified by...............(raf C. Robey

Associate Group Leader, Group 101incoln Laboratory

.<Thesis Supe isor

Certified by ................ L~-~c~ ~Davi H. Staelin

Prfsf 'Electrical EngineeringalThas-Stervisor

Accepted by.............Arthur C. Smith

Chairman, Department Committee on Graduate Students

*This work was sponsored by the Department of the Navy and the Department of the Air Force under

Contract F19682-00-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the

author and not necessarily endorsed by the United States Navy or United States Air Force.

Anti-jamming MTI Radar using Variable Pulse-Codes

by

Kenny Lin

Submitted to the Department of Electrical Engineering and Computer Scienceon May 24, 2002, in partial fulfillment of the

requirements for the degree ofMaster of Science in Electrical Engineering and Computer Science

Abstract

A pulsed Doppler radar is vulnerable to advanced repeat-back jamming techniques. Rapidlyadvancing technology producing inexpensive, high performance commercial off-the-shelf(COTS) components enable the construction of an electronic countermeasure (ECM) sys-tem capable of exploiting this vulnerability. This thesis addresses this threat by examiningthe nature of this vulnerability and developing a modification to the pulsed Doppler/MTIradar system.

Pulsed Doppler radar systems use pulse compression waveforms such as pseudonoise(PN) coded binary phase-modulated sequences. Repeat-back jamming listens, stores, andrepeats back the radar's transmitted signal to block out all other return signals. If a differentPN-code is used for each pulse, the radar receiver will be minimally affected by the jamming.However, a varying PN code creates range sidelobe variation that degrades the integratedsignal-to-clutter ratio by a factor of ,1 where N is the code length. This severely limitsthe ability to perform Doppler and Moving-Target Indication (MTI) processing for cluttersuppression on the radar return.

To recover this performance loss several receiver filtering and digital signal processingtechniques are tested. PN code selection for optimum filter performance is explored resultingin a 7-dB signal-to-clutter performance recovery for a 32-bit code. Digital pulse compression,matched filtering, and adaptive digital equalization filtering methods are applied to theradar return to equalize differences created by variable PN codes. Different equalizationalgorithms with various subsets of PN-codes are presented and simulated with data setsmodelled after existing radar systems. Successful correction reduces clutter, minimizes theperformance degradation to MTI due to variable pulse-codes, and resists some types ofDRFM jamming.

Thesis Supervisor: Frank C. RobeyTitle: Associate Group Leader, Group 101MIT Lincoln Laboratory

Thesis Supervisor: David H. StaelinTitle: Professor

2

Acknowledgments

First and foremost, I would like to thank my advisor, Dr. Michael A. Koerber, for his time

spent teaching and advising me at MIT Lincoln Laboratory. It is through his mentorship,

tireless tutelage and fine example that I have learned so much about signal processing and

engineering research. I would like to thank my thesis supervisors Dr. David H. Staelin and

Dr. Frank C. Robey for their guidance and contribution to my education. I would like

to thank my friends and staff members at Lincoln Laboratory for their assistance and for

making my time here much more enjoyable. Finally, I would like to thank my girlfriend,

Yingli Zhu, and my family, Chiun-wen, Su-ching, and Frank for their love and support in

all my endeavors.

3

Contents

1 Introduction

1.1 Pulsed Doppler Radar ....

1.2 MTI Radar ..........

1.3 Electronic Countermeasures

1.4 Variable Pulse-Code Radar

1.5 Performance Metrics . . . . .

1.6 Contributions of this Thesis

2 Radar System Simulation

2.1 Radar System Model . . . . .

2.2 Target Model . . . . . . . . .

2.3 Environmental Model . . . .

2.3.1 Noise Model . . . . . .

2.3.2 Clutter Model . . . .

2.3.3 Jamming Model . . .

2.4 Return Processing Model

2.4.1 Pulse Compression

2.4.2 Matched Filtering

2.4.3 Weighting . . . . . . .

2.4.4 Doppler Processing . .

2.4.5 MTI Implementation.

2.4.6 Adaptive Digital Equalization Filtering

2.5 Simulation Output . . . . . . . . . . . . . . .

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3 Variable Pulse Code Radar 32

3.1 Doppler Degradation ............................... 32

3.1.1 Target Effects ....... ... ........ . ... .. .. .... . 34

3.1.2 Clutter Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Pulse Compression Filtering 43

4.1 Matched Filter Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Sidelobe Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Pulse Compression Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Pulse Compression Filters . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Doppler and MTI Performance . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 SN R Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Adaptive Digital Equalization Filtering 56

5.1 Filter Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1.1 Time Domain Implementation . . . . . . . . . . . . . . . . . . . . . 57

5.1.2 Frequency Domain Implementation . . . . . . . . . . . . . . . . . . . 57

5.2 Equalization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Doppler and MTI Performance . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.1 SN R Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Anti-Jamming Performance 62

7 Conclusion 64

5

List of Tables

5.1 Comparison of SNR for different filtering methods. . . . . . . . . . . . . . . 61

6

List of Figures

2-1 Radar Simulation Application GUI . . . . . . . . . . . . . . . .

2-2 Simulation Flow and System Block Diagram . . . . . . . . . . .

2-3 Radar System Variables . . . . . . . . . . . . . . . . . . . . . .

2-4 Target Component Magnitude and Phase . . . . . . . . . . . .

2-5 Target Variables . . . . . . . . . . . . . . . . . . . . . . . . . .

2-6 Environment Variables . . . . . . . . . . . . . . . . . . . . . . .

2-7 Complex Gaussian "White" Noise Component . . . . . . . . . .

2-8 Clutter Component . . . . . . . . . . . . . . . . . . . . . . . . .

2-9 Repeat-Back Jammer Block Diagram . . . . . . . . . . . . . . .

2-10 False Targets Created by Repeat-Back Jamming . . . . . . . .

2-11 MTI Filtered Return with Repeat-Back Jamming . . . . . . . .

2-12 Return Processing Variables . . . . . . . . . . . . . . . . . . . .

2-13 Matched Filtering vs. Pulse Compression for a 32-Bit PN Code

2-14 Matched Filter Output with Taylor Weighting . . . . . . . . . .

2-15 Matched Filter Output Weighting Comparison . . . . . . . . .

2-16 Target and Clutter Comparison for Doppler Processing . . . . .

2-17 Doppler Processed Radar Return . . . . . . . . . . . . . . . . .

2-18 MTI Filter Impulse and Frequency Response . . . . . . . . . .

2-19 MTI Processed Radar Return . . . . . . . . . . . . . . . . . . .

2-20 Plotting Options . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-1 Constant and Variable Pulse-Code Comparison . . . . . . . . . . . . . . .

3-2 Matched Filter Output and Doppler Spectrum for a Constant Pulse-Code

3-3 Matched Filter Output and Doppler Spectrum for Variable Pulse-Codes

3-4 Variable Pulse-Code Radar Doppler Return without Clutter . . . . . . . .

7

32

33

. 33

37

3-5 Variable Pulse-Code Radar Doppler Return with Clutter . . . . . . . . . . . 37

3-6 Constant Pulse-Code Radar Doppler Return Profile . . . . . . . . . . . . . 42

3-7 Variable Pulse-Code Radar Doppler Return Profile . . . . . . . . . . . . . . 42

4-1 Variable Pulse-Code Radar Doppler Return using Random Codes . . . . . . 45

4-2 Variable Pulse-Code Radar Doppler Return using ±3 Codes . . . . . . . . . 45

4-3 Mean and Variance of Random and ±3 32-bit PN Codes . . . . . . . . . . . 46

4-4 Variable Pulse-Code Doppler Radar Showing Partial Effectiveness . . . . . 47

4-5 Variable Pulse-Code MTI Radar Showing Partial Effectiveness . . . . . . . 48

4-6 Time Response of Reciprocal Spectrum . . . . . . . . . . . . . . . . . . . . 49

4-7 Time Response of Reciprocal Spectrum - Nearest Root Inside Unit Circle 50

4-8 Time Response of Reciprocal Spectrum - Nearest Root Outside Unit Circle 51

4-9 Doppler Processing with ±3 Variable Pulse-Codes, Pulse Compression . . 52

4-10 MTI Processing with ±3 Variable Pulse-Codes, Pulse Compression . . . . . 52

4-11 Pulse Compression and MTI Filtered Spectrum Profile - Constant Pulse-Code 53

4-12 Pulse Compression and MTI Filtered Spectrum Profile - Variable Pulse-Codes 53

4-13 Matched Filter and MTI Filtered Spectrum Profile . . . . . . . . . . . . . . 54

4-14 Pulse Compression and MTI Filtered Spectrum Profile . . . . . . . . . . . . 55

5-1 Adaptive Equalization Filter System . . . . . . . . . . . . . . . . . . . . . . 57

5-2 Variable Pulse-Code Doppler Radar with Flat Spectrum Equalization Filtering 58

5-3 Variable Pulse-Code MTI Radar with Flat Spectrum Equalization Filtering 59

5-4 Pulse Compression versus Flat Spectrum Equalization . . . . . . . . . . . . 59

5-5 Variable Pulse-Code MTI Radar with 1 st Pulse Equalization Filtering . . . 60

6-1 Constant Pulse-Code MTI Radar in Repeat-Back Jamming Environment . 62

6-2 Variable Pulse-Code MTI Radar in Repeat-Back Jamming Environment . . 63

8

Chapter 1

Introduction

Radar systems in hostile environments face the challenge of detecting targets in the midst

of noise, clutter, and jamming. The magnitudes of these interference signals are typically

many times that of the target signal. Many techniques have been developed to mitigate

this interference and extract the target's parameters.

