RADAR RANGE EQUATION THOUGHTS – M. Budge, Nov 2003
Now that we have completed our study of the radar range equation (RRE)
and detection we want to revisit these topics to discuss the various terms of the
RRE and review some of the conditions associated with the use of the RRE and
detection theory. For our purposes we want to write the RRE as
SNR 
PT GT GR 2 I  n 
 4 
3
R 4kT0 BFL
.
We want to review the properties, characteristics, etc. of the various terms in
the RRE.
PT
PT is termed the peak power and is really the average power of the pulse
during the time the pulse is present. A related power is Pavg , which is the longterm average power. If the radar uses a constant PRI of T and a constant
pulse width of  p then the average and peak power are related by
Pavg   p T  PT . If the PRI and/or pulse width is not constant one would use
the average ratio of  p to T in the previous equation. In the RRE equation, as
written above, PT must have the units of watts.
GT
GT is termed the transmit antenna gain and is used to characterize the
focusing ability of the transmit antenna. We developed several equations for
computing GT . The two that I think are worth remembering are
GT 
25,000
 Az El
and
GT 
4 Ae
2
.
In the above  Az and  El are the azimuth and elevation beam widths expressed
in degrees. The azimuth and elevation notation is a convenient convention that
applies to most antennas. In more general applications, the two beam widths
used in the computation of antenna gain are the major and minor axes lengths
of the ellipse that are used to define the antenna beam. The major and minor
axes don’t need to be oriented up-down and side-to-side. Ae is termed the
effective aperture and is related to the physical aperture, A , of the antenna by
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Ae   A where  is the aperture efficiency of the antenna. A typical value for
 is 0.6.  is termed the radar wavelength and is related to the carrier
frequency, f c , of the radar by   c f c where c is the speed of light. GT has
units of watts/watts when used in the form of the RRE above.
A term related to PT and GT is called effective radiated power and is
given by the equation
ERP 
PT GT
.
Lt
It has units of watts. Lt is the loss between the point where PT is specified and
the antenna feed. The two most common points where PT is specified is at the
power amplifier output and at the antenna, where the latter refers to the
antenna feed. If PT is specified at the antenna, Lt is unity (0 dB). If PT is
specified at the power amplifier output, Lt includes the losses of all devices
between the power amplifier and the antenna feed. Lt has the units of
watt/watt.
An important point is that ERP is not a real power. In other words, the
power out of the antenna is not the ERP ; it is PT Lt . ERP is the power one
would need to transmit from an isotropic antenna in order to get the same
power density on the target that one gets from the actual antenna. ERP has
the units of watts.
GR
GR is the gain of the receive antenna and is calculated using the same
equations as above. GR is used to characterize the ability of the receive
antenna to “capture power”. If the radar uses the same antenna for transmit
and receive then GR  GT . In class we have associated the fact that GR  GT
with a monostatic radar. This is not strictly accurate because a monostatic
radar simply means that the transmit and receive antennas are co-located; the
radar could still use different antennas for transmit and receive.
An important point is implied about the use of GT and GR in the radar
range equation. Namely, it is assumed that the transmit and receive antennas
are pointed at the target. It is more accurate to note that they are the gains of
the transmit and receive antennas in the (angular) direction of the target.

 is defined as the radar cross-section (RCS) of the target and has the
units of m2. It is a single number that is used to represent the ability of the
target to capture and re-radiate power in the (angular) direction of the receive
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antenna. An important issue about RCS is that it is only roughly related to
physical target size. In fact, it is related to target size, the material and coatings
on the target, the target orientation, the target structure and a host of other
factors. Even though we use a single number for RCS in the RRE, we attempt
to account for the RCS variation of the target by using various target models.
The most common models are the Swerling models. These models characterize
RCS as a random process (a RCS that changes randomly with time). The
Swerling models provide for two sizes and rates of RCS variation by using two
pairs of models. The first pair, termed SW1 and SW2, is used to characterize
complex targets such as aircraft, ships, tanks, or other targets with a large
number of facets. This type of target is characterized by fairly large variation in
RCS (because of varying target orientation). The convention used to arrive at
the density function for SW1 and SW2 targets is that they consist of a large
number of equal size scatterers. The SW1 target differs from the SW2 target in
that the RCS of SW1 targets varies slowly with time and the RCS of SW2 targets
varies rapidly with time. The standard convention is that the RCS of SW1
targets changes over periods of seconds and the RCS of SW2 targets changes
over periods of microseconds to milliseconds.
The second pair of Swerling target models is termed SW3 and SW4 and is
used to characterize simple targets such as missiles and streamlined vehicles.
The convention used to arrive at the density function for SW3 and SW4 targets
is that the target consists of a single large scatterer and a large number if equal
size, but smaller, scatterers. The SW3 target differs from the SW4 target in that
the RCS of SW3 targets varies slowly with time and the RCS of SW4 targets
varies rapidly with time. The standard convention is that the RCS of SW3
targets changes over periods of seconds and the RCS of SW4 targets changes
over periods of microseconds to milliseconds.
