3.0 DETECTION THEORY
3.1 INTRODUCTION
In some of our radar range equation problems we looked at finding the detection
range based on SNRs of 13 and 20 dB. We now want to develop some of the theory that
explains the use of these particular SNR values. More specifically, we want to examine
the concept of detection probability, PD . Our need to study detection from a probabilistic
perspective stems from the fact that the signals we deal with are noise-like. From our
studies of RCS we found that, in practice, the signal return looks random. In fact,
Swerling has convinced us that we should use statistical models to represent target
signals. Also, in addition to the target signal we found that the signals in the radar
contain a noise component, which also needs to be dealt with using the concepts of
random variables, random processes and probabilities.
To develop the requisite equations for detection probability we need to develop a
mathematical characterization of the target signal, the noise signal and the target-plusnoise signal at various points in the radar. From the above, we will use the concepts of
random variables and random processes to characterize these quantities. We start with a
characterization of noise and then progress to the target and target-plus-noise signals.
3.2 NOISE IN RECEIVERS
We will characterize noise for the two most common types of receiver
implementations. The first receiver configuration is illustrated in Figure 3-1 and is
termed the IF representation. In this representation, both the matched filter and the
signal processor are implemented at some intermediate frequency, or IF. The second
receiver configuration is illustrated in Figure 3-2 and is termed the baseband
representation. In this configuration, the signal processing is implemented at baseband.
The IF configuration is common in older radars and the baseband representation is
common in modern radars that use digital signal processing.
Figure 3-1 – IF Receiver/Signal Processor Representation
3.2.1 IF Configuration
In the IF representation, the noise is represented by
n IF  t   N  t  cos IF t  φ  t  
©2011 M. C. Budge, Jr
(3-1)
1
where n IF  t  , N  t  and φ  t  are random processes. If we expand Equation (3-1) using
trig identities we get
n IF  t   N  t  cos φ  t  cos IF t  N  t  sin φ  t  sin IF t
(3-2)
 n I  t  cos IF t  nQ  t  sin IF t
where n I  t  and nQ  t  are also random processes. In Equation (3-2), we stipulate that
n I  t  and nQ  t  are joint, wide-sense stationary, zero-mean, equal variance, Gaussian
random processes. They are also such that the random variables n I  n I  t  t t and
1
nQ = nQ  t 
t t1
are independent. The variances of n I  t  and nQ  t  are both equal to  2 .
The above statements mean that the density functions of n I  t  and nQ  t  are equal and
given by
f nI  n   f nQ  n  
2
1
en
 2
2 2
.
(3-3)
We will now show that N  t  is Rayleigh and φ  t  is uniform on   ,   . We
will further argue that the random variables N  N  t  t t and φ  φ  t  t t are
1
1
independent.
From probability and random variables1 if x and y are real random variables,
r  x2  y2
(3-4)
y
φ  tan 1   ,
x
(3-5)
and
where tan 1  y x  denotes the four-quadrant arctangent, then the joint density of r and
φ can be written in terms of the joint density of x and y as
 
f rφ  r,    rf xy  r cos  , r sin   U  r  rect 
 2

.

(3-6)
In our case x  n I , y  nQ , r  N and φ  φ . Thus, we have
N  n2I  nQ2
(3-7)
n 
φ  tan 1  Q 
 nI 
(3-8)
and
1
E.g. Papoulis, A. “Probability, Random Variables, and Stochastic Processes” McGraw-Hill
©2011 M. C. Budge, Jr
2
  
f Nφ  N ,    Nf n I nQ  N cos  , N sin   U  N  rect 
.
 2 
(3-9)
Now, since n I and nQ are independent, Gaussian and zero-mean with equal variance
2
 1
f nI nq  nI , nQ   f nI  nI  f nQ  nQ   
e  nI
  2
2
2
2
 n  n  2
1
e I Q
2
2
2 2
 1
 n2
e Q

   2
2 2


.
(3-10)
If we use this in Equation (3-9) with nI  N cos  and nQ  N sin  we get
N
f Nφ  N ,   
2
e
2
N

2 2
e

 N 2 cos2   N 2 sin 2 

  
U  N  rect 

 2 
2 2
  
U  N  rect 

 2 
.
(3-11)
 N 2 2 2
From random variable theory, we can find the marginal density from the joint density by
integrating with respect to the variable we want to eliminate. Thus,
fN  N   


f Nφ  N ,   d 
N

2
e N
2
U N 
2 2
(3-12)
and
f φ    


f Nφ  N ,  dN 
1
  
rect 
.
2
 2 
(3-13)
This proves the assertion that N  t  is Rayleigh and φ  t  is uniform on   ,   . To
prove that the random variables N  N  t  t t and φ  φ  t  t t are independent we note
1
1
from Equations (3-11), (3-12) and (3-13) that
f Nφ  N ,    f N  N  f φ   ,
(3-14)
which means that N and φ are independent.
Since we will need it later, we want to find the noise power out of the signal
processor. Since n IF  t  is wide-sense stationary we can use Equation (3-2) and write

PnIF  E n 2IF  t   E  n I  t  cos IF t  nQ  t  sin IF t 
 E n 2I  t  cos 2 IF t  E nQ2  t  sin 2 IF t
 2 E n I  t  nQ  t  cos IF t sin IF t
2

