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darpa ltr, 25 jul 1973

00 |<3G-952 OJANUARY 1968

QQopy No. j 3

Technical Memorandum

RADAR SIMULATION AND ANALYSIS BY DIGITAL COMPUTER

by D. M. WHITE

D D C

f\PR 2 9 1968

THE JOHNS HOPKINS UNIVERSITY ■ APPLIED PHYSICS LABORATORY

This document is suHect to special export controls and each transmlttal to foreign governmei s or foreign nationall may be made only with prior approval of Advanced Riuarch Projecti Agency, / f ^f/Q //jUf^Jt /) S* sz2/) •? Si I

FOR OFFICIAL USE ONLY

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BEST AVAILABLE COPY

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TG-952

JANUARY 1968

I

Technical Memorandum

RADAR SIMULATION AND ANALYSIS BY DIGITAL COMPUTER

by D. M. WHITE

SPONSORED BY ARPA UNDER AO #479

THE JOHNS HOPKINS UNIVERSITY ■ APPLIED PHYSICS LABORATORY 8621 Georgia Avenue, Silver Spring, Maryland 20910

Operating under Con ract NOw 62-0604-c with the Department o( the Navy

Thii documant it IU ji«l to ipacial ««port control« ond «och tranuniHal to feroign govarnmanli or foralgn noHenoli may b* mad« only with prior approval of Advanctd RtMOrch Pro|*ctl Agency

FOR OFFICIAL USE ONLY

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TMC JOIIIH HOTKIN» UNIVnWITV APPLIED RiYSiCS LABORATORY

■ILVIH »MIINO. MAÄTUkND

ABSTRACT

A digital computer program is described which simulates the

radar-target engagement providing a representation of the detection,

acquisition, and tracking processes. The program is arranged as a

time simulation of the engagement between a radar and target, taking

into account the detailed characteristics of the target cross section,

radar and target motion throughout the engagement, surface clutter,

atmospheric attenuation, and radar losses. In the output the program

provides the user with target detection probabilities in the presence of

surface clutter as well as receiver noise, radar search and track ac-

curacies, signal-to-noise ratios, target characteristics versus time,

angular and rtJige rates, etc. The input requirements to the program

are: (1) a deck of parameter cards describing the radar parameters,

the clutter environment, and tie initial radar-target geometry, (2) a

deck of cards describing the target motion throughout the engagement,

and (3) a deck of cards describing the target's cross section versus

aspect angld. Many simplifications to the inputs are allowed for

studying and isolating various parts of the radar problem.

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THC JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SILVER 'PRING MARVLANO

PREFACE

Ti is paper describes a digital computer program which

is intended to serve as an aid in the design, analysis, and evaluation

of radar systems. Techniques for analyzing the performance of a

radar without the aid oi a digital computer are well established

and provide sufficiently accurate results if the problem is not

too involved. However, the problems are often quite complicated

which forces the radar analyst to reduce the complexity with

simplifying assumptions, such as, specifying the target's average

cross section as a constant, ignoring the effects of ati/-"»spheric

attenuation and ground clutter, estimating the detection probability,

etc. If numerous, accurate, and detailed analyses are required a

digital computer program must be used.

The purpose of this program is to simulate the radar-

target engagement in the real world in order to provide a representa-

tion of the detection, acquisition, and tracking processes. In the

simulation process it takes into account the detailed characteristics

of the target cross section, radar and target motion through the

engagement, surface and rain clutter, atmospheric attenuation, and

radar losses. In the output the program provides the user with

target detection probabilities in the presence of surface and/or

rain clutter as well as receiver noise, radar search and track

accuracies, signal-to-noise ratios, target characteristics vs

time, angular and range rates, etc.

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The Johnt Hopkln« Onlv«f»ity

The program is made as general purpose as possible by

segmenting it into subprograms and subroutines, and by the liberal

use of data inputs. In its complete form broad examples of its

use are as follows: analyzing the performance of a radar against

different kinds of targets for different sea states; optimizing

the yield of data from a live test by simulating the test before-

hand; optimizing the scanning pattern or any other parameter of a

search radar in various situations; and providing assistance in

determining optimum hardware parameters during the development of

a radar system. If a complete simulation is not desired the target

inputs can be simplified to allow the radar characteristics to be

studied separately.

The program is written in FORTRAN II for the IBM 7094

computer. It compiles in approximately two-tenths of an hour

and computes, for an average number of cases, in two-hundredths

of an hour.

I am indebted to V. Schwab, G. T. Trotter, and

E. Shetland whose contributions represent a significant part of

the computer program.

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ammmMmmmmmm

THE JOHN« HOPKINS UHIVERSITY

APPLIED PHYSICS LABORATORY ■ILVER SPUING, MARYLAND

TABLE OF CONTENTS

II

III

IV.

V.

Page

ix

1

List of Figures

List oi Tables xi

INTRODUCTION

1.1 The Radar Analysis Problem 1 1.2 Summary 4

PROGRAM INPUTS 8

2.1 Target Motion 8 2.2 Target Radar Cross Section 11 2.3 Input Parameter Cards 15 2.4 Summary 30

PROCESSING 32

32 36 39

3.1 Flow Diagram 3.2 Geometry Calculations 3.3 Radar Calculations 3.4 Detection Calculations 43

3.5 Summary 46

PROGRAM OUTPUT 48

4.1 Time Simulation Example 49

4.2 Range Profile Example 66

SUMMARY AND CONCLUSIONS 73

5.1 Summary 73 5.2 Areas for Future Development "5

APPENDIX A ~ LITERATURE REVIEW 78

APPENDIX B — RADAR ANALYSIS PROGRAM IN FORTRAN 83

BIBLIOGRAPHY 113

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THC JOHN* MOPriN» UNIVtmiTV APPLIED PHYSICS LABORATORY

»ILVt» •MINa. MAKTLAND

LIST OF FIGURES

Figure Page

1--1 Radar Analysis Problem in Dynamic Situation 2

1-2 Block Diagram of the Radar Analysis Program 6

2-1 Radar Cross Section of the Target Missile vs Aspect Angle 14

2-2 Initial Radar-Target Geometry 17

2-3 Flow Chart for Selection of Clutter and Noise Input Parameters 25

3-1 Radar Analysis Program Flow Diagram 33

3-2 Radar-to-Target Geometry 37

3-3 Clutter Area 41

4-1 Detection Probability versus Range 71

4-2 Signal-to-Clutter-Plus-Noise Ratio versus Range 72

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVn »PKINO. MAHVLAND

Table

2- -1

2- -2

2 -3

2- ~4

2- -5

2- -6

2 -7

2 -8

4 -1

4 -2

4 -3

4 -4

4 -5

4 -6

4 -7

4 -8

4 -9

4- 10

4- 11

4- 12

LIST OF TABLES

Target Motion Table

Example of Radar Cross Section Input Cards

Initial Radar-to-Target Geometry Input Parameters

Search-Radar Input Parameters

Track-Radar Jnput Parameters

Clutter and Noise Input Parameters

Input Parameters for Program Options

Arrangement of Input Decks for Multiple Runs

Program Output: Target-Motion Input

Program Output: Cross Section Input

Program Output: Input Parameter List

Program Output: Computed Constants

Definitions of Geometry Output Parameters, Part I (Range and Angle Calculations) Part II (Cross Section Calculations)

Program Output: Geometry Range Calculations

Program Output: Geometry Angle Calculations

Program Output: Geometry Cross Section Calculations

Definitions of Detection Probabilities and Radar Accuracy Calculations

Definitions of Signal and Clutter Power Level Calculations

Program Output: Detection Probabilities and Radar Accuracy Calculations

Program Output: Signal and Clutter Power Level Calculations

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27

29

50

51

52

54

55 56

58

59

60

61

62

63

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THI JOHN* HOPKIN« UNIVERSITY APPLIED PHYSICS LABORATORY

»n.vi« mniHa MMTLAMD

Table ■

4-13

4-14

4-15

Program Output: Input Data for Range Profile Run

Program Output: Probabilities and Accuracies versus Range

Program Output: Signal and Clutter Power Levels versus Range

Page

67

68

69

B-l Subroutine Name-a in Program Listing 84

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Th» Johnt Hopkin» Univ«r»ity APPtlCB rmrilCt LABORATORY

Silvnr Spring, M»rvl»nd

■ ■ ■

I. INTRODUCTION

The digital computer program described in this report

was developed to calculate the performance of a radar against a

single target in a dynamic situation. The program is hereinafter

called the Radar Analysis Program for purpose of identification.

1.1 The Radar Analysis Problem

The radar problem that is to be analyzed is illustrated

graphically in Figure Irl. Certain complicating factors such as

time variation of the target and «radar positions, clutter echoes,

antenna sidelobes, and changing situations prompts the radar

analyst to use a digital computer.

Time Variation. The prime complicating characteristic

of the radar analysis problem is the variation of the positions of

radar and target with time. If the target and radar are allowed

to move throughout an engagement then a detection probability

calculation would have to be performed for every increment of

time. This would be necessary to account for changes in the

target's crass section, range to the target, size of the clutter

echoes, and antenna position. The change of target cross section

as a function of time is calculated with a fair amount of precision.

The target cross section as a function of aspect angle is fed into

the program as an input. The actual aspect angle as a function of

time, as calculated during the engagement, is used to find thv

corresponding target cross section by table-look-up.

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THC JOHN* HOFONR UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVIII SPPIINC MARYtANÜ

■f

TARGET.

ATMOSPHERIC EFFECTS , I /'

/ TARGET ' MOTION

r

SCANNING BEAM

y (CHANGING CROSS SECTION)

RECEIVE

ANTENNA SIDELOBES

TRANSMIT

SIGNAL PROCESS

DETECT AND

TRACK

VARYING BACKGROUND

CLUTTER

L RADAR PLATFORM

IN MOTION

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Fig. 1-1 RADAR ANALYSIS PROBLEM IN DYNAMIC SITUATION

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Th« Johns Hopkins Univarsi'y «PPLIID fHVtlC» tA»OH*TOHV

Silv« Spring, Maryland

Surface Clutter, The second most complicating factor In

radar performance calculations is clutter, the reflections from the

background land or sea. The magnitude of the received clutter power

is calculated each increment of time and is added to the receiver

noise power. This total noise power is used to form the signal-to-

noise-plus-clutter-ratio which is the prime quantity for calculating

the detection probability.

The most popular detection theory used for the detection

probability calculation was developed by Marcum and Swerling,1

However this theory assumes that the noise is random from pulse-

to-pulse like receiver noise. If the total noi^e includes clutter

echoes the Marcum-Swerling theory would yield an erroneous prob-

ability of detection since clutter echoes are in general not randomly

distributed from pulse-to-pulse. To improve the calculation of the

detection probability for a target in 4 clutter background a new

theory is used which treats the clutter echoes realistically, i.e.,

some degree of correlation from pulse-to-pulse.8 Both detection

theories are Incorporated as part of the Radar Analysis Program;

their use will be described later.

Antenna Sidelobes. The clatter problem is complicated

further when the antenna sldelobes are included in the simulation.

For example under situations when the main beam of the antenna is

1J.I. Marcum and P, Swerling, "Studies of Target Detection by Pulsed Radar," IRE Transactions on Information Theory. Vol, IT-6 (April, 1960), ~

8E. Shetland, "False Alarm Probabilities for Receiver Noise and Sea Clutter," JHU/APL Internal Memorandum BBD-1387, October, 1964,

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THC JOHN« HOPKIN« .„DIVERSITY

APPLIED PHYSICS LABORATORY •u.vn SrinNa. M«IIYL\ND

pointing at a target at a high altitude the clutter echoes are not

received via the main beam but with the sidelobes. In the present

simulation the sidelobes are considered constant at a level specified

in the program input.

Other facets of the radar-target engagement shown in

Figure 1-1, or otherwise incorporated in the Radar Analysis 1)

Program, will be described in the remainder of the report.

Performance Calculations. Even though the detection

process was emphasized in the above it is not the only performance

characteristic that should be considered. Other performance

characteristics that are included in the program are search and

1.2 Summary

The remaining chapters in this report are arranged

according to the major divisions of the computer program, namely,

■

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:

a track accuracies (in range, doppler, and angle), range resolution,

and target identification times.

All calculations are made as a function of time of the

engagement at any desired time interval. Any number of runs can

be performed in one program deck set-up to analyze radar performance,

or target characteristics, as various parameters (total of 43) are

varied one at a time or together. Examples of parameters that

art1 often varied are clutter reflectivity (or sea state) , radiated

power, antenna gain, sidelobe level, time between false alarms,

target range, and angle of the target's plane of motion.

0 0

3 I 1

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APPLIED PHYSICS LABORATORY SlLVIII SPItlNG. MARVLAND

inputs, processing, and outputs. The Radar Analysis Program is

shown in block diagram form in Figure 1-2. The input parameters

are arranged in five groups as shown and are discussed in

Chapter II. These inputs are fed into the processing part of the

program, which is discussed in Chapter III, resulting in the

program outputs, which are described in Chapter IV.

In Chapter II the input parameters to the Radar Analysis

Program are listed, described, and supported by examples. The

prime problem that will be used as an example throughout Chapter II

and the remaining chapters is the detection and track of a surface-

launched ballistic missile by a radar mounted in an aircraft. This

problem has all the elements of complexity such as rapid target

motion, fluctuating target cross section, high clutter background,

etc. and is the best example for Illustrating what the program can

and cannot do. The changes required on the input cards to analyze

other problems will also be indicated. The inputs required for the

target motion and the target cross section are the most involved and

are arranged in decks (or tables); these are "looked-up" in the

processing operation. The other parameters, which describe

initial radar-to-target geometry, the clutter and noise parameters,

and the radar characteristics, are fed in on one input card to a

parameter for a total of 43.

In Chapter III a description of the processing and

simulation techniques are given. An overall flow diagram of the

program is given, followed by description of the more important

subroutines such as the target and radar motion simulation, the

radar simulation, and the detection routines.

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APPLIED PHYSICS LABORATORY »Lvn amiNa.

INPUT (Chapter II)

Initial Radar-to- Target

Geometry

Target Motion

Target Radar Cross Section

Radar Parameters

Clutter and Noise

Environment

PROCESSING (Chapter III)

Geometry and Target Cross

Section Processing

3 i Radar

Performance Processing

OUTPUT (Chapter IV)

Target/ Geometry Output

Main Radar and Geometry

Output

Radar Only

Figure 1-2

BLOCK DIAGRAM OF THE RADAR ANALYSIS PROGRAM

- 6 -

TMI iOMH» HOf*lM» UNIV«««IT» APPLIEC; PHYSICS LABORATORY

INLvm arm—, MMVLAND

In Chapter IV the output parameters (66 are printed out)

are described followed by the printed and plotted output for the

examples given in Chapter II. Examples will be shown for the full-

scale dynamic situation and the simplified simulation in which the

target motion is static and the target r , section is constant.

In Chapter V the significant developments of this study

will be reviewed along with the limitations of the existing program

and areas for future development.

The complete program, written in FORTRAN II, is given

in Appendix B. A literature review of other digital computer

programs for radar analysis or simulation is given in Appendix A.

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THI JOMH« HOrKIN* UNIVKRSITV APPLIED PHYSICS LABORATORY

II. PROGRAM INPUTS

The program data inputs consist of 43 input-parameter

cards and two dscks. The individual parameter cards describe the

initial radar-to-target geometry, the radar characteristics, the

clutter and noise parameters, and program control. The two decks

of cards form two tables describing the target motion and the

target radar-cross-section. These inputs will be defined and

illustrated by examples. The prime example used will be for a

radar mounted in an aircraft with a surface-launched missile as

the target. The changes required in the inputs for other examples

will be indicated.

The last section in this chapter will describe the

Input parameters and procedures used to control the type of output

desired. Two types are available: the time simulation mode, in

which the radar performance is analyzed against a moving target,

and the detection-probability range-profile mode, in which the

target motion and cross section are unchanging.

2.1 Target Motion

The target motion within a vertical plane is described

with a deck of input cards. On each card in the deck ttere are

eight numbers as listed and defined in Table 2-1. The time value

referred to In Table 2-1 may have a zero or negative value on the

first card and increase from card to card with any desired increment

between 1 and 9999 seconds. The card preceding the target motion

deck has the time increment, total number of cards, and the name

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■

THI JOMM« MO«"M« UHIWWtITY APPLIED PHYSICS LABORATORY

»Lvirn »mi)». M.HTLAMD

Table 2-1

TARGET MOTION TABLE

Card number

Time (T)

Maximum of 322

Time (from zero) in seconds

Down-range position

Horizontal distance i.n feet of target in target plane at time (T) measured front initial target position.

Down range speed

Horizontal speed of target at time (T) in feet per second

Down range acceleration

Horizontal acceleration of target at time (T) in feet per second per second

Altitude poraititm Target altitude (or vertical position in target plane) at time (T) in feet

Altitude speed Vertical velocity of target at time (T) in feet per second

Altitude acceleration Vertical acceleration of target at time (T) in feet per second per second

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TMI JOHN* HOMIM« UNIVHMITV APPLIED PHYSICS LABORATORY

•«.«til ■WIIW. HAKVUND

of the target. The parameters that describe the initial radar-

target geometry, discussed in a subsequent section, assum? their

given values on the time specified on the first card in the target

motion table.

The target motion for the airborne-radar/missile-target

problem is described by a deck of 41 cards using the format in

Table 2-1. The actual table is printed in the first part of the

program output discussed in Chapter IV and will therefore not be

repeated here. It describes, however, 40 seconds of the flight

of a ballistic missile that reaches an altitude, in this time, of

24,000 feet and a horizontal range, from the launch point, of

10,000 feet.

The values for targets other than a surface-launched n

ballistic missile are easily prepared. For a target that does

not move during the engagement, the simplest case, only two cards j

are required: the first at time zero and the second at the maximum

duration of the problem. Soth cards would have the desired range

and altitude values with the velocity and acceleration values equal rT

to zero. On the other extreme a rapidly moving target could be

simulated which could not possibly be achieved by a practical jl

target.

The only restriction which this procedure has is that

the target motion is restricted to a vortical plane. For missile

targets whose trajectories are often in a plane this is not a problem

but it may be for aircraft targets. For example, in the present

version of the program an aircraft target cannot turn or bank. It

may, however, dive or rise in a flight plan confined to a vertical

Plaae. / - 10 -

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Th» Johns Hopklm Univariity ArPLlID PHVtiC» LAMRATORV

Silv« Spring, MiryUnd

2.2 Target Radar Cross Section

The radar cross section of a target is a terra that

relates the power density reflected from a target to the incident

power density from the radar and is defined as follows.1

Power delivered per unit solid angle in the direction of the radar

Ait Power per unit area incident on the target

The value of the radar cross section varies as a function of the

orientation of the target with respect to the radar, the polari-

zation of the radar, the radar wavelength, and the conductivity of

the target's surface. In this program the radar cross section of

the target at the desired radar wavelength is required and is fed

into the computer on a deck of cards.

General Description. The deck of cards contains the

radar cross section of the target as a function of the aspect

angle, for both vertical and horizontal polarization. The aspect

angle is defined as the angle between the imaginary line connecting

the radar to the target and the center line through the target. For

both missiles and eircraft the center line runs from nose to tail

with the nose-on aspect angle taken as zero. The range of aspect

angles covered are from 0 to 180 degrees. For symmetrical targets

the cross section values for the aspect angles between 180 and 360

degrees are the same as the values for the corresponding angles

between 0 to 180 degrees.

1R. S. Berkowitz, ed.. Modern Radar (New York: John Wiley and Sons, 1966), pg. 549.

- 11 -

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TK. John* Honkin» Unlvwiity «PPUIO MiY^tt LAMDATOMV

«Iv« Sprint, «toryland

The radar cross section table is stored in the computer

and is used by the processing part of the Radar Analysis Program

every time a cross section value is desired for a particular

aspect angle. This will be discussed further in the next chapter.

Airborne-radar/missile-target example. The radar cross

section deck for the airborne-radar/missile-target example is

partially shown in Table 2-2 indicating the format arrangement.

The complete table is given in Chapter IV; it was prepared

especially for this report and does not represent the cross section

characteristics of a known missile. The values of the cross

section table were plotted in Figure 2-1 to show the general

characteristics of the target. The fine detail of the cross

section is not shown in this figure because it was plotted in

increments of three degrees. As shown in Table 2-2 the actual

cross section data are fed into the program in .}-degree increments

over aspect angles from 0 to 180° for Vertical and^horizontal

polarizations.

This type of data is available from the Radar Target ■

Scattering Range (RAT SCAT)8, Holloman Air Force Base, New Mexico.