Two classes of radar systems designed for this purpose are the moving target indication

(MTI) radar and the pulsed Doppler (PD) radar. Both systems utilize the Doppler effect

to separate moving targets from relatively stationary clutter. Clutter typically consists of

unwanted radar reflections from the sea, terrain, weather, or chaff.

MTI and PD radars are widely used and play a critical role in many modern radar

systems. Civilian systems rely on MTI and PD radar for air surveillance, especially in bad

weather. Military applications include the detection of low flying aircraft or missiles from

shipboard and airborne radar platforms.

Military radar systems often operate in hostile environments and may be targeted by

electronic countermeasures (ECM) such as repeat-back or digital radio-frequency memory

(DRFM) jamming. DRFM jamming captures radar signals to amplify and repeat-back to

the radar to flood the receiver with erroneous data. Advances in technology have made

inexpensive, high-performance radio frequency parts available so that a DRFM jamming

system can easily be developed to blind a PD radar. This thesis addresses this threat

by examining the nature of this vulnerability and developing a modification to the pulsed

Doppler/MTI radar system to resist some forms of DRFM jamming.

9

1.1 Pulsed Doppler Radar

Pulsed Doppler (PD) radar uses Doppler frequencies to determine a target's range rate by

analyzing the return from two or more transmitted pulses. Because of the Doppler effect,

the radar return of a moving object will be shifted in frequency relative to the frequency of

the transmitted signal and stationary objects. PD radars use this difference in frequency

to detect moving objects and determine their relative speeds [12]. Using this technique,

a moving object with a return signal completely obscured by clutter and noise may still

be detected after Doppler processing if its frequency exceeds the frequency range occupied

by the clutter return. The radar system assumed in this thesis is a shipboard or ground-

based pulsed Doppler/MTI radar system with a phased array of antennas that transmits a

series of narrow band, beam-formed pulses. The transmitter will use binary phase encoded

waveforms [12].

1.2 MTI Radar

Moving Target Indication (MTI) Radar is an older system that uses delay-line cancellers to

filter out clutter. The current radar return is compared with previous returns to isolate dif-

ferences. The relatively stationary clutter is subtracted out and moving targets remain [11].

A newer implementation of MTI modifies the pulsed Doppler radar return by adding a notch

filter around the Doppler frequencies of the clutter in order to improve the signal-to-clutter

ratio [13, 6]. This is the implementation used in this thesis.

1.3 Electronic Countermeasures

In addition to noise and clutter, radar systems in hostile environments may encounter jam-

ming. Jamming, or electronic countermeasures (ECM), is an effort by enemy systems to

degrade the capability of friendly radar systems. It can be passive, in the form of chaff

and decoys, or active, in the form of electromagnetic (EM) transmissions with the intent

to deceive or confuse a radar. Electronic counter-countermeasures (ECCM) are efforts by

10

friendly systems to counter enemy ECM. ECM and ECCM are elements of electronic warfare

(EW) [10]. As military forces become more dependent on electromagnetic systems, EW has

become crucial on the battlefield. The variable pulse-code modification to the radar system

proposed in this thesis can be considered an ECCM system.

1.4 Variable Pulse-Code Radar

PD and MTI radars are susceptible to deceptive ECM (DECM) repeaters. This type of

active jamming system, also known as a repeat-back or DRFM jammer, records and repeats

back a radar's transmitted waveform to flood the receiver with erroneous data [10]. A

possible counter to this system is to use non-repetitive waveforms within each coherent

processing interval (CPI) by varying the waveform transmitted by each pulse. If there is

little correlation between the current and previous waveform, the signal transmitted by the

DRFM jammer will be processed as noise and have minimal effect on the radar return.

If a different pseudo random or pseudonoise (PN) code [3] is used for each pulse, the

radar receiver will be minimally affected by the jamming. However, when the return signal

is Doppler processed, any variation in the filtered output from these PN codes will degrade

the integrated signal-to-clutter ratio. The energy no longer falls in a single Doppler bin

after processing. This will be described in detail in Section 3.1. The strength of the clutter

signal is amplified and the target signal can be obscured. This severely limits the ability

to perform Doppler and MTI processing on the radar return when different PN codes are

used within a CPI.

1.5 Performance Metrics

Radar system performance will be measured by the ability of the radar to detect a target

obscured by interference. The strength of the target's signal-to-noise and signal-to-clutter

ratios will be mathematically predicted and empirically verified. This allows us to quantify

performance degradation or improvement as we modify the system.

11

1.6 Contributions of this Thesis

There is a significant performance loss in the ability to Doppler process or apply an MTI

filter to the return of a variable pulse-code radar. The performance loss was found to be the

result of clutter range sidelobe variation. This thesis presents several receiver filtering meth-

ods by which this performance may be recovered. A simulation framework and graphical

user interface created for PD and MTI radar simulation will be presented. The signal-to-

clutter ratio degradation of the return when using variable code sets and matched filtering

will be quantified. Pulse compression is explored as an alternative. However, pulse com-

pression filters do not meet the signal-to-noise ratio (SNR) optimality criteria of matched

filtering. Adaptive digital equalization filtering methods are applied to the matched filter

output to attempt to recover PD and MTI performance. The SNR loss of applying the

equalization filter will be compared with the SNR loss of a pulse compression filter. Pulse

code selection for maximum filter performance and weighting techniques for further per-

formance improvements will be explored. Finally, the effect of jamming on constant and

variable pulse-code sequences will be shown to test the success of the variable pulse-code

radar solution to repeat-back jamming.

12

Chapter 2

Radar System Simulation

The variable pulse-code Doppler/MTI radar system and all return data were simulated in

Matlab®. The time and cost required to perform the modifications required to test this

system in hardware were impractical for the scope of this thesis.

A graphical user interface (GUI) was created to facilitate testing as seen in Figure 2-

1. The simulation allows the user to recreate the scenarios discussed in this thesis and

interactively reconfigure the system and environmental model to examine the effects of any

changes. Options allow the user to modify the characteristics of the radar transmitter and

receiver, create multiple targets, and selectively introduce noise, clutter, and jamming.

The simulated radar return data is a composite of targets, noise, clutter, and jamming,

which are individually computed based on user specified transmitter characteristics. They

are then additively combined and sampled in a three-dimensional matrix representing a

CPI. Any user selected receiver filtering and digital signal processing is performed on this

matrix. This process is shown in Figure 2-2. The user is able to view the output at each

intermediate stage in the formation of the processed output.

13

Figure 2-1: Radar Simulation Application GUI

Target

Noise Environ Adaptive

Sampling A Filtering a Weighting a Equaliza- 4 D pler/-4 Output

Clutter

Figure 2-2: Simulation Flow and System Block Diagram

2.1 Radar System Model

The radar systems simulated in this thesis are based on modern pulsed Doppler and MTI

radars [11]. The simulation allows user selection of the relevant system characteristics:

carrier frequency, chiprate, pulse repetition frequency (PRF), pulse-code length, pulse-code

type and number of pulses. The default values are shown in Figure 2-3.

CamreiFreq f33 GH2 ChipRate 100MHz

PRF F - KH z PI.ses 10 Pulses

(* Constant PuiseCode Variable Pulse Code

Code Length b2 s Code Type I

F~ Look Code

Figure 2-3: Radar System Variables

The transmitted signal is a binary phase encoded pseudonoise (PN) code. A binary phase

encoded, or binary phase-shift keyed (BPSK) signal is one modulated by a predetermined

sequence. In this case, the sequence is a series of +1's and -1's that appear to be random but

can be deterministically generated [9]. For pulse compression and matched filtering, it is

desirable to select sequences with an auto-correlation function with minimal peak sidelobe

height.

15

2.2 Target Model

Targets are created as point scatterers with specified constant radar cross section (RCS) at

a given range with some Doppler component to simulate velocity from one pulse repetition

interval (PRI) to the next. The magnitude of the Doppler shift, fd, of the target is calculated

by

2v _ 2vfe

A c(2.1)

where v is its velocity, A is signal wavelength, and f, is the carrier frequency. This Doppler

shift is applied incrementally to the scaled return from each pulse to create a moving target

whose target return is modelled by

rT(n) = Oej2rfdnp (2.2)

where UT is the target signal level, n is the PRI number, and Tp is the PRI.

The weighted Doppler shift is then scaled and placed at the specified range to be point-

by-point multiplied with the transmitted waveform envelope to create the target return

signal. Figure 2-4 shows the magnitude and phase plots of the return space with three

targets at difference velocities over one coherent processing interval (CPI). The simulation

allows user selection of velocity, range, and return strength of multiple targets as shown in

Figure 2-5.

Target Magntude

1.4-

1.2

0.8

0.2 -

15

X 1 5 2 0 Pr10e

Range (m) 0 0 pLIIS

Target Phase

20

10

10

-3010000....

40010

4000 62000 2 4

Ranrge (m) 0 0 Pk

Figure 2-4: Target Component Magnitude and Phase

16

Figure 2-5: Target Variables

2.3 Environmental Model

The targets, noise, clutter, and jamming are assumed to be independent. Thus, they can be

combined additively in a matrix representing range and PRI to create the radar's operating

environment. The simulation allows user selection of noise level, noise type, clutter level,

clutter type, clutter doppler, jamming level, and jamming type as shown in Figure 2-6 with

the default values.

