Complex rational orthogonal wavelet and its application in communications

This letter proposes a generalized method to construct complex wavelets under the framework of rational multiresolution analysis, MRA(M) where M is a rational number. Theorems and examples are given for the construction of complex rational orthogonal wavelets (CROWs) whose real and imaginary parts form an exact Hilbert transform pair. Since the classical Mallat's MRA is a special case of the rational MRA(M) with M=2, the theorems hold for the construction of complex dyadic wavelets. Based on a rational MRA(M) with 1 < M < 2, the constructed CROWs not only achieve the benefit by capturing the phase information that the real-valued wavelets are lacking but also have the unique fine rational orthogonal property that is suited for specific application scenarios where binary orthogonality achieved by dyadic wavelets is not sufficient for the scale resolution. The CROWs' application in communications as the modulation signal pulse for PSK/QAM signaling is discussed. Specific communication scenarios that could benefit from the properties of this class of complex wavelets include the communication through a multipath/ Doppler channel and new CROW-based multicarrier modulation (MCM)/ orthogonal frequency division multiplexing (OFDM) systems.