Zak-OTFS and LDPC Codes

TL;DR

This work analyzes Zak-OTFS for 6G-style channels where Doppler spreads are large and channel estimation is challenging. It leverages the crystallization condition ($\tau_p\nu_p=1$) to obtain a predictable, model-free input-output relation and uses LDPC coding with reliability-aware DD-bin allocation to exploit highly reliable regions around a pilot. The main contributions are two DD-bin allocation strategies (RPE-based and strip) that map information bits to more reliable bins, demonstration that LDPC coding extends the Doppler range for reliable communication, and evidence that Zak-OTFS with coding outperforms MC-OTFS under these conditions. Collectively, the results indicate that Zak-OTFS with LDPC coding provides a robust, practical approach for high-Doppler 6G scenarios, reducing sensitivity to transmit filters and enabling near-optimal performance without precise channel knowledge.

Abstract

Orthogonal Time Frequency Space (OTFS) is a framework for communications and active sensing that processes signals in the delay-Doppler (DD) domain. It is informed by 6G propagation environments, where Doppler spreads measured in kHz make it more and more difficult to estimate channels, and the standard model-dependent approach to wireless communication is starting to break down. We consider Zak-OTFS where inverse Zak transform converts information symbols mounted on DD domain pulses to the time domain for transmission. Zak-OTFS modulation is parameterized by a delay period $τ_{p}$ and a Doppler period $ν_{p}$, where the product $τ_{p}ν_{p}=1$. When the channel spread is less than the delay period, and the Doppler spread is less than the Doppler period, the Zak-OTFS input-output relation can be predicted from the response to a single pilot symbol. The highly reliable channel estimates concentrate around the pilot location, and we configure low-density parity-check (LDPC) codes that take advantage of this prior information about reliability. It is advantageous to allocate information symbols to more reliable bins in the DD domain. We report simulation results for a Veh-A channel model where it is not possible to resolve all the paths, showing that LDPC coding extends the range of Doppler spreads for which reliable model-free communication is possible. We show that LDPC coding reduces sensitivity to the choice of transmit filter, making bandwidth expansion less necessary. Finally, we compare BER performance of Zak-OTFS to that of a multicarrier approximation (MC-OTFS), showing LDPC coding amplifies the gains previously reported for uncoded transmission.

Figures (6)

• Figure 1: Heatmap of RPE in the crystallization regime for $\nu_{p}=30$kHz, $\tau_{p} = 1/\nu_{p}$, $B=0.96$MHz, $T=1.6$ms, Veh-A Channel for varying maximum Doppler shifts $\nu_{max}$.
• Figure 2: A DD domain pulse and its time domain/frequency domain realizations referred to as TD/FD pulsone. The TD pulsone comprises of a finite duration pulse train modulated by a TD tone. The FD pulsone comprises of a finite bandwidth pulse train modulated by a FD tone. The location of the pulses in the TD/FD pulse train and the frequency of the modulated TD/FD tone is determined by the location of the DD domain pulse $(\tau_0, \nu_0)$. As the Doppler period $\nu_p\to\infty$, the FD pulsone approaches a single FD pulse which is the FDM carrier. Similarly, as the delay period $\tau_p\to\infty$, the TD pulsone approaches a single TD pulse which is the TDM carrier. Setting $\tau_p\nu_p = 1$, we see that OTFS is a family of modulations parameterized by $\tau_p$ that interpolates between TDM and FDM Zakotfs1.
• Figure 3: The effect of coding with proposed allocation (light blue vs. red and dark blue curves), the effect of coding with standard allocation (black vs. green curves), the effect of RPE allocation on uncoded transmission (black vs. light blue curves), and the effect of proposed allocations on coded transmission (green vs. red and dark blue curves).
• Figure 4: Comparison between all strategies for varying SNR at $\nu_{max}=14500$Hz.
• Figure 5: LDPC coding with RPE and strip allocation in MC-OTFS versus Zak-OTFS.