1. INTRODUCTION TO RADAR SYSTEMS Second Edition Merrill I.
Skolnik McGRAW-HILL BOOK COMPANY Auckland Bogotii Guatemala Hamburg
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2. INTRODUCTIONTO RADAR SYSTEMS International Edition 1981
Exclusive rights by McGraw-Hill Book Co.--Singapore for
manufactureand export.This book cannot be re-exported from the
country to which it is consigned by McGraw-Hill. Copyright @
1980,1962by McGraw-Hill, Inc. All rights reserved. Except as
permitted under the United States Copyright Act of 1976,no part of
thispublication may be reproduced or distributed in any form or by
anymeans, or stored in a data base or retrieval system, without the
prior written permission of the publisher. This book was set in
Times Roman. The editor wasFrank J. Cerra. The production
supervisorwas Gayle Angelson. Library of Congress Cataloging in
PubilcationData Skolnik,Merrill Ivan,date Introduction to radar
systems. Includesbibliographical references and index. 1. Radar. I.
Title. 11. Series. TK6575S477 1980 621.3848 79-15354 ISBN
0-07-057909-1 When orderingthis title use ISBN 0-07-066572- 9
Printed in Singapore
3. CONTENTS Preface 1 The Nature of Radar 1.1 lntroductiorl 1.2
*l'lleSirnple Fortn of the Kadar Equatiorl 1.3 Radar Hlock Diagram
and Operation 1.4 Radar Frequencies 1.5 Radar Dcvcloprnent Prior to
World War I1 1.6 Applications of Kadar References The Radar
Equation Prediction of Range Performance Mirlimurn Detectable
Signal Receiver Noise Probability-density Functions Signal-to-noise
Ratio Integration of Radar Pulses Radar Cross Sectiorl of Targets
Cross-section Fluctuations Transmitter Power Pulse Repetition
Frequency and Range Ambiguities Antenna ~aramete'rs System Losses
Propagation Effects Other Consideratiorls Refererlces 3 CW and
Frequency-Modulated Radar 2.1 Tile Iloppler Effect 3.2 CW Radar 3.3
Frequency-modulated CW Radar
4. Airl>or-neDoppler Navigation Multiple-Frequency CW Radar
References MTI and Pulse Doppler Radar Introd~iction Delay-Line
Cancelers Multiple, or Staggered, Pulse Repetition Freqiirncics
Range-Gated Doppler Filters Digital Signal Processing Other MTI
Delay Lines Example of an MTI Radar Processor Limitations to MTI
Performance Noncoherent MTI Pulse Doppler Radar MTI from a Moving
Platform Other Types of MTI References Tracking Radar Tracking with
Radar Sequential Lobing Conical Scan Monopulse Tracking Radar
Target-Reflection Characteristics and Angular Accuracy Tracking in
Range Acquisition Other Topics Comparison of Trackers Tracking with
Surveillance Radar References Radar Transmitters Introduction The
Magnetron Oscillator Klystron Amplifier Traveling-Wave-Tube
Amplifier Hybrid Linear-Beam Amplifier Crossed-Field Amplifiers
Grid-Controlled Tubes Modulators Solid-State Transnlitters
References Radar Antennas Antenna Parameters Antenna Radiation
Pattern and Aperture Distribution Parabolic-Reflector Antennas
Scanning-Feed Reflector Antennas Lens Antennas
5. CONTENTS vii 7.6 Pattern Sy~~rlicsis 7.7 Cosecarit-Squared
Arttenna Pattern 7.8 i:fTccl of Errors on Radiatiot~Patterns 7.9
Kadomcs 7.10 Stabili7ation of Antcnnas f~cfcrcrlccs 8 l'lle
Electrot~icallySteered Phased Array A t I Radar I r r l r
otlr~ctior~ 1t:tsic ('or~ccl>ts I ' I ~ ~ ~ S C .sl1irlct.s
I,'requc~~cy-Scar1Arritys Array Illcnierits l'cccls for Arrays
Sil~lultarlcousMultil>lc13ea1lisfrom Array Ariterllias Random
Errors in Arrays Computer Control of Phased-Array Radar Otlicr
Array Topics Applications of the Array in Radar Advantages arid
Limitations Kcfcrcl~ccs Receivers, Displays, and Duplexers The
Radar Receiver Noise Figure Mixers Low-Noise Front-Erids [)isplays
1)uplexers and Receiver Protectors References Detectiotl of Radar
Signals in Noise Introductiot~ Matched-Filter Receiver Correlation
Detectiori Detection Criteria Detector Cliaracteristics Performance
of the Radar Operator Automatic Detection Constant-False-Alarm-Rate
(CFAR) Receiver References 11 Extractio~~of Information and
Waveform Design 11.1 Introduction 11.2 Information Available from a
Radar 1 1.3 Theoretical Accuracy of Radar Measurements 1 1.4
Ambiguity Diagram
6. viii CONTENTS 11.5 Pulse Compression 11.6 Classification of
Targets with Radar References Propagation of Radar Waves
Introduction Propagation over a Plane Earth The Round Earth
Refraction Anomalous Propagation Diffraction Attenuation by
Atmospheric Gases Environmental Noise Microwave-Radiation Hazards
References Radar Clutter Introduction to Radar Clutter
Surface-Clutter Radar Equations Sea Clutter Detection of Targets in
Sea Clutter Land Clutter Detection of Targets in Land Clutter
Effects of Weather on Radar Detection of Targets in Precipitation
Angel Echoes References Other Radar Topics Synthetic Aperture Radar
HF Over-the-Horizon Radar Air-Surveillance Radar Height-Finder and
3D Radars Electronic Counter-Countermeasures Bistatic Radar
Millimeter Waves and Beyond References Index
7. PREFACE Although tlie fundamentals of radar have changed
little since the publication of the first edition, there has been
continual development of new radar capabilities and continual im-
provements to the technology and practice of radar. This growth has
necessitated extensive revisions arid tlie introduction of topics
not found in the original. One of the major changes is in the
treatment of MTI (moving target indication) radar (Chap. 4). Most
of the basic MTI concepts Gat have been added were known at the
time of tlie first edition, but they had not appeared in the open
literature nor were they widely used i11 practice. Inclusion in the
first edition would have'been largely academic since the analog
delay-line technology available at that time did not make it
practical to build the sophisticated signal processors that were
theoretically possible. However, subsequent advances in digital
technology, originally developed for applications other than radar,
have allowed the practical implementation of the multiple
delay-line cancelers and multiple pulse-repetition-frequency MTI
radars indicated by the basic MTI theory. Automatic detection and
tracking, or ADT (Secs. 5.10 and 10.7). is another important
evelopment whose basic theory was known for some time, but whose
practical realization ad to await advances in digital technology.
The principle of ADT was demonstrated in the early 1950s. using
vacuum-tube technology, as part .of the United States Air Force's
SAGE air-defense system developed by MIT Lincoln Laboratory. In
this form ADT was physically large, expensive, and difficult to
maintain. The commercial availability in the late 1960sof the
solid-slate minicomputer, however, permitted ADT to be relatively
inexpensive, reliable, and of sniall size so that it can be used
with almost any surveillance radar that requires it. Anotlcr radar
area that has seen much development is that of the electronically
steered ptiased-array antenna. In tlie first edition, the radar
antenna was the subject of a single cliaptcr. I11 tliis edition,
one chapter covers the conventional radar antenna (Chap. 7) and a
separate chapter covers the phased-array antenna (Chap. 8).
Devoting a single chapter to the array antenria is inore a
reflection of interest rather than recognition of extensive
application. The chapter o ~ iradar clutter (Ctiap. 13) has been
reorganized to include methods for the detection of targets in the
presence of clutter. Generally, the design techniques necessary for
ttie detection of targets in a clutter,background are considerably
different from.those necessary for detection in a noise background.
Other subjects that are new or which have seen significant
cliaiiges in the current edition include low-angle tracking,
"on-axis" tracking, solid-state RF ources, the mirror-scan
antet~na,antenna stabilization, computer control of phased arrays,
olid-state duplexers, CFAR, pulse compression, target
classification, synthetic-aperture radar, ver-the-horizon radar,
air-surveillance radar, height-finder and 3D radar, and ECCM. The
bistatic radar and millimeter-wave radar are also included even
though their applications have
8. X PREFACE been limited. Omitted from this second edition is
the chapter on Radar Astronomy since interest in this sub.ject has
dccrcascti with tltc i~vi~ilithilityo f space prolws tlliil cilll
explore ttlc planets at close range. The basic material of the
first edition that covers the radar equation, the detection of
signals in noise, the extraction of information, and the
propagation of radar waves has not changed significantly. The
reader, ttowcvcl., wilt find only a fcw pagcs of the original
edition that have not been modified in some manner. One of the
features of the first edition which Ilas hcen contintled is the
inclt~sionof extensive references at the end of each chapter. These
are provided to acknowlcdgc the sources of material used in the
preparation of the book, as well as to permit the interested reader
to learn more about some particular subject. Some references that
appeared in the first edition have been omitted since they have
been replaced by more current references or appear in publications
that are increasingly difficult to find. The references included in
the first edition represented a large fraction ofthose available at
the time. It woilld have been difficult to add to them extensively
or to include many additional topics. This is not so with the
second edition. The current literature is quite large; and, because
of the limitations of.space, only a milch smaller proportion of
what is available could be cited. In addition to changes in radar
technology, there have been changes also in style and nomenclature.
For example, db has been changed to dB, and Mc is replaced by M i i
~ .Also, t he letter-band nomenclature widely employed by the radar
engineer for designating the common radar frequency bands (such as
L, S, and X) has been officially adopted as a standard by the IEEE.