2.3.1 Noise Model

The noise component of the environment is comprised of both internal receiver noise and ex-

ternal thermal noise [12]. The effects of noise are well described by the Gaussian probability

density function (PDF):

p(x) = 1 e 2 , (2.3)v'27ro.2

where a.2 is the variance of x and xO is the mean value of x. This is sometimes referred to

as Gaussian white noise. For circular symmetric complex Gaussian noise, the PDF is

17

Figure 2-6: Environment Variables

1 Iix~xoII2p(x)= 2 e ( . (2.4)

where |1 |12 is the 2-norm (magnitude squared). For this simulation, we will model the

combined effect using this complex Gaussian PDF. The result is shown in Figure 2-7.

2.3.2 Clutter Model

Clutter is any undesired portion of the radar return created by the environment. This can be

echoes from land, sea, weather, and even wildlife. Chaff, passive reflectors used as a decoy

in EW, is also considered clutter. Clutter is generally too complex to form a uniformly

satisfactory model. Significant efforts have been made in an attempt to better understand

and model clutter returns. This thesis will not focus on these efforts and will instead use

a simplified approach to simulating clutter. An empirical observation can be made relating

radar echo with environmental parameters. The high variability of the clutter return has

been described by Rayleigh, log-normal, and Weibull probability distributions [12]. The

Weibull distribution option in the simulation uses the following PDF,

p(x) = 2(X -')(~1)exp(--((x - P)/a)I), x ;> ';y,a> 0, (2.5)a a

where -y is the shape parameter, p is the location parameter and a is the scale parameter. For

18

Noise

20-

10

0-

110

x 10 65 4

2Range (m) 0 0 Pulse

Figure 2-7: Complex Gaussian "White" Noise Component

comparison between theoretical calculations and empirical results, a Gaussian distribution

is assumed for the clutter. The resultant clutter matrix is shown in Figure 2-8.

2.3.3 Jamming Model

Jamming, or ECM, is the area of electronic warfare designed to exploit and prevent the

effective use of friendly radar systems. Passive jamming techniques such as chaff have little

motion and are thus ineffective against PD and MTI radars. Active jamming techniques are

of greater concern for PD/MTI radars and can be divided into noise jamming and deceptive

electronic countermeasures (DECM).

Noise jamming is similar to raising the level of thermal noise. It attempts to interfere

with the normal operation of a radar by transmitting enough power in the frequency spec-

trum of the receiver to raise the noise floor above the strength of any target signals [10].

This can be simulated by increasing the noise level within a CPI.

Deceptive jamming techniques attempt to deceive a radar by repeating back its trans-

mitted signal to mislead the receiver. These techniques may be used to create false targets,disrupt tracking, and report false positions [10]. For a radar using binary phase-modulated

19

Clutter

70

60

50

S30-

20_-

15

108

x 10 6

2Range (m) 0 0 Pulse

Figure 2-8: Clutter Component

waveforms, a repeater as shown in Figure 2-9, is required to capture the transmitted code

to successfully create a false target [141. This is simulated by using the waveform of the

previous pulse to create 15 false targets. The resulting data is shown in Figure 2-10. Notice

that there is no jamming signal in the first PRI due to the implementation of this type of

repeat-back jamming.

Trigger

Delay Memory Receiver

Antennas

VariableDelay

Stored Amplifier]

Control Pulse

Figure 2-9: Repeat-Back Jammer Block Diagram

When the repeater successfully captures a radar's transmitted waveform, it can create

targets at the receiver input and jamming the radar becomes trivial. The effect of successful

20

Jamming

3.5-

3-

2.5-

15

10 810

X 10 5 4 6

2Range (m) 0 0 Pulse

Figure 2-10: False Targets Created by Repeat-Back Jamming

jamming on an MTI radar is shown in Figure 2-11 where only one of the many targets is real.

The variable pulse-code radar addresses this vulnerability by mitigating the effectiveness of

the repeated signal. This will be further explored in Chapter 6.

21

MTI Filtered Return

80

60

40,

M20,

-20 --

156

x 10 5 42

Range (m) 0 0 Doppler

Figure 2-11: MTI Filtered Return with Repeat-Back Jamming

2.4 Return Processing Model

Radar systems employ many signal detection strategies to distinguish between desired

echoes and interference. Passive and active filtering techniques at the receiver are used

to increase signal-to-noise and signal-to-clutter ratios.

The composite return consisting of the targets and environmental factors are first sam-

pled and filtered using either matched filter or pulse compression filter techniques. An

adaptive digital equalization filter may be added when using variable pulse-codes. The fil-

tered data is then Doppler or MTI processed. The simulation allows the user selection of

sampling rate, receiver filter type, pulse compression filter length, windowing, MTI imple-

mentation, and equalization filtering as shown in Figure 2-12.

2.4.1 Pulse Compression

Pulse compression (PC) and matched filtering (MF) are signal processing strategies for

signal detection improving range resolution and SNR when using waveforms such as binary

phase encoded PN codes. These strategies simultaneously achieve the high output energy

of a long transmit pulse and the range resolution of a short pulse.

22

Figure 2-12: Return Processing Variables

The pulse compression filter is designed to produce a flat frequency response at its output

given the signal for which it was designed at its input [1]. Thus, the frequency response of

the pulse compression filter is the reciprocal of the input signal frequency spectrum. Let

the frequency response of the pulse-code sequence in question, s(n), be given by:

S(f) = DFT(s(n))

where DFT represents the discrete Fourier transform. Then, the pulse compression filter's

frequency response is given by:

1H(f) = (2.6)

SMf)

When multiplied with the input frequency spectrum S(f), the time response of the resulting

flat output spectrum would therefore approximate a dirac delta function, thus minimizing

signal sidelobes:

Y(f) =S(f)H(f),

S(f), = 1,

= 1 < OOt)

thus,

23

y(t) = (t) (2.7)

2.4.2 Matched Filtering

Pulse compression and matched filtering are similar and the terms are often mistakenly

used interchangeably. Their difference is an important distinction in this thesis. Whereas

the pulse compression filter is designed for a flat output spectrum, the matched filter design

criteria is to achieve a maximum signal-to-noise ratio. This criteria is met when the matched

filter is the time reversed input signal [1],

h(t) = s(tm - t). (2.8)

Thus, its frequency response is the complex conjugate of the Fourier transform of the input

signal multiplied by a time shift factor:

H(f) = S*(f)ew't, (2.9)

where tm corresponds to the pulse width of the transmitted signal. A comparison of the

time and frequency responses of a matched filter and pulse compression filter given a 32-bit

PN-code as the input sequence are compared in Figure 2-13. The first two plots show the

magnitude and frequency response of input code sequence. Notice below in the next row

the formation of the matched filter as the time reversed input, and the formation of the

pulse-compression filter as the reciprocal spectrum. The last four plots compare the output

of the matched filter with the pulse compression filter. Notice the flat frequency-spectrum

of the pulse compression filter output and the flat sidelobes in its time response. This

property will become significant in Chapter 4.

2.4.3 Weighting

Weighting, or windowing, is a technique used to reduce sidelobe levels at the expense of

broadening the main-lobe width and a slight reduction in SNR [7]. Sidelobe levels of the

matched filter output will become a problem for the variable pulse-code radar system and

weighting will be explored as a solution.

Weighting is normally applied to a time domain function to perform spectrum shaping.

24

Frequency Specotur

0.5 --- 15 -

0-

-0.5 - o.

-1. -

0 5 10 15 25 35 0 0 20Tinm (step) Frequency

Matched Filter Ttme Flesponse Pulse Compresson Filter Frequency Spectrum5

0.5- -10-

0I

-0.5- -15-

-11

0 5 10 15 20 25 0 35 20 1 203Time (step) Frequency

Matched Fifter Output Pulse Compression Filter Output

30- 30-

25- 25-

20 20,

15 ' - 15-

10 1.

5- -5-

0- .4w 0.5 5

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30Tme (slop) Tme (step)Matched Filler Oulput Frequeny Spectrsmi Pulse Compression Filter Output Specrumn

40

35 -- 35 -

30- 30.

225 5

20 -20-

15- 15-

0 10 20 30 40 50 00 l1o 0 to 20 30 40 50 0Freqency Frequency

Figure 2-13: Matched Filtering vs. Pulse Compression for a 32-Bit PN Code

25

32-81 P- Cods

I AA/Lf~ A(\ /"\h f~/'\1

71-0

70

This is the case when weighting is applied across PRIs in a CPI to shape the Doppler

spectrum. Weighting may also be applied in the frequency domain to reduce sidelobes

in the time domain and shape the waveform of the filtered pulse. Both techniques are

implemented by the simulation. The user is able to choose from uniform (none), Hamming,

Hanning, Chebyshev, Taylor, and Dolph-Chebyshev weighting functions.

Figure 2-14 shows the application of a 63-point Taylor window on the matched filter

output. The window is applied in the frequency domain to reduce sidelobe levels in the

time domain. The result is shown in Figure 2-15. Notice the reduction in sidelobe levels at

the expense of a lower peak value and broadening of the main lobe. The first sidelobe peak

after the apparent null of the weighted output shows a 1.2 dB improvement.

Matthed Filter Oulput Taylor Window Apped m Frequ-ncy

30-.

