2.IO PULSE REPETITION FREQUENCY AND RANG(£ AMBIGUITIES The pulse repetition frequency (prf) is determined primarily by the maximum range at which
2.14 OTHER CONSIDERATIONS
where R,,, .. = maximum radar range, 111 (i
=
antenna gainA
=
antenna aperture, m2 l'a = antenna efficiency11 = number of hits integrated
E;(11) = integration efficiency (less than unity)
L, = system losses (greater than unity) not included in other parameters
rr = radar cross section of target, m2 F" = noise figure
k
=
Boltzmann's constant= 1.38 x 10-23 J/deg T0 = standard temperature= 290 KB = receiver bandwidth, Hz r = pulse width. s
.f,,
= pulse repetition frequency, Hz(SI N)1 = signal-to-noise ratio required at receiver output (based on single-hit detection)
This equation can also he written in terms of energy rather than power. The energy in the transmitted pulse is E, = f'a,/1~ = P1 r and the signal-to-noise power ratio (S/N)1 can be replaced hy the signal-to-noise energy ratio (E/ N 0 ) 1, where E = S/r is the energy in the received signal. and N0 = N/B is the noise power per unit bandwidth, or the noise energy. Also note that Br ~ 1, and T0 Fn = ~ is defined as the system noise temperature. Then the radar equation becomes
R4
=
Er GAp0<rnE;(11)max
(4rr)
2kJ;(E/No)1
Ls (2.55)Although Eq. (2.55) was derived for a rectangular pulse, it can be applied to other waveforms as well, provided matched-filter detection is employed. Most calculations for probability of detection with signal-to-noise ratio as the parameter apply equally well when the ratio of signal-energy lo noise-power-per-hertz is used instead. The radar equation developed in this chapter for pulse radar can be readily modified to accommodate CW, FM-CW, pulse-doppler, MTI. or tracking radar.47·56
·57
Radar performance figure. This is a figure of merit sometimes used to express the relative performance of radar. It is defined as the ratio of the pulse power of the radar transmitter to the minimum detectable signal power of the receiver. It is not often used.
Blip-scan ratio. This is the same as the single-scan probability of detection. It predates the widespread use of the term probability of detection and came about by the manner in which the performance of ground-based search-radars was checked. An aircraft would be flown on a radial course and on each scan of the antenna it would be recorded whether or not a target blip had been detected on the radar display. This was repeated many times until sufficient data was obtained to compute, as a funct_ig1J of r<!_11ge, the ratio of the average num~er of scans the target was seen al a particular range (blips) to the total number of times it could have been seen (scans). This is the blip-scan ratio and is the probability of detecting a target at a particular range. altitude, and aspect. The head-on and tail-on are the two easiest aspects to provide in field measurements. The experimentally found blip-scan ratio curves are subject to many limitations. but it represents one of the few methods for evaluating the performance of an actual radar equipment against real targets under somewhat controlled conditions.
Cumulative probability of detection. If the single scan probability of detection for a surveil- lance radar is Pd, the probability of detecting a target at least once during N scans is called the cumulative probability of detection, and may be written
(2.56) The variation of Pd with range might have to be taken into account when computing P,. The variation of range based on the cumulative probability of detection can be as the third power rather than the more usual fourth power variation based on the single scan probability. 59
The cumulative probability has sometimes been proposed as a measure of the detectabi- lity of a radar rather than the single-scan probability of detection, which is more conservative.
In practice the use of the cumulative probability is not easy to apply. Furthermore, radar operators do not usually use such a criterion for reporting detections. They seldom report a detection the first time it is observed, as is implied in the definition of cumulative probability.
Instead, the criterion for reporting a detection might be a threshold crossing on two successive scans, or threshold crossings on two out of three scans, three out of five, or so forth. In track-while-scan radars the measure of performance might he the probability of initiating a target track rather than a criterion based on detection alone.
Surveillance-radar range equation. The form of the radar equation described in this chapter applies to a radar that dwells on the target for n pulses. It is sometimes called the searchlight range equation. In a· search or surveillance radar there is usually an additional constraint imposed that modifies the range equation significantly. This constraint is that the radar is required to search a specified volume of space within a specified time. Let
n
denote the angular region to be searched in the scan time ts. (For example, Q mjght represent a region 360° in azimuth and 30° in elevation.) The scan time is ts= t0n;n
0 , where t0 is the time on target=11/JP,
andn
0 is the soi id angular beam width which is approximately equal to the product of the azimuth beamwidth 80 times the elevation beamwidth 8e. (This assumes that 0.-t /Oe and 0£ /0~are integers, where O A is the total azimuth coverage and 0£ the total elevation coverage, such that
n
~ 8 Afh .)
The antenna gain can be written as G = 4rr/00 . With the above substitutions into Eq. (2.54) the radar equation for a search radar becomesR4 = PavAeO"Ei(n) ts
max 4rrkT0 Fn(S/N)i Ls
n (2.57)
This indicates that the important parameters in a search radar are the average power and the antenna effective aperture. The frequency does not appear explicitly in the search-radar range equation. However, the lower frequencies are preferred for a search radar since large power and large aperture are easier to obtain at the lower frequencies, it is easier to build a good MTI capability, and there is little effect from adverse weather.
Different radar range equations can .be derived for different applications, depending on the particular constraints imposed. The radar equation will also be considerably different if clutter echoes or external noise, rather than receiver noise, determine the background with which the radar signal must compete. Some of these other forms of the radar equation are given elsewhere in this text.
Accuracy of the radar equation. The predicted value of the range as found from the radar equation cannot be expected to be checked experimentally with any degree of accuracy. It is difficult to determine precisely all the important factors that must be included in the radar equation and it is difficult to establish a set of controlled, realistic experimental conditions in which to test the calculations. Thus it might not be worthwhile to try to obtain too great a
precision in the individual parameters of the radar equation. Nevertheless,
if
a-particular range is required of a radar, the systems engineer must provide it. The safest means lo achieving a specified range performance is to design conservatively and add a safety factor. The inclusion of a safety factor in design is not always appreciated, especially in competitive procurements, but it is a standard procedure in many other engineering disciplines. In the few cases where this luxury was permitted. fine radars were obtained since they accomplished what was needed even under degraded conditions.REFERENCES
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