## 9 Quickest Spectrum Sensing
### Performance Metrics
#### Quickest Detection with Unknown Parameters  
In the standard CUSUM test, it is assumed that the distributions before and after the change are perfectly known to the detector. Unfortunately, in many situations, the distribution after the change in cognitive radio systems is not completely known. For example, if we use received power as theobservation, the exact value of average received power after the change is unknown although we know that the average received power is increased.  
Let us take the detection of pilot in digital TV (DTV) systems for instance. On ignoring the interference leaked from signals in neighboring spectrum, the expressions of received signal are given by  
\[\left\{\begin{array}{ll}H_{0}:&r(t)=n(t)\\ H_{1}:&r(t)=A\sin\left(\omega tT_{O}+\phi\right)+n(t)\end{array}\right. \tag{61}\]  
where \(n(t)\) is additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma_{n}^{2}\) (we assume that \(\sigma_{n}^{2}\) is known to the detector), \(A\) is the received amplitude of pilot, \(\omega\) is the angular frequency of pilot, \(T_{O}\) is the time interval between two consecutive observations and \(\phi\) is the phase. Note that \(\omega\) can be found from the DTV system specification while both \(A\) and \(\phi\) are unknown. Therefore, we cannot apply the CUSUM test directly.  
One approach to tackle the unknown parameter is to apply the generalized likelihood ratio (GLR), in which the unknown parameters are replaced with their maximum likelihood estimation. Another alternative is to adopt the philosophy of multiple random walk approach of CUSUM test [75]. In such an approach, called parallel CUSUM test, we set a family of (say, \(N\)) parameter candidates \(\left\{\left(A_{n},\phi_{n}\right)\right\}_{n=1,...,N}\). We do CUSUM test for each parameter candidate by assuming that the candidate is the true value of the parameter. Then, we obtain a family of stopping times \(\left\{s_{n}\right\}_{n=1,2,...,N}\). The stopping time of the parallel CUSUM test is obtained by choosing the earliest stopping time, i.e.,  
\[s^{*}=\min\left\{s_{n},n=1,2,...,N\right\} \tag{62}\]  
The procedure of parallel CUSUM test is illustrated in Figure 39. The left part shows a possible selection of parameter candidates (we choose a grid in the product space of amplitude and phase). The right part shows the competition of several parameter candidates.  
One disadvantage of the parallel CUSUM test is that the parameter candidates do not change throughout the test. When there are sufficiently many observations (e.g., when the threshold is large), the parameters can be almost perfectly estimated and then be applied to the quickest detection. Therefore, it is more reasonable to estimate the parameters simultaneously and change the set of parameter candidates. As illustrated in Figure 40, when we have more precise estimation for the parameters, we can narrow down the range of parameter candidates. Based on this philosophy, _successive refinement_ is proposed in [152], in which the test is divided into multiple stages. In stage 1, parallel CUSUM test is applied with threshold \(\gamma_{1}\). When the corresponding stopping time is reached, we have some confidence on the  
Figure 39: Illustration of parallel CUSUM test.  
distribution change. Then, we begin stage 2 during which parallel CUSUM test (with threshold \(\gamma_{2}>\gamma_{1}\)) and parameter estimation are simultaneously carried out. When stage 2 is completed (the corresponding stopping time is reached), we use the parameter estimate to narrow down the range of parameter candidates and begin stage 3. Such a test-while-estimate procedure is repeated for certain times and then we claim that the change has happened. Such a successive refinement procedure is illustrated in Figure 41.