©2011 M. C. Budge, Jr 1

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, DP . 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.

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

cosIF IFt t t t n N φ (3-1)

©2011 M. C. Budge, Jr 2

where IF tn , tN and tφ are random processes. If we expand Equation (3-1) using

trig identities we get

cos cos sin sin

cos sin

IF IF IF

I IF Q IF

t t t t t t t

t t t t

n N φ N φ

n n (3-2)

where I tn and Q tn are also random processes. In Equation (3-2), we stipulate that

I tn and Q tn are joint, wide-sense stationary, zero-mean, equal variance, Gaussian

random processes. They are also such that the random variables 1

I I t tt

n n and

1

Q Q t tt

n = n are independent. The variances of I tn and Q tn are both equal to

2 .

The above statements mean that the density functions of I tn and Q tn are equal and

given by

2 221

2I Q

nf n f n e

n n . (3-3)

We will now show that tN is Rayleigh and tφ is uniform on , . We

will further argue that the random variables 1t t

t

N N and 1t t

t

φ φ are

independent.

From probability and random variables1 if x and y are real random variables,

2 2 r x y (3-4)

and

1tan

yφ

x, (3-5)

where 1tan 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

, cos , sin rect2

f r rf r r U r

rφ xy

. (3-6)

In our case Ix n , Qy n , r N and φ φ . Thus, we have

2 2

I Q N n n (3-7)

1tan

Q

I

nφ

n (3-8)

and

1 E.g. Papoulis, A. “Probability, Random Variables, and Stochastic Processes” McGraw-Hill

©2011 M. C. Budge, Jr 3

, cos , sin rect2I Q

f N Nf N N U N

Nφ n n

. (3-9)

Now, since In and Qn are independent, Gaussian and zero-mean with equal variance

2 22 2

2 2 2

22

2

2

1 1,

2 2

1

2

QI

I q I Q

I Q

nn

I Q I Q

n n

f n n f n f n e e

e

n n n n

. (3-10)

If we use this in Equation (3-9) with cosIn N and sinQn N we get

2 2 2 2 2

2 2

cos sin 2

2

2

2

, rect2 2

rect2 2

N N

N

Nf N e U N

Ne U N

Nφ

. (3-11)

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,

2 22

2, NN

f N f N d e U N

N Nφ (3-12)

and

1

, rect2 2

f f N dN

φ Nφ

. (3-13)

This proves the assertion that tN is Rayleigh and tφ is uniform on , . To

prove that the random variables 1t t

t

N N and 1t t

t

φ φ are independent we note

from Equations (3-11), (3-12) and (3-13) that

,f N f N f Nφ N 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 IF tn is wide-sense stationary we can use Equation (3-2) and write

22

2 2

2

cos sin

cos sin

2 cos sin

nIF IF I IF Q IF

I IF Q IF

I Q IF IF

P E t E t t t t

E t t E t t

E t t t t

2 2

n n n

n n

n n

(3-15)

©2011 M. C. Budge, Jr 4

In Equation (3-15), the term on the third line is zero because 1

I I t tt

n n and

1

Q Q t tt

n = n 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

1

2B I Qt t j t n n n (3-16)

where I tn and Q tn are joint, wide-sense stationary, zero-mean, equal variance,

Gaussian random processes. They are also such that the random variables 1

I I t tt

n n

and 1

Q Q t tt

n = n are independent. The variances of I tn and Q tn are both equal to

2 . The constant of 1 2 is included to provide consistency between the noises in the

baseband and IF receiver configurations. The power in B tn is given by (making use of

the properties of I tn and Q tn )

2

1 1

2 2

1 1

2 2

nB B B I Q I Q

I Q

nIF

P E t t E t j t t j t

E t E t

P

2 2

n n n n n n

n n . (3-17)

We note that we can write B tn in polar form as

2

j t

B

tt e

φNn (3-18)

where

©2011 M. C. Budge, Jr 5

2 2

I Qt t t N n n (3-19)

and

1tan

Q

I

tt

t

nφ

n. (3-20)

It will be noted that the definitions I tn . Q tn , tN and tφ 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 tn with ts , tN with tS , tφ with tθ , I tn with

I ts and Q tn with Q ts .