At RAT SCAT the radar cross section of various targets and target

models is measured for any frequency from 150 to 12,000 MHz at

arbitrary polarizations. A program was developed to edit the RAT

SCAT data (on punched paper tape or magnetic tape) and convert it ■

to a deck of cards for direct input to the Radar Ans lysis Program.

8H, C. Marlow, et al. ."The RAT SCAT Cross-Section Facility," Proceedings of the IEEE, Vol. 53, No. 8. Special Issue on Radar Reflectivity, (August 1965), pp. 946*954.

- 12 -

The Johni Hopkim Unlvartity APPLIED PHYSIC» LAtORATORV

Silvw Spring, Maryland

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APPLIED PHYSICS LABORATORY ■HVIN OPNiNa. MARYLAND

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APPLIED PHYSICS LABORATORY SlLVIR SPHtNO. MARYLAND

Other Targets. The cross section of targets other than

missiles are described in the same way with certain limitations.

For an aircraft, for example, the cross section as a function of

azimuthal aspect angles for both horizontal and vertical polari-

zation is not enough to completely describe the target. This is

because the cross section is different for different angles off

the plane of the wings. This is not the case for missiles

(neglecting the effect of fins) and other targets that are basically

symmetrical about their longitudinal axis.

The radar cross section input technique for simple

targets like a sphere, whose cross section is constant as a

function of aspect angle, is accomplished by entering the same

cross section value for zero degrees aspect angle and for 180

degrees. The interpolating routine associated with the part of

the program that "looks-up" this table will then choose this

constant cross section value for all aspect angles. This same

technique can be used to describe a target whose cross section is

constant (or can be assumed constant) over a limited azimuth

sector changing from sector to sector as desired.

2.3 Input Parameter Cards

Following the target motion and the target cross section

decks is a set of 43 input parameter cards. These cards contain

the values of the parameters that describe the Initial radar-to-

target geometry, the characteristics of the radar, the clutter and

noise parameters, and the program options; these parameters are

described in the following sections. '.: It - : ■ ; ^

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TMC JOHN« HOPKIN* UNIVCMITY APPLIED PHYSICS LABORATORY

SlLV» (PKINO. M«B.".»MO

- 16 - "' ~" " ■ ""l l.";1!'!'»!^

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The parameters are arranged one to a card which includes

the parameter number, the FORTRAN symbol, the value, the conversion

factor, and the definition. All cards are printed in the first

pe*t of the output as shown in Chapter IV. Parameters that are

given a value of zero can be omitted from the input parameter

deck^, rr " " ■ li

2.3.1 Initial Radar-to-Target Geometry p,

The initial geometry input parameters specify the

distances and angles between the radar and target at the start

of the engagement, and the velocity vector for the radar motion

during the remainder of the engagement. The geometry is shown in

Figure 2-2.yThe parameters shown are defined in Table 2-3 with

values for the airborne-radar/missile-target example.

As indicated in Table 2-3 the parameters for the airborne- [)

radar/missile-target example describe a radar in an aircraft that

is flying at an altitude of 30,000 feet in level flight. The )

target motion throughout the engagement is confined to a plane

which is positioned at an azimuthal angle of 30° with the reference.

The changes required in the geometric values of the

input parameters for problems other than the airborne-radar/

missile-target problem are apparent. If the radar were on a ship, (J

for example, the altitude ZLOIT would be reduced to zero and the «,

0 ii

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velocity WA reduced to say 70 feet per second. The other angles

and distances could be changed as desired. Note, however, that

in ail cases the target motion (described in Section 2.1) through-

out the remainder of the problem is confined to a vertical plane. 1

which is not a limitation for most problems.

I

iwiwwiii lM^li<l^lw^wf^^ iimimmam»*Bm

THF JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY ''.•.VBR SPRING. MARYLAND

TARGET AT START

(GEOMETRY ORIGIN)

RADAR PLATFORM AT START

RADAR PLATFORM VELOCITY VECTOR / VVA

51 N

a/**0 A2/MUTH TERENCE UNE

A2 = o

,TARGFT PLANE OF MOTION

PROJECTION OF VVA ON AZIMUTH (x-y) PLANE

Fig. 2-2 INITIAL RADAR-TARGET GEOMETRY

17

.——

TM€ JOHN» MOPKIN« UNIWIMITV

APPLIED PHYSICS LABORATORY »ILVl» tmiNa. MARYLAND

Table 2-3

INITIAL RADAR-TO-TARGET GEOMETRY

INPUT PARAMETERS

PARAMETER DEFINITION

XLOIT Distance between radar and target along azimuth reference line, in nautical miles

YLOIT Distance between radar and target, perpendicular to azimuth reference line, in nautical miles

ZLOIT Altitude of radar, in feet

AZTAR Angle between target plane of motion and zero-azimuth reference line, in degrees

WA Velocity vector of radar platform at start and throughout engagement , in feet per second

ELVA Elevation angle of VVA, in degrees

AZVA Azimuth angle of VVA, in degrees

OLDK Switch, 0 or 1, multiplying VVA

EXAMPLE VALUE

30 nm

30 nm

30,000 ft

30°

700 fps

0°

60°

1

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.,,■:. ■ ■ m ^y:-':v;-«rjl s ■ - .

THB JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY •ILVIR tfPRINa. MARYLAND

2.3.2 Radar Characteristics

The radar characteristics are fed into the program on

input cards, one parameter to a card, and can be grouped into

söarch radar and track radar parameters.

Search Radar. The search radar parameters are listed

in Table 2-4 with descriptions and example values. These values

will be used in the airborne-radar/missile-target example but are

otherwise completely arbitrary. As listed these parameters

describe a 5000 MHz (C-band) radar, radiating 500 kilowatts of

power in pulses that are 10 microseconds wide and occur at a

repetition rate of 1000 per second. Pulse compression is used

with a time bandwiuth product of 100. This indicates that the

10 microsecond transmit pulse will be compressed with a receiver

that Is matched to the code within the transmit pulse, to a pulse

that is .1 microsecond wide. The system loss factor is listed as

.1 (-10 dB) and Includes atmospheric loss, radar-line loss, beam-

shape loss, and scanning loss.3

The remaining search parameters describe the antenna

an<? its scanning characteristics. A vertically polarized antenna

was chosen which is 5.6-feet square (generating a 2° pencil beam)

with an average sldelobe level of 30 dB below the mainlobe gain.

The antenna beam will be scanning 26.6° sector in azimuth and a

30° sector in elevation, in one secord. The time to scan the

sector, HTSS, is the program clock and will specify the increment

of time at which the radar calculations are performed by the program.

'D. K. Barton, Radar System Analysis (Englewood Cliffs: Prentice- Hall, Inc., 1964), p. 140.

- 19 -

— ; ' ' ■ ' ' '—-—■"—■ '!' ' _-——,_ —————

Ill II'

THC JOHN* HOFKINC UNIVIMITV APPLIED PHYSICS LABORATORY

•ILVIK »HKINO. MAIIVLANP

Table 2-4

SEARCH-RADAR INPUT PARAMETERS

PARAMETER

IMC

PPEAK

TDWELL

FR

FNOISE

SYSLF

PCN

POL

HAPER

VAPER

AAPEFF

ETA

XIHDEL

XIVDEL

HTSS

DESCRIPTION

Radar frequency in megacycles

Peak power in megawatts

Width of transmit pulse in microseconds

Pulse repetition frequency in Hz

Receiver noise figure

System loss factor

Pulse compression ratio

Antenna polarization; one for horizontal and 0 for vertical*

Horizontal aperture of antenna in feet

Vertical aperture of antenna in Met

Antenna aperture efficiency

Sidelobe to mainlobe antenna gain ratio in decimal form

Azimuth search sector in degrees

Elevation search sector in degrees

Time to scan the sector; also the time increment between all calculations

EXAMPLE

5000 MHz

.5 Mw

10 |1S

1000 Hz

3 (5 dB)

.1

100

1

5.6 ft

5.6 ft

.65

.001

26.6°

30°

1 sec

POL =■ 0. actually indicates that the antenna polarization vector is parallel with the plane of target rotation, and POL = 1 when the polarization vector is perpendicular to the plane of rotation. Therefore for a ballistic missile and a vertically polarized radar POL - 0, while for an aircraft target and a vertically polarized ra^ar POL = 1.

- 20 -

-,—,——i—., ■

*i#«SS*™WWO«W««M(tn*M»i«™-

THt JOHNS HOPKINS UNIVERSITY APPLIED PHYSICS LABORATORY

SILVKR SPRING. MARYLAND

The search radar parameters will be used in the standard

radar equation for calculation of the signal-to-noise ratio. The

parameters may be modified to calculate the performance of radars

with modulations different from the pulse radar depicted here.* A

For example, when analyzing a CW (continuous-wave) radar the peak

power PPEAK is equal to the average power; tue pulse width TDWELL

is equal to the time the beam remains on the target; and the pulse

repetition frequency FR is adjusted to make the number of pulses

that hit the target equal to one.

Track Radar. The track radar input parameters for

the Radar Analysis Program are listed in Table 2-5. These

parameters are used to calculate the angle tracking accuracies

according to equations by Barton.8 The parameters are discussed

completely in Barton6 for monopulse and conical scan trackers.

The values of the parameters indicated in Table 2-5 in the example

column are for a monopulse tracking radar.

2.3.3 Clutter and Noise Parameters

The clutter and noise input parameters are required to

determine the method for calculating the amplitade of the clutter

and the detection theory to be used in calculating the probability

of detection. The six input parameters, with a brief description

and example values for the airborne-radar/missile-target example,

J. J. Bussgang, ■ et al.t. "A unified Analysis of Range Performance of CW, Pulse, and Pulse Doppler Radar," Proceedings of the IRE. Vol. 47, (October 1959), pp. 1753-1762. -----

8Barton, 0£. cit. . p. 279. 6 Ibid. , pp. 263-315.

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THt JOHN« HOmCINS UNIVEMITV

APPLIED PHYSICS LABORATORY ■ILVI* ■nUNO. M»l«YL»NO

Table 2-5

TRACK-RADAR INPUT PARAMETERS

PARAMETER DESCRIPTION

FNTRK Track system factor; correc s for detector and type of tracker;

Monopulse (coherent det) = 1

Monopulse (sq. law det) - 2

Conical scan (linear) = 2

Silent lobing (coherent) = 2 Silent lobing (sq. law) - 4

Beam cross-over loss factor; equals 1 except for conical scan linear detector in which case it equals 2.

Nominal noise bandwidth of angular circuits; /b n

Normalized slope factor; 1,57 for monopulse

Box car bandwidth, cps; PRE for pulse system; bandwidth of doppler filter for pulse doppler system

Angular tracking error due to target motion, in milliradians

Angular tracking error due to platform motion, in milliradians

Angular tracking error due to other causes, in milliradians

CRL

BNSR

FKS

Bl

ATETS

ATEPS

ATEOS

- 22 -

'•",."hv •^■"^

EXAMPLE

2.

1.

1.57

1000.

.5

.5

0

p

i

D 0 i\

0

!i

D 0 0 D 0 I

rnmmtmmmmimM^llWISgimilimimmmtm .-.—. .^n^mm^sfmm,rrti'-m -«,-I,«I»»»«.«,.«,-.

THC JOHN* HOPKINS UNIVIMITY APPLIED PHYSICS LABORATORY

•iLvn •mma. MARYLAND

are given in Table 2-6.

The quantity a listed in Table 2-6 is defined as the

normalized clutter cross section and is equal to the radar cross

section of the clutter medium per unit area intercepted by the

antenna beam. A description of ao can be found in Skölnik7

which includes representative values for different surfaces. A

more detailed treatment of radar backscatter from land, sea, or

atmosphere can be found in the classified literature.*

The selection or specification of the clutter and noise

parameters for any given case is best described with the aid of

Figure 2-3. A flow chart is shown with a box for each of the six

parameters; the double lines between boxes indicate the path taken

by the airborne-radar/missile-target example. The time between

false alarms TAUFA is specified firstc An option for determining

the value of a is then reached. Three options are available: o

a is zero, that is, there is no clutter; a is given, in

which case a value is selected; or <T is to be calculated o

internally. In this latter case the sea state (if in fact the

operation is taking place over the sea) is specified and the value

of a is calculated with an empirical equation based on the sea o

state, angle of incidence of the radar energy to the surface of ■

the sea, and the radar polarization.

(hain clutter, which is added to receiver noise, can be

included by means of the last two parameters in Table 2-6.)

> .I-

7M. I. Skolnik, Introduction to Radar Systems (New York: McGraw- Hill Book Company, Inc., 1962), p. 523.

8F. E. Nathanson, ed., "Report of Radar Clutter Signal Processing Committee: Part I, Radar Clutter Effects (U)," TG 842-1, The Johns Hopkins University Applied Physics Laboratory, September 1966, CONFIDENTIAL.

- 23 -

THC JOHN« HOPKINS UNIVIRIITV

APPLIED PHYSICS LABORATORY ■n.VM •MtlN«. MAIIYtJtND

Table 2-6

\ A+-

PARAMETER

CLUTTER AND NOISE INPUT PARAMETERS

DESCRIPTION EXAMPLE

■"■,■

TAUFA

SIGOPT

-

SIGZ

SEA

-

False alarm time in seconds

a calculation option:

-1 « no clutter 0 = calculate aQ 1 - given as SIGZ

Value of ff in dB when SIGOPT o

■■'

Sea state:

1

FDMS

■ ■'■■

.

CASE

■

■■ ■ ■

■ :

"I ,i IS*.'

^;fr-^;^.^,■• i:.>|^

-

rough sea (Beaufort state 5, 10 ft wave height) calm sea (Beaufort .5, 1 ft wave height)

Frequency diversity:

1 = Use Marcum and Swerling 0 - Use Shotland theory

Swerling target case designation:

0 « nonfluctuating target 1 m scan-to-scan fluctuation

with many nulls 2 « pulse-to-pulse fluctuation

with many nulls 3 = scan-to-scan fluctuation

with a few nulls 4 «f pulse-to-pulse fluctuation

with few nulls . ■

RAIN

SUMSIG

For rain clutter at the target RAIN Otherwise equals zero.

1.

10

0

sec

Backscatter coefficient Za in meter / meter3 in decibels. (See Skolnik, p. 539)

0

,

.

■■■■•

■ :.

24

- ■ ■

■ i. : I

i

THt JOHN* HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVtR SniHa. MARYLAND

'

No (clutter)

fteqUfency diVe^itv

i

Specify TAUFA

(a given)

0 (calculate a )

Specify SEA

state

No frequency diversity

refers to path taken by example

problem

Marcum- Swerling Detection Theory

Shot land Detection Theory

Figure 2-3

FLOW CHART FOR SELECTION OF CLUTTER AND NOISE INPUT PARAMETERS

- 25 -

TNI JOHM» MOntlN« UNIVKIMITV APPLIED PHYSICS LABORATORY

•ILV» amiNa. MAHVUND

The FDMS option is now reached which determines whether

the Marcum-Swerling theory or the Shetland theory is to be used.

In essence the parameter FDMS is not an option bu» a radar parameter

that describes whether frequency diversity, i.e., frequency jumping

from pulse-to-pulse, is used or not. If frequency diversity is

used then the clutter is considered random from jmlse-to-pulse9

and Marcum-Swerling detection theory is used. If the radar does

not use frequency diversity then the clutter is not random from

pulse-to-pulBe which means that the Shetland del otion theory must

be used. Note that if there is no cluti«fr, SIGOPT = -1, then the

FDMS option is bypassed and Marcum-Swerling theory is used in

calculating the detection probability. ■ .

2.3.4 Program Options

There are two options for operating the Radar Analysis

Program which are selected by means of the TARGOP parameter listed

in Table 2-7.

■

Time Simulation (Primary Option). The primary option

starts from TSTART a.id ends at TSTOP and runs as a time simulation

of the radar-target engagement. Calculations of the radar

detection probabilities and other performance characteristics,

are made every time the antenna scans past the target. This

mode is selected when the option TARGOP is equal to zero or

some negative value.

;■/

II

•v. W. Pidgeon, "Time, Frequency, and Spatial Correlation of Radar Sea Return," A Technical Note for Use of Space Systems for Planetary Geology and Geophysics, The American Astronaut leal Society, May 1967.

- 26 -

u

-

THE JOHNS HOPKINS UM.'VERSITY

APPLIED PHYSICS LABORATORY SILVIH SpntMo. hUnrLANo

Table 2-7

INPUT PARAMETERS FOR PROGRAM OPTIONS

PARAMETER DESCRIPTION EXAMPLE

TSTART

TSTOP

TARGOP

BXLOIT

XLOIT

Program starting time in seconds relative to zero time on the target motion input

Program stopping time in seconds relative to zero time on the target motion input

Program option:

-x or 0 - Time simulation +R - Range profiles where R is the range Increment

Maximum range value for performing calculations in the range profile option

Minimum range value for calculations in the range profile option

0 sees

39 sees

(2 n.m. In 2nd run)

78 n.m. (In 2nd run)

30 n.m. (zero n.m. in 2nd run)

i i

- 27 -

■

TMt JOHN« MO! v.iJ« UNIVCRSITY APPLIED PHYSICS LABORATORY

•1LV«» »PHIKO. MARYLAND

n D

0 0

Ranffe Profile (Secondary Option). The inputs described

to this point are those required for the time simulation option.

However, the program can also be used for plotting the radar

performance versus range for a stationary, constant-cross-section

target by simply specifying TARGOP as +R. For a stationary target

with a constant cross section the two input decks are greatly

simplified: the target-motion input deck is reduced to one card ^

indicating the desired altitude, and the cross section input deck

is reduced to one card indicating the desired cross section. The

program then calculates the detection probability and other

parameters as a function of range in increments of R. These

calculations versus range are called range profiles and will be

discussed further in Chapter IV.

Deck Arrangement. The input-deck arrangement for

running the example problems discussed in this chapter is listed

in Table 2-8. Two runs are made: (1) the airborne-radar/missile-

target example in a full time simulation mode, and (2) the airborne

radar versus a stationary target in a range profile run. The

items listed in Table 2-8 were discussed in this chapter except

for the control cards and cover cards. The control cards include

the computer Job cards and loading cards. The cover cards are

associated with the data. There is a target-deck cover card

which indicates the target name, number of cards, and the time

increment. The cross section deck has both cover and trailer

cards. The former indicating the name of the target and the

increment of the aspect angle; the latter indicating the end of

the cross section deck.

- 28 - 'i . '','"°"- ''

THC JOHNS HOPKINS ÜNIVIMITV

APPLIED PHYSICS LABORATORY »ILVi« MRINa. MARYLAND

mm

Table 2-8

ARRANGEMENT OF INPUT DECKS FOR MULTIPLE RUNS

Description of Cards

Control cards

Program

Data (Time simulation run)

Target Motion Deck

Cross Section Deck

Input Parameters

Run 1 End

Run 2 Leader (Range Profile Run)

Stationary Target Card

Cross Section Card

Input Parameters Different from Run 1

End

Approximate Number of Cards

42

300

43

1

1

2

3

3

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i

^ ;•■>"

■

TM« JOHN* HOMCIKW UNIVOMITV

APPLIED PHYSICS LABORATORY ■lUIIH •ntltW. MARTUND

2.4 Summary

The input cards to the Radar Analysis Program have been

described in some detail. The target motion in a 'ertical plane

throughout the engagement and the target's radar cross section

versus aspect angle are described on cards to practically any

degree of detail. For the example problem studied here the

target motion deck comprised of 41 cards describes a missile

launched from the ground and reaching an altitude of 24,000 feet

in 40 seconds. The target's radar cross section is described on

300 cards with values every .1 degree for aspect angles from 0

to 180° for vertical and horizontal polarization. The cross

section values vary over 50 dB from -25 dB to +25 dB relative

to one square meter.

The 43 input parameter cards describe the initial

positions of the radar and target, the radar parameters, the

clutter and noise parameters, and the program mode options. For

the example problem these parameters describe an airplane flying

at an altitude of 30,000 feet away from a missile launched,

essentially straight up, at an initial range of 43 nautical miles.

The radar in the airplane has an antenna 5.6 feet on a side

operating at a'frequency of 5000 MHz, radiating 500 kilowatts

Of power in a 10 {is pulse. The radar employs pulse compression

and frequency jumping from pulse-to-pulse to reduce the effects

of clutter contributed by a rough sea. The antenna is scanning

a sector 26.6° in azimuth und 30° in elevation, centered on the

target, in one second.