(3-15)
2
©2011 M. C. Budge, Jr
3
In Equation (3-15), the term on the third line is zero because n I  n I  t  t t and
1
nQ = nQ  t 
t t1
are independent and zero-mean.
3.2.2 Baseband Configuration
Figure 3-2 – Baseband Receiver/Signal Processor Representation
In the baseband configuration of Figure 3-2 we represent the noise at the signal
processor output as a complex random process of the form
nB t  
1
 n I  t   jn Q  t  
2
(3-16)
where n I  t  and nQ  t  are joint, wide-sense stationary, zero-mean, equal variance,
Gaussian random processes. They are also such that the random variables n I  n I  t  t t
1
and nQ = nQ  t  t t are independent. The variances of n I  t  and nQ  t  are both equal to
1
 . The constant of 1 2 is included to provide consistency between the noises in the
baseband and IF receiver configurations. The power in n B  t  is given by (making use of
2
the properties of n I  t  and nQ  t  )
1
 1

PnB  E n B  t  nB  t   E 
n I  t   jn Q  t  
n I  t   jn Q  t   


2
 2

1
1
 E n 2I  t   E nQ2  t 
.
2
2
  2  PnIF
(3-17)
We note that we can write n B  t  in polar form as
nB t  
N t 
2
e jφt 
(3-18)
where
©2011 M. C. Budge, Jr
4
N  t   n2I  t   nQ2  t 
(3-19)
 n t  
φ  t   tan 1  Q
.
 nI t  
(3-20)
and
It will be noted that the definitions n I  t  . nQ  t  , N  t  and φ  t  are consistent between
the IF and baseband representations. This means that both representations are equivalent
in terms of the statistical properties of the noise. We will reach the same conclusion for
the signal. The ramifications of this are that the detection and false alarm performance of
both types of receiver/signal processor configurations will be the same. Thus, the future
detection and false alarm probability equations that we derive will be applicable to either
receiver configuration.
It should be noted that, if the receiver you are analyzing is not of one of the two
forms indicated above, the ensuing detection and false alarm probability equations may
not be applicable to it. The most notable exception to the two representations above is
the case where the receiver uses only the I or Q channel in baseband processing. While
this is not a common receiver configuration, it is sometimes used. In this case, one would
need to derive a different set of detection and false alarm probability equations that would
be specifically applicable to the configuration.
3.3 SIGNAL IN RECEIVERS
3.3.1 Introduction and Background
We now want to turn our attention to developing a representation of the signals at
the output of the signal processor. Consistent with the noise case, we want to consider
both IF and baseband receiver configurations. Thus, for our analyses we will use Figures
3-1 and 3-2 but replace n  t  with s  t  , N  t  with S  t  , φ  t  with θ  t  , n I  t  with
s I  t  and nQ  t  with sQ  t  .
We will need to develop three signal representations: one for SW0/SW5 targets,
one for SW1/SW2 targets and one for SW3/SW4 targets. We have already
acknowledged that the SW1 through SW4 target RCS models are random process
models. To be consistent with this, and consistent with what happens in an actual radar,
we will also use a random process model for the SW0/SW5 target RCS.
Since the target RCS models are random processes we must also represent the
target voltage signals in the radar (henceforth termed the target signal) as random
processes. To that end, the IF representation of the target signal is
s IF  t   S  t  cos IF t  θ  t    s I  t  cos IF t  sQ  t  sin IF t
(3-21)
where
©2011 M. C. Budge, Jr
5
s I  t   S  t  cos θ  t 
(3-22)
sQ  t   S  t  sin θ  t  .
(3-23)
and
The baseband signal model is
sB t  
1
s I  t   jsQ t   .
2
(3-24)
It will be noted that both of the signal models are consistent with the noise voltage model
of the previous sections.
Consistent with the noise model, we assume that S  S  t  t t and θ  θ  t  t t are
1
1
independent.2
At this point we need to develop separate signal models for the different types of
targets because the signal amplitude fluctuations, S  t  , of each are governed by different
models.
3.3.2 Signal Model for SW0/SW5 Targets
For the SW0/SW5 target case we assume that the target RCS is constant. This
means that the target power, and thus the target signal amplitude, will be constant. With
this, we let
S t   S .
(3-25)
The IF signal model becomes
s IF  t   S cos IF t  θ   S cos θ cos IF t  S sin θ sin IF t
 s I cos IF t  sQ sin IF t
.
(3-26)
We introduce the random variable θ to force s IF  t  to be a random process. We
specifically choose θ uniform on   ,   3. This means that s I and s Q are also random
2
We have made a large number of assumptions concerning the statistical properties of the signal and noise.
A natural question is: Are the assumptions reasonable? The best answer to this question is that we design
radars so that the assumptions are satisfied. In particular, we endeavor to make the receiver and signal
processor linear. Because of this and the central limit theorem, we can reasonably assume that n  t  and
I
n Q  t  are Gaussian. Further, if we enforce reasonable constraints on the bandwidth of receiver components
we can reasonably assume the independence requirements are valid. The stationarity requirements are
easily satisfied if we assume that the receiver gains don’t change with time. We enforce the zero-mean
assumption by using bandpass filters to eliminate DC components. For signals, we won’t need the
Gaussian requirement. However, we will need the stationarity, zero-mean and other requirements. These
are usually satisfied for signals based on the same assumptions as for noise, and by the requirement that the
target RCS is a wide sense stationary process, and that   t  is uniform on   ,   and is wide-sense
stationary. Both of the latter assumptions are valid for practical radars and targets.
©2011 M. C. Budge, Jr
6
variables (rather than random processes). s IF  t  is a random process because of the
presence of the IF t term.
The density functions of s I and s Q are the same and are given by
f sI  s   f sQ  s  
 s 
rect   .
 2S 
 S 2  s2
1
(3-27)
We cannot assert that the random variables s I and s Q are independent because we have
no means of showing that fs s  sI , sQ   fs  sI  fs  sQ  .
I Q
I
Q
The signal power is given by