A common, fifth, target type is a constant RCS target. This is termed a
SW0 target by some and a SW5 target by others. A constant RCS target doesn’t
exist in practice, however it is still used by many radar analysts. The reason for
this is not clear. It is probably a carryover from before Swerling developed his
models.
The density function for the SW1/SW2 target model is
f12   
e   av
 av
u  
and the density for the SW3/SW4 target model is
f 34   
4
 av2
e 2  av u   .
In the above u   is the unit step function.  av is termed the average RCS of
the target and is the value of RCS used in the radar range equation.
The target model is not directly used in the RRE. It comes into play
when one uses the results of the RRE to predict detection range. This will be
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discussed later. It also comes into play in determining I  n  , the integration
gain.
People have attempted to verify that “real” targets actually obey the
Swerling models. To the author’s knowledge, none have been completely
successful. It appears that the best fit of a density function to real targets is a
Gaussian density when the RCS is expressed in dBsm (10log(m2)). This is
called a log-normal target model. The log-normal model is very rarely used in
detection studies. Although the Swerling models don’t perfectly match real
targets they do a good job of predicting detection performance against real
targets.
R
R is the range from the radar to the target. The use of R4 in the RRE
above carries the tacit assumption that the applicable radar is monostatic. If
the radar had been bi-static one would replace R 4 with RT2 RR2 where RT is the
range from the transmit antenna to the target and RR is the range from the
receive antenna to the target. R has the units of meters. The only constraints
on R is that it must be much larger than the size of the antenna. More
specifically one must have R  4 D 2  where D is the largest dimension of the
antenna(s). This constraint is imposed by the antenna and defines the “far
field” of the antenna. Of course, R is also constrained by the waveform and
must satisfy the inequality, R  c p 2 where  p is the uncompressed pulse
width. R could also have an upper bound if the radar is required to operate
unambiguously in range.
k
k is Boltzman’s constant and is equal to 1.38  1023 w  Hz K  .
T0
T0 is a standard temperature of 290° K. It is the standard temperature
used to compute the noise figure.
F
F is the noise figure of the radar receiver and is a dimensionless
quantity. When specifying noise figure, one must specify an input and output
of the receiver. The input is called the reference point. Thus, for example, if
the input is the input to the RF amplifier and the output is the matched filter,
one would say that F is the noise figure at the output of the matched filter
referenced to the RF amplifier input. The two most common reference points
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are the output of the antenna feed (the antenna or the antenna terminals) and
the input to the RF amplifier, if the radar contains an RF amplifier. If the radar
doesn’t contain an RF amplifier, another reference point would be the first
active device, which is usually a mixer. The most common output point is the
matched filter. Sometimes one also specifies the output as the output of the
signal processor. However, in reasonable radar designs, the noise figure at the
output of the signal processor will, for all practical purposes, be the same as
the noise figure at the output of the matched filter. Note: there must be no
active devices between the antenna feed and the reference point for F .
In some applications, one can replace the product of T0 and F with the
receiver noise temperature, TN . Receiver noise temperature would be the
preferred quantity for those cases where the noise temperature at the antenna
terminals is considerably larger than T0 , as would be the case when the
antenna was looking at the sun. It would also be the preferred form for the
case where a low-noise radar is looking into space (which has a temperature
near absolute zero). In most applications, the use of TN or T0 and F is
immaterial. The reader is referred to the discussions on noise figure for more
further study in this area.
L
L is the loss term and accounts for all losses not included elsewhere. It
has the units of w/w. Refer to Skolnik’s text book and handbook for a
discussion of the various things that must be included in L . Some rules of
thumb are as follow: L must include a transmit loss term, Lt , if PT is not
specified at the antenna feed. More specifically, Lt must contain any losses
between the point where PT is specified and the antenna feed. L must include
a receive loss term, LR , if the noise figure is not referenced to the antenna feed.
LR must contain the losses of all devices between the antenna feed and the
reference point used when specifying the noise figure. L is one of the more
difficult quantities to determine accurately. To do so, one must have intimate
knowledge of the radar subsystems, radar operation and the environment. It is
not unusual for the list of the components of L to be very long (10 to 30
entries). Many of the entries in the list will be on the order of tenths of a dB
but, when taken together, can add up to a significant loss.
B
B is the noise bandwidth of the receiver. Throughout this course we
have been careful to define B as the effective noise bandwidth of the receiver.
In this context, B  1  p where  p is the uncompressed pulse width. We have
taken this approach because it helps avoid problems in defining the bandwidth
to use in the RRE. If the radar uses a phase or frequency modulated pulse one
could use the actual waveform bandwidth, Bw , in the RRE (assuming the one
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uses a matched filter). However, one would need to include the compression
gain of the waveform in I  n  . If one uses an LFM (Linear Frequency
Modulation) waveform, the compression gain would be Bw p . If one uses a
phase coded waveform (e.g. Barker coded waveform, Pseudo Random Noise
waveform, random phase coded waveform, Frank polyphase waveform, etc.)
then the compression gain is n , where n is the number of subpulses in the
waveform (we will discuss this further in connection with the ambiguity
function). One would let the waveform bandwidth be Bw  1  sp where  sp is the
subpulse width. It is usually assumed that the subpulses are adjacent to each
other such that  p  n sp .
