DETECTION THEORY
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-plus-noise 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.
NOISE IN RECEIVERS
We will characterize noise for the two most common types of receiver
implementations. The first receiver configuration is illustrated in Figure 1 and
it 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 2 and is termed
the baseband representation. In this configuration, the signal processing is
implemented at baseband. The IF configuration is common to older radars and
the baseband representation is common to more modern radars that use digital
signal processing.
RF Amp
Mixer
IF Amp
Matched
Filter
Signal
Processor
n IF  t 
cos ct  IF t 
Figure 1 – IF Receiver/Signal Processor Representation
In the IF representation, the noise is represented by
n IF  t   N  t  cos IF t  φ  t  
©2005 M. C. Budge, Jr
(1)
1
where n IF  t  n  t  , N  t  and φ  t  are random processes. If we expand (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
(2)
 n I  t  cos IF t  nQ  t  sin IF t
where n I  t  and nQ  t  are also random processes. In (2), 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 
nQ = nQ  t 
t t1
t t1
and
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)
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 t1
and φ  φ  t 
t t1
are
independent.
From probability and random variables1 if x and y are real random
variables,
r  x2  y2
(4)
y
φ  tan 1   ,
x
(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 
(6)
In our case x  n I , y  nQ , r  N and φ  φ . Thus, we have
1
N  n2I  nQ2
(7)
n 
φ  tan 1  Q 
 nI 
(8)
E.g. Papoulis, A. “Probability, Random Variables, and Stochastic Processes” McGraw-Hill
©2005 M. C. Budge, Jr
2
and
 
f Nφ  N ,    Nf n I nQ  N cos  , N sin   U  N  rect 
 2

.

(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
 nI2  nQ2  2 2
1
e
2 2
2 2
 1
 n2
e Q

   2
2 2


.
(10)
If we use this in (9) with nI  N cos  and nQ  N sin  we get
N
f Nφ  N ,   
2
2
N

2 2
e
e

 N 2 cos2   N 2 sin 2 

  
U  N  rect 

 2 
2 2
  
U  N  rect 

 2 
.
(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
(12)
and
f φ    


f Nφ  N ,  dN 
1
 
rect 
2
 2

.

(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
1
1
we note from (11), (12) and (13) that
f Nφ  N ,    f N  N  f φ   ,
(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 (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

(15)
2
©2005 M. C. Budge, Jr
3
In (15), the term on the third line is zero because n I  n I  t 
t t1
and nQ = nQ  t 
t t1
are independent and zero-mean.
cos IF t
Matched
Filter
Signal
Processor
nB t 
sin IF t
Figure 2 – Baseband Receiver/Signal Processor Representation
In the baseband configuration of Figure 2 we represent the noise at the
signal processor as a complex random process of the form
nB t  
1
 n I  t   jn Q  t  
2
(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 t1
and nQ = nQ  t 
t t1
are independent. The variances of
n I  t  and nQ  t  are both equal to  2 . The constant of 1
2 is included to
provide consistency between the noises in the baseband and IF receiver
representations. The power in n B  t  is given by (making use of 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
(17)
We note that we can write n B  t  in polar form as
nB t  
N t 
2
e jφt 
(18)
where
©2005 M. C. Budge, Jr
4
N  t   n2I  t   nQ2  t 
(19)
 n t  
φ  t   tan 1  Q
.
n
t


 I

(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.
SIGNAL IN RECEIVERS
We now want to turn our attention to developing a representation of the
signals at the output of the signal processor of the receiver. Consistent with the
noise case, we want to consider both IF and baseband receiver configurations.
Thus, for our analyses we will use Figures 1 and 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, 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
(21)
where
©2005 M. C. Budge, Jr
5
s I  t   S  t  cos θ  t 
(22)
sQ  t   S  t  sin θ  t  .
(23)
and
The baseband signal model is
sB t  
1
s I  t   jsQ t   .
2
(25)
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 t1
and
θ  θ  t  t t are independent.2
1
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.
SW0/SW5
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. This means that we let
S t   S .
(26)
With this 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
.
(27)
We have made a large number of assumptions concerning the statistical properties of the signal
and noise. A natural questions 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
2
theorem, we can reasonable assume that n I  t  and n Q  t  are Gaussian. Further, if we enforce
reasonable constraints on the bandwidth of receiver components we can reasonable assume the
independence requirements. 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 processes 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 stationary process and that φ  t  is uniform on   ,   and
is wide-sense stationary. Both of the latter assumptions are valid for practical radars and
targets.
©2005 M. C. Budge, Jr
6
Consistent with the constant RCS assumption we have chosen the phase, θ , as
a random variable (still uniform on   ,   ). This means that s I and s Q are
also random 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
(28)
We note that the random variables s I and s Q are not independent since they
are “tied” together through the random variable θ .
The signal power is given by