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

cos cos sinIF IF I IF Q IFt t t t t t t t s S θ s s (3-21)

where

©2011 M. C. Budge, Jr 6

cosI t t ts S θ (3-22)

and

sinQ t t ts S θ . (3-23)

The baseband signal model is

1

2B I Qt t j t s s s . (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 1t t

t

S S and 1t t

t

θ θ are

independent.2

At this point we need to develop separate signal models for the different types of

targets because the signal amplitude fluctuations, tS , 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

t SS . (3-25)

The IF signal model becomes

cos cos cos sin sin

cos sin

IF IF IF IF

I IF Q IF

t S t S t S t

t t

s θ θ θ

s s. (3-26)

We introduce the random variable θ to force IF ts to be a random process. We

specifically choose θ uniform on , 3. This means that Is and Qs 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 I

tn and

Q

tn 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 7

variables (rather than random processes). IF ts is a random process because of the

presence of the IFt term.

The density functions of Is and Qs are the same and are given by

2 2

1rect

2I Q

sf s f s

SS s

s s . (3-27)

We cannot assert that the random variables Is and Qs are independent because we have

no means of showing that ,I Q I QI Q I Qf s s f s f ss s s s

.

The signal power is given by

22

2 2 2 2 2 2

cos cos sin sin

cos cos sin sin

2 cos sin cos sin

sIF IF IF IF

IF IF

IF IF

P E t E S t S t

S E t S E t

SE t t

s θ θ

θ θ

θ θ

. (3-28)

In the above we can write

2 2 2

2

1cos cos cos rect

2 2

1 1cos

2 2

E f d d

d

θθ

. (3-29)

Similarly, we get

2 1sin

2E θ (3-30)

and

cos sin 0E θ θ . (3-31)

Substituting Equations (3-29), (3-30) and (3-31) into Equation (3-28) results in

2

2 2 2 21 1cos sin 2 0 cos sin

2 2 2sIF IF IF IF IF

SP S t S t S t t

. (3-32)

From Equation (3-24) the baseband signal model is

1

cos sin2 2

B B I Q

St j j s s s s θ θ . (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 8

2

cos sin cos sin22 2

sB B B sIF

S S SP E E j j P

s s θ θ θ θ . (3-34)

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

1

AV

AV

f e U

σ

. (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

1

Sp P

S

f p e U pP

P

(3-36)

where

2

3 44

T T RS AV

P G GP

R L

(3-37)

From random variable theory it can be shown that the signal amplitude, tS , is

governed by the density function

2 2 SS P

S

Sf S e U S

P

S

. (3-38)

Which is recognized as a Rayleigh density function. This, combined with the fact that

tθ in Equation (3-21) is uniform, and the assumption that 1t t

t

S S and 1t t

t

θ θ

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

cos cos sinIF IF I IF Q IFt t t t t t t t s S θ s s (3-39)

where tS is Rayleigh and tθ is uniform on , . If we adapt the results from our

noise study we arrive at the conclusion that I ts and Q ts are Gaussian with the

density functions

2 21

2

S

I Q

s P

S

f s f s eP

s s . (3-40)

Furthermore, 1

I I t tt

s s and

1Q Q t t

t

s s are independent.

The signal power is given by

©2011 M. C. Budge, Jr 9

22

2 2 2 2

cos sin

cos sin

cos sin

sIF IF I IF Q IF

I IF Q IF

I Q IF IF

P E t E t t t t

E t t E t t

E t t t t

s s s

s s

s s

. (3-41)

Invoking the independence of 1

I I t tt

s s and

1Q Q t t

t

s s and the fact that I ts and

Q ts are zero mean and have equal variances of SP leads to the conclusion that

sIF SP P . (3-42)

The baseband representation of the signal is

1

2 2

j t

B I Q

tt t j t e

θSs s s (3-43)

where the various terms are as defined above. The power in the baseband signal

representation can be written as

2 2

1 1

2 2

1

2

sB B B I Q I Q

I Q S

P E t t E t j t t j t

E t E t P

s s s s s s

s s

(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

2

2

4AV

AV

f e U

σ . (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

2

2

4Sp P

S

pf p e U p

P

P (3-46)

where

2

3 44

T T RS AV

P G GP

R L

(3-47)

From random variable theory it can be shown that the signal amplitude, tS , is

governed by the density function

©2011 M. C. Budge, Jr 10

2

3

2

2SS P

S

Sf S e U S

P

S

. (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

cos cos sinIF IF I IF Q IFt t t t t t t t s S θ s s (3-49)

and

1

2 2

j t

B I Q

tt t j t e

θSs s s . (3-50)

However, we don’t know the form of I ts and Q ts . Furthermore, deriving its form

has proven very laborious and elusive.

