## 10 Soft Decision Cognitive Radio and Hybrid Overlay/Underlay Cognitive Radio Waveform Design
### Underlay CR and Overlay CR  
Figures 45 and 46 illustrate the concepts of underlay CR transmission and overlay CR transmission.  
We start with the famous Shannon channel capacity equation:  
\[C=W\log\left(1+\frac{S}{N}\right) \tag{81}\]  
It is well known that to increase the channel capacity in a communication system, we need to increase the SNR (signal to noise ratio) \(\frac{S}{N}\) or the bandwith \(W\), or both.  
In underlay CR transmission, a very large contiguous bandwidth is used for secondary user's transmission, with the primary users operating within the same bandwidth. In this way, we maximize the bandwidth \(W\) in (81). However, to avoid interferences to primary (licensed) users, the underlay CR transmission has to limit its transmission power density at a very low level. Hence, the channel capacity of underlay CR transmission is extremely limited although it has maximized the transmission bandwidth. Specifically, the channel capacity of underlay CR transmission is  
\[C_{Underlay}=W\log\left(1+\frac{\Phi_{Underlay}W}{n_{0}W+\sum_{i=1}^{M}\Phi_{ pi}W_{pi}}\right) \tag{82}\]  
Figure 45: Underlay CR transmission.  
Figure 46: Overlay CR transmission.  
where \(n_{0}\) is the additive Gaussian noise power spectrum density, \(\Phi_{Underlay}\) is the average power spectrum density of underlay CR transmission, \(M\) is the total number of primary users operating within the bandwidth \(W\), \(\Phi_{p_{i}}\) is the average power spectrum density of the \(i^{th}\) narrowband primary user's transmission and \(W_{p_{i}}\) is the corresponding bandwidth of \(i^{th}\) primary user. Notice that underlay CR transmission suffers interference from all primary users, which is characterized by \(\sum_{i=1}^{M}\Phi_{p_{i}}W_{p_{i}}\). Since (1) the power spectrum density of underlay CR transmission \(\Phi_{Underlay}\) is very low and (2) the primary narrowband users have much higher power spectrum density \(\Phi_{p_{i}}\), the signal to interference and noise ratio (SINR) is significantly decreased. As a direct result, even though underlay CR transmission has a very large bandwidth, its channel capacity is very limited.  
On the other hand, overlay CR transmission finds the unused frequency bands and only transmits over those bands, totally avoiding interference to primary users. The channel capacity of cognitive radio transmission is characterized as  
\[C_{Overlay}=\sum_{k=1}^{N}W_{u_{k}}\log\left(1+\frac{\sum_{k=1}^{N}\Phi_{CR_{ k}}W_{u_{k}}}{n_{0}\sum_{k=1}^{N}W_{u_{k}}}\right) \tag{83}\]  
where \(N\) is the total number of unused bands in the entire bandwidth \(W\), \(W_{u_{k}}\) is the bandwidth of the \(k^{th}\) unused band, \(\Phi_{CR_{k}}\) is the power spectrum density of cognitive radio transmission on the \(k^{th}\) unused band. It is evident that in cognitive radio transmission, the total bandwidth exploited is less than the total bandwidth \(W\). However, since (1) there is no interference from primary users to cognitive radio and (2) there is no limit in the cognitive radio transmission power spectrum density \(\Phi_{CR_{k}}\), the signal to noise ratio is much improved (compared to underlay CR transmission). As a direct result, the channel capacity of cognitive radio is much higher than that of underlay CR transmission.