repeated for each value of)' and 11.13 This is not done here. Instead, for simplicity, an efficiency will be defined which is the ratio of the average signal-to-noise ratio for the exponential integrator to the average signal-to-noise ratio for the uniform integrator. For a dumped integrator, one which erases the contents of the integrator after II pulses and starts over, the efficiency is 1 4
tanh
(11y/2)
p=
11
tanh (1,/2)(2.35a)
An example of an integrator that dumps is an electrostatic storage tube that is erased whenever it is read. The efficiency of an integrator that operates continuously without dumping is
[l - exp (-ny)]2
p
= -- - --· ---
" tanh (r/2)
(2.351,)
The maximum efficiency of a dumped integrator occurs for}'= 0, but for a continuous integra- tor the maximum efficiency occurs for II}'= 1.257.
N
I=:: t:l
'-..
1.0
~ 0.1
,,:;
.,
a.
b"'
0.01
Mie or resonance
region Optical region
0.001 ..__ _ _ _,___-'--...,_...,_..._ ... ...,_ _ _ __. _ __.__J__J...J...J....-L...L-L-_ _ _ J
0.1 0.2 0.3 0.4 0.5 0.8 1.0 2 3 4 5 6 8 10 20 Circumference/ wavelength = 2 rr a/ X
Figure 2.9 Radar cross section of the sphere. a
=
radius; l = wavelength.interest to the radar engineer because the cross sections of raindrops and other meteorological particles fall within this region at the usual radar frequencies. Since the cross section of objects within the Rayleigh region varies as l -4, rain and clouds are essentially invisible to radars which operate at relatively long wavelengths (low frequencies). The usual radar targets are much larger than raindrops or cloud particles, and lowering the radar frequency to the point where rain or cloud echoes are negligibly small will not seriously reduce the cross section of the larger desired targets. On the other hand,
if
it were desired to actually observe, rather than eliminate, raindrop echoes, as in a meteorological or weather-observing radar, the higher radar frequencies would be preferred.At the other extreme from the Rayleigh region is the optical region, where the dimensions of the sphere are large compared with the wavelength (2na/l ~ 1). For large 2rca/l, the radar cross section approaches the optical cross section na2• In between the optical and the Rayleigh region is the
Mie,
orresonance,
region. The cross section is oscillatory with frequency within this region. The maximum value is 5.6 dB greater than the optical value, while the value of the first null is5.5
dB below the optical value. (The theoretical values of the maxima and minima may vary according to the method of calculation employed.) The behavior of the radar cross sections of other simple reflecting objects as a function of frequency is similar to that of the sphere.15-25Since the sphere is a sphere no matter from what aspect it is viewed, its cross section will not be aspect-sensitive. The cross section of other objects, however, will depend upon the direction as viewed by the radar.
Figure 2.10 is a plot of the backscatter cross section of a long thin rod as a function of aspect. 26 The rod is 39.t long and .t/4 in diameter, ~nd is made of silver.
If
the rod were of steel instead of silver, the first maximum would be about5
dB below that shown. The radar cross section of the thin rod (and similar: objects) is small when viewed end-on (0 = 0°) since the physical area is small. However, at near end-on, waves couple onto the scatterer which travel16
E 11.5
0 ::, CT
V, I
0 7 - I
CV > I
0 I
.0
0 I
0 2 -l
QJ
L. l
0
0 I
.c u
w -3
I
•
I,
I !.1
t I I I I
I
,
...I I I
20
-r--
Measured Calculated
30 40 50 60
Angular orientalion B 70 80 90
Figure 2.IO Backscatter cross section
or
a long thin rod. (From l;eters,26 IRE Trans.)down the length of the object and reflect from the discontinuity at the far end. This gives rise to a traveling wave component that is not predicted by physical optics theory.26•35
An interesting radar scattering object is the cone-sphere, a cone whose base is capped with a sphere such that the first derivatives of the cone and sphere contours are equal at the join between the two. Figure 2.11 is a plot of the nose-on radar cross section. Figure 2.12 is a plot as a function of aspect. The cross section of the cone-sphere from the vicinity of the nose-on direction is quite low. Scattering from any object occurs from discontinuities. The discontinui- ties, and hence the backscattering, of the cone-sphere are from the tip and from the join between the cone and the sphere. There is also a backscattering contribution ftom a" creeping
0
er =0.4 >..2
§
li;
-201---1-~~~~~~~-+~~~~~~~~~~~
er
=
0.01 X2V>
l
~ -.30--
e
uOiomefer (Wovefengths)
Figure 2.11 Radar cross section of a cone sphere with 15° half angle as a function of the diameter in wavelengths. (After Blore,21 IEEE Trans.) ·
2700 90° (al
160°
90° (b)
Figure 2.12 Measured radar cross section (u/A2 given in dB) of a large cone-sphere with 12.5° half angle and radius of base= 10.4A. (a) horizontal (perpendicular) polarization, (b)vertical (parallel)polarization.