- 30 -

■—f I) '—— ~ ". . ■ ""■ —; ;- ■ ■

——i—mmimimi» n «

THE JOHN* HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVIR •PRIMa. MARYLAND

The calculations performed on these inputs will be

described in the next chapter. The resulting outputs are

discussed in Chapter IV. /

ii

■ ..'■

■

- 31 -

TM« JOHNS HOTKINS UNIVWKITY APPLIED PHYSICS LABORATORY

siLvt» umma. M»»vtANi

III. PROCESSING

i';"'1^

B 0 D 0

The processing part of the Radar Analysis Program

includes those calculations and analytical models required to ij

determine target cross section characteristics versus time, ■

detection probabilities, time required for detection, tracking

accuracies, and search accuracies. The calculations have been

divided into three groups for purposes of discussion: geometry

calculations, radar calculations, and detection calculations. 11

These three blocks of calculations are snown in Figure 3-1, the

Radar-Analysis-Program flow diagram, which is discussed next.

3.1 Flow Diagram jj

The simplified flow diagram. Figure 3-1, indicates the

general calculating procedure for the time simulation or main mode

of operation. The range-profile mode of operation is a special

case and can also be described with the aid of Figure 3-1.

- 32 -

D 0

D

0 0 Q Time Simulation. The main mode of operation reads SJ

the input cards described in Chapter II, stores this information,

and prints it out. The geometry calculations are performed next

at time TSTART. In these calculations the target-motion input (J

deck and the initial radar-to-target geometry parameters are

used to calculate the ranges, angles, and range rates between

the radar, target, and other points of interest. The aspect fl

angle determined in this set of calculations allows the cross

section of the target to be determined by looking up the value |

Ö

I

0


APPLIED PHYSICS LABORATORY SILVn SFRINO. MARYLAND

-> Start

READ INPUTS

1. 2. 3.

Read new inputs

and repeat

NO

this\ ^the last^-^"

vrun?.

Target motion deck Target radar-cross section deck Parameter cards

GEOMETRY CALCULATIONS

Aspect angles Range Range rates, etc.

Look up target cross section

RADAR CALCULATIONS

Clutter power Signal-to-noise ratios Tracking accuracies, etc

Increase time by sector- scan time and repeat

DETECTION CALCULATIONS

Marcum-Swerling or

Shetland

PRINT OUTPUT

Figure 3-1

RADAR ANALYSIS PROGRAM FLOW DIAGRAM - 33 -

in iiiirini»miiii i in

THC JOHNS HOPKINS UNIVCRSITY

APPLIED PHYSICS LABORATORY SiLvn SraiNO. MARYLAND

in the radar-cross-section input table. These geometry and cross-

section values are used in the next set of calculations, called

radar in Figure 3-1, to determine the signal-to-noise ratios and

the tracking accuracies (to A>ame a few) . The signal-to-noise

ratio and the clutter and noise input parameters are then used

in the detection calculations for finding the single-scan detection

probabilities.

The time step of this set of calculations is then

compared with the last time for calculations, TSTOP. If TSTOP

has not been reached the time variable is increased by the

increment of time required to scan a sector, HTSS, and the

calculations are repeated. When TSTOP is finally reached the

output is printed signifying the end of one run. Then if a new

set of input parameters exists in the deck they are read and the

entire processing operation is repeated.

Range Profiles. If the range profile option is selected,

by specifying TARGOP as +R, the processing operation is somewhat

simplified and the format of the output is different. Referring

to Figure 3-1, the input is read as before except now the target-

motion deck and the cross-section deck consist of only one card

each. The parameter cards are essentially the same as for the

time simulation except that the parameter TARGOP equals +R.

(Those input parameters which do not apply to the range-profile

case, such as TSTART and TSTOP, need not be changed or omitted

since they are automatically avoided by the program.) This set

of inputs, as described in Chapter II, specify a target whose

- 34 -

•w


APPLIED PHYSIC'ä LABORATORY SILVER 8PI«IN3. MARYLAND

altitude is constant, and whose cross section is constant and

independent of aspect angle.

The processing for the range-profile option omits

most of the geometry calculations and performs the radar and

detection calculations as a function of range. Instead of the

comparison of time with TSTOP as shown in Figure 3-1 the range-

profile processing compares the range with the maximum specified

range BXLOIT. If the range is smaller than BXLOIT it is incremented

by +R and the calculations are repeated. If the range equals

BXLOIT the run is printed out and the next run is processed.

Subroutines. The overall processing as described above

is controlled by the main program. All the calculations are

performed by subroutines. When the calculations are completed

control is returned to the main program. The main program is

then used for printing and plotting the output data.

There are two subroutines that are associated with the

main program that do not perform calculations. These are TARGIN

and CROSIN which are used in the first blovk shown in Figure 3-1

to read in the target-motion and cross-section input decks. They

also are responsible for printing out this input data in the first

part of the output. The main program and the subroutines are

given in their FORTRAN II language in the Appendix B. The first

page of Appendix B lists the parts of the program in the order

called.

- 35 -

TMC JOHNS HOPKINS UNIVtKSITV APPLIED PHYSICS LABORATORY

■ILV» SmitM. MARTkAND

The actual calculations performed will be discussed in

the next three sections. The final form of the output can be

found in Chapter IV. ■

3.2 Geometry Calculations ■

The geometry calculations are performed by the subroutine

GEOM and its associated subroutines: TARGET, AIRCFT, five vector

subroutines, and RATSCT. (See Appendix B, Section 2.0, for a

listing of these subroutines.)

The geometry subroutine is called by the main program

each increment of time to solve the geometry illustrated in Figure

3-2. The geometry, as indicated, is solved in three dimensions i

over a spherical earth.

Pour points in space are involved in the calculations:

the radar location, the target location, the point of reflection ■ -

on the earth's surface, and the clutter spot. The procedure

Involved in calculating the pertinent ranges, range rates, and

angles between these points is as follows. The target position

and velocity at the time of interest are gotten from the target-

motion input table, using interpolation if required; this is

performed by subroutine TARGET. The radar position at the

calculating time is determined by the subroutine AIRCFT, which

determines the distance traveled due to the input velocity WA

specified for the radar platform. Then the ranges, range rates,

and angles connecting the radar and target can be computed. They

XA 4/3 earth radius is used to account for refraction.

- 36


APPLIED PHYSICS LABORATORY SlLVIR SPRING MARYLAND

MM mm MHMM

ELEVATION PLANE

RADAR ALTITUDE

REFLECTION POINT DEPRESSION

ANGLE

TARGET

. CLUTTER SPOT ' (AT SAME

RANGE AS TARGET)

AZIMUTH PLANE

REFLECTION POINT

ZERO AZIMUTH LINE

Fig. 3-2 RADAR-TO-TARGET GEOMETRY

TARGET VELOCITY VECTOR

CLUTTER SPOT

■

: tt

i

-

- 37

TMi JOHN* HomciNS \jnivin-.-.i APPLIED PHYSICS LABORATORY

tiLViR ■mnNa. MABVI-AND

are computed using subroutines for each of the vector operations

involved; namely, finding the cross product between two vectors

(CROSS), finding the dot product between two vectors (DOT), reducing

a vector to its unit vector and magnitude (UNIT), multiplying a

vector by a scalar (MULT), finding the angle of a vector given

the sine and cosine (ANGLE), and finding the elevation and

azimuth angle of a vector with reference to a given coordinate

system (TRIAD).

The same procedure is used to find the ranges, range

rates, and angles between the radar and the reflection point, and

the radar and the clutter spot. The reflection point, located

at the surface of the earth, comes into play when the radar's

antenna is broad enough to permit energy to be reflected off the

surface of the earth and up to the target. Two types of reflections

are considered: single oounce, where the radar energy follows

the reflected path to the target and the direct path back, and

double bounce, where the energy travels the reflected path to

and from the target. The clutter spot is simply the point

on the surface at a range equal to the range to the target.

The aspect angles, also shown in Figure 3-2, are

calculated by the geometry subroutine and are used directly in

finding the target's cross section at this instant of time.

Subroutine RATSCT is used to look up the cross section table.

The cross section value selected is the value opposite the closest

angle to the desired angle.

- 38 -

-

THI JOHN» HOPKINS UNIVMIITY APPLIED PHYSICS LABORATORY

SILVIR SPHINO. MARYLAND

There are a total of 31 parameters (many with three

components) which are calculated and stored by the geometry

subroutine. However, at the present time only the most important

ones are printed as described in Chapter IV on output.

3.3 Radar Calculations

The radar calculations are performed each time step

by subroutine DAVE following the execution of the geometry

subroutine. These calculations W'll be briefly described here

with the details being available in Section 3.0 of Appendix B.

Antenna. The physical size of the antenna, the efficiency,

and the radar frequency are used to calculate the antenna gain and

the beamwidth. The value for the bearawidth is then used 1th the

given values for the search sector and the time allowed to scan

the sector to determine the scan rate. The scan rate is used

with the given value for the pulse repetition period to calculate

the number of pulses received from the target in one scan. The

antenna gain and the number of received pulses per scan are prime

parameters in the subsequent calculations of the signal power and

the detection probability.

The antenna equations as they now exist can only be

used for radars that use the same antenna for both transmit and

receive. This, however, is the most common case.

Signal. The signal power is calculated with the standard

range equation using the pertinent parameters supplied by the input

cards. The signal power is determined for the direct signal from

- 39 -

». . -* iiiii———

TMl JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVIII «M NO MAKVUtM»

the radar to the target, the single bounce signal which takes one

path to the target via a surface reflection, and the double bounce

signal which reflects off the surface to and from the target. The

appropriate antenna gain is used in each calculation. That is,

in many cases the angle taken by the reflected-signal path is

greater than the half-beamwidth of the antenna which puts it in

the antenna's sidelobes. In this case the signal power for the

single-bounce and double-bounce signals are reduced from the

mainlobe signal by the given antenna sidelobe ratio (squared).

Clutter and Receiver-Noise Power. The signal power

will be divided by the clutter and receiver-noise power to form

the "signal-to-noise" ratio used in the detection-probability

calculations. The receiver noise is calculated using the standard

equation involving Boltzman's constant, the signal bandwidth, and

the receiver noise figure; rain backacatter is added to receiver noise

The clutter power, on the other hand, involves numerous

parameters such as tbe antenna beamwidth and the geometrical

quantities shown in Figure 3-3. The clutter power is calculated

the same way as the target signal power, i.e., using the radar

range equation, but using the clutter cross section instead of the

target cross section. The clutter cross section is equal to the

clutter area times the normalized clutter cross section a , which o was discussed in Chapter II.

As indicated in Figure 3-3 the clutter area to be used

in the calculation depends on the range to the target. The reason

for this is that the region on the earth's surface that must be

- 40 -

TH( JOHN« MOPKIN» UNIVIM«ITV

APPLIED PHYSICS LABORATORY •ILVIR ■PRINS M»«YL«ND

ELEVATION PLANE

■

'■

UNIFORM SIDELOBES

ANGE CASE 1

/ / / /

mmmmmmlhmllmmm

/ CLUTTER INTERCEPTED BY PULSE WIDTH

NOTBi CASE 1 TARGET COMPETES WITH MAINLOBE AND SIDELOBE CLUTTER

CASE 2 TARGET COMPETES WITH SIDELOBE CLUTTER

AZIMUTH PLANE

RADAR

\ V \ SIDELOBE \ \r CLUTTER

CLUTTERX v AREA

\ \ CASE 2

\ \ \ \

I /

/

/ /

CASE 1

\ \

\ r \ \

SIDELOBE .CLUTTER

+ MAINLOBE CLUTTER

/ / ■

SEMICIRCULAR RINGS

/ /

Fig. 3-3 CLUTTER AREA

■

* II II

THE JOHN« HOPKINS UNIVimiTV APPLIED PHYSICS LABORATORY

»u.««» •mnm. M*im.*ric

used in calculating the clutter power is that region which is

at the same range from the radar as the target. When the target

is low in altitude and at a long range then the main beam intercepts

both the target and the surface of the earth. In this instance,

called Case 1 in Figure 3-3, the clutter area is made up of two

pieces, one in the mainlobe of the antenna, the other in the

sidelobe. In either case the clutter area is bounded by the

pulse width in range.

When the target is close to the radar the clutter

region may be at an elevation angle that is out of the main beam

of the antenna, such as Case 2 in Figure 3-3. Even though the

physical size of the clutter area for this case may be comparable

to that of Case 1 the resultant clutter power received by the

radar receiver is considerably reduced due to the low gain of the

sidelobes.

Accuracy Calculations. The previous calculations can

be combined to form the signal-to-noise ratio which is important

in determining the radar's measurement accuracy as well as the

detection probability. Once the target is detected the radar

must determine the target's location, in range and angle, and its

doppler velocity or range rate. Since the target echo exists in

the presev>ce of receiver noise or clutter the ability of the radar

to extract the information is a problem in the statistical

estimation of parameters. The general form of the error for

measurements in range, range rate, or angle are of the form3

'D. K. Barton, Radar System Analysis. (Englewood Cliffs: Prentice- Hall, Inc., 1964), pp. 38-63.

- 42 -

D

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0

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8

TWI JOHN» HOPKINI UNIVKRSITV

APK' 1ED PHYSICS LABORATORY ■ILVIR »WtlHO HAmLAUO

error "

W^ „here E/N0 Is equal to the slgnal-to-nolse energy ratio which

includes the observation time. The factor y changes according

to whether range, doppler velocity, or angle is being measured

and according to other factors, such as the type of signal

modulation. For the range measurement the y factor is basically

equal to the bandwidth of the signal; for doppler velocity it is

equal to the length of tiie signal; for angle measurements it is

equal to the inverse of the beamwidth.

Thus for accurate measurements of the thrte parameters

it would be desirable to have a wide bandwidth signal, transmitted

over a long time duration, and using an antenna with a very narrow

beamwidth. In addition the signal-to-noise ratio mvst be high.

The mathematical form of the accuracy equation is the

same for both the search accuracies and the tracking accuracies.

The prime difference is that the tracking accuracies are usually

better due to the longer time of the measurement. The search

accuracies are naturally lower then the tracking accuracies since

the time for measurement is short due to the finite time that the

target remains in the scanning beam.

3.4 Detection Calculations

The detection problem was discussed briefly in Chapters

I and II. It was indicated (See Figure 2-3) that two detection

- 43 -

THC JOHN« HOmim UNIVKMITV APPLIED PHYSICS LABORATORY

»Lvn »niNa MADVLANO

theories are used: Marcum-Swerling3 and Shetland* theory. The

Marcum-Swerling theory is a generalized theory for calculating

the probability of detecting a target, with a fluctuating cross

section, in the presence of receiver noise. It was put into a

form, suitable for calculation by a digital computer, by Fehlner5

and is incorporated in this program as subroutine MARCUM. The

Shetland theory is an extension of the Marcum-Swerling theory to

include the offeet of clutter as an interfering noise source as

well as receiver noise. The Shetland theory is programmed as

subroutine EDDIE.

Generalized Detection Calculations. The basic parameters

required to calculate the detection probability are the false-alarm

time from the input cards, and the signal-to-neise ratio and number

of received pulses per scan from the radar calculations. The

detection of a target is a decision procesb and is based on

establishing a threshold level at the output of the receiver. A

target is assumed to be prei 3nt if the signal out of the receiver

is large enough to exceed th threshold. Occasionally noise alone

will exceed the threshold generating a false alarm. The threshold

level is calculated as a function of the noise power, the bandwidth,

a J. I. Marcura and P. Swerling, "Studies of Target Detection by Pulsed Radar," IRE Transactions on Information Theory, vol. IT-6 (April, 1960).

4E. Shetland, "False Alarm Probabilities for Receiver Noise and Sea Clutter," JHU Applied Physics Laboratory, BBD-1387,October 27, 1964.

8L. F. Fehlner, "Marcura's and Swerling's Data on Target Detection by a Pulsed Radar," JHU Applied Physics Laboratory, TG-451, July 2, 1962.

„ 44 _

THE JOHN« HOPKINS UNIVEMITV

APPLIED PHYSICS LABORATORY •ILVRR •PNIM. MARYLAND

and the number of pulses, to produce the false alarm time specified

by the program input. The probability of detecting a target is

then the probability that the target signal exceeds this pre-

determined threshold level which varies up or down according to

the level of the noise power. The procedure for calculating the

detection probabilities involves evaluating a series of incomplete

Gamma functions, where each function is expressad as a series.8

Target and Clutter Statistics. The two detection theories

provided in this program provide a fairly flexible arrangement for

calculating the detection probability for targets and clutter with

various statistical characteristics.

In the Marcum-Swerling theory incorporated in this

program five target models may be specified. The first, called

Case 0, is a target with a constant cross section area. The

targets in the other four cases have fluctuating cross sections.

Case 1 and 2 targets have a Rayleigh amplitude distribution and

apply to targets that can be represented as a number of independently

fluctuating reflectors of about equal echoing area. Case 1 and 2

are often used for aircraft targets. Cases 3 and 4 apply to

targets that can be represented as one large reflector with a

number of small reflectors, such as a missile. Cases 1 and 3

apply foi targets whose cross section fluctuates slowly, or with

a period equal to the scan time. Cases 2 and 4 are for targets

whose cross section fluctuates rapidly, i.e. from pulse-to-pulse.

3 Ibid., pp. 21-32.

7Marcum and Swerling, loc. cit.

- 45 -

,—_. ..-

■

TMI JOHN» HOPKINt UNIVIIWITV APPLIED PHYSICS LABORATORY

•tLvm UMuna. MAMVLANO

3.5 Summary

The processing part of the program contains the

calculations performed on the Input data to ascertain the

performance of the proposed radar In the given situation. The

calculations are divided Into three groups on geometry, radar,

and detection.

- 46 -

"■"■'^ '',/■ .'mmmmmammmMmmmrc: ^ . . ,-? u " ■ - """"" Ul -■■■■-■ "—'i'" —*~— ■■■■■ ■ ,-— —

D

D 0

0 0

Since the detection probability Is higher for targets

which fall In Cases 2 or 4 schemes are often devised to Insure

that the cross section fluctuates rapidly. One way of doing

this Is to employ frequency jumping vhlch Is accomplished by

varying the transmitted frequency from pulse-to-pulse or within

a pulse.

In the Shetland theory, as used In this program,both

the target and the clutter are of the Case 1 variety, slowly

fluctuating with Raylelgh amplitude distributions. The clutter

does fluctuate somewhat faster than the target depending on the

width of the clutter spectrum (calculated In the radar subroutine).

Under these conditions, however, very large slgnal-to-nolse ratios

are required for detection. On the other hand, If frequency

jumping is used with a sufficiently wide frequency range the

target changes to a Case 2 or 4 target and in addition the clutter

spectrum is widened causing the clutter to fluctuate randomly

like receiver noise. Under these conditions the Marcum-Swerling

theory can be used which will yield a much higher probability of

detection.

0 D n

0 0 0 y o a i

■'

THI JOHNS MOPR1U« UNIVCRtlTV

APPLIED PHYSICS LABORATORY SlLVn Sr-IINO. MARYLAND

As indicated in the program flow diagram the output for

a particular run is stored in an array and is not printed until the

calculations for that run are completed. This procedure has the

advantage of assuring that all the results of calculations in

one subroutine are available to other subroutines. In addition,

it allows all of the output commands and formats to be placed in

the main program.

The calculations and processing techniques discussed in

this Chapter are used to produce the program output described in

the next Chapter.

- 47 -

■■■ ■-n «W III II I i I 1 ■ ■ -"——

■

TNt JOHN« MOWdN« UMIVIMITV AWJfO PHVtlC» LABORATORY

•«.Via •»■«IN« MMtUkNO

IV. PROGRAM OUTPUT

The results of the calculations discussed in Chapter III

are provided as output in either a printed or digitally plotted

format. The bulk of the output is printed by the main program,

as previously mentioned, except for the target motion and cross

section decks which are read and printed by separate subroutines.

Two samples of the program output ara included in this Chapter,

one on the time simulation of the airborne-radar/missile-target

example, proposed in Chapter II, and the other on the range profile

for the same radar.

In review of the inputs proposed in Chapter II the ■ ■

atrborne-radar/missile-target example to an airborne radar at

an altitude of 30,000 feet and at a range of approximately 43

nautical miles (diagonal distance of 30 nautical miles on a

square) from a ballistic missile launched from the surface. The

radar operates at a nominal frequency of 5000 megacycles per

second with a peak power of 500 kilowatts and employs frequency

jumping from pulse-to-pulse. The ballistic missile flies in a

vertical plane for 41 seconds reaching an altitude of 24,000 feet.

The output of the program in the time simulation mode will cover

39 seconds of the engagement and will indicate the following: the

success or failure of detecting the target, the length of time from

missile launch required for detection, the accuracies of locating

the target, etc.