The material in this book has been used as the basis for a graduate
course in radar taught by the author at the Johns Hopkins
.University Evening College and, before that, at several other
institutions. This course is different from those usually found in
most graduate electrical engineering programs. Typical EE courses
cover topics related to circuits, components, de- vices, and
techniques that might make up an electrical or electronic system;
but seldom is the student exposed to the system itself. It is the
system application (whether radar, communica- tions, navigation,
control, information processing, or energy) that is the raison
d'itre for the electrical engineer. The course on which this book
is based is a proven method for introducing the student to the
subject of electronic systems. It integrates and applies the basic
concepts found in the student's other courses and permits the
inclusion of material important to the practice of electrical
engineering not usually found in the traditional curriculum.
Instructors of engineering courses like to use texts that contain a
variety of problems that can be assigned to students. Problems are
not included in this book. Althoirgh the author assigns problems
when using this book as a text, they are not considered a major
learning technique. Instead, the comprehensive term paper, usually
involving a radar design problem or a study in depth of some
particular radar technology, has been found to be a better means
for having the student reinforce what is covered in class and in
the text. Even more important, it allows the student to research
the literature and to be a bit more creative than is possible by
simply solving standard problems. A book of this type which covers
a wide variety of topics cannot be written in isolation. It would
not have been possible,without'themany contributions on radar that
have appeared in the open literature and which have been used here
as the basic source -material. A large measure of gratitude must be
expressed to those radar engineers who have taken the time anci
energy to ensure that the results :of their work were made
available by publication ill recognized journals. I . On a more
personal note, neither edition of this book could have been written
without the complete support and patience of my wife Judith and my
entire family who allowed me tllc time necessary to undertake this
work. Merrill 1. Skolrlik
9. CHAPTER ONE THE NATURE RADAR 1.1 INTRODUCTION Radar is an
electromagnetic system for the detection and location of objects.
It operates by transmitting a particular type of waveform, a
pulse-modulated sine wave for example, and detects the nature of
the echo signal. Radar is used to extend the capability of one's
senses for observing the environment, especially the sense of
vision. The value of radar lies not in being a si~hstitutefor the
eye, but in doing what the eye cannot do-Radar cannot resolve
detail as well the eye, nor is it capable of recognizing the
"color" of objects to the degree of sophistication which the eye is
capable. However, radar can be designed to see through those
conditions irnpervioris to normal human vision, such as darkness,
haze, fog, rain, and snow. In addition, radar has the advantage of
being able to measure the distance or range to the object. This is
probably its most important attribute. An elementary form of radar
consists of a transmitting antenna emitting electromagnetic
radiation generated by an oscilIator of some sort, a receiving
antenna, and an energy-detecting device. or receiver. A portion of
the transmitted signal is intercepted by a reflecting object
(target)and is reradiated in all directions. 1.t is the energy
reradiated in the back direction that is of prime interest to the
radar. The receiving antenna collects the returned energy and
delivers it to a receiver, where it is processed to detect the
presence of the target and to extract its location and relative
velocity. The distance to the target is determined by measuring the
time taken for the radar signal to travel to the target and back.
The direction, or angular position, of the target may be determined
from the direction of arrival of the reflected wave- front. The
usual method of measuring the direction of arrival is with narrow
antenna beams. If relative motion exists between target and radar,
the shift in the carrier frequency of the reflected wave (doppler
effect) is a measure of the target's relative (radial) velocity and
may be used to distinguish moving targets from stationary objects.
In radars which continuously track the movement of a target, a
continuous indication of the rate of change of target position is
also available. 1
10. 2 INTRODUCTION TO RADAR SYSTEMS The name radar reflects the
emphasis placed by the early experimenters on a device to detect
the presence of a target and measure its range. Radar is a
contraction of the words radio detection and ranging. It was first
developed as a detection device to warn of the approach of hostile
aircraft and for directing antiaircraft weapons. Although a
well-designed modern radar can usually extract more information
from the target signal than merely range, the measure- ment of
range is still one of radar's most important functions. There seem
to be no other competitive techniques which can measure range as
well or as rapidly as can a radar. The most common radar waveform
is a train of narrow, rectangular-shape pulses modu- lating a
sinewave carrier. The distance, or range, to the target is
determined by measuring the time TRtaken by the pulse to travel to
the target and return. Since electromagnetic energy propagates at
the speed of light c = 3 x 10' m/s, the range R is The factor 2
appears in the denominator because of the two-way propagation of
radar. With the range in kilometers or nautical miles, and TRin
microseconds, Eq. (1.1) becomes Each microsecond of round-trip
travel time corresponds to a distance of 0.081 nautical mile, 0.093
statute mile, 150 meters, 164 yards, or 492.feet. Once the
transmitted pulse is emitted by the radar, a sufficient length of
time must elapse to allow any echo signals to return and be
detected before the next pulse may be transmitted. Therefore the
rate at which the pulses may be transmitted is determined by the
longest range at which targets are expected. If the pulse
repetition frequency is too high, echo signals from some targets
might arrive after the transmission of the next pulse, and
ambiguities in measuring I I . . . Pulse repetition' frequency, Hz
Figure 1.1 Plot of maximum unambiguous range as a function of the
pulse repetition frequency.
11. THE NATURE OF RADAR 3 range might result. Echoes that
arrive after the transmission of the next pulse are called
secorrd-tinte-arotrrtd (or multiple-time-around) echoes. Such an
echo would appear to be at a much shorter range than the actual and
could be misleading if it were not known to be a second-time-around
echo. The range beyond which targets appear as second-time-around
echoes is called the rna.uintttrn trr~arnhigtrousrattge and is
where./, = pulse repetition frequency, in Hz. A plot of the maximum
unambiguous range as a function of pulse repetition frequency is
shown in Fig. 1.1. Although the typical radar transmits a simple
pulse-modulated waveform, there are a number of other suitable
modulations that might be used. The pulse carrier might be
frequency- or phase-modulated to permit the echo signals to be
compressed in time after reception. This achieves the benefits of
high range-resolution without the need to resort to a . short
pulse. The technique of using a long, modulated pulse to obtain the
resolution of a short pulse, but with the energy of a long pulse,
is known as pulse compression. Continuous waveforms (CW) also can
be used by taking advantage of the doppler frequency shift to
separate the received echo from the transmitted signal and the
echoes from stationary clutter. Unmodulated CW waveforms do not
measure range, but a range measurement can be made by applying
either frequency- or phase-modulation. 1.2 THE SIMPLE FORM OF THE
RADAR EQUATION The radar equation relates the range of a radar to
the characteristics of the transmitter, receiver, antenna, target,
and environment. It is useful not just as a means for determining
the maximum distance from the radar to the target, but it can serve
both as a tool for under- standing radar operation and as a basis
for radar design. In this section, the simple form of the radar
equation is derived. I f the power of the radar transmitter is
denoted by P,, and if an isotropic antenna is used (one which
radiates uniformly in all directions), the power density (watts per
unit area) at a distance R from the radar is equal to the
transmitter power divided by the surface area 4nR2of an imaginary
sphere of rapius R, or pt Power density from isotropic antenna = -
4nR2 Radars employ directive antennas to channel, or direct, the
radiated power Pt into some particular direction. The gain G ofan
antenna is a measure of the increased power radiated in the
direction of the target as compared with the power that would have
been radiated from an isotropic antenna. It may be defined as the
ratio of the maximum radiation intensity from the subject antenna
to the radiation intensity from a lossless, isotropic antenna with
the same power input. (The radiation intensity is the power
radiated per unit solid angle in a given direction.) The power
density at the target from an antenna with a transmitting gain G is
Pt GPower density from directive antenna = - 4nR2 The target
intercepts a portion of the incident power and reradiates it in
vqrious directions.
12. 4 INTRODUCTION TO RADAR SYSTEMS The measure of the amount
of incident power intercepted by the target and reradiated back in
the direction of the radar is denoted as the radar cross section
a,and is defined by the relation P,G a Power density of echo signal
at radar = ---- - 4nR24nR2 The radar cross section a has units of
area. It is a characteristic of the particular target and is a
measure of its size as seen by the radar. The radar antenna
captures a portion of the echo power. If the effective area of the
receiving antenna is denoted A., the power P, received by the radar
is The maximum radar range Rmaxis the distance beyond which the
target cannot be detected. It occurs when the received echo signal
power P, just equals the minimum detectable signal S,,, . Therefore
1 This is the fundamental form of the radar equation. Note that the
important antenna par- ameters are the transmitting gain and the
receiving effective area. Antenna theory gives the relationship
between the transmitting gain and the receiving effective area of
an antenna as Since radars generally use the same antenna for both
transmission and reception, Eq. (1.8)can be substituted into Eq.
(1.7), first for A, then for G, to give two other forms of the
radar equation These three forms (Eqs. 1.7, 1.9, and 1.10)
illustrate the need to be careful in the inter- pretation of the
radar equation. For example, from Eq. (1.9) it might be thought
that the range of a radar varies as All2,but Eq. (1.10) indicates a
1-'12 relationship, and Eq. (1.7) shows the range to be independent
of 1.The correct relationship depends on whether it is assumed the
gain is constant or the effective area is constant with wavelength.
Furthermore, the introduc- tion of other constraints, such as the
requirement to scan a specified volume in a given time, can yield a
different wavelength dependence. These simplified versions of the
radar equation do not adequately describe the perfor- mance of
practical radar. Many important factors that affect range are not
explicitly included. In practice, the observed maximum radar ranges
are usually much smaller than what would be predicted by the above
equations, sometimes by as much as a factor of two. There are many
reasons for the failure of the simple radar equation to correlate
with actual performance, as discussed in Chap. 2. _ , 9 , 1 1 '
.
