CFAR Thesis - James Jen - PDF

A STUDY OF CFAR IMPLEMENTATION COST AND PERFORMANCE

TRADEOFFS IN HETEROGENEOUS ENVIRONMENTS

A Thesis

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

In

Electrical and Computer Engineering

By

James J. Jen

2011

SIGNATURE PAGE THESIS: A STUDY OF CFAR IMPLEMENTATION COST AND PERFORMATIVE TRADEOFFS IN HETEROGENEOUS ENVIRONMENTS AUTHOR: James J. Jen DATE SUBMITTED: Spring 2011 Electrical and Computer Engineering Dr. Zekeriya Aliyazicioglu _________________________________________ Thesis Committee Chair Electrical and Computer Engineering College of Engineering Dr. H.K Hwang _________________________________________ Electrical and Computer Engineering College of Engineering Dr. James Kang _________________________________________ Electrical and Computer Engineering College of Engineering

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ACKNOWLEDGEMENTS

I recall clearly last year when, at the end of a Signal Processing lecture, Profes‐

sor Hwang asked for interest in graduate research. My time at Cal Poly Pomona, up un‐

til that point, had been rocky and undistinguished. Tentative, I almost didn’t approach

the professor that. I’m glad I did. The last year of graduate research with Professors

Hwang and Zeki had meaningfully colored and enriched my graduate studies.

Much thanks to Professor H.K. Hwang and Zekeriya Aliyazicioglu: firstly, for the

opportunity for research and their many Friday mornings spent on us students, and

secondly, for their invaluable guidance and encouragement throughout.

Appreciation too to Thales‐Raytheon for supporting the research, and particu‐

larly to Tom Nichols Walker Birrell for their expertise and input.

Thanks too to Nellie Qian, who, for most of the past year, had been my partner

in crime in MATLAB ventures from Minimax beamforming, to antenna array interfer‐

ence suppression, and finally to this CFAR thing that had become our thesis.

And thanks, most sincerely, to my mom and dad— Emily and Chin‐Ping Jen—

for their support, encouragement, and patience.

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ABSTRACT

CFAR— Constant False Alarm Rate— is a critical component in RADAR detec‐

tion. Through the judicial setting of detection threshold, CFAR algorithms allow RADAR

systems to set detection thresholds and reliably differentiate between targets of inter‐

est and interfering noise or clutter.

In many operating conditions, noise and clutter distributions may be highly het‐

erogeneous— with sudden jumps in clutter power or with the presence of multiple tar‐

gets in close proximity. A good CFAR algorithm must reliably operate in these condi‐

tions but without prohibitively high implementation costs.

In this thesis, we investigate a number of CFAR algorithms— new and old— all

the while assessing their operational flexibility and cost of operation.

As balance of performance and implementation cost, two algorithms stood out

as desirable: Variability Index (VI) CFAR— a procedure that allowed for dynamically se‐

lection between the leading, lagging, or whole of the reference windows— and Switch‐

ing (S) CFAR— a test cell technique that allowed for the selection of representative

subsets of the reference cell as compared to the cell under test. We conclude with de‐

veloping a new algorithm: Switching Variability‐Index (SVI) CFAR that combines the ad‐

vantages of both VI and S CFAR.

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TABLE OF CONTENTS

Signature Page …………………………………………………………………………………………………………

Acknowledgements ………………………………………………………………………………………………...

Abstract …………………………………………………………………………………………………………………..

Table of Contents …………………………………………………………………………………………………….

List of Tables …………………………………………………………………………………………………………...

List of Figures …………………………………………………………………………………………………………..

1 Introduction ………………………………………………………………………………………………..

1.1 Background ……………………………………………………………………………………………

1.2 Objective ……………………………………………………………………………………………….

1.3 Investigation Tool ………………………………………………………………………………….

2 Background ………………………………………………………………………………………………….

2.1 Introduction …………………………………………………………………………………………..

2.2 The Radar System ………………………………………………………………………………….

2.3 Doppler Processing ………………………………………………………………………………..

2.4 Radar Signals………………………………………………………………………………………….

2.4.1 Target ……………………………………………………………….…………………….

2.4.2 Clutter …………………………………………………………………………………….

2.4.3 Noise ………………………………………………………………………………………

2.4.4 Square Law Detector …………………………………………………..………….

2.5 Radar Signal Environment ……………………………………………………………………..

2.5.1 Homogeneous ………………………………………………………………………….

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2.5.2 Multiple Targets ………………………………………………………………………

2.5.3 Clutter Wall ……………………………………………………………………………..

2.6 About CFAR …………………………………………………………………………………………...

2.6.1 Probability of False Alarm ………………………………………………………..

2.6.2 The CFAR Window ……………………………………………………………………

2.6.3 Probability of Detection …………………………………………………………..

2.6.4 CFAR Loss …………………………………………………………………………………

2.7 Monte Carlo Simulations ……………………………………………………………………….

3 Cell Averaging CFAR ………………………………………………………………………………….

3.1 Implementation …………………………………………………………………………………….

3.2 Homogeneous Environment…………………………………………………………………..

3.2.1 Probability of False Alarm…………………………………………………………

3.2.2 Detection………………………………………………………………………………….

3.3 Multiple Targets………………………………………………………………………………………

3.4 Clutter Wall……………………………………………………………………………………………

3.4.1 Clutter Wall False Alarm……………………………………………………………

3.4.2 Clutter Wall Detection………………………………………………………………

3.5 Summary: CA‐CFAR Advantages and Disadvantages………………………………..

3.6 A Combat for Multiple Targets: Smallest‐Of Cell Averaging CFAR…...……..

3.7 A Remedy for Clutter Walls: Greatest‐Of Cell Averaging CFAR ………………

4 Variability‐Index CFAR ……………………………………………………………………………

4.1 Implementation …………………………………………………………………………………….

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4.2 VI‐CFAR Decision Logic……………………………………………………………………………

4.3 Homogeneous Environment……………………………………………………………………


4.3.2 Probability of Detection……………………………………………………………

4.4 Masking Targets………………………………………………………………………………………

4.5 Clutter Wall……………………………………………………………………………………………..


4.5.2 Probability of Detection……………………………………………………………

5 Ordered Statistics CFAR …………………………………………………………………………….

5.1 Implementation …………………………………………………………………………………….

5.2 Homogeneous ……………………………………………………………………………………….

5.2.1 Probability of False Alarm ……………………………………………………….

5.2.2 Detection…………………………………………………………………………………


5.5 Clutter Wall …………………………………………………………………………………………..

6 Ordered Statistics Greatest‐Of CFAR …………………………………………………………

6.1 Implementation …………………………………………………………………………………….

6.2 Homogeneous ……………………………………………………………………………………….

6.2.1 Probability of False Alarm ……………………………………………………….

6.2.2 Detection…………………………………………………………………………………


6.5 Clutter Wall …………………………………………………………………………………………..

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7 Conclusions and Final Assessment……………………………………………………………….

References ………………………………………………………………………………………………………………

Appendix A MATLAB Functions …………………………………………………………………………..

A.1 Radar Return Generation ……………………………………………………………………..

A.2 Monte Carlo Simulations ………………………………………………………………………

A.3 Neyman Pearson Detection ………………………………………………………………….

A.4 Cell Averaging CFAR ………………………………………………………………………………

A.5 Greatest‐Of Cell Averaging CFAR …………………………………………………………..

A.6 Smallest‐Of Cell Averaging CFAR ……………………………………………………………

A.7 Variability‐Index CFAR ……………………………………………………………………………

A.8 Ordered Statistics CFAR ………………………………………………………………………...

A.9 Ordered Statistics Greatest‐Of CFAR ……………………………………………………..

A.10 Switching CFAR …………………………………………………………………………………….

A.11 Switching Variability‐Index CFAR ………………………………………………………….

Appendix B MATLAB Scripts ………………………………………………………………………………..

B.1 Probability of False Alarm in Homogeneous Environment………………………

B.2 Probability of False Alarm in Clutter Wall Environment………………………….

B.3 Probability of Detection in Homogeneous/ Multiple Target Environment.

B.4 Probability of Detection in Clutter Wall Environment……………………………..

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LIST OF TABLES

Table 2.1 Swerling Targets ……………………………………………………………………………………….

Table 2.2 Clutter Types ……………………………………………………………………………………………

Table 4.1 VI‐CFAR Decision Logic………………………………………………………………………………

Table 7.1 CFAR Comparison………………………………………………………………………………………

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LIST OF FIGURES

Figure 2.1. Radar Block Diagram.……………………………………………………………………………..

Figure 2.2. Doppler Processing…………………………………………………………………………………

Figure 2.3. Scattering for Different Swerling Targets………………………………………………..

Figure 2.4. Diode Conductance Characteristic.………………………………………………………….

Figure 2.5. Radar Return Statistics Before and After Square Law Detector ………………

Figure 2.6. Homogeneous Environment……………………………………………………………………

Figure 2.7. Multiple Targets. ……………………………………………………………………………………

Figure 2.8. Clutter Edge……………………………………………………………………………………………

Figure 2.9. Exponential Distribution and Thresholding.…………………………………………….

Figure 2.10. CFAR Window……………………………………………………………………………………….

Figure 2.11. The Threshold. ……………………………………………………………………………………..

Figure 2.12. A Typical Probability of Detection Curve illustrating CFAR Loss……………..

Figure 2.13. Monte Carlo with Exponential Distribution.………………………………………….

Figure 2.14. Monte Carlo Weight Function for Exponential Distribution…………………..

Figure 3.1. Cell Averaging CFAR Block Diagram.…………………..……………………………………

Figure 3.2. CA‐CFAR Threshold in Homogeneous Environment…………………………………

Figure 3.3. CA‐CFAR Theoretical v. Experimental PFA.………………………………………………

Figure 3.4. CA CFAR Homogeneous Probability of Detection.……………………………………

Figure 3.5. CA CFAR Homogeneous CFAR loss…………………………………………………………..

Figure 3.6. CA‐CFAR Threshold with 1 Masking Target……………………………………………..

Figure 3.7. CA‐CFAR Probability of Detection with 1 to 3 masking targets………………..

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Figure 3.8. CA‐CFAR CFAR Loss with 1 to 3 masking targets…………………………………….

Figure 3.9. CA‐CFAR Threshold in Clutter Wall Transition…………………………………………

Figure 3.10. CA‐CFAR Clutter Wall Probability of False Alarm.………………………………….

Figure 3.11. CA‐CFAR Clutter Wall Probability of Detection………………………………………

Figure 3.12. SOCA‐CFAR Block Diagram…………………………………………………………………….

Figure 3.13. SOCA‐CFAR Experimental v. Theoretical PFA…………………………………………

Figure 3.14. SOCA‐CFAR Homogeneous PD.……………………………………………….……………..

Figure 3.15. SOCA‐CFAR Homogeneous CFAR Loss …………………………………………………..

Figure 3.16. SOCA‐CFAR 1 Masking Target Probability of Detection…………………………

Figure 3.17. SOCA‐CFAR 1 Masking Target CFAR Loss……………………………………………….

Figure 3.18. SOCA‐CFAR Clutter Wall Probability of False Alarm………………………………

Figure 3.19. GOCA‐CFAR Block Diagram……………………………………………………………………

Figure 3.20. GOCA‐CFAR Homogeneous Probability of False Alarm…………………………

Figure 3.21. GOCA‐CFAR Homogeneous Probability of Detection…………………………….

Figure 3.22. GOCA‐CFAR Homogeneous CFAR Loss…………………………………………………..

Figure 3.23. GOCA‐CFAR Probability of Detection with 1 Masking Target…………………

Figure 3.24. GOCA‐CFAR CFAR Loss with 1 Masking Target………………………………………

Figure 3.35. GOCA‐CFAR Clutter Wall PFA...……………………………………………………………….

Figure 4.1. VI‐CFAR Block Diagram……………………………………………………………………………

Figure 4.2. VI‐CFAR ‐ Case I………………………………………………………………………………………

Figure 4.3. VI‐CFAR ‐ Case II………………………………………………………………………………………

Figure 4.4. VI‐CFAR ‐ Case III…………………………………………………………………………………….

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Figure 4.5. VI‐CFAR ‐ Case IV…………………………………………………………………………………….

Figure 4.6. VI‐CFAR ‐ Case V……………………………………………………………………………………..

Figure 4.7. VI‐CFAR Threshold in Homogeneous Environment………………………………….

Figure 4.8. VI‐CFAR Theoretical v. Experimental Homogeneous PFA …………………………

Figure 4.9. VI‐CFAR Probability of Detection Curve ………………………………………………….

Figure 4.10. VI‐CFAR CFAR Loss in Homogeneous Environment………………………………..

Figure 4.11. VI‐CFAR Threshold 1 Masking Target…………………………………………………….

Figure 4.12. VI‐CFAR Probability of Detection with 1 Masking Target……………………….

Figure 4.13. VI‐CFAR PD Curve with varying Numbers of Masking Target………………….

Figure 4.14. VI‐CFAR CFAR Loss Curve with varying Numbers of Masking Target………

Figure 4.15. Three Targets………………………………………………………………………………………..

Figure 4.16. VI‐CFAR Probability of Detection with Three Masking Targets………………

Figure 4.17. VI‐CFAR Threshold in Clutter Wall Transition………………………………………

Figure 4.18. VI‐CFAR Clutter Wall Probability of False Alarm..…………………………………..

Figure 4.19. VI‐CFAR Clutter Wall Probability of Detection..……………………………………..

Figure 5.1. OS‐CFAR Block Diagram……………………………………..……………………………………

Figure 5.2. OS‐CFAR Homogeneous PFA.…………………………………..……………………………….

Figure 5.3. OS‐CFAR Threshold in Homogeneous Environment.……………………………….

Figure 5.4. OS‐CFAR Homogeneous Probability of Detection……………………………………

Figure 5.5. OS‐CFAR Homogeneous CFAR Loss …………………………………………………………

Figure 5.6. OS‐CFAR Resistance to Multiple Targets …..……………………………………………

Figure 5.7. OS‐CFAR Threshold with 1 Masking Target.…………………………………………….

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Figure 5.8. OS‐CFAR Probability of Detection with 1 Masking Target.……………………….

Figure 5.9. OS‐CFAR CFAR Loss with 1 Masking Target …………………………………………….

Figure 5.10. OS‐CFAR Probability of Detection with 3 Masking Target.……………………

Figure 5.11. OS‐CFAR CFAR Loss with 3 Masking Target …………………………………………

Figure 5.12. OS‐CFAR in Clutter Wall Threshold.……………………………………………………….

Figure 5.13. OS‐CFAR Clutter Wall Probability of False Alarm……………………………………

Figure 5.14. OS‐CFAR Clutter Wall Probability of Detection……………………………………..

Figure 6.1. OSGO‐CFAR Block Diagram.…………………………………………………………………….

Figure 6.2. OSGO‐CFAR Threshold in Homogeneous Environment……………………………

Figure 6.3. OSGO‐CFAR Homogeneous PFA.………………………………………………………………

Figure 6.4. OSGO‐CFAR Homogeneous PD………………………………………………………………...

Figure 6.5. OSGO‐CFAR Homogeneous CFAR Loss…………………………………………………….

Figure 6.6. OSGO‐CFAR Resistance to Multiple Targets

Figure 6.7. OSGO‐CFAR Threshold with 1 Masking Target…………………………………………

Figure 6.8. OSGO‐CFAR PD with 1 Masking Target……………………………………………………

Figure 6.9. OSGO‐CFAR CFAR loss with 1 Masking Target…………………………………………

Figure 6.10. OSGO‐CFAR PD with 3 Masking Target…………………………………………………

Figure 6.11. OSGO‐CFAR CFAR loss with 3 Masking Target………………………………………

Figure 6.12. OSGO‐CFAR Threshold in Clutter Wall Transition………………………………….

Figure 6.13. OSGO‐CFAR Clutter Wall PFA………………………………………………………………….

Figure 6.16. OSGO‐CFAR Clutter Wall PD…………………………………………………………………..

Figure 7.1 CFAR Obstacle Course……………………………………..……………………………………….

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CHAPTER 1

Introduction

1.1 Background

Radar return is often a mix of noise; clutter; and, if present, targets. A key com‐

ponent in RADAR processing is the setting of detection thresholds. These thresholds

differentiate between targets of interest and unwanted radar returns. As operating en‐

vironments and conditions change, the amount and nature of noise and clutter also

change. For accurate and reliable detection, the threshold must self‐adjust dynamically

and intelligently.

CFAR, constant false alarm rate, represents a key technique in adaptively set‐

ting target detection threshold [1]. Employing a moving window, across range bins of

data, CFAR algorithms look at neighborhoods of power returns in the estimate of the

noise or clutter mean. By scaling the estimated mean with a pre‐calculated multiplier,

the threshold is set as to limit false alarms to a desired rate.

CFAR algorithms are assessed for their abilities to maintain desired probabilities

of detections (PD) and their probabilities of false alarm (PFA). The probability of detec‐

tion describes the chances of successfully declaring a target, when a target is actually

present. The probability of false alarm describes the odds of incorrectly declaring a tar‐

get, when the signal is, in actuality, noise or clutter.

Developed CFAR algorithms must be able to operate in a variety of canonical

environments: homogeneous, multiple targets, and clutter wall. In the homogeneous

environment, a single target exists in a sea of clutter or noise of uniform noise. In the

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multiple target exists, several targets exist in close proximity to one another. In the

clutter wall environment, noise and/or clutter power experiences sudden, discontinu‐

ous increases or decreases.

1.2 Objective

The goal of this study is to investigate and assess the variety of existing CFAR

algorithms and judging their performance in homogeneous, multiple target, and clut‐

ter wall environments. Towards this, the Ordered Statistics Greatest‐Of CFAR (OSGO‐

CFAR), Variability Index CFAR (VI‐CFAR), and Switching CFAR (S‐CFAR) are studied.

While Cell Averaging CFAR (CA‐CFAR) and Ordered Statistics CFAR (OS‐CFAR) are also

looked at, they are used primarily as basis of comparisons for the algorithms of inter‐

est. We end with development of our own CFAR technique, Switching Variability Index

CFAR (SVI‐CFAR) that represents a hybrid approach between VI‐CFAR and S‐CFAR.