- 48 -

TMt JOHN« ^C.-KINII uNivmaiTv APPLIED PHYSICS LABORATORY

■ILVIK «MlltM. MaHTLAND

For an example of the range profile option the target

is assumed to be at a stationary altitude of 30,000 feet and

have a constant cross section of .1 square meters. The range

from the target is decreased in increments of 2 nautical miles

from a range of 78 nautical miles with detection and radar

calculations being made each increment. The single-scan detection

probability and the signal-to-noise ratio are plotted versus

range by the CALCOMP Plotter.

4.1 Time Simulation Example

The output for the time simulation of the airborne-

radar/missile-target example is divided into four parts for

discussion purposes. The first part is a listing of the input

data. The remaining three parts give the output data arranged

according to computed constants, geometry calculations, and

radar and detection calculations. ■

4.1.1 Input Data

The data inputs are printed out in three parts: target

motion deck, target cross section deck, and input parameter cards.

The target motion for the ballistic missile is reproduced in

Table 4-1. The cross section of the ballistic missile is given

in Table 4-2. The cross section values are listed in decibels

above a one square meter reference, for angles between 0 and

36.6 degrees.1 The values are arranged in the format previously

described in Table 2-2.

Remainder of table to 180° aspect angle not shown.

- 49 -

_^_

THE JOHN« HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY llLVIR »P-«,NO MAKrUkNO

Table 4-1

PROGRAM OUTPUT: TARGET-MOTION INPUT

EXAMPLb TAKG büÜST TkAJtLTÜKY COÜRUINATCS

UÜWN KANGi:

AKO TIME PDilTIÜN SPELU ACCELEK

1 0. 0. 0. 0. 2 1. 0. 0. 0. J 2. 0. -0. 0.01 ^ 3. 0. -0. 0.02 i) 4. 0. 0.1 0.03 6 i. 0. 0.1 0.03 7 ü. 0. 0.1 0.04 8 7. 0. 1.4 3.49 9 8. 5.0 7.3 8.13

10 9. 17.0 16.5 9.55 11 10. 38.0 26.6 10.73 12 11. 70.0 37.9 11.84 13 12. 114.0 50.2 12.85 14 13. 171.0 63.5 13.74 15 14. 242.0 77.7 14.53 16 13. 327.0 92.6 15.23 17 16. 427.0 108.1 15.63 lö 17. 543.0 124.2 16.33 19 ia. 675.0 • 141.1 17.51 20 19. 825.0 159.2 18.70 21 20. 994.0 178.5 19.69 22 21. 1183.0 199.0 21.11 23 22. 1392.0 220.7 22.31 24 23. 1624.0 243.6 23.52 25 *4. 1880.0 267.7 24.72 26 25. 2160.0 293.1 25.86 27 26. 2466.0 319.5 26.99 28 27. 2799.0 347.0 28.03 29 28. 3160.0 375.6 29.01 30 29. 3550.0 405.0 29.84 31 30, 397C.0 435.1 30.26 32 31. 4420.0 465.4 30.12 33 32. 4900.0 495.8 30.89 34 33. 5411.0 527.2 31.85 35 34. 5954.0 559.7 33.13 36 35. 6530.0 593.6 34.54 37 36. 7141.0 628.9 36.03 38 37. 7787.0 6ö5.7 37.61 39 38. 6471.0 704.2 39.28 40 39. 9194.0 744.3 40.87 41 40. 9959.0 786.0 42.60

ALTITUDE

PUSIT1ÜN SPEED ACCELERATIUN

0. 13.0 5?,0

11 .0 21u.O 330.0 478.0 654.0 858.0 1091.0 1353.0 1643.0 1963.0 2313.0 2693.0 3103.0 3545.0 4017.0 4522.0 5C59.0 5628.0 6230.0 6866.0 7535.0 8239.0 8976.0 9748.0 10554.0 11394.0 12268.0 13176.0 14115.0 15083.0 16081.0 17108.0 18165,0 19251.0 20368.0 21517.0 22698.0 23914.0

0. 25.9 52.3 79.1

106.3 133.8 161.8 190.1 218.6 247.2 276.1 305.3 334.8 364.7 395.0 425.7 456.9 488.5 520.6 553.0 585.3 618.9 652.5 686.3 720.3 754.6 768.9 823.2 857.2 890.9 923.6 954.4 983.5

1012.4 1041.4 1071.1 1101,6 1132,9 1165.1 1198,3 1232,3

25.66 26.14 26.59 27.01 27.40 27.74 26.14 28.45 26.47 28.77 29.05 29.36 29.73 30.08 30.48 30.93 31.39 31.90 32.27 32.62 32.96 33.34 33,64 33.93 34.19 34.27 34.33 34.16 33.92 33.33 31.94 29.49 28.99 28.69 29.37 30.07 30.87 3U75 32.67 33.57 34. 58

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51

THE JOHNS HCPKIN« UNIVERilTV

APPLIED PHYSICS LABORATORY SILVER SPRING MARVLAND

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52

!

fl

.1

THI JOHN* HOI-KIN» UNIVIRIITV

APPLIED PHYSICS LABORATORY ■ILVI* Srmma ttAKYLAMO

0 The input parameters that are used to describe the

ft initial radar-target geometry, the clutter and noise conditions,

the radar characteristics, and the output options, are reproduced

in Table 4-3. These parameters, which were described in Chapter

n n> are given the values listed in the nominal-value column. The

nominal vilue, are multiplied by the number given in the conversion-

[J factor column to agree with the units in the definition colu«n.

41-2 Computed Constants

The first set of output parameters are constant with

|] respect to time and are therefore calculated only once during

each run. The results of these calculations are listed with

definitions in Table ,4-4. Note that the horizontal scan rate of

the antenna was computed as 800 degrees per second. This implies

that the airborne radar must employ an electronically scanned

|] antenna since mechanical scanning cannot be performed at an 800

degree/second rate

0 4-1-3 Geometry Output

li The result* of the geometry calcUations ocoUr on

three pages of the printed output. The first page is for the

U ranges ana range rates „et.een the radar, target, refiection point,

J and clutter spot, the second page is for the angle calculations;

and the third page is for the cross-section calculations.

The definitions of all the geometry output para-eters are listed i„ Table ,., in tBO ^ ^ , ^ ^ ^ ^

angle calculations and part ,1 for the cross-section calculations. ■

- 53 -

i— —-—; ... —: , I.,.. ■..■■,...■■,,

THt JOHN« HOFKIN» UNIVIHSiTY

APPLIED PHYSICS LABORATORY SILVH •ntiNa. M«IIVLAND

Name

LAMBDA

AR

GAIN

BETAH

BETAV

FSCH

TAU

N

PO

PAVG

PNOISE

PFA

Table 4-4

PROGRAM OUTPUT: COMPUTED CONSTANTS FOR AIRBORNE-RADAR/MISSILE-TARGET EXAMPLE

Definition

Wavelength in feet

Effective area of antenna in square feet

Mainlobe gain in dB

Horizontal beamdwidth in degrees

Vertica^ oeamwidth in degrees

Horizontal scan rate in degrees/sec

Compressed pulse length in seconds

Number of hits/scan

Power received at initial range in watts/square feet

Average transmit power in watts

Noise power in dB above one milliwatt plus rain backseatter power when RAIN «■ 1.

False alarm probability

::

D ■

[ Computed 'Value 1

.1967

20.66 :

38

2.0

2.0 i 800

1.0 x 10~7 j

5

2.0 x 10"12

5 x 103

-99 0 1.4 x 10~8

:!

i !

D

- 54 -

0 a o

THE JOHNS HOPKINI UNIVIRIITY APPLIED PHYSICS LABORATORY

■ILVBN SPRINO. MARYLAND

Table 4-5

DEFINITIONS OF GEOMETRY OUTPUT PARAMETERS, PART I

1. Page 1: Geometry Range Calculations

R

Rl

R2

RDOT R1D0T R2D0T

TDIR

DELT

Radar-to-target range in feet

Radar-to-reflection-point range in feet

,Target-to-reflection-point range in feet

Respective range rates in feet/sec.

Propagation time of direct signal in sees.

Time difference between one-way direct signal and one-way reflection signal in sees

Page 2: Geometry Angle Calculations

ELAAT Elevation angle of radar/target line in degrees

ELAAB Elevation of radar/reflection-point line in degrees

ELAAC Elevation of radar/clutter-point line in degrees

AZAAT Azimuth angle of radar/target line in degrees

ALPHAB Angle of incidence of radar/reflection-point line with respect to tangent at reflection point, in degrees

ALPHAC Angle of incidence with respect to tangent at clutter point, in degrees

THETA(l) Angle .between target plane of rotation and radar polarization plane in degrees

- 55 -

THI JOHNS HOPKINS UNIVSRSITV

APPLIED PHYSICS LABORATORY SILVI* SPHIIM. MtRVLAND

3.

Table 4-5

DEFINITIONS OF GEOMETRY OUTPUT PARAMETERS,, PART II

and vertical polarization, in square feet

ALP(2) Single-bounce-path aspect angle in degrees

SIG(2,1) Target cross section for single-bounce path and horizontal polarization, in square feet

SIG(2,2) Target cross section for single-bounce path and vertical polarization, in square feet

ALP(3) Double-bounce path aspect angle in degrees

SIG(3,1) Target cross section for double-bounce path and horizontal polarization, in square feet

- 56 -

11

U

0 D

Page 3; Geometry Cross-Sections Calculations

ALP(l) Direct-path aspect angle in degrees

SIG(1,1) Target cross section for direct path and horizontal polarization, in square feet (POL = 0)

SIG(1,2) Target cross section for direct signal nnH v^r+.ioal nolari^ation. in sauare feet

D 0

o SIG(3,2) Target cross section for double-bounce path ^

and vertical polarization, in square feet

0 0 0 0 0 0 0

-

1 THE JOHN« HOFHINS UNIVERSITY

APPLIED PHYSICS LABORATORY Suvtl» tPHINa. MARVLAMD

The geometrical qua^ntitites that these definitions describe were

previously illustrated in Figure 3-2.

The actual output for the geometry range calculations

for the airborne-radar/missile-target example is reproduced in

Table 4-6. This Table indicates that the range R changes only

16,000 feet (2.7 nautical miles) from the initial range, increasing

from zero to 36 seconds and decreasing thereafter.

The geometry angle calculations are shown in Table 4-7.

Note that all of the elevation angles are negative indicating

that the antenna beam is pointing down throughout the engagement.

The last page of the geometry output showing the target

cross-section values for the different signal paths is reproduced

in Table 4-8.

4.1.4 Radar and Detection Output

The results of the radar and detection calculations are

presented on two pages. The first page includes tho output

parameters associated with the detection probabilities and the

accuracy calculations which are defined in Table 4-9. The

second page presents the output parameters associated with the

signal and clutter power level calculations which are defined in

Table 4-10. The corresponding two pages of output for the airborne-

radar/missile-target example are shown in Tables 4-11 and 4-12.

Some of the most important conclusions revealed by these two

tables are indicated below.

The first measurable single scan de\ tion probability

PD1, equal to .403, occurs 11 seconds after the missile launch.

- 57 -

THt JOHN» HOPKINS UNIVIRSITV

APPLIED PHYSICS LABORATORY SltVIM SPRtNG MAIIVLAND

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Table 4-9

DEFINITIONS OF DETECTION PROBABILITIES AND RADAR ACCURACY CALCULATIONS

PD1 Single-scan detection probability for the direct path signal

PDCUM1 Cumulative detection probability

PDT Detection probability for direct and reflected paths (when coincide)

PCUM?1 Cumulative detection probability of combined signal

CASE Marcum-Swerling fluctuating-target model (0-4)

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S/N Signal-to-(receiver) noise ratio in dB

C/N Clutter-to-(receiver) noise ratio in dB

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DOPDIF Difference in doppler frequencies between direct and reflected paths, in cps

TDIF Time difference between direct and reflected signals in microseconds

DELR Search range accuracy in feet

DELD Search doppler accuracy in cps

DELA Search angular accuracy in milliradians

SIGNV Angular tracking accuracy in vertical plane

SIGNH Angular tracking accuracy in horizontal plane

SIGT Total angular tracking accuracy in milliradians ■

ZT Altitude of target in thousands of feet

- 61

THC JOHN« HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SlLVM MWINa. MANVLAND

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DEFINITIONS OF SIGNAL AND CLUTTER POWER LEVEL CALCULATIONS

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Reliable detection of the missile, however, does not occur until

19 seconds when the detection probability equals unity. From 19

seconds until the end of the engagement the detection probability

fluctuates between unity and .03. The latter probability value

occurs at 36 seconds where the missile cross section drops to its

smallest value of .42 square feet.

The missile is not detected reliably until 19 seconds

because of the high clutter conditions. As shown in Table 4-12

the clutter power level CLO is composed of clutter echoes entering

the mainlobe of the antenna until 18 seconds. Aftt 18 seconds

the clutter level decreases abruptly because at this point the

clutter echoes enter via the antenna sidelobes only. Thus the

airborne radar in this particular engagement cannot detect the

missile in a background of mainlobe clutter, but must wait until

the missile is high enough for reliable detection in sidelobe

clutter.

The effect of the high mainlobe clutter return is also

reflected in the signal-to-noise-plus-clutter ratio, S/(C+N) in

Table 4-12. As shown the ratio is less than unity below 19

seconds and greater than unity with fairly high values above 19

seconds.

After the radar detects the missile at 19 seconds its

location is determined in the search mode to an accuracy of two

feet in range, 358 KHz in doppler, and four milliradians in angle.

The poor doppler accuracy would force this particular radar to

determine velocity by measuring the range rate. The angle accuracy

- 65

1H( JOHN« MO^KIN» UNIVERdTY APPLIED PHYSICS LABORATORY

■u vn •raiMo. MMTUND

of the search radar in this case is high enough for the radar to

begin tracking with an angular accuracy equal to one milliradian.

4.2 Range Profile Example

The time simulation mode analyzed the performance of

the airborne radar in detail in detecting and locating the missile,

launched approximately 43 nautical miles from the radar. The

range profile output discussed here will indicate the detection

performance of the same airborne radar but this time against a

simplified, generalized target.

The target for the range profile run is described by

the input parameters listed in Table 4-13. As shown the target

analyzed will remain at an altitude of 30,000 feet and will have

a constant cross section of .1 square meters (-10 dB/square meters). ■

The three input parameters shown are the only ones that are changed

from the previous time-simulation run. They specify calculations

to be performed every two nautical miles from zero to a maximum of

78 nautical miles.

The results are shown in Tables 4-14 and 4-15 in the

same format used in the time simulation output except that the

target range is incremented instead of the time variable. The

radar itself is operating in the same mode of operation as used

in time simulation run, that is, searching for the target over

a 26 by 30 degree sector in one second. This produces five return

pulses which are integrated by the radar to produce the detection

probabilities listed in Table 4-14. The corresponding signal-to-

noise-plus-clutter ratio is listed versus range in Table 4-15.

- 66 - 0

APPLIED PHYSICS LABORATORY SILVER SCHICS MARYLAND

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TMC JOHN« HOmtlN« UNIVtMITV APPLIED PHYSICS LABORATORY

■nvm srnw MiunuuiD

The detection probability PD1 and signal-to-noise-plus-

clutter ratio are generally the most important quantities in the

range profile ontion and are therefore plotted with the CALCOMP

plotter as shown in Figure 4-1 end 4-2. These figures indicate

that the nJrborne radar would reliably detect a .1 square meter

target at an altitude of 30,000 feet out to a range of about 40

nautical miles. The target altitude of 30,000 feet is significant

in that the clutter returns enter the receiver via the antenna

sidelobes. If the target was much lower in altitude the clutter

would be in the mainlobe of the antenna which would preclude

target detection.

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THE JOHNS HOPKINS UNIVERSITV APPLIED PHYSICS LABORATORY

Sn vtR Sfrtm, MARYLAND

HimiOMMrkUiMMftm.'M**!....' nm^jt

1.0

^ ,

0.3

\

I >-

\ 1- I U I to < 0.6 m

1 1

o 1 OS 1 a. 1 z 1 o 1 i- 1 u 1 1- ! UJ 1 a 1 z i < 1 o 1 If) iu 0.4 -i

j 1

u 1 z 1 M

\

NOTES: RADAR ALTITUDE ■ 30 KFT TARGET ALTITUDE = 30 KFT

n 7 TARG ET RCS= 0.1 SQ. MET 22dB §0H. MILE

ER 0°= -

= -38dB @78 N. MILES PFA= 1.4 x ID"8

HITS/. SCAN - S ■

0 1 1 1 J i i 1 i -LJ_± 1 1 1 1 J _L_l,_l..J i i i i MVU

,

60 70 30 RANGE (n. miles)

■

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Fig. 4-1 DETECTION PROBABILITY VERSUS RANGE

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»Iliiii — in li'i" ■

THC JOHN* HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY «11««« SMtlNO M»«YI »NO

RANGF (n, miles)

Fi9. 4-2 SIGNAL-TO-CLUTTER-PLUS-NOISE RATIO VERSUS RANGE

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THE JOHN! HOPKINI UNIVERSITY ^APPLIED PHYSICS LABORATORY

SILV1R tPRINa. MARYLAND

V. SUMMARY AND CONCLUSIONS

The digital computer program for radar analysis and

simulation has been described. In this chapter some of the high-

lights will be summarized and some areas for future development

will be discussed.

5.1 Summary

The report started with a description of the basic

radar problem to be analyzed in this study: the detection,

acquisition and tracking performance of a generalized radar

against a single moving target in a clutter environment.

Chapters II through IV described the program inputs, the process-

ing, and the program outputs, respectively. Chapter II on inputs

descritied the manner in which the radar problem is defined by the

user. A flexible but detailed arrangement was indicated for

describing target moLion, target cross section, initial radar-

to-target geometry, radar characteristics, clutter and noise

environment, and program options Any of the input parameters

my be changed for performing multiple runs on any one job.

The processing or calculations performed by the program,

as described in Chapter III, were shown to be arranged as a time

simulation of the radar-target engagement. The geometry calcu-

lations are performed with vector algebra permitting an accurate

simulation of the radar and target motion, and calculation of the

pertinent geometrical quantities between the radar, clutter, and

reflection points. The radar calculations are performed each

time step yielding signal-to-noise ratios, clutter characteristics,

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THt rjHK» HomiNi UNIVHISITV

APPLIED PHYSICS LABORATORY ■a.v(n ■ram*. MARYLAND

.

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and search and tracking accuracies. The detection probabilities

are calculated for many target and noise models, and include the

cumulative detection probabilities. A program option permits the

target model to be greatly simplified for performing radar

comparisons with the target motion and cross section held

constant.

A few samples of the program's output were presented

in Chapter IV. In the time simulation mode the output is arranged

as a function of time with calculations being made every time the

radar antenna scans past the target. The output parameters are '

quite extensive and include: single-scan and cumulative detection

probabilities for the direct and surface-reflected signals; search

accuracies in range, doppler, and angle; angular tracking accuracies;

clutter statistics and power levels; cross section variations;

signal-to-noise ratios; and many geometrical parameters.

In the range profile mode the target's motion and

cross section are held invariant to permit comparisons of radar

performance„ The target model is grossly oversimplified in this

mode which is necessary to analyze the performance of a radar,

or radars, for a broad range of target cross sections, target

statistics, clutter conditions, and distances from the radar.

That is, the boundaries of the radar's performance can be found

by operating in the range profile mode with calculations being

made for all probable cross section values, clutter conditions,

etc., without regard to one particular radar-target engagement.

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THI JOHN« HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY ■ILVH «PHINO, MARYLAND

Thus, the range-profile mode is used for calculating

general radar performance, while the time-simulation mode is for

detailed radar performance.

The actual program was written in FORTRAN II In a manner

to achieve considerable flexibility and growth capability. This

was accomplished by dividing the program into 22 subroutines

controlled by one main program. Thus, if any changes are required,

or desired, they may be inserted into the subroutine in which they

apply without affecting the status of the other subroutines.

5•2 Areas for Future Development

The program as it now exists is a fairly complete

representation of the radar-target engagement. There are some

areas, however, which deserve future attention; some of these

were indicated in previous chapters.

Antenna. There are two antenna characteristics that

are simplified in the present program that could be changed to

achieve a more accurate representation of an actual antenna. They

are the antenna beam pattern and the antenna beam motion.

Presently, the antenna beam pattern has a constant

mainlobe gain ever the solid angle defined by the half power

points, and a constant sidelobe gain for all other angles. In

the future, it may be desirable to modify the radar-calculation

subroutine, or add an additional subroutine, to incorporate

antenna gain variations with angle.