13. T H E N A T U R E OF RADAR 5 1.3 RADAR BLOCK DIAGRAM AND
OPERATION Ttle operation of a typical pulse radar may be described
with the aid of the block diagram shown in Fig. 1.2. Tlle
transtnitter may be an oscillator, such as a magnetron, that is "
pulsed" (turned on and on) by the rnodulator to generate a
repetitive train of pulses. The magnetron has prohnhly been the
most widely used of the various microwave generators for radar. A
typicrtl radar for tile dctcction of aircraft at ranges of 100 or
200 nmi might employ a peak power of the order of a megawatt, an
average power of several kilowatts, a pulse width of several
microseconds, and a pulse repetition frequency of several hundred
pulses per second. The waveform generated by the transmitter
travels via a transmission line to the antenna, where it is
radiated into space. A single antenna is generally used for both
transmitting and receiving. The receiver must be protected from
damage caused by the high power of the transmitter. This is the
function of the duplexer. The duplexer also serves to channel the
returned echo signals to the receiver and not to the transmitter.
The duplexer might consist of two gas-discharge devices, one known
as a TR (transmit-receive) and the other an ATR
(anti-transmit-receive). The TR protects the receiver during
transmission and the ATR directs the echo signal to the receiver
during reception. Solid-state ferrite circulators and receiver
protectors with gas-plasma TR devices and/or diode limiters are
also employed as duplexers. The receiver is usually of the
superheterodyne type. The first stage might be a low-noise RF
amplifier, such as a parametric amplifier or a low-noise
transistor. However, it is not always desirable to employ a
low-noise first stage in radar. The receiver input can simply be
the mixer stage, especially in military radars that must operate in
a noisy environment. Although a receiver with a low-noise front-end
will be more sensitive, the mixer input can have greater dynamic
range, less susceptibility to overload, and less vulnerability to
electronic interference. The mixer and local oscillator (LO)convert
the RF signal to an intermediate frequency (IF).A " typical" IF
amplifier for an air-surveillance radar might have a center
frequency of 30 or 60 MHz and a bandwidth of the order of one
megahertz. The IF amplifier should be designed as a
n~atcltedfilter; i.e., its frequency-response function H ( f )
should maximize the peak-sigtial-to-mean-noise-powerratio at the
output. This occurs when the magnitude of the frequency-response
function 1H ( f ) ( is equal to the magnitude of the echo signal
spectrum IS(.f')1, and the phase spectrum of the matched filter is
the negative of the phase spectrum of the echo signal (Sec. 10.2).
In a radar whose signal waveform approximates a rectangular pulse,
the conventional IF filter bandpass characteristic approximates a
matched filter when the product of the IF bandwidth B and the pulse
width r is of the order of unity, that is, Bt -1. After maximizing
the signal-to-noise ratio in the IF amplifier, the pulse modulation
is extracted by the second detector acd amplified by the video
amplifier to a level where it can be Tronsrnitler Pulse modulalor 4
Low - noise RF Mixer amplifier I.. Figure 1.2 Block diagram of a
pulse radar.
14. Figure 1.3 (a) PPI presentation displaying range vs. angle
(intensity modulation); ( h ) A-scope presenta- tion displaying
amplitude vs. range (deflection modulation)."4 properly displayed,
usually on a cathode-ray tube (CRT).Timing signals are also
supplied to the indicator to provide the range zero. Angle
information is obtained from the pointing direction of the antenna.
The most common form of cathode-ray tube display is the plan
position indicator, or PPI (Fig. 1.3a), which maps in polar
coordinates the location of the target in azimuth and range. This
is an intensity-modulated display in which the amplitude of the
receiver output modulates the electron-beam intensity (z axis)as
the electron beam is made to sweep outward from the center of the
tube. The beam rotates in angle in response to the antenna
position. A B-scope display is similar to the PPI except that it
utilizes rectangular, rather than polar, coordinates to display
range vs. angle. Both the B-scope and the PPI, being intensity
modulated, have limited dynamic range. Another form of display is
the A-scope, shown in Fig. 1.3b, which plots target .amplitude (y
axis) vs. range (x axis), for some fixed direction. This is a
deflection-modulated display. It is more suited for tracking-radar
applica- tion than for surveillance radar. The block diagram of
Fig. 1.2 is a simplified version that omits many details. I t does
not include several devices often found in radar, such as means for
automatically compensating the receiver for changes in frequency
(AFC)or gain (AGC),receiver circuits for reducing interfer- ence
from other radars and from unwanted signals, rotary joints in the
transmission lines to allow movement of the antenna, circuitry for
discriminating between moving targets and unwanted stationary
objects (MTI),and pulsecompression for achievingthe resolution
benefits of a short pulse but with the energy of a 'long pulse. If
the radar is used for tracking, some means are necessary for
sensing the angular location of a moving target and allowing the
antenna automatically to lock-on and to track the target.
Monitoring devices are usually included to ensure that the
transmitter is delivering the proper shape pulse at the proper
power level and that the receiver sensitivity has not degraded.
Provisions may also be in- corporated in the radar for locating
equipment failures so that faulty circuits can be easily found and
replaced. Instead of displaying the "raw-video" output of a
surveillance radar directly on the CRT, it might first be processed
by an'autornaticdetection and tracking (ADT)device that quantizes
the radar coverage into range-azimuth resolution cells, adds (or
integrates) all the echo pulses received within each cell,
establishes a threshold (on the basis of these integrated pulses)
that permits only the strong outputs due to target echoes to pass
while rejecting noise, establishes and maintains the tracks
(trajectories) of each target, and displays the processed
information
15. THE NATURE OF RADAR 7 to the operator. These operations of
an ADT are usually implemented with digital computer techriology. A
common form of radar antenna is a reflector with a parabolic shape,
fed (illuminated) from a point source at its focus. The parabolic
reflector focuses the energy into a narrow beam, just as does a
searchlight or an automobile headlamp. The beam may be scanned in
space by mechanical pointing of the antenna. Phased-array antennas
have also been used for radar. In a pllascd array, tllc bcam is
scanned by electronically varying the phase of the currents across
the aperture. 1.4 RADAR FREQUENCIES Conventional radars generally
have been operated at frequencies extending from about 220 MHz to
35 GHz, a spread of more than seven octaves. These are not
necessarily the limits, . , since radars can be, and have been,
operated at frequencies outside either end of this range. Skywave H
F over-the-horizon (OTH)radar might be at frequenciesas low as 4 or
5 MHz, and groundwave H F radars as low as 2 MHz. At the other end
of the spectrum, millimeter radars have operated at 94 GHz. Laser
radars operate at even higher frequencies. The place of radar
frequencies in the electromagnetic spectrum is shown in Fig.
1.4.Some of the nomenclature employed to designate the various
frequency regions is also shown. Early in the development of radar,
a letter code such as S, X, L, etc., was employed to
desigr~ateradar frequency bands. Although its original purpose was
to guard military secrecy, the designations were maintained,
probably out of habit as well as the need for some conven- ient
short nomenclature. This usage has continued and is now an accepted
practice of radar engineers. Table 1.1 lists the radar-frequency
letter-band nomenclature adopted by the IEEE.' These are related to
the specific bands assigned by the International Telecommunica-
tions Union for radar. For example, although the nominal frequency
range for L band is 1000 to 2000 MHz, an L-band radar is thought of
as being confined within the region from 1215to 1400 MHz since that
is the extent of the assigned band. Letter-band nomenclature is not
a Wovelenath 3 0 H z 3 0 0 H z 3kHz 30kHz 3 0 0 k H z 3 M H z 3 0 M
H z 3 0 0 M H z 3GHz 30GHz 300GHz 3,000 GHr Frequency 10 km l k m
100 m 1 0 m Im lOcm t c m 1mm O . l n m Figure 1.4 Radar
frequencies and the electromagnetic spectrum. UHF---c+SHF-+k Super
high frequency Centimetric woves I- V L F Very low frequency I
Myriometric woves -+HF-t-VHF-+ High frequency Decometric waves 4 E
H F --+ Extremely hrgh frequency Millimetric waves --LF-+-MF Low
frequency Kilometric woves 1 I I I Bond 7 Very high frequency
Metric woves Decimilli- metric woves Medium frequency Hectometric
woves Video frequencies Ultrohigh frequency Decimetric wovcs Bond 4
Audio frequencies c Bond 8 I Bond 9 Bond 10 Bond l l mdgf r w Bond
12 Submillimeler Bond 5 IFor in f m m Bond 6 t------* IOTH rodor I
Letter designotions L S C X Ku Ka * I Microwove region I
16. 8 INTRODUCTION TO RADAR SYSTEMS Table 1.1 Standard
radar-frequency letter-band nomenclature Specific radiolocatio~l
Band Nominal (radar) bands based on designation frequency range ITU
assignments for region 2 HF VHF UHF K K , rnrn 138-144 MHz 216-225
420-450 MHz 890-942 1215-1400 MHz 2300-2500 MHz 2700-3 700
5250-5925 MHz 8500- 10,680 MHz 13.4-14.0 GHz 15.7- 17.7 24.05-24.25
GHz 33.4-36.0 GHz substitute for the actual numerical frequency
limits of radars. The specific numerical frequency limits should be
used whenever appropriate, but the letter designations of Table 1.1
may be used whenever a short notation is desired. 1.5 RADAR
DEVELOPMENT PRIOR TO WORLD WAR I1 Although the development of radar
as a full-fledged technology did not occur until World War 11, the
basic principle of radar detection is almost as old as the subject
of electromagnetism itself. Heinrich Hertz, in 1886,experimentally
tested the theories of Maxwell and demonstrated the similarity
between radio and light waves. Hertz showed that radio waves could
be reflected L ,I by metallic and dielectric bodies. It is
interesting to note that although Hertz's experiments were
performed with relatively short wavelength radiation (66 cm), later
work in radio engin- eering was almost entirely at longer
wavelengths. The shorter wavelengths were not actively used to any
great extent until the late thirties. In 1903a German engineer by
the name of Hiilsmeyer experimented with the detection of radio
waves reflected from ships. He obtained a patent in 1904 in several
countries for an obstacle detector and ship navigational d e ~ i c
e . ~His methods were demonstrated before the German Navy, but
generated little interest. The state of technology at that time was
not sufficiently adequate to obtain ranges of more than about a
mile, and his detection technique was dismissed on the grounds that
it was little better than a visual observer. Marconi recognized the
potentialities of short waves for radio detection and strongly
urged their use in 1922 for this application. In a speech delivered
before the Institute of Radio Engineers, he said:' As was first
shown by Hertz, electric waves can be completely reflected by
conducting bodies. In some of 'my tests I have noticed the effects
of reflection and detection of these waves by metallic objects
miles away. It !eems to me that it should be possible to design
apparatus by means of which a ship could
17. T H E N A T U R E OF R A D A R 9 radiate or project a
divergent beam of these rays in any desired direction, which rays,
if coming across a metallic object, such as another steamer or
ship, would be reflected back to a receiver screened from the local
transmitter on the sending ship, and thereby, immediately reveal
the presence and bearing of the other ship in fog or thick weather.