1.3 Investigation Tool

Radar signals and CFAR algorithms are assessed through Monte Carlo, numeri‐

cal experiments. MATLAB was chosen as the investigation tool (Version 7.10.0,

R2010a).

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CHAPTER 2

Background

2.1 Introduction

Since their development during World War II, radar had grown to occupy criti‐

cal civilian and military roles. An introductory understanding of radar processing and

signals is necessary before getting into the analysis and assessment of CFAR algo‐

rithms.

2.2 The Radar System

Originally acronymic as RADAR— for RAdio Detection And Ranging, common

usage had established the word in its own right, as radar— a free‐standing noun. The

original word, though, helps remind of the technology’s dual purpose: To detect tar‐

gets of interests and to ascertain their distance.

Figure 2.1. Radar Block Diagram.

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Figure 2.1 shows a basic block diagram of a RADAR system. The particular sys‐

tem is monostatic— where a single antenna is multiplexed for both transmission and

reception [2].

The waveform generator forms a timing pulse to trigger the transmitter to send

an electromagnetic pulse at a designed frequency. As this pulse is sent it, it will en‐

counter different objects at varying distances, and these objects will scatter back some

of these pulses back to the antenna.

The timing and frequency of the return pulse corresponds to the relatives

speeds and distance of the reflected target. The subsequent processing blocks— the

A/D converter, the pulse compression, and the Doppler processor all helps extract the

relevant information.

Pulse repetition interval (PRI) refers to the period of time before the successive

radar pulses. Pulse repetition frequency (PRF) is simply the inverse of the PRI. During

the period between successive pulses, during a given pulse repetition interval, the ra‐

dar system waits for back‐echoed pulses. Detected echoes are detected, binned and

stored.

The amount of time a sent pulse is reflected back to the radar antenna is di‐

rectly proportional to the distance of the reflector:

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: time

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2.3 Doppler Processing

With doppler processing, a radar system can not only detect target range and

location, but also the radial velocity, the range rate.

A single target may be looked at several times over several pulse intervals . The

change in power over a given range bin, or distance, across many pulse intervals may

give a sense of the doppler shift, and hence, the speed of the target (Figure 2.2).

When discrete Fourier transform (DFT), through fast fourier transform (FFT), is

performed on each range sample over its many range intervals, the pulse intervals are

converted to doppler intervals (also called doppler bins).

Figure 2.2. Doppler Processing.

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2.4 Radar Signals

Radar signals are often composed of unwanted noise, undesired clutter returns,

and a target of interest. Over a short period of time, each signal elements possesses its

own stochastic distribution with a given power. Statistical characteristics of the signals

change as they pass through a square law detector.

2.4.1 Target

The power of a radar return depends on the distance and the effective radar

cross section (RCS) of the target. Over distance r, the power of the signal diminishes by

a factor r4. Many factors of the radar, the target, and the operating environment come

together to determine the target’s RCS: the illumination angle, the frequency, the po‐

larization of the transmitted wave, target motion, vibration, and kinematics of the ra‐

dar itself. Rather than hopelessly describing a target’s RCS deterministically, they’re

characterizing stochastically.

Building upon Marcum’s pioneering studies, Peter Swerling developed four ca‐

nonical stochastic models: Swerling I, II, III, and IV. With these Swerling models, the re‐

Table 2.1. Swerling Targets

Model Distribution Notes

Swerling I Rayleigh Scan‐to‐scan Characterizes well targets like planes, with many independent reflectors Swerling II Pulse‐to‐Pulse

Swerling III Chi‐Squared (Four degrees of freedom)

Scan‐to‐scan

Swerling IV Pulse‐to‐Pulse

Swerling V Non‐fluctuating Also called Swerling 0 or Marcum’s Model

Characterizes well targets like frigates, with one strong reflector and many smaller, weaker inde‐pendent reflectors

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turn power of a target will fluctuate— randomly increase and decrease— with a char‐

acteristic probability density function (PDF) with either quickly or slowly (Table 2.1).

The Swerling I and II models both characterized with a Rayleigh distribution

and assumed to be composed of many independent small reflectors; and with none

dominating, they each return radar signals of about the same power. Swerling I models

fluctuate slowly— from scan‐to‐scan (Figure 2.3a). Each pulse return is highly corre‐

lated. Swerling II models, meanwhile, fluctuate quickly— varying from pulse‐to‐pulse. It

is claimed that observed data on airborne aircraft targets agree with Swerling Models I

and II.

Swerling III and IV models are described by the Chi‐squared distribution with

four degrees of freedom. They are conceptualized as many independent reflectors of

roughly equal amplitude with one dominating reflector (Figure 2.3b). The Swerling III

model varies slowly— from scan to scan while the Swerling IV model fluctuates

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Figure 2.3. Scattering for Different Swerling Targets. The Swerling models assumes each tar‐get composed of a large number of independent scatters. (a) Swerling I/II. For Swerling targets I and II, none of the independent scattering elements dominates:

their reflected powers is comparable to one another. (b) Swerling III/IV. For Swerling targets III and IV, there is a single dominant non‐fluctuating scatter

together with other smaller scatters.

: Many scatter elements; each of approximately equal amplitudes.

: One scattering element producing an amplitude much larger than other elements.

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quickly— from pulse to pulse. The Swerling III/IV model is said to well‐characterize a

frigate or corvette with one large target but with complex superstructures.

2.4.2 Clutter

Clutter is unwanted radar returns. They may be a echoes from the environ‐

ment— as from ground, mountain, sea— or objects of no importance— as a flock of

geese. Designation of clutter changes with the aims and designs of the radar system. A

doppler weather system, for instance, may regard radar returns from a storm cloud

with prime importance while the same cloud may only be interference to a civilian air

traffic control radar. The stochastic characteristic of clutter varies with the operating

parameters of the radar as well as the nature of the clutter source (Table 2.2).

Older radar systems had longer pulse repetition intervals and larger range bins.

It was observed that ground clutter was well characterized with a Gaussian distribu‐

tion. However, as technology improved, and shorter pulse repetition intervals gave

greater range resolution, large spikes in amplitude were observed in radar data. The

Table 2.2. Clutter Types

Model Application Equation

Gaussian Clouds

Weibull Clutter Land Clutter

K‐distribution Sea Clutter

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Gaussian distribution no longer fitted.

Studies during the late 70’s and early 80’s found that ground clutter were more

accurately characterized with a Weibull distribution. Unlike the Gaussian distribution,

the Weibull is characterized by two variables: a shaping coefficient and a mean coeffi‐

cient.

In contrast with ground clutter, sea clutter is highly correlated. The two‐

parameter k‐distribution well‐characterizes this.

2.4.3 Noise

Gaussian white noise results from random thermal motion in electronic sys‐

tems. Unlike target and clutter returns, noise is always present. As implied by its

name, Gaussian noise is characterized by the bell‐shaped Gaussian distribution.

2.4.4 Square Law Detector

The stochastic characteristic of received radar signals may change at the diode

detector. The diode conductance characteristic is shown figure 2.4. Depending on the

strength of the received signal, the detector may either operate in the conductance

square or in the linear region. Each offers its advantages. Marcum showed that if the

signal is low, lying below the conductance knee, the square law detector gave the best

results. Linear detectors, meanwhile, offers the greatest dynamic range and is best for

high signals. Mathematically, in analysis, the square law detector offers additional ad‐

vantages.

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Both Rayleigh and Gaussian PDF’s, passing through the square law detector,

become exponential distributions (Figure 2.5). Given the relative simplicity of integra‐

tion of the exponential function, this fact comes in handy in analytical treatments of

target detection and false alarm rates.

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Figure 2.5. Radar Return Statistics Before and After Square Law Detector . The square law detector changes stochastic PDF of the signal.. (a) Gaussian Distribution. For Swerling targets I and II, none of the independent scattering elements

dominates: their reflected powers is comparable to one another. (b) Exponential Distribution. For Swerling targets III and IV, there is a single dominant non‐fluctuating

scatter together with other smaller scatters.

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Linear Law

Figure 2.4. Diode Conductance Characteristic. Radar detection can either occur in the linear or the square region of the diode.

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2.5 Radar Signal Environment

A number of different simplified RADAR environments have been used to assess

and judge the efficacy and applicability of the various CFAR algorithm. Although no

one model captures the full complexity and nuance of actual operating conditions, they

each test important qualities of the CFAR algorithm: The homogeneous environment,

the multiple target environment, and the clutter edge environment.

2.5.1 Homogeneous

The homogeneous environment is the most straight‐forward: It assumes a sin‐

gle target against an independent and identically distributed (iid) noise or clutter

(Figure 2.6).

A homogeneous environment may arise when, for instance, the radar antenna

makes no detection, and noise of a uniform power dominates the range bins; or when

(a) (b)

Figure 2.6. Homogeneous Environment (a) There is just one target of power 25dB at range gate 37. The noise power is uniform at 5dB. (b) MATLAB simulation of homogeneous environment.

12

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Range Gates

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Target 1

Target 2

Clutter

(a) (b)

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100

Range Gates

Mag

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the clutter return over a great stretch of distance is of a constant environment (say, a

forest, mountain, or ocean).

The homogeneous model assesses the ability of various CFAR algorithms as the

simplest test for properly assessing its ability to estimate the noise/clutter mean in set‐

ting the threshold of detection.

2.5.2 Multiple Targets

Targets of interest, however, don’t always exist in isolation along the range

gates of a given doppler bin. By chance or design, targets may be within close proximity

of one another (Figure 2.6).

Some CFAR algorithms suffer serious performative decline in multiple target

cases. When the targets are within half a window length of one another, their high

powers, improperly elevates the estimated mean of the background noise/clutter.

Figure 2.7. Multiple Targets. (a) At range gates 37 and 42, there are targets of power 25dB. (b) MATLAB simulation of multiple targets with exponential distribution.

13

0 10 20 30 40 50 60 70 80 90 1000

5

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35

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Range Gates

Mag

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Clutter

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80

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Range Gates

Mag

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(a) (b)

When the resulting thresholds are set, they are higher than they lead to be, and result

in losses in probability of detection (PD).

2.5.3 Clutter Wall

In addition to one‐target assumption, the iid assumption of the background

noise/clutter may also be violated. Radar environments often undergo abrupt changes

in power— such as transitioning from clear land to a forest, or from clouds to the clear

sky. These sudden jumps in mean power may throw off the CFAR algorithm, and intol‐

erably increase the probability of false alarm following the upward clutter wall transi‐

tion or decrease the probability of detection right before the clutter wall (Figure 2.8).

Figure 2.8. Clutter Edge. (a) At range gate 41, there is a sudden jump in clutter power; from 5dB to 10dB. (b) MATLAB simulation of a clutter edge with exponential distribution.

14

2.6 About CFAR

The goal of CFAR— constant false alarm rate— algorithms is to set thresholds

high enough to limit false alarms to a tolerable rate, but low enough to allow target

detection. In the past, this differentiation between target and background noise and

clutter were done by human radar operators upon a radar display screen. However,

with the advent of computer technology, detection thresholding was able to be auto‐

mated, and thus bypassing much of the cost, inefficiencies, and limitations of a human

operator.

2.6.1 Probability of False Alarm

The probability of false alarm, PFA, is the chance that spikes in noise or clutter is

mistaken by the CFAR algorithm as a target. As noise and clutter distributions are con‐

tinuous, extending from amplitudes very close to zero to amplitudes extending infi‐

nitely outwards. No matter how high thresholds are set, there will always be a finite

chance of random noise or clutter exceeding that threshold. So rather than eliminating

false alarms all together (this would be impossible), the goal of CFAR algorithms is to

reliably estimate the mean noise, and scaling the estimated mean by a multiplier to

obtain the threshold set high enough to limit false alarm rate to a tolerably small rate.

In many operating environments, the shape of the noise/clutter is known a

prior. From off‐line computation or from the experience of the engineering, the prob‐

ability density function (PDF) of the noise can be known ahead of time. For instance,

land clutter is often‐characterized by the Weibull distribution and Clouds by the Gaus‐

15

sian distribution.

Knowing the distribution, then, a measurement of the mean may be properly

scaled with a threshold multiplier for a threshold that yields the desired probability of

false alarm (Figure 2.9)

So the question becomes: How to estimate the mean noise or clutter over sev‐

eral range bins? The answer: The CFAR window

x

Threshold Multiplier

Threshold

Noise Mean

(a)

(b)

Threshold Mean

Figure 2.9. Exponential Distribution and Thresholding. (a) Exponential PDF. (b) To get the threshold, the mean is scaled by a threshold multiplier.

Noise Power

Probab

ility Distrib

utio

n

Probability of False Alarm

16

2.6.2 The CFAR Window

To estimate the mean noise/clutter present in a specific range bin, other local

bins are used. Towards this end, most CFAR algorithms utilizes a moving window.

There are several components to the CFAR window (Figure 2.10).

The cell under test (CUT)— also called the test cell— is the range bin with which

the mean noise/clutter is estimated and with which the threshold is set.

To estimate the mean, references cells— cells local to the CUT— are used. Dif‐

ferent CFAR algorithms utilize different mathematical assessments and different deci‐

sions logical towards this end. Cell averaging CFAR, for instance, takes the mean of the

reference cells while Ordered Statistics CFAR orders the reference cells from smallest

to largest to take the kth largest as representative of the average noise.

Guard cells are optional and may be variable in length. A target, if present in

the CUT, may straddle consecutive; and this would lead to inaccurate estimates of the

noise. Guard cells then, would guarantee then, guard against this. Typically, one guard

cell is used to either side of the CUT.

Leading Window Lagging Window

Cell Under Test

Guard Cells

Figure 2.10. CFAR Window

17

2.6.3 Probability of Detection

The CFAR reference window is used to set the threshold (Figure 2.11). The

threshold helps differentiate between target a noise. If the signal power, for a given

range cell, lies underneath the threshold, a target is deemed absent. If the same signal

power exceeds the height of the threshold, then a target is declared present.

Since target powers are typically stochastic— with a Swerling I/II or Swerlign

III/IV distribution— the probability of detection is also stochastic. Around a mean

value, the received target power would fluctuate to larger than the mean or smaller

than the mean according to its characteristic PDF. Sometimes, it would flucturate as

low as to miss the mean. So regardless to how well the threshold is set for a given

probability of false alarm, there will always be the chance of missing a target, if it’s pre‐

sent. The probability of detection helps characterize the odds of detection over many

trial runs or over many looks.

0 10 20 30 40 50 60 70 80 90 1000

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30

Range Gates

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dB)

Threshold

Recieved Waveform

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5

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15

20

25

30

Range Gates

Mag

nitu

de (

dB)

Threshold

Recieved Waveform

Missed Detection

Detection!

(a) (b)

Figure 2.11. The Threshold. (a) Missed detection. The target power falls underneath the threshold. (b) Successful detection. Target power greater than threshold.

18

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Probability of Detection - Homogeneous Environment

Neyman-Pearson

CA-CFAR

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Probability of Detection - Homogeneous Environment

CFAR Loss

Figure 2.12. A Typical Probability of Detection Curve illustrating CFAR Loss

2.6.4 CFAR Loss

Although the PD is a good measure of CFAR performance, it is relative to operat‐

ing conditions like the signal to noise ratio (SNR). So rather than PD, CFAR loss is the

usual ruler used to judge CFAR performance.

Figure 12.12 shows a typical probability of detection curve. As the SNR in‐

creases, the probability of detection also increases. The Neyman‐Pearson detector

represents the theoretical best detection— if perfect knowledge of the mean noise is

known a priori. It is against the Neyman‐Pearson detector that the CFAR loss is calcu‐

lated.

CFAR loss is the SNR difference, for a given PD between the Neyman‐Pearson

detector and the CFAR algorithm in interest. For instance, in Figure 12.12, for a prob‐

ability of detection of 0.4, the CFAR loss is close to 1dB of SNR.

19

2.7 Monte Carlo Simulations

Frequently, desired probability of false alarms are very small— on the order of

10‐5 or 10‐6. Then, in MATLAB, during a CFAR numerical experiment, a great number of

trial runs— on the order of 108 or 109— are necessary before a reliable experimental

probability of false alarm can be ascertained. This often intolerably increases MATLAB

run‐time or computer memory demands. Monte Carlo Importance Sampling repre‐

sents a class of numerical techniques used to reliably simulate very rare events [3].

Rather than using the original distribution to simulate events, with Monte Carlo

Importance Sampling, a similar function but with a different mean is used, but with the

same threshold (Figure 2.13). Any successful hits, then, with the new similar function,

rather than having a weight of one, is fractionally weight with a pre‐derived equation

(Figure 2.14).

Th

2 2

1exp

xf x

2 2

1expm

m m

xf x

610 10FA

Th

Q f x dx P

Figure 2.13. Monte Carlo with Exponential Distribution.

20

2

2 2 2

2

2

1 1exp

: The weighing function for successful Monte Carlo "hits"

: Random varaible with uniform distribution

: Variance of the estimated exponential distribution

: V

m

m

m

w x x

w x

x

ariance of the Monte Carlo exponential distribution

Figure 2.14. Monte Carlo Weight Function for Exponential Distribution.

21

Chapter 3

Cell Averaging CFAR

Cell Averaging CFAR (CA‐CFAR) was developed early on, in 1947, by Howard

Finn [1]. Of all the CFAR algorithms, CA‐CFAR works best in a homogeneous environ‐

ment. With increasing window size, the probability of detection for CA‐CFAR ap‐

proaches that of the optimal Neyman‐Pearson detector.

However, when the homogeneous assumptions are violated, the performance

of CA‐CFAR rapidly deteriorates [4]. As a starting point in algorithmic development

and as a basis of comparison, the algorithm is studied here.

3.1 Implementation

In Cell Averaging CFAR, every reference cell is added together and multiplied to

the threshold multiplier (Figure 3.1). For Gaussian noise and a square law detector, the

Figure 3.1. Cell Averaging CFAR Block Diagram.