The motion of the antenna beam is presently arranged

to scan the given angular sector in the allotted scan time with

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THC JOHN* HOmilNS UNIVHWITV

APPLIED PHYSICS LABORATORY •14-VKK tnilNa. MARYLAND

the target assumed to be in the center of the sector. A more

accurate representation of the antenna beam motion can be incor-

porated by having the antenna beam follow a predetermined scan

pattern starting from an initial position specified on the input

or chosen randomly by Monte Carlo techniques. This method would

change the time increment of the output. That is, the calculations

of the radar detection probabilities would be performed only at

those times in which the target is in the antenna beam. This may,

or may not, occur once each sector-scan time as in the current

program.

Acquisition Mode. Presently the transition from the

search mode to the tracking mode is assumed to occur instantaneously.

This somewhat unrealistic assumption can be corrected by calculating

the time required to point the antenna at the target. The ca.~M-

lation can be programmed using techniques by Barton1 and quantities

available in the present program; namely, the time required to

scan the search sector, the ratio between the size of the search

sector and the tracking beam, and the signal-to-noise ratio.

Jamming. The present program does not consider jamming.

However, radar performance in the presence of certain types of

Jamming, such as wide-band noise jammers, could be added in a

straightforward manner by modifying the signal-to-clutter-plus-

noise ratio to include the noise power received from a jammer.

1D. K. Barton, Radar System Analysis. Prentice-Ha11, Inc., Englewood Cliffs, New Jersey, 1964, Section 14.2.

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APPLIED PHYSICS LABORATORY SlLVIR SPRINO. MARYLAND

Other Areas. There are evidently other areas of the

program that could be improved or other simulations that could

be added, to represent more accurately the real situation. Some

of these areas are an improved simulation of the tracking mode,

clutter model for land (sea clutter model and rain clutter already

included), multiple target detection, sequential detection, and

target motion not confined to a vertical plane.

(

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THI JOHN* HOPKINS UNIVIBSITY APPLIED PHYSICS LABORATORY

atLvn trmNa. MAIIVLANO

APPENDIX A

LITERATURE REVIEW

The literature was searched for digital computer

programs and radar simulation techniques applicable to this

study. The pertinent literature can be conveniently grouped

into the following categories: war game simulations, computer

programs for radar analysis, and detection-probability algorithms.

War Game Simulations. Numerous programs exist which

simulate the radar only to the extent of determining the outcome

of a war game. Examples of war game simulations involving radars

are programs written by Andrus 1 , and Kennard8. The program by

Andrus is designed to simulate the interactions between surface-

to-air missile systems and aircraft. The different events of

the game are assigned probability values and during the operation

of the program the occurrence of an event is determined by comparing

the given probability with an internally generated random number.

The gross parameters of the system are fed into the program along

with the number and positions of the aircraft. The program

determines missile firing doctrines, optimum number and placement

of the radars, and target kill probabilities.

A. F. Andrus, "A Computer Simulation for the Evaluation of Surface-to-Air Missile Systems in a Clear Environment," Naval Postgraduate School, TR/RP No. 67, June 1966.

'P. H. Kennard, "Effectiveness Studies of Manned Airborne Weapon Systems by Digital Simulation". Abstract: Instruments and Control System, vol. 32 (July, 1961), pp. 2171-1272. "~

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APPLIED PHYSICS LABORATORY SlLVtn SFKINd. MAIIVLANO

These types of war game simulations are considerably

different from the simulation in the Radar Analysis Program. The

only radar parameters involved in the Andrus3 program are the

search radar maximum range, the probability of detecting a. target,

and the time required to acquire a target with the tracking radar.

These are performance parameters of a. radar and in the Radar

Analysis Program they are calculated, as opposed to being provided

as inputs in war game simulations.

Computer Programs for Radar Analysis. Three programs

were found. They will be discussed in order of their increasing

similarity to the Radar Analysis Program.

The first program by White and James* is an example of

how a digital computer program can be used to simulate or model

a radio communication system, which is in many ways similar to a

radar system. Their specific problem was to model the transmitter,

receiver, and atmospheric phenomena in order to study interference

problems. This involved deriving mathematical equations to describe

the operations of each significant parameter of the transmitter,

receiver, and propagation phenomena. Some of the techniques which

are used for modeling the antenna gain pattern and the propagation

phenomena are the same as those used^for the radar case and can

be incorporated into the Radar Analysis Program if desired.

3Andrus, OJD. cit. , p. 4,

4 D. R. J. White and W. G. James, "Digital Computer Simulation for Prediction and Analysis of Electromagnetic Interference," IRE Transactions on Communications Systems, vol. CS-9 no 2 (June, 1961), pp. 148-159. "

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APPLIED PHYSICS LABORATORY tMLVI» «PHINO. MAHVLAND

In the second example of a program for radar analysis,

Wan briefly describes a completely analytical approach to a

radar program that he says will be developed.6 Wan's technique

is to isolate parts of the problem, such as, the transmitter,

receiver, antenna, noise power, weather, and target and to write

a mathematical model of each in signal-space notation. The

transmitter, for example, can be modeled strictly in the signal

or frequency domain by representing it by the Fourier transform

of the transmitted time waveform. The model for the target, on

the other hand, is described in both signal and space notation;

the cross section as a function of the transmitted frequency

describes the target in the signal (or frequency) domain, while

the range of the target from the radar describes the target in

the space domain. These mathematical models, or transfer functions,

of the different parts of the problem are combined to simulate a

particular radar-target engagement. For example, if the radar is

to operate in rain the transfer function which synthesizes the

rain would be included.

The last program discussed in this section has the most

similarity to the Radar Analysis Program. This program, by Boothe,

is used to determine the performance of an acquisition radar through

application of radar detection probability theory.8 Both programs

L.A. Wan, "Tactical System Radar Signal-Space Model," Ninth Conference on Military Electronics. September 22-24, 1965, sponsored by IEEE, pp. 13-17.

8R. R. Boothe, "A Digital Computer Program for Determining the Performance of an Acquisition Radar through Application of Radar Detection Probability Theory," Advanced Systems Laboratory, U. S. Army Missile Command, No. R|ferTR-64-2, December 31, 1964.

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THE JOHN« HOPKINS UNIVERSirv

APPLIED PHYSICS LABORATORY SlLVtR tPRlNa. MARYLAND

are arranged as time simulations of a single radar, single

target engagement. In addition both programs simulate the target

motion and cross section by means of external input for maximum

flexibility and use the standard radar equation for calculating

signal-to-noise ratio. Some of the notable differences between

the Boothe program and the Radar Analysis Program are in the

calculation of the noise power from all sources and in the

calculation of the detection probability. The Boothe program

ignores clutter but does have an option for including jammer

noise. The Boothe program incorporates a detection theory

developed by Marcum7 in 1947 which is a method for calculating

the detection probability for a constant cross section target

in a background of receiver noise. In 1960 Marcum in conjunction

with Swerling extended the theory to include fluctuating targets.8

This theory is used in the Radar Analysis Program in a computer

program developed by Fehlner.8

Target Detection Programs. Two other sources were found

which contain digital computer programs for calculating detection

probability. Both of these programs use Marcum and Swerling

detection theory.

7J. I. Marcum, "A Statistical Theory of Target Detection by Pulsed Radar," The Rand Corporation, RM-754, December 1, 1947.

8J, I. Marcum and P. Swerling, "Studies of Target Detection by Pulsed Radar," Institute of Radio Engineers Transactions on Information Theory. Vol. IT-6. (April 1960).

SL. F. Fehlner, "Marcum's and Swerling's Data on Target Detection ' by Pulsed Radar," The Johns Hopkins University Applied Physics Laboratory, TG-451, July 2, 1962.

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TMC JOHNS MOPKIN» UNIVnWITV

APPLIED PHYSICS LABORATORY •ICVSM l»IIIWa. MAnvLANO

The first program, by Kirkwood, calculates the cumulative

detection probability for targets approaching a scanning radar.10

In Kirkwood's program only two out of the four Marcum-Swerling

target models are used, and the single scan detection probabilities

are not available. The program is available in the reference and

is written in FORTRAN.

The other computer program for calculating detection

probability, developed by Nolen, et al.,11 is very similar to the

work performed by Fehlner. Both of these programs were produced

for the prime purpose of generating a set of general charts to

be used by radar analysts for calculating the detection probability

manually. The programs were run for a large number of cases to

cover most conditions. The charts produced by the Molen program,

however, are only available for 20 integrated pulses while Fehlner's

charts are available for up to 3000 integrated pulses. Tfe« program

developed by Fehlner was arranged in a subroutine and is the main

detection probability algorithm in the Radar Analysis Program.

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10JP. K. Kirkwood, "Radar Cumulative Detection Probabilities for Radial and Nonradian Target Approaches," The Rand Corporation, RM 4€;3-PR, September 1965.

lxJ. C. Nolen, et al.."Statistics of Radar Detection," Th2 Bendix Corporation, Baltimore, Maryland, February 1966..

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1 THt JOHN! HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY SILVIR »rnma. M»«TU«NO .

APPENDIX B

RADAR ANALYSIS PROGRAM IN FORTRAN

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APPLIED PHYSIOS LABORATORY ■H.V» »KINO. MAIIVUtNB

Table B-l

SUBROUTINE NAMES IN PROGRAM LISTING

MAINP

TARGIN CROSIN

GEOM

TARGET AIRCFT UNIT CROSS TRIAD MULT DOT ANGLE RATSCT

DAVE

EDDIE

PFA(f) PPAC(f) MARCUM

DGAM(f) DEVAL(f) GAM(f) EVAL(f) SUMLOG(f)

Section No

1.0

1.1 1.2

2.0

2.1 2.2 2.3 2,4 2.5 2.6 2.7 2.8 2.9

3.0

4.0

4.1 4.2 4.3

4.3.1 4.3.2 4.3.3 4.3.4 4.3.5

(f) Purctlon Subroutine

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LKAÜ4K C i KAüAK ANALYSIS PKLGKAM

Dl^LIVSiüM Ülij) bi^LNSlLh bPK(iU) Ul^cNSlUN XXAA(/), YXAX(<!i)t XY ÜlfDSSlUN XPhit), YHHii), XPV(

AX(2)f YYW^J

L»ift'\SlUN £;Ph(3ii,0) tCKü.*)tC( Idül, ^i^til ( ÖÜ,321 tH(80f 40 ? tCÜM( 100) OUMMth TIML,KbTi:PjKHT,tPH,C«CSbL»>ÖTfH,CüM NUIK = ÜU y Uu 5 i=i,lüO ^

5 CLMD = KJ,

KtNG = 1 twCPLT = i+6htNüPLr» KHtÜ = 0 CALL IMKGIM TJVIAX)

CALL CKUilN 9 HCAÜ iNPul JAPt b,iZ,NUOMA,iNüüM&,KC,SYM,Aföf (0(I),I = 1,6)

\l t-UKMAl ( iiti2f iiiA6|2eib.öf6A6j IFlNüJMA) 101,102,101

iol CALL TAKOirt(TMAXJ lüü iF(KüUMO) loi,10^,ii.3 lui CALL OKUSIN

Gu TC «i loH IKIMOUMA) 9,lu3,S 105 i«-(KC) 13,1ö,1Uü 106 IFKhtÜ) 13,lu7,lJ 107 wkiTt ÜUTPOT 7APt 6,10

10 l-UHHAT 11H1 5HX 22HALÜ1S INPOT PARAMETtRS //18H II 12 13 SYMBOL 1 7X i3HNüMII\AL VALUt 3X 17HC0NVEKSIÜN FACTOR 8X 2 loHuuFirMinuN ///) KHtÜ = 1

li «KlTt UUIPUT TAPfc o,14,^00(1A,NOOMb,KCfSYMfA,Bt(ü(I),1 = 1,6» 14 FÜHMAT (2X,Ii,I3,I4,2XtA6,2E2C.Ö,2X,6AoJ

001» (KL) = A*B üü ro s

15 COMiNOL iF(SVM-tMOPLT) 17,10,17

16 KtNC = 2 17 KHfcO = 0

Krir = o OUMI31) CÜM(3^) 00^(33) Cuf(34) CU^(33) lF(C0rt(6l))

ld TIMt = TMAX SNAML = (+5HRAN0t) TMAiSt = {+4HK Ml) iMHT = (CüMto2)-C0Mi5))/CCM(61) NKtM = NHT

19 iFliMRtM-NÜlM) 21,21,22 ^il NSTtP ■ NRfcM

NKtM = 0 00 TO 23

22 NSTtP = NOIM NRfcM = NRtM-NOIM

23 00 20 K=1,NSTEP

CUM(2)*CüSF(CüM{3))*C0SF(CaM(4)) CUK(2)«CüSF(C0Mi3))*SINF(CÜM(4)} CüM(2)*SIi^F(COM(3)) SllVf (CCMUO)) CüSF(COM(iO))

2ö,2ö,lö

♦ l.ül

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3t C c c

tUMci) = CüM(olJ*FLÜATF(Kril) + CüMCil KilEP = K KHT = KHT+1 CALL GtUM CALL üAVt CALL büUlk HiKti) ■ C0H(o3)/öo8U. Nt « 4 GU Tu iul CÜM63) = CUW{5) SNAML = (+4HTIME) iNAMt = (♦3HStC) HTSS = C0h<29) NHT » MINlFCTMAX,C0rt(bü)-CÜM(65n/HTSS NAtW « NHT If(NhtM-WÜiM) 3i,3it32 NSTtP = NREM NktM • 0 GU TL 33 NS1EP = NLilM M\tn = NKtM-NUiM UU 3t K=ltM>TfcP TIME = HTSi*fLÜATF(KHT) + CCM(65) KiTtP = K KHI » KHT+1 CALL GtUM CALL UAVE CALL tüüit

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NL = 1 LT » 1 LP = o Gü TL MK1TL GO 14 WiUTL Gu TC MK1TL GU TU MRiTb Gu TC WKlTt LL » v. K.G = L GU TC WKilt

I GU TC GPKdJ bPK(^) GPH(3) GPH(4) UPK(S) GPKibl GPK(7)

GU 531. UPK( 1)

(HUit4C2f4v3t3U3f304) tiSL UUTPUT TAPE bföul

uuTPCT TAtn, 6f8üi 3U5 OUTPUT TAPE ötbU3 30 5 UUTPul TAPt 6i3yUiSNAKEfTNAME JU5 UUTPUT TAPE 6|3yifSÂÊ,TÂrt£

T (5Uif5t2fi)i,3,3oöt307),l\L UUTPUT TAPE o,SwitGT(KGii)f (GHKGfl )fI^6»U) tGT(KG,23)f

GT(KG.2<t) job • GT(KUt2l • GTtKGf4) • bT(KG(5) = GT(KG»3) ■ oTCKGfi<:) • GT(KGfl3) • GT(KGf2E)

2 I=it7 « UPK(i)*57.2S577*!>

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hkilt üuTHuT IAPL 6»9(/2fGT(Kbf i) »(GPR(l»f I»if7) Gu TC jcb

kvKllt üülPüI JAPL o,9u3füT(KÜ»i)»{üPR(l),GnKÜ|H-l6)fGT(KGf I + I9»t i 1=1,3)

bü TL WRITL

Go 1L his lib LF =

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31i

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Öoi. KOKMAI

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i

job uüTPOT

UUTPOl LT+i

TAPL c,39^,iH(LT,I),l=l»2ü)

lAPÜ ü,393,H(LT,l),{H(LT,I),I=21,3i)

JO9 ,309,^12 IKLT-ivSIfcP) LP = LP+1 LflLf-HV) 3iC,3o2r3oi: LL = LL+1 UiLL-'ji 3i)3,jii»3ii INK Alt lJÜl>ÜT TAPt ü,394 Ob lo Jos KL = UL+i iP(KL-t>) 3ol.3oi,jl4 CCi\Tli\Ut hUKMAI (bhilli-lt. IK IHK i4A 2HKi 13X ^HK2 12X ^HRÜÜT iOX ÖHR1D0T

ioX JHK^üüT xiX 4HTUlk ilX 4HDfcLT /5HÜ SfcC 7X 2HFT 13X <Lhi-T 13X 2rifT ii.X 6HFT/StC 9X 6HFl/StC 9X ÖHFT/SEC IOX 3HStL 12X 3hSbC //)

(35HiTlML LLAAl fcLAAü LLAAC A2AAT ALPHAB i9H ALPMAC IHLTA(i) /3Ho 5tC 4X 7(3HDtG 7X) //)

OijhlllMt ALP(i) SIGd.l) SlGli,2) ALP(2) SIG(2,U 39h bib(2,2) ALP(3) SIG(3,i) SIG(3,2) /5H0 SEC 3(4X 3Hütb oX i)HSg FT 5X ?HSg FT 2X) //)

(IHi AüftöriPDi PCUMi PUT PCUMT CASE VB S/N i)t)H KC Fül UOPDIF TUIF üELR ÜELD UELA 1.9h SIGi^H ilGT ZT /IHO A6,3öX,2HUB 5X 2H0Ö 13X ^X 3HCPi> 4X 2HUS 4X 2HFT 3X 37hNCPS *llLb MILS MILS MILb KFT //)

39c FüRMAl i 2 ->

H

391 FUKHAT 1 l 3

9ol FüKNAT 9u^ FuKriMl ^cJ MJKMAT jVi l-LiKhAl

1 j9i FüKMAT 394 Fühil^Mr

C/N SIGNV

3HCPS

I 1H1 30H Ihv

i 1h ( IH ( 1h ( In

PhA A0,4Bh

CPS

Süü HHAütL ÜÖM

CHS

bbd

b^O

a"* 1

SCB SÖii CLO^ CLUS CLO SIGÜ HEFL UELFC S/(C+N) //

ÜBH ÜBM ÜBM DBM ÜBM DB CPS ÜB // I

Ir is. i,dEii>.b) tb. l,F9.^,öFlü.2) F3,l ,Fü.2fbF10.2) r'j. if4Fc.3»F4. o,F6.0,3F7.1,F9,c,Fö«LfF5.1,F<>.0iF8. I«

Fo.ot^tFi.i) (111 »-5.i,Fo.j,&F7.üfF9.1,F7.1»F6.3,Fb.0,F7.1) ( iH )

1FICLH(59H büo,b9l,ötib DU dbV N=l,WSrbP oTtKt^w) = lo.*LooloF(oT(K,2o)/lo.7d3b7f GI(K,1H) = bTiN,iH)*b7.^9^7795 PUNCH ö9C, IGUKj i) ,bT(K,l*) ,GT (K, 2ü J t K=l ,USTEP ) l-URMAi (//(F6.i,F9.1,FiU.in OUl\l iNOt

o C C

PLLlfl.\G FUH VtKSIUM 11.

lFlCLh(6l)) ^0,90.700 UC li. K = l,WSThP hlK',2) = H(Kf2)*iü,-

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—r-

-

71u HCKfj.:) = (h(K,Ji:)+^ü.>/3. XHÄX ■ CüM(ö^)/t)CÖ(J. üfcLTAA = XMAX/o.JJ IcHLX = -UtLTAX XYAXti) = Ü. XYAX(i:) = w. YYAXili = O. YYAX(^) ■ iü. XXAX(i) ■ XiMAX XXAX(^} h o. XPH(i) = -UcLTAX/lo. XPhiti = ucLlAA/iJ. l>Xi = -.i*ÜtLTAX DX^ = -.2*UtLrAX ÖXJ = -.J*üELfAX ÜXM = -.^ufcLTAX ÜX5 = -.ö*ÜELTAX NPV = uoM(o2)/CuM<bi) ■♦• .Oi CALL PL INI (-l»6HüWHlTfct-6,i,ÜX4,i,ZI:KÜXrUhLTAX,XMAX,9.9,l,3Hl.Ü) CALL PLALP (-Oi^tUXSfT.y.ZH.b) CALL PLALP (-OKifUXJO.yfZH.ö) CALL PLALP (-Of2füX3f3.9,2H.^) CALL PLALP l-6»2fÜX3il.9t2H.2> CALL PLALP (-6fItÜXli-.liIHÜ) CALL PLl-UlM (2,UfXYAX,^,YYAX,lf 1) Üu 72t; I = iflC YPh(iJ = lÜ.-FLüATf(I-l) YPh(2) « YPH(i)

72t CALL PLPUN <2tÜiXPHf2IYPHI1,l)

XPV(i) » CCMl6i)*FCÜATF(I)/608ü. XPV(2) - XPV(l) IKXMUÜPilfäH 721f722i721

7*1 YPV(i) = -.Üb Üü TC 723

7«12 TtMP = FPbCDKXPV) YPVdl » -.1 CALL PLALP (-4,OfXPV*UXf>,-..^,TeMP)

723 YPV42) = -YPV(l) CALL PLFJN UfOtXPVt2fYPVtl<'Jl)

72b CüiMTliMOk YXAX(l) = 0. YXAX<2) = ü. CALL PLrUN(2fÜtXXAXt2tYXAXfi»i)

PLFUN (2fu,H(itl)»,MSTfcP,H(l,2),l,l? (1) (-l,6HÜWHITtf-6,2f'JX3ilfZERÜXf0eLTAX,XMAX,9.9fi,2H30) (-b»2fJX3t7.9»2H2g)

CALL CALL CALL CALL CALL CALL CALL CALL CALL Du 73C YPMl)

PLtNÜ PLiNi PLALP PLALP PLALP PLALP FLALP PLFUN

(-ö,2tüX3f5.9,2HiÜ) (-o,ltJX2t3.9,lHÜ) {-o,3füX4fi.9,3H-lü» (-/ai3fUX"*,-.if3ri-20) 12»0fXYAX,2fYYAX,l,l)

» IC.-FCUATFd-i)

73w YPF(i) « YPH(l) CALL PLFUN (2,ÜfXPH,2,YPH,1,i) DO lib I«liNPV XPVti) ■ CCM(öl)*FLÜATF(n/ü08Ü.

