Zak-OTFS for Faster-Than-Nyquist Signaling in the Presence of Mobility & Delay Spread

Orthogonal signaling limits the number of information symbols transmitted in bandwidth $B$ and time $T$ to be $BT$. This corresponds to the Nyquist signaling and is achieved by mounting information symbols on $BT$-dimensional basis spanning the $BT$-dimensional space spaced $\frac{1}{B}$ and $\frac{1}{T}$ apart. Faster-than-Nyquist signaling involves transmitting more than $BT$ informational symbols in a $BT$-dimensional space. This leads to loss of orthogonality. This is achieved by time and/or bandwidth expansion resulting from packing more information symbols in the same $BT$-dimensional space (spacing less than $\frac{1}{B}$ and/or $\frac{1}{T}$). In this paper, we take a different approach to faster-than-Nyquist signaling. We propose to superimpose the information symbols on one another maintaining the original spacing in the Nyquist signaling. We carry this out in the delay-Doppler (DD) domain using Zak-transform based orthogonal time frequency space (Zak-OTFS) modulation. In Zak-OTFS, the channel varies slowly. Further Zak-OTFS also allows construction of mutually unbiased bases the interference between which appear like Gaussian noise. The proposed scheme leverages the slow variation in the DD channel to construct a precoder that mitigates the effect of the doubly-spread channel. Further, in the proposed scheme we mount information symbols on two mutually unbiased bases which allows superposition of information symbols. This simplifies receiver processing to detection in Gaussian noise since each basis appears to the other as Gaussian noise. This reduction makes it possible to use trellis coded modulation to enhance bit-error performance. Numerical results demonstrate that the faster-than-Nyquist signaling scheme achieves similar uncoded performance as that of Nyquist signaling and with coding the performance is better than Nyquist signaling at high signal-to-noise ratios.