(From Pannell et a/.61)
36
wave" which travels around the hasc of the sphere. The nose-on radar cross section is small and decreases as the square of the wavelength. The cross section is small over a relatively large angular region. A large specular return is obtained when the cone-sphere is viewed at near perpendicular incidence to the cone surface, i.e., when O
=
90 - a, where a= cone half angle.From the rear half of the cone-sphere, the radar cross section is approximately that of the sphere.
The nose-on cross section of the cone-sphere varies, but its maximum value is approxi- mately 0.412 and its minimum is 0.0U.2 for a wide range of half-angles for frequencies above the Rayleigh region. The null spacing is also relatively insensitive to the cone half-angle. If a
"typical" value of cross section is taken as
0. L.t2,
the cross section at S band (.,l= 0.1
m) is 10--3 m2, and at X band (,l = 3 cm), the cross section is approximately 10-4 m2. Thus, in theory. the cone-sphere can have very low backscatter energy. Suppose, for example, that the projected area of the cone-sphere were 1 m2• The radar cross section of a sphere, with the same projected area, at S band is about 30 dB greater. A corner reflector at S band, also of the same projected area, has a radar cross section about 60 dB greater than the cone-sphere. Thus, objects with the same physical projected area can have considerably different radar cross sections.In order to realize in practice the very low theoretical values of the radar cross section for a cone-sphere, the tip or the cone must be sharp and not rounded, the surface must be smooth (roughness small compared to a wavelength), the join between the cone and the sphere must have a continuous first derivative, and there must be no hoJes, windows,
or
protuberances on the surface. A comparison of the nose-on cross section of several cone-shaped objects is givenin Fig. 2.13. .
Shaping of the target, as with the cone-sphere, is a good method for reducing the radar cross section. Materials such as carbon-fiber composites, which are sometimes used in aero- space applications, can further reduce the radar cross section of targets as compared with that produced by highly reflecting metallic materials.62
bl':'<
0
r:: Q u
(I)
"'
"'
Ill ~ u
~
0 0 0 0::
+10
0
10
Cone Sphere
-20 ,
\ I
I 1-.
\ 1 · Double - rounded cone -30
40 -
M
""-- Circular ogive-so---~~'--~-'-~~~~~---'~~--'~~ ... ~~--'
0 0 4 0 8 12 1 6 2.0 2.4
Oiomefer in wavelengths
Figure 2.13 Radar cross section of a set of 40° cones, double-backed cones, cone-spheres, double- rounded cones, and circular ogives as a function of diameter in wave- lengths. (From Blore,21 IEEE Trans.)
Complex targets.32·33 The radar cross section of complex targets such as ships, aircraft, cities, and terrain are complicated functions of the viewing aspect and the radar frequency. Target cross sections may be computed with the aid of digital computers, or they may be measured experimentally. The target cross section can be measured with full-scale targets, but it is more convenient to make cross-section measurements on scale models at the proper scaled frequency. 63
A complex target may be considered as comprising a large number of independent objects that scatter energy in all directions. The energy scattered in the direction of the radar is of prime interest. The relative phases and amplitudes of the echo signals from the individual scattering objects as measured at the radar receiver determine the total cross section. The phases and amplitudes of the individual signals might add to give a large total cross section, or the relationships with one another might result in total cancellation. ln general, the behavior is somewhere b~tween total reinforcement and total cancellation. If the separation between the individual scattering objects is large compared with the wavelength-and this is usually true for most radar applications-the phases of the individual signals at the radar receiver will vary as the viewing aspect is changed and cause a scintillating echo.
