Echo strength in free space
4.3 Passive reflector; radar cross section, radar range equation
4.3.1 Radar cross section
How well a passive object reflects energy back towards the radar depends on its size, shape, aspect angle to the illuminating radar and material (e.g. whether conduc- tive metal or insulating plastic), as well as the radar wavelength and polarisation.
A sphere has the unique property of looking the same from any aspect. Unlike almost all other shapes, its radar cross section (RCS), radar echoing area (REA) or [radar] cross-sectional area (CSA) is uniform, not fluctuating with the angle of aspect. Unusually, the proportion of incident energy reflected does not depend on wavelength, provided the sphere is in the 'optical region' where it is at least several wavelengths in circumference. These properties suit the sphere for use as a reference reflector and lead to the definition of RCS of an object: 'RCS is the proportion of incident energy reflected back in the direction of the source, relative to the proportion reflected by a replacement perfectly conducting sphere of cross-sectional area I m2' . Spheres reflect isotropically (in all directions). RCS, cr, equals the cross-sectional (silhouette) area:
a=nr2m2. (4.2a)
From the mechanism causing a sphere to reflect, detailed in Chapter 7, RCS can also be defined as
(power per unit solid angle scattered in a specified direction and polarisation)
o = . (power per unit area incident on the scatterer from the
specified direction and polarisation)
(4.2b)
RCS of practical targets often fluctuate, for example as they roll in sea waves.
Fluctuation is considered in Chapter 12. Meanwhile, unless stated otherwise, we assume RCS and echo strength are the average values presented to the radar.
More complicated shapes have RCS only very roughly similar to their silhouette.
For example, RCS of a flat conductive plate falls rapidly when turned oblique to the radar. Much time and computing power is needed to calculate RCS of practical targets of even moderately complex shape. RCS of point targets is discussed in Chapter 7, with ships and other extended targets in Chapter 10.
4.3.2 Two-way free space radar range equation On the receive leg, total power reflected towards the source is
The power density, d\ reaching the scanner mouth is calculated by considering the target as a transmitter of power Prefl. Applying Eq. (4.1):
, _ ^ _ JaPGnAnR2Lt)] _ PGa 2
" 4nR2 "" 4TTR2 " (4jr)2R*Lt ' ( j We denote the echo power delivered to the receiver by the scanner as Se(FS 12) > subscript (FS) indicating free-space conditions and (12) that both transmit and receive legs are included. The signal fed to the receiver is aperture x density
Se(FS12) = Ad'. (4.5) Substituting in Eq. (2.7e) for A, in Eq. (4.4) for dr and adding receiver losses L1
**»«> = ^ W ^ L , = PG^\AnT'R-\LtL^ W. (4.6a) This form of the radar range equation connects the radar parameters (such as trans- mitter power, scanner gain, received signal strength) with target and external factors (such as RCS and range). There are numerous variants. For example, the equation can be written to give R in terms of receiver bandwidth, noise figure, probability of detection, etc. To be exact, Eq. (4.6a) is one form of the two-way radar range equation for a passive target in free space. As expected from considerations of conservation of energy, it confirms Section 4.1 that echo power follows an inverse fourth power (R~4) law. Note that for a given signal strength Se(FSl2)> operation at longer range demands an increase of radar parameters P or G, or of target size a:
• doubling P or a increases R by 21Z4 = 19 per cent
• doubling G (e.g. by doubling scanner area A, say by doubling aperture width) increases R by 21//2 = 41 per cent, because the benefits of high gain accrue on both transmit and receive legs.
Eq. (4.6a) can be written in terms of aperture area, A, rather than gain:
Se(FSi2) = o A2 P (47T)-1 X -2^ -4I L t L1] "1. (4.6b)
This highlights that for constant aperture, doubling wavelength k reduces R by 41 per cent for a given signal strength. In practice, a tends to fall at long wavelength and multipath further also affects performance.
The range equation predicts that halving scanner gain, G, reduces echo strength to one quarter. However, low G means more solid angle is illuminated, in general raising azimuth beamwidth, giving more echoes per scan. Chapter 3, Section 3.6, indicated that this improves signal to clutter ratio, partly recouping the reduction in echo strength. We shall encounter several similar secondary effects through the book.
The range equation merely indicates echo strength, not echo detectability - which also involves competition from noise and clutter - and is just the first step of the road leading to calculation of whether a given target will be seen by a particular radar under particular circumstances.
In basic radar theory, detectability is ultimately dependent on the mean power transmitted, Pm, independent of the form of modulation. The reasoning is as follows.
We rewrite Eq. (4.6a) as
^
4 =PG^X\Anr\ULA-\
( 4 6 c ) Se(FS12)We replace Se(FSi2) by Si, the minimum single pulse which can be detected and put Si = nkTBs with the meanings of Chapter 3, Section 3.3.2, Eq. (3.2a);,? = minimum SNR for detection. If there are Af pulses in the packet and these are integrated within a coherent system, the minimum packet power which can be detected, SN , is inversely dependent on N:
Si nkTBs
SN =
^
=IT'
If pulselength = r and pulse repetition frequency = F , and assuming a matched filter so B = 1/r,
P1n = P r F , so P = ^ .
Substituting in Eq. (4.6c) for maximum detectable range Rmax
The terms in bold type are independent of the form of modulation, and N oc F, so the B, F and N terms cancel and
4 PmG2<rA2(47r)-3[L,Lr]-1
* -
a^fS '
(4-
6d)which is dependent on the mean power but does not contain terms in P , F , r or B which depend on the form of modulation. Whether Pm (for marine radars
PM ~ 10 W) is radiated continuously as in a broadcast radio transmitter or is concentrated in short pulses as in marine radar is basically a matter of practical convenience. It is possible to make continuous-wave radars, primarily measuring radial velocity by Doppler frequency shift, rather than range by time delay as in marine radar. Range is then determined by the phase shift of a video-frequency amplitude or phase modulation superimposed on the transmission. Such radars are ill-suited to marine use and will not be discussed until Chapter 16. Nevertheless, it is well to be aware that the current marine/VTS radar format is neither essential nor some unique arrangement ordained by heaven. The format could be replaced should need be, for example, as pressure intensifies on the electromagnetic spec- trum. Indeed, telecommunications users might like to consign pulse radars with their extreme EIRP (equivalent isotropic radiated power, Section 4.5.2) to a place far from heaven.