SEARCH RADAR RANGE EQUATION
We now want to study an extension to the radar range equation that is used to analyze and design search radars. Its most common use is in gross sizing search radars in terms of power and physical size. In fact, the measure of performance that is usually used to characterize these types of radars is power-aperture product, , which is the product of the average power and the effective antenna area of the radar.
To start, we assume that the radar must search an angular region, or sector, which we denote by . has the units of rad2, or steradians. If the azimuth and elevation extents of the sector to be searched is small,
(44)
where and are the azimuth and elevation extents of the search sector, in radians. The implication of (44) is that the search sector is a rectangle in azimuth and elevation space. In fact, the area can be any shape and represented in any appropriate coordinate system. One of the more common search sectors is section of the surface of a sphere bounded by some elevation and azimuth extents. An example of such a surface is shown in Figure 5. In this figure the azimuth extent is and the elevation extent goes from to . As shown in Appendix 1, the angular area of this search sector is
(45)
where and and all angles are in radians.
Earlier it was shown that the angular area of the beam could be written as
.(46)
Dividing by results in an angular beam area of
.(47)
The number of beams required to cover the search sector is
.(48)
Equation (48) is ideal in that it essentially assumes a rectangular search sector and rectangular beams. In practice, the number of beams required to fill a search sector is given by
(49)
where is a packing factor that accounts for the fact that the beams are actually circular (or elliptic) and that there is some overlap of the beams. A typical value for the packing factor for the case of hexagonal packing with 3-dB beam overlap is .
Recall that one of the parameters of the search radar range equation is the average power, . If a radar has a pulse width of and a PRI of the average power is
(50)
where d is defined as the duty cycle of the radar.
On of the requirements imposed on a search radar is that it cover the search sector in seconds. This means that the radar must process signals from n beams in seconds. With this, the time allowed to each beam is
.(51)
If we allow one PRI per beam we get
.(52)
From (48) we have which, with (52) results in
.(53)
Finally, using results in
.(54)
If we substitute (54) and (50) into the radar range equation we get
.(55)
For the last step we use (7) and (27) in (55) to arrive at the final search radar range equation of
(56)
We note that (56) does not contain an explicit dependence upon operating frequency (via ), antenna gain or pulse width, as does the standard radar range equation. This can be of value in performing first-cut search radar designs in that we needn’t specify a lot of parameters. This can be done after the first-cut design is completed.
EXAMPLE
As an interesting example we consider a requirement placed on search radars used for ballistic missile defense. Specifically, the SALT I treaty specifies that the power aperture product be limited to 3×106 w-m2. Given this limitation, we wish to perform a first cut design of a radar to be used for ballistic missile search.
We start by assuming that we want to search a region of space that extends from 0 to 45 degrees in elevation and 30 degrees in azimuth. Further, we want to cover the search sector in 10 seconds. The targets of interest have a RCS of -10 dBsm and we need to achieve a SNR of 13 dB to declare a detection. Current technology can support a noise figure of 4 dB and total losses of 6 dB. We want to determine the detection range of the radar.
Solving (56) for R results in
.(57)
All of the parameters except have been specified. We can compute from (45) as
.(58)
With this we find
.(59)
Let’s carry the above further and see if we can establish some more of the characteristics of this radar. We start by placing the requirement that the radar operate unambiguously in range. The means that we need to choose the PRI, T, to satisfy
.(60)
We will choose .
If we devote 1 PRI per beam then, over 10 seconds we would need to transmit and receive
.(61)
If we assume a circular beam we can use (48) to calculate the beam width from
.(62)
We choose to operate the radar at L-band. This gives a wavelength of
.(63)
From (12) we have
(64)
We can next use (27) to write
.(65)
If we assume an efficiency of 60%, we get a physical area of
.(66)
Finally, if we assume a circular aperture we obtain an antenna diameter of
(67)
which is approximately the height of a 7 story building.
As a final calculation, we want the peak power of the radar. We assume that we want a range resolution of 150 m, which translates to a pulse width of 1 µs. With the computed PRI of 6.5 ms we get a duty cycle of
,(68)
From (65) and the given average power aperture of 3×106 w-m2 we compute an average power of
(69)
From this and (68) we compute a peak power of
(70)
which is larger than desired. One way to reduce the peak power requirement would be to use a longer pulse and employ pulse compression. If we would go to a 100 µs we would reduce the peak power to 534 Kw, which is a much more reasonable value.
At this point we have a preliminary design for a search radar. In practice this would serve as a starting point for a much more detailed design.
