Pd IMPROVEMENT TECHNIQUES
In the previous discussions we derived equations for detection probability
((106), (109), (110)) and showed that the use of a matched filter will provide the
maximum SNR and Pd that can be obtained for a given set of radar parameters
and a given, single transmitted pulse. We term the resultant SNRs and Pd s
single pulse SNR and Pd . We now want to address the gains in Pd that we can
obtain by using multiple transmit pulses. We will examine three techniques
1. Coherent Integration
2. Non-coherent Integration
3. Cumulative Probability
COHERENT INTEGRATION
With coherent integration we insert a coherent integrator, or signal
processor, between the matched filter and amplitude detector as shown in
Figure 14. This signal processor adds returns (thus the word integrator) from
N pulses. After it accumulates the N pulse sum it performs the amplitude
detection and threshold check. In practice the process of forming the N -pulse
sum is somewhat complex. In one implementation, the signal processor
samples the return from each transmit pulse at a spacing equal to the range
resolution of the radar. Thus, for example, if we were interested in a range
window from 5 to 80 Km and had a range resolution of 150 m the signal
processor would form 75,000/150 or 500 samples for each pulse. The signal
processor would then accumulate (add) each of the 500 samples in 500
summers. After the signal processor has summed the first N pulses it would
begin dropping older pulses off of the accumulator as new pulses arrive. Thus,
the signal processor will add the returns from the most recent N pulses. In
analog processors the integration (summation, accumulation) is accomplished
by filters. It is accomplished by FFTs or other digital signal processors in digital
signal processors.
Matched
Filter
Coherent
Integrator
Amplitude
Detector
Threshold
Device
Detect
No Detect
(Signal
Processor)
Figure 14 – Location of the Coherent Integrator
As we did in our previous studies we will separately consider the signal
and noise. For the signal, we assume that the amplitude of the signal on pulse
k is given by (note, we assume that we are looking at the specific range cell –
out of 500 for the previous example – that contains the target return)
©2005 M. C. Budge, Jr
37
vs  k   VS e j .
(172)
The formulation of vs  k  in (172) carries several assumptions about the target.
Specifically, it implies that the amplitude and phase of the signal returned from
the target is constant, at least over the N pulses that are to be summed. This
means that we are assuming that the target is SW0/SW5, SW1 or SW3. It does
not admit SW2 or SW4 targets. As we will show later, coherent integration
offers no benefit for SW2 and SW4 targets.
The second assumption is that there is no Doppler on the target return.
If the target is moving, and thus will have a Doppler frequency, this Doppler
frequency must be removed by the signal processor before the summation takes
place. In digital signal processors that use FFTs, Doppler removal is
accomplished by the FFT.
If we sum over N pulses the output of the summer will be
vsout   vs  k   N VS e j .
(173)
N
If the signal power at the input to the summer is
Psin  VS  PS
2
(174)
the signal power at the output of the summer will be
Psout  N 2 VS  N 2 PS .
2
(175)
We can write the noise at the input to the coherent integrator as
vn  k  
1
 vnI  k   jvnQ  k   .
2
(176)
If we sum over N pulses, the noise at the output of the summer will be
v nout   v n  k  
N
1 

  v nI  k   j  v nQ  k    v noutI  jv noutQ .
2 N
N

(177)
The noise power at the output of the summer will be
Pnout  E v nout vnout   E v 2noutI   E v 2noutQ  .
(178)
We now need to compute the two terms on the right of (178). We write
E v
2
noutI
 1 N

 1 N
  E  2  v nI  k    2  v nI  l   
k 1
l 1



.
1
1
2
  E v nI  k  
 E v nI  k  v nI  l 
2 N
2 l ,k1, N 
(179)
l l
Since v nI  k  is WSS and zero-mean
©2005 M. C. Budge, Jr
38
E v 2nI  k  
2
2
k .
(180)
We also assume that the noise samples are uncorrelated from pulse to pulse.
This means that v nI  k  and v nI  l  are uncorrelated k  l . Since v nI  k  and
v nI  l  are also zero-mean we get
E v nI  k  v nI  l   0 l  k .
(181)
If we use (180) and (181) in (179) we get
E v 2noutI  
N 2 NPnin

