Cognitive radio is envisioned to be an enabling radio technology for future wireless networks. In this paper, we study the capacity scaling laws for cognitive radio ad hoc networks (CRNs), i.e., how each individual node's capacity scales as the number of nodes in the network increases. This effort is critical to the fundamental understanding of the scalability of such network. However, due to the heterogeneity in available frequency bands at each node, the asymptotic capacity is much more difficult to develop than prior efforts for other types of wireless networks. To overcome this difficulty, we introduce two auxiliary networks $zeta$ and $alpha$ to analyze the capacity upper bound and lower bound. We derive the capacity results under both the protocol model and the physical model. Further, we show that the results developed by Gupta and Kumar for the simple single-channel single-radio (SC-SR) networks are special cases under the results for CRNs.
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