PsIF  E s2IF  t   E  S cos θ cos IF t  S sin θ sin IF t 
2
 S 2 E cos2 θcos2 IF t  S 2 E sin 2 θsin 2 IF t

.
(29)
 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 
 2
 2



  d

.
(30)
1
2
 cos  d  2
Similarly, we get
E sin 2 θ 
1
2
(31)
and
E cos θ sin θ  0 .
(32)
Substituting (30), (31) and (32) into (29) 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
(33)
From (25) the baseband signal model is
sB  t   sB 
1
S
s I  jsQ  
 cos θ  j sin θ .

2
2
(34)
The signal power is
S
S2
 S
PsB  E sBsB   E 
 cos θ  j sin θ  cos θ  j sin θ    PsIF .
2
 2
 2
©2005 M. C. Budge, Jr
(34)
7
SW1/SW2
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   .
(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 (35). That is
fP  p  
where PS 
1  p PS
e
U  p
PS
PT GT GR 2
 4 
3
R4 L
(36)
 AV
(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
(38)
Which is recognized as a Rayleigh density function. This, combined with the
fact that θ  t  in (21) is uniform, and the assumption that S  S  t 
and
t t
1
θ  θ  t  t t are independent, leads to the interesting observation that the signal
1
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
(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
f s I  s   f sQ  s  
2
1
e  s 2 PS .
2 PS
Furthermore, s I  s I  t  t t and sQ  sQ  t 
1
(40)
t t1
are independent.
The signal power is given by
©2005 M. C. Budge, Jr
8

PsIF  E s 2IF  t   E  s I  t  cos IF t  sQ  t  sin IF t 
 E s 2I  t  cos 2 IF t  E sQ2  t  sin 2 IF t
2

.
(41)
 E s I  t  sQ  t  cos IF t sin IF t
Invoking the independence of s I  s I  t 
t t1
and sQ  sQ  t 
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 .
(42)
The baseband representation of the signal is
sB  t  
S  t  jθ t 
1
s I  t   jsQ  t   
e

2
2
(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


(44)
as expected.
SW3/SW4
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   .
(45)
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 (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
(46)
(47)
From random variable theory it can be shown that the signal amplitude, S  t  ,
is governed by the density function
©2005 M. C. Budge, Jr
9
fS  S  
2 S 3  S 2 PS
e
U S  .
PS2
(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
(49)
and
sB  t  
S  t  jθ t 
1
s I  t   jsQ  t   
e .

2
2
(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
.
(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.
SIGNAL-PLUS-NOISE IN RECEIVERS
Now that we have characterizations for signals and for 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  .
(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   ,
(53)
 V  t  cos IF t  ψ  t  
and if we are using the baseband representation we would write
v B  t    s I  t   n I T    j  s I  t   n I  t  
 v I  t   jv Q  t 

V t 
2
©2005 M. C. Budge, Jr
e
.
(54)
ψt 
10
In either representation, the primary variable of interest is the magnitude of the
signal-plus-noise voltage, V  t  , since this is the quantity used in computing
detection probability. We will compute the other quantities as needed.
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.
SW1/SW2
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 .
(55)
With this we get
f vI  v  
1
2  PS   2 
e

 v 2 2 PS  2

.
(56)
By similar reasoning we get
1
f vQ  v   f v I  v  
2  PS  
2

e

 v 2 2 PS  2

.
Since s I  s I  t  t t , n I  n I  t  t t , sQ  sQ  t 
1
1
(57)
t t1
and nQ  nQ  t 
mutually independent, v I  t   v I  t  t t and vQ  t   vQ  t 
1
t t1
t t1
are
are independent.
This, with the 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  
©2005 M. C. Budge, Jr
(58)
11
SW0/SW5
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   .
(59)
We then use
f V V  
 

f Vψθ V , ,  d d
(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  
(61)
where we have made use of (27). If we expand (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 .
(62)
According to the conditional density of (59) we want to consider (62) for
the specific value of θ   . If we do this we get
v IF  t    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
.
(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 

1

t t1
are
independent  S cos  n I  and S sin   nQ are also independent. With this we
can write


f v I vQ vI , vQ θ   
1
2

  vI  S cos    vQ  S sin 

2
e
2

2

2 2
.
(64)
If we invoke the discussions on page 2 (Equations (4), (5) and (6)) we can write
©2005 M. C. Budge, Jr
12

f Vψ V , θ     Vf v I vQ V cos ,V sin θ   U V  rect 
 2

.