2
2
(182)
where Pnin is the noise power at the output of the matched filter.
By similar reasoning we have that
N 2 NPnin

2
2
(183)
Pnout  E v 2noutI   E v 2noutQ   NPnin .
(184)
E v 2noutQ  
and, from (178)
If we combine (175) and (184) we find that the SNR at the output of the
coherent integrator is
SNRout 
N 2 Psin
 N  SNR 
NPnin
(185)
where SNR is the SNR at the output of the matched filter; or the SNR given by
the radar range equation. Thus, we conclude that the coherent integrator
provides a factor of N gain in SNR, where N is the number of pulses
integrated.
In the above development we have made the assumption that the signal
level (in voltage and power) at the input to the coherent integrator was constant
from pulse to pulse. This is indicative of a SW0/SW5 target, a SW1 target or a
SW3 target. The specific amplitude over the N pulses integrated by the
coherent integrator is governed by the probability density function for the
specific target type. What this says is that we can essentially consider the
output of the coherent integrator as the return from a single pulse whose SNR
is N times the SNR provided by the radar range equation.
The noise on each pulse is zero-mean and Gaussian. The noise samples
from each pulse are uncorrelated (and thus independent since they are
Gaussian) and the I and Q samples are also Gaussian. Thus the noise out of
the coherent integrator has the same statistical properties of the noise out of
the matched filter.
©2005 M. C. Budge, Jr
39
The consequence of the above discussions is that, when computing Pfa
and Pd for the case where the radar uses a coherent integrator (and for
SW0/SW5, SW1 or SW3 targets) we use the same Pfa and Pd equations we used
before; that is (105), (106), (109) and (110). However, in place of SNR we use
N  SNR  where N is the number of pulses that are coherently integrated and
SNR is the SNR at the output of the matched filter.
If the target is SW2 or SW4, coherent integration of no help. This stems
from the fact that, for SW2 and SW4 targets, the signal is not constant from
pulse to pulse but, instead, behaves like noise. This means that we must treat
the target signal the same as we do noise. Thus in place of (173) we would
write
v sout   v s  k  
N
1 