Consider the scattering from a relatively" simple" complex target consisting of two equal, isotropic objects (such as spheres) separated a distance I (Fig. 2.14). By isotropic scattering is meant that the radar cross section of each object is independent of the viewing aspect. The separation I is assumed to be less than cr/2, where c is the velocity of propagation and r is the pulse duration. With this assumption, both scatterers are illuminated simultaneously by the pulse packet. Another restriction placed on I is that it be small compared with the distance R from radar to target. Furthermore, R 1 ~ R2 ::::: R. The cross sections of the two targets are assumed equal and are designated a0 • The composite cross section a, of the two scatterers is
a, [
(4rr/ . i) l
a
O
= 2 1
+
cos-y-
sm (, (2.37)The ratio a,/a0 can be anything from a minimum of zero to a maximum of four times the cros~
section of an individual scatterer. Polar plots of
a,/a
0 for various values of//J...
arc shown in Fig. 2.15. Although this is a rather simple example of a "complex" target, it is complicated enough to indicate the type of behavior to be expected with practical radar targets.The radar cross sections of actual targets are ~ar more complicated in structure than the simple two-scatterer target. Practical targets are composed of many individual scatterers, each with different scattering properties. Also, interactions may occur between the scatterers which affect the resultant cross section.
An example of the cross section as a function of aspect angle for a propeller-driven
Rodar Figure 2.14 Geometry or the two-scatterer complex target.
r
\\'\.
'
\
\.,
\
oo
I
:J \ ~
----\-
--~\~,.,.-,- \
\ \ CTr/CTo __
2 3 4 900
I
~-1-
--7;_-
/ '
/ \
--
\00
I
•
•k 4A-..1
(c)
00
I
•
•~ 2A k
(b)
Figure 2.15 Polar plots of a,/a0 for the two-scatterer complex target (Eq. (2.37)]. (a) I= ,l.; (b) l = 2-l.;
(c) l = 4)..
aircraft28 is shown in Fig. 2.16. The aircraft is the B-26, a .World War II medium-range two-engine bomber. The radar wavelength was 10 cm. These data were obtained exper- imentally by mounting the aircraft on a turntable in surroundings free from other reflecting objects and by observing with a nearby radar set. The propellers were running during the measurement and produced a modulation of the order of 1 to 2 kHz. The cross section can change by as much as 15 dB for a change in aspect of only
1°.
The maximum echo signal occurs in the vicinity of broadside, where the projected area of the aircraft is largest.It is not usually convenient to obtain the radar cross section of aircraft by mounting the full-size aircraft on a rotating table. Measurements can be obtained with scale models on a pattern range. 29 An example of such model measurements is given by the dashed curves in Fig. 2.17. If care is taken in the construction of the model and in the pattern-range instrumentation, it is possible to achieve reasonably representative measurements.
The radar cross section of an aircraft can also be obtained by computation.1 7 The target is broken up into a number of simple geometrical shapes, the contribution of each (taking
35 dB
Figure 2.16 Experimental cross section of the B-26 two-engine bomber at 10-cm wavelength as a function of azimuth angle. (From Ridenour,28 courtesy McGraw-Hill Book Company, Inc.}
account of aspect changes and shadowing of one component by another) is computed and the component cross sections are combined to yield the composite value. The" theoretical" values of Fig. 2.17 for B-47 were obtained by calculation.
The most realistic method for obtaining the radar cross section of aircraft is to measure the actual target in flight. There is no question about the authenticity of the target being measured. An example of such a facility is the dynamic radar cross-section range of the U.S.
Naval Research Laboratory.30 Radars at L, S, C and X bands illuminate the aircraft target in flight. The radar track data is used to establish the aspect angle of the target with respect to the radar. Pulse-to-pulse radar cross section is available, but for convenience in presenting the data the values plotted usually are an average of a large number of values taken within a 10 by 10° aspect angle interval. Examples of such data are given in Figs. 2.18 to 2.20. The radar cross section of the T-38 aircraft at head-on incidence is shown in Table 2.1. This data was also obtained from an aircraft in flight. (The T-38 is a twin-jet trainer with a 7.7 m wing span and a
14 m length.)
Ll ..
2 - 1,000
8
4 - Peok due to leading 2 . edge cl
Wtng \ I
100 8. ."- I
,, I
·- I 1
" 0
'o I
\ I
2 10'
e\
5 \
\ 4 \
\ I
\ I
\ I
V
2
·-
10 8 6 4 2
-... __
--r -T- ~--1··--r·--r---i-1-~--
,,
I ' I '
I \
I I
I I
Peak due to / \ - - Peak due gos lank ----1--- ...