Σ Σ

Σ

x


Threshold

1

1NFAP

22

threshold multiplier is given by:

Variable α is the threshold‐multiplier, N is the total number of reference cells, and PFA

is the desired probability of false alarm.

Typically, PFA is designed to be very low, on the order of 10‐5 or 10‐6. The refer‐

ence window length, N, must be selected by the designer as a balance of performance

in homogeneous and heterogeneous environments.

A cell averaging threshold with PFA = 10‐4 and N = 32 is simulated for a single‐

target in a homogeneous environment (Figure 3.2). A single target Swerling I/II target is

in range bin 37 in homogeneous noise. SNR is set to 15dB. We see that the threshold,

1

1NFAP

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

Recieved Waveform

Figure 3.2. CA‐CFAR Threshold in Homogeneous Environment.

23

across the noise‐only range bin, makes good clearance from the noise. There is a suc‐

cessful detection at the target. To either sides of the target in range bin 37 are elevated

threshold plateau— these are artifacts from target presence. This is a characteristic of

CA‐CFAR. Although it poses no problems in the single‐target homogeneous environ‐

ment case, with multiple targets, it negatively effects PD.

3.2 Homogeneous Environment


The primary function of any CFAR is to set the threshold to maintain the desired

probability of false alarm. To confirm the workability of the algorithm in a homogene‐

ous environment, we performed 10,000 trials for various designed PFA (Figure 3.2). The

experimental PFA closely matched the expected theoretical results.

2 4 6 8 10 12 14 16 18 20 2210

-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

Theoretical PFA

Experimental PFA

Figure 3.3. CA‐CFAR Theoretical v. Experimental PFA.

24

3.2.2 Detection

Cell Averaging CFAR successfully sets thresholds to keep to desired false alarm

rates, but does this same threshold allow for good probability of detection? We first

look at CA‐CFAR in a homogeneous environment— with a single target in a background

of noise or clutter with uniform mean . The threshold was set for a single trial run in

figure 3.1. In figure 3.4, we see the probability of detection profile over a range of SNR.

For this data set, the CA‐CFAR window size was set to 32, and a million trial per SNR

was performed. The CA‐CFAR PD curve closely matches to the contours of the Neyman‐

Pearson detector. In figure 3.5, we see that the CFAR loss is much less than 1dB.

25

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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Neyman-Pearson

CA-CFAR

0 5 10 15 20 250

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5 10 15 200

0.5

1

1.5

SNR (dB)

CF

AR

Los

s (d

B)

CA-CFAR

5 10 15 200

0.5

1

1.5

SNR (dB)

CF

AR

Los

s (d

B)

Figure 3.4. CA‐CFAR Homogeneous Probability of Detection.

Figure 3.5. CA‐CFAR Homogeneous CFAR loss

26

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

Recieved Waveform

Figure 3.6. CA‐CFAR Threshold with 1 Masking Target

3.2.2 Multiple Targets

Typical to CA‐CFAR, as seen in figure 3.1, to either side of a detected targets are

elevated threshold plateaus. When another target falls within half a window length of

it, each target’s probability of detection significantly decreases (Figure 3.6). This phe‐

nomena is referred to as “target masking.”

Detection performance diminishes drastically with numbers of interfering tar‐

gets (Figure 3.7). For CA‐CFAR with N=32, at SNR=20dB, the PD for a single target is 0.9.

With one masking target, the PD drops to 0.7, and with three masking targets PD sinks

to 0.4. The masking effect of multiple targets worsens with increasing signal to noise

ratio (Figure 3.8). Whereas the CFAR loss for one interfering target is less than 2dB at

27

5 10 15 200

5

10

15

CF

AR

Los

s (d

B)

SNR (dB)

5 10 15 20

0

5

10

15

CF

AR

Los

s (d

B)

SNR (dB)

homogeneous

one masking targettwo masking targets

three masking targets

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0

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Neyman Pearson

homogeneous



Figure 3.7. CA‐CFAR Probability of Detection with 1 to 3 masking targets

Figure 3.8. CA‐CFAR CFAR Loss with 1 to 3 masking targets

28

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15

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25

30

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

Recieved Waveform

Figure 3.9. CA‐CFAR Threshold in Clutter Wall Transition.

SNR=10dB, at SNR=20dB, the CFAR loss increases to 6B.

3.4 Clutter Wall

The clutter wall poses two different, but complementary challenges for CA‐

CFAR. Target masking occurs at the base of the clutter edge, and probability of false

alarm rates increase drastically following a clutter wall edge.

3.4.1 Clutter Wall False Alarm

Figure 3.10 summarizes the false alarm rates for various distances before and

after the clutter wall. The pink dash line at 10‐4 indicates the designed false alarm rate.

29

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Number of Reference Cells in Clutter

Pro

babi

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of F

alse

Ala

rm

CA-CFAR

Figure 3.10. CA‐CFAR Clutter Wall Probability of False Alarm.

Simulated for N=32, the blue line at 16 demarcates when half the reference cells are

immersed in clutter.

PFA decreases drastically in the cells before the clutter wall. Having a false alarm

rate smaller than the designed rate, at these regions, isn’t a concern. The problem oc‐

curs following the clutter wall, at reference cells 16 and outwards in Figure 3.10. At ref‐

erence cell 16, PFA reaches 10‐2; a false alarm rate nearly 100 times greater than the

desired PFA at 10‐4! This degree of deviance from desired PFA is usually intolerable for a

radar system’s designs; and other CFAR techniques were designed, in part, to remedy

this deficiency.

30

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CA-CFAR

N/2 cells in clutter

Figure 3.11. CA‐CFAR Clutter Wall Probability of Detection

3.4.2 Clutter Wall Detection

The detection at clutter walls is complementary to its false alarm effects. With

CA‐CFAR, there is a gradual decrease of PD from before the wall to after it (Figure 3.11).

Before the wall, from reference cells 1 to 16, the diminishment in PD can be attributed

to a masking effect by the clutter region. The decrease in PD after the clutter wall can

be accounted by the clutter to noise ratio (CNR) being smaller than the SNR.

3.5 Summary: CA‐CFAR Advantages and Disadvantages

Although CA‐CFAR performs very well in homogeneous environments, its detec‐

tion performances suffers drastically with closely spaced multiple targets. At the clutter

wall, false alarm probabilities increase by as much as an order of 102. This renders the

algorithm all but useless in clutter wall environments.

31

3.6 A combat for Multiple Targets: Smallest‐Of Cell Averaging CFAR

Smallest‐Of Cell Averaging (SOCA) CFAR was developed to remedy Cell‐

Averaging CFAR’s deficiencies in multiple target situations. If a secondary target intol‐

erably increases the average power of the leading or lagging window, then SO‐CFAR

simply takes the smaller of the two. [5]. SOCA‐CFAR works by selecting the smaller of

the sum of the leading and lagging windows (figure 3.12). This procedure requires a

threshold multiplier different from CA‐CFAR:

As there is no analytical solution for αSO, it is to be solved analytically from the above

equation.

In the homogeneous case, the algorithm can successfully hold for the designed

probability of false alarm (Figure 3.13). Its Probability of Detection and CFAR loss is

only slightly worse than CA‐CFAR (Figures 3.14, 3.15). The difference can be accounted

Σ Σ

x


Threshold

Min

Figure 3.12. SOCA‐CFAR Block Diagram

1/2 2

0

112 22

2 / 2 / 2

NN k

SO SOFA

k

Nk

PN N

k

32

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10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

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alse

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rm

p FA

Theoretical PFA

Experimental PFA

by SOCA‐CFAR using only half its reference windows (the smaller of the lead and lag) to

estimate the noise or clutter mean.

In probability of detection, for the multiple‐target case, SOCA‐CFAR out‐

performs CA‐CFAR (Figure 3.16). SOCA‐CFAR follows the shape of the Neyman‐Pearson

detector. Improvement over CA‐CFAR begins past SNR = 10dB for SOCA‐CFAR. In both

PD and CFAR loss, SOCA‐CFAR is similar to Ordered Statistics CFAR (OS‐CFAR) in its ad‐

vantages over CA‐CFAR (Figure 3.17).

Figure 3.13. SOCA‐CFAR Experimental v. Theoretical PFA

33

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CA-CFAROS-CFAR

SOCA-CFAR

0 5 10 15 20 250

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CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFARSOCA-CFAR

5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

Figure 3.14. SOCA‐CFAR Homogeneous PD.

Figure 3.15. SOCA‐CFAR Homogeneous CFAR Loss

34

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CA-CFAROS-CFAR

SOCA-CFAR

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6

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9

10

CF

AR

Los

s (d

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SNR (dB)

CA-CFAR

OS-CFARSOCA-CFAR

5 10 15 200

1

2

3

4

5

6

7

8

9

10

CF

AR

Los

s (d

B)

SNR (dB)

Figure 3.16. SOCA‐CFAR 1 Masking Target Probability of Detection

Figure 3.17. SOCA‐CFAR 1 Masking Target CFAR Loss

35

Whereas SOCA‐CFAR showed resistance to target masking, it is especially vul‐

nerable to the clutter wall environment (Figure 3.18). Near the clutter edge transition,

the probability of false alarm is elevated by over 103 times over that of the design

point! This is over 10 times worse than CA‐CFAR.

In practice, this clutter wall weakness of SOCA‐CFAR renders it unpractical.

However, as we will see in the next chapter, the philosophy of SOCA‐CFAR may be use‐

ful if utilized in a environment‐specific fashion.

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

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alse

Ala

rm

CA-CFAR

OS-CFARSOCA-CFAR

Figure 3.18. SOCA‐CFAR Clutter Wall Probability of False Alarm

36

3.7 A Remedy for Clutter Walls: Greatest‐Of Cell Averaging CFAR

Greatest‐Of Cell Averaging (GOCA) CFAR was a method designed to avoid the

elevated false alarm problem of clutter walls [6]. The algorithm works by summing

each of the leading and lagging window and taking the larger of the two to be used

with the threshold multiplier (Figure 3.19).

The threshold multiplier can be solved from this equation:

Since an analytical equation doesn’t exist for αGO, it must be solved numerically.

In a homogeneous environment, the threshold multiplier was shown to be suc‐

cessfully maintaining the desired false alarm rate (Figure 3.20).

The probability of detection profile for GOCA‐CFAR in the homogeneous envi‐

ronment closely matches CA‐CFAR (Figure 3.21). The small additional CFAR loss of

Σ Σ

x


Threshold

Max

Figure 3.19. GOCA‐CFAR Block Diagram

1/2 /2 2

0

11 11 2 22

2 / 2 2 / 2 / 2

NN N k

GO GO GOFA

k

Nk

PN N N

k

37

2 4 6 8 10 12 14 16 18 2010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

FA

Theoretical PFA

Experimental PFA

Figure 3.20. GOCA‐CFAR Homogeneous Probability of False Alarm

GOCA‐CFAR can be accounted for by the fact that the CFAR only averages from half the

available reference windows (the leading or lagging) as opposed to CA‐CFAR that

makes use of the full set of reference windows.

In the masking target case, GOCA‐CFAR performs worse than CA‐CFAR (Figure

3.23). The algorithm suffers as much as a 2dB CFAR loss over CA‐CFAR (Figure 3.24). If a

masking target exists, it will either elevate the average of the leading or lagging win‐

dow. Since GOCA‐CFAR takes the larger of the two windows, it will always take the win‐

dow artificially elevated by the masking target.

It is in the clutter wall environment that GOCA‐CFAR vast outperforms that of

CA‐CFAR (Figure 3.25). At the clutter wall, GOCA‐CFAR only suffers around a 10‐fold

increase in PFA while CA‐CFAR suffers a 100‐fold increase. Over CA‐CFAR, this translates

38

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CA-CFAROS-CFAR

GOCA-CFAR

0 5 10 15 20 250

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CF

AR

Los

s (d

B)

SNR (dB)

g

CA-CFAR

OS-CFARGOCA-CFAR

5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

g

Figure 3.21 GOCA‐CFAR Homogeneous Probability of Detection

Figure 3.22 GOCA‐CFAR Homogeneous CFAR Loss

39

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CA-CFAROS-CFAR

GOCA-CFAR

0 5 10 15 20 250

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5 10 15 200

2

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10

12

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFARGOCA-CFAR

5 10 15 200

2

4

6

8

10

12

CF

AR

Los

s (d

B)

SNR (dB)

Figure 3.23. GOCA‐CFAR Probability of Detection with 1 Masking Target

Figure 3.24. GOCA‐CFAR CFAR Loss with 1 Masking Target

40

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

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of F

alse

Ala

rm

CA-CFAR

OS-CFARGOCA-CFAR

Figure 3.25. GOCA‐CFAR Clutter Wall PFA

to roughly a 10‐fold improvement for GOCA‐CFAR.

GOCA‐CFAR is, in many ways, the complement to SOCA‐CFAR. Whereas SOCA‐

CFAR improves over CA‐CFAR in multiple targets but suffers in clutter wall, GOCA‐CFAR

suffers in multiple targets but improves in clutter wall.

41

Chapter 4

Variability‐Index CFAR

Variability‐Index CFAR (VI‐CFAR) dynamically switches between CA‐CFAR, SOCA‐

CFAR , and GOCA‐CFAR depending on the mean and the distribution of cells in the lead‐

ing and the lagging windows [7]. This approach captures particular advantages of each

of the three CFAR approaches while avoiding their respective weaknesses.

4.1 Implementation

For each of the leading window and the lagging window, the mean and the vari‐

ability index is computed.

The variability index (VI) is a second‐order statistics that is closely related to an

estimate of the shape parameter. The greater the variability index, the greater the vari‐

ability in relatives magnitudes of the reference cells. When, for instance, a leading

window contains the high powers of targets, the variability would differentially weigh

Figure 4.1. VI‐CFAR Block Diagram

Σ Σ

x


Threshold

Σ(·)2

Logic

Σ(·)2

42

the target power over that of the lower powers of the noise and magnitude, resulting

in a larger variability index.

The variability index may be calculated by this equation:

Here, Xi is the power of reference cell i. The summation is taken over all cells Xi of ei‐

ther the leading or the lagging window.

Once the VI is computed, the algorithm then makes a determination of whether

the leading and/or the lagging windows are variable. If the variability index is smaller

than variability index constant KVI, then the window is declared not variable. If the vari‐

ability index is larger than KVI, then the window is declared variable:

The Mean Ratio (MR) is a measure of how different the leading and lagging win‐

dow means are. It is computed as a ratio of the summations of the reference cells of

each window:

If the MR falls within bounds defined by the mean ratio constant, KMR, then the leading

and lagging window means are declared the same:

ii AA

B ii B

XX

MRX X

2

12

1

n

ii

n

ii

XVI n

X

Not Variable

VariableVI

VI

VI K

VI K

1 Same MeanMR MRK MR K

43

Leading Window Variable?

Lagging Window Variable?

Different Mean?

VI‐CFAR Adaptive Threshold

Equivalent CFAR Method

No No No αN ∙ ΣAB CA CFAR

No No Yes αN ∙ max(ΣA,ΣB) GOCA CFAR

Yes No — αN ∙ ΣB CA CFAR

No Yes — αN ∙ ΣA CA CFAR

Yes Yes — αN ∙ min(ΣA,ΣB) SOCA CFAR

Table 4.1. VI‐CFAR Decision Logic

Illustrative Figure

Figure 4.2 Case I

Figure 4.3 Case II

Figure 4.4 Case III

Figure 4.5 Case IV

Figure 4.6 Case V

If not, then they are declared as different means:

Once the variability and mean determinations had been made, then, according

to the logic of Table 4.1, CFAR is performed with either both the leading and lagging

windows, one of them, the smaller of the two, or the larger of the two.

If only a half‐window is chosen— either only the leading or lagging window—

then the threshold multiplier becomes

If both leading and lagging windows are chosen, as in case I, then the threshold multi‐

plier becomes:

A little more can be explained of the different cases of Table. 4.1

1 or Different MeanMR MRMR K MR K

1

/2/2 1N

N FAP

1

1NN FAP

44

Figure 4.2. VI‐CFAR ‐ Case I

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Range Gates

Pow

er (

linea

r)

Target 1

ClutterCell Under Test

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Range GatesP

ower

(lin

ear)

Signal

Cell Under Test

Lagging Window: ‐ Not Variable ‐ Mean similar to Lagging

Leading Window: ‐ Not Variable ‐ Mean similar to Leading

CFAR to use:

Cell Averaging CFAR with all reference cells

Situation:

Homogeneous Environment

4.2 VI‐CFAR Decision Logic

Case I: Same means, both nonvaraible

When both the leading and lagging windows are nonvaraible, they aren’t likely

to contain targets; and when both windows have similar means, then the reference

window probably doesn’t straddle a clutter edge. The cell under test, then, is most

likely in a homogeneous environment. VI‐CFAR, thus, uses CA‐CFAR.

The threshold is set by summing all reference cells and using the CA‐CFAR

threshold multiplier for length N:

1

1NN FAP

45

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

160

180

200

Range GatesP

ower

(lin

ear)

Signal

Cell Under Test

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

Range Gates

Pow

er (

linea

r)

Target 1

Target 2Target 3

Cell Under Test

Lagging Window: ‐ Not Variable ‐ Mean similar to Leading

Leading Window: ‐ Variable ‐ Mean similar to Lagging

CFAR to use:

Cell Averaging CFAR with leading window only

Example Situation:

Interfering targets in the leading window.

Case II: Leading Window Variable, Lagging Window not Variable

A variable leading window is indicative of high spikes of power, and with a rea‐

sonable signal to noise ratio, these spikes are most likely the result of targets. The av‐

erage power, then, of the leading window wouldn’t be indicative of the noise/clutter

contained. In this case, VI‐CFAR switches to using CA‐CFAR with the lagging window

alone.

As the lagging window is nonvaraible, it is likely to contain noise/clutter only. To

set the threshold, all cells in the lagging window is summed and multiplied to the

threshold multiplier for window length N/2:

Figure 4.3. VI‐CFAR ‐ Case II

1

/2/2 1N

N FAP

46

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

160

180

200

Range Gates

Pow

er (

linea

r)

Signal

Cell Under Test

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

Range Gates

Pow

er (

linea

r)

Target 1

Target 2Target 3

Cell Under Test

Lagging Window: ‐ Not Variable ‐ Mean similar to Leading


CFAR to use:

Cell Averaging CFAR with lagging window only

Example Situation:

Interfering targets in the lagging window.

