and (3.8) for the case of same‐channel in‐band local decision fusion building on Equations (3.19)–(3.22). Does the signal overlay model simplify this ROC model? Why?3 A set of nodes in a cognitive MANET are moving in tandem while using directional antennas, as shown in the figure below. This could be a row of vehicles, illustrated by the oval shapes. Assume that this convoy is driving along a road in a straight line with the distance between each two vehicles being the same, D. Let us assume the road width is W. Let us also assume that the frequency used for communication between vehicles needs to be used again after a distance W away from the road, as shown in the figure, i.e. the spatial separation is 2W. Consider a directional beam such that spectrum emission would reach the vehicle in front and the vehicle at the back and then fade away to the side of the road. Spectrum analysis of the directional beam power spectral density shows that D1 has to be 1.1 × D. The distance D1 defines the spatial separation regions such that when a node is communicating to the node ahead of it, the spectral density further away from D1 is negligible to allow for spectrum reusability.If we use sectored antennas for the vehicles, find the minim number of sectors per antenna we need in order to meet the above spectral pattern design. Note that you would need to round up your calculation to an even number of antenna sectors to find a symmetrical division of the full circle. Note that this will require some geometric analysis.Assume that we can apply power control on the antenna spectrum emission such that we can use a frequency after n hops. Find the number of frequencies we need for any number of nodes driving on this road.Based on the nulling matrix explained in Section 3.3.1.2, do you recommend a higher number of sectors than estimated by (a) above? Why?
4 A cognitive military MANET is implementing a directional antenna technique with a sectored antenna with N equal sectors. Let us assume that there is no overlapping between the sectors. This directionality technique is evaluated based on its ability to avoid enemy eavesdropping nodes. Let us assume that we have node a that is communicating to node b. The maximum area of coverage (AoC) for node a is defined by a circle of diameter d where the radius R is the maximum distance for node a reachability to and beyond node b, as shown in the figure below. As the figure shows, at a certain point in time, the distance between nodes a and b was L.What is the sector angle of this sectored antenna as a function of N?Let us assume that the enemy is able to put eavesdropping nodes randomly within the AoC of node a. What is the probability of the enemy succeeding in receiving the radio signal if the radio uses an omnidirectional antenna with the same AoC?If the enemy succeeded in deploying a single eavesdropping node randomly in the AoC, what is the probability of the enemy succeeding in receiving the military signal if the military node used described the directional antenna with a single active sector?If the enemy was able to deploy 1, 2, 3, and 4 eavesdropping nodes randomly within the AoC, create a table showing the probability of the enemy succeeding in receiving the radio signal for each of these four cases if the radio used the prescribed directional antenna with a single active sector.If the number of sectors used by the military node N = 12, find the number of eavesdropping nodes the enemy would have to drop randomly so that the probability of the enemy succeeding in receiving the radio signal approach 0.5.
5 Refer to Appendix 3A. Let us assume we have two ROC curves as shown in the figure below. Are both curves in the “use” region? Based on these intersected curves, will you consider the area under the curve a good metric to assess a ROC curve? Why?
6 In military communications, is DSA alone sufficient to overcome an enemy's follower jammer? Why?
Appendix 3A: Basic Principles of the ROC Model
This appendix describes the ROC model in simple terms to give the reader who is not familiar with the different statistical decision concepts related to ROC models a basic understanding of the model characteristics. The ROC plots are well studied in multiple fields where two basic evaluation measures are needed. These evaluation measures are referred to as sensitivity and specificity in some fields. With DSA, sensitivity is known as the probability of detection of the sensed signal while specificity is known as the probability of false alarm indication of the sensed signal. Any DSA design has to consider a tradeoff between these two evaluation measures. Signal measurements in some cases can lend higher probability of detection at lower probability of false alarm, but the tradeoff always exist.
Figure 3A.1 Ideal labeling of a dataset.
Let us consider a dataset where the values in the dataset can be classified as positive (P) or negative (N). As shown in Figure 3A.1, all the values in the dataset ideally can be classified as P or N.
An observer classifying the dataset into P or N may create four outcomes, as shown in Figure 3A.2. The four outcomes are true positive (TP), true negative (TN), false positive (FP), and false negative (FN). Notice that the observer hypothesizes the positive and negative value creating the FP and FN events.
Figure 3A.2 A classifier outcome of the dataset.
The idea behind the ROC model is to create plots such that one axis specifies the false positive rate while the other axis specifies the false negative rate,26 as shown in Figure 3A.3.
Figure 3A.3 Specifying FP and FN rates.
Let us assume that our ROC model plots the false positive (specificity) rate as the x axis and false negative (sensitivity) rate as the y axis. We refer to this ROC model as the ROC space and it is a two‐dimensional space that allows us to create the tradeoff needed in DSA design. A DSA decision‐making process is a classifier in the ROC space.
The ROC Curve as Connecting Points
An ROC point is a point in the ROC space with x and y values where x is the probability of false alarm and y is the probability of detection. Each of the x and y axes spans from 0 to 1. Let us use an example of ROC curves in the ROC space where we simplify the curves by linearly connecting adjacent points. Let us assume that for an example dataset as explained above, we have four possibilities to classify the dataset:
1 Achieve a probability of detection equal to zero at a probability of false alarm equal to zero.
2 Achieve a probability of detection equal to 0.5 at a probability of false alarm equal to 0.25.
3 Achieve a probability of detection equal to 0.75 at a probability of false alarm equal to 0.5.
4 Achieve a probability of detection equal to 1 at a probability of false alarm equal to 1.
These four possibilities become four points in a ROC curve in the ROC space as shown in