-r
lo fuselageI I
cl I
I I
I I
0
!
1I I I
I I I
\ / I
I I I
I o I
I / I
I I I I
I I I I
\ I
I IPeak due to lroling edge cl wing
I f I·
I I.
I I
I I I I
I J I I
\ / /1 I
--o--o
I I I '°\ I
\ I I I / ,,
lf I I / \
I o" \ f
J \
I
,,,/ '\, I
\ ---·/ • 'J
...
,,,,.,-· \
\ /
\ I
\-~··
Azimuth angle -degrees
Trailing edge of horizontal toil
I
Figure 2.17 Comparison of the theoretical and model-measurement horizontal-polarization radar cross sections of the B-47 medium bomber jet aircraft with a wing span of 35 m and a length of 33 m. Solid curve is the average of the computed cross sections obtained by the University of Michigan Engineering Research Institute at a frequency of 980 MHz. Dashed curves are model measurements obtained by the Ohio State University Antenna Laboratory at a frequency of 600 MHz. Open circles are the maximum values averaged over 10° intervals: solid circles are median values. Radar is assumed to be in the same plane as the aircraft.64
It can be seen that the radar cross section of an aircraft is difficult to specify concisely.
Slight
changes in viewing aspect or frequency result in large fluctuations in cross section.Nevertheless, a single value of cross section is sometimes given for specific aircraft targets for use in computing the radar equation. There is no standard, agreed-upon method for specifying the single-valued cross section of an aircraft. The average value or the median might be taken.
Sometimes it is a "minimum" value, perhaps the value exceeded 99 percent of the time or 95 percent of the time. It might also be the value which when substituted into the radar equation assures that the computed range agrees with the experimentally measured range.
Table 2.2 lists "example" values of cross section for various targets at microwave frequencies. Note that only a single value is given even though there can be a large variation.
They should not be used for design purposes when actual data is available for the particular targets of interest.
A military propeller aircraft such as the AD-48 has a cross section of about 20 m2 at L band, but a 100 m2 cross section at VHF. The longer wavelengths at VHF result in greater
j
C
<O :!:!
b
• CTHH
o'~---
o 20 40 - ~ 8 0 - 100 120 140 160 180(b)
Az,muth aspect angle {deg)
!al
30
20
10
·--·-===--
c_.---o--0
A=IBO
0o -10 -20 -30 -40
£1evolian aspect ong\e (degl kl
Figure 2.18 (a) Azimuth variation of radar cross section of a C-54 aircraft with constant elevation angle of - 10". (The C-54 is the military version of the four-piston-engined DC-4 commercial aircraft with a wing span of 36 m and a length of 29 m.) Each point represents the average of medians obtained from samples within a lO by 10" aspect cell.
Frequency is 1300 MHz (L band) with linear polarization. V V
=
vertical polarization, H H = horizontal polarization. {b) Eleva- tion-variation nose-on (azimuth = O") (c) Elevation-variation tail-on {azimuth = 180°). ( From Olin and Queen. ;o)cross section than microwaves because the dimensions of the scattering objects are compar- able to the wavelength and produce resonance effects.
An example of th~ measured radar cross section of a large ship (16,000 tons) is shown in Fig. 2.21. The aspect is at grazing incidence. When averages of the cross section are taken about the port and starboard bow and quarter aspects of a number of ships (omitting the peak at broadside), a simple empirical expression is obtained for the median (50th percentile) value of the cross section:
a = 52f i /'1. D312
Oo 20 40 60 80 100 120 140 . 160 180
Az11nuth os.:,ec1 ongle(degl (al
4 0 ~ - - - ~ 4 0 r - - - = , . , . . - - ,
,£' 30
b 10
A• o•
-10 -20 (bl
30
20
10 A• 1so•
·30 -40 °i) -10--=-20 -30 -40 Elevatoon os;pec1 11119le( <1c9 I le I
(2.38)
Figure 2.19 Same as Fig. 2.18 except fre- quency is 9225 MHz (X band). VH and HV represent cross•polarized components.
(From Olin and Queen.30)
,/
iIn!
I
~-:: 1)" !
