Zak-OTFS for Integration of Sensing and Communication

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. The filter taps can simply be read off from the response to a single Zak-OTFS point (impulse) pulsone waveform, and the I/O relation can be reconstructed for a sampled system that operates under finite duration and bandwidth constraints. Predictability opens up the possibility of a model-free mode of operation. The time-domain realization of a Zak-OTFS point pulsone is a pulse train modulated by a tone, hence the name, pulsone. The Peak-to-Average Power Ratio (PAPR) of a pulsone is about $15$ dB, and we describe a general method for constructing a spread pulsone for which the time-domain realization has a PAPR of about 6dB. We construct the spread pulsone by applying a type of discrete spreading filter to a Zak-OTFS point pulsone. The self-ambiguity function of the point pulsone is supported on the period lattice ${\Lambda}_{p}$, and by applying a discrete chirp filter, we obtain a spread pulsone with a self-ambiguity function that is supported on a rotated lattice ${\Lambda^*}$. We show that if the channel satisfies the crystallization conditions with respect to ${\Lambda^*}$ then the effective DD domain filter taps can simply be read off from the cross-ambiguity between the channel response to the spread pulsone and the transmitted spread pulsone. If, in addition, the channel satisfies the crystallization conditions with respect to the period lattice ${\Lambda}_{p}$, then in an OTFS frame consisting of a spread pilot pulsone and point data pulsones, after cancelling the received signal corresponding to the spread pulsone, we can recover the channel response to any data pulsone.

Zak-OTFS and LDPC Codes

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 $\tau_{p}$ and a Doppler period $\nu_{p}$, where the product $\tau_{p}\nu_{p}=1$. 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.

OTFS -- Predictability in the Delay-Doppler Domain and its Value to Communication and Radar Sensing

In our first paper [2] we explained why 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. We argued that a communication system should operate within the crystalline regime. In this paper, we provide an explicit formula for reconstructing the Zak-OTFS I/O relation from a finite number of received pilot symbols in the delay-Doppler (DD) domain. This formula makes it possible to study predictability of the Zak-OTFS I/O relation for a sampled system that operates under finite duration and bandwidth constraints. We analyze reconstruction accuracy for different choices of the delay and Doppler periods, and of the pulse shaping filter. Reconstruction accuracy is high when the crystallization condition is satisfied, implying that it is possible to learn directly the I/O relation without needing to estimate the underlying channel. This opens up the possibility of a model-free mode of operation, which is especially useful when a traditional model-dependent mode of operation (reliant on channel estimation) is out of reach (for example, when the channel comprises of unresolvable paths, or exhibits a continuous delay-Doppler profile such as in presence of acceleration). Our study clarifies the fundamental origins of predictability by revealing how non-predictability appears as a consequence of aliasing in the DD domain. This perspective leads to a canonical decomposition of the effective DD channel as a sum of predictable and non-predictable components, which we refer to as the crystalline decomposition.

OTFS -- A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain

Orthogonal time frequency space (OTFS) is a framework for communication and active sensing that processes signals in the delay-Doppler (DD) domain. This paper explores three key features of the OTFS framework, and explains their value to applications. The first feature is a compact and sparse DD domain parameterization of the wireless channel, where the parameters map directly to physical attributes of the reflectors that comprise the scattering environment, and as a consequence these parameters evolve predictably. The second feature is a novel waveform / modulation technique, matched to the DD channel model, that embeds information symbols in the DD domain. The relation between channel inputs and outputs is localized, non-fading and predictable, even in the presence of significant delay and Doppler spread, and as a consequence the channel can be efficiently acquired and equalized. By avoiding fading, the post equalization SNR remains constant across all information symbols in a packet, so that bit error performance is superior to contemporary multi-carrier waveforms. Further, the OTFS carrier waveform is a localized pulse in the DD domain, making it possible to separate reflectors along both delay and Doppler simultaneously, and to achieve a high-resolution delay-Doppler radar image of the environment. In other words, the DD parameterization provides a common mathematical framework for communication and radar. This is the third feature of the OTFS framework, and it is ideally suited to intelligent transportation systems involving self-driving cars and unmanned ground/aerial vehicles which are self/network controlled. The OTFS waveform is able to support stable and superior performance over a wide range of user speeds.

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.

Spectral Efficiency of OTFS Based Orthogonal Multiple Access with Rectangular Pulses

In this paper we consider Orthogonal Time Frequency Space (OTFS) modulation based multiple-access (MA). We specifically consider Orthogonal MA methods (OMA) where the user terminals (UTs) are allocated non-overlapping physical resource in the delay-Doppler (DD) and/or time-frequency (TF) domain. To the best of our knowledge, in prior literature, the performance of OMA methods have been reported only for ideal transmit and receive pulses. In [20] and [21], OMA methods were proposed which were shown to achieve multi-user interference (MUI) free communication with ideal pulses. Since ideal pulses are not realizable, in this paper we study the spectral efficiency (SE) performance of these OMA methods with practical rectangular pulses. For these OMA methods, we derive the expression for the received DD domain symbols at the base station (BS) receiver and the effective DD domain channel matrix when rectangular pulses are used. We then derive the expression for the achievable sum SE. These expressions are also derived for another well known OMA method where guard bands (GB) are used to reduce MUI (called as the GB based MA methods) [19]. Through simulations, we observe that with rectangular pulses the sum SE achieved by the method in [21] is almost invariant of the Doppler shift and is higher than that achieved by the methods in [19], [20] at practical values of the received signal-to-noise ratio.