We can find the power in the signal from

22

2

cos

1

2

sIF IF IF

sB B B S

P E t E t t t

P E t t E t P

s S θ

s s S

. (3-51)

We will need to deal with the inability to characterize I ts and Q ts 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

t t t v s n . (3-52)

If we are using the IF representation we would write

cos cos

cos

IF IF IF

IF IF

IF

t t t

t t t t t t

t t t

v s n

S θ N φ

V ψ

, (3-53)

and if we are using the baseband representation we would write

©2011 M. C. Budge, Jr 11

2

B I I Q Q

I Q

t

t t t j t t

t j t

te

ψ

v s n s n

v v

V

. (3-54)

In either representation, the primary variable of interest is the magnitude of the signal-

plus-noise voltage, tV , 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 tV . Since I ts and I tn are Gaussian, I tv will

also be Gaussian. Since I ts and I tn are zero-mean, I tv will also be zero-mean.

Finally, since I ts and I tn are independent, the variance of I tv will equal to the

sum of the variances of I ts and I tn . That is

2 2

v SP . (3-55)

With this we get

2 22

2

1

2

S

I

v P

S

f v eP

v . (3-56)

By similar reasoning we get

2 22

2

1

2

S

Q I

v P

S

f v f v eP

v v . (3-57)

Since 1

I I t tt

s s ,

1I I t t

t

n n , 1

Q Q t tt

s s and

1Q Q t t

t

n n are mutually

independent, 1

I I t tt t

v v and

1Q Q t t

t t

v v are independent. This, with the

above and our previous discussions of noise and the SW1/SW2 signal model, leads to the

observation that tV is Rayleigh. Thus the density of tV is

2 22

2

SV P

S

Vf V e U V

P

V . (3-58)

©2011 M. C. Budge, Jr 12

3.4.3 Signal-plus-Noise Model for SW0/SW5 Targets

Since I ts and Q ts are not Gaussian for the SW0/SW5 case when we add

them to I tn and Q tn the resulting I tv and Q tv will not be Gaussian. This

means that directly manipulating I tv and Q tv to obtain the density function of

tV will be difficult. Therefore, we take a different tack and invoke some properties of

joint and marginal density functions. Specifically, we use

, , ,f V f V f Vψθ Vψ θθ . (3-59)

We then use

, ,f V f V d d

V Vψθ (3-60)

to get the density function of tV . 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

cos cosIF IF IFt S t t t t v θ N φ (3-61)

where we have made use of Equation (3-26). If we expand Equation (3-61) and group

terms we get

cos cos sin sinIF I IF Q IFt S t t S t t v θ n θ n . (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

cos cos sin sin

cos sin

cos

IF I IF Q IF

I IF Q IF

IF

t S t t S t t

t t t t

t t t

θv n n

v v

V ψ

. (3-63)

With this we note that cos IS t n and sin QS t n are Gaussian random

variables with means of cosS and sinS . They also have the same variance of 2 .

Further more, since 1

I I t tt

n n and

1Q Q t t

t

n n are independent cos IS n and

sin QS n are also independent. With this we can write

22 2cos sin 2

2

1,

2

I Q

I Q

v S v S

I Qf v v e

v v θ . (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 13

, cos , sin rect2I Q

f V Vf V V U V

Vψ v vθ θ . (3-65)

If we substitute from Equation (3-64) we get

2 2 2cos cos sin sin 2

2, rect

2 2

V S V SVf V e U V

Vψ θ . (3-66)

We can manipulate the exponent to yield

2 2 22 cos 2

2, rect

2 2

V S VSVf V e U V

Vψ θ (3-67)

Finally we can use

1

rect2 2

f

θ

(3-68)

along with Equation (3-59) to write

2 2 22 cos 2

2 2, , rect rect

2 22

V S VSVf V e U V

Vψθ. (3-69)

For the next step we need to integrate , ,f V Vψθ with respect to and to

derive the desired marginal density, f VV

. That is (after a little manipulation)

2 2 2

2

2

2

2 cos 2

2

1rect rect

2 22

V S

VS

Vf V e U V

e d d

V

. (3-70)

We want to first consider the integral with respect to . That is,

2 22 cos 2 2 cos 21 1, rect

2 2 2

VS VSS V e d e 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

22

2 cos 21,

2

VSS V e d

. (3-72)

If we make the change of variables the integral becomes

2

2

cos

0 2

0

1,

2

VS VSS V e d I

(3-73)

©2011 M. C. Budge, Jr 14

where 0I x is a modified Bessel function of the first kind.

If we substitute Equation (3-73) into Equation (3-70) the latter becomes

2 2 2

2 2 2

2

02 2

2

02 2

1rect

2 2

V S

V S

V VSf V e U V I d

V VSI e U V

V

(3-74)

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 tV .