Although Marconi predicted and successfully demonstrated radio
communication be- tween continents, he was apparently not
successful in gaining support for some of his other ideas involving
very short waves. One was the radar detection mentioned above; the
other was the suggestion that very short waves are capable of
propagation well beyond the optical line of sight-a phenometlon now
know11as tropospheric scatter. He also suggested that radio waves
be used for the transfer of power from one point to the other
without the use of wire or other trat~smissiot~lit~cs. In the
aututnrl of 1922 A. ti. I'aylor arid L. C. Young of tile Naval
Research Laboratory detected a wooden ship using a CW
wave-interference radar with separated receiver and transmitter.
The wavelerlgth was 5 m. A proposal was submitted for further work
but was not accepted. The first application of the pulse technique
to the measurement of distance was in the basic scientific
investigation by Breit and Tuve in 1925 for measuring the height of
the i~nosphere.~.'~However, more than a decade was to elapse before
the detection of aircraft by pulse radar was demonstrated. The
first experimental radar systems operated with CW and depended for
detection upon the interference produced between the direct signal
received from the transmitter and the doppler-frequency-shifted
signal reflected by a moving target. This effect is the same as the
rhythmic flickering, or flutter, observed in an ordinary television
receiver, especially on weak stations, when an aircraft passes
overhead. This type of radar originally was called CW
wqaoe-irtfer-erenceradar. Today, such a radar is called a bistatic
CW radar. The first experimen- tal detections of aircraft used this
radar principle rather than a monostatic (single-site) pulse radar
because CW equipment was readily available. Successful pulse radar
had to await the development of suitable components, especially
high-peak-power tubes, and a better under- standing of pulse
receivers. The first detection of aircraft using the
wave-interference effect was made in June, 1930,by L. A. tlyland of
the Naval Research Laboratory.' It was made accidentally while he
was working with a direction-finding apparatus located in an
aircraft on the ground.The transmit- ter at a frequency of 33 MHz
was located 2 miles away, and the beam crossed an air lane from a
nearby airfield. When aircraft passed through the beam, Hyland
noted an increase in the received signal. This stimulated a more
deliberate investigation by the NRL personnel, but the work
continued at a slow pace, lacking official encouragement and funds
from the govern- nrent. although it was fully supported by the NRL
administration. By 1932the equipment was demonstrated to detect
aircraft at distances as great as 50 miles from the transmitter.
The NRL work on aircraft detection with CW wave interference was
kept classified until 1933, when several Bell Telephone
Laboratories engineers reported the detection of aircraft during
the course of other experiments.' The NRL work was disclosed in a
patent filed and granted to Taylor, Young, and Hyland6 on a "System
for Detecting Objects by Radio." The type of radar described in
this patent was a CW wave-interference radar. Early in 1934, a
60-MHz CW wave-interference radar was demonstrated by NRL. The
early CW wave-interference radars were useful only for detecting
the preserrce of the target. The problem of extracting
target-position information from such radars was a difficult one
and could not be readily solved with the techniques existing at
that time. A proposal was made by NRL in 1933 to errlploy a chain
of transmitting and receiving stations along a line to be guarded.
for the purpose of obtaining some knowledge of distance and
velocity. This was
18. 10 INTRODUCTION TO RADAR SYSTEMS never carried out,
however. The limited ability ofCW wave-interferenceradar to be
anything more than a trip wire undoubtedly tempered what little
official enthusiasm existed for radar. It was recognized that the
limitations to obtaining adequate position information coiild be
overcome with pulse transmission. Strange as it may now seem, in
the early days pulse radar encountered much skepticism.
Nevertheless,an effort was started at NRL in the spring of 1934 to
develop a pulse radar. The work received low priority and was
carried out prin- cipally by R. M. Page, but he was not allowed to
devote his full time to the effort. The first attempt with pulse
radar at NRL was at a frequency of60 MHz. According to Guerlac,'
the first tests of the 60-MHz pulse radar were carried out in late
December, 1934, and early January, 1935.These tests were
"hopelessly unsuccessful and a grievousdisappoint- ment." No pulse
echoes were observed on the cathode-ray tube. The chief reason for
this failure was attributed to the receiver's being designed for CW
communications rather than for pulse reception. The shortcomings
were corrected, and the first radar echoes obtained at NRL using
pulses occurred on April 28, 1936, with a radar operating at a
frequency of 28.3 MHz and a pulse width of 5 ,US. The range was
only 24 miles. By early June the range was 25 miles. It was
realized by the NRL experimenters that higher radar frequencies
were desired, especially for shipboard application, where large
antennas could not be tolerated. However, the necessary components
did not exist.The success of the experiments at 28 MHz encouraged
the NRL experimenters to develop a 200-MHz equipment. The first
echoes at 200 MHz were received July 22, 1936,less than three
months after the start of the project. This radar was also the
first to employ a duplexing system with a common antenna for both
transmitting and receiving. The range was only 10to 12miles. In the
spring of 1937it was installed and tested on the destroyer Leary.
The range of the 200-MHz radar was limited by the transmitter. The
development of higher-powered tubes by the Eitel-McCullough
Corporation allowed an improved design of the 200-MHz radar known
as XAF. This occurred in January, 1938. Although the power
delivered to the antenna was only 6 kW, a range of 50 miles-the
limit of the sweep-was obtained by February. The XAF was tested
aboard the battleship New York, in maneuvers held during January
and February of 1939,and met with considerable success. Ranges of
20 to 24 kiloyards were obtained on battleships and cruisers. By
October, 1939, orders were placed for a manufactured version called
the CXAM. Nineteen of these radars were installed on major ships of
the fleet by 1941. The United States Army Signal Corps also
maintained an interest in radar during the early 1930s.' The
beginning of serious Signal Corps work in pulse radar apparently
resulted from a visit to NRL in January, 1936.By December of that
year the Army tested its first pulse radar, obtaining a range of 7
miles.The first operational radar used for antiaircraft firecontrol
was the SCR-268,available in 1938,' The SCR-268 was used in
conjunction with searchlights for radar fire control. This was
necessary because of its poor angular accuracy. However, its range
accuracy was superior to that obtained with optical methods. The
SCR-268 remained the standard fire-control equipment until January,
1944,when it was replaced by the SCR-584 microwave radar. The
SCR-584 could control an antiaircraft battery without the necessity
for searchlights or optical angle tracking, In 1939the
Armydeveloped the SCR-270,a long-range radar for early warning. The
attack on Pearl Harbor in December, 1941,was detected by an
SCR-270, one of six in Hawaii at the time.' (There were also 16
SCR-268s assigned to units in Honolulu.) But unfortunately, the
true significanceof the blips on the scope was not realized until
after the bombs had fallen. A modified SCR-270 was also the first
radar to detect echoesfrom the moon in 1946. The early developments
of pulse radar were primarily concerned with military applica-
tions. Although it was not recognized as being a radar at the time,
the frequency-modulated
19. THE NATURE OF RADAR 11 aircraft radio altimeter was
probably tlie first commercial application of tlie radar principie.
The first equipments were operated in aircraft as early as 1936and
utilized the same principle of operation as the FM-CW radar
described in Sec. 3.3. In the case of the radio altimeter, the
target is tlie ground. 111 13rit.aiti [lie development of radar
began later than it1 the United States.'-'' But because they felt
the nearness of war more acutely and were in a more vulnerable
position with respect to air attack, the British expended a large
amount of efforton radar development. By the time the United States
entered tlie war, the British were well experienced in the military
applications of radar. British interest in radar began in early
1935,when Sir Robert Watson- Watt was asked about the possibility
of producing a death ray using radio waves. Watson- Watt concluded
that this type of death ray required fantastically large amounts of
power and could be regarded as not being practical at that time.
Instead, he recotnmended that it would be more promising to
investigate means for radio detection as opposed to radio
destruction. (The only available means for locating aircraft prior
to World War IE were sound locators whose maximum detection range
under favorable conditions was about 20 miles.) Watson- Watt was
allowed to explore the possibilities of radio detection, and in
February, 1935, he issued two memoranda outlining the conditions
necessary for an effectiveradar system. In that same month the
detection of an aircraft was carried out, using 6-MHz communication
equip- ment, by observing tlie beats between the echo signal and
the directly received signal (wave interference). The technique was
similar to the first United States radar-detection experiments. The
transmitter and receiver were separated by about 5.5 miles. When
the aircraft receded froin the receiver, it was possible to detect
the beats to about an 8-mile range. By June, 1935, the British had
demonstrated the pulse technique to measure range of an aircraft
target. This was almost a year sooner than the successful NRL
experiments with pulse radar. By September, ranges greater than 40
miles were obtained on bomber aircraft. The frequency was 12 MHz.