:20 -

10-

5 -

-40 -30 -20 -10 0 10 20 30T,, (Stp)

Malthed FAKe Output wraylor Westaow

-40 -30 -20 -10 0 10 20 30Ti-. (step

Figure 2-14: Matched Filter

0-

-20 -

-30-

-70-

-W0

0 10 20 3 0 5 0 70Teie (step)

Output with Taylor Weighting

2.4.4 Doppler Processing

Doppler processing separates signals with different range rates within the radar return. This

allows us to separate the relatively stationary clutter from moving targets. The motion

26

-30 -

1-35-40 -

55 -

10 20 30 40 50 60 70

FreqaoncyTaylor WMM arform M mTo1 0111102

40

0 0

Matched Filter Output Comparison

-+-No Weighting0 Taylor Weighting

30-

25

5 20

S15-

10-

5-

C0

-40 -30 -20 -10 0 10 20 30 40Time (step)

Figure 2-15: Matched Filter Output Weighting Comparison

contributes a frequency shift to the reflection in proportion to its velocity. By comparing

the received signals coherently across several pulses, the frequency shift and therefore range

rate, can be observed. The relation between target velocity and Doppler frequency shift is

shown in Equation 2.1.

The amount of target gain due to Doppler processing can be calculated. Suppose we

have a noisy input signal,

x(n) = e2" + v(n).

The discrete Fourier transform (DFT) is then performed to Doppler process the signal. The

result is

2wrknX(k) = x(n)e N , or,

X(k) = ei(a- " + v(n)e--2 .n

The power of this signal is defined by

P(k) = E{x(k)x*(k)}

= Z E{eJ(a-T iAn--m + v(n)v*(m)e-3A(-"0N + cross terms}.n,m

27

If we assume Gaussian white noise, we can ignore the cross terms since the noise has zero

mean. In addition, the noise is uncorrelated from sample-to-sample and from pulse-to-

pulse. Therefore, the expected value is nonzero only for n = m. Our equation may now be

simplified to

P(k) = N eNa Y )(-") + S a2 6(n - m)e-j2 (N-". (2.10)n,m n,m

If = 27, then for the first term,

eAa-)(n-m) = N 2 , (2.11)n,rna 2-rk

which is the peak energy of the target signal after Doppler processing. For the second term,

5 a2 6(n - m)e-i("- 2jO = Na2, (2.12)n,m n

which is noise energy level after Doppler processing. From Equation 2.11 and Equation 2.12,

we can calculate the Signal-to-Noise (SNR) ratio,

N2 NSNR ~ N2 - N -N-SNRO.

Na2 U2

Therefore, the Doppler processing gain is 10 logio(N) where N is the number of pulses in

the CPI. For 10 samples, the Doppler processing gain is 10dB.

The advantage of Doppler processing is the ability to detect targets buried within a

significantly stronger clutter return, as long as there is a difference in velocity. This case is

taken to the extreme in Figure 2-16 where we see the clutter dominating the target signal by

approximately 100 dB. Figure 2-17 shows the return after Doppler processing. The clutter

energy has been transformed into the zero Doppler bin and the target can now be clearly

seen in the last Doppler bin.

2.4.5 MTI Implementation

Moving Target Indication (MTI) radar uses the same physical phenomenon as pulsed

Doppler radar. Doppler frequency shifts over multiple pulses are used to separate sig-

nals by velocity. MTI adds the additional objective of removing clutter to improve the

28

Target Magrntude

130

120

110,-

100

x 1001

X10

S2

Rn () 0 0PUSS

Figure 2-16: Target and Clutter Comparison for Doppler Processing

Doppler Processed Retum

200

150 -

100 -4)

V

50

0-1

-5015

X10

Range (m)Doppler

Figure 2-17: Doppler Processed Radar Return

29

1 _

0.8-

0.6

04.

0.2,

1

x15 100

Range (m) 0 0Pulse

Ckuter

g

-10

5 0 0 246 8

10

signal-to-clutter ratio [11]. Early MTI systems use delay-line cancellers and were limited

in complexity by the capability of analog acoustic devices. The availability of digital tech-

nologies has significantly enhanced MTI processing.

The implementation of MTI used in this thesis applies a three-point digital low-pass

filter to the Doppler frequency spectrum. This filter, whose time samples and frequency

response is shown in Figure 2-18, attenuates all low frequency signals including clutter and

leaves only detected targets. The result after MTI processing is shown in Figure 2-19.

UTI FilEW IM R eSp M . MTI Filter Frnquency Responm3 20

2- 10-

1 0 -

2- -10

-2-

-20-r

-2 L I 1 30L4

0 5 1 15 2 25 3 35 4 0 5 10 15 F 2 0 35 40 45 50ml.w Fesqlmmy

Figure 2-18: MTI Filter Impulse and Frequency Response

2.4.6 Adaptive Digital Equalization Filtering

When using a variable pulse-code radar, the output of the Doppler filter will be degraded

beyond use due to the differences between codes. This is fully explored in Chapter 3. An

adaptive digital equalization filter is applied to the matched filter output as one approach

to recover this performance. This type of filter attempts to equalize an input to a given

function. It will change its coefficients to minimize the error between the input sequence

and that function.

For this thesis, the implementation of the equalization filter will be performed digitally

in the frequency domain. The details of the process and results are discussed in Chapter 5.

30

MTI Filtered Return

60

40

S20

0

-20

-40 ->15-

A810 7

X 10 55

32

Range (m) 0 J Doppler

Figure 2-19: MTI Processed Radar Return

2.5 Simulation Output

The individual elements in the formation of the simulated radar return may all be plotted

along with the intermediate and final results of any return processing. The user is able to

plot the target, noise, clutter, jamming, composite return, filtered return, weighted return,

Doppler processed return, MTI processed return, and adaptively equalized returns as shown

in Figure 2-20.

Plotting Options

Noised MTI E qualze /T

Figure 2-20: Plotting Options

31

Chapter 3

Variable Pulse Code Radar

The variable pulse-code radar modifies the transmitted waveform of the pulsed Doppler

or MTI radar. Instead of selecting one PN code sequence, the variable pulse-code radar

transmits a newly selected code with every PRI as shown in Figure 3-1. The transmitted

waveform is then no longer predictable and thus difficult to jam.

Ccnstant 32-4it Pulse-Cods, 10 Pulse CPI Varable 32-bit Pulse-Code, 10 Pulse CP

30 10-

-0 8 -P

-0 4

Tim (stop) 0 0 PRI

1.5,-

_1

30 10

206

10 4

Tre0 0Pl

Figure 3-1: Constant and Variable Pulse-Code Comparison

3.1 Doppler Degradation

The use of variable pulse-codes presents several problems. The matched filter output from

the radar echoes of each pulse will have different range sidelobes. When the return is Doppler

processed, the effect of changing sidelobe magnitude and phase results in a spreading effect

thereby creating false signals in the Doppler spectrum. Over multiple clutter points, this

effect significantly degrades the signal-to-clutter ratio.

32

me (sep) PRI

For example, let us examine the Doppler spectrum of the matched filter output given a

point scatterer. When the same pulse-code is used for each PRI, the matched filter output

in that range bin will be constant and contribute to only one Doppler bin in the Doppler

spectrum as we see in Figure 3-2.

Matched Filter Output with a Constant Pulse-Code Doppler Spectrumr (Constant Code)

25,s.-

-5, .--

- 60 10 --.400

Ran08 (M) 0 0

350 .

300,

250 .

200 , --

50 - .. -.

-50

-100 - -- -

60 10

40 6

0 0Plse Repetition Interval ~Doppler Frequency

Figure 3-2: Matched Filter Output and Doppler Spectrum for a Constant Pulse-Code

However, when a different pulse-code is used for each PRI, the sidelobes of the matched

filter outputs will differ due to the variations between pulse-codes. These variations will

proportionally contribute to different Doppler bins. The Doppler spectrum of the outputs

will be spread across all Doppler bins. For this paper this will be referred to as a false

Doppler effect since it appears that the energy is in the wrong Doppler bin. We see this

spreading in Figure 3-3.

Matched Filter Output wilh Variable Puise-Codes Doppler Spectrui (Vanable Codes)

40, 350 .

3- 250-

10-20

-00-

-20 -80 50

80 10 010

Ranga (M) 0 0 u R Inl Range (M) 0 0 Doppler Frequency

Figure 3-3: Matched Filter Output and Doppler Spectrum for Variable Pulse-Codes

33

3.1.1 Target Effects

Fortunately, the peak value of the pulse-compressed target return signal is minimally af-

fected by changing from constant to variable pulse-codes. In the absence of clutter, the false

Doppler effect created by varying sidelobes is insignificant for the relatively low number of

individual targets. Given that the length of every different pulse-code remains the same,

the magnitude and phase of the zero-lag matched filter output will remain the same and

thus the return signal can be successfully Doppler processed.

This is proven by calculating the variance of the matched filter output at zero-lag given

a point scatterer. In the absence of noise and clutter, the return signal is r(n) = s(n - m),

where s(n) is a randomly selected, 32-bit PN code and m denotes some time delay. At

m = 0, given a sampled signal, we can define the matched filter output as

min(N,N-1)

y(n) = E s(n - k)s(N - 1 - k), or,k=max(,n-(N-1))

min(N-1,N-1+1)

= E s(N - 1 + l - k)s(N - 1 - k), (3.1)k=max(0,l)

where 1 = n - (N - 1) and 1 > 0 advances the return signal through the filter. Thus, the

peak output is at (0) = y(N - 1) and the first sidelobes' value is at (1) = y(N). From

this, we can compute the matched filter's range sidelobe performance. In order to perform

this analysis, we will model Sn, a randomly chosen 32-bit code sequence, as a stochastic

process [8]. This allows computation of the mean of the matched filter output's range

sidelobes.

