Low-complexity Equalization of Zak-OTFS Via Neumann Series

Abstract

We describe a general method for selecting an orthonormal basis of carrier waveforms that aligns the basis with delay / Doppler characteristics of a wireless channel. We show that our method enables low-complexity equalization for two channels of practical interest. The first is satellite communication and the second is communication from a ground station to an unmanned aerial vehicle (UAV). After Doppler compensation both scenarios are characterized by a first line of sight (LOS) path with zero delay and zero Doppler shift, and a second weaker path. We show that our method is robust to fractional delay and Doppler shifts. We consider orthonormal bases of carrier waveforms that are obtained from the pulsone basis of Zak-OTFS carrier waveforms by applying a generalization of the discrete affine Fourier transform (GDAFT). This family of waveforms includes AFDM and other modulations proposed for 6G. What distinguishes these bases is that the carrier waveforms are common eigenvectors of some maximal commutative subgroup S of a Heisenberg-Weyl group of discrete delay and Doppler shifts. We describe how to choose S to mitigate the damaging effects of interference between carriers. We represent the wireless channel by delay-Doppler taps and observe that a channel tap located within S multiplies every waveform by a complex phase. If all channel taps are located within S, then the channel multiplies every waveform by a complex phase, and a single tap equalizer supports reliable communication. This is the case for a linear time-invariant (LTI) channel, where S is the group of discrete time shifts and the carrier waveforms are discrete tones (OFDM). In general, we choose the subgroup S to maximize the diagonal component of the channel energy.