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In this paper we demonstrate the operation of a wavelet packet based multi-carrier modulation (WP-MCM) scheme in the context of cognitive radio. The wavelet packets are derived from multistage tree-structured paraunitary filter banks. The emphasis is on the design and development of maximally frequency selective wavelets derived from a modified Remez exchange algorithm. To enable the WP-MCM cognitive radio system to co-exist with other licensed users a common spectrum pool is maintained and the WP-MCM transmission waveform characteristics are shaped to communicate in the idle time-frequency gaps of the licensed user. This is achieved by dynamically deactivating wavelet packet carriers in and near the region of the licensed user spectrum. Through simulation results, we demonstrate the efficacy of the proposed wavelet packet based mechanism in seamlessly cohabiting with licensed users. The bit error rate (BER) performance is shown to be comparable, and even at times better, to conventional Fourier based OFDM system.