• i

- 88 -

r~

mmmmmmmm

Til

lit

733

iy.

APV(2) = XPV(i) I^iXFUUKI.S)) 731f/32,731 YPv(i) = J.95 Gü TC 73i Tcl^P = l-PbCüF(XPV) YPV(l) = 3.v CALL PLALP i-H,o,APV+ÜX3t3.75fTLMP) YPV(2) = b.-YPVtl» -ALL PL1-UM (i,ü,XPV,2,YPV,lf i) CLM1NUE YXAXii) = 4. YXAXm = 4. CALL CALL CALL

PLFUN (^,UfXXAXf2,YXAX,i,l} PLI-UN (2iüiH(if ll,NSTtPfHii»32)flfX) PLENü iKhl\ü)

C C

90 91

Ih(lMKLil) S,9,9i lf(CCi4(6l)) 29,29,19 ENC

- 89 -

l.i

<lüu2

SüfcKüü

TAHGtl HUKMAT FUKMA1 FOKMAT FUHMAT FUkMAT

l)t//) FUKMAT

liNt lAKolMTMAX) ION TiM/WcU) lüN EPHiJ^ü.ö)tCKUS6C{lb01,2)tüT(ÖQ,32),H(8Ü,4Ü),CÜM(10Ü) TihtfKSTtP,KHT,t-PH,CkUStC,GTtH,CüM INHLT <6X,2Aot2X>I3t2X,lF3.Ü)

( IJfiX,lF'«.Ü»2(3X,lF7.C,4Xf 1F7.1,2X,1FÖ.2)) (IHi. f2Aöf2^ri öliüST TKAJLCTJRY CÜURUlNATtS,//» ( IH ,2<»X,10HÜUWN RANGE,26X,ÖHALTlTUDfc,//) (iH »lOHCARD nMt,2(4X,31HPUSITIÜN SPEED ACCELtRATIÜN,

2ü^ FUKMAT iih , 14,2X,F<*.üf2(4X,F8.1t2X,F7.i,3X,F6.2,4X1)

IC

U

;>vj'

1.2

lote 4t tC 4Üu2

TMAX = KtAO 1 WRITt MRITE WRITE ÜU 11 KEAU 1 TMAX = MRITE

CUK(S) RETURN ENC

u • nPUl TAPE b,iüül,(TNAMt(J»,J=1,2),^MAX,TAU UUTPOT TAPE ö,2uul,(INAMEiJ),J=i,2) üUTPOT TAPE ö,2uo2 ÜOTPOT TAPE o,2uu3 i<=l,i\MAX IMFUT TAPE b,lJ02,L,T,(EPH(L,J),J = i,6) MAXlFiTjTMAX)

ÜUTPLT lAPt 6,2üÜ4,L,T,(EPh(L,J),J=l,6) = TAU

SOuRLUTiNt CROSiN

ÜlMENSiuN t:PH(3<.2,0) ,CKÜSbC( ItiÜl, 2 ), GT( du ,32) ,H( 80,40 ) ,CüM( 100 ) COWMEN Tii'ic,KöTtP,KHT,EPH,C«tSEC,OT,H,CüM CRObi-iEOl 1U IlMPtT FORMAT (l'jFb.i) FüiuVAT (1F6.2,12Aü)

Fu^riAT tiH ,1J<F6.1,JX),2X) FOKMAT {1H1,12Aü)

Ü^Mc^Siü^ EXtl3),KNAME<12)

3u KEAO IKPUT TAPt 3,ijaö,0ELALP,(RNAME(K),K=l,l2) WKilE OOTPUT TAPE b,4üw^,(RNAME(L),1=1,12) R=L

31 REAO luPUl TAPt b,1JU3,I EX(J),J»l,13) Ir(cX(l)-iöO.) J2,3t,34

t2 WKiTE uOTPOT TAPE 6,4uuü,(EX(L),L=l,13) Üü Jj J=l,ö Lri = «:*J LV=LF+l LE=K+J ORCStOlLE,!)

33 CR0SEC(LE,2) R»K*c Oü TC ui

34 OüM li^ut: CRCSEtlK,!) = CKLSfcu(N-i,i) CKL^L01K,2) • CKcSEC(K-i,2) OüMil) ■ ÜcLALP KE]OK^ ENO

tXPFJ (cX( Lh)/10. )*2.3025öt)l)*iJ.76387 EXPFnEX(LV)/lO.)*2.3Ü25bt>l)*10.76387

- 90 -

[ £.U

C L C

ülfthSL LÜMMLN

6v/,3ü) fHCüM(öC,40) tCÜM( lüO) HCÜMfCO«

K = K^TtP UVfcKK iJ=o.

UVkKTi3)=l. Kc=t7ö7b4üC. PIsH^ATANFd. ) KAlüCt=lbc./Pi CLlohT = u.Vo357iuut+Li9 WA = CUM12J ULLALP = CLM(II)

c c TAKCi c

CALL CALL CALL CALL IFd

ki.3 DU 2 UVCü

^25 UKCü ^2^ COM

OG 2 KUAT RÜTA

22C VüAI CALL CALL CALL LALL CALL CALL DO 2 TKIA

22 i TK1A CALL CALL DO 2

icl TKIA CALL

CtCMfcTKY

TAKCLT AND AIKCKAFT

TAKütT (TIMEfKüÜTfVOOTI A1KCFT (Tii«it,RuüAfVCUA) UNIT (ROOT,UKÜOT,KÜT) uiMlT UCüTtÜVbüTiVVT)

ii»IL) /24,2^J,224 ^i> J = l,3 T(JI = UVtKT(J) T(J) = UVLKT(J)

INI/t 20 J=l,3 (J)=KOCTU)-l<U0A(J) (J)=-KOATU) (J)=VbLT(J)-VÜuA(JI ONIT (ROATtOKOAT,KAT) UNIT (VOATfUVGAIjVAT) UNIT (ROTA,UKUTA,SCALAR) UNIT (VOuA,LVOOA,VVA) CROSS (LVEKTfOVOOAfVkCTOR) UNIT (VtCTOKiVtCTOR,SCALAR)

21 J=ifi LA(l,J)=UVüUA(J) LA(2fJ)=vECTCR(o)

CROSS CJVOOA,VtCTORtVECTCR) UNIT (VECrOR.VECTORtSCALAR)

ci J=lf3 ÜA(jlJ)=VcCTCR(J)

TRIAD (TRlADAiORUATtELAATtAZAAT) . • i

- 91 -

C ÖUUNtt PCiiMT - KLJND tAr<TH kITH Ktf-KACTIQN

< S = bwivT»-1 kuA T ( i ) **£;+RüAT (2 ) *♦<:) A=l.

C=-KL*(RüLl(3)+KLüA(il )+u. iJ*RS*KS

ki)i = (iNUUÄ(iJ*KS)/(RüoT(3) + RLUA(3) ) üji F = käi»(RLi*<8i>i«A+ü)+C) + Ü

i-H =Kbi«( Ri>iî .»A-*«:.*ti)+t üLLRSA=-l-/fP IFUbbhCuLLRSiJ-i. ) ^ji.^jî^j'!

<;3i Kii = RSl + üLLRSl tiü iL i*l

eis A4.HHAa=KuüA(i}»/Ki)l-KSl/(2.«KE)

hl=KSi«KSi/(ü.»kc) kS*=kS-Ri>l k=iiWkl>( iKuaMl3)-KüOr(3)+H)*«^+Ki)**^) ki=bwKT»-( (KUCAl j)+Hi)**2 + RSl**2) R2 = b(1ikT»-((RüOT(j)-H+Hl)**2*RS<i**2) ALPHAC = AkCSlN(küJA<3)/RAT - (KAr**2-RuUA<3)**i)/(2.*RAT*kh))

C C TIML ucKlVAIIVtS uF RS,k,Ri ANÜ R2 C

Ri>KSuT=KUMT( l»*\/ÜAT( i)+kUAT(2)*VüAT(2) KSi:Ll=Ri»R!>üT/RS £JutT=-i.5«RiÜüT Cut1=-Rt«iVÜÜT(3)+VÜÜA(j)»♦RSRSDT üül* a)st*(kSULT*HUjAi3i+RS*VLCA(3)) KSiuUl=-(KSl»(k!>i*büüT+CÜüT)+ÜCüT)/(RSl«(RSi*3.*A+2.*B)+C)

■ tNo2üLT=kSüür-kilüUT

kKüuT = (RüijA(3)-RuUr(3i+H)*(VüüA(3)-V0UT(3»4-(RSRSür/REn*l',SRS0T KUCT=KHJüT/R Rikii;T = {RiJÜA(3)+Hi)*(VÜüA(3) + (RSl*RSiDUT/RE) )+RSl*RS100T kitüJ=RlkiüI7Rl

R2k2Üf=lAüuT(3)-H*.U)«(VÜUT(3)'RSRSüT/RC+lRSX*RSiüOT/RE))+ i kS2*RS2üür R2CüT=K2k2UT/K2 TüiK={2.*kl/CLlühT TüLUHM2.>MRl+R2))/CLiGHT TSiKü=o.i)*(TCiR+TDOUb} üi:LT=ri.üüb-ISlNG UiJr'ÜlRs2.«'RüCT üüPi=KüUT+RlLUT+RÔÜT L>üP^»c.*(RlüLT+R2ÜüT)

C t ÖuLNCE ANÜ CLorrtR PÜlNTS IN HAT tARTH COÜRDlNATeS C

HAH li=RU<jT(i)-RüÜA(l) HATI2)=RüüT(2>-kCUA(2) HAT(3)-U. CALL UHIT (HATtHATtSCALARi CALL MULT(RSitHArfHAd) RÜCb(X)sRÜCIA(l)t-HAö(I) KUÜb(2)sRÜoA(2)^HAÜ(2) RCJLb(3)3C. Xc » iWRTF(RAr**2 - RÜÜA(3)**2) CALL NCLTUEfhATtHAC)

~ " 92 "

■

c c c

KüCCm = KULAm + HA0(i) KULC(^) = HLÜAiZ) + HAC(2) KuLLiJ)=C.

Üü .û J = l,i kUAb( JJ=KIJüB(J)-KüÜA( J) RÜAC(J)=KGüC{JJ-KüüA(J) KÜ IC{J) = KÜUl-l J)-KüUr( J)

Zni KülüiJ)=küGb(J)-RüüT(J) tALt UiMlT IKuAüf ÜKüAÜ.HAb) CALL uiU T (K(JAC,ÜKÜACfRAC) CALL UiMiT (RUTb,ÜKCrb»l<TB) CALu UNIT <RÜTC,ÜKÜTC»RTC) CALL TRlAü (TKlAüAfURUAb,tLAAB»AZAAö) CALL TKlAü (TKiAUA,üKUACitLAACfAZAAC)

DlkLcT, SlNGLc Aj\J üüüöLE ÜCCNCt: SIGNALS

C C G

CALL L)i-T (UVCGitURUTAfCuSALP) CALL CKÜib (UVÜGT.URUTAfVtCTUK) CALL ülMir (VLCTGR,VtCTUK,SllNiALP) CALL AisGLE ( M iÂLP »CÜSALPt ALP ( i) ) GALL bGT (UVGGi.UKUrdfCUSALP» CALL CKüSS (GVGüTiUKüforVtCTGRJ LALL Ul\iT ( VLcTüKf VCCTiJKfilNALP» CrtLL AiV.GLL (SiKALPfGÜSALP,ALP(3) ) Ih(TiMt) t'iZit'ilfZ'tl

î ALP(i) = AL'Mi) dtui CUIWINUL

CALL CRGSS (LVüüTfKÜTAfVtCTl) CALL CRuSS (LVhril.KüTAtVfcCr^) CALL UiMlT (VLLll fVtCri fSCALAR) CALL JlUT (VtC12fVtCTZ,SCALAR) LALL JGT ( VLCUt VC:GT2,CÜSTH)

CALL C.<GSi> 1 VtCflt VtCrîVhClCR) CALL Gi\iIT i VLCTLiK.VtCrURf Sli\Th) CALL AUGLL ( SIiNfHiCuSTHiTricTi Üü ZH-J J = l,3

<.HC. ThcIAl J)=1>ILT

ALPU)=ALPm + U.5*(ALPm-ALP( li )

ALt:P = MLP( J)«KATGDL

LALL KATSLT (ALPP.SIGMAH,SIOMAV.ULLALP)

ail\TH=SlwF<ThtTAiJ)) CüSrt- = cüSi-{ fhcTAUJ j SlC:cKü( Ji i.) = SlbMAH«uGbTrt**^ + SlGHAV<tSI^TH**ii

Zt>i* SiLGRüC J,^) = SIürlAM*6liîH*«i: + SIGi-1Av*CÜSlH*«^

STcfNtu OCUMEIKIL VARIABLES

bT(Kf i.J oT(ls,.i) GI (K, i I GT(Kf*t J GTiK»J)

GT(N iu) GflN,/) 01 (K ( b ) G 1 IK t <> > (JT(RI ic)

UME tLAAT A/AAT LLAAO

LLAAC

R Rl K2 RüLT

' Rli.'GT

■

- 93 -

■

^.1

UT U,ii)

bT(Kf i<«) bl(KtlS) GTlKt xoi Gl IKtx7)

GT(KT4:o) GT(K^.i) GT(Ki^^) GT(Kt<:i) GT(Kt^H) GKK«:^) GTiK^c) G1 1H|27I G1 (Nt^ö)

Kt lUi<l\

K^ÜüT ALHnAO ALPHAC ALFiiJ ALP(2» ALP(J) SiGCKUitl)

SJGCKU(Jti)

Si[GLRC(2,2) SiaCRC(3»21 TCik UtLT ÜLPÜIK ÜCPi ÜLP2 fHtlAd) KLLTij)

SubKtibTiWt TAKotT (TlrttfRfV) UlHtN^Iui^ A( 3) fV(3) Ui^tNSiüN tPh(3^2,6),C«OSfcG<ibüi,2l,GT(3U,32),H(dÜ,40),COM(lÜÜ) CÜFMLI\i üUMr,KiTEPtKHT,C:PMfCÄüJ>ECfGT,H,CÜM TAU = GÜfUSI K = liMt/TAÜ + i.ol TI»TAÜ*hLliATf-l!K-i) TU»! iMt-JI K(i) = ti,hJKflJ*CüMi35) K<i) = LPh(K,i)#tüMl3'») K(2J = cPhiK.t) vm * LPH(K,^)*COMI^!>) V(i) = ».PhlK,2)»CüM(34) Vli) = LPhlKo) lf(CLM(6l)) i,3,2 iKTC) «.»2,1 ThLlA=fu/TAU rhtTA2=lhtTA«»<; TMtTA3=ThLrA2»THtTA A=i.*ThtlM3-3,*THtTA2*l. b»-2.»ThfclAj+3.»THtTA2 C=IAG«(rht]A3-^.*THtTA2+THLTA) u=TAü*(rHtTA3-rhtTA2»

Kl*) = A»t»iHlKt^)+b*ePH(Kif<t)+C«tPHCKt5)+U*tPH(Klt5» = A*tPHlKf J)'J*8*tPHtM,5)*C*tPH(K,6)+a<'tPH(Ki,<>) A»LHH(Kfi)+o*tPHiKl,l)*C*fcPH(Ki2)*U*tPH(Kl,2) A*tPhiK,i)*ii*EPH(Kl,2)+C*tPH(Kl3)*U*tPh(Kl,3) = XK*CüM(3i>J » XK»GÜM<34) = XV»tL^(35) = XV*tUW(j^)

V(i) XR > XV = Ml) K<2) • Vli) = V(<) ■ KtlUKN CNL.

<•<:

- 94 -

0 D

ö 0 0 Q

0 0 0 ö

0 B

-~~ --■"■" •.-I "'■". —T"-

_

2.3

4i*H

i.Ö

CüKMLfv. V( xi = VU) = V(J) = ThLTA i<{ i) = RC«:) = K(3) = Kt7UKi\

TiKt A1KC1-T (flMt»R»V) 1UJ\ h( 3) fV(3) lüh tPhij^f 6) tCKüSt:L( ifaül i^) iGTCdu ,32 I »H(Öu 14ü) f CQM< 1001 üUMTiKoTtPfKriTttPHtCKÜbtCtOTfrifCüM CUK(ji» CuM(3iJ Cüh(i3)

= CLM(4i)i*(TlMt-CÜM{b) ) CCMtoj) + lritTA*V(l) (..ÜM<ö} + lHtTA*Vl^) CüM(7)+lMtfA*V(5)

-

: :

SubKLljllNt üNIT iXX,Y)rv/ZZ) DlffeN^IuN XX(i),YY(3) i:zz=SUKTF (xx (i) *»<:+xx (i) **2+xx (3 j »»^) ifUZt) itlti

t. 0.

YYiiJ YY(2I YY(3) üu Tu YYii) = AXili/lll YY12)=XX(2)/ZZZ YY(3)=AX(3)/^Z ktlUKlM tNÜ

iUürtüCTINt CwÜSS {XX,YY,ZZ) OlKcNSlOM XX(3lfYY(3)fZZ(3) A = XXl2)*YYlj) - YYC2i*XX(3) b = XX(3)*YY(i> - YY(3)«XX(il ZZm = XX(1)»YY<2) - YY{i)*XX(2) zzm ■ A ZZ12) = ü KtTtiKN

i

■

SüuKüuTliMt TMAD ( TK I »VtCt tL f AZI uliiUviSlüiM Tki(3f3) ,VtC(3) »ÜVtC(3) CALL ÜMT (VtLfUVbCfSCALAK) SUM = L.

DU I J«l«3 sur* i=i>oM t+T« i j i, j j ♦uvecc j i iJK2=äoM2+TKI(2,Ji*ÜVfcC(J) SOW3=i>UM3+TK1 ( 3, J) »UVtCi J) CüSLL = i)tiRTHSUMi**2*SUM2»*2J SINÄZ=SUK2/CLSEL CUliAZ = ÜüMl/CCSEL tL = ATANFtSUrtä/CüSeD CALL ANGLt (SINAZ»CÜSAZ,AZ> KEIURN ENC

■

- 95 -

-TT-

2.6

<*7

SUdKLuTiUt MLLT (XXXtYY,Zi)

ZZti)=AXX«YY(i) iZ(üf=XAX*YYlt) /.Z(3) = XXX*YY(^) ktlUkU LUC

bObKLUU^L uCT iXXfYY,ZZ2) uiJ'tKiiibN XA(3|fYY(3{ /.Z^ = AX(l)*YY(iJ+XX(i)»YY(2)*XX(3)*YY(3) Kil ibKN

2.Q SubKuUilNt ANGLt ISiNCHI.CQSCh Pi • 3.iî5927 ksigHUMSiNChlî+C'JSCHl**^) SINCI-I=SINCHi/H CüiCH»tLStHl/K IMCLSCH1) S,l,<r

1 IKblrtCHlJ 2,2,^ k CMi * (3.*PIJ/2.

2 Chi = Pi/2, GÜ JL it

^ CHiPsAfANf«SiNCHl/CUSCni ) ii-(SiiMChi) y,!3,t) ■

B iMcbSCril) 7»0|0 £ Chl=LhiP 'i".■-'■■.m:.'-* ':„..-"

&Ü IC iu 7 CHi=CHlP+Pl

GU IL IU £ IFRCbCHi) 7,^,9

^■'-^^.^ •»■■

S CHi « CHiP+2.*Pi iC CüMlNbt

RETURN , ^r^.- Vw?, „ kliÜ ,

l.Lhl)

2.4 bObKLlUriNE KATSCT (AWÜLfc ,ÜIÜÂH, SiGMAVfDtLALP) ÜIMLNSJUN tPh(322fö)fCKÜS£C(i8Clf2lfGT«a0i32)tH(8ü,40)iCOM(100) CuMKUv TiMtfkSTt»-fKHT|EPH,CHCSfcCfGT,H,CQM Kl • ANoLfc/UhLALP ♦ JL.5 iIGMAH = CKÜSEC(Kl,il ÜÜMAV = CKüScCtKi,2) KfcTUKN tNC

- 96 -

n

ö 0 B B

I 1

J.o

C c c

1 0

Ö

D :!