To simplify analysis, let S,, s(N - 1 - n), where N is the number of samples. The

elements, Sn, belong to the set S = {-1, +1} with equal probability, thus having zero mean.

Furthermore, Si is assumed orthogonal to Sj for all i 54 j. Based on these assumptions, the

autocorrelation of Sn is

R(k+l,k) = E{Sk+jSk},

= ; =0 (3.2)0 ; 4i0

34

Returning to equation 3.1 and substituting Sk for s(N - 1

(l) =min(N-1,N-1+1)

E Sk+lSk,k=max(0,1)

from which the mean value of the sidelobe level quickly follows via Equation 3.2

min(N-1,N-1+l)

= Yk=max(0,1)

min(N-1,N-1+l)

= E(k=max(O,i)

N

E {Sk+lSk}, and

R(k+l,k)

1=0

l0(3.4)

where N is the number of samples of S,.

To calculate the variance of p(l), we must first compute the second moment of y(l);

E{y2(l)}= IEE{Si+,SiSj+Sj}, or

i= j

= Y:EfSi2±si~+ EES+iSi}ESISjjli=j

For the peak, zero-lag value where 1 = 0,

E{ 2(0)} = E{S4}i=j

+ E E{sf }E{Sh},i#j

Efy2(0)} = N + N(N - 1) = N 2 . (3.6)

For the sidelobe, non-zero-lag values where 1 $ 0, return to Equation 3.5 and compute

N - l| non-zero terms

E= E{Sf S+ 1} +0, ori=j

= N - l. (3.7)

Let us define N = N - 1l. Then from Equation 3.6 and Equation 3.7, we find that the

second moment is

35

(3.3)

E{Q(l)}

i#j(3.5)

or,

k) yields

E 2(1) = , (3.8)N ; ?if0

where N is the number of terms in the overlapped region and equal to N - 111. From

Equation 3.4 and Equation 3.8, we see that the variance of the matched filter's range

sidelobes, y(l), is

Var{y(l)} ={ ' (3.9)>-111 1=4

Thus, given any PN code sequence, the peak (zero-lag) value of the matched filter output

has no intrinsic variation with code, but the non-zero-lag values have a non-zero variance.

The implication is that given a set of random sequences of length N, the peak values of

the matched filter output will produce an accurate Doppler frequency estimate. However,

the Doppler spectrum from the sidelobe non-zero-lag values will not be resolved due to the

variations from pulse to pulse.

This result is also verified by simulation. The match filtered target return in the absence

of clutter shows a constant peak value but different sidelobes. A plot of the Doppler

processed return from a variable pulse-code radar in the absence of clutter is shown in

Figure 3-4.

3.1.2 Clutter Effects

When clutter is reintroduced, it acts as multiple point scatterers, each with varying sidelobes

as we saw in Equation 3.9. This variance gives each sidelobe a potentially non-zero value in

the target's range bin and final Doppler spectrum. The aggregate spreading effect of these

false Doppler values created by the range sidelobe leakage of clutter thus effectively buries

the target signal. This can be seen in Figure 3-5.

To quantify this effect, we need to look at the Doppler spectrum. Let x, (t) represent

the target in MF gate 1. The Fourier transform of the 11h gate is

P-1

Yi(w) = l ip(O~xi(Ae -jwp. (3.10)p=o

where yp(l), the ideal MF output of Equation 3.1, acts as a "weighting function" from PRI

to PRI. For the 1 = 0 case the mean and second moment can be computed as

36

Doppler Processed Return

60-

50 --7 Target40

30-

20-.0

0 10 -

4 -

-10 -

-30-

x151

5 4

Range (m) 0 0 Doppler

Figure 3-4: Variable Pulse-Code Radar Doppler Return without Clutter

Doppler Processed Return

110 Cut

100

90-

50

40

3015-

o 70

x 101

Range (m) 0 0 Doppler

Figure 3-5: Variable Pulse-Code Radar Doppler Return with Clutter

37

P-1

E{Yo(w)} = E E{pp(O)}xo(p)e-iwPp=o

Substituting with Equation 3.4,

P-1E{Yo(w)} = N E xo(p)ei-P.

p=o

E{1Yo(w)j2} Z E{yp()Q*(O)}xO(p)x*(q)e-j,(P-q)pAq

= Z E{y2(O)}xo(p)x4(q)e-jw(P-q) + E E{yp(O)}E{yq(0)XO(p)4(q)e-j,(p-q)p=q p9q

Thus,

E{jY(w)2} = N 2 E xo(p)x*(q)e-3W(P~-).

p,q

From Equation 3.11

|E{Yo(w)12 = N 2 5 XO(p)X*(q)e-jw(p-q) .p,q

Combining Equation 3.12 and Equation 3.13 we see that the variance is

Var{Yo(w)} = 0;

(3.12)

(3.13)

(3.14)17 0.

This is as we expected; using variable pulse-codes will not affect the peak value of the MF

output. For the 1 / 0 case the mean, second moment and variance are

E{Y(w)} =P-1

E{yp(l)}x(p)e-iwP,p=o

1 = 0. (3.15)

38

(3.11)

E{IYi(w)12} = ZE{Qp(l)pq(l)}x(p)x*(q)e-jw(P~q)p,q

= (Np=q

- j1j)xi(p)x*(q)e-jw(P-q) + E E{Qip(l) }E{yq(l)} 1(p)X* (q)e~-w(P-q)

ppq

P-1

E{IYi(w) 12} = (N - 111) E IXI(p)12.p=o

P-1

Var{YijLl} = (N - Ill) Z IX(p)12 .p=O

(3.16)

(3.17)

Thus, the variation in range sidelobe levels causes a rise in Doppler filtered output sidelobes.

Now suppose that in MF bin 1 = 0 we have

X0 (p) = xt(p) + xc(P). (3.18)

where xO is our output composed of a target, xt, and clutter, x,. For analysis, we can assume

that the target and clutter Doppler are easily separated. Substitution in Equation 3.11 yields

P-1o(L,) = N E (xt (p) + xc(p))e-'P

p=O(3.19)

where P is the number of pulses in the CPI. If we let Xt = -tejwP, the Doppler spectrum

becomes

Y0(w) = N E atej('t-w)P + N E Zxc(p)e-jP,

P(3.20)

P

Now we can compute the final signal-to-clutter ratio. With wt = w, from Equation 3.20, we

see that the target signal energy, Et is,

Et = (NPat)2 .

The clutter signal energy, Ec, is computed by

Ec = N 2 E{xc(p)x*(q)}ejw(P-q)P~q

39

(3.21)

We assume xc(p) and xc(q) are independent. For all p 5 q, the expected value of xc(p)

and xc(q) is zero for a Gaussian distribution. Thus, in the remaining case where p = q, the

energy is

Ec = N 2 Z E{xc(p) 12}P

Using Parseval's theorem and Raleigh's energy theorem which equates energy in the time

domain with energy in the frequency domain, we calculate the clutter signal energy as

Ec = BcN 2No, (3.22)

where Be is the bandwidth of the clutter and No is the spectral level of the clutter. From

Equation 3.21 and Equation 3.22, the computed signal-to-clutter ratio is

(put)2 (3.23)

BcNO

To compute the effect of the clutter sidelobes on the target signal, suppose we have

some type of clutter signal present in the previous range bin. The 1 = 1 correlation lag of

this clutter signal will affect the 1 = 0 correlation lag of the target signal. It will cause an

increase in spectral power as per Equation 3.17.

P-1

Var{Yjwj} = (N - 1l) E E{|xi(p)j 2}.p=0

Using Parseval's theorem this becomes

Var{Yj(w)} = (N - 1lI)NoBc. (3.24)

Summing over all non-zero-lag values, 1, the total contribution to clutter from these cells

becomes

N-1

Z Var{Y(w)} = NoBc2 Z(N-l)I =1

= NoBcN(N -1)

40

Thus,

Var{Y(w)} = NoBc(N - 1)N. (3.25)

Let us compare this to the non-overlapping sidelobe Doppler spectrum that we calculated

in Equation 3.23. When we introduce random MF range sidelobes, we see that our noise

power level has increased by a N(N-1) factor that is not separable from the target.

(Pa-t) 2SNR o - .pN (3.26)

BcNo

SN R0SNRL = N . (3.27)N(N - 1)'

This leads to approximately a N 2 reduction in SNR. For a 32-bit PN code where N = 32,

this is approximately a -30dB (1Olog 32x 31 ) loss in SNR performance. Figures 3-6 and 3-7

rotate our three-dimensional plots to show the profile of the Doppler spectrum for a pulsed

Doppler radar using constant versus variable pulse-codes. We see that there is in fact a

30dB clutter sidelobe ceiling over the target in the Doppler spectrum due to the MF in a

variable pulse-code radar.

This empirically verifies the calculated performance degradation associated with matched

filtering in a variable pulse-code radar. We have proven that zero-lag peak values have no

variance when matched filtered. However, the variance in clutter sidelobe strength across all

non-zero-lag values create a false Doppler signal. This results in a significant SNR loss and

buries the target in the Doppler spectrum. Thus, any attempt to recover this performance

will need to address the strength of these clutter sidelobes. This can be done directly, which

is explored in Chapter 4, or indirectly, which is explored in Chapter 5.