  v sI  k   j  v sQ  k    v soutI  jv soutQ .
2 N
N

(186)
If we follow the procedure we use for the noise case we would have that
E v 2soutI   E v 2soutQ  
NPsin
2
(187)
and
Psout  E v 2soutI   E v 2soutQ   NPsin .
(188)
This, with (184) would lead to the result
SNRout 
NPsin
 SNR .
NPnin
(189)
In other words, the SNR at the coherent integrator output would be the same as
the SNR at the matched filter output and the coherent integrator would offer no
integration gain. For the SW4 case there is an additional complication that
indicates that we may not be able to easily determine the density function of the
signal-plus-noise at the output of the coherent integrator (we can determine
that the density function will be Gaussian for the SW2 target). This could
further complicate the computation of Pd . The summary consequence for SW2
and SW4 type targets (targets whose RCS changes from pulse to pulse) is that
coherent integration offers no benefits in terms of increasing Pd .
In the above development we made some ideal assumptions concerning
the target based on the fact that we were collecting, and summing, returns from
a sequence of pulses. In particular we assumed that the target amplitude was
constant from pulse to pulse. Further, we assumed that we sampled the output
of the matched filter at its peak. In practice neither of these is strictly true.
First, we really can’t expect to sample the matched filter output at the peak of
the target return. Because of this, the SNR in (185) will not be the SNR at the
matched filter output (the SNR given by the radar range equation). It will be
some smaller value. We usually account for this by degrading SNR by a factor
©2005 M. C. Budge, Jr
40
we call range straddling loss. If the sample period is the pulse width, the range
straddling loss is usually taken to be 3 dB.
There are other reasons that the signal into the coherent integrator will
vary. One is target motion. This will create a Doppler frequency which will
cause amplitude variations from pulse to pulse. It the Doppler frequency is
large enough so as to cause large amplitude variations the gain of the coherent
integrator will be nullified. In general, if the Doppler frequency is greater than
about PRF N the coherent integration gain will be nullified. In fact, the
coherent integration will most likely result an a SNR reduction. Doppler
frequency offsets can be circumvented by using banks of coherent integrators
that are tuned to different Doppler frequencies.
Another degradation that is related to Doppler is termed range gate walk.
Because of Doppler, the target signal will move relative to the location of the
various samples that are fed to the coherent integrator. This means that, over
the N pulses, the signal amplitude will change. As indicated above, this could
result in a degradation of SNR at the output of the coherent integrator. In
practical radars, designers take steps to avoid range walk by not integrating too
many pulses. Unavoidable range walk is usually accounted for by including a
small (less than 1 dB) SNR degradation (SNR loss).
Still another factor that causes the signal amplitude to vary is the fact
that the coherent integration may take place while the radar scans its beam
across the target. The scanning beam will cause the GT and GR terms in the
radar range equation to vary across the N pulses that are coherently
integrated. As before, this will degrade the SNR and its effects are included in
what is termed a beam scanning loss. This loss, or degradation, is usuall 1 to 3
dB in a well designed radar.
Phased array radars have a similar problem. For phase array radars the
beam doesn’t move continuously (in most cases) but in discrete steps. This
means that the phased array radar may not point the beam directly at the
target. This means, in turn, that the GT and GR of the radar range equation
will not be their maximum values. As with the other cases, this phenomena is
accommodated through the inclusion of a loss term called, in this case, beam
shape loss. Typical values of beam shape loss are 1 to 3 dB.
NON-COHERENT INTEGRATION
We now want to discuss non-coherent, or post-detection, integration.
The name post-detection integration derives from the fact that the integrator, or
summer, is placed after the amplitude or square law detector as shown in
Figure 15. The name non-coherent integration derives from the fact that, since
the signal has undergone amplitude or square law detection, the phase
information is lost. The non-coherent integrator operates in the same fashion
as the coherent integrator (see the discussion at the beginning of the previous
section) in that it sums the returns from N pulses before performing the
threshold check.
©2005 M. C. Budge, Jr
41
Matched
Filter
Amplitude
Detector
Non-coherent
Integrator
Threshold
Device
Detect
No Detect
Figure 15 – Location of the Non -coherent Integrator
A non-coherent integrator can be implemented in several ways. In older
radars it was implemented via the persistence on displays plus the integrating
capability of a human operator. These types of non-coherent integrators are
very difficult to analyze and will not be considered in this course. The reader is
referred to the text and the Radar Handbook by Skolnik.
A second implementation is termed an m-of-n detector and uses more of
a logic circuit rather than a device that integrates. Simply stated, the radar
examines the output of the threshold device for n pulses. If a DETECT is
declared on any m of those n pulses the radar declares a target detection. This
type of implementation is also termed a dual threshold detector. Again, we will
not discuss this type of non-coherent integrator in this class. Its analysis is
reasonably straight forward for SW0/SW5, SW1 and SW3 targets but becomes
somewhat more difficult for SW2 and SW4 targets.
The third type of non-coherent detector is implemented as a summer or
integrator. In older radars low-pass filters were used to implement them. In
newer radars they are implemented in special purpose hardware or the radar
computer as digital summers. They operate as described in the beginning of
the discussion of coherent integrators.
For SW0/SW5, SW1 and SW3 targets the main advantage of a noncoherent integrator over a coherent integrator is hardware simplicity. As
indicated in earlier discussions, coherent integrators must contend with the
effects of target Doppler. In terms of hardware implementation this usually
translates to increased complexity of the coherent integrator. Specifically, it is
usually necessary to implement a bank of coherent integrators that are tuned to
various ranges of Doppler frequencies. Because of this one will need a number
of integrators or summers equal to the number of range cells in the search
window multiplied by the number of Doppler bands needed to cover the Doppler
frequency range of interest. Although not directly stated earlier, this will also
require a larger number of amplitude (or square law) detectors and threshold
devices.
Since the non-coherent integrator is placed after the amplitude detector,
it does not need to accommodate multiple Doppler frequency bands. This lies
in the fact that the amplitude detection process recovers the signal (plus noise)
amplitude without regard to phase (i.e. Doppler). Because of this, number of
integrators is reduced; it is equal to the number of range cells in the search
window.
It will be recalled that coherent integration offers no improvement in SNR
for SW2 or SW4 targets. If fact, it can degrade SNR relative to that that can be
obtained from a single pulse. In contrast, non-coherent integration can offer
significant improvement in SNR relative to a single pulse. In fact, contrary to
most people’s intuition, non-coherent integration can offer SNR improvement
©2005 M. C. Budge, Jr
42
factors that are greater than the number of pulses integrated! It is interesting
to note that some radar designers are using various schemes, such as
frequency hopping, to force targets to exhibit SW2 or SW4 characteristics so as
to exploit the significant SNR improvement offered by non-coherent integration.
Analysis of non-coherent integrators is much more complicated that
analysis of coherent integrators because the integration takes place after the
non-linear process of amplitude or square law detection. From our previous
work we note that the density functions of the noise and signal-plus-noise are
somewhat complex. More importantly, they are not Gaussian. This means that
when we sum the outputs from successive pulses we cannot conclude that the
density function of the sum of signals will be Gaussian (as we can if the density
function of each term in the sum was Gaussian). In fact, the density functions
become very complex. This has the further ramification that the computation of
Pfa and Pd become very complicated. Analysts such as DiFranco and Rubin,
Marcum, Swerling, and Mayer and Meyer have devoted considerable energy to
analyzing non-coherent integrators and documenting the results of these
analyses. We will not attempt to duplicate the analyses here. Instead, we
present the results of their labor. The equations below are those that one could
use to compute Pfa , and Pd for each of the five types of Swerling targets.
The equation for Pfa at the output of the N -pulse non-coherent
integrator is
12
 TNR N 1lnTNR N  
 N  e
Pfa  
.
 