Case III: Lagging Window Variable, Leading Variable Nonvariable

This is the converse situation of Case II. When only the lagging window is vari‐

able and not the leading, the lagging window is likely to contain targets and the leading

noise/clutter only.

In this case, the threshold is set by summing only the reference cells of the

leading window and multiplied to a CA‐CFAR threshold multiplier for window length

N/2:

Figure 4.4. VI‐CFAR ‐ Case III

1

/2/2 1N

N FAP

47

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

160

180

200

Range GatesP

ower

(lin

ear)

Signal

Cell Under Test

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

Range Gates

Pow

er (

linea

r)

Target 1

Target 2Target 3

Cell Under Test

Lagging Window: ‐ Variable ‐ Mean similar to Leading


CFAR to use:

Smallest of Cell Averaging

Example Situation:

Interfering targets in both leading and lagging window

Case IV: Both Windows Variable

When both the leading and lagging windows are variable, then both windows

very likely contains targets. Whichever window used is likely to result in target mask‐

ing. VI‐CFAR, then, seeks to minimize the masking by taking the smaller of the sums of

the leading and the lagging windows. The threshold is set by multiplying this smaller

sum to a CA‐CFAR threshold multiplier for window length N/2:

Figure 4.5. VI‐CFAR ‐ Case IV

1

/2/2 1N

N FAP

48

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Range GatesP

ower

(lin

ear)

Noise/Clutter

Cell Under test

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

Range Gates

Pow

er (

linea

r)

Noise/Clutter

Cell Under test

Leading Window: ‐ Not Variable ‐ Mean different from leading

Leading Window: ‐ Not Variable ‐ Mean different from lagging

Use:

Greatest of Cell Averaging

Example Situation:

Near clutter wall. Can be on either edge of the clutter edge.

Case V: Not Variable but Different Mean

When both the leading and lagging variables are nonvariable but different in

mean, then the cell under test is very likely near a clutter wall. The goal for VI‐CFAR , in

this case, is to set the threshold by the higher edge of the clutter wall.

The threshold is set by taking the larger of the sums of the leading and lagging

window and multiplied to the CA‐CFAR threshold multiplier for window length of N/2:

Figure 4.6. VI‐CFAR ‐ Case

1

/2/2 1N

N FAP

49

4.3 Homogeneous Environment


In a homogeneous environment, VI‐CFAR largely reverts to CA‐CFAR and uses

the CA‐CFAR threshold multiplier (Figure 4.8). In the threshold plot, we see that the

threshold of VI‐CFAR almost entirely overlaps with that of CA‐CFAR. The discrepancy

near and around the target— with CA‐CFAR displaying its distinctive elevated threshold

plateaus near the target and VI‐CFAR with no such display— can be accounted for by

the decision logic of VI‐CFAR. The CUTs N/2 cells in front of and N/2 cells behind the

target will interpret the target as a masking target, and switch to using only the lagging

and the leading windows respectively in the computation of CA‐CFAR.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFARVI-CFAR

Recieved Waveform

Figure 4.8. VI‐CFAR Threshold in Homogeneous Environment.

50

2 4 6 8 10 12 14 16 18 20 2210

-7

10-6

10-5

10-4

10-3

10-2

10-1


Pro

babi

lity

of F

alse

Ala

rm

FA

Theoretical PFA

Experimental PFA

Figure 4.9. VI‐CFAR Theoretical v. Experimental PFA in Homogeneous Environment

Not surprisingly, in the homogeneous environment, VI‐CFAR successfully holds

the probability of false alarm like CA‐CFAR (Figure 4.9).


Across various signal to noise ratios, we see that in the homogeneous environ‐

ment, CA‐CFAR consistently slightly outperforms VI‐CFAR (Figure 4.10). This slight dis‐

crepancy is to be expected since, due to the random distribution of the noise/clutter,

VI‐CFAR may very rarely misinterpret the homogeneous environment as a Type II, III,

IV, or V environment. In these cases, VI‐CFAR switches to using only its leading or lag‐

ging windows in the estimation of the noise.

In CFAR loss, this translates to about a 0.25 dB loss across all SNR’s.

51

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Neyman-Pearson

CA-CFAROS-CFAR

VI-CFAR

0 5 10 15 20 250

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etec

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5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFARVI-CFAR

5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

Figure 4.9. VI‐CFAR Probability of Detection Curve

Figure 4.10. VI‐CFAR CFAR Loss in Homogeneous Environment

52

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFARVI-CFAR

Recieved Waveform

Figure 4.11. VI‐CFAR Threshold 1 Masking Target.

4.4 Masking Targets

With the variability index, VI‐CFAR may reliably predict the presence of masking

targets in the leading window, the lagging window, or both windows and appropriately

adjust to use of the lagging window or the leading window.

The VI‐CFAR threshold was simulated for a two‐target environment (Figure

4.11). It successfully detects both targets. In the graph, OS‐CFAR also allowed for de‐

tection of both targets. CA‐CFAR, in contrast, detects only the bin 42 target; setting the

threshold too high for the bin 37 target.

VI‐CFAR’s resistance to target masking in two‐target environments is clear in its

probability of detection curve (Figure 4.12). For most signal to ratio values, the prob‐

ability of detection for VI‐CFAR is only slightly lower than that of OS‐CFAR. CA‐CFAR,

53

0 5 10 15 20 250

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0.5

0.6

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Neyman-Pearson

CA-CFAROS-CFAR

VI-CFAR

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

Figure 4.12. VI‐CFAR Probability of Detection with 1 Masking Target

meanwhile, shows it usual loss in probability of detection wit target masking.

VI‐CFAR itself, however, begins to suffer masking effects when there are more

than one interfering target (Figure 4.13). Whereas the CFAR loss of either homogenous

and one‐interfering VI‐CFAR is less than 1dB, the CFAR loss for two and three interfer‐

ing targets increase to as much as 10dB (Figure 4.14). How can we account for such dis‐

parate results?

When two masking targets are randomly placed around a CUT, part of the time,

there will be one in the leading window and one in the lagging window. In this case, no

matter which window is selected, there will be a masking effect for the CUT (Figure

4.15). In this case, VI‐CFAR only mollifies the effect by switching to SOCA‐CFAR.

On the other hand, when both interfering targets are together, either in the

54

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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0 5 10 15 20 250

0.1

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0.5

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of D

etec

tion


Neyman Pearson

homogeneous



5 10 15 200

2

4

6

8

10

12

CF

AR

Los

s (d

B)

SNR (dB)

5 10 15 20

0

2

4

6

8

10

12

CF

AR

Los

s (d

B)

SNR (dB)

homogeneous



Figure 4.13. VI‐CFAR Probability of Detection Curve with varying Numbers of Masking Target

Figure 4.14. VI‐CFAR CFAR Loss Curve with varying Numbers of Masking Target

55

leading or the lagging window, VI‐CFAR can entirely avoid their masking effects by se‐

lecting the other window. The differences can be profound. In Figure 4.16, the prob‐

ability of detection is computed for different distributions of two masking targets.

56

0 5 10 15 20 250

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Neyman-Pearson

Mean PD

Targets 1,3

Target 2

Figure 4.16. VI‐CFAR Probability of Detection with Three Masking Targets

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

Range Gates

Pow

er (

linea

r)

Target 3: When the CUT reaches here, VI‐CFAR takes only the leading win‐dow for CA‐CFAR

Target 1: When the CUT reaches here, VI‐CFAR takes only the lagging window for CA‐CFAR

Target 2: When the CUT reaches here, VI‐CFAR uses SOCA‐CFAR.

Figure 4.15. Three Targets

57

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFARVI-CFAR

Recieved Waveform

Figure 4.17. VI‐CFAR Threshold in Clutter Wall Transition.

4.5 Clutter Wall


When VI‐CFAR detects a clutter wall— a type V case— it switches to using to

GOCA‐CFAR. The algorithm uses the greater of the sum of the leading and lagging win‐

dow in its cell averaging CFAR computation.

In setting the threshold this way there are two consequences: Leading into the

clutter wall, the VI‐CFAR threshold slopes upwards earlier and more steeply than CA‐

CFAR, and after the clutter wall, the VI‐CFAR has already stopped sloping and achieved

a sufficiently high threshold (Figure 4.17).

This effects translates to good maintenance of the probability of false alarm

(Figure 4.18). We see that VI‐CFAR suffers less than a 10‐fold increase in PFA.‐‐ from 10‐4

Greater threshold clearance for VI‐CFAR VI‐CFAR threshold increases

earlier and more steeply than CA‐CFAR or OS‐CFAR

58

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

CA-CFAR

OS-CFARVI-CFAR

Figure 4.18. VI‐CFAR Clutter Wall Probability of False Alarm.

to less than 10‐3. Cell averaging and Ordered Statistics CFAR, in contrast, suffers a 100‐

fold increase in PFA.


The drawback of VI‐CFAR’s clutter wall strategy is a greater drop probability of

detection than CA‐CFAR or OS‐CFAR (Figure 4.19). We see that for all cells under clut‐

ter, the probability of detection for VI‐CFAR is less than that of CA‐CFAR and OS‐CFAR.

This is a result of two separate effects: threshold set too high before the clutter wall,

and threshold set correctly after the clutter wall.

Due to VI‐CFAR’s GOCA‐CFAR clutter wall strategy, the threshold set before the

clutter wall is set by the greater of the leading and lagging window sums. This means

59

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

CA-CFAR

OS-CFAR

VI-CFAR

N/2 cells in clutter

Figure 4.19. VI‐CFAR Clutter Wall Probability of Detection.

that the set threshold will be based, in part, on the larger powers of the clutter region.

This elevates the threshold much higher than the level necessary to maintain the de‐

sired PFA of 10‐4.

After the clutter wall, when more than 16 cells are under clutter, all three

CFARs experience a low probability of detection. This is to be expected. In these re‐

gions of the clutter wall, instead of noise, the target is in a clutter background. Since

the clutter to noise ratio (CNR) is much smaller than the signal to noise ratio (SNR), the

probability of detection for any CFAR would be lower.

The detection of CA‐CFAR and OS‐CFAR is higher than VI‐CFAR after the clutter

wall because their thresholds were set too low— so low that the PFA is much greater

than the designed 10‐4 point.

60

1 2 3 k Nx x x x x

ˆ

OS kT a x

CHAPTER 5

ORDERED STATISTICS CFAR

Cell‐averaging CFAR was the first of the CFAR methods. Assuming homogene‐

ous Gaussian white noise, through a sliding window, CA‐CFAR estimates the noise level

of the cell under test, CUT, through an arithmetic sum of N adjacent cells . In heteroge‐

neous environments, however, CA‐CFAR performance rapidly degrades.

Ordered Statistics CFAR, OS‐CFAR, had been developed to compensate for

much of CA‐CFAR’s pitfalls. Rather than taking the arithmetic mean of adjacent cells,

OS‐CFAR ranks the N adjacent cells from smallest to largest and multiplies the kth larg‐

est cell to the appropriate threshold multiplier to set the threshold level.

5.1 Implementation

Each reference cell is sorted from smallest to biggest:

The kth‐largest cell is multiplied with the corresponding threshold multiplier to get the

desired threshold:

The kth‐ranked cell could be the median ranked value or even the largest value. As

simulations will show, the larger x(k), the smaller the CFAR‐loss in homogeneous envi‐

ronment but the less the resistance to target masking and clutter‐edge transition.

61

Sort

1

2

k

N

x

x

x

x

x


Threshold

Figure 5.1. OS‐CFAR Block Diagram

!

! !

N N

k k N k

/2

/2

!

! !

COS

FA COS

N kNP

k N k M

1D OSOS

Through a square‐law detector and in a background with Gaussian White Noise,

the threshold‐multiplier αOS may be found from this equation:

With the binomial given by:

Here, N is the total number of reference cells; k is the rank of the representative cell

used, PFA is the desired probability of false alarm, αOS is the threshold multiplier and Γ(∙)

is the Gamma function.

The theoretical probability of detection is given by:

62

5.2 Homogeneous


Using a window size of 32 and a kth rank of 22 (k chosen for 0.7∙N), thresholds

multipliers were used to set and simulated for probability of false alarms between 10‐1

and 10‐7 (Figure 4.2). The experimental probability of false alarm conformed with theo‐

retical expectation (Figure 5.2).

5.2.2 Detection

The ordered statistics threshold, in general, is higher than the cell averaging

threshold for the same designed probability of false‐alarm (Figure 5.2). This higher

2 4 6 8 10 12 14 16 18 20 2210

-7

10-6

10-5

10-4

10-3

10-2

10-1


Pro

babi

lity

of F

alse

Ala

rm

FA

Theoretical PFA

Experimental PFA

Figure 5.2. OS‐CFAR Homogeneous PFA.

63

threshold results from slightly smaller reliability of an ordered statistics estimations as

versus a mean estimation of the average noise (Figure 5.3). In the probability of detec‐

tion curve, this translates to a slightly smaller PD (Figure 5.4).

Amongst ordered statistics curves of the same window length, In the CFAR loss

curve, the kth ranked cell chosen for the threshold multiplier also affects performance.

In a homogeneous environment, in general, the higher the kth value used, the slightly

better the probability of detection and the smaller the CFAR loss.

In the probability of detection curve, for a window size of 32, k=0.9N (or k=29)

gave a lower CFAR loss about half a dB lower than that of k=0.5N (Figure 5.5). The re‐

sults for 0.9N was also about 0.1 dB better than that of 0.7N. All results, however, had

at least half a dB of CFAR loss as much as the cell averaging CFAR of the same window

length.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFARRecieved Waveform

Figure 5.3. OS‐CFAR Threshold in Homogeneous Environment.

64

Figure 5.4. OS‐CFAR Homogeneous Probability of Detection

Figure 5.5. OS‐CFAR Homogeneous CFAR Loss

5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFAR k=0.5 NOS-CFAR k=0.7 N

OS-CFAR k=0.9 N

5 10 15 200

0.5

1

1.5

CF

AR

Los

s (d

B)

SNR (dB)

0 5 10 15 20 250

0.1

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0.7

0.8

0.9

1

Neyman-Pearson

CA-CFAR


OS-CFAR k=0.9 N

0 5 10 15 20 250

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1

65

kth cell

N‐k‐1 buffer cells

smallest largest

N reference cells

Figure 5.6. OS‐CFAR Resistance to Multiple Targets

5.3 Multiple Targets

Ordered statistics CFAR offers natural resistance to target masking. With cell

averaging CFAR, any target within the reference cell gets added to be multiplied with

the threshold multiplier. If the signal to noise ratio is 10dB, the estimated noise may

be as much as 50% greater.

With ordered statistics, since only the ranked kth value is selected, interfering

targets don't’ directly affect the threshold multiplier. Since target powers are typically

much greater than noise or clutter power, when ranked, they find themselves below

the k‐value. The cells larger than the kth cell, in the reference then, act as a kind of

buffer against target masking. If the number of buffer cells exceed the number of inter‐

fering targets, then OS‐CFAR can escape directly influence of masking targets. Also,

even when there are less targets than buffer cells, the less targets there are less indi‐

rect influence the kth cell will feel from the target.

Figure 5.7 shows an example of a OS‐CFAR threshold in the multiple target case.

As is evident, OS‐CFAR draws a threshold low enough to detect both targets. CA‐CFAR,

66

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Range Gates

Mag

nitu

de (

dB)

CA-CFAR


Figure 5.7 OS‐CFAR Threshold with 1 Masking Target.

on the other hand, misses detection of one target.

With one masking target, the probability of detection curve of OS‐CFAR follows

the contour of the Neyman‐Pearson detector (Figure 5.8). The cell averaging detection

begins to suffer major detection degradation beyond an SNR of about 10 dB. The selec‐

tion of different k values, 0.5N, 0.7N, or 0.9N (or k of 16, 22, and 29 respectively for a

window size of N=32) led to no major differences in detection probability. The CFAR

losses for each k value selection were no more than 0.5dB different from one another

(Figure 5.9).

In contrast to the homogeneous case, k=0.7N produced better detection and

CFAR loss values than k=0.9N. The reason, most likely is that the buffer cells allowed by

k=0.7N were more adequately insulating than that of k=0.9N.

67

Figure 5.8. OS‐CFAR Probability of Detection with 1 Masking Target

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of Detection

CF

AR

Los

s (d

B)

Neyman-Pearson

CA-CFAR


OS-CFAR k=0.9 N

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.9. OS‐CFAR CFAR Loss with 1 Masking Target

5 10 15 200

0.5

1

1.5

2

2.5

3

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFAR k=0.5 N

OS-CFAR k=0.7 N

OS-CFAR k=0.9 N

5 10 15 200

0.5

1

1.5

2

2.5

3

CF

AR

Los

s (d

B)

SNR (dB)

68

With three masking targets, the probability of detection for k=0.9N became

noticeably worse than that of k=0.9N while k=0.5N continued to give only a very

slightly less detection curve than k=0.9N (Figure 5.10). In CFAR loss, k=0.9N got to as

much as nearly 2dB worse for SNR of 25dB than k=0.7N. The CFAR loss difference be‐

tween 0.7N and 0.5N was about 0.25dB throughout the range of SNRs simulated.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)

Neyman-Pearson

CA-CFAR


OS-CFAR k=0.9 N

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.10 OS‐CFAR Probability of Detection with 3 Masking Target

69

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFAR k=0.5 N

OS-CFAR k=0.7 N

OS-CFAR k=0.9 N

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CF

AR

Los

s (d

B)

SNR (dB)

Figure 5.11. OS‐CFAR CFAR Loss with 3 Masking Target

70

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Range Gates

Mag

nitu

de (

dB)

CA-CFAR


Figure 5.12. OS‐CFAR in Clutter Wall Threshold.