3.4.4 Signal-plus-Noise Model for SW3/SW4 Targets

As with the SW0/SW5 case, I ts and Q ts are not Gaussian for the SW3/SW4

case. Thus, when we add them to I tn and Q tn the resulting I tv and Q tv will

not be Gaussian. This means that directly manipulating I tv and Q tv to obtain the

density function tV 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

cos cos

cos

IF IF IF

IF

t t t t t t t

t t t

v S θ N φ

V ψ. (3-75)

In this case we will need to find the joint density of tV , tS , tψ and tθ and

perform the appropriate integration to get the marginal density of tV . More

specifically, we will find

, , , , , ,f V S f V S f S VSψθ Vψ Sθθ S (3-76)

and

, , ,f V f V S d d dS

V VSψθ . (3-77)

We can draw on our work from the SW0/SW5 case to write

2 2 22 cos 2

2, , rect

2 2

V S VSVf V S e U V

Vψ S θ . (3-78)

Further, since tS and tθ are, by definition, independent, we can write

2

3

2

2 1, rect

2 2SS P

S

Sf S f S f e U S

P

Sθ S θ . (3-79)

©2011 M. C. Budge, Jr 15

If we substitute Equations (3-78) and (3-79) into Equation (3-76) we get

2 2 2

2

2 cos 2

2

3

2

, , , rect2 2

2 1rect

2 2S

V S VS

S P

S

Vf V S e U V

Se U S

P

VSψθ

. (3-80)

From Equation (3-77) we can write

2 2 2

32

2 2

2,V S

S

V Sf V e e S V U S dS U V

P

V (3-81)

where

2

1 1

2SP

(3-82)

and

22 cos 2

2

1, rect rect

2 22

VSS V e 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

0 2, ,

VSS V S V I

(3-84)

and

2 2 2

2 2 2

32

02 2 2

32

02 2 2

0

2

2

V S

S

V S

S

V S VSf V e e I U S dS U V

P

V S VSe e I dS U V

P

V

. (3-85)

To complete the calculation of f VV

we must compute the integral

23

0

0

2 ss e I s ds

(3-86)

where

2V . (3-87)

It turns out that Maple was able to compute the integral as

2

24

2

11

4e

. (3-88)

©2011 M. C. Budge, Jr 16

With this f VV

becomes

2 2 2

22 4

2 2 2

2 11

4

V

S

Vf V e e U V

P

V (3-89)

which, after manipulation can be written as

2 2222

2 22

22

22

SV PS

SS

V PVf V e U V

PP

V. (3-90)

Now that we have completed the characterization of noise, signal and signal-plus-

noise 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 17

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 2 tN when only noise is present at the signal processor output and 2 tV when

signal-plus-noise is present at the signal processor output. For the linear detector the

output will be tN when only noise is present at the signal processor output and tV

when signal-plus-noise is present at the signal processor output.

3.5.3 Detection Logic

Since both tN and tV 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 18

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

- detection probability target presentdP P T V (3-91)

and

- false alarm probability target not presentfaP P T N . (3-92)

where 1t t

t

V V is the signal-plus-noise voltage evaluated at a specific time and

1t t

t

N N is the noise voltage evaluated at a specific time.

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 dP if the signal contains a target signal and faP 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

dP P T V (3-93)

and

faP P T N (3-94)

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

2 2

dP P T V (3-95)

and

2 2

faP P T N . (3-96)

©2011 M. C. Budge, Jr 19

3.5.4 Calculation of dP and faP

From probability theory we can write

d

T

P f v dv

V or 2

2

d

T

P f v dv

V (3-97)

and

fa

T

P f n dn

N or 2

2

fa

T

P f n dn

N (3-98)

In the above T is the threshold voltage level and 2T is the threshold expressed as

normalized power.

To avoid having to use two sets of dP and faP 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 0y then

22f x xf xx y. (3-99)

If we write

d

T

P f v dv

V (3-100)

we can use Equation (3-99) to write

2

22d

T T

P f v dv vf v dv

V V. (3-101)

If we make the change of variables 2x v we can write

2

2

d

T T

P f v dv f x dx

V V. (3-102)

Similar results apply to faP and indicate that one can use either form to compute

detection and false alarm probability.

If we examine the equations for dP and faP we note that both are integrals over

the same limits. This integration is illustrated graphically in Figure 3-5. It will be noted

that dP and faP 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 0faP and 1dP . However this is not

4 Papoulis, A. “Probability, Random Variables, and Stochastic Processes” McGraw-Hill

©2011 M. C. Budge, Jr 20

possible and we therefore usually choose the threshold as some sort of tradeoff between

dP and faP . In fact, what we actually do is choose the threshold to achieve a certain faP

and find other means of increasing dP .

