Zak-OTFS: Pulse Shaping and the Tradeoff between Time/Bandwidth Expansion and Predictability

TL;DR

The Gaussian filters considered in this paper are a particular case of factorizable pulse shaping filters in the DD domain, and this family of filters may be of independent interest.

Abstract

The Zak-OTFS input/output (I/O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which we refer to as the crystallization condition. When the crystallization condition is satisfied, we describe how to integrate sensing and communication within a single Zak-OTFS subframe by transmitting a pilot in the center of the subframe and surrounding the pilot with a pilot region and guard band to mitigate interference between data symbols and pilot. At the receiver we first read off the effective channel taps within the pilot region, and then use the estimated channel taps to recover the data from the symbols received outside the pilot region. We introduce a framework for filter design in the delay-Doppler (DD) domain where the symplectic Fourier transform connects aliasing in the DD domain (predictability of the I/O relation) with time/bandwidth expansion. The choice of pulse shaping filter determines the fraction of pilot energy that lies outside the pilot region and the degradation in BER performance that results from the interference to data symbols. We demonstrate that Gaussian filters in the DD domain provide significant improvements in BER performance over the sinc and root raised cosine filters considered in previous work. We also demonstrate that, by limiting DD domain aliasing, Gaussian filters extend the region where the crystallization condition is satisfied. The Gaussian filters considered in this paper are a particular case of factorizable pulse shaping filters in the DD domain, and this family of filters may be of independent interest.

Figures (11)

• Figure 1: Non-predictability results from aliasing in the DD domain illustrated as overlap between the quasi-periodic aliases of the received pulse (whose support are depicted as ellipses). Aliasing occurs when the effective channel spreads are larger than the delay-Doppler periods.
• Figure 2: The impact of pulse shaping on the support of the effective channel. When the effective channel spread exceeds a DD domain period, aliasing leads to non-predictability. Here, the effective channel spread for the sinc pulse is larger than that for the Gaussian pulse (green vs. purple ellipses), resulting in Doppler domain aliasing and therefore non-predictability with the sinc pulse. There is no aliasing with the Gaussian pulse, hence the channel interaction is predictable.
• Figure 3: Zak-OTFS transceiver processing.
• Figure 4: Pulse magnitude $10 \log_{10}\vert w(\tau) \vert$ in dB as a function of the normalized delay $B \tau$. Gaussian filters (red) exhibit better localization than sinc filters (green) and RRC filters (black) in the case of no time/bandwidth expansion.
• Figure 5: The fundamental region ${\mathcal{D}}_0$ of the Zak-OTFS modulation tiled with pilot, guard and data regions. The pilot is located at $(k_p, l_p) = (M/2, N/2)$ and the information symbols modulate Zak-OTFS carriers located in the data regions. The pilot region contains carriers located in the region ${\mathcal{S}} + (k_p, l_p)$ where ${\mathcal{S}}$ is the support of the effective channel. It extends to the left of the pilot location illustrating a physical channel with a zero delay path and a shaping filter that leaks energy outside the main lobe.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2