Also, in that month, the first radar measurement of the height of
aircraft above ground was made by measuring the elevation angle of
arrival of the reflected signal. In March, 1936, the range of
detection had increased to 90 miles and the frequency was raised to
25 MHz. A series of CH (Chain Homej radar stations at a frequency
of 25 MHz were successfully demonstrated in April, 1937. Most of
the stations were operating by September, 1938, and plotted the
track of the aircraft which flew Neville Chamberlain, the British
Prime Minister at that time, to Munich to confer with Hitler and
Mussolini. In the same month, the CH radar stations began 24-hour
duty, which continued until the end of the war. The British
realized quite early that ground-based search radars such as CH
were not sufficiently accurate to guide fighter aircraft to a
complete interception at night or in bad weather. Consequently,
they developed, by 1939,an aircraft-interception radar (AI),
mounted on an aircraft, for the detection and interception of
hostile aircraft. The A1 radar operated at a frequency of 200 MHz.
During the development of the A1 radar it was noted that radar
could be used for the detection of ships from the air and also that
the character of echoes from the ground was dependent on the nature
of the terrain. The former phenomenon was quickly exploited for the
detection and location of surface ships and submarines. The latter
effect was not exploited initially, but was later used for airborne
mapping radars. Until the middle of 1940 tlie development of radar
in Britain and the United States was carried out independently of
one another. In September of that year a British technical mission
visited the United States to exchange information concerning the
radar developments in the two countries. The British realized the
advantages to be gained from the better angular resolution possible
at the microwave frequencies, especially for airborne and naval
applica- tions. They suggested that the United States undertake the
development of a microwave A1
20. 12 INTRODUCTION TO RADAR SYSTEMS radar and a microwave
antiaircraft fire-control radar. The British technical mission
demonstrated the cavity-magnetron power tube developed by Randell
and Boot and furnished design information so that it could be
duplicated by United States manufacturers. The Randell and Boot
magnetron operated at a wavelength of 10 cm and produced a power
output of about 1 kW, an improvement by a factor of 100 over
anything previously achieved at cen- timeter wavelengths. The
development of the magnetron was one of tile most important
contributions to the realization of microwave radar. The success of
microwave radar was by no means certain at the end of
1940.Therefore the United States Service Laboratories chose to
concentrate on the development of radars at the lower frequencies,
primarily the very high frequency (VHF) band, where techniques and
components were more readily available. The exploration of the
microwave region for radar application became the responsibility of
the Radiation Laboratory, organized in November, 1940, under the
administration of the Massachusetts Institute of Technology. In
addition to the developments carried out in the United States and
Great Britain, radar was developed essentially independently in
Germany, France, Russia, Italy, and Japan during the middle and
late thirties.12The extent of these developments and their
subsequent military deployment varied, however. All of these
countries carried out experiments with CW wave interference, and
even though the French and the Japanese deployed such radars opera-
tionally, they proved of limited value. Each country eventually
progressed to pulse radar operation and the advantages pertaining
thereto. Although the advantages of the higher frequencies were
well recognized, except for the United States and Great Britain
none of the others deployed radar at frequencies higher than about
600 MHz during the war. The Germans deployed several different
types of radars during World War 11. Ground- based radars were
avgilable for air search and height finding so as to perform ground
control of intercept (GCI).Coastal, shipboard, and airborne radar
were also employed successfully in significant numbers. An
excellent description of the electronic battle in World War I1
between the Germans and the Allies, with many lessons to offer, is
the book " It~strtlrnerttsof Dcrrkt~ess" by Price.I3 The French
efforts in radar, although they got an early start, were not as
energetically supported as in Britain or the United States, and
were severely disrupted by the German occupation in 1940.12The
development of radar in Italy also started early, but was
slow.There were only relatively few Italian-produced radars
operationally deployed by the time they left the war in September,
1943. The work in Japan was also slow but received impetus from
disclosures by their German allies in 1940and from the capture of
United States pulse radars in the Philippines early in 1942. The
development of radar in the Soviet Union was quite similar to the
experience elsewhere. By the summer of 1941they had deployed
operationally a number of 80-MHz air-search radars for the defense'
of Moscow against the German invasion.14Their indigenous efforts
were interrupted by the course of the war. Thus, radar developed
independently and simultaneously in severalcountries just prior to
World War 11. It is not possible to single out any one individual
as the inventor; there were many fathers of radar. This was brought
about not only by the spread of radio technology to many countries,
but by the maturing of the airplane during this same time and the
common recognition of its military threat and the need to defend
against it. . ' 1.6 APPLICATIONS OF RADAR Radar has been employed
on'the ground, in the air, on the sea, and in space. Ground-based
radar has been applied chiefly to'the detection, location, and
tracking of aircraft or space targets. Shipboard radar is used as a
navigation aid and safety device to locate buoys, shore
21. lines, and other ships. as well as for observing aircraft.
Airborne radar may be used to detect other aircraft, ships, or land
vehicles, or it may be used for mapping of land, storm avoidance,
terrain avoidance, and navigation. In space, radar has assisted in
the guidance of spacecraft and for the remote sensing of the land
and sea. The major user of radar, and contributor of the cost of
almost all of its development, has been the military: although
there have been increasingly important civil applications, chiefly
for niaririe and air tiavigation. The niajor areas of radar
application, in no particular order of irnpo~ta~icc,are Ijriefly
described below. Air. Trclffic Corrtrol ( ATC). Radars are employed
throughout the world for the purpose of safely coritrollit~gair
traffic en route and in tlic vicinity of airports. Aircraft and
ground vcllicular traffic st large airports are monitored by
tliearis of high-resolution radar. Radar has been used with GCA
(ground-control approach) systems to guide aircraft to a safe
landing in bad weather. In addition, the microwave landing system
and the widely used ATC radar-beacon system are based in large part
on radar technology. Aircv-aft Nac~iqatiotl.The weather-avoidance
radar used on aircraft to outline regions of preci- pitation to the
pilot is a classical form of radar. Radar is also used for terrain
avoidance and terrain following. Although they may not always be
thought of as radars, the radio altimeter (either FM/CW or pulse)
and the doppler navigator are also radars. Sometimes ground-mapping
radars of moderately high resolution are used for aircraft
navigation purposes. Ship Safety. Radar is used for enhancing the
safety of ship travel by warning of potential collision with other
ships, and for detecting navigation buoys, especially in poor
visibility. I11 terms of numbers, this is one of the larger
applications of radar, but in terms of physical size and cost it is
one of the smallest. It has also proven to be one of the most
reliable radar systems. Automatic detection and tracking equipments
(also called plot extractors) are commercially available for use
with such radars for the purpose of collision avoi- dance.
Shore-based radar of moderately high resolution is also used for
the surveillance of liarbors as an aid to navigation. Space.. Space
vehicles have used radar for rendezvous and docking, and for
landing on the moon. Some of the largest ground-based radars are
for the detection and tracking of satellites. Satcllitc-borne
radars have also been used for remote sensing as meritioried below.
Rer~roteSetrsirrg. A11 radars are remote sensors; however, as this
term is used it implies the sensing of geophysical objects, or the
"environment." For some time, radar has been used as a remote
sensor of the weather. It was also used in the past to probe the
moon and the planets (radar astronomy).The ionospheric sounder, an
important adjunct for HF (short wave) communications, is a radar.
Remote sensing with radar is also concerned with Earth resources,
which includes the measurement and mapping of sea conditions, water
resources, ice cover, agriculture, forestry conditions, geological
formations, and environ- niental pollution. The platforms for such
radars include satellites as well as aircraft. La~vErfircentenr. In
addition to the wide use of radar to measure the speed of
automobile traffic by highway police, radar has also been employed
as a means for the detection of intruders. Alilitnrv. Many of the
civilian applications of radar are also employed by the military.
The traditional role of radar for military application has been for
surveillance, navigation, and for the control and guidance of
weapons. It represents, by far, the largest use of radar.
22. 14 INTRODUCTION TO RADAR SYSTEMS REFERENCES 1. Guerlac, H.
E.: "OSRD Long History," vol. V, Division 14, "Radar," available
from Office of Technical Services, U.S. Department of Commerce. 2.
British Patent 13,170,issued to Christian Hiilsmeyer, Sept. 22,
1904,entitled " Hertzian-wave Project- ing and Receiving Apparatus
Adapted to Indicate or Give Warning of the Presence of a Metallic
Body, Such as a Ship or a Train, in the Line of Projection of Such
Waves." 3. Marconi, S. G.: Radio Telegraphy, Proc. IRE, vol. 10,
no. 4, p. 237, 1922. 4. Breit, G., and M. A. Tuve: A Test of the
Existence of the Conducting Layer, Phys Rev., vol. 28, pp. 554-575,
September, 1926. 5. Englund, C. R., A. B. Crawford, and W. W.
Mumford: Some results of a Study of Ultra-short-wave Transmission
Phenomena, Proc. IRE, vol. 21, pp. 475-492, March, 1933. 6. CIS.
Patent 1,981,884,"System for Detecting Objects by Radio," issued to
A. H. Taylor, L. C. Young, and L. A. Hyland, Nov. 27, 1934. 7.
Vieweger, A. L.: Radar in the Signal Corps, IRE Trans., vol. MIL-4,
pp. 555-561, October, 1960. 8. Origins of Radar: Background to the
Awards of the Royal Commission, Wireless World, vol. 58, pp. 95-99,
March, 1952. 9. Wilkins, A. F.: The Story of Radar, Research
(London), vol. 6, pp. 434-440, November, 1953. 10. Rowe, A. P.:
"One Story of Radar," Cambridge University Press, New York, 1948. A
very readable .i description of the history of radar development at
TRE (Telecommunications Research Establish- ment, England) and how
TRE went about its business from 1935 to the end of World War 11.
11. Watson-Watt, Sir Robert: "Three Steps to Victory," Odhams
Press, Ltd., London, 1957;"The Pulse of Radar," The Dial Press,
Inc., New York, 1959. 12. Susskind, C.: "The Birth of the Golden
Cockerel: The Development of Radar," in preparation 13. Price, A.:
"Instruments of Darkness," Macdonald and Janes, London, 1977. 14.