PsIF  E s2IF  t   E  S cos θ cos IF t  S sin θ sin IF t 
 S 2 E cos2 θcos2 IF t  S 2 E sin 2 θsin 2 IF t
2

.
(3-28)
 2 SE cos θ sin θ cos IF t sin IF t
In the above we can write
E cos θ 
2

 cos
2
 fθ   d 

1

2

 cos

2
 1
  
rect 
  d
 2  
 2


.
(3-29)
1
 cos  d  2
2
Similarly, we get
E sin 2 θ 
1
2
(3-30)
and
E cos θ sin θ  0 .
(3-31)
Substituting Equations (3-29), (3-30) and (3-31) into Equation (3-28) results in
S2
1 2
21
2
.
PsIF  S   cos IF t  S   sin IF t  2  0 S cos IF t sin IF t 
2
2
2
2
(3-32)
From Equation (3-24) the baseband signal model is
sB  t   sB 
1
S
s I  jsQ  
 cos θ  j sin θ .

2
2
(3-33)
The signal power is
3
This model is actually very consistent with what happens in the actual radar. Specifically, the phase of
the signal is random.
©2011 M. C. Budge, Jr
7
S
S2
 S
PsB  E sBsB   E 
 cos θ  j sin θ  cos θ  j sin θ    PsIF . (3-34)
2
 2
 2
3.3.3 Signal Model for SW1/SW2 Targets
For the SW1/SW2 target case we have already stated that the target RCS is
governed by the density function
f σ   
1
 AV
e   AV U   .
(3-35)
Since the power is a direct function of the RCS (from the radar range equations), the
signal power at the signal processor output has a density function that is the same form as
Equation (3-35). That is
fP  p  
where PS 
1  p PS
e
U  p
PS
PT GT GR 2
 4 
3
R4 L
(3-36)
 AV
(3-37)
From random variable theory it can be shown that the signal amplitude, S  t  , is
governed by the density function
fS  S  
S  S 2 2 PS
e
U S  .
PS
(3-38)
Which is recognized as a Rayleigh density function. This, combined with the fact that
θ  t  in Equation (3-21) is uniform, and the assumption that S  S  t  t t and θ  θ  t  t t
1
1
are independent, leads to the interesting observation that the signal model for a
SW1/SW2 target is of the same form as the noise model. That is, the IF signal model for
a SW1/SW2 target is of the form
s IF  t   S  t  cos IF t  θ  t    s I  t  cos IF t  sQ  t  sin IF t
(3-39)
where S  t  is Rayleigh and θ  t  is uniform on   ,   . If we adapt the results from our
noise study we arrive at the conclusion that s I  t  and sQ  t  are Gaussian with the
density functions
2
1
e  s 2 PS .
2 PS
f s I  s   f sQ  s  
(3-40)
Furthermore, s I  s I  t  t t and sQ  sQ  t  t t are independent.
1
1
The signal power is given by
©2011 M. C. Budge, Jr
8

PsIF  E s 2IF  t   E  s I  t  cos IF t  sQ  t  sin IF t 
2

 E s 2I  t  cos 2 IF t  E sQ2  t  sin 2 IF t
.
(3-41)
 E s I  t  sQ  t  cos IF t sin IF t
Invoking the independence of s I  s I  t  t t and sQ  sQ  t 
1
t t1
and the fact that s I  t  and
sQ  t  are zero mean and have equal variances of PS leads to the conclusion that
PsIF  PS .
(3-42)
The baseband representation of the signal is
sB  t  
S  t  jθt 
1
s I  t   jsQ  t   
e

2
2
(3-43)
where the various terms are as defined above. The power in the baseband signal
representation can be written as
1
 1

PsB  E s B  t  sB  t   E 
s I  t   js Q  t  
s I  t   js Q  t   


2
 2

1
 E s 2I  t   E sQ2  t   PS
2


(3-44)
as expected.
3.3.4 Signal Model for SW3/SW4 Targets
For the SW3/SW4 target case we have already stated that the target RCS is
governed by the density function
f σ   
4
2
 AV
e 2  AV U   .
(3-45)
Since the power is a direct function of the RCS (from the radar range equation), the
signal power at the signal processor output has a density function that is the same form as
Equation (3-45). That is
fP  p  
where PS 
4 p 2 p PS
e
U  p
PS2
PT GT GR 2
 4 
3
R4 L
 AV
(3-46)
(3-47)
From random variable theory it can be shown that the signal amplitude, S  t  , is
governed by the density function
©2011 M. C. Budge, Jr
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fS  S  
2 S 3  S 2 PS
e
U S  .
PS2
(3-48)
Unfortunately, this is about as far as we can carry the signal model development for the
SW3/SW4 case. We can invoke the previous statements and write
s IF  t   S  t  cos IF t  θ  t    s I  t  cos IF t  sQ  t  sin IF t
(3-49)
and
sB  t  
S  t  jθt 
1
s I  t   jsQ  t   
e .