5.5 Clutter Wall

Ordered Statistics CFAR performs similarly to cell averaging CFAR at the clutter

wall. Both algorithms only gradually slope upwards their thresholds approaching the

clutter wall, and past the clutter edge, the thresholds of both algorithms get as low as

to frequently allow false alarms (Figure 5.12).

OS‐CFAR algorithms selected for different ranked kth values perform differently

at the clutter wall. In general, the greater the k, the less the probability of false alarm

will increase at the clutter edge. Cell averaging CFAR for window length 32, at the clut‐

ter wall, suffers roughly a 100‐fold increase in false alarm rate: from the designed 10‐4

to nearly 10‐2 (Figure 5.13). Ordered Statistics CFAR for k=0.5*N (k=16) and k=0.7*N

71

Figure 5.13. OS‐CFAR Clutter Wall Probability of False Alarm

(k=22) actually perform worse than CA‐CFAR— suffering a false alarm rate that’s nearly

a thousand times more than what was desired! OS‐CFAR for k=0.9*N, in contrast, im‐

proved upon CA‐CFAR, with a false alarm rate that’s only between 10 times and 100

times the designed rate of 10‐4.

Probability of detection performance at the clutter wall for the various k value

is reversed from that for probability of false alarm. The smaller the k value, the less the

target masking effect before the clutter wall. K=0.5N gave better detection results than

k=0.7N, and k=0.9N.

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

CA-CFAR

OS-CFAR k=0.5*NOS-CFAR k=0.7*N

OS-CFAR k=0.9*N

72

Figure 5.14. OS‐CFAR Clutter Wall Probability of Detection

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

CA-CFAR

OS-CFAR k=0.5*N

OS-CFAR k=0.7*N

OS-CFAR k=0.9*N

73

Chapter 6

Ordered Statistics Greatest‐Of CFAR

Although OS‐CFAR works well in avoiding the drawbacks of target masking, it

costs more than CA‐CFAR to implement and it suffers more false alarms than CA‐CFAR

at the clutter wall. Ordered Statistics Greatest‐Of CFAR (OSGO‐CFAR) represents an at‐

tempt to kill both problems with one stone.

6.1 Implementation

OSGO‐CFAR operates similarly to OS‐CFAR in its sorting of reference cells. The

difference, though, is that in OSGO CFAR, the N reference cells of the leading window

and the lagging window are sorted independently, into two N/2 sublists (Figure 6.1).

The kth largest cells of the two sublists are compared, and the larger of the two are

Rank Order

x


Threshold

Rank Order

1

2

/2

k

N

y

y

y

y

1

2

/2

k

N

x

x

x

x

Max

Figure 6.1. OSGO‐CFAR Block Diagram

22

2 22

0 0

1 12 2 2 2

12

N Nk k N k j i

OSGOFA

j i

N N NN j ik k

P kN N j iik j i

used with the threshold multiplier to set the threshold.

The threshold multiplier is given by this equation:

As no analytical expression for threshold multiplier α exists, it must be solved numeri‐

cally.

Figure 6.2. OSGO‐CFAR Threshold in Homogeneous Environment.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFAROSGO-CFAR

Recieved Waveform

6.2 Homogeneous

In the homogeneous environment, OSGO‐CFAR sets its threshold very similarly

to OS‐CFAR. Neither method gives the elevated threshold plateaus around targets.

75

0 5 10 15 20 2510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

p FA

Theoretical PFA

Experimental PFA

Figure 6.3. OSGO‐CFAR Homogeneous PFA.


To verify that the OSGO CFAR works in limiting noise/clutter to the desired

probability of false alarm, the algorithm was simulated to a number of desired PFA—

from 10‐1 to about 10‐7. The measured probability of false alarm conformed well with

expected results.

6.2.2 Detection

In probability of detections, OSGO‐CFAR performs comparatively with OS‐CFAR

(Figure 6.4). Like OS‐CFAR, OSGO‐CFAR gives a CFAR loss greater than that of OS‐CFAR.

For CFAR algorithms of equal window length, OSGO‐CFAR always gave higher CFAR loss

76

Figure 6.4. OSGO‐CFAR Homogeneous PD

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)

Neyman-PearsonCA-CFAR

OS-CFAR k=0.7*N

OSGO-CFAR k=0.5*N/2

OSGO-CFAR k=0.7*N/2OSGO-CFAR k=0.9*N/2

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

than OS‐CFAR. Also like OS‐CFAR, the greater the ranked kth value taken for the thresh‐

old multiplier, the less the CFAR loss (Figure 6.5). The CFAR loss for OSGO‐CFAR

k=0.7*N/2 and k=0.9*N/2 were within 0.2 dB of the CFAR loss for k=0.7*N of OS‐CFAR.

77

Figure 6.5. OSGO‐CFAR Homogeneous CFAR Loss

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5C

FA

R L

oss

(dB

)

SNR (dB)

CA-CFAR

OS-CFAR k=0.7*N


OSGO-CFAR k=0.9*N/2

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5C

FA

R L

oss

(dB

)

SNR (dB)

78

kth cell

N/2‐k‐1 buffer cells

smallest largest

N/2 reference cells

Figure 6.6. OSGO‐CFAR Resistance to Multiple Targets

6.3 Multiple Targets

Using ordered statistics, OSGO‐CFAR, like OS‐CFAR, offers strong resistance

against target masking. However, due to the kth cell being taken from the larger of the

sorted leading or lagging window, the amount of multiple target resistance is dimin‐

ished (Figure 6.6). Since OSGO‐CFAR only takes k from half a full window, there is only

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFAROSGO-CFAR

Recieved Waveform

Figure 6.7. OSGO‐CFAR Threshold with 1 Masking Target.

79

Figure 6.8. OSGO‐CFAR PD with 1 Masking Target

N/2‐k‐1 buffer cells— as opposed to N‐k‐1 for OS‐CFAR.

The threshold set for OSGO‐CFAR looks a bit like that of OS‐CFAR (Figure 6.7).

Like OS‐CFAR, OSGO‐CFAR avoids much of the masking problem of CA‐CFAR.

The probability of detection across SNR=2 and 24 shows little difference be‐

tween OS‐CFAR and the test OSGO‐CFAR algorithms. In terms of CFAR loss, we see that

the OS‐CFAR and OSGO‐CFAR algorithms, in the single interfering target environment,

performs within 0.5dB of one another.

One thing to note, though, is that OSGO‐CFAR for k=0.9*N/2 begins to suffer

greater CFAR loss than OSGO‐CFAR for k=0.7*N/2 beyond an SNR of about 5dB. It is

clear that for a window size of 32, the number of buffer cells offered by k=N/2*0.9 only

offers enough masking resistance for one interfering target.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)


OS-CFAR k=0.7*N

OSGO-CFAR k=0.5*N/2


0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)

80

Figure 6.9. OSGO‐CFAR CFAR loss with 1 Masking Target

5 10 15 200

0.5

1

1.5

2

2.5C

FA

R L

oss

(dB

)

SNR (dB)

CA-CFAR

OS-CFAR k=0.7*N


OSGO-CFAR k=0.9*N/2

5 10 15 200

0.5

1

1.5

2

2.5C

FA

R L

oss

(dB

)

SNR (dB)

At three masking targets, we see that the probability of detection for OSGO‐

CFAR of k=0.9*N/2 decreases precipitously (Figure 6.10). Though slightly lower than

the OS‐CFAR curve, the detection curve for both OSGO k=0.7*N/2 and k=0.5*N/2 both

followed the contour of the Neyman‐Pearson detector. The CFAR loss for these two

curves were about 2 dB— about half a dB more than OS‐CFAR. (Figure 6.11).

81

Figure 6.10. OSGO‐CFAR PD with 3 Masking Target

Figure 6.11. OSGO‐CFAR CFAR loss with 3 Masking Target

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CF

AR

Los

s (d

B)

SNR (dB)

CA-CFAR

OS-CFAR k=0.7*N


OSGO-CFAR k=0.9*N/2

5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CF

AR

Los

s (d

B)

SNR (dB)

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)


OS-CFAR k=0.7*N

OSGO-CFAR k=0.5*N/2


0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CF

AR

Los

s (d

B)

82

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Range Gates

Mag

nitu

de (

dB)

CA-CFAR

OS-CFAROSGO-CFAR

Recieved Waveform

Figure 6.12. OSGO‐CFAR Threshold in Clutter Wall Transition.

6.5 Clutter Wall

In previous chapters, we’ve seen how cell averaging CFAR and ordered statistics

CFAR both suffer from elevated false alarm rates near the clutter wall. Their PFA be‐

come elevated by as much as 100 times the set PFA. Can OSGO‐CFAR do better?

Figure 6.12 shows the OSGO‐CFAR threshold at a clutter wall. compared to CA‐

CFAR and OS‐CFAR, OSGO‐CFAR has a much better clearance at the clutter edge. This

translates to significant improvements for keeping desired false alarm rates (Figure

6.12).

The probability of false alarm were generated over a million trials for OSGO‐

CFAR of window length N=32 and kth rank of N/2*0.5, N/2*0.7, and N/2*0.9. At worst,

83

Figure 6.12. OSGO‐CFAR Clutter Wall PFA

0 5 10 15 20 25 3010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Pro

babi

lity

of F

alse

Ala

rm

CA-CFAROS-CFAR k=0.5*N

OS-CFAR k=0.7*N

OSGO-CFAR k=0.5*N/2


OSGO‐CFAR only suffers a ten‐fold increase form the designed PFA of 10‐4. This is a ten‐

fold improve over cell‐averaging CFAR. These results are also comparable to those of

the clutter wall false alarm performances of VI‐CFAR (chapter 4).

The clutter wall probability of detection performance for OSGO‐CFAR can be

found in figure 6.13.

84

Figure 6.13. OSGO‐CFAR Clutter Wall PD

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of D

etec

tion

CA-CFAR

OS-CFAR k=0.5*N

OS-CFAR k=0.7*NOSGO-CFAR k=0.5*N/2

OSGO-CFAR k=0.7*N/2

OSGO-CFAR k=0.9*N/2

85

Chapter 7

Conclusions and Final Assessment

In the course of our research, we’ve studied a number of CFAR algorithms—

each approaching the same problem of noise/clutter mean estimation and threshold

setting through different techniques. Cell Averaging CFAR (CA‐CFAR) though optimal in

the homogeneous environment, suffers major performative drop‐off in the heteroge‐

neous environments: with multiple targets and the clutter wall environment.

Although Ordered Statistics CFAR (OS‐CFAR), Smallest‐Of Cell Averaging CFAR

(SO‐CFAR), and Greatest‐Of Cell Averaging CFAR (GOCA‐CFAR) each remedy one of the

two heterogeneous conditions, they fail to fully remedy the other.

Two algorithms, Ordered Statistics Greatest‐Of CFAR (OSGO‐CFAR) and Variabil‐

ity Index CFAR (VI‐CFAR), balance performance in each environment— homogeneous,

multiple targets, and clutter wall— to be practically feasible.

Figure 7.1 illustrates a practical CFAR environment. From it, we can see that

only OSGO‐CFAR and VI‐CFAR detect the homogeneous target, the multiple target, and

make the clutter wall without a false alarm.

86

87

88

5010

015

020

025

030

00510152025

Cel

l Ave

ragi

ng

Ord

ered

Sta

tistic

s

Var

abili

ty In

dex

Ord

ered

Sta

tistic

s G

reat

est O

f

Figure 7.1. C

FAR Obstacle Course

REFERENCES

[1] Finn, H.M., and Johnson, R.S.: ‘Adaptive detection mode with threshold control as a

function of spatially sampled clutter estimates’, RCA Rev., 1968, vol.29, no.3, pp.

414–464

[2] Richards, M.A. Fundamentals of Radar Signal Processing. New York; McGraw‐Hill.

2005

[3] Kang, K.E. Radar System: Analysis, Design, and Simulation. Chapter 9: Monte Carlo

Method and Function Integration. Pages 247‐260. Artech House 2008

[4] Gandhi, P.P.; Kassam, S.A.; , "Analysis of CFAR processors in homogeneous back‐

ground," Aerospace and Electronic Systems, IEEE Transactions on , vol.24, no.4,

pp.427‐445, Jul 1988

[5] Trunk, G.V.; , "Range Resolution of Targets Using Automatic Detectors," Aerospace

and Electronic Systems, IEEE Transactions on , vol.AES‐14, no.5, pp.750‐755, Sept.

1978

[6] Hansen, V.G.; Sawyers, J.H.; , "Detectability Loss Due to "Greatest Of" Selection in a

Cell‐Averaging CFAR," Aerospace and Electronic Systems, IEEE Transactions on ,

vol.AES‐16, no.1, pp.115‐118, Jan. 1980

[7] Smith, M.E.; Varshney, P.K.; , "Intelligent CFAR processor based on data variability,"

Aerospace and Electronic Systems, IEEE Transactions on , vol.36, no.3, pp.837‐847,

Jul 2000

[8] Rohling, H.; , "Radar CFAR Thresholding in Clutter and Multiple Target Situations,"

Aerospace and Electronic Systems, IEEE Transactions on , vol.AES‐19, no.4, pp.608‐

621, July 1983

[9] Elias‐Fuste, A.R.; de Mercado, M.G.G.; de los Reyes Davo, E.; , "Analysis of some

modified ordered statistic CFAR: OSGO and OSSO CFAR," Aerospace and Electronic

Systems, IEEE Transactions on , vol.26, no.1, pp.197

[10]Cao, T.‐T.V.; , "Constant false‐alarm rate algorithm based on test cell information,"

Radar, Sonar & Navigation, IET , vol.2, no.3, pp.200‐213, June 2008

89

Appendix A

MATLAB Functions

A.1 Radar Return Generation

function [P]=simRangeData(N,iN,S,iS) % [P]=simRangeData(N,iN,S,iS) produces a vector P that corresponds to % the power over range gates with complex gaussian white noise and % Swerling I/II targets. % %Vector N determines the power of the complex gaussian white noise across % Range Cells controlled by indices iN. % % Vector S determines the power of a Swerling I/II target in Range Cells % dictated by vector iS pwrSig=10.^(S/10); %Converting signal power from dB to linear pwrNoise=10.^(N/10); %Converinting noise power from dB to linear nRangeGate=iN(end); %The total number of range gates %%GENERATING NOISE voltNoise_I=zeros(nRangeGate,1); %Pre-allocating for speed voltNoise_Q=zeros(nRangeGate,1); %Pre-allocating for speed I=length(iN); %Number of range gate segments with different noise power iLead=1; %This loop produces the voltage waveform for the gaussian white noise. %Every range gate segment between iLead and iLag will have its own power of %N(i) for i=1:I iLag=iN(i); voltNoise_I(iLead:iLag)=sqrt(pwrNoise(i)/2)*randn(iLag-iLead+1,1); %voltNoise_I Channel voltNoise_Q(iLead:iLag)=sqrt(pwrNoise(i)/2)*randn(iLag-iLead+1,1); %voltNoise_Q Channel iLead=iLag; end noise=voltNoise_I+j*voltNoise_Q; %Voltage waveform of the noise %%GENERATING TARGET RETURN signal=zeros(size(noise)); %Pre-allocating for speed I=length(iS); %The total number of signal returns for i=1:I %Power of the signal follows an exponential distribution pwrSwerling=exprnd(pwrSig(i),1,1); %Voltage waveform fo the signal signal(iS(i))=sqrt(pwrSwerling/2)+j*sqrt(pwrSwerling/2); end

90

%%GENERATING OVERALL SIGNAL data=signal+noise; %Overall voltage waveform P=abs(data).^2; %Power of the overall return

91

A.2 Monte Carlo Simulations

function [expPfa]= monteCarloFalseAlarm(T,noisePwr) % This function performs a Monte Carlo Importance Sampling to determine % experimental probabilities of false alarm % % INPUTS % T - This is the threshold % % noisePwr - the noise power in linear scales % % OUTPUTS % expPfa: The experimental probability of false alram % % ASSUMPTIONS % Exponential distribution of noise/clutter % % AUTHORSHIP % James Jen % Cal Poly Pomona % % UPDATE HISTORY % May 28th, 2011 - created % nCFAR=size(T,2); nRangeGate=length(T); %The number of range gates monteCarloPwr=noisePwr*20; %Monte Carlo mean 20 times greater iCFARwin=1:nRangeGate; iCFARwin=iCFARwin(T(:,1)~=0); %indicies for non-zero thresholds nTrial=length(iCFARwin); %Exponential distribution with the Monte carlo power pwrMonteCarlo=exprnd(monteCarloPwr,nRangeGate,nCFAR); %The Monte carlo weight weight=monteCarloPwr/noisePwr*exp(-(1/noisePwr-1/monteCarloPwr)*pwrMonteCarlo); %with iCFAR win, we avoid the zero thresholsd at the ends of the range gate weight=weight(iCFARwin,:); T=T(iCFARwin,:); pwrMonteCarlo=pwrMonteCarlo(iCFARwin,:); %Logical indicies for when Monte Carlo power greater than threshold iFalseAlarm=T<pwrMonteCarlo; %When monte carlo weight greater, we use the corresponding weights to %determine false alarm. falseAlarm=zeros(size(weight)); falseAlarm(iFalseAlarm)=weight(iFalseAlarm); %The overall probability of false alarm expPfa=sum(falseAlarm)/nTrial;

92

function [Pd]=Pd_inf(Pfa,SNRdB) SNR=10.^(SNRdB./10); Pd=Pfa.^(1./(1+SNR));

A.3 Neyman Pearson Detector

93

A.1 Cell Averaging CFAR

function [T,iCFARwin]=CA_CFAR1(varargin) % [T,iCFARwin]=CA_CFAR(Pfa,N,nGuardCell, signal) produces a vector T, % that is the threshold produced by the Cell-Averaging CFAR (CA-CFAR) % method from the vector, signal, that is a one-dimension of range % power. % % % [T,iCFARwin]=CA_CFAR(Pfa,N,signal) produces a vector T, that is the % threshold produced by the Cell-Averaging CFAR (CA-CFAR) method from the % vector, signal, that is a one-dimension of range power. nGuard-Cell % defaults to one % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % nGuardCell- the number of guard cells to either side of the test cell % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detec-tor, will % have an exponential distribution % % SOURCE: % Threshold Multiplier equation taken from Chapter 7 of M.A. Rich-ard's % "Fundamentals of Radar Signal Processing" %