Lobanov, M. M.: "Iz Proshlovo Radiolokatzii" (Out of the Past of
Radar), Military Publisher of the Ministry of Defense, USSR,
Moscow, 1969. 15. IEEE Standard Letter Designations for
Radar-Frequency Bands, IEEE Std 521-1976, Nov. 30, 1976. 16.
Villard, 0.G., Jr.: The Ionospheric Sounder and Its Place in the
History of Radio Science, Radio Science, vol. 11, pp. 847-860,
November, 1976.
23. TWO THE RADAR EQUATION 2.1 PREDICTION OF RANGE PERFORMANCE
The simple form of the radar equation derived in Sec. 1.2expressed
the maximum radar range R,,, in terms of radar and target
parameters: where P, = transmitted power, watts G = antenna gain A,
= antenna emective aperture, m2 a = radar cross section, m2 Smin=
minimum detectable signal, watts All the parameters are to some
extent under the control of the radar designer, except for the
target cross section a. The radar equation states that iflong
ranges are desired,the transmitted power must be large, the
radiated energy must be concentrated into a narrow beam (high
transmitting antenna gain), the received echo energy must be
collected with a large antenna aperture (also synonymous with high
gain),and the receiver must be sensitiveto weak signals. In
practice, however, the simpleradar equation does not predict the
range performance of actual radar equipments to a satisfactory
degree of accuracy. The predicted values of radar range are usually
optimistic. In some cases the actual range might be only half that
predicted.' Part of this discrepancy is due to the failure of Eq.
(2.1) to explicitly include the various losses that can occur
throughout the system or the loss in performance usually
experienced when electronic equipment is operated in the field
rather than under laboratory-type conditions. 4nother important
factor that must be considered in the radai equation is the
statistical or unpredictable nature of several ofthe parameters.
The minimum detectable signalS,,, and the target cross section cr
are both statistical in nature and must be expressed in statistical
terms.
24. 16 INTRODUCTION TO RADAR SYSTEMS Other statistical factors
which do not appear explicitly in Eq. (2.1)but which have an
effecton the radar performance are the meteorological conditions
along the propagation path and thc performance of the radar
operator, if one is employed. The statistical nature of these
several parameters does not allow the maximum radar range to be
described by a single number. Its specification must include a
statement of the probability that the radar will detect a certain
type of target at a particular range. In this chapter, the simple
radar equation will be extended to include most of the impor- tant
factors that influence radar range performance. If all those
factors affecting radar range were known, it. would be possible, in
principle, to make an accuratc prediction of radar perforpance.
But, as is true for most endeavors, the quality of the prediction
is a function of the amount of effort employed in determining the
quantitative effects of the various pa- rameters. Unfortunately,
the effort required to specify completely the effects of all radar
pa- rameters to the degree of accuracy required for range
prediction is usually not economically justified. A compromise is
always necessary between what one would like to have and what one
can actually get with reasonable effort. This will be better
appreciated as we proceed through the chapter and note the various
factors that must be taken into account. J A complete and detailed
discussion of all those factors that influence the prediction of
radar range is beyond the scope of a single chapter. For this
reason many subjects will appear to be treated only lightly. This
is deliberate and is necessitated by brevity. More detailed
information will be found in some of the subsequent chapters or in
the references listed at the end of the chapter. ' . 6 The ability
of a radar receiver to detect a weak echo signal is limited by the
noise energy that occupies the same portion of the frequency
spectrum as does'the signal energy. The weakest signal the receiver
can detect is called the minimum detectable signal. The
specification of the minimum detectable signal is sometimes
difficult because of its statistical nature and because the
criterion for deciding whether a target is present or not may not
be too well defined. Detection is based on establishing a threshold
level at the output of the receiver. If the receiver output exceeds
the threshold, a signal is assumed to be present. This is called
threshold detection. Consider the output of a typical radar
receiver as a function of time (Fig. 2.1). This might represent one
sweep of the video output displayed on an A-scope. The envelope has
a fluctuating appearance caused by the random nature of noise. If a
large signal is present such as at A in Fig. 2.1, it is greater
than the surrounding noise peaks and can be recognized on the basis
of its amplitude. Thus, if the threshold level were set
sufficiently high, the envelope would not generally exceed.the
threshold if noise alone were present, but would exceed it if a
strong signal were present. If the signal were small, however,it
would be more difficult to recognize its presence. The threshold
level mustbe low if weak signals are to be detected, but it cannot
be so low that noise peaks cross the threshold and give a false
indication of the presence of targets. The voltage envelope :of.
Fig. 2.1 , is assumed to be from a matched-filter receiver (Sec.
10.2). A matched filter is one designed to maximize the output peak
signal to average noise (power) ratio. It has a frequency-response
function which is proportional to the complex conjugate of the
signa1,spectrum.(This is not the same as the concept of" impedance
match " of circuit theory.) The ideal matched-filterreceiver cannot
always be exactly realized in prac- tice, but it is possible to
approach.itwith practical receiver circuits. A matched filter for a
radar transmitting a rectangular-shaped .pulse is usually
characterized by a bandwidth B approxi- mately the reciprocal of
the pulse width 7, or Br = 1.The output of a matched-filter
receiver is
25. Threshold level , A Time - Figure 2.1 Typical envelope of
tile radar receiver output as a function of time. A , and B, and C
represent signal plus noise. ,4 arid B would be valid detections,
but C is a missed detection. the cross correlation between the
received waveform and a replica of the transmitted waveform. Hence
it does not preserve the shape of the input waveform. (There is no
reason to wish to preserve the shape of the received waveform so
long as the output signal-to-noise ratio is maximized.) Let us
return to tlie receiver output as represented in Fig. 2.1. A
threshold level is estab- lished, as shown by the dashed line. A
target is said to be detected if the envelope crosses tlie
thresliold. if the sigrial is large such as at A, it is not
difficult to decide that a target is present. I3ut consider tlie
two signals at B and C, representing target echoes of equal
atnplitudc. 'I'lic noise voltage accompanying the signal at B is
large enough so that the combination of signal plus noise exceeds
the tlireshold. At C the noise is not as large and the resultant
signal plus rioise does not cross the tlireshold. Thus the presence
of noise will sometimes enhance' the detection of weak signals but
it may also cause the loss of a signal which would otherwise be
detected. Weak signals such as C would riot be lost if the
threshold level were lower. But too low a tlireshold increases the
likelihood that noise alone will rise above the threshold and be
taken for a real signal. Such an occurrence is called afalse alarm.
Therefore, if the threshold is set too low, false target
indications are obtained, but if it is set too high, targets might
be missed. The selection of the proper threshold level is a
compromise that depends upon how important it is if a mistake is
made either by (1) failing to recognize a signal that is present
(probability of z miss) or by (2) falsely indicating the presence
of a signal when none exists (probability of a false alarm). When
the target-decision process is made by an operator viewing a
cathode-ray-tube display, it would seem that the criterion used by
the operator for detection ought to be arialogous to the setting of
a threshold, either consciously or subconsciously. The chief
differ- ence between tlie electronic and the operator thresholds is
that the former may be determined with some logic and can be
expected to remain constant with time, while the latter's threshold
might be difficult to predict and may not remain fixed.The
individual's performance as part of the radar detection process
depends upon the state of the operator's fatigue and motivation, as
well as training. The capability of the human operator as part of
the radar detection process can be determined only by experiment.
Needless to say, in experiments of this nature there are likely to
be wide variations between different experimenters. Therefore, for
the purposes of the preserit discussion, the operator will be
considered the same as an electronic threshold detec- tor, an
assumption that is generally valid for an alert, trained operator.
The signal-to.noise ratio necessary to provide adequate detection
is one of the important
26. parameters that must be determined in order to comptite the
minimum detectable signal. Although the detection decision is
usually based on measurements at the video otrtput, it is easier to
consider maximizing the signal-to-noise ratio at the output of the
IF amplifier rather than in the video. The receiver may be
considered linear irp to the output of the IF. It is shown by Van
Vieck and Middleton3 that maximizing the signal-to-noise ratio at
the output of the IF is equivalent to maximizing the video output.
The advantage of considering the signal-to-noise ratio at the IF is
that the assumption of linearity may be made. It is also assumed
that the IF filter characteristic approximates the matched filter,
so that the oirtput signal-to-noise ratio is maximized. 2.3
RECEIVER NOISE Since noise is the chief factor limiting receiver
sensitivity, it is necessary to obtain some means of describing it
quantitatively. Noise is unwanted electromagnetic energy which
interferes with the ability of the receiver to detect the wanted
signal. It may originate within the receiver itself, or it may
enter via the receiving antenna along with the desired signal. If
the radar were to operate in a perfectly noise-free environment so
that no external sources of noise accompanied the desired signal,
and if the receiver itself were so perfect that it did not generate
any excess noise, there would still exist an unavoidable component
of noise generated by the thermal motion of the
conduction'electrons in the ohmic portions of the receiver input
stages. This is called thermal noise, or Johnson noise, and is
directly proportional to the temperature of the ohmic portions of
the circuit and the receiver band~idth.~'The available
thermal-noise power generated by a receiver' of bandwidth B, (in
hertz) at a temperature T (degrees Kelvin) is equal to Available
thermal-noise power = kTB, (2.2) where k = Boltzmann's constant =
1.38 x J/deg. If the temperatiire T is taken to be 290 K, which
corresponds approximately to room temperature (62"F), the factor kT
is 4 x lo-" W/Hz of bandwidth. If the receiver circuitry were at
some other temperature, ttie thermal-noise power would be
correspondingly different. A receiver with a reactance input such
as a parametric amplifier need not have any ::! significant ohmic
loss. The limitation in this case is the thermal noise seen by the
antennii and the ohmic losses in the transmission line. For radar
receivers of the superheterodyne type (the type of receiver used
for most radar applications), the receiver bandwidth is
approximately that of the intermediate-freqire~lcy stages. It
should be cautioned that the bandwidth B, of Eq. (2.2) is not the
3-dB, or half-power, bandwidth commonly employed by electronic
engineers. It is an integrated bandwidth and is given by where H(f
) = frequency-response characteristic of I F amplifier (filter) and
fo = frequency of maximum response (usually occurs at midband).
