Transmission over wireless channels is subject to time dispersion due to multipath propagation as well as frequency dispersion due to the Doppler effect. Standard OFDM systems using a guard time interval or cyclic prefix combat intersymbol interference but provide no protection against intersymbol interference. This drawback has lead to the introduction of pulse-shaping OFDM systems. In the first part of the paper we present a general framework for pulse shape design. Our analysis shows that certain pulse shapes proposed in the literature are in fact optimal in a well-defined sense. Furthermore our approach provides a simple way to adapt the pulse shape to varying channel conditions. In the second part of the paper we show that (pulse-shaping) OFDM systems based on rectangular time-frequency lattices are not optimal for time- and frequency dispersive wireless channels. This motivates the introduction of Lattice-OFDM (LOFDM) systems which are based on general time-frequency lattices. Using results from sphere packing theory we show how to optimally design LOFDM systems (the lattice and the pulse shape) for time- and frequency dispersive channels in order to minimize the joint ISI/ICI. Our theoretical analysis is confirmed by numerical simulations, showing that LOFDM systems outperform traditional pulse-shaping OFDM systems with respect to robustness against ISI/ICI.
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