2
2
(3-50)
However, we don’t know the form of s I  t  and sQ  t  . Furthermore, deriving its form
has proven very laborious and elusive.
We can find the power in the signal from

PsIF  E s2IF  t   E S  t  cos IF t  θ  t  

2
1
 PsB  E s B  t  s  t   E S2  t   PS
2
.
(3-51)

B
We will need to deal with the inability to characterize s I  t  and sQ  t  when we consider
the characterization of signal-plus-noise.
3.4 SIGNAL-PLUS-NOISE IN RECEIVERS
3.4.1 General Formulation
Now that we have characterizations for the signal and noise we want to develop
characterizations for the sum of signal and noise. That is, we want to develop the
appropriate density functions for
v t   s t   n t  .
(3-52)
If we are using the IF representation we would write
v IF  t   s IF  t   n IF  t 
 S  t  cos IF t  θ  t    N  t  cos IF t  φ  t   ,
(3-53)
 V  t  cos IF t  ψ  t  
and if we are using the baseband representation we would write
©2011 M. C. Budge, Jr
10
v B  t    s I  t   n I  t    j  sQ  t   nQ  t  
 v I  t   jv Q  t 

V t 
2
.
(3-54)
eψt 
In either representation, the primary variable of interest is the magnitude of the signalplus-noise voltage, V  t  , since this is the quantity used in computing detection
probability. We will compute the other quantities as needed, and as we are able.
We will begin the development with the easiest case, which is the SW1/SW2
case, and progress through the SW0/SW5 case to the most difficult, which is the
SW3/SW4 case.
3.4.2 Signal-plus-Noise Model for SW1/SW2 Targets
For the SW1/SW2 case we found that the real and imaginary parts of both the
signal and noise were zero-mean, Gaussian random processes. Since Gaussian random
processes are relatively easy to work with we will use the baseband representation to
derive the density function of V  t  . Since s I  t  and n I  t  are Gaussian, v I  t  will
also be Gaussian. Since s I  t  and n I  t  are zero-mean, v I  t  will also be zero-mean.
Finally, since s I  t  and n I  t  are independent, the variance of v I  t  will equal to the
sum of the variances of s I  t  and n I  t  . That is
 v2  PS   2 .
(3-55)
With this we get
1
f vI  v  
2  PS  
2

e

 v 2 2 PS  2

.
(3-56)
By similar reasoning we get
1
f vQ  v   f v I  v  
e
2  PS   2 

 v 2 2 PS  2

.
(3-57)
Since s I  s I  t  t t , n I  n I  t  t t , sQ  sQ  t  t t and nQ  nQ  t  t t are mutually
1
1
1
1
independent, v I  t   v I  t  t t and vQ  t   vQ  t  t t are independent. This, with the
1
1
above and our previous discussions of noise and the SW1/SW2 signal model, leads to the
observation that V  t  is Rayleigh. Thus the density of V  t  is
f V V  
V 2 2 PS  2 
V
e
U V  .
2
PS  
©2011 M. C. Budge, Jr
(3-58)
11
3.4.3 Signal-plus-Noise Model for SW0/SW5 Targets
Since s I  t  and sQ  t  are not Gaussian for the SW0/SW5 case when we add
them to n I  t  and nQ  t  the resulting v I  t  and vQ  t  will not be Gaussian. This
means that directly manipulating v I  t  and vQ  t  to obtain the density function of
V  t  will be difficult. Therefore, we take a different tack and invoke some properties of
joint and marginal density functions. Specifically, we use
f Vψθ V , ,    f Vψ V , θ    f θ   .
(3-59)
We then use
f V V  
 

f Vψθ V , ,  d d
(3-60)
 
to get the density function of V  t  . This procedure involves some tedious math but it is
math that can be found in many books on random variable theory.
To execute the derivation we start with the IF representation and write
v IF  t   S cos IF t  θ   N  t  cos IF t  φ  t  
(3-61)
where we have made use of Equation (3-26). If we expand Equation (3-61) and group
terms we get
v IF  t    S cos θ  n I  t   cos IF t   S sin θ  nQ  t   sin IF t .
(3-62)
According to the conditional density of Equation (3-59) we want to consider
Equation (3-62) for the specific value of θ   . If we do this we get
v IF  t  θ   S cos   n I  t   cos IF t   S sin   nQ  t   sin IF t
v I  t  cos IF t  vQ  t  sin IF t
.
(3-63)
 V  t  cos IF t  ψ  t  
With this we note that  S cos   n I  t   and  S sin   nQ  t   are Gaussian random
variables with means of S cos and S sin  . They also have the same variance of  2 .
Further more, since n I  n I  t  t t and nQ  nQ  t  t t are independent  S cos  n I  and
1
 S sin   n  are also independent.
Q


f v I vQ vI , vQ θ   
1
2
1
With this we can write
2
e
2

  vI  S cos    vQ  S sin 


2

2 2
.
(3-64)
If we invoke the discussions related to Equations (3-4), (3-5) and (3-6), we can write
©2011 M. C. Budge, Jr
12
 
f Vψ V , θ     Vf v I vQ V cos ,V sin θ   U V  rect 
.
 2 
(3-65)
If we substitute from Equation (3-64) we get
f Vψ V , θ    
V
2
e
2
 
