Delay-Doppler Signal Processing with Zadoff-Chu Sequences

TL;DR

The paper develops a delay–Doppler signal-processing framework for uplink grant-free access by embedding Zadoff-Chu preambles into Zak-OTFS. It derives DD-domain pilot expressions, establishes crystallization conditions to avoid aliasing, and shows how to detect multiple preambles and read the effective I/O channel directly in the DD domain. It integrates sensing of the I/O relation with data transmission in a single subframe, aided by turbo processing to mitigate pilot interference, and demonstrates robust multi-user detection using OST in a doubly-dispersive channel. The results indicate practical potential for 2-step RACH extension to 3MTC and unsourced random access in dynamic, high-Doppler environments. The approach leverages the mathematical properties of ZC sequences to enable low-complexity, model-free DD-domain processing with direct channel readout from pilots.

Abstract

Much of the engineering behind current wireless systems has focused on designing an efficient and high-throughput downlink to support human-centric communication such as video streaming and internet browsing. This paper looks ahead to design of the uplink, anticipating the emergence of machine-type communication (MTC) and the confluence of sensing, communication, and distributed learning. We demonstrate that grant-free multiple access is possible even in the presence of highly time-varying channels. Our approach provides a pathway to standards adoption, since it is built on enhancing the 2-step random access procedure which is already part of the 5GNR standard. This 2-step procedure uses Zadoff-Chu (ZC) sequences as preambles that point to radio resources which are then used to upload data. We also use ZC sequences as preambles / pilots, but we process signals in the Delay-Doppler (DD) domain rather than the time-domain. We demonstrate that it is possible to detect multiple preambles in the presence of mobility and delay spread using a receiver with no knowledge of the channel other than the worst case delay and Doppler spreads. Our approach depends on the mathematical properties of ZC sequences in the DD domain. We derive a closed form expression for ZC pilots in the DD domain, we characterize the possible self-ambiguity functions, and we determine the magnitude of the possible cross-ambiguity functions. These mathematical properties enable detection of multiple pilots through solution of a compressed sensing problem. The columns of the compressed sensing matrix are the translates of individual ZC pilots in delay and Doppler. We show that columns in the design matrix satisfy a coherence property that makes it possible to detect multiple preambles in a single Zak-OTFS subframe using One-Step Thresholding (OST), which is an algorithm with low complexity.

Figures (9)

• Figure 1: A DD domain pulse and its TD/FD 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 $(\tau_0, \nu_0)$. The time duration ($T$) and bandwidth ($B$) of a pulsone are inversely proportional to the characteristic width of the DD domain pulse along the Doppler axis and the delay axis, respectively. The number of non-overlapping DD pulses, each spread over an area $B^{-1}T^{-1}$, inside the fundamental period $\mathcal{D}_0$ (which has unit area) is equal to the time-bandwidth product $BT$ and the corresponding pulsones are orthogonal to one another, rendering OTFS an orthogonal modulation that achieves the Nyquist rate. 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.
• Figure 2: Zak-OTFS transceiver processing.
• Figure 3: TD realization of (a) point pulsone located at $(k_p, l_p) = ((M + 1)/2, (N + 1)/2)$, (b) Zak-OTFS spread pilot with slope $q=3$, and (c) ZC pilot with root $u=23$. In each case, $M = 31, N = 37$ for a Zak-OTFS grid with Doppler period $\nu_p 30$ KHz. RRC pulse shaping with $\beta_\tau = \beta_\nu = 0.6$. Only part of the TD pulsone is shown, with samples taken every $1/4B$ seconds where the bandwidth $B = M\nu_p = 930$ KHz.
• Figure 4: Blue dots mark the support of the self-ambiguity function of the Zadoff-Chu spread pilot with $u = 11$, $M = 31, N=37$. The self-ambiguity function is supported on the line $l=-11k$.
• Figure 5: Magnitude of the cross-ambiguity function. (a) Here $M = 35, N = 39$ are odd and $u = 11, w = 13$ and the cross-ambiguity is flat ($\vert A_{u, w}[k, l] \vert = 1/\sqrt{MN}$). (b) Here $M=32$ is even, $N=37, u = 11, w = 13$ and the cross ambiguity is no longer flat.