 2   TNR  N  1 
(190)
Because of arithmetic problems it is often better to use the natural log of (190)
or


ln  Pfa   12 ln  2N   N 1  ln TNR N    TNR  ln TNR  N  1
(191)
As with the single pulse case, one normally specifies Pfa and then use (190) or
(191) to compute TNR to use in the Pd equations below.
The Pd equation for the five target types we have studied are
SW0/SW5:

Pd  Pd 1 10log  N  SNR   , e TNR
e
 TNR  N  SNR 
 TNR 



r  2  N  SNR  
N

 r 1
2

I r 1 2
TNR  N  SNR  
(192)
SW1:
©2005 M. C. Budge, Jr
43
Pd  1   TNR, N  2 

 1
1
N  SNR 

N 1

 TNR 1 N  SNR 
TNR

, N  2e
 1  1  N  SNR  



(193)
SW2:
Pd  1    1TNR
SNR , N  1
(194)
SW3:


2
Pd   1 

N  SNR  

N 2

2  N  2   TNR 1 N  SNR  2
TNR

1 
e
 1  N  SNR  2 N  SNR  
(195)
SW4:
SNR
Pd  1   SNR
2 

N
In the above Pd 1 S , Pfa
N

k 0
N!
k ! N  k !
 SNR
2 
k
2TNR
  SNR
2 , 2 N  1  k 
(196)
 is the single pulse, SW0/SW5 detection probability
equation defined earlier (Equation (106)). I r  x  is the modified Bessel function
of the first kind and order r, and,
a
  a, N   
0
x N e x
dx
N!
(197)
is the incomplete gamma function. The SNR values in the above equations are
the single-pulse SNR values defined by the radar range equation. Also, the
above equations are based on the assumption that the amplitude detector is a
square law detector. It turns out that they also apply well to the case where the
amplitude detector is a magnitude detector.
In many applications it is cumbersome to implement the above Pfa and
Pd equations. As a result of this we often resort to graphical techniques and
rules-of-thumb developed from the graphical techniques.
©2005 M. C. Budge, Jr
44
Figure 16 – Curves Used for Graphical Analysis of Non-Coherent Integration
An example of a figure we use for the graphical technique is shown in
Figure 16. This figure was taken from an earlier edition of your text and is a
©2005 M. C. Budge, Jr
45
plot if improvement factor, I i  n  , of the non-coherent integrator versus number
of pulses integrated, n . (Note that this figure uses n instead of the N we have
been using.). To relate I i  n  to our previous work, we would set
I i  n   10log n
(198)
for a coherent integrator. Thus, I i  n  is the effective increase in SNR, in dB,
afforded by an n -pulse non-coherent integrator and the type of target specified.
It will be noted that the lower right of the figure has the notation
n f  108 . This means that the curves apply to a Pfa of
Pfa 
0.693
 0.693  108 .
nf
(199)
It turns out that the curves also apply to other values of Pfa within the range we
normally encounter in practical radar applications.
To use the curves one would use the following procedure

Decide on a desired Pfa , target type and number of pulses integrated,
and an estimated Pd .