0 0 0 0 1

3

c c c

SUüKUüTlNL UAVfc ÜlfLNSiÜN TPK(i2) LiiPchSIUN tPht3<i*»6)iCRüStC(l8(;if2lf(iTI8üi3ai »HU0t40) tCOM( CüfMliV, TIMt,Ki>rtPiKHT,h*>H,CI<USECfGTtH,CUM K = K5ItP

Gtl INPUTS FRUW CUMMÜN

VVA • CüM(2l l-MC = CuM( 12) HAPtK = CLM(lil VAPHK = CüM(141 AAPtfh = CCM(15J tT/l = COM (16) pc^ = cuMiüo) PPtAK = tüMiüiJ PK = CÜMi<.4) TukcLL = CüM(^i») bYSLf = CuMtZü) TZetAM = CCMt27) FNLlüfc = L'..M(2ö) HTii = CÜMt2S) VbC = U0M(3U) AiHÜtL = CCM(j7) XiVüfcL = Cb^(3ä) Xlt-Z = CüMUS) X1VZ = C»jM(tÜ) XIH2 = cOM(4i) XlVi ■ CtJMI42) ScA = CÜMCO) PÜL = CUM(44) ♦-NTKK = CLM(46) CRL = UüM(47) b.MSK = bwKTF(l,0M(4b)) F^S = Cum 4«*) bi = CÜ>1(5v,» ATLTS = CCM{bx)«*2 AlE^S = CbM(5^)»*<c ATLLS = CCM(t.J)»*2 SIGUPI = CCMtiio) SlüZ « lü.**(CCM37)/iC.) RljLA^^ = CüM(ö4) RAIN ■ CülSi^i) SÜf'SIG = lC.**(CÜrt(22)/iü.i

&LT STUFF FROM CCüM

12

LH

IOC)

'%it

L » K KANbt = CT(L»o) Kl = ÜML»7) R2 ■ GT(Lfd) IIIPLL} 14,1^,1H SIÖÜC • ül(Lii7) siccu ~ GT(LfIö: Sliiöti = GTiLiiS) Gü TC 16 SIGuL ■ GT(Lf2w)

i

1

- 97 -

ic ALPhAL = OTiLtii) ALPMAü = AUi>F(GUi.f UM Xi vt = oTlL, r;)

uUHi = üTlL,2u)

tL^AT = üliL.iJ fcLAAU = bT(Lf^) OLLI = üTR.i^J

C t C

I

W-

5u2

C = S.öjbfi td

bbLU = A.ido«! L-2J SltMP = i<iQ,

ANlfcKNA

AUAKEA = AAPtHf;*HAPck*VAPtK AuAlNA = ^.*PI*Al\iAHcA/(FLAMti**2) HtlAh = "FLAMb/HAPtK bcTAV = FLAMb/VAPtK hbtAMH » 1, VbtAMH ■ 1.

ii-(MßSJ (cLAAT-tLAAöi-btTAV/i.) !)wi,5Ui,302 bütAMP = '1,

odtA^P = tTA CulNl iitUE

FSCh = XihÜtL*Ai VücL/iHrsS»üL-TAV) IüI»P = »-itH*(Tirtt-TZbcAM) ♦ XIHZ AIJ-O = TtHH - XIhJfcL*iNTF(TfcMP/XIH2) Ir(VSC) Siw,i:iotS2ü

dlC XlvG • XIVZ

^ i^K ' ilV« + ttbTAV*lwTP<<PSCh*{T|Mg-Ti8eAM)*XlHZJ/XIH2) XiVb = TtMP - XlVUtL*INTF(TtMP/XIV2)

FÜR NÜW, Life XIhT, X1VT XiFb = ÜT(L,i) XiVÜ = (,r(L,2)

PH.lVt = XlVC-XiVÜ

KtFLtCnUN CtiLFFKIti^T

IFlPLLi 72ü,7iCi72o IF(ALPhAb-i,0*t/2J 711,711,712 KtFL = i.-ALPHAü*#6/Pi CU TL /äU RtFL = ,6 tu Tb 73ü IF(MLPhAb-.Oä727) 7^1,721,722 HtK = l.-ALPHAtt*i2.4/PI GO TL 73L ktFL » .ö-.y2b36*tAPF(-10.g8#ALPHAö/Pn iFiSEA) 73^,731,732

- 98 -

7iL 711

712

/2C 721

722 73L

il 3

0 i I I

■f

mmmm»

c c c

c c c

c

c

c c c

73i IbVP = i, üü TC 7J.J

Ilk TtMH = lo. /3J ivtFL = Kcl-L*LXfF(-ü.*<Pi*TtMP*ALPMAB/FLAHÜ)»«2»

SiCNAl. PRtXtuSINu

Pti = PCN

Al^LSPHtKlC ATTtNOATiuN

ATfLDU = 1. AlMtb = i. ATI^LCU = i.

TKAwbMiTTtK

PZ = lPPtAK«ANAKtA*«2)/U.*PI*FLArtö»»2»KANÜi.*«<») PAVü = PPtAN*TülNtLL*Fk PoLbt = TiJWtLL/PC(\i

blGivAL

FLLäi = i»YJ>LF*PCS Tt^p = PZ»FLI;SS

SüL = ItiMP»SlüOL'«ATMLüü*H|t>tAMP**i*VBEAMP**2 iUU = «cMP*SlGüb*(KAi^fc/(ki + K2n«*2*ÄtFt**2*ATMLÜB*HBEAMP**2

i »ViitAMP«BBL;AMP

ibb = ltMP«ilOÜÜ*(KAN*fc/(kl + k2))**4*RtFL**^«AmBB*BBeAMP**2 IFtucLT-PULSt) 21Uf2iu,22u

■

C i, c

c

^iü IFiihA) <i<:Üf2ilf22U 211 Stl ■ iüDtSBb*4,*ÜUü

üu (L 2iO

23u CüM iNüfc i

iyükPLtk .......

fut = ULPÜIK/FLAMU Fu*b = üüPi/FLAMb Fbb * vüP2/FLAMb

Ph*St Üii-FbRtiMCb

.. C i. .■ i .

>

i '.

c c c

PHA * (Kl+K2-rtANÜc)/FLAMB IF(^LL) t'jiic'sZtc'sl

cüi lF(ALP»-MB-,ob727) 2a2f252f2£3 ib2 Pn/i = PHA+.a 26i HlAuLL = (ÜüP2-uCPl)«btTAH/(FtAMb*FSCH)

■ ■

■y :

■X'rW-"-

• ,^■j^•■■ . .

CbLfTtK

IF(ALPhAL) 2S*if2'»<»f29!J ^S4 S1GZ s ü,

GL TL J32 2*5 ir(ALHhAo-Pi/2.i Z*»7,2S7,Z9b 2St AbPhAC « Hi-/>LPHAu ^S/ ir(AbiF(ALPHAt-Pl/2.)-i.t-ä) 2Sü»29y,299 ^i^b ALIrHAb = PI/2.-l.fc-a

- fUiJ

- 99 -

Htm

i».l,jo<i,332

■i.i:+.£)«aSfcA)

i2t

i3t 33^

CLLTTK = c. &o TL J^C

iF<FMC-3c.tC.) JWCJoiogJ IKStAJ Jwi,3o<i, jw3 bStA = .b <JU TL JLü bitA = i«

SiGt ■ ^.»SIKf(ALPHAC)*iü.**( t»U TC jj«: iKSfcA) 3iC,3iO,3^t

s • öö iL 33c = .lööö » 2.ob

SiuZ » '•l*UL?HAC**C2)/(FLAMb**t3) L6f.*^^üwft,-^^-^ÜSFUiPW>J

TtMH»PI * SIGZ^MCM » SIGZ«ACS ^2* SiGCM^YSLH <;>Z*i>IÜCi>*cTA*»2*SYSLF

" CLS

Gü Li

ACH 3 ACi = Site« SitLS CLC = CLS = CLLTIK

.»3a 340 330

C c c

c c c

c c c

c c c

ifUdLrtiviK) 350,33d,35^

CLLTTK = CLOTTK ♦ CLrt - ETA*-*2*CLM TtMP = AiHG ♦ btfAHÎüNH1.fxiriG)/4.

i /FL;Ma'A*AaSF(CQSHXIVÜ,*iIWF(TeMP^SI^^ÂH/4.n*8.)

NÜiSfc PLOb HAIN CLOTTtK

PNLJSL » tlÜLTZ*Sr2MP*l-i\lUISE*PCN/TOh!ELL

PrtLiSc = PNUiSt ♦ CLRAIN ■

TKACKlNi, ACCOKACY

L"NH'/?tTMP:BtT:SHW/'fKS,SUR'HiÜ0*Bl/"'WIS'=*CLurTR)).

SiL!\iV • TtMP+beTAV TKKtRR » SOHTFUIGWH*^ * SIGNV2 ♦ ATETS ♦ ATEPS + ATfcO.)

SfcAKCH ACCURACY

tH » lNTF(btTAH*Fk/FSCH ♦ .S) TEMP = SöKlF(tN*SüÜ/<PNQiS£*CLOTTk)J ütLu « 1.732t51*FLAMÖ/(2.«PI*TüW£LL«TcMP) OtLK = i.732ci>l*P0LSE*C/(2.*PI»TeMP) ULLA - ^TKK^CKL^SWÄTFiBETAH.^^bETAVÎ/iFKSnEMPI

STCKE UtTfcCTIÜN PAHAMfcTERS

Q

Q

0

n

o

a a o i

»v—immmmmmiimmm* , _:.'

CC^(iLü) = PNGISt CUM99) = CLBh CJK(SäJ = BtTAH CJM(97) « HULSt Cüf(96) = FStH Cü^(95l « iTT CUMSA) ■-> SUU C0^(9J) = CLtTTk

c C SILKt ÜUTPOTS c

. h<K,l) = TiMfc

I H(K(12) « FüU-f-üb

j H(K, it)) = utLD HCK, 16) = UtLA

. H(K,17) = SlUiMV H(H,lti} = bUsNH H(K,19) = ]FiKtRK 00 ÖC I*i6,lS

öC H(Kf 1) = H(KfI)«i,t3 > HiN.^tJ = ÜT(i.,^<»)*i,t-3

H(K,ii) = iüü I hU,i2) * Süb

H(Kf«:^) = CLM , H(Kfü) = CLS

HU,^6) = CLLTTR Üü d^ 1=21,26

Öi HU,!! = lU.*LUGicF(H(K,lJ*l.fc3j H(K,27) = iü.*LÜÜioF(SIÜZJ H(Kf28) = PHA HU,29) = PHAuLL H(K,3o) = KtfL MiK,Jl) « tLÖh

H{K,J2) = 10.*iuüluF(SrT/{PWüISt+CtUTIR)) lF(Kh7-l) 90,90,99

9C IPR(i) * FLAMb TPK12) ■ ANARfcA rPK(3) = xü.*l.ü01uF(AGAINA) TPRi^) a BeTAH*57,2957795 TPH(5) = beTAV«57.2957795 TPB(6) « FSCh*57.2957795 TPR(9) » PZ IPK( 1U} = PAVÜ TPM7) • PULSt IPK(il) = iü.^LÜÜiüFiPNüISfcn.fci) TPK(d) ■ tN hKnt ÜUTPUT TAPfc ö,91,(TPK(n,l=l,ll)

91 Fü^MAi (Ihi 54X IbHCÜMPUTfcU CC.^TAr4TS ///9h LAMBOA « E13.4.4H FT / 1 9hu Ak » ti3.4,üH iUFT. 5X öHGAIN = bl3.4,3H ÜB 8X 2 dhdfcTAri - ti3.4,5H ü£G. 6X öHbtTAV = £13,4,5H ÜEG. / 3 9FuFSCH = tl3.4,4H 0/S 7X bHTAU - fcl3.4.5H SEC. 6X t ^ AfT 9„Li-*''*''*HWO • £13.4,7H W/SWFT 4X QHPAVÜ = E13.4. i> ö»-_»«AfTS aX dHPNüISE » E1J.^,4H ÜbM ) C1J.4,

RETURN

9S CulNTINtt

RET END

- 101 -

s.u

C C C

c c c

c c c

c c c

ÖC

d^

u

SUbKLüTlNt tüÜlfc Ul^tN^iUN XbAK(<:)fSUNL(2) tPÜ(2) üll'kNiiÜW c£Jh{i^,.,6) fChUStC(idul,2),OT{bü,32) ,H(8U,4Ü),CüM(loC) Cü^ML.M TiMtt KSTEPtKHTi bPHiCKüStC»ÜTfH,CUM

GfcT STUFF FKLM ÜAVL

K ■■ KSTtP IF(KHT-i) bOf8u,e2 PCLM1 = ü. PLLMT = Ü. üü b 1 = 1, «i SoNL(l) « Cürt(l+9iJ CLL1TK = LCrtCSi) PNLISL • CLMJIOO) UbLFC = LUrt(«i9) Öt-lAh = CuMiSb) PULl»t = ct-ms?) FiCH = Cüi-ii^t)

Utl 6TUFF FR CM LüttMUN

FK « CuMU4) TALFA = cbMl3o) KASt « LCM(P5)*.5

IFtXASc) 3,3,4 KASc = 1 CüMltNUfc FJ^i» = LUMlöo» L = K ALFriAb = üT(L,li)

ScT UuTtLTlÜN PAKAMtTtkS

N = btTAH*FR/FSch ♦ ,5» NC ■ btTAH*ütLFC/FSLH + .ü IFtNC) 7,7,b NC = 1 UM 1 N eint = NC

FAF = l:N*PüLSt*.fcyjl47ia/TAUFA FAHL = LÜÜlUF(FAP) IbfiK = tLUTTK/FNClSt UlFGiii) bt,c,5C IF(ZfcAR-.l) i:w,^,S IF(^fcAK-lü.) lu,lü,Jü

l^tlTFtR LLüTTtR WCR NülSfc OUMINANT—fcUS. lb-i7A.

CLMlNUt LMCZ = tMC*ZbAR Vi = 7.»tN*ZcAR Pi.^ = LübiuF(PFA(N,N(.,ZbAK,VZ) ) VI » VZ ♦ .v>l*VZ*SIyNF{l.,PLZ-FAPL) PL1 ■ LbOlwF(PFA(N,NC,ZBAKfVin IFlblbuFi1.,PLZ-FAPL)-SIGNF(1,,FAPL-PL1» » i£,14t12

- 102 -

.

c c c

c c c

- LU^RA^) - (LiMC-l.)*LüGF(Ai» - VÖ/A3

1^ VZ = Vi PLl = PLi Oü TC 11

iM Vu = VZ + iVi-VZ)*(PLZ-<:APLi/(PLZ-PLi) Uü ZK, i=lt^ Xd^rttl) = büNLd )/PWÜiSt Al = tN*XüAH(J) A2 ■ cViLZ ♦ Al Aj» = 1, + Ac b = ciW^LüOf-(Aj) Pul 1) = tXPF(d)

üü LühlliiLt HtK.fc) = c. Gü TC 99

CLUTtK ÜLMNANT — t(JS. 18-19

3c VZ = (cNC+lu«)*cKC PLZ = Lüoll<F(Pf AC(NfNL,VZi »

il Vl = VZ ♦ .01*VZ*SlUiMF(i«/HLZ-FAPL) PLl = LüülwMPf AUi>ifNC,Vin IF(6lü^F(l,fPLZ-FAPL)-SIGNF(U,FAPL-PLl)) 32,34,32

32 VZ = VI PL2 = PLl ÜU TL 31

3^1 Vti « vZ ♦ (Vl-Vi)«(PLZ-FAPL»/(PLZ-PLl) Du AL 1=1,^ XbAhCn = iGNLiD/CLÜTTR IF(XtiAK(I)-l.t-2) Hl,42,<t2

ti pum = c. GU TL 4Ü

42 Al = tNC*XbAR(n A2 = (l.+AU/Al A3 = Vb/tMC IF(i\C-l) 3S,3i,37

35 PÜ(i) = tXPF(-A3/<i.>Ai)l Gü TG tG

3 7 SGI» = t, JMAX = NC-i DO 3ö J=1,JMAX R = C« DG 36 M=1,J K = k ♦ tVAL{A3,M-l) SUM » iü« + fcXPF{FLUATF(NC-l-J)»LÜGF(A2l+L0GF(R)-LQGFC Pü(l) = SGM ♦ fcXPF(-A3/(l.+Al)+(tNC-l.|*LüGFCA2n CÜMJNUfi H(K,6) = -1. GU TL 99

3c 3b

40

AU)

bo

6L

99

NU ISt GÜMIMANT—MARCUM-SWERLIIVG

CGNTINUt FAN • LüGluF(TAüFA/(eiM*PULSt)) Du bC 1=1*2 XdAK(l) = SGNL(iJ/(CLUTTR+PNQISfc) CALL MARCUM (NVFAN,XBAR(11,KASE,PD{I)yVBt CGNTINÜ6 H(K,t) ■ KASL CuNTINLt

■

.

- 103 -

m -k

IKALHhAu) ISV,^^^ i'.S PJll) = C.

PüUi = «,. 2SS CCM llNiLb

PCUMl = PCUMi ♦ <i.-PCUMi)»pL;(l) PLLrtT = PCLMT + (l.-PCUMT)*PU(i:) h(h,i) = HÜ(1) h{K,3) = PCUKl H(K,î = f'DIi) H(Kfj3) = PCJMT HJK,71 = VÜ M(K,b) = It-^LüOlOfiSÜNLt^J/FNClSc) H(K,SJ = lC.*Lu'-lwF<ZÖAKI H(Kflu) = tMC IKKhT-i) iUciitüiiw^

iut WKITL üUTPüT TAPt 6flüi»FAP iüi ♦-üHÂT (SihCPPA = tli.-t) IC« CCMlNüt

Kt TURIN

i« 1 FUNOTiü^ PFA(N,i>«LfZüAKtVb)

c CLMPUltl) FALSt-ALAKM PhCBAblUTY UF RECeiVfck NÜISE c ,

ANü OLJTTtK ÜNLY. N IS NG. UF PULStS, NC IS NO. OF c INUtPtNOtNT CtuTTtK oRÜUPS, ibAR IS AVERAGE CLUTTER TO c iMOlit RAIIU, V6 IS THE ThRtSHOLU» SEE BBD-1387,

FMC « FLGATFC^J/FLOATFCINC) FrtCZ = FMC*^bAK FrtCZU = l.+Ff-OZ t\HL • N-NC FNNC » FLOATFINiMC» SUW = U. W » FNNC*LÜGF(FMCZU/FMC2) - SUWLUG(NNC-l) ÜÜ SS i»ifNC NU « I-l FNL » FLGAlFdMOJ R « C. KK = NG-NU ÜU 4 K=liKK

M R * R ♦ £VAL(VÖ/FMGZÜ,K-l) IF(R) 99,99,5

c TcRrt = a+SUMLGG(NNG-l+NU)-SUMLOG(NÜ|-FNU*LÜGF(FMCZ)*LOGF(R) TERM » EXPFITERMi

10 TERM = -TERM 20 SU« = SUM+TERil »| CGMINUt

PF* « SUM RETURN ENC

4« 2

c c

FU/\GTIG.N PÂC(N,^CtYC)

FALSE ALARM PKÜbAbILITY, SIMPLIFIED FUR THE CASE WHERE c CLLTTtR IS UCM1NANT. (GLa-4-U7, EU. 18) c .