When H(f) is normalized .to unity at midband (maximum-response
frequency), H(fo) = 1.The bandwidth Bnis called the noise bandwidth
and is the bandwidth of an equiva- lent rectangular filter whose
noise-power output is the same as the filter with
characteristic
27. THE RADAR EQUATION 19 I ! ( / ) '1 lic 3-ti13 I ~ i ~ r ~ t
l w i t l t l iis tlcfirictl as tlic scparntioti it1 licrtz
betwceri tlie poitits oti tlic frequericy-resi~otisccliaractcristic
wliere the response is reduced to 0.707 (3 dB) fro111its r~iaxi-
nlilm valric. Tllc 3-dl3 t~i~ndwicithis widely i~sed,since it is
easy to measure. The meastire~nent of rioisc t)aridwicftli.
I~owcvcr,irivolves a coriiplete knowledge of tlie resporrse
cliaractet.istic N(/). Tlie rreqiicncy-response cliaracteristics of
many practical radar receivers are such that tlic 3-dl3 i~ricitlic
tioisc I~nt~tlwidtlistlo riot differ appreciably. Tlierefore tlie
3-dl3 I~itnciwidtli rnay be used in niatiy cases as an
approximation to the rioise bandwidth.' The noise power in
practical receivers is often greater than can be accounted for by
thertnal noise alone. The additional noise cotnpotlents are due to
mechanisms other than the tlierrnal agitation of tlie conduction
electrons. For purposes of the present discussion, tiowever, the
exact origin of tlie extra noise components is not important except
to know that it exists. No matter whether the noise is generated by
a thermal mechanism or by some other mechanism. tile total tloise
at tlie output of the receiver may be considered to be equal to the
thermal-noise power obtained from an " ideal " receiver multiplied
by a factor called the iroise fig~rre.The noise figure Fn of a
receiver is defined by the equation i NI: = ----"-.. - tloise out
of practical receiver - " kToBnG, noise out of ideal receiver at
std temp To (2.40) where No = rioise output from receiver, and G, =
available gain. The standard temperature To is taken to be 290 K,
according to the Institute of Electrical and Electronics Engineers
definition. 'Tlie noise No is measured over the linear portion of
the receiver input-output characteristic, usually at the output of
tlie IF amplifier before the nonlinear second detector. 'The
receiver bandwidth Bn is that of tlie IF aniplifier in most
receivers. The available gain G, is tlie ratio of the signal out
Soto the signal in Si,and kToBn is the input noise Niin an ideal
receiver. Equation (2.40) may be rewritten as The noise figure may
be interpreted, therefore, as a measure of the degradation of
signal-to- noise-ratio as the signal passes through the receiver. j
Rearranging Eq. (2.417). the input signal may be expressed as If
the minimum detectable signal S,,, is that value of SIcorresponding
to the minimum ratio of output (IF)signal-to-noise ratio ( S o / N
o ~ i nnecessary for detection, then Substituting Eq. (2.6) into
Eq. (2.1) results in the following form of the radar equation:
Before continuing the discussion of the factors involved in the
radar equation, it is necessary to digress and review briefly some
topics in probability theory in order to describe the
signal-to-noise ratio in statistical terms.
28. 20 INTRODUCTION TO RADAR SYSTEMS 2.4 PROBABILITY-DENSITY
FUNCTIONS The basic concepts of probability theory needed in
solving noise problems may be found in any of several In this
section we shall briefly review probability and the
probability-density function and cite some examples. Noise is a
random phenomenon. Predictions concerning the average performance
of random phenomena are possible by observing and classifying
occurrences, but one cannot predict exactly what will occur for any
particular event. Phenomena of a random nature can be described
with the aid of probability theory. Probability is a measure of the
likelihood of occurrence of an event. The scale of probabil- ity
ranges from 0to 1.t An event which is certain is assigned the
probability 1. An impossible event is assigned the probability
0.The intermediate probabilities are assigned so that the more
likely an event, the greater is its probability. One of the more
useful concepts of probability theory needed to analyze the
detection of signals in noise is the probability-density function.
Consider the variable x as representing a typical measured value of
a random process such as a noise voltage or current. Imagine each x
to define a point on a straight line corresponding to the distance
from a fixed reference point. The distance of x from the reference
point might represent the value of the noise current or the noise
voltage. Divide the line into small equal segments of length Ax and
count the number of times that x falls in each interval. The
probability-density function p(x) is then defined as (number of
values in range AXat x)/Ax p(x) = lim (2.8) AX-o total number of
values = N N-rm The probability that a particular measured value
lies within the infinitesimal width ds centered at x is simply p(x)
dx. The probability that the value of x lies within the finite
rangz from xl to x2 is found by integrating p(x) over the range of
interest, or X2 Piobability (x, < x < x2)= 1 p(x) dx X I By
definition, the probability-density function is positive. Since
every measurement must yield some value, the integral of the
probability density over all values of x must be equal to unity; .j
that is, The average value of a variable function, +(x), that is
described by the probability-density function, p(x), is This
follows from the definition of an average value and the
probability-density function. The mean, or average, value of x is t
Probabilities are sometimes expressed in percent (0 to 100) rather
than 0 to 1.
29. and tlie mean square value is THE RADAR EQUATION 21 'Tlie
quantities in, and l,lz are sometimes called the first and second
moments of the random variable .u.I f .urepresents an electric
voltage or current, inl is the d-c component. It is the value read
by a direct-curretlt voltmeter or ammeter. The mean square value
(nt,) of the current wl~erirrlllltiplied by tile resistaricet gives
the mean power. The mean square value of voltage times tlie
conductance is also the mean power. The variarlce is defined as The
variance is tile meall square deviation of x about its mean and is
sometimes called the secorld ceiltral rnonrcllt. If the random
variable is a noise current, the product of the variance '/a and
resistance gives the mean power of the a-c component. The square
root of the variance o is called the stclrldard deviatioit and is
the root-mean-square (rms) value of the a-c component. We shall
consider four examples of probability-density functions: the
uniform, gaussian, Rayleigh, and exponential. The uniform
probability-density (Fig. 2 . 2 ~ )is defined as Ik f o r a < x
< a + b /I(.Y) = O for .w < a and x > a + b t 111
~ioisetheory it is customary to take the resistance as 1 ohm or the
conductance as 1 mho. Figure 2.2 Examples of probability-density
functions. (a) Unlform; (6)Gaussian; (c) Rayleigh (voltage); ( d )
Rayleigtl (power) or exponential.
30. 22 INTRODUCTION TO RADAR SYSTEMS where k is a constant. A
rectangular, or uniform, distribution describes the phase of a
random sine wave relative to a particular origin of time; that is,
the phase of the sine wave may be found, with equal probability,
anywhere from 0 to 2n, with k = 1121s.It also applies to the
distribution of the round-off (quantizing) error in numerical
computations and in analog-to- digital converters. The constant k
may be found by applying Eq. (2.10); that is, The average value of
x is This result could have been determined by inspection. The
second-moment, or mean square, value is and the variance is a =
standard deviation = b rJj The gaussian, or normal, probability
density (Fig. 2.2b) is one of the most important in noise theory,
since many sources of noise, such as thermal noise or shot noise,
may be represented by gaussian statistics. Also, a gaussian
representation is often more convenient to manipulate
mathematically. The gaussian density function has a bell-shaped
appearance and is defined by where exp [ ] is the exponential
function, and the parameters have been adjusted to satisfy the
normalizing condition of Eq. (2.10). It can be shown that * - m -02
The probability density of the sum of a large number of
independently distributed quanti- ties approaches the gaussian
probability-density function no matter what the individual dis-
tributions may be, provided that the contribution of any one
quantity is not comparable with the resultant of all others. This
is the central limit theorem. Another property of the gaussian
distribution is that no matter how large a value x we may choose,
there is always some finite probability of finding a greater value.
If the noise at the input of the threshold detector were truly
gaussian, then no matter how high the threshold were set, there
would always be a chance that it would be exceeded by noise and
appear as a false alarm. However, the probability diminishes
rapidly with increasing x, and for all practical purposes the
probability of obtaining an exceedingly high value of x
,isnegligibly small. The Rayleigh probabi~itydensit~function is
also of special interest to the radar systems
31. erigirice~ 11 tlescr il~cstl~ceriveloi~cof
(lie~ioiscoutlxrt fro111ii rlar r.owbatld filtcr (sucli as tile IF
filter iri a sr~pcrheterotlynereceiver). tile cross-section
fluctuatioris of certain types of conlplex radar targets. arid
rnariy kinds of clutter arid weather echoes. The Rayleigh density
function is Tliis is plottetf iri Fig. 2.2~.Tlie parameter .u might
represent a voltage, and (.u2),, the mean, or average, valirc of
tlic voltage squar-ed.If .uZis replaced by w, wlierc kc represents
power instead of voltage (assuming the resistance is I ohm), Eq.
