Waveforms for xG Non-stationary Channels

Waveform design aims to achieve orthogonality among data signals/symbols across all available Degrees of Freedom (DoF) to avoid interference while transmitted over the channel. In general, precoding decomposes the channel matrix into desirable components in order to construct a precoding matrix, which is combined with the data signal to orthogonality in the spatial dimension. On the other hand, modulation uses orthogonal carriers in a certain signal space to carry data symbols with minimal interference from other symbols. However, it is widely evident that next Generation (xG) wireless systems will experience very high mobility, density and time-varying multi-path propagation that will result in a highly non-stationarity of the channel states. Conventional precoding methods using SVD or QR decomposition, are unable to capture these joint spatio-temporal variations as those techniques treat the space-time-varying channel as separate independent spatial channel matrices and hence fail to achieve joint spatio-temporal orthogonality. Meanwhile, the carriers in OFDM and OTFS modulations are unable to maintain the orthogonality in the frequency and delay-Doppler domain respectively, due to the higher order physical variation like velocity (Doppler effect) or acceleration (time-varying Doppler effect). In this article, we review a recent method called High Order Generalized Mercer's Theorem (HOGMT) for orthogonal decomposition of higher dimensional, non-stationary channels and its application to MU-MIMO precoding and modulation. We conclude by identifying some practical challenges and the future directions for waveform design for MU-MIMO non-stationary channels based on HOGMT.