41

Constant Pulse-Code Radar Doppler Return Profile120 r

Cluttr->

-. .....- ' * * '- * --

-.. .....

4 5 6 7 8 9 10Doppler

Figure 3-6: Constant Pulse-Code Radar Doppler Return Profile

Variable Pulse-Code Radar Doppler Processed Profile

Clutter-

-/

--x

.. .. ... .. N.

1 2 3 4 5 6 7 8 9 10Doppler

Figure 3-7: Variable Pulse-Code Radar Doppler Return Profile

42

Target-

- .-- --.-. ..

100 F --.

80k-

4) 60 - - .- .- .-.

Range (Tm)

40 - - .

20~77

/

0 1 I,2 3

Range i)

---. ... -. . -. --. . -.

120

100

80

E

N60

40

20

0

--..-.--.-.

........... ...........

Chapter 4

Pulse Compression Filtering

In the previous chapter, we discovered that for a variable pulse-code radar, Doppler and

MTI processing performance is significantly degraded by clutter sidelobe strength at the

matched filter output. Using a different pulse-code for each PRI results in a different

matched filter for each PRI. When the clutter component of the radar return is matched

filtered, the output response is different for each PRI. Though the zero-lag peak values

remain unchanged, the sidelobe values differ significantly. These differences spread energy

across the Doppler spectrum. Over many clutter points, the aggregate effect is a loss of the

target signal-to-clutter ratio.

To mitigate this Doppler spread effect caused by using variable pulse-codes, we can

either reduce clutter sidelobe strength at the matched filter output or process the return

signal to undo the negative effects of variable pulse-codes. Performance recovery using

digital signal processing techniques will be discussed in the following chapter.

4.1 Matched Filter Waveforms

Clutter sidelobe strength at the matched filter output is dependent on the variance of the

non-zero-lag values of the transmitted waveform's autocorrelation function as we saw in

Equation 3.9. Its contribution to the Doppler spectrum is shown in Equation 3.17. Thus,

the magnitude of the Doppler effect is dependent on the transmitted waveform which is

determined by the selected pulse-code sequence. By selecting code sequences with minimal

non-zero-lag variance, we can attempt to mitigate the false Doppler spread effect.

43

4.1.1 Sidelobe Reduction

The study of code sequences and their properties is a significant field of study in digital

communications and signal processing. To recover Doppler performance, our code selection

will focus on minimizing the output sidelobes of the matched filter. Barker codes are a

small family of codes, none longer than length N = 13 [3]. They are characterized by

R, N T )0 (4.1)±1,0, r 0

where Rc(T) is the autocorrelation value at lag T. Notice that all sidelobe values are within

±1. This is a desirable property given our requirements, but these codes are unsuitable for

the application at hand. With a maximum length of 13-bits and such a small set of codes

with this ±1 sidelobe property, the waveforms from a radar using Barker codes would be

highly predictable and susceptible to jamming.

For the 32-bit codes used in this simulation, we can apply the same selection concept

by relaxing the criteria posed in Equation 4.1. A search through all 32-bit codes yielded no

codes for whom all sidelobe values are within ±2. However if we allow

Re(T) =NIT = 0 .(4.2)±3,±2,i1,0, T 4 0.2

then there are 3,376 codes to choose from. Figure 4-1 shows the Doppler spectrum profile

when using randomly generated pulse-codes and Figure 4-2 shows the Doppler spectrum

when using our set of ±3 codes. The comparison shows approximately a 7 dB drop in the

clutter floor when using the ±3 codes.

Let us return to our variance calculation in Section 3.1.2 to verify our theory against

the empirical results. As in Section 3.1.1, we treat the codes as a stochastic process.

Equation 3.9 gives us the variance of the non-zero-lag values of the matched filter output.

Figure 4-3 shows the results of an exact analysis [4] of the variance of the 32 bit PN

sequences' auto-correlation lags, i.e., range-sidelobes for both the unfiltered case and the

case where the 32 bit sequences were filtered to include only those sequences for which the

range-sidelobes were within a ±3 bin range. The mean of these range-sidelobes is zero.

The average of the variance over non-zero-lag values of the unfiltered sequences is 16.5

44

Doppler Processed Return using Random Pulse-Codes

Range

-

-...-...-.

Clutter

-/

/

- -

- -'-

//

1 2 3 4 5 6 7 8 9 10Doppler

Figure 4-1: Variable Pulse-Code Radar Doppler Return using Random Codes

Doppler Processed Return - +/-3 MF Sidelobe Codes110 r

105 F - -- -

Range

/

Clutter-

- >

-. ...

I -.. ....

-.. .. -.

1 2 3 4 5 6 7 8 9 10Doppler

Figure 4-2: Variable Pulse-Code Radar Doppler Return using ±3 Codes

45

110 r-

105 -

100 - --

95 F -

a4) 90

85

80

75

70

100 l - -

95 F --v4)~0 90 F

'W85

80

75

I

ru

-

using Equation 3.9. The variance for the censored data is seen to alternate between 4.75

and 2.6 with an average over the non-zero-lag values of 3.43. This is close to a variance

of 3.0 which would be arrived at by assuming uniform distribution of the non-zero-lag value

range-sidelobes between ±3. Based on the exact analysis, there is a 10log( 6) = 6.8219

dB improvement which matches the empirical result. Although 6.8 dB is a significant

improvement, it is not nearly enough to even begin to recover the target SNR.

Sidelobe Variance versus Lag35

Var Uncensored.... ........................................V a r + /- 3 ~ x

10)

10

0 5 10 15 20 25 30

Lag Value

Figure 4-3: Mean and Variance of Random and ±3 32-bit PN Codes

4.2 Pulse Compression Waveforms

We have verified that selecting codes with a limited matched filter output sidelobe level

improves our SNR. Lower sidelobe levels will result in lower sidelobe variance and therefore,

less false Doppler spreading effect. If we develop this idea to the extreme and tighten the

criteria to specify no sidelobes, we are left with a function that approaches a dirac delta (6o)

function as N -+ inf. Again, using Equation 3.9 and Equation 3.17, we see that if there is

zero variance in the non-zero-lag values of the ME output, then there is zero contribution

to the Doppler spectrum.

Clutter sidelobe strength at the matched filter output is dependent on the level of the

46

non-zero-lag values as we saw in Equation 3.9. Its contribution to the Doppler spectrum is

shown in Equation 3.17.

From Section 2.4.1 we know that the output time response of the ideal pulse compressor

is also a dirac delta function. Therefore, a pulse compression filter should be ideal for

recovering variable pulse-code Doppler performance loss. Figures 4-4 and 4-5 show this to

be only partially true. In fact, pulse compression often fails when using randomly selected

codes or even the ±3 sidelobe set of codes.

Doppler Processed Return - Variable Pulse-Codes, Pulse Compression

80>

60-

40,.-

20-

-20

-4015

10 8 10

X10 56

2Range (m) 0 0 Doppler

Figure 4-4: Variable Pulse-Code Doppler Radar Showing Partial Effectiveness

The reason why is that the pulse compression filter problem is ill-posed since there may

be zeros in the input spectrum. Thus, the reciprocal spectrum of the 32-bit pulse-codes will

have problems: it may not be fully defined and a pulse compression filter generated from it

could be numerically unstable.

If we examine the frequency spectrum of each pulse-code, we see that failure to generate

an output is caused by inability to form a valid pulse compression filter. Equation 2.6

defines the pulse compression filter spectrum as the reciprocal of the input spectrum. If the

input spectrum contains any zero values, the pulse compression filter will have values of oc

and therefore not be realizable.

47

MTI Filtered Return - Variable Pulse-Codes, Pulse Compression

60

40

20 - -

10 64

x 10 105 42

Range (m) 0 0 Doppler

Figure 4-5: Variable Pulse-Code MTI Radar Showing Partial Effectiveness

In cases where a pulse compression filter may be formed but the output less than ideal

as seen in Figures 4-4 and 4-5, we see that the clutter floor has risen and we can hypothesize

that the performance degradation is due to a numerically unstable filter or one with a poor

transient response.

4.2.1 Pulse Compression Filters

To guarantee the formation of a valid pulse compression filter we must create a subset of

codes whose frequency spectrums have no zero-values. The ±3 subset of codes were used

as a starting point.

We can hypothesize that the ±3 subset of codes will create better pulse compression

filters than codes selected at random from the pool of all possible 32-bit codes. The ±3

codes form better matched filters and the MF output shares greater similarity with the

time response of the pulse compression filter output. It is reasonable to believe that a pulse

compression filter formed by these codes will result in less SNR loss when compared to the

optimal matched filter SNR.

The 3,376 codes in the ±3 subset of 32-bit codes are filtered to ensure formation of

48

a valid pulse compression filter. Two quick tests are first performed. Codes whose values

summed to zero have a DC value of zero and can be discarded since the spectrum would be

non-invertible. Codes that, when multiplied by the sequence defined by - 1 N, sum to zeros

also have a zero value at f, = N/2 and can be discarded. However, these two criteria do

not account for zeros anywhere else in the spectrum so further filtering is required. At this

point, there are 3,072 codes remaining.

