Dynamic Spectrum Access Decisions. George F. Elmasry

      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 of detection and the probability of a false alarm for a variety of different thresholds. Based on given requirements and machine learning techniques that count for the dynamics of the sensed environments, a close‐to‐optimal threshold can be reached. Figure 3.2 exemplifies different ROC curves for different SNIR values using the equations above. SNIR is defined as the ratio of the sensed signal power to noise power .

Figure 3.2 Different ROC curves for different SNIR (not to scale).

One can see from Figure 3.2 that as SNIR increases,⁶ we can achieve higher P_D at lower P_F. While one can see the same tendency in demodulation and decoding of a signal where higher SNIR results in less symbol error probability, it is critical to understand that this energy detection process relies on estimating the noise variance. Noise power estimation error can cause significant performance loss (significant shift in λ_E in Figure 3.1). Noise level can be estimated dynamically and more accurately if the spectrum sensor is able to separate the noise subspace from the signal subspace. Some sensors estimate noise variance as the smallest eigenvalue of the sensed signal's autocorrelation. This estimated noise variance can then be used to find the decision threshold λ_E that satisfies the requirements for a given false alarm rate. This noise estimation algorithm is applied iteratively, where N can be a moving average window that continuously normalizes the noise power.

      In addition to noise power estimation, a machine learning technique⁷ that uses the ROC model can leverage the following techniques to tune the decision threshold:

      1 Measure the success of its own decisions.8

      2 Take into consideration external variables such as emitter power, emitter distance to the sensor, terrain, rain, and fog that can affect SNIR.

      3 Increase accuracy by increasing the number of decision samples. Cooperative distributed DSA and centralized DSA techniques can be looking at more comprehensive information than a single node to make the ROC estimation more accurate.

      The purpose of using the above three techniques is to make the DSA system able to adapt the decision threshold to adhere to the same P_D at the same given requirement of P_F even with the increase of uncertainty.

      Example: Evaluation Metrics and ROC Design for Different Applications

      Equations (3.5) and (3.6) express the probability of detection and the probability of false alarm, respectively, for a single threshold ROC model. A third probability calculation could be the probability of misdetection.