(2.17) becomes 1 P ( w )= exp (- ,v 2 o "'0 where H., is tlie
average power. This is tlie exponential probability-density
function, but it is sornetiriie5 called the Rayleigh-power
probability-density function. It is plotted in Fig. 2.2d. , ?'lie
starid:trd dcviatiori of the Kayleigli density of Eq. (2.17) is
equal to J(4/n) - 1 times tlie mean valuc, arid for tile
cxponeritial density of Eq. (2.18)tlie standard deviation is equal
to w o . 'l'licrc ate otlicr pr ot~ability-densityfunctions of
interest in radar, such as the Rice, log norrnal, arid tlie chi
square. I'liese will be introduced as needed. Ariotlicr
rriathcrnatical description of statistical phenomena is the
probability distrihrrtiot~ Jirr~c.tir,r~f'(u), dclined as tile
probability tliat tile value x is less than some specified value
111 sorrie cases, tlie distribution function may be easier to
obtain from an experimental set of data tlian the derisity
function. Tlie density function may be found from the distribution
futiction by dilferentiatiori. 2.5 SIGNAL-TO-NOISE RATIO : In this
section tile results of statistical noise theory will be applied to
obtain the signal-to-noise ratio at the output of the IF amplifier
necessary to achieve a specified probability of detection without
exceeding a specified probability of false alarm. The output
signal-to-noise ratio thus obtained rnay be substituted into Eq.
(2.6) to find the minimum detectable signal, which, in turn. is
used in the radar equation, as in Eq. (2.7). Corisider an IF
amplifier with bandwidth BIFfollowed by a second detector and a
video arnplificr witli baridwidth B,.(Fig. 2.3). Tlie second
detector and video amplifier are assumed to form an envelope
detector, that is, one which rejects the carrier frequency but
passes the niodulation envelope. To extract the modulation
envelope, the video bandwidth must be wide enough to pass the
low-frequency components generated by the second detector, but not
so wide as to pass the high-frequency components at or near the
intermediate frequency. The video bandwidth B,,must be greater than
BIF/2in order to pass all the video modulation. Most radar
receivers used in conjunction with an operator viewing a CRT
display meet this condition and Second detector - V~deo amplifier
(Bv) . : Figure 2.3 Envelope detector.
32. 24 INTRODUCTION TO RADAR SYSTEMS may be considered envelope
detectors. Either a square-law or a linear detector may be assumed
since the effect on the detection probability by assuming one
instead of the other is iisually small. The noise entering the I F
filter (the terms filter and amplifier are used interchangeably) is
assumed to be gaussian, with probability-density function given by
where p(v) do is the probability of finding the noise voltage v
between the values of 11 and v +dl,, $o is the variance, or
mean-square value of the noise voltage, and the mean value of 11 is
taken to be zero. If gaussian noise were passed through a
narrowband IF filter-one whose bandwidth is small compared with the
midfrequency-the probability density of the envelope of the noise
voltage output is shown by Riceg to be where R is the amplitude of
the envelope of the filter output. Equation (2.21) is a form of the
Rayleigh probability-density function. The probability that the
envelope of the noise voltage will lie between the values of V, and
V2 is v 2 R Probability (V, < R < V2)= - exp (- 5)d R v, +o
2+0 The probability that the noise voltage envelope will exceed the
voltage threshold V,- is " R Probability (VT< R < m) = [ -
exp (- c)dR v7.$0 2$ 0 = exp (- 2)= P,, Whenever the voltage
envelope exceeds the threshold, a target detection is considered to
have -? occurred, by definition. Since the probability of a false
alarm is the probability that noise will cross the threshold, Eq.
(2.24) gives the probability of a false alarm, denoted PI,. The
average time interval between crossings of the threshold by noise
alone is defined as the filse-alarm time 3,, I N Ta= lim -- N - + W
N k = ~ where & is the time between crossings of the threshold
VT by the noise envelope, when the slope of the crossing is
positive. The false-alarm probability may also be defined as the
ratio of the duration of time the envelope is actually above the
threshold to the total time it coirld have been above the
threshold, or N-
33. Time - Figure 2.4 Envelope of receiver output illustrating
false alarms due to noise. where t, and & are defined in Fig.
2.4. The average duration of a noise pulse is approximately the
reciprocal of the bandwidth B, which in the case of the envelope
detector is BIF.Equating Eqs. (2.24) and (2.25) we get 1 v", 3,=
-exp - BIF 21//0 A plot of Eq. (2.26) is shown in Fig. 2.5, with
V;/2t,bo as the abscissa. If, for example, the bandwidth of the IF
amplifier were 1 MHz and the average false-alarm time that could be
tolerated were 15 nlin, the probability of a fdse alarm is 1.11 x
lo-'. From Eq. (2.24) the threshold voltage necessary to achieve
this false-alarm time is 6.45 times the rms value of the noise
voltage. The false-alarm probabilities of practical radars are
quite small.The reason for this is that the false-alarm probability
is the probability that a noise pulse will cross the threshold
during an interval of time approximately equal to the reciprocal of
the bandwidth. For a 1-MHz bandwidth, there are of the order of 106
noise pulses per second. Hence the false-alarm probability of any
one pulse must be small ( < if false-alarm times greater than 1
s are to be obtained. The specification of a tolerable false-alarm
time usually follows from the requirements desired by the customer
and depends on the nature of the radar application. The exponential
relationship between the false-alarm time Tfa and the threshold
level VT results in the false- alarm time being sensitive to
variations or instabilities in the threshold level. For example,
if' the batidwidtll were 1 MHz, a value of 10 log (V$/2t,bo)= 12.95
dB results in an average false-alarm time of 6 min, while a value
of 14.72 dB results in a false-alarm time of 10,000 h. Thus a
change in the threshold of only 1.77 dB changes the false-alarm
time by five orders of magnitude. Such is the nature of gaussian
noise. In practice, therefore, the threshold level would probably
be adjusted slightly above that computed by Eq. (2.26), so that
instabilities which lower the threshold slightly will not cause a
flood of false alarms. If the receiver were turned off (gated) for
a fraction of time (as in a tracking radar with a servo-controlled
range gate or a radar which turns off the receiver during the time
of transmis- sion), the false-alarm probability will be increased
by the fraction of time the receiver is not operative assuming that
the average false-alarm time remains the same. However, this is
usually not important since small changes in the probability of
false alarm result in even smaller changes in the threshold level
because of the exponential relationship of Eq. (2.26). Thus far, a
receiver with only a noise input has been discussed. Next, consider
a sine-wave signal of amplitude A to be present along with noise at
the input to the IF filter. The frequency
34. 26 INTRODUCTION TO RADAR SYSTEMS 1 yeor 6 months 30 doys 2
weeks l week 15 rnin Figure 2.5 Average time between false alarms
as a function of the threshold level I/, and thc receiver bandwidth
8;(I/, is the mean square noise voltage. 3 of the signal is the
same as the IF midband frequencyhF.The outpul or the envelope
detector has a probability-density function given by9 where lo(Z)
is the modified Bessei function of zero order and argument Z. For Z
large, an asymptotic expansion for i o ( Z )is When the signal is
absent, A = 0 and Eq. (2.27)reduces to Eq. (2.21),the
probability-density function for noise alone. Equation (2.27) is
sometimes called the Rice probability-density function. The
probability that the ;ignal will be detected (which is the
probability ofdetenio,,)is the same as the probability that the
envelope R will exceed the predetermined threshold VT.The
35. probability' of detection f', is therefore This cannot he
evaluated by sirnple nleans, and numerical techniques or a series
approxima- tion must be used. A series approximation valid when
RA/$o % 1, A 9 IR - A 1, and terms in A - and beyond can be
neglected is9 VT - A 1 + (VT - A)2/$oX [ I - 4 ~ - -+ - - . . .
8A2/rC/0 where tlie error fu~lctionis defined as erf Z = - A
graphic illustratior~of the process of threshold detection is shown
in Fig. 2.6. The probability density for noise alone [Eq. (2.21)]
is plotted along with that for signal and noise [Eq. (2.27)Jwith
/I/(/:!* = 3. A tl~resl~oldvoltage VT/$;12= 2.5 is shown. The
crosshatched area to the right of I ~ , / $ : ' ~under the curve
for signal-plus-noise represents the probability of detection.
while the double-crosshatched area under the curve for noise alone
represents the probability of a false alarm. If v ~ / $ ; ' ~is
increased to reduce the probability of a false alarm, the
probability of detection will be reduced also. Equation (2.29)may
be used to plot a family of curves relating the probability of
detection to the threshold voltage and to the amplitude of tlie
sine-wave signal. Although the receiver designer prefers to operate
with voltages, it is more convenient for the radar system engineer
to ernploy power relationships. Equation (2.29) may be converted to
power by replacing the signal - to -rrns-noise-voltageratio with
the following: .I signal amplitude fi(rms signal voltage). - . --
-- = ( 2 signal power) ' I 2 = (:) - ' 1 2 d,:,I2 - rrns noise
voltage rms noise voltage noise power We shall also replace
If;/2rC/, by In (l/P,,)[from Eq. (2.24)). Using the above
relationships, the probability of detection is plotted in Fig. 2.7
as a function of the signal-to-noise ratio with the probability of
a false alarm as a parameter. Figure 2.6 Probability-density
function for noise alone arid for signal-plus-noise, illustrating
the process of tl~resl~olddetection.
36. 28 INTRODUCTION TO RADAR SYSTEMS 4 6 8 " 10 12 14 16 18 20
(S/N ), , signal-to-noise ratio, dB Figure 2.7 Probability of
detection for a sine wave in noise as a function of the
signal-to-noise (power) ratio and the probability of false alarm.
.I Both the false-alarmtime and the detection probability are
specified by the system require- ments. The radar designer computes
the probability of the false alarm and from Fig. 2.7 determines the
signal-to-noise ratio. This is the signal-to-noise ratio that is
used in tht: eqlta- tion for minimum detectable signal [Eq.
(2.6)].The signal-to-noise ratios of Fig. 2.7 apply to a single
radar pulse. For example, suppose that the desired false-alarm time
was 15 min and tllc IF bandwidth was 1 MHz. This gives a
false-alarm probability of 1.11 x lo-'. Figure 2.7 indicates that a
signal-to-noise ratio of 13.1 dB is required to yield a 0.50
probability of detection, 14.7dB for 0.90, and 16.5 dB for 0.999.
There are several interesting facts illustrated by Fig. 2.7. At
first glance, it m