Use the appropriate curve to compute the appropriate I i  n  .

Increase the single-pulse SNR of the radar range equation by I i  n  .

Use the resulting increased SNR and desired Pfa in (105) and (106),
(109) or (110) as appropriate to compute the actual Pd . If the actual
Pd differs from the estimated Pd by a significant amount it may be
necessary to estimate a new Pd and repeat the process.
Example
As an example of how one might compute the effects of coherent and
non-coherent integration we will consider a practical example. The radar in
this example employs a continuously rotating antenna that completes a rotation
in Tscan seconds. The radar operates with a fixed PRI of T seconds. Finally, the
antenna has an azimuth beamwidth of  AZ degrees. We will not be directly
concerned with the specific elevation beamwidth. We will state that the
elevation beamwidth and target elevation position are such that the antenna
beam will be roughly centered on the target, in elevation, as the beam scans by
it.
We want to incorporate a coherent or non-coheent integrator that will
integrate target returns as the beam scans by the target. In order to determind
©2005 M. C. Budge, Jr
46
the integration gain offered by the integrators we need to determine the number
of target return pulses the radar will receive as the beam scans by the target.
Since the antenna moves 360º in Tscan seconds the antenna angular rate
is

360
deg/sec .
Tscan
(200)
The reciprocal of the angular rate is
 
1


Tscan
sec/deg .
360
(201)
We are interested in the time it takes the beam to move an azimuth
beamwidht, or  AZ degrees. We denote this  dwell and write
 dwell    AZ 
Tscan
 AZ sec/bw .
360
(202)
 dwell is often called the dwell time or the time on target and is the length of time
that the antenna beam is “looking at the target”.
As the beam scans by the target the radar will receive a target return
pulse every PRI, or T seconds. Thus, the number of target return pulses
received during  dwell will be
N PulInt 
 dwell
T