U V  rect 
.
 2 
2
2
 V cos  S cos    V sin  S sin    2 2


(3-66)
We can manipulate the exponent to yield
f Vψ V , θ    
V
2
e
2
 
U V  rect 

 2 
 V 2  S 2  2VS cos   2 2


(3-67)
Finally we can use
f θ   
1
  
rect 

2
 2 
(3-68)
along with Equation (3-59) to write
f Vψθ V , ,   
V
 2  
2
e
2
 V 2  S 2  2VS cos   2 2



U V  rect 
 2

 
 rect 

 2

 . (3-69)

For the next step we need to integrate f Vψθ V , ,  with respect to  and  to
derive the desired marginal density, f V V  . That is (after a little manipulation)
 V V 2  S 2 
f V V    2 e


1
 
e
2
 2   

U V  

2 2
2VS cos   2 2

rect 
 2

 
 rect 

 2

 d d

.
(3-70)
We want to first consider the integral with respect to  . That is,
1
  S ,V  
2

e
2VS cos   2 2

1
 
rect 
 d 
2
 2 

2VS cos  
e
2 2
d
(3-71)

We recognize that the integrand is periodic with a period of 2 and that the integral is
performed over a period. This means that we can evaluate the integral over any period.
Specifically, we will choose the period from  to 2   . With this we get
1
  S ,V  
2
2 

e2VS cos  
2 2
d .
(3-72)
If we make the change of variables      the integral becomes
  S ,V  
©2011 M. C. Budge, Jr
1
2
2
e
0
VS cos   2
 VS 
d  I 0  2 
 
(3-73)
13
where I 0  x  is a modified Bessel function of the first kind.
If we substitute Equation (3-73) into Equation (3-70) the latter becomes
 V V 2  S 2 
f V V    2 e

  VS   1
U V   I 0  2  
     2
2 2
 VS  V 2  S 2 
 2 I0  2  e

 
V

 
 rect  2
 
 d 
 
(3-74)
2 2
U V 
where the last step derives from the fact that the integral with respect to  is equal to
one. Equation (3-74) is the desired result, which is the density function of V  t  .
3.4.4 Signal-plus-Noise Model for SW3/SW4 Targets
As with the SW0/SW5 case, s I  t  and sQ  t  are not Gaussian for the SW3/SW4
case. Thus, when we add them to n I  t  and nQ  t  the resulting v I  t  and vQ  t  will
not be Gaussian. This means that directly manipulating v I  t  and vQ  t  to obtain the
density function V  t  will be difficult. Based on our experience with the SW0/SW5
case, we will again use the joint/conditional density approach. We note that the IF
signal-plus-noise voltage is given by
v IF  t   S  t  cos IF t  θ  t    N  t  cos IF t  φ  t  
 V  t  cos IF t  ψ  t  
.
(3-75)
In this case we will need to find the joint density of V  t  , S  t  , ψ  t  and θ  t  and
perform the appropriate integration to get the marginal density of V  t  . More
specifically, we will find
f VSψθ V , S , ,    f Vψ V , θ   , S  S  f Sθ  S ,  
(3-76)
and
f V V  
  

f VSψθ V , S , ,  d d dS .
(3-77)
  
We can draw on our work from the SW0/SW5 case to write
f Vψ V , S  S , θ    
V
2
2
e

U V  rect 
 2
 V 2  S 2  2VS cos   2 2



.

(3-78)
Further, since S  t  and θ  t  are, by definition, independent, we can write
 2S 3 2
 1
 
fSθ  S ,   fS  S  f θ     2 e  S PSU  S   
rect 
 2
 PS
  2
©2011 M. C. Budge, Jr

 .

(3-79)
14
If we substitute Equations (3-78) and (3-79) into Equation (3-76) we get
f VSψθ V , S , ,   
V
2
2
e
 
U V  rect 

 2 
 V 2  S 2  2VS cos   2 2


 2 S 3  S 2 PS
 1
 
 2 e
U S 
rect 
 2
 PS
  2



.
(3-80)
From Equation (3-77) we can write
f V V  
V
2
e
  2S 3  S 2

  2 e   S ,V U  S  dS  U V 
  PS

V 2 2 2
(3-81)
where

1
1

PS 2 2
(3-82)
and
  S ,V  
 
1
e
 2   
2VS cos   2 2
2
 

rect 
 2

 
 rect 

 2

 d d .

(3-83)
We recognize Equation (3-83) as the same double integral of Equation (3-70). Thus,
using the discussions related to Equation (3-73) we get
 VS 
  S ,V     S ,V   I 0  2 
 
(3-84)
and
f V V  

V

eV
2
V
2
2
eV
2
  2S 3  S 2  VS 

  2 e I 0  2  U  S  dS  U V 
 
  PS

.