Since each code is a finite sequence, the discrete fourier transform (DFT) is continuous

in frequency domain. Thus, testing the frequency transform for zero values is incomplete

since there could exist zeros that were not sampled. To thoroughly filter the codes, we must

find and factor the Z-transform of each code to check for zeros in the frequency spectrum.

Any root on the unit circle will indicate a zero in the frequency spectrum.

Now that we have a set of codes able to form valid pulse compression filters, we would

like to further filter the set for codes which have a good transient response to form a cleaner

filter. A heuristic approach may be taken to select codes with a better transient response.

Examining each code sequence, we notice that for some codes, the impulse response of the

resulting pulse-compression filter are oscillatory or have unsatisfactorily long settling times.

An algorithm was devised to select for codes with less oscillation. The impulse response

was time shifted to center the peak. The resulting function was scaled and treated as

a probability density function. The standard deviation was computed and any code not

within a threshold value of 50 time samples was discarded. Figure 4-6 shows a discarded

code with a high level of oscillation and one whose signal energy had a standard deviation

< 50. The result was a subset of 568 codes that formed clean pulse compression filters.

Time Response of Reciprocal Spectrum Time Response of Reciprocal Spectrum0.06

0.05

0.04

0.03

0.02

0.01

0.07

0 .0 4 - - - - - - - - - - - - - -

0.03 - - - - - -

0.02 --- -

0.01 - - - - -

0 . 6 ....... ....... ........ ..... ....... ........................... ..... . ..... ........ ........

0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200Sample Time Sample Time

Figure 4-6: Time Response of Reciprocal Spectrum

Another approach is to return to the Z-transform and examine the proximity of the

49

-... ..... .... .... ... ... ... ........ .... .. .... .... .. ...... .. ... .....

- .. .... .... - .. ... ... -... .. .... .. ... . .. .. -... ... ...

..... .. .... ... ....... ... ... .. .... ... ... ... ... ... .... ... .. .... .. ......

roots of each code to the unit circle. For our binary phase-modulated sequences, Jury's

test [5] tells us that if there are roots outside the unit circle, the filter will be unstable. This

was found to be true for every sequence. We also found that for every code sequence, there

exist roots inside the unit circle as well. Let us consider the fact that zeros on the unit

circle result in an invalid pulse compression filter since the reciprocal of any zeros in the

frequency spectrum are undefined. It would then follow that as you move away from the

unit circle and thus away from the zero value frequency, a better pulse compression filter

will result.

Figure 4-7 shows the reciprocal filter time response of each code in our filtered code

subset versus the distance of the nearest zero to the inside of the unit circle. Figure 4-8

shows the reciprocal filter time response versus the distance of the nearest zero to the outside

of the unit circle. Darker bands and contrast changes along the vertical axis indicate greater

oscillation and worse pulse compression filters. From this it would appear that codes with

zeros further away outside of the unit circle form better pulse compression filters.

Reciprocal Filter Time Response

0.05120

100 0.04

80 0.03

600.02

40

20 0.01

0 00.015 0.0125 0.01 0.0075 0.005 0.0025

Distance Inside UC

Figure 4-7: Time Response of Reciprocal Spectrum - Nearest Root Inside Unit Circle

50

Reciprocal Filter Time Response

0.05120-

100 -0.04

0.03

600.02

40

20

0 &-J 00.0025 0.005 0.0075 0.01 0.0125 0.015

Distance Outside UC

Figure 4-8: Time Response of Reciprocal Spectrum - Nearest Root Outside Unit Circle

4.3 Doppler and MTI Performance

The radar simulation using variable pulse-codes, the filtered 568 code subset, and pulse

compression filters successfully and consistently performed Doppler and MTI processing.

The strategy of minimizing filter output sidelobes to achieve phase alignment was successful

as seen in Figures 4-9 and 4-10.

Doppler and MTI processing are minimally affected by using variable pulse-codes when

a pulse compression filter is used for signal detection. There is negligible degradation in

the Doppler response of PC filter outputs within a coherent processing interval (CPI) when

compared to using constant pulse-codes. However, this does not address the SNR loss

associated with the pulse compression filter which is discussed in the following section.

Figure 4-11 presents the profile of the MTI filtered spectrum of a constant pulse-code radar

using a PC filter. Figure 4-12 presents the profile of a variable pulse-code radar using a

PC filter. Notice the PC filter ring-up and ring-down and that there is no apparent loss in

using variable pulse-codes when using pulse compression.

For the ideal case, there are no sidelobes in the time response output of the PC filter

or in the Doppler spectrum regardless of the pulse-code selected, given that the pulse-code

is able to form a valid PC filter. However, the ideal case is a Jo which is not physically

51

Doppler Processed Return - +/-3 Variable Pulse-Codes, Pulse Compression

80,

60,

CL

40

20

0O

-20,

-40 3J

-6015

10

x 10 5106

4

Range (m) 0 0Doppler

Figure 4-9: Doppler Processing with ±3 Variable Pulse-Codes, Pulse Compression

MTI Filtered Return - =/-3 Variable Pulse-Codes, Pulse Compression

20

0,

-20

xl0

10 -

x10 6 7

3 4

Range (m)

8

0 1Doppler

Figure 4-10: MTI Processing with ±3 Variable Pulse-Codes, Pulse Compression

52

60

401

50

45

40

35

-30

- 25

~20

15

10

5

0

Figure 4-11:

Doppler-I

MTI Filtered Return - Constant Pulse-Code, Pulse Compression

......- ...... ........ ........-- -- - -

<- Target

S .......... - -

S.. -

0 5 10 15Range (m) x 10

Pulse Compression and MTI Filtered Spectrum Profile - Constant Pulse-Code

50

45

40

35

-30

25

20

15

MTI Filtered Return - Variable Pulse-Code, Pulse Compression

-~. ....- - --. .-- -. .-.-. - - -----

<- Filter Ring-up/down

<- Target

Doppler

0 5 10 15Range (m) x 10

Figure 4-12: Pulse Compression and MTI Filtered Spectrum Profile - Variable Pulse-Codes

53

realizable. Implementation in an actual system will result in some small performance loss

dependent on the quality of the PC filter output.

4.3.1 SNR Loss

The pulse compression filter performs equivalently with variable or constant pulse-code

radar system. However, its primary disadvantage is a SNR loss when compared to the

optimum SNR of the matched filter output. Figure 4-13 shows the profile of the Doppler

processed spectrum of a constant pulse-code radar using matched filtering. Figure 4-14

shows the profile of the Doppler processed spectrum of a constant pulse-code radar using

pulse compression. Looking at the target in Doppler bin ten, notice the slight difference in

SNR between the two Figures. This will be empirically quantified in Section 5.3.1.

Doppler Processed Return using Matched Filtering Profile100-

Clutter-8 0 - -. .. -.. --. .. .-. -- --

70 -

6 0 - - -- .- -..-70-

Range Target

30-

20 :

10--

1 2 3 4 5 6 7 8 9 10Doppler

Figure 4-13: Matched Filter and MTI Filtered Spectrum Profile

54

Doppler Processed Return using Pulse Compression Profile

Clutter-5 0 - - - -.-.-. .....

40 -

30CO

20Range Target -

*10 /0 - - - -- -. ..

-10

-20

1 2 3 4 5 6 7 8 9 10Doppler

Figure 4-14: Pulse Compression and MTI Filtered Spectrum Profile

55

Chapter 5

Adaptive Digital Equalization

Filtering

In Chapter 4 we showed that a variable pulse-code radar was able to recover the ability

to perform Doppler and MTI processing performance by using a pulse compression filter

in place of the matched filter. We found this to be the result of the near-zero sidelobe

level of the pulse compression filter output. With no sidelobe energy between PRIs, phase

alignment was achieved and the sidelobes did not significantly contribute to the Doppler

spectrum. However, the pulse compression filter lacks the maximum SNR ratio criteria of

the matched filter.

It is desirable to maintain the maximum SNR of the matched filter with the ability to

perform Doppler and MTI processing of the pulse compression filter. This would require

either realigning or removing the range sidelobes of the matched filter output before Doppler

processing. An adaptive digital equalization filter may be added to the matched filtering

in an attempt to force phase alignment. If the signal degradation of the equalization filter

is less than that of the pulse compression filter, then a performance improvement will be

achieved.

5.1 Filter Implementation

The adaptive equalization filters used in this thesis are finite-duration impulse response

(FIR) filters. The equalization filter attempts to minimize the total squared error between

two inputs and adapt to varying input sequences. The problem presented by matched filter

56

outputs produced by non-similar input PN code sequences is non-standard as depicted

in Figure 5.1. Each unique PN code requires a unique matched filter. Thus, we have

two distinctly different inputs, xo(n) and xi(n), applied to two distinctly different system

responses, ho(n)and hi(n).

X(o(ng y. (n)-h(n) - (n)

- -- + h (n) g, (n)

Figure 5-1: Adaptive Equalization Filter System

5.1.1 Time Domain Implementation

The traditional time domain FIR adaptive equalization filtering technique [2] was tested

and compared with the frequency domain implementation discussed in the next section.

Both produced equivalent results.

5.1.2 Frequency Domain Implementation

For the radar system simulation developed for this thesis, the equalization filtering was

performed digitally in the frequency domain. As we saw in Figure 2-13, shaping the fre-

quency domain spectrum of the output signal will affect its time domain impulse response.

Therefore, we can equalize the time domain matched filter output of multiple distinct input

pulses by forcing their output spectrums to match.

