Next generation cognitive radios will benefit from the capability of transmitting and receiving communications waveforms across many disjoint frequency channels spanning hundreds of megahertz of bandwidth. The information theoretic advantages of multi-channel operation for cognitive radio (CR), however, come at the expense of stringent linearity requirements on the analog transmit and receive hardware. This paper presents the quantitative advantages of multi-channel operation for next generation CR, and the advanced digital compensation algorithms to mitigate transmit and receive nonlinearities that enable broadband multi-channel operation. Laboratory measurements of the improvement in the performance of a multi-channel CR communications system operating below 2 GHz in over 500 MHz of instantaneous bandwidth are presented.