3

2 
2
2
S
VS


2
 S
  2 e I 0  2  dS  U V 
P
  
0 S
2 2
(3-85)
To complete the calculation of f V V  we must compute the integral

  2 s 3e s I 0   s  ds
2
(3-86)
0
where
 V 2 .
(3-87)
It turns out that Maple was able to compute the integral as

1 
 2  2
1

e
 2  4 
©2011 M. C. Budge, Jr
4
.
(3-88)
15
With this f V V  becomes
2
2V
f V V   2 2 e V
PS 
2 2
1 
 2  2
 2 1 
e
   4 
4

 U V 

(3-89)
which, after manipulation can be written as
f V V  
 2
2V
2
 PS 
2
 2
PSV 2  V 2  2 2  PS 
U V  .
 2 
e
 2 2  PS  

(3-90)
Now that we have completed the characterization of noise, signal and signal-plusnoise we are ready to attack the detection problem.
3.5 DETECTION PROBABILITY
3.5.1 Introduction
A functional block diagram of the detection process is illustrated in Figure 3-3. It
consists of an amplitude detector and a threshold device. The amplitude detector
determines the magnitude of the signal coming from the signal processor and the
threshold device is a binary decision device that outputs a detection declaration if the
signal magnitude is above some threshold, or a no-detection declaration if the signal
magnitude is below the threshold.
Figure 3-3 – Block Diagram of the Detector and Threshold Device
3.5.2 Amplitude Detector Types
The amplitude detector can be a square-law detector or a linear detector. Both
variants are illustrated functionally in Figure 3-4 for the IF implementation and the
baseband implementation. In the IF implementation, the detector consist, functionally, of
a diode followed by a low-pass filter. If the circuit is designed such that it uses small
voltage levels, the diode will be operating in its low signal region and will result in a
square-law detector. If the circuit is designed such that it uses large voltage levels the
diode will be operating in its large signal region and will result in a linear detector.
For the baseband case, the digital hardware (which we assume in the baseband
signal processing case) will actually form the square of the magnitude of the complex
signal out of the signal processor by squaring the real and imaginary components of the
©2011 M. C. Budge, Jr
16
signal processor output and then adding them. The result of this operation will be a
square-law detector. In some instances the detector also performs a square root to form
the magnitude.
Figure 3-4 – IF and Baseband Detectors – Linear and Square Law
In either the IF or baseband representation the output of the square-law detector
will be N2  t  when only noise is present at the signal processor output and V 2  t  when
signal-plus-noise is present at the signal processor output. For the linear detector the
output will be N  t  when only noise is present at the signal processor output and V  t 
when signal-plus-noise is present at the signal processor output.
3.5.3 Detection Logic
Since both N  t  and V  t  are random processes we must use concepts from
random processes theory to characterize the performance of the detection logic. In
particular, we will use probabilities to characterize the performance of the detection logic.
Since we have two signal conditions (noise only and signal-plus-noise) and two outcomes
from the threshold check we have four possible events to consider:
1. signal-plus-noise  threshold – detection
2. signal-plus-noise < threshold – missed detection
3. noise  threshold – false alarm
4. noise < threshold – no false alarm
©2011 M. C. Budge, Jr
17
Of the above, the two desired events are 1 and 4. That is, we want to detect targets when
they are present and we don’t want to detect noise when targets are not present. Since
events 1 and 2 are related and events 3 and 4 are related we only find probabilities
associated with events 1 and 3. We term the probability of the first event occurring the
detection probability and the probability of the third even occurring the false alarm
probability. In equation form
Pd - detection probability  P  V  T target present 
(3-91)
Pfa - false alarm probability  P  N  T target not present  .
(3-92)
and
where V  V  t  t t is the signal-plus-noise voltage evaluated at a specific time and
1
N  N  t  t t is the noise voltage evaluated at a specific time.
1
The above definition carries some subtle implications. First, when one finds
detection probability it is tacitly assumed that the target return is present at the time the
output of the threshold device is checked. Likewise, when one finds false alarm
probability it is tacitly assumed that the target return is not present at the time the output
of the threshold device is checked.
In practical applications it is more appropriate to say: At the time the output of the
threshold device is checked the probability that there will be a threshold crossing is equal
to Pd if the signal contains a target signal and Pfa if the signal does not contain a target
signal. In typical applications the output of the threshold device will be checked at times
separated by a pulse width and will result in many checks per PRI.
It will be noted that the above probabilities are conditional probabilities. In
normal practice we don’t explicitly use the conditioning and write
Pd  P  V  T 
(3-93)
Pfa  P  N  T 
(3-94)
and
and recognize that we should use signal-plus-noise when we assume the target is present
and noise only when we assume that the target is not present.
The above assumes that the detector preceding the threshold device is a linear
detector. If the detector is a square law detector the appropriate equations would be
Pd  P  V 2  T 2 
(3-95)
Pfa  P  N 2  T 2  .
(3-96)
and
©2011 M. C. Budge, Jr
18
3.5.4 Calculation of Pd and Pfa
From probability theory we can write

Pd   f V  v  dv or Pd 
T


T
f V2  v  dv
(3-97)
2
and

Pfa   f N  n  dn or Pfa 
T


f N2  n  dn
(3-98)
T2
In the above T is the threshold voltage level and T 2 is the threshold expressed as
normalized power.
To avoid having to use two sets of Pd and Pfa equations we will digress to show
that we can compute them using either of the integrals of Equations (3-97) and (3-98).
It can be shown4 that if x  y and y  0 then
f x  x   2 xf y  x 2  .
(3-99)
If we write