94

% AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment % May 30th, 2011 - Vectorize operations switch nargin case 3 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; nGuardCell=1; case 4 Pfa=varargin{1}; N=varargin{2}; nGuardCell=varargin{3}; signal=varargin{4}; otherwise error('Unexpected Number of Inputs') end %CFAR Computations and Parameters a=N*(Pfa^(-1/N)-1); %Threshold multiplier %Number of Range Gates L=length(signal); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iRefWin=index; i=N/2+(1:2*nGuardCell+1); iRefWin(:,i)=[]; %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i);

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%Summing the content of each reference window B=sum(signal(iRefWin),2)/N; %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=a*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experi-mental % trial with one test cell else %Row index for the Cell Under Test (CUT) iCUT=N/2+nGuardCell+1; %Rwo index cor the Reference Cells iRefCell=1:N+2*nGuardCell+1; notRefCell=N/2+(1:1+2*nGuardCell); iRefCell(notRefCell)=[]; %Calculating the threshold B=sum(signal(:,iRefCell),2)/N; %sum of each cell in refer-ence win T=a*B; %Multiplying, a, the thresh-old mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin]=GOCA_CFAR1(varargin) % [T,iCFARwin]=GOCA_CFAR(Pfa,N,k,signal) produces a vector T, that is the % threshold produced by the Greatest-Of Cell-Averaging CFAR (GOCA-CFAR) % method from the vector, signal, that is a one-dimension of range power. % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Threshold Multiplier equation taken from Chapter 7 of M.A. Richard's % "Fundamentals of Radar Signal Processing" % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment % switch nargin case 3 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; nGuardCell=1; case 4 Pfa=varargin{1}; N=varargin{2}; nGuardCell=varargin{3}; signal=varargin{4}; otherwise error('Unexpected Number of Inputs') end L=length(signal); %Total number of range bins %There is no analytical function to directly calculate the

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%thershold-multiplier, a. Instead, we will compute a look-up table and %interpolate the value. a_set=0.1:0.05:50; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); sum_term=zeros(N/2,1); for k=0:N/2-1 sum_term(k+1,1)=nchoosek(N/2-1+k,k)*(2+a/(N/2))^-k; end sum_term=sum(sum_term); P(i,1)=2*((1+a/(N/2))^(-N/2)-(2+a/(N/2))^(-N/2)*sum_term); end %Threshold Multiplier: a=interp1(P,a_set,Pfa); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iLead=index(:,1:N/2); iLag=index(:,N/2+2*nGuardCell+2:end); %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %Performaing SO-CFAR leadWin=signal(iLead); sumLead=sum(leadWin,2); lagWin=signal(iLag); sumLag=sum(lagWin,2); B=max(sumLead,sumLag)/(N/2); %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=a*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else %Rwo index cor the Reference Cells

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leadWin=signal(:,1:N/2); sumLead=sum(leadWin,2); lagWin=signal(:,N/2+2*nGuardCell+2:end); sumLag=sum(lagWin,2); B=max(sumLead,sumLag)/(N/2); %Calculating the threshold T=a*B; %Multiplying, a, the threshold mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin]=SOCA_CFAR1(varargin) % [T,iCFARwin]=SOCA_CFAR(Pfa,N,nGuardCell, signal) produces a vector T, % that is the threshold produced by the Smalelst-Of Cell-Averaging CFAR % (SOCA-CFAR) method from the vector, signal, that is a one-dimension of % range power. % % % [T,iCFARwin]=SOCA_CFAR(Pfa,N,signal) produces a vector T, that is the % threshold produced by the Smallest-Of Cell-Averaging CFAR (SOCA-CFAR % method from the vector, signal, that is a one-dimension of range power. % nGuardCell defaults to one % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % nGuardCell- the number of guard cells to either side of the test cell % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Threshold Multiplier equation taken from Chapter 7 of M.A. Richard's % "Fundamentals of Radar Signal Processing" % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment % May 30th, 2011 - Vectorize operations switch nargin case 3 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; nGuardCell=1; case 4 Pfa=varargin{1}; N=varargin{2};

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nGuardCell=varargin{3}; signal=varargin{4}; otherwise error('Unexpected Number of Inputs') end %Number of Range Gates L=length(signal); %Finding the threshold multiplier numerically a_set=0.1:0.1:150; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); sum_term=zeros(N/2,1); for k=0:N/2-1 sum_term(k+1,1)=nchoosek(N/2-1+k,k)*(2+a/(N/2))^-k; end sum_term=sum(sum_term); P(i,1)=2*(2+a/(N/2))^(-N/2)*sum_term; end a=interp1(P,a_set,Pfa); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iLead=index(:,1:N/2); iLag=index(:,N/2+2*nGuardCell+2:end); %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %Performaing SO-CFAR leadWin=signal(iLead); sumLead=sum(leadWin,2); lagWin=signal(iLag); sumLag=sum(lagWin,2); B=min(sumLead,sumLag)/(N/2); %Outuptted thershold

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T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=a*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else %Rwo index cor the Reference Cells leadWin=signal(:,1:N/2); sumLead=sum(leadWin,2); lagWin=signal(:,N/2+2*nGuardCell+2:end); sumLag=sum(lagWin,2); B=min(sumLead,sumLag)/(N/2); %Calculating the threshold T=a*B; %Multiplying, a, the threshold mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin,selection,selectionName]=VI_CFAR1(varargin) % [T,iCFARwin]=VI_CFAR(Pfa,N,signal,K_VI,K_MR) produces a vector T, that % is the threshold produced by the Varaibility Index CFAR (CA-CFAR) method % from the vector, signal, that is a one-dimension of range power. % % [T,iCFARwin]=VI_CFAR(Pfa,N,signal) defaults to pre-determiend K_VI and % K_MR values % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % K_VI - Variability Index Constant % % K_MR - Mean Ratio Constant % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Based on "Intelligent CFAR Process Based on Data Variability" by M.E. % Smith and P.K. Varshney published in IEEE Transactions on Aerospace and % Electronic Systems Vol. 36 No. 30 July 2000 % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % Updated: % May 23rd, 2011 - Fixed code mistakes % May 28th, 2011 - Code Comment % May 30th, 2011 - Vectorization, Speedup % switch nargin %User may use pre-defined varaibilty index and mean ratio constants case 3 Pfa=varargin{1}; %desired probability of false alarm N=varargin{2}; %total number of reference windows signal=varargin{3}; %vector of range bin to be thresholded K_VI=4.76; %default variaiblity index constant K_MR=1.806; %default mean ration constant nGuardCell=1;

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%User may use pre-defined varaibilty index and mean ratio constants case 4 Pfa=varargin{1}; %desired probability of false alarm N=varargin{2}; %total number of reference windows nGuardCell=varargin{3}; signal=varargin{4}; %vector of range bin to be thresholded K_VI=4.76; %default variaiblity index constant K_MR=1.806; %default mean ration constant %User may enter varaibility index and mean ratio constants case 5 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; K_VI=varargin{4}; %user-entered K_MR=varargin{5}; %user-entered nGuardCell=1; case 6 Pfa=varargin{1}; N=varargin{2}; nGuardCell=varargin{3}; signal=varargin{4}; K_VI=varargin{5}; %user-entered K_MR=varargin{6}; %user-entered otherwise error('Unexpected Number of Inputs'); end %Number of Range Gates L=length(signal); %Finding the threshold multiplier numerically a_N=(Pfa^(-1/N)-1); %Threshold multiplier a_N2=(Pfa^(-1/(N/2))-1); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iLead=index(:,1:N/2); iLag=index(:,N/2+2*nGuardCell+2:end); %Linear index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %Summing and Squaring leadWin=signal(iLead); sumLead=sum(leadWin,2); sumSquareLead=sum(leadWin.^2,2);

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lagWin=signal(iLag); sumLag=sum(lagWin,2); sumSquareLag=sum(lagWin.^2,2); %VARIABILITY INDEX leadVI=N/2*sumSquareLead./(sumLead.^2); lagVI=N/2*sumSquareLag./(sumLag.^2); leadVariable=leadVI>K_VI; lagVariable=lagVI>K_VI; %MEAN RATIO meanRatio=sumLead./sumLag; diffMean=(K_MR^-1>meanRatio)|(meanRatio>K_MR); oneVariable=xor(leadVariable,lagVariable); bothVariable=leadVariable & lagVariable; noneVariable=(~leadVariable) & (~lagVariable); B=zeros(nCUT,1); %LEAD WINDOW iLead=oneVariable & lagVariable; B(iLead)=sumLead(iLead)*a_N2; %LAG WINDOW iLag=oneVariable & leadVariable; B(iLag)=sumLag(iLag)*a_N2; %SO-CFAR iSO=bothVariable; B(iSO)=min(sumLead(iSO),sumLag(iSO))*a_N2; %GO-CFAR iGO=diffMean & noneVariable; B(iGO)=max(sumLead(iGO),sumLag(iGO))*a_N2; %CA-CFAR iCA=~diffMean & noneVariable; B(iCA)=(sumLead(iCA)+sumLag(iCA))*a_N; %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else nTrial=length(signal); iLead=1:N/2; iLag=N/2+2+2*nGuardCell:N+1+2*nGuardCell; %Summing and Squaring leadWin=signal(:,iLead); sumLead=sum(leadWin,2); sumSquareLead=sum(leadWin.^2,2);

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lagWin=signal(:,iLag); sumLag=sum(lagWin,2); sumSquareLag=sum(lagWin.^2,2); %VARIABILITY INDEX leadVI=N/2*sumSquareLead./(sumLead.^2); lagVI=N/2*sumSquareLag./(sumLag.^2); leadVariable=leadVI>K_VI; lagVariable=lagVI>K_VI; %MEAN RATIO meanRatio=sumLead./sumLag; diffMean=(K_MR^-1>meanRatio)|(meanRatio>K_MR); oneVariable=xor(leadVariable,lagVariable); bothVariable=leadVariable & lagVariable; noneVariable=(~leadVariable) & (~lagVariable); T=zeros(nTrial,1); %LEAD WINDOW iLead=oneVariable & lagVariable; T(iLead)=sumLead(iLead)*a_N2; %LAG WINDOW iLag=oneVariable & leadVariable; T(iLag)=sumLag(iLag)*a_N2; %SO-CFAR iSO=bothVariable; T(iSO)=min(sumLead(iSO),sumLag(iSO))*a_N2; %GO-CFAR iGO=diffMean & noneVariable; T(iGO)=max(sumLead(iGO),sumLag(iGO))*a_N2; %CA-CFAR iCA=~diffMean & noneVariable; T(iCA)=(sumLead(iCA)+sumLag(iCA))*a_N; end selectionName{1}='CA'; selectionName{2}='GO'; selectionName{3}='SO'; selectionName{4}='lead'; selectionName{5}='lag'; selection=zeros(L,1); selection(iCA)=1; selection(iGO)=2; selection(iSO)=3; selection(iLead)=4; selection(iLag)=5; %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

106

function[T,iCFARwin]=OS_CFAR1(varargin) % [T,iCFARwin]=OS_CFAR(Pfa,N,k,nGuardCell signal) produces a threshold T, % from signal to limit to probability of false larm Pfa using Ordered % Statistics CFAR (OS-CFAR). N is the number of reference cells in the % CFAR window and k is the cell rank used to generate the threshold % multiplier. nGuardCell is the number of guard cells to either side of % CUT % % % [T,iCFARwin]=OS_CFAR(Pfa,N,k,nGuardCell signal) produces a threshold T, % from signal to limit to probability of false larm Pfa using Ordered % Statistics CFAR (OS-CFAR). N is the number of reference cells in the % CFAR window and k is the cell rank used to generate the threshold % multiplier. nGuardCell, the number of guard cells to either side of CUT % defaults to 1. % % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % k - the index used in computation of the threshold multiplier used in % computing the threshold, T. % % nGuardCell - the number of guard cells to either side of CUT % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Threshold Multiplier equation taken from Chapter 7 of M.A. Richard's % "Fundamentals of Radar Signal Processing" % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment % switch nargin %If "k" is omitted as an input, it defaults to round(0.7*N)

A.8 Ordered Statistics CFAR

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case 4 Pfa=varargin{1}; N=varargin{2}; k=varargin{3}; signal=varargin{4}; nGuardCell=1; %If "k" is entered as an input: case 5 Pfa=varargin{1}; N=varargin{2}; k=varargin{3}; nGuardCell=varargin{4}; signal=varargin{5}; otherwise error('Unexpected Number of Inputs'); end nLead=N/2; %size of lead window nLag=N/2; %size of lag window %Number of Range Gates L=length(signal); %The proper threshold multiplier is determined numerically: a_set=0.1:0.05:100; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); P(i)=k*nchoosek(N,k)*gamma(a+N-k+1)*gamma(k)/gamma(a+N+1); end a=interp1(P,a_set,Pfa); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iRefWin=index; i=N/2+(1:2*nGuardCell+1); iRefWin(:,i)=[]; %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %sorting and taking the kth value sortedPwrSignal=sort(signal(iRefWin),2); B=sortedPwrSignal(:,k);

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%Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=a*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else %Matrix index for the Cell Under Test (CUT) iCUT=N/2+nGuardCell+1; %Rwo index cor the Reference Cells iRefCell=1:N+2*nGuardCell+1; notRefCell=N/2+(1:1+2*nGuardCell); iRefCell(notRefCell)=[]; %sorting and taking the kth value sortedPwrSignal=sort(signal(:,iRefCell),2); B=sortedPwrSignal(:,k); T=a*B; %Multiplying, a, the threshold mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin]=OSGO_CFAR1(varargin) % [T,iCFARwin]=OSGO_CFAR(Pfa,N,k,signal) produces a vector T, that is the % threshold produced by the Order Statistics Greatest-OF CFAR (OSGO-CFAR) % method from the vector, signal, based on the k-th ranked index. % % [T,iCFARwin]=OSGO_CFAR(Pfa,N,k,signal) produces a vector T, that is the % threshold produced by the Order Statistics Greatest-OF CFAR (OSGO-CFAR) % method from the vector, signal. "k" is defaulted to round(0.6*N/2) % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % k - the index used in computation of the threshold multiplier used in % computing the threshold, T. % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Elias-Fuste, A.R.; de Mercado, M.G.G.; de los Reyes Davo, E.; , % "Analysis of some modified ordered statistic CFAR: OSGO and OSSO CFAR," % Aerospace and Electronic Systems, IEEE Transactions on , vol.26, no.1, % pp.197-202, Jan 1990 % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment % May 30th, 2011 - Vectorized switch nargin %If "k" is omitted as an input, it defaults to round(0.7*N) case 4 Pfa=varargin{1}; N=varargin{2}; k=varargin{3}; signal=varargin{4}; nGuardCell=1; %If "k" is entered as an input: case 5

A.9 Greatest‐Of Ordered Statistics CFAR

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Pfa=varargin{1}; N=varargin{2}; k=varargin{3}; nGuardCell=varargin{4}; signal=varargin{5}; otherwise error('Unexpected Number of Inputs'); end %Number of Range Gates L=length(signal); %Numerically determining the threshold multiplier, a: a_set=1:0.05:100; Pfa_set=ones(size(a_set)); i=(0:N/2-k)'; j=(0:N/2-k)'; I=i*ones(1,length(i)); J=ones(length(j),1)*j'; for ia=1:length(a_set) a=a_set(ia); A=2*k^2*nchoosek(N/2,k)^2; B=factorial((N/2-k)*ones(length(j)))./factorial(J)./factorial(N/2-k-J); C=factorial((N/2-k)*ones(length(j)))./factorial(I)./factorial(N/2-k-I); D=(-1).^(N-2*k-J-I)./(N/2-I); E=gamma(N-J-I).*gamma(a+1)./gamma(N-J-I+a+1); Pfa_set(ia)=A*sum(sum(B.*C.*D.*E)); end a=interp1(Pfa_set,a_set,Pfa); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iRefWin=index; i=N/2+(1:2*nGuardCell+1); iRefWin(:,i)=[]; iLead=iRefWin(:,1:N/2); iLag=iRefWin(:,N/2+1:N); %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %sorting and taking the kth value

111

sortedLead=sort(signal(iLead),2); sortedLag=sort(signal(iLag),2); B=max(sortedLead(:,k),sortedLag(:,k)); %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=a*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else %Matrix index for the Cell Under Test (CUT) iCUT=N/2+nGuardCell+1; %Rwo index cor the Reference Cells iLead=1:N/2; iLag=N/2+2*nGuardCell+2:N+1+2*nGuardCell; sortedLead=sort(signal(:,iLead),2); sortedLag=sort(signal(:,iLag),2); B=max(sortedLead(:,k),sortedLag(:,k)); T=a*B; %Multiplying, a, the threshold mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin,A,B]=S_CFAR(varargin) % [T,iCFARwin]=S_CFAR(Pfa,N,signal,N_T,a,B_CA,B_0,B_1) produces a vector % T, that is the threshold produced by the Switching CFAR (S-CFAR) method % from the vector, signal, that is a one-dimension of range power. % % [T,iCFARwin]=S_CFAR(Pfa,N,signal,N_T,a) defaults to CA-CFAR threshold % multiplier for window size N for B_CA, B_0, and B_1 % % [T,iCFARwin]=S_CFAR(Pfa,N,signal) defaults to CA-CFAR threshold % multiplier for window size N for B_CA, B_0, and B_1; and to N_T of N/2 % and a=0.5 % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % N_T - S0 Threshold. When the number of members of set S0 exceed N_T, the % members of S0 are used in cell averaging. % % a - Test cell multiplier used in comparison and sorting between set S0 % and set S1 % % B_CA - Threshold multiplier used when switched to Cell Averaging % % B_0 - Threshold multiplier used for set S0 % % B_1 - Threshold multiplier used for set S1 % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Cao, T.-T.V.; , "Constant false-alarm rate algorithm based on test cell % information," Radar, Sonar & Navigation, IET , vol.2, no.3, pp.200-213, % June 2008 % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % UPDATE HISTORY: % May 28th, 2011 - Code Comment %