RADAR CROSS SECTION
At this time we want to turn our attention to some of the other parameters of the radar range equation. In particular, we want to discuss radar cross section, or RCS. RCS is defined by
.(71)
The units of the numerator of (71) are watts while the units of the denominator are watts/m2. Thus, the units of RCS is m2, as discussed earlier. Two of the key phrases in the definition of (71) are “reradiated” and “toward the source”. This says that the RCSparameter attempts to capture, in a single number, the ability of the target to capture energy from the radar and reradiate it back toward the radar.
In general, computation of target RCS is very complicated. In fact, except for some very simple surfaces, RCS can only be approximately computed. In fact, there is a large amount of current research in methods to more reliably predict the RCS characteristics of practical targets.
In general, the RCS of a target depends upon its physical size. However this is not always the case. An example of the case where RCS depends upon physical size is a sphere. Specifically, the RCS of a sphere of radius r is
(72)
provided .
A case where RCS does not depend upon physical size is a cone where the nose of the cone is facing toward the radar, as shown in Figure 6. For the case of Figure 6 the RCS is given by
.(73)
In this case it will be noted that the RCS is proportional to wavelength but is not dependent on the overall size of the cone. If the cone had any other orientation relative to the line-of-sight (LOS) to the radar (see Figure 6 for a definition of LOS), its RCS would depend upon the length of the cone and the diameter of bhe base. Also, if the point of the cone was not perfectly sharp, as shown in the case of Figure 7, the RCS will depend upon the size of the nose.
In most cases, the RCS is dependent on both the size of the object and the radar wavelength. Examples of other simple shapes and their RCSs are contained in Figure 8. For the case of the chaff dipole, it appears as if the RCS depends only upon wavelength. However, as implied in the figure, the given equation for RCS applies only to the case where the length of the chaff dipole is equal to the wavelength. If one were to consider all orientations of a wavelength chaff dipole, the average RCS would be .
A classical plot in RCS theory is shown in Figure 9. This figure contains a plot of normalized RCS versus normalized radius for a sphere. It provides an illustration that the RCS of an object is generally a complicated function of both the size of the object and the wavelength of the electromagnetic wave that impinges on the object.
If the object size is less than a wavelength we say that the object is in the Rayleigh region of the incident electromagnetic wave. In this region, the RCS of the object is a function of the size of the object relative to a wavelength. As an example, for the sphere case illustrated in Figure 9 the sphere RCS in the Rayleigh region is given by
.(74)
The most common example of the case where an object size is on the order of a wavelength is the case of rain or clouds. Another example would be insects.
The center region of Figure 9 is termed the resonance, or Mie, region. The Mie designation is in honor or the person that developed the equation for the curve of Figure 9. In this region the object size is on the order of a wavelength and the the RCS is transitioning from being dependent upon both object size and wavelength to being dependent mainly on object size. As indicated in this figure, the RCS can often appear to be larger than dictated by object size. Typical objects that could be in the resonance region would be birds, bullets, artillery shells, some missiles and very small aircraft.
The third RCS region is termed the optical region and is where most objects fall. As indicated, it this region the object is much larger than a wavelength. Further, the RCS is (or can be) a strong function of the size of the object.
In general, the RCS of an object depends upon the orientation of the object relative to the LOS. As an example, the RCS of the flat plate illustrated in Figure 10 is given by
(75)
where .
A plot of versus is also shown in Figure 10 for a flat plate with and and (L-band). As can be seen, the RCS varies significantly as the angle of the LOS changes. It will also be noted that the peak RCS is significantly larger than the 1-m2 area of the plate.
Most targets of interest are not the simple shapes indicated thus far. In fact, targets such as airplanes consist of many different shapes that are in different orientations. Further, as the targets move relative to the radar LOS the relative orientations of the various shapes change significantly. As a result, a typical plot of target RCS versus orientation relative to the LOS is a very complex looking plot. The classical plot that illustrates this variation of RCS is given in Figure 2-15 of the text, which is duplicated in Figure 11 below. This figure shows the measured variation in RCS of a B-26 bomber as a function of azimuth orientation relative to the LOS. As can be seen, the RCS varies by quite a large amount and in a random looking fashion. If one considers that the orientation of the aircraft will change continually as the aircraft flies toward the radar, the angular variation in RCS will translate to a time variation of RCS that would lookrandom.
In an attempt to capture target RCS fluctuation effects in a mathematical model that could be easily used in detection studies, Peter Swerling developed statistical representations of RCS that are commonly referred to as the Swerling RCS models. There are four Swerling models termed Swerling 1, Swerling 2, Swerling 3 and Swerling 4. Many radar analysts refer to a fifth Swerling model that is termed Swerling 0 or Swerling 5. The fifth Swerling model is defined as a target that has a constant RCS. This Swerling model would be representative of a sphere since the ideal RCS of a sphere is constant over orientation angle and time.