Pd   f V  v  dv
(3-100)
T
we can use Equation (3-99) to write


T
T
Pd   f V  v  dv   2vf V2  v 2  dv .
(3-101)
If we make the change of variables x  v 2 we can write

Pd   f V  v  dv 
T

 f  x  dx .
V2
T
(3-102)
2
Similar results apply to Pfa and indicate that one can use either form to compute
detection and false alarm probability.
If we examine the equations for Pd and Pfa we note that both are integrals over
the same limits. This integration is illustrated graphically in Figure 3-5. It will be noted
that Pd and Pfa are areas under their respective density functions to the right of the
threshold value. Thus, increasing the threshold decreases the probabilities and decreasing
the threshold increases the probabilities. This is not exactly what we want. Ideally, we
want to select the threshold so that we have Pfa  0 and Pd  1 . However this is not
4
Papoulis, A. “Probability, Random Variables, and Stochastic Processes” McGraw-Hill
©2011 M. C. Budge, Jr
19
possible and we therefore usually choose the threshold as some sort of tradeoff between
Pd and Pfa . In fact, what we actually do is choose the threshold to achieve a certain Pfa
and find other means of increasing Pd .
If we refer to Equation (3-12) the only parameter that affects f N  n  is the noise
power,  2 . While we have some control over this via noise figure and effective noise
bandwidth, executing this control can be very expensive. On the other hand, f V  v  is
dependent upon both PS and  2 . Thus, this gives us some degree of control. In fact,
what we usually try to do is affect both f N  n  and f V  v  by increasing PS and
decreasing  2 . The net result of this is that we try to maximize SNR.
Figure 3-5 – Probability Density Functions for Noise and Signal-plus-noise
3.5.4.1 False Alarm Probability
If we use Equation (3-12) in Equation (3-98) we get


Pfa   f N  n  dn  
T
©2011 M. C. Budge, Jr
T
n

2
en
2
2 2
dn  e T
2
2 2
.
(3-103)
20
In this equation we define
T2
TNR 
2 2
(3-104)
as the threshold-to-noise ratio. As indicated earlier, we usually select a desired Pfa and,
from this derive the required TNR as
TNR   ln Pfa .
(3-105)
3.5.4.2 Detection Probability
We can compute detection probability for the three target classes by substituting
Equations (3-58), (3-74) and (3-90) into Equation (3-102). The results for SW0/SW5
targets is
Pd 

1
1  erf
2

TNR  SNR

2
 
1 2
e
1  TNR  SNR 

4  SNR 
4 SNR



TNR  SNR

TNR  SNR
16SNR

2

.



(3-106)
where
SNR 
PS
2

S2
2 2
(3-107)
is the signal-to-noise ratio that one would compute from the radar range equation and
erf  x  
2

x
e
u2
du
(3-108)
0
is one form of the error function.5
The detection probability equations for the SW1/SW2 case and for the SW3/SW4
case are, respectively
Pd  eTNR  SNR1
(3-109)
 2  SNR TNR   2TNR  2 SNR 
Pd  1 
.
e
2
 2  SNR  