5.2 Equalization Strategies

To recover Doppler and MTI performance when using matched filters we can equalize the

matched filter output to an ideal pulse compression filter output spectrum. We know that

the pulse compression filter can successfully be used with a variable pulse-code radar. Thus,

the output of the equalized signal will contain no range sidelobes in the ideal case.

57

An equalization filter is formed for each pulse in the frequency domain by taking the

reciprocal of the matched filter output spectrum. As with the pulse compression filter, the

output of each filter is a dirac delta function when the sequence from which it was derived

is the input. The absence of sidelobes would result in zero sidelobe variance and therefore

no false Doppler.

The equalization filter is then multiplied by the spectrum of the matched filtered return

signal for its PRI. This is equivalent to clocking the signal through an FIR equalization filter

in the time domain. With the exception of filter ring-up and ring-down at the range fringes,

the result is a clean Doppler spectrum. Figure 5-2 shows the Doppler spectrum of a return

that has been matched filtered and equalized to a flat frequency spectrum. Application of

the MTI filter will not be affected by equalization. Figure 5-3 shows the MTI filter output.

Equalized Doppler Processed Return

80

40-

-20

-40

15

10 8 10100

4

2Range (m) 0 0 Doppler

Figure 5-2: Variable Pulse-Code Doppler Radar with Flat Spectrum Equalization Filtering

The performance of using a pulse compression filter can be compared with that of the

combination equalization and matched filter. From Figure 5-4, we observe that despite

the maximum SNR of the matched filter output, losses through the equalization filter have

brought the performance of the system down to the same level as pulse compression by

itself. This result is not surprising as the equalization filter was designed to meet the same

58

Equalized MTI Processed Retum

80-

60,

40,

' 20-

0

-20

-40

-60,15

x 10

Doppler

Figure 5-3: Variable Pulse-Code MTI Radar with Flat Spectrum Equalization Filtering

criteria as the pulse compression filter.

MTI Filered Return using Piuse Comeression Filfting

60-

0

05 10Range (m)

MTI Processed Retum using Equization to a Flat Spectrun0

60

40

-Targe

-20

-- F"-201 0 510x 10 Range (m)

Figure 5-4: Pulse Compression versus Flat Spectrum Equalization

We can attempt to improve upon this performance by returning to the concept of phase

alignment across pulses. As in the case of the traditional pulsed Doppler radar, zero matched

filter output sidelobe is not a requirement for Doppler processing. However, to successfully

perform Doppler processing, zero variation and phase alignment, across PRIs in the sidelobes

of the matched filter output is a requirement.

59

15

10 86

5 42

Ranne (m) 0 0

Therefore, it would follow that if the each equalization filter were formed such that its

output was the same as some reference matched filter output, then both phase alignment

and minimum sidelobe level variation could be achieved.

Figure 5-5 shows the successful result of equalizing the spectrum of all matched filter

outputs to the spectrum of the first matched filter output. The equalization filter for the

first pulse is Go = 1 and the remaining equalization filters were formed by

Gn(f) == H ,(f) (5.1)Hn(f)'

where n is the PRI number.

MTI Processed Return using 1st Pulse Spectrum Equalization

100

80Target

60-

S 40

-20-

-40 -15-

10--84 6

x 10 5 42

Range (m) 0 0 Doppler

Figure 5-5: Variable Pulse-Code MTI Radar with 1' Pulse Equalization Filtering

5.3 Doppler and MTI Performance

The adaptive digital equalization filtering was successful in mitigating the Doppler spread

effect created by using variable pulse-codes. The flat spectrum equalization algorithm uses

the same strategy as the pulse compression filter and has the same performance. Minimizing

the sidelobe levels of the matched filter output achieves phase alignment within a CPI and

60

minimizes sidelobe level variation, the cause of the Doppler spreading effect derived in

Section 3.1. Phase alignment is also achieved with the 1 " pulse spectrum equalization

algorithm.

5.3.1 SNR Loss

The first-pulse equalization algorithm in Equation 5.1 will result in less SNR loss than

equalizing to a flat spectrum. We can verify this result empirically. Since an equalization

filter is a random process, theoretical computation of SNR loss is impractical given that any

code may follow any other code when the codes are randomly selected. Thus, a simulation

was created to compute the SNR of both equalization algorithms at the filter output. The

SNR of matched filtering alone and pulse compression will also be computed for comparison

using our filtered set of 568 codes whose signal energy had a standard deviation < 50.

Although the gain of the target in the Doppler spectrum will remain unchanged as we

discovered in Section 3.1.1, the noise floor will vary with the filter technique used, thus

changing the SNR. The results are shown in Table 5.3.1.

Table 5.1: Comparison of SNR for different filtering methods.

Filter Matched Pulse Flat Spectrum 1 st PulseMethod Filtering Compression Equalization Equalization

SNR Loss (dB) 0 -2.43 -2.43 -2.37

Notice that the SNR for equalizing to a flat spectrum and pulse compression filtering are

the same. This is expected since both filters are designing to the same criteria. We also

see that equalizing to the spectrum of the first matched filtered pulse offers some improve-

ment in performance. However, this is not considered a significant improvement over pulse

compression.

61

Chapter 6

Anti-Jamming Performance

The final test for the variable pulse-code radar is its effectiveness in a hostile jamming

environment. Jamming was created as described in Section 2.3.3. Figure 6-1 shows the

MTI filter output of a constant pulse-code radar. The jamming environment contains 15

false targets, each with a different position and velocity. Figure 6-2 shows the MTI filter

output of a variable pulse-code radar. Notice all of the false targets have been rejected to

below the noise level and the true target revealed.

MTI Filtered Return with Repeat-Back Jamming

80

40

240>

15-

10 ---- 84 6

X 10 5 42

Range (m) 0 0 Doppler

Figure 6-1: Constant Pulse-Code MTI Radar in Repeat-Back Jamming Environment

62

Equalized MTI Processed Return with Repeat-Back Jamming and Variable Pulse-Codes

10 86

5 42

Range (m) 0 0Doppler

Figure 6-2: Variable Pulse-Code MTI Radar in Repeat-Back Jamming Environment

63

100-

80 -

60 -

4)

ft!

40

20

0-

-20,

-4015

4X 10

Chapter 7

Conclusion

In this thesis, a new modification to the Pulsed Doppler and Moving Target Indication radar

systems was explored. The variable pulse-code radar system was proposed in response to

a potential vulnerability to repeat-back jamming in current systems. This new system

varies the codes within a coherent processing intervals by changing the PN code sequence

transmitted with each PRI. This is highly effective in resisting some types of repeat-back

jamming. However, the use of variable pulse-codes results in a significant performance loss

in the ability to Doppler process or apply an MTI filter to the return.

The performance loss was investigated and it was discovered that while the zero-lag

peak values of the return signal were unaffected by using variable pulse-codes, there was a

significant amount of clutter sidelobe interference during matched filtering. This interference

disrupted amplitude and phase alignment across PRIs which resulted in a degradation of

the integrated signal-to-clutter ratio by a factor of 1 where N is the code length.

Several options to mitigate this performance loss were explored. Code selection of PN

codes able to form matched filters with minimal sidelobe variation improved performance

by approximately 7 dB. Pulse compression was considered as an alternative to matched

filtering. The pulse compression filter was successful due to the absence of sidelobe energy

in the filtered output. However, the pulse compression filter suffers from a 2.5 dB SNR

loss when compared to the matched filter. Further filtering of codes was performed by

examining time responses of the reciprocal spectrum of code sequences to create a subset

of 32-bit codes which optimized the pulse compression filter.

The maximum SNR matched filter was revisited with the addition of adaptive digital

64

equalization filters designed to recover pulsed Doppler and MTI performance based on

the lessons learned previously. A flat spectrum equalization filter, similar in function to

the pulse compression filter, was tested and found to be successful in recovering some

Doppler performance. To attempt to combine the advantages of the matched filter and

pulse compression filter, a 11 pulse spectrum equalization filter was designed to maximize

Doppler performance using variable pulse-codes. This too was found to be successful.

Finally, the variable pulse-code radar was placed in a hostile jamming environment to

test the effectiveness of these processing techniques. A repeat-back jammer was created

which introduced false targets using captured pulse-codes from each previous PRI. The

variable pulse-code radar successfully mitigated the effects of the jamming and rejected all

false targets. There exists much room for future work to perform a more rigorous study of

this radar system. The environmental factors can be modelled in greater detail and a more

complex jamming environment created to test the robustness of this system. The variable

pulse-code radar has met its performance goals in this thesis and shows promise for future

development.

65

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[3] Leon W. Couch II. Digital and Analog Communications Systems. Prentice-Hall, Inc.,

New Jersey, 2001.

[4] Michael A. Koerber. Personnel Correspondance, May 2002.

[5] Benjamin C. Kuo. Digital Control Systems. Hold, Rinehart, and Winston, Inc., New

York, 1980.

[6] Hamish Meikle. Modern Radar Systems. Artech House, Boston, 2001.

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[8] Athanasios Papoulis. Probability, Random Variables, and Stochastic Processes.

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[10] Michael B. Szymanski Robert N. Lothes and Richard G. Wiley. Radar Vulnerability to

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[11] Curtis D. Schleher. MTI and Pulsed Doppler Radar. Artech House, Boston, 1991.

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66

[13] Merrill I. Skolnik. Introduction to Radar Systems. McGraw-Hill, New York, third

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67