If we refer to Equation (3-12) the only parameter that affects f nN 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 vV is

dependent upon both SP and 2 . Thus, this gives us some degree of control. In fact,

what we usually try to do is affect both f nN and f vV

by increasing SP 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

2 2 2 22 2

2

n T

fa

T T

nP f n dn e dn e

N . (3-103)

©2011 M. C. Budge, Jr 21

In this equation we define

2

22

TTNR

(3-104)

as the threshold-to-noise ratio. As indicated earlier, we usually select a desired faP and,

from this derive the required TNR as

ln faTNR P . (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

2 2

11 erf

2

1 21

164 4

d

TNR SNR

P TNR SNR

TNR SNRe TNR SNR

SNRSNR SNR

. (3-106)

where

2

2 22

SP SSNR

(3-107)

is the signal-to-noise ratio that one would compute from the radar range equation and

2

0

2erf

x

ux e du

(3-108)

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

1TNR SNR

dP e

(3-109)

and

2 2

2

21

2

TNR SNR

d

SNR TNRP e

SNR

. (3-110)

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 22

3.5.5 dP Behavior vs. Target Type

Figure 3-6 contains plots of dP versus SNR for the three target types and 610faP , a typical value. It is interesting to note the dP behavior for the three target

types. In general, the SW0/SW5 target provides the largest dP for a given SNR, the

SW1/SW2 target provides the lowest dP 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 0.5dP , with 610faP , on a

SW1/SW2 target is about 13 dB. This same SNR gives a 0.9dP on a SW0/SW5 target.

To obtain a 0.9dP 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 23

Figure 3-6 - dP vs. SNR for Three Target Types and 610faP

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), ln faTNR P , where faP 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 24

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 faP . However, we can’t set faP

to an arbitrarily small value because this will increase TNR and reduce detection

probability, dP (see Equations (3-106), (3-109) and (3-110)). As a result we set faP to

provide an acceptable number of false alarms within a given time period. This last

statement provides the criterion normally used to compute faP . Specifically, one states

that faP is chosen to provide an average of one false alarm within a time period that is

termed the false alarm time, faT . faT is usually set by some criterion that is driven by

radar resource limitations.

The classical method of determining faP 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 1t , 2t and 3t . Further, the spacings between threshold

crossings are 1T and 2T . Since a threshold crossing constitutes a false alarm one can say

that over the interval 1T false alarms occur for a period of 1t . Likewise, over the interval

2T false alarms occur for a period of 2t , and so forth. If we were to average all of the kt

we would have the average time that the noise is above the threshold, kt . Likewise, if we

were to average all of the kT we would have the average time between false alarms; i.e.

the false alarm time, faT . To get the false alarm probability we would take the ratio of kt

to faT , i.e.

kfa

fa

tP

T . (3-111)

Figure 3-7 – Illustration of False Alarm Time

©2011 M. C. Budge, Jr 25

While faT is reasonably easy to specify, the specification of kt is not obvious.

The standard assumption is to set kt 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 faP 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 faP 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 faP easier, and more intuitively appealing, in that one can

deal with discrete events. With modern radars one computes the number of false alarm

chances, faN , within the desired false alarm time, faT , and computes the probability of

false alarm from

1

fa

fa

PN

. (3-112)

To compute faN 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

R

R

TN

. (3-113)

Each of the range samples provides a chance that a false alarm will occur.

In a time period of faT the radar will transmit

fa

pulse

TN

T (3-114)

pulses. Thus, the number of range samples (and thus chances for false alarm) that one

has over the time period of faT is

fa R pulseN N N . (3-115)

©2011 M. C. Budge, Jr 26

In some radars, the signal processor consists of several ( DopN ), parallel Doppler

channels. This means that it will also contain DopN amplitude detectors. Each amplitude

detector will generate RN range samples per PRI. Thus, in this case, the total number of

range samples in the time period faT would be

fa R pulse DopN N N N . (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 400 sT . It uses a 50 s pulse with linear frequency modulation

(LFM) where the LFM bandwidth is 1 MHz. With this we get 1 sR . 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 300 sT . The signal processor is not a multi-channel Doppler processor. The

radar has a search scan time of 1 sST and we desire no more than one false alarm every

two scans.

From the last sentence above we get 2 2 sfa ST T . If we combine this with the

PRI we get

6

25000

400 10

fa

pulse

TN

T

. (3-117)

From T and R we get

300 s

3001 s

R

R

TN

. (3-118)

This results in

6300 5000 1.5 10fa R pulseN N N (3-119)

and

7

6

1 16.667 10

1.5 10fa

fa

PN

. (3-120)