(3-110)
and
In Equations (3-109) and (3-110) SNR is the signal-to-noise ratio computed from the
radar range equation.
5
This Equation (3-106) should only be used for cases where SNR is larger than TNR.
©2011 M. C. Budge, Jr
21
3.5.5 Pd Behavior vs. Target Type
Figure 3-6 contains plots of Pd versus SNR for the three target types and
Pfa  106 , a typical value. It is interesting to note the Pd behavior for the three target
types. In general, the SW0/SW5 target provides the largest Pd for a given SNR, the
SW1/SW2 target provides the lowest Pd and the SW3/SW4 is somewhere between the
other two. With some thought this makes sense. For the SW0/SW5 target model the
only thing affecting a threshold crossing is the noise (since the RCS of the target is
constant). For the SW1/SW2 the target RCS can fluctuate considerably, thus both noise
and RCS fluctuation affects the threshold crossing. The standard assumption for the
SW3/SW4 model is that it consists of a predominant (presumably constant RCS) scatterer
and several smaller scatterers. Thus, the threshold crossing for the SW3/SW4 target is
affected somewhat by RCS fluctuation, but not to the extent of the SW1/SW2 target.
It is interesting to note that the SNR required for Pd  0.5 , with Pfa  106 , on a
SW1/SW2 target is about 13 dB. This same SNR gives a Pd  0.9 on a SW0/SW5 target.
To obtain a Pd  0.9 on a SW1/SW2 target requires a SNR of about 21 dB. These
numbers are the origin of the 13 dB and 20 dB SNR numbers we used in our radar range
equation studies.
©2011 M. C. Budge, Jr
22
Figure 3-6 - Pd vs. SNR for Three Target Types and Pfa  106
3.6 DETERMINATION OF FALSE ALARM PROBABILITY
One of the parameters in the detection probability equations is threshold-to-noise
ratio, TNR . As indicated in Equation (3-105), TNR   ln Pfa , where Pfa is the false
alarm probability. False alarm probability is set by system requirements.
In a radar, false alarms result in wasted radar resources (energy, timeline and
hardware) in that every time a false alarm occurs the radar must expend resources
determining that it did, in fact, occur. Said another way, every time the output of the
amplitude detector exceeds the threshold, T , a detection is recorded. The radar data
processor does not know, a priori, whether the detection is a target detection or the result
©2011 M. C. Budge, Jr
23
of noise (i.e. a false alarm). Therefore, the radar must verify each detection. This usually
requires transmission of another pulse and another threshold check (an expenditure of
time and energy). Further, until the detection is verified, it must be carried in the
computer as a valid target detection (an expenditure of hardware).
In order to minimize wasted radar resources we wish to minimize the probability
of a false alarm. Said another way, we want to minimize Pfa . However, we can’t set Pfa
to an arbitrarily small value because this will increase TNR and reduce detection
probability, Pd (see Equations (3-106), (3-109) and (3-110)). As a result we set Pfa to
provide an acceptable number of false alarms within a given time period. This last
statement provides the criterion normally used to compute Pfa . Specifically, one states
that Pfa is chosen to provide an average of one false alarm within a time period that is
termed the false alarm time, T fa . T fa is usually set by some criterion that is driven by
radar resource limitations.
The classical method of determining Pfa is based strictly on timing. This can be
explained with the help of Figure 3-7 which contains a plot of noise at the output of the
amplitude detector. The horizontal line labeled “Threshold, T” represents the detection
threshold voltage level. It will be noted that the noise voltage is above the threshold for
three time intervals of length t1 , t2 and t3 . Further, the spacings between threshold
crossings are T1 and T2 . Since a threshold crossing constitutes a false alarm one can say
that over the interval T1 false alarms occur for a period of t1 . Likewise, over the interval
T2 false alarms occur for a period of t2 , and so forth. If we were to average all of the tk
we would have the average time that the noise is above the threshold, tk . Likewise, if we
were to average all of the Tk we would have the average time between false alarms; i.e.
the false alarm time, T fa . To get the false alarm probability we would take the ratio of tk
to T fa , i.e.
Pfa 
tk
.
T fa
(3-111)
Figure 3-7 – Illustration of False Alarm Time
©2011 M. C. Budge, Jr
24
While T fa is reasonably easy to specify, the specification of tk is not obvious.
The standard assumption is to set tk to the range resolution expressed as time,  R . For
an unmodulated pulse,  R is the pulse width. For a modulated pulse,  R is the
reciprocal of the modulation bandwidth.
It has been the author’s experience that the above method of determining Pfa not
very accurate. While it would be possible to place the requisite number of caveats on
Equation (3-111) to make it accurate, with modern radars this is not necessary.
The previously described method of determining Pfa was based on the
assumption that detections were recorded via hardware operating on a continuous-time
signal. In modern radars, detection is based on examining signals that have been
converted to the discrete-time domain by sampling or by and analog-to-digital converter.
This makes determination of Pfa easier, and more intuitively appealing, in that one can
deal with discrete events. With modern radars one computes the number of false alarm
chances, N fa , within the desired false alarm time, T fa , and computes the probability of
false alarm from
Pfa 
1
.
N fa
(3-112)
To compute N fa one needs to know certain things about the operation of the
radar. We will outline some thoughts along this line.
In a typical radar, the return signal from each pulse is sampled with a period equal
to the range resolution,  R , of the pulse. As indicated above, this would be equal to the
pulse width for an unmodulated pulse and the reciprocal of the modulation bandwidth for
a modulated pulse. These range samples are usually taken over the instrumented range,
T . In a search radar, T might be only slightly less than the PRI, T . However, for a
track radar, T , may be significantly less than T . With the above, we can compute the
number of range samples per PRI as
NR 
T
 R
.
(3-113)
Each of the range samples provides a chance that a false alarm will occur.
In a time period of T fa the radar will transmit
N pulse 
T fa
T
(3-114)
pulses. Thus, the number of range samples (and thus chances for false alarm) that one
has over the time period of T fa is
©2011 M. C. Budge, Jr
25
N fa  N R N pulse .
(3-115)
In some radars, the signal processor consists of several ( N Dop ), parallel Doppler
channels. This means that it will also contain N Dop amplitude detectors. Each amplitude
detector will generate N R range samples per PRI. Thus, in this case, the total number of
range samples in the time period T fa would be
N fa  N R N pulse N Dop .
(3-116)
In either case, the false alarm probability would be give by Equation (3-112).
3.6.1 Example
To illustrate the above, we consider a simple example. We have a search radar
that has a PRI of T  400 s . It uses a 50 s pulse with linear frequency modulation
(LFM) where the LFM bandwidth is 1 MHz. With this we get  R  1 s . We assume
that the radar starts its range samples one pulse-width after the transmit pulse and stops
taking range samples one pulse-width before the succeeding transmit pulse. From this we
get T  300 s . The signal processor is not a multi-channel Doppler processor. The
radar has a search scan time of TS  1 s and we desire no more than one false alarm every
two scans.
From the last sentence above we get T fa  2TS  2 s . If we combine this with the
PRI we get
N pulse 
T fa
T

2
 5000 .
400  106
(3-117)
From T and  R we get
300 s
 300 .
1 s
(3-118)
N fa  N R N pulse  300  5000  1.5  106
(3-119)
NR 
T
 R

This results in
and
Pfa 
1
1

 6.667  107 .
6
N fa 1.5  10
©2011 M. C. Budge, Jr
(3-120)
26