Delay-Doppler Signal Processing with Zadoff-Chu Sequences

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.

Zak-OTFS with Interleaved Pilots to Extend the Region of Predictable Operation

When the delay period of the Zak-OTFS carrier is greater than the delay spread of the channel, and the Doppler period of the carrier is greater than the Doppler spread of the channel, the effective channel filter taps can simply be read off from the response to a single pilot carrier waveform. The input-output (I/O) relation can then be reconstructed for a sampled system that operates under finite duration and bandwidth constraints. We introduce a framework for pilot design in the delay-Doppler (DD) domain which makes it possible to support users with very different delay-Doppler characteristics when it is not possible to choose a single delay and Doppler period to support all users. The method is to interleave single pilots in the DD domain, and to choose the pilot spacing so that the I/O relation can be reconstructed by solving a small linear system of equations.

Low Complexity Channel Estimation for OTFS Modulation with Fractional Delay and Doppler

We consider the problem of accurate channel estimation for OTFS based systems with few transmit/receive antennas, where additional sparsity due to large number of antennas is not a possibility. For such systems the sparsity of the effective delay-Doppler (DD) domain channel is adversely affected in the presence of channel path delay and Doppler shifts which are non-integer multiples of the delay and Doppler domain resolution. The sparsity is also adversely affected when practical transmit and receive pulses are used. In this paper we propose a Modified Maximum Likelihood Channel Estimation (M-MLE) method for OTFS based systems which exploits the fine delay and Doppler domain resolution of the OTFS modulated signal to decouple the joint estimation of the channel parameters (i.e., channel gain, delay and Doppler shift) of all channel paths into separate estimation of the channel parameters for each path. We further observe that with fine delay and Doppler domain resolution, the received DD domain signal along a particular channel path can be written as a product of a delay domain term and a Doppler domain term where the delay domain term is primarily dependent on the delay of this path and the Doppler domain term is primarily dependent on the Doppler shift of this path. This allows us to propose another method termed as the two-step method (TSE), where the joint two-dimensional estimation of the delay and Doppler shift of a particular path in the M-MLE method is further decoupled into two separate one-dimensional estimation for the delay and for the Doppler shift of that path. Simulations reveal that the proposed methods (M-MLE and TSE) achieve better channel estimation accuracy at lower complexity when compared to other known methods for accurate OTFS channel estimation.