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we could create the following three metrics:

(3.10)

(3.11)

      (3.12)

      where P_D is the probability of hypothesizing the presence of the sensed signal given that the sensed signal is present, P_F is the probability of hypothesizing the presence of the sensed signal given that the sensed signal was not present, and P_m is the probability of hypothesizing the absence of the sensed signal given that the sensed signal was present.

Notice that there is a fourth possibility that is irrelevant to performance evaluation. Table 3.1 shows the four cases of signal presence and absence versus hypotheses with the “N, N” case (the signal is not present and the sensor did not detect it) being irrelevant.⁹

Table 3.1 Signal presence versus hypotheses.

Signal presence	Hypotheses	Evaluation metric
Y	Y	P D
Y	N	P F
N	Y	P m
N	N	N/A

      The P_D, P_F, and P_m metrics can be used to measure the efficiency of the decision‐making process given some design requirements. Notice that:

(3.13)¹⁰

      Although¹⁰ Equations (3.10)–(3.13) can apply to different systems, the system under design should influence how a machine‐learning algorithm would estimate λ_E. Let us consider the following two cases:

       Case 1: A commercial communication system of a secondary user attempting to opportunistically use the primary user spectrum. In this case, a higher probability of false alarm can be acceptable as the higher probability of misdetection can cause the secondary user to interfere with the primary user.11 With this case, the design of the machine learning algorithm would accept a higher probability of false alarm to minimize the probability of misdetection.

       Case 2: A military MANET system that can operate in an antijamming mode and the formed MANET can switch to a different waveform type only if the interference level is too high. With this case, a higher probability of misdetection may be acceptable since the antijamming waveform can operate in the presence of some level of interference. With this case, the design of the machine learning algorithm may target a higher probability of misdetection to minimize the probability of false alarm.

3.2 Adapting the ROC Model for Same‐channel in‐band Sensing

Same‐channel in‐band sensing use of the ROC model has some factors that can make the hypothesizing process more accurate but also has its own challenges. As mentioned in Chapter 2, with same‐channel in‐band sensing, the receiver can have a clear dwell time in the presence of the communications signal and a clear dwell time in the absence of the communication signal. For example, if we have a time‐domain preamble and a fixed over‐the‐air frame size, a sampling point can be an instant detection of energy within the length of the frame and N in Equation (3.2) can be selected for the number of samples (instants) collected during the frame time. N should be large enough to smooth the effect of noise spikes. This sampling can occur when the preamble is acquired and when the preamble is not acquired separately.

      The goal of same‐channel in‐band sensing has two folds. The first is to hypothesize for the presence of an interfering signal plus noise or hypothesize the presence of only noise when the communications signal is known to be absent. The second is when the communications signal is known to be present, the ROC model would then hypothesize if the interfering signal and noise power are too high to warrant a change of the operating frequency. The ROC model explained in the previous section has to be adapted to consider two different thresholds instead of one threshold. With same‐channel in‐band sensing, Equation (3.1) becomes:

3.14

      where r(n) is the interfering signal.

      When the same‐channel in‐band demodulation finds a preamble and it is demodulating and decoding an over‐the‐air frame, the same‐channel in‐band energy detection process is faced with the following two hypotheses:

3.15

3.16

      When the same‐channel in‐band demodulation does not find a preamble and it is not demodulating and decoding an over‐the‐air frame, the same‐channel in‐band energy detection process is faced with the following two hypotheses:

3.17

3.18

      Same‐channel in‐band sensing and the presence of the communications signal's marks such as preambles should lead the ROC model to minimizing P_F, with very low probability that the receiver falsely decides that the preamble exist when it does not exist.¹² With same‐channel in‐band detection, P_F is the probability of deciding an interfering signal r exists when it does not exist. The key here is to have a good estimation of the noise floor energy and a good estimation of the communication signal energy in order to effectively hypothesize the presence or the absence of an interfering signal.

Let us consider one of the most suitable signal types for same‐channel in‐band sensing with minimal computational power requirements. This is the n‐ary phase shift keying (PSK) signal type¹³. This signal can be used with OFDM as 4‐ary PSK when noise level is high to increase range and reduce data rate, or it can be used as 8‐ary PSK when noise level is low and the signal needs to achieve a higher transmission rate.¹⁴ This signal is depicted in Figure 3.3 with the 4‐ary case encoding two bits per symbol and the 8‐ary case encoding three bits per symbol.

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