Dynamic Spectrum Access Decisions. George F. Elmasry

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the same frequency band. This can mislead the decision‐making process and result in increasing the probability of false alarm and of misdetection.²

      With DSA, the ROC methodology can be implemented in different approaches depending on the sensing metrics and where the decision is made. This chapter will start with the generic aspects of the ROC hypothesizing process in DSA applications and present simple ROC‐based decision fusion cases while gradually moving to the harder cases. Statistical decision models in modulation and coding are well studied and well presented in textbooks. This chapter covers decision models for spectrum sensing pointing to the similarities and difference with modulation and coding models. While a demodulator may use fixed thresholds and rely on well‐known statistical models such as AWGN and the communication signal known power spectral density characteristics to decode a symbol, spectrum sensing models use adaptive thresholds and machine learning techniques to account for the many factors that can compound the spectrum sensing hypotheses. If the reader is not familiar with the ROC models, the reader is encouraged to refer to Appendix A of this chapter to get some basic understanding of the ROC methodology.

3.1 Basic ROC Model Adaptation for DSA

This ROC model is the most basic model where a sensor is probing a frequency band to check for the presence or absence of a communications signal. This basic model relies on energy detection and can assume that the noise is AWGN such that the signal received by the sensor can be expressed as follows:

3.1

      In Equation (3.1), s(n) is the sensed signal, w(n) is the AWGN, and n is the sampling index. If the sensed frequency band has no signal occupying it, then s(n) = 0 and the sensing process will detect the energy level of the AWGN.

      The sensed signal energy can be expressed as a vector of multiple sampling points as follows:

3.2

      where N is the size of the observation vector.

      The value of N and the definition of sampling points can differ from one sensor to another and any pre‐knowledge of the sensed signal waveform characteristics can guide the sensor into creating more optimal sampling points.

      The energy detection process can compare the decision metric M from Equation (3.2) against a fixed threshold λ_E. This processes needs to distinguish between two hypotheses, one hypothesis is for the presence of only noise and the other hypothesis is for the presence of signal and noise. These two hypotheses are:

      3.3

      3.4

      The spectrum sensor detection algorithm can successfully detect the sensed frequency with probability P_D and the noise variance can cause a false alarm³ with a probability of P_F. The detection problem can be expressed as:

3.5

3.6

Equations (3.5) and (3.6) can be illustrated as shown in Figure 3.1 where selecting an energy threshold λ_E can deviate from the optimum threshold. The optimum threshold is not known at any given instant and would have resulted in P_D = 1 and P_F = 0. The estimated λ_E can either be intentionally shifted to the right or shifted to the left as shown by the arrows at the bottom of Figure 3.1. If it is shifted to the left, the ROC model would increase the probability of hypothesizing , leading to a higher false alarm probability. If it is shifted to the right, the ROC model would increase the probability of hypothesizing , leading to a higher misdetection probability. This intentional shifting of the threshold depends on the communications system being designed. The system requirements can lead to shifting the threshold either to the left or to the right within a range, as indicated by the dashed arrow at the top of Figure 3.1.

Figure 3.1 Single‐threshold ROC model leading to false alarm and misdetection.

Maximum likelihood decisions can be applied to the decision threshold λ_E. A key factor in selecting λ_E is the estimation of noise power. Also, estimating the signal power by the sensor can be difficult since it can change due to propagation environments, and the distance between the transmitting node and the sensor. A good approach to select λ_E is to balance P_D and P_F based on given requirements. The threshold λ_E can be chosen to meet a given false alarm rate that can be deemed acceptable for the system under design. This makes it sufficient to model the noise variance, assuming a zero‐mean Gaussian random variable with variance . With this noise model, we can express the noise as . We can also simplify how we model the signal s(n) in order to make this analysis possible.⁴ We can assume that there is no fading and express the signal as a zero‐mean Gaussian variable making ⁵ These assumptions allows us to express the decision metric M as a Chi‐square distribution with 2N degree of freedom giving:

3.7

      Thus, the ROC model can calculate P_D and P_F as follows:

3.8

      3.9

      where Γ(a, x) is the incomplete gamma function and L_f and L_t are the associated Laguerre polynomials.

The DSA decision fusion process can use the ROC model to compare the performances for different threshold values. ROC models are a set of convergence curves that explore the relationship between the probability

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