A.10 Switching CFAR

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switch nargin case 3 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; N_T=N/2; a=0.5; aCA=N*(Pfa^(-1/N)-1); a0=aCA; a1=aCA; case 5 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; N_T=varargin{4}; a=varargin{5}; aCA=N*(Pfa^(-1/N)-1); a0=aCA; a1=aCA; case 7 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; N_T=varargin{4}; aCA=varargin{5}; a0=varargin{6}; a1=varargin{7}; aCA=a0; otherwise error('Unexpected Number of Inputs'); end nGuardCell=1; %size of the guard cell L=length(signal); T=zeros(size(signal)); if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iRefWin=index; i=N/2+(1:2*nGuardCell+1); iRefWin(:,i)=[]; %Lienar index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); cCUT=signal(iCUT); %Summing the content of each reference window mRefWin=signal(iRefWin);

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sumRefWin=sum(mRefWin,2)/N; %Subset iS0=a*repmat(cCUT,1,N)>mRefWin; S0=zeros(nCUT,N); S0(iS0)=mRefWin(iS0); nS0=sum(iS0,2); sumS0=sum(S0,2)./nS0; cBigNT=nS0>=N_T; cSmallNT=~cBigNT; B=zeros(nCUT,1); B(cSmallNT)=sumRefWin(cSmallNT); B(cBigNT)=sumS0(cBigNT); %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=aCA*B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else %Row index for the Cell Under Test (CUT) iCUT=N/2+nGuardCell+1; %Rwo index cor the Reference Cells iRefCell=1:N+2*nGuardCell+1; notRefCell=N/2+(1:1+2*nGuardCell); iRefCell(notRefCell)=[]; %Calculating the threshold mRefCell=signal(:,iRefCell); sumRefCell=sum(signal(:,iRefCell),2)/N; %sum of each cell in refer-ence win nTrial=length(signal); cCUT=signal(:,iCUT); iS0=a*repmat(cCUT,1,N)>mRefCell; S0=zeros(nTrial,N); S0(iS0)=mRefCell(iS0); nS0=sum(iS0,2); sumS0=sum(S0,2)./nS0; iBigNT=nS0>=N_T; iSmallNT=~iBigNT; B(iBigNT)=sumS0(iBigNT); B(iSmallNT)=sumRefCell(iSmallNT); T=aCA*B; %Multiplying, a, the threshold mult end %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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function[T,iCFARwin,selection,selectionName]=VI_CFAR1(varargin) % [T,iCFARwin]=VI_CFAR(Pfa,N,signal,K_VI,K_MR) produces a vector T, that % is the threshold produced by the Varaibility Index CFAR (CA-CFAR) method % from the vector, signal, that is a one-dimension of range power. % % [T,iCFARwin]=VI_CFAR(Pfa,N,signal) defaults to pre-determiend K_VI and % K_MR values % % INPUT: % Pfa - the desired probability of False Alarm % % N - the total number of CFAR windows-- with N/2 in the leading window % and N/2 in the lagging window. % % Signal - the power of the recieved waveform after having passed through % a square law detector. It is upon this that the threshold is set. % % K_VI - Variability Index Constant % % K_MR - Mean Ratio Constant % % OUTPUT: % T is the threshold set according to the functional inputs. % % iCFARwin are the indicies corresponding to the range gates where a CFAR % threshold is produced. When the first few and last few range bins are % used as test cells, they don't allow enough room for the entire CFAR % reference window. Subsequently, they are ignored and scipped. % % ASSUMPTION: % signal noise/clutter, after passing through a square law detector, will % have an exponential distribution % % SOURCE: % Based on "Intelligent CFAR Process Based on Data Variability" by M.E. % Smith and P.K. Varshney published in IEEE Transactions on Aerospace and % Electronic Systems Vol. 36 No. 30 July 2000 % % AUTHORSHIP: % James Jen % Cal Poly Pomona % % Updated: % May 23rd, 2011 - Fixed code mistakes % May 28th, 2011 - Code Comment % May 30th, 2011 - Vectorization, Speedup % switch nargin %User may use pre-defined varaibilty index and mean ratio constants case 3 Pfa=varargin{1}; %desired probability of false alarm N=varargin{2}; %total number of reference windows signal=varargin{3}; %vector of range bin to be thresholded K_VI=4.76; %default variaiblity index constant K_MR=1.806; %default mean ration constant nGuardCell=1;

A.11 Switching Variability‐Index CFAR

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%User may use pre-defined varaibilty index and mean ratio constants case 4 Pfa=varargin{1}; %desired probability of false alarm N=varargin{2}; %total number of reference windows nGuardCell=varargin{3}; signal=varargin{4}; %vector of range bin to be thresholded K_VI=4.76; %default variaiblity index constant K_MR=1.806; %default mean ration constant %User may enter varaibility index and mean ratio constants case 5 Pfa=varargin{1}; N=varargin{2}; signal=varargin{3}; K_VI=varargin{4}; %user-entered K_MR=varargin{5}; %user-entered nGuardCell=1; case 6 Pfa=varargin{1}; N=varargin{2}; nGuardCell=varargin{3}; signal=varargin{4}; K_VI=varargin{5}; %user-entered K_MR=varargin{6}; %user-entered otherwise error('Unexpected Number of Inputs'); end %Number of Range Gates L=length(signal); %Finding the threshold multiplier numerically a_N=(Pfa^(-1/N)-1); %Threshold multiplier a_N2=(Pfa^(-1/(N/2))-1); %If range gate inputs is a vector, then perform CFAR along entire range %gate: if min(size(signal))==1 %Organizing linear index into a matrix iWindow=1:N+1+2*nGuardCell; nCUT=L-N-2*nGuardCell; shiftCFAR=(0:nCUT-1).'; index=repmat(iWindow,nCUT,1)+repmat(shiftCFAR,1,N+1+2*nGuardCell); %Linear Index for the reference window iLead=index(:,1:N/2); iLag=index(:,N/2+2*nGuardCell+2:end); %Linear index for the CUT i=N/2+nGuardCell+1; iCUT=index(:,i); %Summing and Squaring leadWin=signal(iLead); sumLead=sum(leadWin,2); sumSquareLead=sum(leadWin.^2,2);

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lagWin=signal(iLag); sumLag=sum(lagWin,2); sumSquareLag=sum(lagWin.^2,2); %VARIABILITY INDEX leadVI=N/2*sumSquareLead./(sumLead.^2); lagVI=N/2*sumSquareLag./(sumLag.^2); leadVariable=leadVI>K_VI; lagVariable=lagVI>K_VI; %MEAN RATIO meanRatio=sumLead./sumLag; diffMean=(K_MR^-1>meanRatio)|(meanRatio>K_MR); oneVariable=xor(leadVariable,lagVariable); bothVariable=leadVariable & lagVariable; noneVariable=(~leadVariable) & (~lagVariable); B=zeros(nCUT,1); %LEAD WINDOW iLead=oneVariable & lagVariable; B(iLead)=sumLead(iLead)*a_N2; %LAG WINDOW iLag=oneVariable & leadVariable; B(iLag)=sumLag(iLag)*a_N2; %SO-CFAR iSO=bothVariable; B(iSO)=min(sumLead(iSO),sumLag(iSO))*a_N2; %GO-CFAR iGO=diffMean & noneVariable; B(iGO)=max(sumLead(iGO),sumLag(iGO))*a_N2; %CA-CFAR iCA=~diffMean & noneVariable; B(iCA)=(sumLead(iCA)+sumLag(iCA))*a_N; %Outuptted thershold T=zeros(L,1); %For CUT too small or too big for complete %reference window, we set T to zero T(iCUT)=B; %Multiplying threshold multiplier %If range gate input is a matrix, then treat each row as one experimental % trial with one test cell else nTrial=length(signal); iLead=1:N/2; iLag=N/2+2+2*nGuardCell:N+1+2*nGuardCell; %Summing and Squaring leadWin=signal(:,iLead); sumLead=sum(leadWin,2); sumSquareLead=sum(leadWin.^2,2);

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lagWin=signal(:,iLag); sumLag=sum(lagWin,2); sumSquareLag=sum(lagWin.^2,2); %VARIABILITY INDEX leadVI=N/2*sumSquareLead./(sumLead.^2); lagVI=N/2*sumSquareLag./(sumLag.^2); leadVariable=leadVI>K_VI; lagVariable=lagVI>K_VI; %MEAN RATIO meanRatio=sumLead./sumLag; diffMean=(K_MR^-1>meanRatio)|(meanRatio>K_MR); oneVariable=xor(leadVariable,lagVariable); bothVariable=leadVariable & lagVariable; noneVariable=(~leadVariable) & (~lagVariable); T=zeros(nTrial,1); %LEAD WINDOW iLead=oneVariable & lagVariable; T(iLead)=sumLead(iLead)*a_N2; %LAG WINDOW iLag=oneVariable & leadVariable; T(iLag)=sumLag(iLag)*a_N2; %SO-CFAR iSO=bothVariable; T(iSO)=min(sumLead(iSO),sumLag(iSO))*a_N2; %GO-CFAR iGO=diffMean & noneVariable; T(iGO)=max(sumLead(iGO),sumLag(iGO))*a_N2; %CA-CFAR iCA=~diffMean & noneVariable; T(iCA)=(sumLead(iCA)+sumLag(iCA))*a_N; end selectionName{1}='CA'; selectionName{2}='GO'; selectionName{3}='SO'; selectionName{4}='lead'; selectionName{5}='lag'; selection=zeros(L,1); selection(iCA)=1; selection(iGO)=2; selection(iSO)=3; selection(iLead)=4; selection(iLag)=5; %Range Cells where Thrsholds are set: iCFARwin=1:L; iCFARwin(T==0)=[];

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Appendix B

MATLAB Scripts

B.1 Probability of False Alarm in Homogeneous Environment

% % =============================================== % Probability of False Alarm - Homogeneous % =============================================== % % Computes the Probability of False Alarm for different CFAR % algorithms in the homogeneous conditiosn with various "desired" % Probability of false alrms % % ASSUMPTION % Swerling I/II Target % Exponential Noise/Clutter % % AUTHORSHIP % James Jen % Cal Poly Pomona % [email protected] % % %UPDATES % May 28th, 2011 Created % May 31st, 2011 Updated for faster CFAR functions % clear all close all clc % ######################### % % Simulation Parameters % % ######################### % %CFAR numbering CA=1; OS=2; GOCA=3; SOCA=4; OSGO=5; VI=6; S=7; SVI=8; CFARselection=VI; Pfa_set=10.^-(1:0.25:7); %Desired probability of false alarm nPfa=length(Pfa_set); nRangeGate=10^5; %Number of trials to run N=32;

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nGuardCell=1; %number of Guard Cells noisePwr=1; % ################### % % CFAR Parameters % % ################### % %CFAR window size k_OS=round(N*0.7); %OS-CFAR k-rank k_OSGO=round(0.6*N/2); %k_OSGO k_rank % ############################## % % CFAR selection and setting % % ############################## % %Legend name CFARtype{CA}='CA-CFAR'; CFARtype{OS}='OS-CFAR'; CFARtype{GOCA}='GOCA-CFAR'; CFARtype{SOCA}='SOCA-CFAR'; CFARtype{OSGO}='OSGO-CFAR'; CFARtype{S}='S-CFAR'; CFARtype{VI}='VI-CFAR'; %CFAR plot color % #################################### % % Theoretical Threshold Multiplier % % #################################### % pwrSignal=exprnd(noisePwr,nRangeGate,1); a=zeros(size(Pfa_set)); switch CFARselection case CA clr =[0 0 0 ]/255; %black a=N*(Pfa_set.^(-1/N)-1); case OS k=k_OS; clr=[255 0 0 ]/255; %red a_set=0.1:0.05:100; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); P(i)=k*nchoosek(N,k)*gamma(a+N-k+1)*gamma(k)/gamma(a+N+1); end for i=1:length(Pfa_set) Pfa=Pfa_set(i); a(i)=interp1(P,a_set,Pfa); end

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case SOCA clr=[0 153 204]/255; %light blue a_set=0.1:0.1:150; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); sum_term=zeros(N/2,1); for k=0:N/2-1 sum_term(k+1,1)=nchoosek(N/2-1+k,k)*(2+a/(N/2))^-k; end sum_term=sum(sum_term); P(i,1)=2*(2+a/(N/2))^(-N/2)*sum_term; end for i=1:length(Pfa_set) Pfa=Pfa_set(i); a(i)=interp1(P,a_set,Pfa); end case GOCA k=k_OSGO clr=[153 153 0 ]/255; %brown a_set=0.1:0.05:50; P=zeros(length(a_set),1); for i=1:length(a_set) a=a_set(i); sum_term=zeros(N/2,1); for k=0:N/2-1 sum_term(k+1,1)=nchoosek(N/2-1+k,k)*(2+a/(N/2))^-k; end sum_term=sum(sum_term); P(i,1)=2*((1+a/(N/2))^(-N/2)-(2+a/(N/2))^(-N/2)*sum_term); end for i=1:length(Pfa_set) Pfa=Pfa_set(i); a(i)=interp1(P,a_set,Pfa); end case OSGO clr=[0 204 51 ]/255; %orange k=k_OSGO; %Numerically determining the threshold multiplier, a: a_set=1:0.05:100; P=zeros(size(a_set)); i=(0:N/2-k)'; j=(0:N/2-k)'; I=i*ones(1,length(i)); J=ones(length(j),1)*j'; for ia=1:length(a_set) a=a_set(ia);

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A=2*k^2*nchoosek(N/2,k)^2; B=factorial((N/2-k)*ones(length(j)))./factorial(J)./factorial(N/2-k-J); C=factorial((N/2-k)*ones(length(j)))./factorial(I)./factorial(N/2-k-I); D=(-1).^(N-2*k-J-I)./(N/2-I); E=gamma(N-J-I).*gamma(a+1)./gamma(N-J-I+a+1); P(ia)=A*sum(sum(B.*C.*D.*E)); end for i=1:length(Pfa_set) Pfa=Pfa_set(i); a(i)=interp1(P,a_set,Pfa); end case VI clr=[204 0 204]/255; %purple a=N*(Pfa_set.^(-1/N)-1); case S clr=[255 153 0 ]/255; %green case SVI end %for CFARselection % ##################### % % CFAR Thresholding % % ##################### % threshold=zeros(nRangeGate,nPfa); for i=1:nPfa stopBar= progressbar(i/nPfa,0); if(stopBar) break; end Pfa=Pfa_set(i); switch CFARselection case CA [threshold(:,i) iCFAR]=CA_CFAR1(Pfa,N,pwrSignal); case SOCA [threshold(:,i) iCFAR]=SOCA_CFAR1(Pfa,N,pwrSignal); case GOCA [threshold(:,i) iCFAR]=GOCA_CFAR1(Pfa,N,pwrSignal); case OS [threshold(:,i) iCFAR]=OS_CFAR1(Pfa,N,k,pwrSignal); case OSGO [threshold(:,i) iCFAR]=OSGO_CFAR1(Pfa,N,k_OSGO,pwrSignal); case VI [threshold(:,i) iCFAR]=VI_CFAR1(Pfa,N,pwrSignal); case SVI [threshold(:,i) iCFAR]=SVI_CFAR1(Pfa,N,pwrSignal); case S [threshold(:,i) iCFAR]=S_CFAR1(Pfa,N,pwrSignal); end %switch CFARselection end %for i=1:length(Pfa_set) close expPfa=monteCarloFalseAlarm(threshold(iCFAR,:),noisePwr);

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% ############ % % Plotting % % ############ % figure semilogy(a,Pfa_set,'k-*','linewidth',2); hold on semilogy(a,expPfa,'color',clr,'linestyle','-','linewidth',2); legend('Theoretical P_F_A','Experimental P_F_A','location','SouthWest') title(['\fontsize{11}Theoretical v. Experimental P_F_A - \color{magenta}' CFARtype{CFARselection}]); xlabel('Threshold Multiplier'); ylabel('Probability of False Alarm')

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% % =============================================== % Probability of False Alarm - Clutter Wall % =============================================== % % Computes the Probability of False Alarm with varying clutter edge % interference for various CFAR % % ASSUMPTION % Swerling I/II Target % Exponential Noise/Clutter % % AUTHORSHIP % James Jen % Cal Poly Pomona % [email protected] % % %UPDATES % May 28th, 2011 Created % May 31st, 2011 Updated for faster CFAR functions % clear all close all clc % ######################### % % Simulation Parameters % % ######################### % CNR_dB=10; Pfa=10^-4; %Desired probability of false alarm nTrial=1.5*10^6; %Number of trials to run N=32; nGuardCell=1; %number of Guard Cells % ################### % % CFAR Parameters % % ################### % %CFAR window size k_OS=round(N*0.7); %OS-CFAR k-rank k_OSGO=round(0.6*N/2); %k_OSGO k_rank % ############################## % % CFAR selection and setting % % ############################## % %CFAR numbering

B.2 Probability of False Alarm in Clutter Wall Environment

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CA=1; OS=2; GOCA=3; SOCA=4; OSGO=5; VI=6; S=7; SVI=8; %Total number of CFAR's nCFAR=6; %Legend name CFARtype{CA}='CA-CFAR'; CFARtype{OS}='OS-CFAR'; CFARtype{GOCA}='GOCA-CFAR'; CFARtype{SOCA}='SOCA-CFAR'; CFARtype{OSGO}='OSGO-CFAR'; CFARtype{S}='S-CFAR'; CFARtype{VI}='VI-CFAR'; %CFAR plot color clr=zeros(nCFAR,3); clr(CA,:) =[0 0 0 ]/255; %black clr(OS,:) =[255 0 0 ]/255; %red clr(GOCA,:)=[153 153 0 ]/255; %brown clr(SOCA,:)=[0 153 204]/255; %light blue clr(OSGO,:)=[0 204 51 ]/255; %orange clr(S,:) =[255 153 0 ]/255; %green clr(VI,:) =[204 0 204]/255; %purple % ############################### % % CFAR Environment Generation % % ############################### % CNR=10^(CNR_dB/10); noisePwr=1; %Noise Power clutterPwr=noisePwr*CNR; nWindow=N+2*nGuardCell+1; pwrNoise=exprnd(noisePwr,nTrial,nWindow); pwrClutter=exprnd(clutterPwr,nTrial,nWindow); iCUT=N/2+nGuardCell+1; iLead=1:N/2; iTest=N/2+nGuardCell+1; iLag=N/2+2*nGuardCell+2:nWindow; iClutterWall=[0 iLead iTest iLag]; nClutterWall=length(iClutterWall); % ############## % % Simulation % % ############## % threshold=zeros(nTrial,nCFAR); expPfa=zeros(nClutterWall,nCFAR);