The four models Swerling attempt to represent differences in both statistical and temporal variations in RCS. The statistical properties of Swerling 1 and Swerling 2 RCS variations (which we will refer to Swerling 1 or Swerling 2 targets, or SW1 and SW2 targets) are the same and are governed by the density function
(76)
where is the unit step function. Equation (76) is the equation for an exponential density function. is the average RCS of the target and is the value that would be used in the radar range equation.
The statistical properties of SW3 and SW4 targets are also the same and are governed by the density function
.(77)
Equation (77) is a Chi-squared, two degree of freedom density function. Again, is the average RCS of the target.
The difference between a SW1 and SW2 target lies in the difference in the time variation of RCS. Likewise for the difference between a SW3 and SW4 target. With a SW1 or SW3 target the RCS fluctuates slowly over time and with a SW2 or SW4 target the RCS fluctuates rapidly over time. In the classical definitions, SW1 and SW3 targets are said to change RCS on a scan-to-scan basis and SW2 and SW4 targets are said to change RCS on a pulse-to-pulse basis.
Plots of the density functions of (76) and (77) are shown in Figure 12. The plots of Figure 12 indicate that the RCS values for SW1 and SW2 targets are concentrated at small values (values below ) whereas the RCS values for SW3 and SW4 targets are concentrated at values fairly close to the average RCS. This is further illustrated in Figure 13 which contains plots of RCS versus dimensionless time for SW2 and SW4 targets. As will be noted, the RCS values for the SW2 case tend to vary significantly and stay well below the average RCS of 0 dBsm (1 m2) whereas the RCS values for the SW4 case tend to cluster around a value slightly below the average RCS of 0 dBsm.
As was indicated earlier, the difference between SWodd (SW1, SW3) targets and SWeven (SW2, SW4) targets lies in the rate at which the RCS is assumed to vary. It was stated that the SWodd model assumes that the RCS changes on a scan-to-scan basis. In the search RRE discussions we referred to a search volume (see Figure 5) and indicated that the search radar covers the search volume within a certain time we termed . This process of covering the search volume is termed a scan and is termed the scan time. If we were using a SWodd target model in the search radar analysis we would assume that the RCS changed from scan to scan but stayed constant during the scan. Thus, we would assume that the RCS changed every seconds, but stayed constant over any specific interval. If we were using a SWeven target model we would assume that the target RCS changed every PRI, or every seconds. Thus, the SWeven RCS models imply rapid RCS fluctuation whereas the SWodd models imply slow RCS fluctuation. This difference in RCS fluctuation models is illustrated in Figure 14, which is a plot of RCS versus pulse number, or PRI for the two cases. For the SW1 model the RCS changes every 50 pulses and the RCS changes every pulse for the SW2 model.
The concept of SWodd and SWeven represents an idealization that is not achieved in practice. Actual targets exhibit RCS variations that lie somewhere between SWodd and SWeven. How close the fluctuation lies to either model depends upon the complexity of the target, the operating frequency of the radar and the time between RCS observations. As an example, we consider the 5-point target model of Homework 5. In this example we “fly” the target model toward the radar with a constant x-velocity of 150 m/s and a y-velocity of zero. The center starts at x=20,000 m and y=5,000 m. We assume that all of the scatterers have the same RCS (1 m2) and compute the composite RCS (the total RCS of all five scatterers) as a function of time.
Figure 15 contains plots of composite RCS over a 3-second interval for the case where the carrier frequency is 8.136 GHz (low X-band) and 97.632 GHz (W-band or high MMW).
For the X-band case, the RCS remains fairly constant for time periods of 10’s of milliseconds. However, over periods of seconds the RCS variation becomes unpredictable (i.e. random). Thus, at X-band, this target exhibits a RCS behavior that is consistent with a SWodd target.
For the MMW case the RCS variation (with time) is much more rapid so that the RCS varies significantly over time intervals of 10’s of milliseconds. In this case, it might be appropriate to represent the target with a SWeven model.
To understand the above relation between RCS variation rate and operating frequency we need to consider how the signals from the scatterers combine to form the composite signal in the radar. We start by considering the unmodulated pulse we discussed earlier. For this case we can write the voltage pulse at the transmitter output as
.(78)
This voltage is converted to an electric field by the antenna and propagates to the target which creates another electric field. The electric field created by the target propagates back to the radar where the antenna converts it to a voltage. If the target is a sphere (a point scatterer) the voltage at the antenna output, or some point in the receiver (before the matched filter), can be written as
.(79)