## 10 Soft Decision Cognitive Radio and Hybrid Overlay/Underlay Cognitive Radio Waveform Design
### Hybrid Overlay/Underlay Waveform Design for Soft Decision Cognitive Radio
Two cases of under utilized spectrum are demonstrated: (1) when the spectral assignment is based on a binary decision, the bands adjacent to the primary users are unavailable to overlay-CR users and (2) primary users bands below the IT are unavailable to CR users. A soft decision CR (SDCR) will be able to exploit these _underused_ frequency bands to improve spectral efficiency and increase channel capacity. To support the envisioned SDCR system, the original SMSE framework is extended to account for both _unused_ and _underused_ frequency bands. The proposed SD-SMSE framework is first illustrated using Figures 50 and 51, then the design variables are re-defined to extend the SMSE expression to account for both unused and _underused_ spectrum. Figure 50a,b shows how the current CR framework identifies the used and unused spectrum based on binary decisions. Figure 50c shows the weighted spectrum estimation resulted from spectrum sensing block in Figure 51. The weighted spectrum estimate (WSE) (**a**) is further processed by taking into account inputs from the IT estimator, primary users, other secondary users requirements and channel conditions. Specifically, the weighted spectrum estimate provides a metric of the allowable transmission power density at each and every frequency component in the entire bandwidth. Hence, the WSE divides the entire bandwidth into unused (**u**) and _underused_ (**b**) frequency components and both the unused and _underused_ spectrum can be exploited. Notice in Figure 50 that different _underused_ frequency  
Figure 49: Identification of primary users, unused and underused spectral region. components have different allowable CR transmission power densities. It is envisioned that a CR-based SDR will have the option to choose an overlay-CR, underlay-CR or hybrid overlay/underlay waveform to improve performance based on the scenario, situation and need. The first step in SD-SMSE framework development is to re-examine the design variables in the original SMSE framework. For the SD-SMSE development, frequency related factors are termed primary variables, whereas amplitude and phase related factors are termed secondary variables. Since the objective here is to optimize the spectrum usage, only frequency components related design variables are considered. From this point forward the SD-SMSE framework development is based on the scenario depicted in Figure 50. As shown in Figure 50c, the weighted spectrum estimate represents all frequency components, which can be utilized for secondary user applications. It is represented by variable \(\mathbf{a}\) with the range changed from binary values (hard decision) to real values (soft decision), i.e.,  
\[\mathbf{a}\ =\ [a_{0},a_{1},\ldots,a_{N_{f}-1}],0\leq a_{m}\leq 1 \tag{90}\]  
Figure 51: Block diagram representation of SD-SMSEframework [167]. Figure 50: Spectrum parsing using weighted spectrum estimation in realization of SD-SMSE waveform. From the weighted spectrum estimate \(\mathbf{a}\), the _unused_ spectrum vector \(\mathbf{u}\) can be derived as  
\[S_{k}[m]=\ a_{m}u_{m}c_{m}d_{m,k}w_{m}e^{j(\theta_{d_{m,k}}+\theta_{c_{m}}+ \theta_{w_{m}}+\theta_{o_{m,k}})} \tag{88}\]0  
where,  
\[u_{m}=\left\{ \begin{array}{ll}1&ifa_{m}=1\\ 0&else\end{array} \right.\;m=0,1,\cdots N_{f}-1 \tag{92}\]  
The original SMSE hard decision CR design transmits over the unused spectrum specified by \(\mathbf{u}\). Now introducing a new design variable \(\mathbf{b}\) to account for the _underused_ spectrum,  
\[\mathbf{b}\;=\;[b_{0},b_{1},\ldots,b_{N_{f}-1}] \tag{93}\]  
where  
\[b_{m}=\left\{ \begin{array}{ll}0&a_{m}=1\\ a_{m}&a_{m}\
eq 1\end{array} \right. \tag{94}\]  
for \(m=0,1,\cdots,N_{f}-1\). Note that when \(a_{m}=1\) the value of \(b_{m}=0\). This is because when \(a_{m}=1\), the spectral component is _unused_ and accounted for in the assignment of \(u_{m}\). It is obvious that if one frequency component is _underused_, it cannot also be counted as unused and vice versa, i.e., \(u_{m}=0\) if \(b_{m}>0\) and \(b_{m}=0\) if \(u_{m}=1\).