# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 2 Superimposed Training System Model and Detection Problem
### Detection Problem for SUs in Cognitive Radio  
In order to determine the available frequency bands for opportunistic transmissions, SUs rely on the spectrum sensing functionality. For the CR scenario analyzed here, it is considered that in a frequency band of the PU, each SU independently performs the spectrum sensing task over \(N\) received samples, with \(N/P\) being an integer. Hence, the vector \(\mathbf{x}=[x[0],\ x[1],\ \cdots,x[N-1]]^{\mathrm{T}}\) denotes the sequence periodically collected by the SU's receiver, with each sample of the set expressed as:  
\[x[n]=\theta e^{i\phi}s[n-\tau]+w[n],\ \ \ n=0,\ 1,\ \ldots,\ N-1, \tag{2}\]  
where \(\theta\) and \(\phi\) represent the unknown amplitude and phase shift of a flat fading channel, respectively. Furthermore, \(s[n-\tau]\) denotes the independent and identically distributed (i.i.d.) ST PU's signal samples given by Equation (1) with power \(\sigma_{s}^{2}\), where \(\tau\) is used to express the lack of time synchronization between the PU's transmitter and the SU's receiver. Then, it is assumed that the transmitted data sequence \(d[n]\) is unknown to the SU and the samples are modeled as i.i.d. circularly symmetric complex Gaussian random variables with variance \(\sigma_{d}^{2}\), i.e., \(d[n]\sim\mathcal{CN}(0,\ \sigma_{d}^{2})\) and the ST sequence \(t[n]\) is known to the SU. Similarly, it is considered that the noise samples \(w[n]\) are i.i.d. circularly symmetric complex Gaussian random  
Figure 1: Comparison of frame structures for the time domain multiplexed training (TDMT) scheme (**top**) and the superimposed training (ST) scheme (**bottom**).  
variables with zero mean and variance \(\sigma^{2}\), i.e., \(w[n]\sim\mathcal{CN}(0,\,\sigma^{2})\). Therefore, the detection problem consists in deciding between the following two hypotheses:  
1. The PU is not using the frequency band of interest (i.e., hypothesis \(\mathcal{H}_{0}:\,\theta=0\)),
2. the PU is transmitting data (i.e., hypothesis \(\mathcal{H}_{1}:\,\theta>0\)).  
Nevertheless, deciding if the PU is active from the received sequence \(\{x[n]\}_{n=0}^{N-1}\) can be very tough when random channel gains are considered. Especially in a very low SNR region (i.e., when \(\theta\to 0\)), which results in collisions with the PUs (caused by missed detections) or in the loss of opportunities for SUs' transmissions (caused by false alarms) due to the imperfect spectrum sensing. Additionally, exploiting the ST information for spectrum sensing requires the synchronization of the SUs in the CR network with the ST sequence transmitted by the PU, i.e., \(\tau\) in Equation (2) must be estimated at each SU's receiver. These aspects are considered in the next section, where a new superimposed training combined approach for a reduced phase (SCAR-Phase) of spectrum sensing is presented. This approach provides a feasible solution to the synchronization issue for spectrum sensing algorithms that take into account the training sequence to improve the detection performance in CR networks.