With some simple analyses, it is easy to show that if Bw and the
accompanying compression gain are used in the RRE then the RRE can still be
reduced to the form where the effective noise bandwidth, B , is used. Having
said this, we need to point out an exception. If one uses a phase coding and
leaves a gap between the subpulses then one should use a noise bandwidth of
Bw  1  sp and include the compression gain in I  n  . As an alternative, one
could use an equivalent pulse width of  pe  n sp and define the effective noise
bandwidth as B  1  pe .
A final note on noise bandwidth is related to Doppler signal processors;
signal processors that contain narrow-band band-pass filters. In this case,
some people propose using the band-pass filter bandwidth, Bbp , as the noise
bandwidth in the radar range equation. If one uses this approach one must
include a factor to account for the increase in noise power spectral density
caused by the sampling effect of the range gates in the receiver. Specifically,
one must increase the noise power spectral density from kT0 F to kT0 F T  p


where T is the PRI and  p is the pulse width. The reason for this will be
discussed in EE 725. An alternate approach is to define use the standard
approach of using effective noise bandwidth (i.e., B  1  p ) and including the
integration gain of the Doppler processor in I  n  . The integration gain would
be PRF Bbp .
I n 
I  n  is used to represent the actual or effective SNR gain associated with
the signal processing that occurs after the matched filter. (As discussed above,
it can also be used to account for the integration gain of the matched filter.) If
the signal processor is a coherent integrator then I  n  represents the actual
gain in SNR that the signal processor provides. Recall that if the coherent
integrator sums the returns from n pulses (integrates n pulses), then I  n   n .
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If one implements the coherent integrator with a FFT then n is the number of
taps in the FFT. If one implements the coherent integrator as a band-pass filter
then I  n   B BBPF where B is the effective noise bandwidth discussed above
and BBPF is the bandwidth of the band-pass filter.
If the signal processor consists of a non-coherent integrator (an
integrator or a summer that operates on the target returns after they have gone
through the magnitude or square-law detector) then I  n  is an effective
integration gain. It does not represent the actual increase in SNR afforded by
the non-coherent integrator. It is a gain that allows us to use the single-pulse
false alarm and detection probability equations with the SNR that comes from
the RRE when I  n  is included. The values of I  n  depends upon the target
model being considered and the range of detection and false alarm probabilities
being considered. Fortunately, for SW0, SW1 and SW3 targets we can
approximate I  n  by the equation I  n   n 0.8 . This equation is valid for all
detection and false alarm probabilities that are usually of interest in radar
design and analysis. There is no set approximate equation for SW2 and SW4
targets. As a first approximation one can use the curves provided in class. If
one desires a more accurate relation between I  n  and n, Pd and Pfa one
should refer to the equations or curves in Meyer and Mayer’s book.1 If the
radar being analyzed contains a coherent integrator that integrates n pulses
and a non-coherent integrator that integrates m groups of n pulses then the
integration gain is I  n, m   n  m0.8 . It is important to note that this equation
applies only to SW0, SW1 and SW3 targets. For SW2 or SW4 targets I  n   1

and I  n, m would become I  n, m   f m, Pd , Pfa

as discussed above. In either
case, I  n, m is the effective integration, or SNR, gain.
SNR
SNR is the signal-to-noise power ratio at a certain point in the receiver.
When we first derived the RRE, without I  n  , we formulated the derivation so
that SNR was the signal-to-noise measured at the output of the matched filter.
Furthermore, we showed that it was the SNR measured when the signal level
out of the matched filter was at its peak value. Because of this we often refer to
SNR as the peak signal - to - average noise power ratio. For further discussion
Meyer, Daniel P. and Mayer, Herbert A., “Radar Target Detection – Handbook of Theory and Practice”,
Academic Press, San Diego, CA, 1973 ISBN 0-12-492850-1
1
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of this refer to your notes on the matched filter. We also termed this SNR the
single-pulse SNR.
When we consider coherent integration and include I  n  in the RRE, the
SNR is the signal-to-noise power ratio (peak signal - to - average noise power
ratio) at the output of the coherent integrator. In terms of detection, we treat
the single pulse SNR and the SNR at the coherent the same way. That is, we
use the same set of equations for Pfa and Pd . This stems from the fact that, for
the SW0 through SW4 target models, coherent integration does not change the
form of the noise, signal and signal-plus-noise density functions at the output
of the magnitude or square law detector.
When we consider non-coherent integration and include I  n  in the
RRE, the SNR is not the signal-to-noise power ratio at the output of the noncoherent integrator. It is an effective SNR that we use to account for the effects
of the non-coherent integrator in computing detection probability. Essentially,
it allows us to use the same set of Pfa and Pd equations that we use when we
have only a matched filter (single pulse) or a coherent integrator.
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