## 2 Summary of the Paper
### Quickest Detection for Spectrum Sensing  
Spectrum sensing is the (instead of 'a') key problem in cognitive radio systems. Secondary users need to monitor the spectrum occupancy in order to use the spectrum when there is no primary user or quit the spectrum when primary users emerge. Like all other detection (or equivalently, hypotheses testing) problems, the spectrum sensing needs to find a tradeoff between miss detection (not detecting the primary user when it emerges) and false alarm (claiming that a primary user exists when there is actually no primary user). Due to the requirements of tolerable violation to primary users and tolerable interruption to the communication of secondary users, the spectrum sensing has substantial impact on the overall performance of cognitive radio systems.  
In most existing publications on spectrum sensing, traditional block detection is used, in which the observations are grouped into blocks and decision is made at the end of each block based on the observations of the corresponding block (illustrated in Figure 4 (a)). The advantage of such a block detection is easy implementation. However, it is difficult to determine the size of each block: if the block size is small, the decision will be unreliable; if the block size is large, then the detection delay may be large (e.g., a primary user emerges at the beginning of a block and can be detected only at the end of the block; then the delay will be the block duration).  
Therefore, a more suitable framework for spectrum sensing is sequential detection (illustrated in Figure 4 (b)), in which the decision could be made when each new observation arrive. Moreover, we notice that the observation distribution could change (e.g., the average received power is increased when primary user emerges), which is different from traditional detection problems (the distribution is static). Hence the _spectrum sensing is essentially to detect the change of observation distribution_. A powerful tool for such a problem of change detection is _quickest detection_ (also called quickest change detection or  
Figure 4: Illustrations of block and sequential detections.  
abrupt change detection). As illustrated in Figure 5, we assume that the original observation distribution is \(P_{0}\) and the distribution is changed to \(P_{1}\) at a time unknown to the detector. The job of quickest detection is to detect the change as quickly as possible under the constraint of tolerable false alarm rate. We coin the spectrum sensing within the framework of quickest detection as _quickest spectrum sensing_.  
Research on quickest detection dates back to 1931 [47] and has attracted substantial research since then. Quickest detection is useful in tasks like remote sensing [48, 49, 50], financial decision making [51, 52, 53, 54, 55], medical diagnosis [56, 57, 58, 59], signal segmentation [60, 61, 62, 63], environmental monitoring [64, 65], and network security [66, 67, 68, 69, 70, 71]. In recent years, decentralized quickest detection has received plenty of studies [72, 73, 74]. Comprehensive introductions on quickest detection can be found in [75] and [76].  
The quickest spectrum sensing can be categorized according to the following criterions:  
* Bayesian vs. non-Bayesian: the _a priori_ probability of primary user activity is available [77] in the former case and unavailable in the latter case [78];
* Single channel vs. multiple channels: whether the secondary user could monitor one channel or multiple channels in the frequency spectrum;
* Completely observable vs. partially observable: this applies to only the case of multiple channels; in the former situation, the secondary user can get observations from all channels whereas it can monitor only a subset of channels in the latter case.  
In the remainder of Section 9., we will discuss two typical scenarios of quickest spectrum sensing: single channel with non-Bayesian detection and multiple channels with Bayesian detection based on partial observations.