Tscan AZ
pulses/bw .
360T
(203)
N PulInt is the number of pulses that can be coherently or non-coherently
integrated.
As a specific example we consider a radar that has a scan period of
Tscan  0.5 sec , an azimuth beamwidth of  AZ  1.5 deg and a PRI of T  600 s .
With this we get
N PulInt 
0.5  1.5
 3.47 or 3 pulses/bw
360  600  106
(204)
If we were to coherently integrate the 3 pulses we would get a coherent
integration gain of 3, or about 4.8 dB (this assumes we have a SW0/SW5, SW1
or SW3 target). For the non-coherent integration gain we would use the curves
of Figure 16. For a SW0/SW5, SW1 or SW3 target and a desired Pd of 0.9, the
non-coherent integration gain would be about 4 dB. If the target was SW2 (and
the desired Pd was 0.9, the integration gain wild be about 10 dB. For a SW4
target the non-coherent integration gain would be about 8 dB.
We point out that the radar will not realize all of the integration gain
specified above. The reason for this is that not all of the pulses are at the peak
SNR predicted by the RRD. Over the beamwidth, the SNR can vary by 3 dB. To
account for this we incorporate a loss term of 1.6 dB. This would reduce the
©2005 M. C. Budge, Jr
47
effective integration gain by 1.6 dB. If we account for the tact that the target
may, in fact, not be centered in the elevation beam we would need to
incorporate another loss of 1.6 dB for a total loss of 3.2 dB. Thus, the coherent
gain above would be, effectively, 1.6 dB. The non-coherent gain for the
SW0/SW5/SW1/SW3 target would be 0.8 dB and the integration gains for the
SW2 and SW4 targets would be, respectively, 6.8 and 4.8 dB.
If the radar uses a phased array antenna we would compute N PulInt
somewhat differently. With phased array antennas the beam (usually) doesn’t
move continuously. Instead, it moves in steps, dwelling at particular beam
positions for  dwell seconds. In this case the number of pulses that could be
integrated would equal the number of pulses per dwell, or
N PulInt 
 dwell
T
.
(205)
An interesting observation of the above example is that for the SW2 case
the non-coherent integration gain is 10 w/w and for the SW4 case it is 6.3 w/w.
Both if these non-coherent integration gains are greater than the number of
pulses. This is an interesting feature of non-coherent integration gains for SW2
m
and SW4 targets. That is, the non-coherent integration varies with N PulInt
where
m is a number that can be significantly larger than unity. In fact, for SW2 type
targets m varies between about 0.8 and 2.5 depending upon N PulInt and Pd . For
SW4 targets it varies between about 0.75 and 1.7.
Sometimes it is convenient to have a rule of thumb for computing noncoherent integration gain, instead of having to rely on figures like Figure 16. A
good rule of thumb for SW0/SW5, SW1 and SW3 targets is that the integration
0.8
gain is equal to N PulInt
.
CUMULATIVE DETECTION PROBABILITY
The third technique we examine for increasing detection probability is
through the use of multiple detection attempts. Although we group it with
integration techniques, we don’t analyze it in terms of its effect on SNR. We
focus more on its effects in terms of Pd , and Pfa . The premise behind the
multiple detection attempt approach is that if we attempt to detect the target
several times, we will increase the overall detection probability. We can
formally state the multiple detection problem as follows.
If we check for threshold crossing several on several occasions, what is
the probability that the signal-plus-noise voltage for a target will cross the
threshold at least once. Thus, suppose, for example, we check for a threshold
crossing on 3 occasions. We want to determine the probability of a threshold
crossing on any 1, 2 or 3 of the occasions.
To compute the appropriate probabilities we must use probability theory.
To develop the technique we start by considering the events of the occurance of
a threshold crossing on two occasions. We denote these two events as
©2005 M. C. Budge, Jr
48
D1 - Threshold crossing on occasion 1
D2 - Threshold crossing on occasion 2.
If we form the event
D = D1 D2
(206)
where
denotes the union then D is the event consisting of a threshold
crossing on occasion 1, or occasion 2, or occasions 1 and 2. Since D is the
event of interest to us we want to find the probability that it will occur. That is
we want
P  D   P  D1
D2  .
(207)
From probability theory we can write
P  D   P  D1
D2   P  D1   P  D2   P  D1 D2 
(208)
where D1 D2 represents the intersection of D1 and D2 and is the event
consisting of a threshold crossing on occasion 1 and 2. The first two probability
terms on the right side, P  D1  and P  D2  are computed using (106), (109),
(110), (192), (193), (194), (195) or (196) depending upon the target type and
whether or not non-coherent integration is used.