(65)
If we substitute from (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



.

(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


(67)
Finally we can use
f θ   
1
  
rect 

2
 2 
(68)
along with (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

.

(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

.
(70)
We want to first consider the integral with respect to  . That is,
1
  S ,V  
2

e
2VS cos   2 2


rect 
 2
1

 d 
2


2VS cos  
e
2 2
d
(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 .
(72)
If we make the change of variables      the integral becomes
  S ,V  
©2005 M. C. Budge, Jr
1
2
2
 VS 
VS cos   2
e
d  I 0  2 
0
 
(73)
13
where I 0  x  is a modified Bessel function of the first kind.
If we substitute (73) into (70) the latter becomes
 V V 2  S 2 
f V V    2 e

 VS  V
 2 I0  2  e

 
V
  VS   1
U V   I 0  2  
     2
2 2
2
S
2

2

  

 rect  2  d 

(74)
U V 
2
where the last step derives from the fact that the integral with respect to  is
equal to one. Equation (74) is the desired result, which is the density function
of V  t  .
SW3/SW4
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 signal, we will again use the
joint/conditional density approach. In this case we note that the if signal-plusnoise 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  
.
(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 ,  
(76)
and
f V V  
  

f VSψθ V , S , ,  d d dS .
(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



.

(78)
Further, since S  t  and θ  t  are, by definition, independent, we can write
©2005 M. C. Budge, Jr
14
 2S 3 2
 1
  
fSθ  S ,   fS  S  f θ     2 e  S PSU  S   
rect 
 .
 2  
 PS
  2
(79)
If we substitute (78) and (79) into (76) we get
f VSψθ V , S , ,   
V
2
e
2
 
U V  rect 

 2 
 V 2  S 2  2VS cos   2 2


 2S 3 2
 1
  
  2 e  S PSU  S   
rect 

 2  
 PS
  2
.
(80)
From (77) we can write
f V V  
V

e V
2
2
  2 S 3  S 2

  2 e   S ,V  U  S  ds  U V 
  PS

2 2
(81)
where

1
1

PS 2 2
(82)
and
  S ,V  
 
1
e
 2   
2VS cos   2 2
2
 

rect 
 2

 
 rect 

 2

 d d .

(83)
We recognize (83) as the same double integral of (70). Thus, using the work
from pages 13 and 14 we get
 VS 
  S ,V     S ,V   I 0  2 
 
(84)
and
f V V  

V

e V
2
V
2
e V
2
2 2
2
2
2
  2 S 3  S 2  VS 

  2 e I 0  2 U  S  ds  U V 
 
  PS

  2 S 3  S 2  VS  
  2 e I 0  2  ds  U V 
  
 0 PS
.
(85)
To complete the calculation of f V V  we must compute the integral

  2 s 3e s I 0   s  ds
2
(86)
0
where
 V 2 .
(87)
It turns out that Maple was able to compute the integral as
©2005 M. C. Budge, Jr
15

1 
 2  2
1

e
 2  4 
4
.
(88)
With this f V V  becomes
f V V  
2V V 2
e
PS2 2
2 2
1 
 2  2
 2 1 
e
   4 
4

 U V 

(89)
which, after manipulation can be written as
f V V  
 2
2V
2
 PS 
2
 2
PSV 2  V 2  2 2  PS 
2


U V  .

e
2
2


P



S 

(90)
Now that we have completed the characterization of noise, signal and
signal-plus-noise we are ready to attack the detection problem.
DETECTION PROBABILITY
A functional block diagram of the detection process is illustrated in
Figure 3. It consists of a magnitude detector and a threshold. The magnitude
detector determines the magnitude of the signal coming from the signal
processor and the threshold is a binary decision that outputs a detection signal
if the signal magnitude is above some threshold or a no-detection if the signal
magnitude is below the threshold.
Signal
Processor
v t 
n t 
Amplitude
Detector
V  t  or V 2  t 
N  t  or N 2  t 
Threshold
Device
Detect
No Detect
Figure 3 – Block Diagram of Detector and Threshold
The magnitude detector can be a square-law detector or a linear detector.
Both variants are illustrated functionally in Figure 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 signal processor output and then adding them.
©2005 M. C. Budge, Jr
16
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.
Signal
Processor
x
v t 
2
V2 t 
V t 
IF Detector
vI t 
x2
Signal
Processor
V2 t 

vQ  t 
x2
x
V t 
Baseband Detector
Figure 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 N 2  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.
Since both N  t  and V  t  are random processes we must use concepts
from random process 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 or 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
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
©2005 M. C. Budge, Jr
17
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 
(91)
Pfa - false alarm probability  P  N  T target not present  .
(92)
and
where V  V  t 
and N  N  t 
t t1
t t1
is the signal-plus-noise voltage evaluated at a specific time
is the noise voltage evaluated at a specific time. This 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.
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   T 
(93)
Pfa  P  N  t   T 
(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   T 2 
(95)
Pfa  P  N 2  t   T 2  .
(96)
and
From probability theory we can write

