In this paper, a new matrix-based characterization ofgeneralized-frequency-division-multiplexing (GFDM) transmitter matrices isproposed, as opposed to traditional vector-based characterization withprototype filters. The characterization facilitates deriving properties of GFDM(transmitter) matrices, including conditions for GFDM matrices beingnonsingular and unitary, respectively. Using the new characterization, thenecessary and sufficient conditions for the existence of a form oflow-complexity implementation for a minimum mean square error (MMSE) receiverare derived. Such an implementation exists under multipath channels if the GFDMtransmitter matrix is selected to be unitary. For cases where thisimplementation does not exist, a low-complexity suboptimal MMSE receiver isproposed, with its performance approximating that of an MMSE receiver. The newcharacterization also enables derivations of optimal prototype filters in termsof minimizing receiver mean square error (MSE). They are found to correspond tothe use of unitary GFDM matrices under many scenarios. The use of such optimalfilters in GFDM systems does not cause the problem of noise enhancement,thereby demonstrating the same MSE performance as orthogonal frequency divisionmultiplexing. Moreover, we find that GFDM matrices with a size of power of twoare verified to exist in the class of unitary GFDM matrices. Finally, while theout-of-band (OOB) radiation performance of systems using a unitary GFDM matrixis not optimal in general, it is shown that the OOB radiation can besatisfactorily low if parameters in the new characterization are carefullychosen.
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