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Cognitive radio is considered as one of the main enablers for provisioning dynamic and flexible spectrum/channel allocation in wireless communications. On the other hand several physical layer mechanisms such as adaptive modulation, multiple-input multiple output systems, advanced channel coding and/or combinations of them enhance the capacity of wireless networks. However little effort has been put till now in studying the performance gains of physical layer mechanisms with the presence of cognition capabilities. The incorporation of cognitive mechanisms demands more detailed studies for assessing the impact on the spectral efficiency. To this direction, cross-layer combination of such a physical layer with upper layers should be also considered as a case study in a cognitive wireless environment. In this work we present a study on the spectral efficiency of adaptive modulation and coding which is one of the most promising schemes of applying cognitive radio at the physical layer. Besides, we study a cross-layer combination of adaptive modulation with upper layers in the same cognitive context. We prove that the performance gain of cognitive radio over such a physical layer is not negligible.