Pd   f V  v  dv or Pd 
T
©2005 M. C. Budge, Jr


T
f V2  v  dv
(97)
2
18
and

Pfa   f N  n  dn or Pfa 
T

 f  n  dn
(98)
N2
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 (97) and (98).
In homework 7 you showed that if x  y then
f x  x   2 xf y  x 2  .
(99)
If we write

Pd   f V  v  dv
(100)
T
we can use (99) to write


Pd   f V  v  dv   2vf V2  v 2  dv .
T
(101)
T
If we make the change of variables x  v 2 we can write

Pd   f V  v  dv 
T

 f  x  dx .
V2
(102)
T2
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 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 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 (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
©2005 M. C. Budge, Jr
19
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 5 – Probability Density Functions for Noise and Signal-plus-noise
If we use (12) in (98) we get


Pfa   f N  n  dn  
T
T
n

en
2
2
2 2
dn  e T
2
2 2
.
(103)
In this equation we define
TNR 
T2
2 2
(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 .
(105)
We compute detection probability for the three target classes by
substituting (58), (74) and (90) into (102). The results for SW0/SW5 is
©2005 M. C. Budge, Jr
20
Pd 

1
1  erf
2

TNR  SNR
 TNR  SNR 
e

4  SNR
2


1 2
1  TNR  SNR 

4 SNR


TNR  SNR
16SNR

2

.



(106)
where
SNR 
PS
2

S2
2 2
(107)
is the signal-to-noise ratio that one would compute from the radar range
equation and
erf  x  
2

x
e
u2
du
(108)
0
is one form of the error function.
The detection probability equations for the SW1/SW2 case and for the
SW3/SW4 case are, respectively
Pd  eTNR  SNR1
(109)
 2  SNR TNR   2TNR  2 SNR 
.
Pd  1 
e
2
 2  SNR  

(110)
and
In (109) and (110) SNR is the signal-to-noise ratio computed from the radar
range equation.
Figure 6 contains plots of Pd versus SNR for the three target types with
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
©2005 M. C. Budge, Jr
21
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.
0.9999
0.9995
0.999
0.998
0.995
0.99
0.98
0.95
P
d
0.9
0.8
0.7
SW0/SW5
0.6
SW1/SW2
0.5
SW3/SW4
0.4
0.3
0.2
0.1
0.05
5
10
15
20
25
SNR (dB)
30
35
40
Figure 6 – Pd versus SNR for Three Target Types and Pfa=10-6
©2005 M. C. Budge, Jr
22
COMPUTATION OF Pfa
One of the parameters in the detection probability equations is
threshold-to-noise ratio, TNR . As indicated in (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 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 (a expenditure of hardware).
In order to minimize wasted radar resources we wish to minimize the
possibility 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 (106), (109) and (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
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
©2005 M. C. Budge, Jr
(111)
23
t1
t3
t2
Threshold, T
v t   N t 
T1
T2
time
Figure 3-7 – Illustration of False Alarm Time
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 modulation bandwidth.
Pfa
It has been the author’s experience that the above method of determining
not very accurate. While it would be possible to place the requisite number
of caveats on (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 base 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
(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
©2005 M. C. Budge, Jr
24
a duration, T , that is less than the PRI, T . In a search radar, T might be
only slightly less than 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
.
(113)
Each of the range samples provides represents a chance that a false alarm will
occur.
In a time period of T fa the radar will transmit
N pulse 
T fa
(114)
T
pulses. Thus, the number of range samples (and thus chances for false alarm)
that one has over the time period of T fa is
N fa  N R N pulse .
(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 .
(116)
In either case, the false alarm probability would be give by (112).
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
(117)
From T and  R we get
©2005 M. C. Budge, Jr
25
300 s
 300 .
1 s
(118)
N fa  N R N pulse  300  5000  1.5  106
(119)
NR 
T
 R

This results in
and
Pfa 
1
1

 6.667  107 .
6
N fa 1.5  10
©2005 M. C. Budge, Jr
(120)
26