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for i=1:nClutterWall clutterWall=iClutterWall(i); stopBar= progressbar(i/nClutterWall,0); if(stopBar) break; end if clutterWall==0 pwrSignal=pwrNoise; pwrCUT=noisePwr; elseif clutterWall==nWindow pwrSignal=pwrClutter; pwrCUT=clutterPwr; else iClutter=1:clutterWall; iNoise=clutterWall+1:nWindow; pwrSignal(:,iNoise)=pwrNoise(:,iNoise); pwrSignal(:,iClutter)=pwrClutter(:,iClutter); if clutterWall>=iCUT pwrCUT=clutterPwr; else pwrCUT=noisePwr; end end threshold(:,CA)=CA_CFAR1(Pfa,N,pwrSignal); threshold(:,OS)=OS_CFAR1(Pfa,N,k_OS,pwrSignal); threshold(:,SOCA)=SOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,GOCA)=GOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,OSGO)=OSGO_CFAR1(Pfa,N,k_OSGO,pwrSignal); threshold(:,VI)=VI_CFAR1(Pfa,N,pwrSignal); expPfa(i,:)=monteCarloFalseAlarm(threshold,pwrCUT); end close %close progress bar % ############ % % Plotting % % ############ % cellLead=1:N/2+1; iLead=0:N/2; cellLag=N/2+2:N+2; iLag=N/2:N; xMin=0; xMax=N; yMin=10^-8; yMax=10^-0; %PROBABILITY OF DETECTION CURVE for iCFAR=1:nCFAR figure if iCFAR==CA semilogy(iLead,expPfa(cellLead,CA),'color',clr

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(CA,:),'linewidth',1.5) hold on elseif iCFAR==OS semilogy(iLead,expPfa(cellLead,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') hold on semilogy(iLead,expPfa(cellLead,OS),'color',clr(OS,:),'linewidth',1.5) else semilogy(iLead,expPfa(cellLead,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--'); hold on semilogy(iLead,expPfa(cellLead,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle','--') semilogy(iLead,expPfa(cellLead,iCFAR),'color',clr(iCFAR,:),'linewidth',1.5) end %Line marking the designed Probability of False Alarm line([xMin xMax], [Pfa Pfa],'color','m','linestyle',':','linewidth',1.5); %Line marking where the half-window point is line([N/2 N/2], [yMin yMax],'color','b','linestyle',':','linewidth',1.5); if iCFAR==CA legend([CFARtype([CA])],'location','SouthEast'); semilogy(iLag,expPfa(cellLag,CA),'color',clr(CA,:),'linewidth',1.5) elseif iCFAR==OS legend([CFARtype([CA OS])],'location','SouthEast'); semilogy(iLag,expPfa(cellLag,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') semilogy(iLag,expPfa(cellLag,OS),'color',clr(OS,:),'linewidth',1.5) else legend([CFARtype([CA OS iCFAR])],'location','SouthEast'); semilogy(iLag,expPfa(cellLag,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--'); semilogy(iLag,expPfa(cellLag,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle','--') semilogy(iLag,expPfa(cellLag,iCFAR),'color',clr(iCFAR,:),'linewidth',1.5) end ylabel('Probability of False Alarm'); xlabel('Number of Reference Cells in Clutter'); axis([xMin xMax yMin yMax]); % grid on % set(gca,'Xcolor',[0.5 0.5 0.5]); % set(gca,'Ycolor',[0.5 0.5 0.5]); % Caxes = copyobj(gca,gcf); % set(Caxes, 'color', 'none', 'xcolor', 'k', 'xgrid', 'off', 'ycolor','k', 'ygrid','off'); end

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% % ============================================================ % Probability of Detection - Homogeneous/ Multiple Targets % ============================================================ % % This script plots the SNR v. Probabilty of Detection % and SNR v. CFAR loss for various CFAR algorithms % % ASSUMPTION % Swerling I/II Target % Exponential Noise/Clutter % % AUTHORSHIP % James Jen % Cal Poly Pomona % % %UPDATES % May 28th, 2011 Created % May 31st, 2011 Updated for faster CFAR functions % clear all close all clc % ######################### % % Simulation Parameters % % ######################### % iMasking=[5]; %location of masking targets Pfa=10^-4; %Desired probability of false alarm nTrial=10^6; %Number of trials to run SNRdB_set=2:2.5:26; %SNR's to be simulted and plotted % ################### % % CFAR Parameters % % ################### % N=32; %CFAR window size k_OS=round(N*0.7); %OS-CFAR k-rank k_OSGO=round(0.6*N/2); %k_OSGO k_rank nGuardCell=1; %number of Guard Cells % ############################## % % CFAR selection and setting % % ############################## % %CFAR numbering CA=1; OS=2;

B.3 Probability of Detection in Homogeneous/ Multiple Target Environment

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GOCA=3; SOCA=4; OSGO=5; VI=6; S=7; SVI=8; %Total number of CFAR's nCFAR=6; %Legend name CFARtype{CA}='CA-CFAR'; CFARtype{OS}='OS-CFAR'; CFARtype{GOCA}='GOCA-CFAR'; CFARtype{SOCA}='SOCA-CFAR'; CFARtype{OSGO}='OSGO-CFAR'; CFARtype{S}='S-CFAR'; CFARtype{VI}='VI-CFAR'; %CFAR plot color clr=zeros(nCFAR,3); clr(CA,:) =[0 0 0 ]/255; %black clr(OS,:) =[255 0 0 ]/255; %red clr(GOCA,:)=[153 153 0 ]/255; %brown clr(SOCA,:)=[0 153 204]/255; %light blue clr(OSGO,:)=[0 204 51 ]/255; %orange clr(S,:) =[255 153 0 ]/255; %green clr(VI,:) =[204 0 204]/255; %purple % ############################### % % CFAR Environment Generation % % ############################### % SNR_set=10.^(SNRdB_set/10); nSNR=length(SNRdB_set); noisePwr=1; %Noise Power nWindow=N+2*nGuardCell+1; iCUT=N/2+nGuardCell+1; nMasking=length(iMasking); iTarget=[iMasking iCUT]; nTarget=length(iTarget); pwrSignal=exprnd(noisePwr,nTrial,nWindow); % ############## % % Simulation % % ############## % threshold=zeros(nTrial,nCFAR); Pd=zeros(nSNR,nCFAR); for iSNR=1:nSNR SNR=SNR_set(iSNR); targetPwr=noisePwr*SNR;

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pwrSignal(:,iTarget)=exprnd(targetPwr+noisePwr,nTrial,nTarget); stopBar= progressbar(iSNR/nSNR,0); if(stopBar) break; end threshold(:,CA)=CA_CFAR1(Pfa,N,pwrSignal); threshold(:,OS)=OS_CFAR1(Pfa,N,k_OS,pwrSignal); threshold(:,SOCA)=SOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,GOCA)=GOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,OSGO)=OSGO_CFAR1(Pfa,N,k_OSGO,pwrSignal); threshold(:,VI)=VI_CFAR1(Pfa,N,pwrSignal); nDetection=sum(repmat(pwrSignal(:,iCUT),1,nCFAR)>threshold); Pd(iSNR,:)=nDetection/nTrial; end close %close progress bar % ############ % % Plotting % % ############ % SNRdB_set=SNRdB_set(:); %PROBABILITY OF DETECTION CURVE for iCFAR=1:nCFAR figure hold on %Neyman-Pearson plot(SNRdB_set,Pd_inf(Pfa,SNRdB_set),'k--'); if iCFAR==CA plot(SNRdB_set,Pd(:,CA),'color',clr(CA,:),'linewidth',1.5) legend(['Neyman-Pearson' CFARtype([CA])],'location','SouthEast'); elseif iCFAR==OS plot(SNRdB_set,Pd(:,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle',':') plot(SNRdB_set,Pd(:,OS),'color',clr(OS,:),'linewidth',1.5) legend(['Neyman-Pearson' CFARtype([CA OS])],'location','SouthEast'); else plot(SNRdB_set,Pd(:,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle',':'); plot(SNRdB_set,Pd(:,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle',':') plot(SNRdB_set,Pd(:,iCFAR),'color',clr(iCFAR,:),'linewidth',1.5) legend(['Neyman-Pearson' CFARtype([CA OS iCFAR])],'location','SouthEast'); end ylabel('Probability of Detection'); xlabel('Signal to Noise Ratio (dB)'); %Simulation Title if nMasking==0 title('\fontsize{11}Probability of Detection - \color{blue}Homogeneous Environment'); else

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title(['\fontsize{11}Probability of Detection - \color{blue}' num2str(nMasking) ' Masking Target']); end grid on; set(gca,'Xcolor',[0.5 0.5 0.5]); set(gca,'Ycolor',[0.5 0.5 0.5]); Caxes = copyobj(gca,gcf); set(Caxes, 'color', 'none', 'xcolor', 'k', 'xgrid', 'off', 'ycolor','k', 'ygrid','off'); end %CFAR LOSS CURVE SNRdB_set=SNRdB_set(:); CFARloss=zeros(nSNR,3); for iCFAR=1:nCFAR figure hold on if iCFAR==CA iPd=Pd(:,CA); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,CA)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,CA),'color',clr(CA,:),'linewidth',1.5) legend([CFARtype([CA])],'location','NorthWest'); elseif iCFAR==OS iPd=Pd(:,CA); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,CA)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') iPd=Pd(:,OS); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,OS)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,OS),'color',clr(OS,:),'linewidth',1.5) legend([CFARtype([CA OS])],'location','NorthWest'); else iPd=Pd(:,CA); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,CA)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') iPd=Pd(:,OS); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,OS)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle','--') iPd=Pd(:,iCFAR); SNRoptimal=OptSNR(iPd,Pfa); CFARloss(:,3)=SNRdB_set-SNRoptimal; plot(SNRdB_set,CFARloss(:,3),'color',clr(iCFAR,:),'linewidth',1.5) legend([CFARtype([CA OS iCFAR])],'location','NorthWest'); end

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ylabel('CFAR Loss (dB)'); xlabel('SNR (dB)'); title('CFAR Loss'); xMin=min(SNRdB_set); xMax=max(SNRdB_set); yMin=0; yMax=ceil(2*max(max(CFARloss)))/2; axis([xMin xMax yMin yMax]); if nMasking==0 title('\fontsize{11}CFAR Loss - \color{blue}Homogeneous Environ-ment'); else title(['\fontsize{11}CFAR Loss - \color{blue}' num2str(nMasking) ' Masking Target']); end grid on; set(gca,'Xcolor',[0.5 0.5 0.5]); set(gca,'Ycolor',[0.5 0.5 0.5]); Caxes = copyobj(gca,gcf); set(Caxes, 'color', 'none', 'xcolor', 'k', 'xgrid', 'off', 'ycolor','k', 'ygrid','off'); end

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B.4 Probability of Detection in Clutter Wall Environment

% % =============================================== % Probability of False Alarm - Clutter Wall % =============================================== % % Computes the Probability of False Alarm with varying clutter edge % interference for various CFAR % % ASSUMPTION % Swerling I/II Target % Exponential Noise/Clutter % % AUTHORSHIP % James Jen % Cal Poly Pomona % [email protected] % % %UPDATES % May 28th, 2011 Created % May 31st, 2011 Updated for faster CFAR functions % clear all close all clc % ######################### % % Simulation Parameters % % ######################### % CNR_dB=10; SNR_dB=15; Pfa=10^-4; %Desired probability of false alarm nTrial=1.5*10^6; %Number of trials to run N=32; nGuardCell=1; %number of Guard Cells % ################### % % CFAR Parameters % % ################### % %CFAR window size k_OS=round(N*0.7); %OS-CFAR k-rank k_OSGO=round(0.6*N/2); %k_OSGO k_rank % ############################## % % CFAR selection and setting % % ############################## %

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%CFAR numbering CA=1; OS=2; GOCA=3; SOCA=4; OSGO=5; VI=6; S=7; SVI=8; %Total number of CFAR's nCFAR=6; %Legend name CFARtype{CA}='CA-CFAR'; CFARtype{OS}='OS-CFAR'; CFARtype{GOCA}='GOCA-CFAR'; CFARtype{SOCA}='SOCA-CFAR'; CFARtype{OSGO}='OSGO-CFAR'; CFARtype{S}='S-CFAR'; CFARtype{VI}='VI-CFAR'; CFARtype{nCFAR+1}='N/2 cells in clutter'; %CFAR plot color clr=zeros(nCFAR,3); clr(CA,:) =[0 0 0 ]/255; %black clr(OS,:) =[255 0 0 ]/255; %red clr(GOCA,:)=[153 153 0 ]/255; %brown clr(SOCA,:)=[0 153 204]/255; %light blue clr(OSGO,:)=[0 204 51 ]/255; %orange clr(S,:) =[255 153 0 ]/255; %green clr(VI,:) =[204 0 204]/255; %purple % ############################### % % CFAR Environment Generation % % ############################### % SNR=10^(SNR_dB/10); CNR=10^(CNR_dB/10); noisePwr=1; %Noise Power clutterPwr=noisePwr*CNR; targetPwr=noisePwr*SNR; nWindow=N+2*nGuardCell+1; pwrNoise=exprnd(noisePwr,nTrial,nWindow); pwrClutter=exprnd(clutterPwr,nTrial,nWindow); iCUT=N/2+nGuardCell+1; iLead=1:N/2; iTest=N/2+nGuardCell+1; iLag=N/2+2*nGuardCell+2:nWindow; iClutterWall=[0 iLead iTest iLag]; nClutterWall=length(iClutterWall); % ############## % % Simulation % % ############## %

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threshold=zeros(nTrial,nCFAR); expPd=zeros(nClutterWall,nCFAR); for i=1:nClutterWall clutterWall=iClutterWall(i); stopBar= progressbar(i/nClutterWall,0); if(stopBar) break; end if clutterWall==0 pwrSignal=pwrNoise; pwrCUT=noisePwr; elseif clutterWall==nWindow pwrSignal=pwrClutter; pwrCUT=clutterPwr; else iClutter=1:clutterWall; iNoise=clutterWall+1:nWindow; pwrSignal(:,iNoise)=pwrNoise(:,iNoise); pwrSignal(:,iClutter)=pwrClutter(:,iClutter); if clutterWall>=iCUT pwrCUT=clutterPwr; else pwrCUT=noisePwr; end end threshold(:,CA)=CA_CFAR1(Pfa,N,pwrSignal); threshold(:,OS)=OS_CFAR1(Pfa,N,k_OS,pwrSignal); threshold(:,SOCA)=SOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,GOCA)=GOCA_CFAR1(Pfa,N,pwrSignal); threshold(:,OSGO)=OSGO_CFAR1(Pfa,N,k_OSGO,pwrSignal); threshold(:,VI)=VI_CFAR1(Pfa,N,pwrSignal); if i<=(nClutterWall/2) pwrTarget=exprnd(noisePwr+targetPwr,nTrial,nCFAR); else pwrTarget=exprnd(clutterPwr+targetPwr,nTrial,nCFAR); end iDetection=pwrTarget>threshold; nDetection=sum(iDetection,1); expPd(i,:)=nDetection/nTrial; end close %close progress bar % ############ % % Plotting % % ############ % cellLead=1:N/2+1; iLead=0:N/2; cellLag=N/2+2:N+2;

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iLag=N/2:N; xMin=0; xMax=N; yMin=0; yMax=1; %PROBABILITY OF DETECTION CURVE for iCFAR=1:nCFAR figure if iCFAR==CA plot(iLead,expPd(cellLead,CA),'color',clr(CA,:),'linewidth',1.5) hold on elseif iCFAR==OS plot(iLead,expPd(cellLead,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') hold on plot(iLead,expPd(cellLead,OS),'color',clr(OS,:),'linewidth',1.5) else plot(iLead,expPd(cellLead,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--'); hold on plot(iLead,expPd(cellLead,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle','--') plot(iLead,expPd(cellLead,iCFAR),'color',clr(iCFAR,:),'linewidth',1.5) end %Line marking where the half-window point is line([N/2 N/2], [yMin yMax],'color','m','linestyle',':','linewidth',1.5); if iCFAR==CA legend(CFARtype([CA end]),'location','NorthEast'); plot(iLag,expPd(cellLag,CA),'color',clr(CA,:),'linewidth',1.5) elseif iCFAR==OS legend([CFARtype([CA OS end])],'location','NorthEast'); plot(iLag,expPd(cellLag,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--') plot(iLag,expPd(cellLag,OS),'color',clr(OS,:),'linewidth',1.5) else legend([CFARtype([CA OS iCFAR end])],'location','NorthEast'); plot(iLag,expPd(cellLag,CA),'color',clr(CA,:),'linewidth',1.5,'linestyle','--'); plot(iLag,expPd(cellLag,OS),'color',clr(OS,:),'linewidth',1.5,'linestyle','--') plot(iLag,expPd(cellLag,iCFAR),'color',clr(iCFAR,:),'linewidth',1.5) end ylabel('Probability of Detection'); xlabel('Number of Reference Cells in Clutter'); axis([xMin xMax yMin yMax]); % grid on % set(gca,'Xcolor',[0.5 0.5 0.5]); % set(gca,'Ycolor',[0.5 0.5 0.5]); % Caxes = copyobj(gca,gcf); % set(Caxes, 'color', 'none', 'xcolor', 'k', 'xgrid', 'off', 'ycolor','k', 'ygrid','off'); end

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