!=,~F~I
?P'"
""1r1
,,, ~( I
---·
0 0
·~.,:::·::::::-::=·----
· - - - - ~ ~ ; ~
'
---
,1 - 1nn"
'.o ·- j -'1('
(t,) fll",<111r"1n,;prrlo11'1t"{drfll (r.l
Figure 2.20 Same as Fig. 2.19, but for circu- lar polarization. RR = right-hand polariza- tion, LL
=
left-hand polarization; RL an<l LR are cross-polarized components. (From Olin and Q11ee11. 30 )Table 2. la Radar cross section (square meters) of the T-3866
Head-on aspect ( ± 1.0 degree)
x,.,.
X I.R Svv LvvPercentile 20 50 80 20 50 80 20 50 80 20 50 80
Landing
gear up 0.33 0.83 l.7 0.21 0.53 1.2 1.6 3.1 4.7 1.5 1.8 2.2 Landing
gear extended 0.53 J .6 3.5 0.24 0.80 2.1 1.1 2.3 4.4 0.99 2.6 4.5
Table 2.lb Median cross section (X ,.,_) for aspects near nose-on
Azimuth angle (degrees) Azimuth angle (degrees)
Elevation Elevation
angle 0 2 5 7 angle 0 2 5 7
() lUn 16 0.72 0.45 0.90 1.2
(i 0.68 0.61 0.99 0.82 18 . 0.43 0.44 1.3 0.84
8 1.2 0.94 1.7 2.2 20 0.43 0.52 0.63 0,64
10 1.4 1.6
:u
I .4 22 0.45 0.66 l.l l.112 U.70 1.0 1.6 2.1 24 0.65
14 0.79 0.60 0.63 1.5
Xu.: Transmit left circular polarization, receive left circular (X band). XLR: Transmit left circular polarization, receive right circular (X band). Svv: Transmit vertical polarization, receive vertical (S band).
Ln- : Same for L band.
Table 2.2 Example radar cross sections at microwave frequencies
Conventional, unmanned winged missile Small, single engine aircraft
Small fighter, or 4-passenger jet Large fighter
Medium bomber or medium jet airliner Large bomber or large jet airliner Jumbo jet
Small open boat Small pleasure boat Cabin cruiser
Ship at zero grazing angle Ship at higher grazing angles
Pickup truck Automobile Bicycle Man Bird Insect
Square meters 0.5
I 2 6 20 40 100
0.02 2 10
See Eq. {2.38)
Displacement tonnage expressed in m2 200
100 2 1
o.oi
10-s
where er= radar cross section in square meters,!= radar frequency in megahertz, and Dis the ship's (full load) displacement in kilotons.31 This expression was derived from measurements made at X, S, and L bands and for naval ships ranging from 2000 to 17,000 tons. Although it is probably valid outside this size and frequency range, it does not apply to elevation angles other than grazing inc;idence. At higher elevation angles, as might be viewed from aircraft, the cross sections of ships might be considerably less than at grazing incidence, perhaps by an order of magnitude. When no better information is available, a very rough order of magnitude estimate of the ship's cross section at other-than-grazing incidence can be had by taking the ship's displacement in tons to be equal to its cross section in square meters. The average cross section of small pleasure boats 20 to 30 ft in length might have a radar cross section in the vicinity of a few square meters at X band.68 Boats from 40 to 50 ft in length might have a cross section of the order of 10 square meters.
The radar cross section of an automobile at X band is generally greater than that of an aircraft or a boat. From the front the cross section might vary from 10 to 200 m2 at X band, with 100 m2 being a typical value.65 The cross section increases with increasing frequency (up to 60 GHz, the range of the measurements).
The measured radar cross section of a man has been reported32 to be as follows:
Frequency, MHz 410 1,120 2,890 4,800 9,375
a, m2 0.033-2.33 0.098-0.997 0.140-1.05 0.368-1.88 0.495-1.22
The spread in cross-section values represents the variation with aspect and polarization.
IPORTl
18d'
(a)
160W)
d
1ac!
(b)
figure 2.21 Azimuth v:iriation of the radar cross-section
or
a large Naval Auxiliary Ship at (a) S band (2800 MHz) and (h) X band (9225 MHz). both with horizontal polarization.The cross-section data presented in this section lead to the conclusion that it would not be appropriate simply to select a single value and expect it to have meaning in the computation of the radar equation without further qualification. Methods for dealing with the cross sections of complicated targets are discussed in the next section.