Detection Theory
6.3 SIGNAL IN RECEIVERS
We write nB(t) in polar form as
where
and
We note that the definitions of nI(t), nQ(t), N(t), and φ(t) are consistent between the IF and baseband representations. This means the two representations are equivalent in terms of the statistical properties of the noise. We will reach the same conclusion for the signal. As a result, the detection and false alarm performances of both types of receiver configurations are the same. Thus, the detection and false alarm probability equations we derive in the future will apply to either receiver configuration.
If the receiver being analyzed is not of one of the two forms indicated above, the detection and false alarm probability equations derived herein may not apply. A particular example 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 specifically applicable to the configuration.
We now turn our attention to developing a representation of the signals at the input to the detection logic. Consistent with the noise case, we consider both IF and baseband receiver configurations. Thus, we will use Figures 6.1 and 6.2, but replace n(t) with s(t), N(t) with S(t), φ(t) with θ(t), nI(t) with sI(t), and nQ(t) with sQ(t).
We will 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 maintain consistency with this idea, and consistency with what happens in an actual radar, we also use a random process model for the SW0/SW5 target.
In Chapter 3, we learned the SW1 and SW2 targets share one RCS fluctuation model and the SW3 and SW4 targets share a second RCS fluctuation model. The difference between SWodd (SW1, SW3) and SWeven (SW2, SW4) was in how their RCS varies with time. SWodd targets have an RCS that is constant from pulse to pulse, but varies from scan to scan. SWeven targets have an RCS that varies from pulse to pulse. All cases assumed the RCS did not vary during a PRI. Because of this assumption, the statistics for SW1 and SW2 targets are the same on any one pulse. Likewise, the statistics for SW3 and SW4 targets are the same on any one pulse.
Consequently, in terms of single pulse probabilities, we can combine SW1 and SW2 targets and we can combine SW3 and SW4 targets. This accounts for our use of the terminology
“SW1/SW2 targets” and “SW3/SW4 targets” when discussing single pulse detection probability. In Chapter 8, we will develop separate equations for each of the Swerling target types, since we will base detection decisions on the results from processing several pulses.
Since the target RCS models are random processes, we 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
where
and
The baseband signal model is
We note that both of the signal models are consistent with the noise model of the previous sections. We assume S = S (t)|t=t
1 and θ = θ(t)|t=t
1 are independent.
We have made many assumptions concerning the statistical properties of the signal and noise. A natural question is: are the assumptions reasonable? The answer is that radars are usually designed so that the assumptions are satisfied. In particular, designers endeavor to make the receiver and matched filter linear. Because of this and the central limit theorem, we can reasonably assume nI(t) and nQ(t) are Gaussian. Further, if we enforce reasonable constraints on the bandwidth of receiver components, we can reasonably assume the validity of the independence requirements. The stationarity requirements are easily satisfied if we assume the receiver gains and noise figures do not change with time. We enforce the zero-mean assumption by using AC coupling and bandpass filters (BPFs) to eliminate DC components. For signals, we will not need the Gaussian requirement. However, we will need the stationarity, zero-mean, and other requirements. These constraints are usually satisfied for signals by using the same assumptions as for noise, by requiring a WSS random process for the target RCS, and by requiring θ(t) be wide sense stationary and uniform on (–π, π]. The latter two assumptions are valid for practical radars and targets.
At this point, we need to develop separate signal models for the different types of targets because each signal amplitude fluctuation, S(t), is governed by a different model.
6.3.2 Signal Model for SW0/SW5 Targets
For the SW0/SW5 target case, we assume a constant target RCS. This means the target power, and thus the target signal amplitude, is constant. With this assumption, we let
The IF signal model becomes
We introduce the random variable θ to force sIF(t) to be a random process. We specifically choose θ to be uniform on (–π,π]. This makes sI and sQ random variables, rather than random processes. sIF(t) is a random process because of the presence of the ωIFt term. This model is actually consistent with what happens in an actual radar. Specifically, the phase of the signal is random for any particular target return.
The density functions of sI and sQ are the same and are given by [13]:
We cannot assert the independence of random variables sI and sQ because we have no means of showing fs
IsQ (SI,SQ) = fs
I (SI) fs
Q (SQ).
The signal power is given by
In the above, we can write
Similarly, we get
and
Substituting (6.31), (6.32), and (6.33) into (6.30) results in
From (6.26), the baseband signal model is
and the signal power is
6.3.3 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 (see Chapter 3)
Since the power is a direct function of the RCS (from the radar range equation), the signal power at the detection logic input has a density function identical in form to (6.37). That is,
where
Random variable theory shows the signal amplitude, S(t), governed by the density function,
which is recognized as a Rayleigh density function [13]. This, combined with the fact that θ(t) in (6.21) is uniform, and the assumption of the independence of random variables S = S (t)|t=t
1
and θ = θ(t)|t=t
1, leads to the interesting observation that the signal model for a SW1/SW2 target takes the same form as the noise model. That is, the IF signal model for a SW1/SW2 target takes the form
where S(t) is Rayleigh and θ(t) is uniform on (–π,π]. If we adapt the results from our noise study, we conclude that sI(t) and sQ(t) are Gaussian with the density functions
Furthermore, sI = sI (t)|t=t
1 and sQ = sQ(t)|t=t
1 are independent.
The signal power is given by
Invoking the independence of sI = sI (t)|t=t
1 and sQ = sQ (t)|t=t
1, and the fact that sI(t) and sQ(t) are zero mean and have equal variances of PS, lead to the conclusion that
The baseband representation of the signal is
where the various terms are as defined above. The power in the baseband signal representation can be written as
as expected.
6.3.4 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 (see Chapter 3)
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 takes the same form as (6.47).
That is,
where PS is defined earlier in (6.37). From random variable theory it can be shown that the signal amplitude, S(t), is governed by the density function
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
and
However, we do not know the form of the densities of sI(t) and sQ(t). Furthermore, deriving their form has proven very laborious and elusive.
We can find the power in the signal from
We will need to deal with the inability to characterize sI(t) and sQ(t) when we consider the characterization of signal-plus-noise.