To compute the third term, P  D1
D2  , we need to make an assumption
about the events D1 and D2 . Specifically, we must assume that they are
independent. This, in turn places a limitation on when we can use cumulative
detection concepts. Specifically, we should only use cumulative detection
concepts on a scan-to-scan basis. If we do, we are assured that the detection
events on successive scans are independent. If we try to use cumulative
detection concepts on a pulse-to-pulse basis we must be careful. For a
SW0/SW5, signal-plus-noise will be independent on a pulse-to-pulse basis
because the “randomness” is due solely to noise, which is indeed independent
on a pulse-to-pulse basis. Signal-plus-noise will also be independent on a
pulse-to-pulse basis for SW2 and SW4 targets since both the signal and noise
are independent on a pulse-to-pulse basis (by definition). Signal-plus-noise will
not be independent on a pulse-to-pulse basis for SW1 and SW3 targets. This
stems from the fact that the signal is random but is correlated on a pulse-topulse basis. Thus, the signal-plus-noise will be correlated on a pulse-to-pulse
basis. Since the signal-plus-noise is correlated on a pulse-to-pulse basis it
can’t, by definition, be independent on a pulse-to-pulse basis. Further, since
the events D1 and D2 depend upon signal-plus-noise, they are also not
independent on a pulse-to-pulse basis.
If D1 and D2 are independent we can write
P  D1
D2   P  D1  P  D2 
(209)
and
©2005 M. C. Budge, Jr
49
P  D   P  D1   P  D2   P  D1  P  D2  .
(210)
As an example, suppose P  D1   P  D2   0.9 . Using (210) we would obtain
P  D   P  D1   P  D2   P  D1  P  D2 
 0.9  0.9  0.9  0.9  1.8  0.81  0.99
.
(211)
While (210) is reasonably easy to implement for two events, its extension
to many events is difficult. In order to set the stage for such an extension we
consider a different means of determining P  D  . We begin by observing that
S  Di
Di
(212)
where S is the universe and Di is the complement of Di . Di contains all
elements that are in S but not in Di . By the definition of Di we note that Di
and Di are mutually exclusive. We also note that P  S   1 . With this we get
P  S   1  P  Di
Di   P  Di   P  Di 
(213)
and
P  Di   1  P  Di  .
(214)
To proceed with the derivation we let
D  D1 D2
(215)
and
S D
D
D   D1 D2 
1
D2  .
(216)
From (213) we get
1  P  D1
D2   P  D1 D2 
(217)
By DeMogan’s Law we can write
D
1
D2   D1 D2
(218)
and
1  P  D1
D2   P  D1
D2  .
(219)
Now, since D1 and D2 are independent so are D1 and D2 . If we use this along
with (215) we can write
P  D   1  P  D1  P  D2  .
(220)
Finally, making use of (214) we obtain
©2005 M. C. Budge, Jr
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2
P  D   1  1  P  D1   1  P  D2    1   1  P  Dk  
(221)
k 1
We can now generalize (221) to any number of events. Specifically, if
D  D1 D2
where D1 , D2 , D3
D3
DN
(222)
DN are independent then
N
P  D   1   1  P  Dk  
(223)
k 1
As an example of the use of (221) or (223) we consider the previous example
wherein P  D1   P  D2   0.9 . With this we get
2
P  D   1   1  P  Dk    1  1  P  D1   1  P  D2  
k 1
(224)
 1  1  0.9 1  0.9   1  0.1  0.1  0.99
We now want to restate (224) in terminology more directly related to
detection probability. To that end we write
N
Pdcum  1   1  Pdk 
(225)
k 1
where Pdcum is the cumulative detection probability over N pulses and Pdk is the
detection probability on the kth scan.
In addition to increasing detection probability, the use of cumulative
detection techniques also increases false alarm probability. In fact, if we
consider the false alarm case we can express (225) as
Pfacum  1   1  Pfak 
N
(226)
k 1
In the case of false alarm probability we usually have that Pfak  Pfa k  1, N 
and write
Pfacum  1  1  Pfa 
N
If we further recall that Pfa
(227)
1 we can write
Pfacum  NPfa .
(228)
Equation (228) tells us that when we use cumulative detection concepts we
should compute the individual Pdk ’s using Pfak  Pfacum N where Pfacum is the
desired false alarm probability.
As a rule of thumb, you should not invoke cumulative detection concepts
in a fashion that allows any Pdk to be such that the SNR per scan falls below 10
©2005 M. C. Budge, Jr
51
to 13 dB. If the SNR is below 10 to 13 dB the radar will likely not be able to
establish track on a detection. If this is stated in terms of Pdk , you should never
let any Pdk to fall below about 0.5.
RADAR LOSSES
For our last radar range equation-related topic we want to address radar
losses. In our previous discussions we have indicated various situations that
cause radar losses. We now want to summarize some of these. We will not
address all possible loss terms since their number can be very large. See the
text and Skolnik’s Radar Handbook for a more complete discussion of losses

