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, . 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
(3-1)
where , and are random processes. If we expand Equation (3-1) using trig identities we get
(3-2)
where and are also random processes. In Equation (3-2), we stipulate that and are joint, wide-sense stationary, zero-mean, equal variance, Gaussian random processes. They are also such that the random variables and are independent. The variances of and are both equal to . The above statements mean that the density functions of and are equal and given by
.(3-3)
We will now show that is Rayleigh and is uniform on . We will further argue that the random variables and are independent.
From probability and random variables[1] if and are real random variables,
(3-4)
and
,(3-5)
where denotes the four-quadrant arctangent, then the joint density of and can be written in terms of the joint density of and as
.(3-6)
In our case , , and . Thus, we have
(3-7)
(3-8)
and
.(3-9)
Now, since and are independent, Gaussian and zero-mean with equal variance
.(3-10)
If we use this in Equation (3-9) with and we get
.(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,
(3-12)
and
.(3-13)
This proves the assertion that is Rayleigh and is uniform on . To prove that the random variables and are independent we note from Equations (3-11), (3-12) and (3-13) that
,(3-14)
which means that and are independent.
Since we will need it later, we want to find the noise power out of the signal processor. Since is wide-sense stationary we can use Equation (3-2) and write
(3-15)
In Equation (3-15), the term on the third line is zero because and 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
(3-16)
where and are joint, wide-sense stationary, zero-mean, equal variance, Gaussian random processes. They are also such that the random variables and are independent. The variances of and are both equal to . The constant of is included to provide consistency between the noises in the baseband and IF receiver configurations. The power in is given by (making use of the properties of and )
.(3-17)
We note that we can write in polar form as
(3-18)
where
(3-19)
and
.(3-20)
It will be noted that the definitions . , and 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 with , with , with , with and with .
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
(3-21)
where
(3-22)
and
.(3-23)
The baseband signal model is
.(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 and are independent.[2]
At this point we need to develop separate signal models for the different types of targets because the signal amplitude fluctuations, , 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
.(3-25)
The IF signal model becomes
.(3-26)
We introduce the random variable to force to be a random process. We specifically choose uniform on [3]. This means that and are also random variables (rather than random processes). is a random process because of the presence of the term.
The density functions of and are the same and are given by
.(3-27)
We cannot assert that the random variables and are independent because we have no means of showing that .
The signal power is given by
.(3-28)
In the above we can write
.(3-29)
Similarly, we get
(3-30)
and
.(3-31)
Substituting Equations (3-29), (3-30) and (3-31) into Equation (3-28) results in
.(3-32)
From Equation (3-24) the baseband signal model is
.(3-33)
The signal power is
.(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
.(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
(3-36)
where (3-37)
From random variable theory it can be shown that the signal amplitude, , is governed by the density function
.(3-38)
Which is recognized as a Rayleigh density function. This, combined with the fact that in Equation (3-21) is uniform, and the assumption that and 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
(3-39)
where is Rayleigh and is uniform on . If we adapt the results from our noise study we arrive at the conclusion that and are Gaussian with the density functions
.(3-40)
Furthermore, and are independent.
The signal power is given by
.(3-41)
Invoking the independence of and and the fact that and are zero mean and have equal variances of leads to the conclusion that
.(3-42)
The baseband representation of the signal is
(3-43)
where the various terms are as defined above. The power in the baseband signal representation can be written as
(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
.(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
(3-46)
where (3-47)
From random variable theory it can be shown that the signal amplitude, , is governed by the density function
.(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
(3-49)
and
.(3-50)
However, we don’t know the form of and . Furthermore, deriving its form has proven very laborious and elusive.
We can find the power in the signal from
.(3-51)
We will need to deal with the inability to characterize and 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
.(3-52)
If we are using the IF representation we would write
,(3-53)
and if we are using the baseband representation we would write
.(3-54)
In either representation, the primary variable of interest is the magnitude of the signal-plus-noise voltage, , 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 . Since and are Gaussian, will also be Gaussian. Since and are zero-mean, will also be zero-mean. Finally, since and are independent, the variance of will equal to the sum of the variances of and . That is
.(3-55)
With this we get
.(3-56)
By similar reasoning we get
.(3-57)
Since , , and are mutually independent, and are independent. This, with the above and our previous discussions of noise and the SW1/SW2 signal model, leads to the observation that is Rayleigh. Thus the density of is
.(3-58)
3.4.3 Signal-plus-Noise Model for SW0/SW5 Targets
Since and are not Gaussian for the SW0/SW5 case when we add them to and the resulting and will not be Gaussian. This means that directly manipulating and to obtain the density function of will be difficult. Therefore, we take a different tack and invoke some properties of joint and marginal density functions. Specifically, we use
.(3-59)
We then use
(3-60)
to get the density function of . 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
(3-61)
where we have made use of Equation (3-26). If we expand Equation (3-61) and group terms we get
.(3-62)
According to the conditional density of Equation (3-59) we want to consider Equation (3-62) for the specific value of . If we do this we get
.(3-63)
With this we note that and are Gaussian random variables with means of and . They also have the same variance of . Further more, since and are independent and are also independent. With this we can write
.(3-64)
If we invoke the discussions related to Equations (3-4), (3-5) and (3-6), we can write
.(3-65)
If we substitute from Equation (3-64) we get
.(3-66)
We can manipulate the exponent to yield
(3-67)
Finally we can use
(3-68)
along with Equation (3-59) to write
.(3-69)
For the next step we need to integrate with respect to and to derive the desired marginal density, . That is (after a little manipulation)
.(3-70)
We want to first consider the integral with respect to . That is,
(3-71)
We recognize that the integrand is periodic with a period of 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 . With this we get
.(3-72)
If we make the change of variables the integral becomes
(3-73)
where is a modified Bessel function of the first kind.
If we substitute Equation (3-73) into Equation (3-70) the latter becomes
(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 .
3.4.4 Signal-plus-Noise Model for SW3/SW4 Targets
As with the SW0/SW5 case, and are not Gaussian for the SW3/SW4 case. Thus, when we add them to and the resulting and will not be Gaussian. This means that directly manipulating and to obtain the density function 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
.(3-75)
In this case we will need to find the joint density of , , and and perform the appropriate integration to get the marginal density of . More specifically, we will find
(3-76)
and
.(3-77)
We can draw on our work from the SW0/SW5 case to write
.(3-78)
Further, since and are, by definition, independent, we can write
.(3-79)
If we substitute Equations (3-78) and (3-79) into Equation (3-76) we get
.(3-80)
From Equation (3-77) we can write
(3-81)
where
(3-82)
and
.(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
(3-84)
and
.(3-85)
To complete the calculation of we must compute the integral
(3-86)
where
.(3-87)
It turns out that Maple was able to compute the integral as
.(3-88)
With this becomes
(3-89)
which, after manipulation can be written as
.(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 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 when only noise is present at the signal processor output and when signal-plus-noise is present at the signal processor output. For the linear detector the output will be when only noise is present at the signal processor output and when signal-plus-noise is present at the signal processor output.