. 104 -

SUf « ü. V = YC»FUiATf(NC)/FLÜATMN) DU iu i=i,i\C

lü Süf = bUM ♦ fcVALCYil-l) PFAC * SUM

RtTURN fcNC

>

- 105 -

■

■

41.3

C C C c c

c c c u

c c c 0 Ü

c u c

SUBKUUTlNt MAKCUH {N.FAM.SNK,KASe,PNfBIAS)

CUFPUTt MAKCLM-ShtKLlNG ÜtTtCTIUN PROßAbILITItS

TtST INPUTS

IKM S9,SS,2 2 IfitAHl 99v4«»3 i iFIKASbi 99f4t4 <t IF(KASfc-4) OfbtS*

tSIIMAIt blAS LtVcL

t fcNFK = 0. CI\IPK = HAN tN = N YbPK = 0. HMN-i2) 7,7,0

7 YiiPK = tN*<i.*i.2*tMPK/EN«*(i;,/-J. + ,0iI>*tNPRI ) Gü IC i.i

| YbHn = tN«(l.+l.3*tNPk/cN*«(.5+.üll*tNPk))

Cü^Pült dlAS LeVfcL

ii tNPh = iJ.«*t,\PK OAKPK » UoAM(YbPK,N-ll PYt = .t5*»Ii./tNPki ii-lu*.«Piv-PYb) ii.,ia,ii

IC H = .ti Gü 1L l«f

C ii H « -.ul C i^ Yu « YoPK U LSJ * üLVAL(YO^-i) 0 iü Yi * YC+H C ti = ücVALlYl,N-l) C SltP = OAf-iPK ♦ h*iLo-»-tiJ/2. ü iF(SAGi\Ki.,bTEP-PYb)-SiGWFa.,H)) iö,2t,lÖ £ it Vu = Yi li tw = ti U GANPH ' STLP

Go R io Ü iC iF(h). 22,24, JH

C 22 Yb = Yl - H*(HYb-STtP)/(GAMPK-SrL-r) Gu IL it

L «:<i Yb = Yu + h*(PYb-GAMPK|/(STtP-CAMPK) JC ölAS ~ Yb

C c iti.ttr M-s tASfe

c c c

K • KASu*i Gu it (itv. ,2Cu,3ÜC,<t0u,500l , K

cAit u

iti* Sjr = L. '

- 106 -

,

P = tN«X lf(Yt:-P-tK) OfLilCitHJl

isji. Kb = -lLi\l+i.)/2. + ;>Ji<TmifcM-I.)/2.»**Ü>P*¥BI KS = XliAXuHlKStO) ob = l,-C»AM( Yö,KS+N-l»TN) Tb = tVAUPtKS)*^ ü = US» K = KS TtHM - TS TL = TN

lie TcNP = SUH+TLKM IFiSOW-TLMP) U^,U6tiio

Hi SJF - TtMP IHK) i.i6»il6,IlH

ii^ TtKM ■ ftK«*FLLATF(K)*lü-rLI/(P»vi) Q = G-TL K = K-i TL = 1L*I;:LCATFJK*N)/Yb ftj IC 110

116 TL = iN^Vb/FLUATHKS+N) K = KS+1 G = Gi+TL TLKM = TS*P*G/US*J-LüATFiKn

IIQ ItfP = SUM+TtKM IFlSüM-TfcMP) UithiütiSi,

iZ'i SUM ■ TEMP TL = TL*Yv./FLUATF(K+NJ K = K+l Tfc«M = TfcKM*P*(G+Ti.jy(G*FLUAIF|K)i G = G^TL GU TL 120

Ibu KS = ~i. - fcN/2. + S0i<TFCeN**2M.+P*YB) KS =XMAXGF(KS*Ö) GS s GAMtrbfKS*^-i»TN) IF(GSJ 17<ifl7^fl55

US TS = tVAL<P,KSI*GS G s GS TtHM = TS K = KS Tt = TN

loC TE^P = SUM+TtRM IFiSUN-TcMP» i62flö6eloö

lo2 SUM = TtMP IFtK) icü,lbfc.l6^

let ItiW = f£RM*FLÜATFiKi*<G*Iil/<P*GI G - G+TL TL = TL»»-LÜATF«K+N-1)/Ya K - K-l GO TC 16U

löfc Tl. = T.\*^ü/FLGATFiKS*N) K = KS+1 G = GS-TL TtKM = rS»P*G/(GS*Fl.ÖATF(Kn

17L Tfc^P = SOM + TCK« IFiSOM-TtMPJ i72,174fl74

172 SüK « TtMF TL = TL*Vb/FLüATF(K+Wl TtKM • n:KM*P*(G-TL)/(G*FLOATF(M-in G « G-TL

fA m ■' ■ ■ ■ -; ■ ' ■ .■ . ' ■ ■ ■ •. ■ •

NMMaMntwMMMB

i>P*YBI

' i.

• '

■

■i ■

i

'

■

,' ^fr^ 'fc#;fi.

■

i

-. - 107 -

—■■■■—-'-

c c c

c c c

N « IV+i Gu \L 17c

174} Suf = l.-bUM

GU TL «iu

cAic 1

^1C PN = t-XPK-YB/d.^XI ) üu TC So

PiM = i. - GA>i(Yet!\.-2tJUf1) + tXPH (t:N-l.i*I.ÜGF(TfcMP)-Y8/(l.-»-tN*xn I *GAM(Yb/T6rtP,iM-2,l)UM)

Gb TL VQ

CA^t ^

301 ir(N-i) 31v.f51w»32o iiw Prt » tXPF(-Yt/(i.+xn

Gu TC Su 3io PIM = 1. - G.AM(YD/(1.+X),IW,G0M)

Gu TL SO C C LASt * C

*H,L IFIN-«:) <»iO,^iüf 43o

'♦ic F.x = U.+t,»X#Yb/(X+2.)««*2)*£XPH-2,*Y8/i2r + Xn Go 1L SO

^•20 Pi^ ' (l.+Yb/(l. + X))*tXPF(-Yb/(l.+X)) GL TC 91

^iO C « 2»/lt,*Lh*X) ü ' i,-L ie(Ye#Ü-tN) ^4u»45o,^5U

^4o SoK » 0. ItK^ » i. J = N

4^2 TL^H = bLM+TfckM IF(SLM-TtMP) ^44,44öf4<«6

•♦*«<• SUf /■ ItWP TtKM = TLKM*Yb<'D/FLUATF< J) j » J+l Go TC 442

^c Piv » 1. - GAM(Yi>n>*-2»JUM» ♦ C*Ya«tVAL(YbiN-2) i ♦ u*tVAL(Y3,N-l)*(i.+C*Yä-|tN-2.)#C/D)*SUM

GÜ iC 90 4bc PN « i, - GArt(Yb.N-3tDUM) ♦ Yb»tVALiY6tN-3)*C/Ü

1 ♦ fcXPF(-C*Yb-(cN-2.i*LüGF(ün*a.+C»YB-(EN-2,)*C/0) 2 ♦OAW(YB*Ü,N-3,0U«)

GÜ TC 9u C o C

CASt 4

ÖOV. i>u^ • 0. C * 2./(2.<-Xi 0 « l.-C 0 » c/o P » C*Yb

Ki »= (J.*tN+(Yb*0))/2.-Süi<T'-((EW~l, + <YB*D))**2/4. + (Yß*Ül*(fcN^l.n

- 108 -

■

0

Ö

8

imf'w 'mb—•*******■ w*

KS ■ XMlNbF(KStN) Kb ■ *MAX0F(KStu) K « KS J * N-KS FKb = KS K * XMlNüMKSfM if (Yb-tN*(l.+ü)) *)5u,a01,50l

5ÜI OS » i. - ÜA^(P,2*i^i-l-KS,TN» IHGS) 5^6»5i6»l>C2

3ui Ti> = tXPI-(HKS*LÜüF(C) + (£N-FKSI*LOGF(t))"l-SUMLOG(N>-SüMLüG<KS) 1 -SbMLÜG(J)+LCGF(GS))

G = GS JckM = TS TL = TN

iiv. TEMF = SUM + rtKM IF(i^-TtMP) 512,516,516

i)i2 SUJ^ = TtMP IFtK) 510,516,51^.

51^» TL = TL»P/FLOATF(i*N-K) TtKM ^ TtHM«FLÜATF(K)*(G+TL)/(Ci*FLÜATF(N-K+l)«G) b * Ü+TL K = K-l GÜ TC 510

51c 1FUS-N) 518,526,526 516 FtKivi = Tü*u*FLUATF(^-KS)*(GS-TN)/(FLÜATF(KS+l>*GS)

G = Gä-TISl TL = Ti\*FLuATF(2*N-l-KS)/P K = KS+i

52C TEN** = SUh + TfcHM IF(bLM-TbMP) 522,520,520

522 SUM ■ TtMP iF(K-i\.) 52^,5i6,52o

52^ TcKM = Tt»<rt»»,*FLüATF(N-K)*(G-TL)/(FLUATF(K+l)*G) G = G-TL TL * 1L*FLCATF(2*N-1-KI/P K = K+l Gu TG 52G

54.6 P^ = iUM GU TC Su

55i> G5 = GAMtP,2*N-l-KS, JN) IFtGS) 576,57o,552

bJc TS = LXPF(FKS*LUoF(C)*(tN-FKS)«LÜGF(ü)*SUMLUÜ(N|-SÜMLüG(KS) 1 -Sü^LüG(J)+LCGF(GS)l ü = US TtKM = Tu TL = TN

5oG TefP = jUM + TfcKJ«. IF(SüM-TfcMP) 562,560,560

562 SUM ■ TLMK IKN) 5O6,5O6,5O4

50^ TL = TL»P/FLÜAIF(2«N-K) TtRH « TEkM*t-LCATF(K)*(b-TL)/(t»FLOATF<N-K+l)*G)

. .

..

■

G « G-TL K * K-i GU TC 56U

500 IF(KS-N) 5fcb,576,576 5o£ ItkM r. fiWü*FLÜArF(N-Ki)««Gi+TN>/(FLUATF(KS*l)*GS)

o = Gb+TN TL = TN«FLüATF{i«N-l-KS)/P

I* 109 -

■

■

c c c

c c c

K = KS+1 57c TL^P * iUM+TEKM

IKSLM-TLMP) iJ72,57o,b7o 572 61^ = UMH

iFiK-N) b7<t,i57t,676 57^» TERM = ItKM*W*fLuATI-(N-K)*(G+TL)/(FLOATF(K+l)*G)

Ü « G+TL 1L = lL*FLCATFU*N-i-iU/P K « K+i GJ FL 57U

576 PN = I.-SUM Gb TL 90

StiT FKOLAblLiTY

90 iF(PN) S/l,94f>»2 91 PN = c.

GU 7L 9*t 92 IFiPivi.) 9^,9^,93 93 PtM = i. 9^ KcTUkN

LRkLh MtSSAGh Füh üAü INPUTS

99 WK17L OUTPUT TAPfc 0,9,^,FA^,S^K,KASb 9 FUMAJ (iho /bOh Ui\KtAi.üt>JABLE CALL StUUENCt Tu MAKCUM, ZtRÜ RESULT i 7hS GIVEN //4H N = ibf5X,5HFAN = tlö.8,5X,5HSNk = 2 L-1ü.O,3X,6HKASC = U<)

PW = U. lilAb = u* KClOKiN

4.J.1 FUM, IIUU ÜGAMb.M

C IMtOKAL C SUK • 0. G K » B

IF(K-h) iCO,200f2UO iut J = N+i

L TLhM =ühVALJß,J) G iO It>P « SUM+TfckM C lF(SOK-lt.MF) 15,20,20 C 15 f;»«l = TtMP

J i J*l L FJ = J t TtRi^i = Tt<M"l*B/FJ

GÜ 1L lu ü 2c boAH ■ SOri

RfcTOKN 2oo J ■ f\

C TchN -ülVAL(B,JJ U 30 It^P = SUh+TthM L lF(iOM-TtMPJ i3,HC,^0 0 it» Süf - ItrtK

iHJ-i) *.v,it,3c G JC hJ » J 0 ^tHil » ll;KM*FJ/u

J » J-l

1-(SUM, J=0 TO N, ÜF EXPFtJOLOGFJB)-^-LOGF(NFAC))

«

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KüTüAN

fcNC

■HIWIII.IIIIII—«—w»

ü

0 C ^o

^UIXUTILN UEVAHY»I\)

XPCI\ = -Y iF(iM) 2w»it»lÜ t\\ = w XPClM » XPoN+Ei^LüoPm-SUMLCMM ütVAL » cXPFiXPLN)

tNÜ

C

it

15

3C

35

36

<»0

füKwTIütNi üAiM(OfNiTN) SINWLE PKEUSICN VLKSiOW UF UOAM

SUI" = u. K = H IF(K-N) iuCiiow»2üC J * N+i TtHM = tVAUb,JJ TU * TtKM*FLCATF(J)/ii TcNP = SUf'+TtRM iFlbUrt-TtMP) it),iUfiü SU^ = TtMP,. J » J+l FJ * J TtM » ftKM*6/FJ Gü IC iO GAF » SUM KcTUKN J » N TEhM = ty/AL(B,J) liN » TcRM TEMP ■ SUM + TtkiM IF(l>LM-TfcMF) 3t>.40f4c SÜ^ « ThMP iF(J-l) ^G|36i3t FJ « J FtKM = TtRM*Fj/d J « J-l Gü TL 3C GAM » l.-SüM KETüKN ENC

IC

2L

FONCTILN fcVAL(YfN) XPCN = -Y lF(i\) 2iitibtlH EiM ■ N XPLN = XPÜN+EN*LÜGF(Y;)-iUMLOG(N) EVAL * tXPFIXPCN) RETURN ENC

4.3.ä

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u

Ü

0

^UNd ILK i»üMLLO(K)

HMM = 2t'U iH(üLlHA-UUMli ) ^fK|2u

10 ÜUI'A = i. üÜfti = u. NLAS1 » i Ail) • u.

«^C wW s XAuS* dv)

Jt SUf'LLL = W« HbTLkN

tt IKNK-NLAST) iJC.DCoo äl bü^LLG = A(NK)

HIETUHN ot K » MAbl + i

IF(tMK-NMAA) 7c»7v>f8j 7C Jü 7ü i = N,M 72 AU) = A(i-i) + LüoMFLUATFUn

NLAbI • Wi\i ÜL Tc 5Ü

bU IKKLAOT-KMAXJ 82t9u,9u 62 DO 04 I=K,(V1AX b<i All) » A(i-iJ + LUGMJ-LiJATFU))

NLA ill = i\MAX SiL p = AfUftAM)

K » MAX + i öü Vi iaK,NN

Vi li * b ♦ LLLFiFLuATHI) ) SL^LLÜ = b KL1LKi\ bNL

B 0 0

D ] D ]

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- 112 -

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THC JOHN* HC>KIN» UNIVIRSITV

APPLIED PHYSICS LABORATORY ■ILVH «FOPNO. MkHYkAND

BIBLIOGRAPHY

[1] Andrus, Alvin F., "A Computer Simulation for the Evaluation of Surface-to-Air Missile Systems in a Clear Environment," U.S. Naval Postgraduate School, Monterey, California, TR/RP No. 67, June I960.

[2] Barton, David K., Radar System Analysis. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964.

[3] Berkowitz, Raymond S. (ed.). Modern Radar: Analysis. Evaluation, and System Design. John Wiley and Sons, Inc., New York, 1966.

[4] Boothe, Robert R., "A Digital Computer Program for Determining the Performance of an Acquisition Radar through Application of Radar Detection Probability Theory," Army Missile Command, Huntsville, Alabama, December 31, 1964.

[5] Bussgang, J. J. , et al..t "A Unified Analysis of Range Performance of CW, Pulse, and Pulse Doppler Radar," Proceedings pf x.he Institute of Radio Engineers, Vol. 47, (October lnöü), pp. 1753-1962; corrections in Proceedings of the Institute of Radio Engineers. Vol. 48, May 1960, p. 931, and Vol. 48, October I960, p. 1755.

r6] Fehlner, Leo F., "Marcum's and Swerling's Data on Target Detection by a Pulsed Radar," The Johns Hopkins University Applied Physics Laboratory, TG-451, July 2, 1962.

[7] Johnson, R. D., "Simulation: Key to Nike-Hercules System Testing," Bell Laboratories Record. Vol. 41, November 1963, pp. 388-393.

[8] Kennard, P. H., "Effectiveness Studies of Manned Airborne Weapon Systems by Digital Simulation," Abstract: Instruments and Control Systems. Vol. 34, July 1961, pp. 1272-

[9] Kirkwood, Patricia K., "Radar Cumulative Detection Probabilities for Radial and Nonradial Target Approaches," The Rand Corporation, RM-4643-PR, September 1965, AD-623~277.

[10] Manske, R. A., "Detection Probability for a System with Instantaneous Automatic Gain Control," Proceedings of the Ninth Conference on Military Electronics. Septetdber 22-24. 1965, pp. 28-31.

[11] Marcum, J. I., "A Statistical Theory of Target Detection by Pulsed Radar," The Rand Corporation, Santa Monica, California, RM-753, December 1, 1947.

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-7T-

__ ,

TM« JOHN» HOMIN* UMiVmaiTV APPLIED PHYSICS LAt ORATORY

■«.vn •«»■Na M*IIYUUID

[12] Marcum, J. I. and Swerllng, P., "Studies of Target Detection Pulsed Radar," Institute of Radio Engineers Transactions on Information Theory, Vol. IT-6. Special Monograph Issue. April 1960.

[13] Nathanson, F. E., ed., "Report of Radar Clutter Signal Processing Committee: Part I, Radar Clutter Effects (U)," The Johns Hopkins University Applied Physics Laboratory, TG 842-1, September 1966, CONFIDENTIAL.

[14] Nolen, J. C., et al.."Statistics of Radar Detection," The Bendix Corporation, Baltimore, Md., February 1966.

[15] Oshiro, Fred, et al.."Calculation of Radar Cross Section," Torthrop Corporation Norair Division, NOR 66-320, October 3 966.

[16] Shotland, Edwin, "False Alarm Probabilities for Receiver Noise and Sea Clutter," The Johns Hopkins University Applied Physics Laboratory, Internal Memorandum BBD-1387, October 1964.

[17] Skolnik, Merrill I., Introduction to Radar Systems, McGraw-Hill Book Company, Inc., New York, 1962, 648 pp.

[18] Wan, Lawrence A., "A Tactical System Radar Signal-Space Model," Proceedings of the Ninth Conference on Military Electronics. September 22-24. 1965. pp. 13-17.

[19] White, D. R.,J. and James, W. G,, "Digital Computer Simulation for Prediction and Analysis of Electromagnetic Interference," Institute of Radio Engineers Transactions on Communications Systems. Vol. CS-9" No. 2. June 1961. pp. 148-1507

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TTNrT.ASStgIED Socurity Cl«»wific»tion

DOCUMENT CONTROL DATA - k i, D iScctirliy clatnltlctllon ol lltlo, body ol »btltarl mnd Indtxlnj mnolallon muni bt »nland when th» ontalt rtport /« rla»»lll»d)

i oniciNATiNG *c r i vi T v (Cr>rpor«(e «i mor;

The Johns Hopkins Univ. Applied Physics Lab. 8621 Georgia Avenue Silver Spring. Md.

2«. REPORT JECURl TV C L AStl PIC A TION

Unclassified 2b. SNOUP ————^.^—.

3 REPORT TITLE

Radar Simulation and Analysis by Digital Computer

4. DESCRIPTIVE NOTES (Typ» of »port and inclusive dale*)

Technical Memorandum S AUTHORISI (F'rsl name, mlddla initial, last name)

D. M. White 6 REPORT DATE

January 1968 7«. TOTAL NO. OP PAGE«

114 (Ö. NO. OF REFS

30 f-a. CONTRACT OR GRANT NO.

NOw 62-0604-c b. PROJECT NO.

e. Task Assignment Z12

M. ORIGINATOR-S REPORT NUMBERIS)

TG-952

»6. OTHER REPORT NOdl (Any othat numb*» Mat may bo maalanad Oil» raport) ' ~

10. DISTRIBJ-ION SVATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Advanced Research Projects Agency.

II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Advanced Research Projects Agency

13. ABSTRACT

—^ A digital computer program is described which simulates the radar-target engagement providing a representation of the detection, acquisition, and tracking processes. The program is arranged as a time simulation of the engagement between a radar and target, taking into account the detailed characteristics of the target cross section, radar and target motion throughout the engagement, surface clutter, atmospheric attenuation, and radar losses. In the output the program provides the user with target detection probabilities in the presence of surface clutter as well as receiver noise, radar search and track accuracies, signal-to-noise ratios, target characteristics vs time, angular and range rates, etc. The input requirements to the program are: (1) a deck of parameter cards describing the radar parameters, the clutter environment, and the initial radar-target geometry, (2) a deck of cards describing the target motion throughout the engagement, and (3) a deck of cards describing the target's cross section vs aspect angle. Many simplifications to the Inputs are allowed for studying and isolating various parts of the radar problem. I ) *

DD FORM 1473 mm.

UNCLASSIFIED B«curlty ClauincaUon

UNCLASSIFIED Security Classification

iif— 14.

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■

—i. mi i i

KEY WORDS

Radar analysis Computer simulation Digital computer program Marcum-Swerling FORTRAN

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.

UNCLASSIFIED Security Classification

unclassified ad number limitation changes · search radar in various situations; and providing...

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