Transmit Losses – Typically associated with the waveguides and
other components between the power amplifier and the antenna.
These are typically 1 to 2 dB in a well-designed radar.

Receive Losses – Typically associated with the waveguides and other
components between the antenna and RF amplifier. These are also
typically 1 to 2 dB for a well-designed radar. If the noise figure is
referenced to the antenna terminals, receive losses are included in the
noise figure.

Atmospheric Losses – These are losses due to atmospheric
absorption by the atmosphere. They are dependent upon the radar
operating frequency, the range to the target and the elevation angle of
the target relative to the radar. Your author has graphs depicting
atmospheric losses. So does Skolnik’s Radar Handbook.

Scanning or Beamshape Loss – This loss term accounts for the fact
that, as the beam scans across the target, the signal amplitudes of
the pulses coherently or non-coherently integrated varies. Because
of the, the full integration gain of the integrator can’t be realized.
From the Skolnik Radar Handbook typical values are
o
1.6 dB for a scanning, fan beam radar
o
3.2 dB for a thinner beam, scanning radar
o
3.2 dB for a phased array radar wherein the beams of a search
sector overlap at the 3-dB beam positions.

Range-Gate Straddling Loss – If the radar samples in range at a rate
of once per range resolution cell the loss is usually taken to be 3 dB.

Doppler Straddling Loss – The loss associated with forming the
Doppler dimension of a range-Doppler map. Its particular value
depends upon the specific Doppler processor implementation but
typical values are 1 to 2 dB.

Collapsing Loss – If the coherent or non-coherent integrator
integrates only noise over some if its integration time (due to the fact
that the beam has moved fairly far off of the target) the radar will
incur a loss that is given by
©2005 M. C. Budge, Jr
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Lc 
nm
n
where n is the number of pulses containing signal-plus-noise and m
is the number of pulses containing only noise.

Signal Processing Loss – If the radar uses an MTI with a staggered
PRF waveform, and a good MTI and PRF stagger design, it will suffer 1
to 2 dB signal processing loss.

Miscellaneous Loss – Radar designers and analysts usually include
an additional 1 to 2 dB loss to account for various factors they forgot
to consider.
As an example of the above losses we will consider the 13 dB loss from
Homework 4.

Transmit Losses – We will be conservative an assume a transmit loss
of 2 dB

Receive Losses – Since the noise figure was large at 8 dB we will
assume that the receive losses are included in the noise figure.

Atmospheric Losses – We will be conservative on the atmospheric
losses and consider the losses at 0º elevation. From Figure 8.18a of
your text (page 523) the losses at about 75 Km and a frequency of 5
GHz are about 2.2 dB.

Scanning or Beamshape Loss – Our elevation beamwidth of 5º is on
the border of what we could call a fan beam. Therefore, we will be
conservative and not consider it a fan beam. In this case, we will
include scanning losses of 3.2 dB.

Range-Gate Straddling Loss – We will include the standard rangegate straddling loss of 3 dB.

Doppler Straddling Loss – The radar we have specified thus far only
uses single pulse processing with a narrow pulse. Therefore, we will
ignore Doppler straddling loss.

Collapsing Loss – In Homework 10 we limited ourselves to noncoherently integrating only the number of pulses we expect in a
beam. Thus, we can assume no collapsing loss.

Signal Processing Loss – Again, since we have assumed only single
pulse processing, we will have no signal processing loss

Miscellaneous Loss – We will assume a miscellaneous loss factor of 1
dB.
The losses above are summarized in Table 1. As can be seen, the total losses
sum to 11.4 dB instead of the 13 dB we assumed. We will use the reduction in
losses to help us reduce the peak transmit power.
©2005 M. C. Budge, Jr
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Table 1 – Loss Budget for Homework 4
Loss Term
Value
Transmit
2.0
Receive
0.0
Atmospheric
2.2
Scanning
3.2
Range Gate Straddling
3.0
Doppler Straddling
0.0
Collapsing
0.0
Signal Processing
0.0
Miscellaneous
1.0
Total
11.4
This completes our study of the radar range equation. As was
indicated at the start of this class, the radar range equation is an equation that
appears to be very simple but, in fact, is very complex. In closing this part of
the course we will summarize some of the observations associated with the
radar range equation
RADAR RANGE EQUATION THOUGHTS
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.
©2005 M. C. Budge, Jr
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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
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.
©2005 M. C. Budge, Jr
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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
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.
©2005 M. C. Budge, Jr
56
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

2
av
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
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.
©2005 M. C. Budge, Jr
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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
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
©2005 M. C. Budge, Jr
58
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
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,
©2005 M. C. Budge, Jr
59

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 .
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/SW5, SW1 and SW3 targets we can
approximate I  n  by the equation I  n   n0.8 . 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
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
©2005 M. C. Budge, Jr
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applies only to SW0/SW5, 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
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/SW5 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.
©2005 M. C. Budge, Jr
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