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Orthogonal Time Frequency Space (OTFS)

Orthogonal Time Frequency & Space (OTFS) is an innovative modulation scheme (air interface) that delivers a multi-dimensional characterization of the wireless channel to comprise a complete, continuous and reliable representation of the wireless channel.

OTFS is a 2D modulation technique that modulates information (that is, QAM) symbols onto a set of two-dimensional (2D) orthogonal basis function that spans the bandwidth and time duration of the transmission burst or packet. This 2D modulation technique transforms data carried in a delay-Doppler coordinate system to the familiar time-frequency domain utilized by traditional modulation schemes, such as OFDM, CDMA and TDMA. From a broader perspective, OTFS establishes a conceptual link between radar and communication.

OTFS overcomes inherent issues of wireless communication far more effectively than current modulation schemes, such as TDMA and OFDM, to dramatically increase bandwidth and throughput. Since OTFS is a multi-dimensional scheme, it completely changes how a wireless channel is seen – from the transmitter to the receiver and everything in between. OTFS manages signal defraction, reflection and absorption inherent in wireless transmission to virtually eliminate signal fading, capacity loss, and other issues to ensure excellent signal reception throughout coverage areas while maximizing throughput.

OTFS method of transforming the time-varying multipath channel into a time-invariant delay-Doppler two-dimensional convolution channel helps eliminate the difficulties in tracking time-varying fading, for example in high speed vehicle communication. Moreover, OTFS increases the coherence time of the channel by orders of magnitude. It simplifies signaling over the channel using well studied AWGN codes over the average channel SNR (signal-to-noise ratio). More importantly, it enables linear scaling of throughput with the number of antennas in moving vehicle applications due to the inherently accurate and efficient estimation of channel state information (CSI). In addition, since the delay-Doppler channel representation is very compact, OTFS enables massive MIMO and beamforming with CSI at the transmitter for four, eight, and more antennas in moving vehicle applications. The CSI information needed in OTFS is a fraction of what is needed to track a time varying channel.

Introduction

OTFS modulation scheme multiplexes QAM information symbols in a signal representation called delay-Doppler. In mathematical literature, the delay-Doppler representation is sometimes referred to as the lattice representation of the Heisenberg group. The structure was later rediscovered by physicists who refer to it as the Zak representation.^[1] Delay-Doppler representation generalizes time and frequency representations, rendering OTFS as a far-reaching generalization.

OTFS creates a waveform that optimally couples with the wireless channel to capture the physics of the channel. This yields a high-resolution delay-Doppler radar image of the constituent reflectors. This results in a simple symmetric coupling between the channel and the information carrying QAM symbols. The symmetry manifests itself through three fundamental properties:

Invariance
Separability
Orthogonality

Invariance is the coupling pattern that is the same for all QAM symbols (that is, all symbols experience the same channel, or the coupling is a translation invariant). Separability (also known as, “hardening”) means that all diversity paths are separated from one another, which makes each QAM symbol experience all the diversity paths of the channel. Finally, orthogonality means that the coupling is localized, which implies that each QAM symbol remains roughly orthogonal to one another at the receiver. The orthogonality property should be contrasted with conventional PN (pseudonoise) sequence-based CDMA modulations, where every codeword introduces a global interference pattern that affects all the other codewords. The invariance property should be contrasted with TDM and FDM, where the coupling pattern vary significantly among different, time-frequency coherence intervals.

A variant of OTFS can be architected over an arbitrary multicarrier modulation scheme by means of a two-dimensional (symplectic) Fourier transform between a grid in the delay-Doppler plane and a grid in the reciprocal time-frequency plane. The Fourier relation creates a family of orthogonal 2D basis functions on the time-frequency grid, where each function can be viewed as a codeword that spreads over multiple tones and multiple multicarrier symbols. This interpretation renders OTFS as a time-frequency spreading technique that generalizes CDMA.

Background

To begin to understand OTFS, consider the foundation of signal processing, which at its core revolves around two basic signal principles: time and frequency representation. Using Fourier transform, these two representations are interchangeable and complement one another.

Specifically, if a signal is localized in time, then it is non-localized in frequency. Equivalently, if a signal is localized in frequency, then it is non-localized in time (shown in the following figure). This mathematical fact hides a deeper truth. As it turns out, there are signals that behave as if they are simultaneously localized to any desired degree both in time and in frequency; that is, a property that renders them optimal both for delay-Doppler radar multi-target detection and for wireless communication (two use cases are strongly linked).

Delay-Doppler variables are commonly used in radar and communication theory. In radar, they are used to represent and then separate moving targets by both delay (range) and Doppler (velocity). In communication, they are used to represent channels by the superposition of time and the frequency of shift operations. The delay-Doppler channel representation is important in wireless communication, since it coincides with the delay-Doppler radar image of the constituent reflectors. The following figure shows an example of the delay-Doppler representation of a specific channel. It is composed of two main reflectors that share similar delay (range), but differ in their Doppler characteristic (velocities).

The use of the delay-Doppler variables to represent channels is well known. However, what is less known is the fact that these variables can also be used to represent information-carrying signals in a way that is harmonious with the delay-Doppler representation of the channel. The delay-Doppler signal representation is mathematically subtler and requires the introduction of a new class of functions called quasiperiodic functions. Therefore, the delay period is represented by 𝜏_𝜏 and the Doppler period is 𝜈_𝜏 satisfying the condition 𝜏_𝜏 ∙ 𝜈_𝜏 = 1 and, thus, defining a box of unit area (as shown in the following figure). A delay-Doppler signal is a function of 𝜙 𝜏, 𝜐 that satisfies the following quasiperiodic condition:

𝜙 (𝜏 + 𝑛𝜏_𝜏, 𝜐 + 𝑚𝜐_𝜏) = 𝑒^j2π(nvr_r-mrv_r) 𝜙 (𝜏, 𝜐)

In summary, there are three fundamental ways to represent a signal:

A function of time
A function of frequency
A function of aquasi-periodic of delay-Doppler.

These three fundamentals are interchangeable by means of canonical transforms. The conversion between the time and frequency representations is carried through the Fourier transform. The conversion between delay-Doppler and the time-frequency representations is carried by the Zak transforms. The Zak transforms are realized by means of periodic Fourier integration formulas:

Specifically, the Zak transform’s time representation is given by the inverse Fourier transform along a Doppler period; and (equally) the Zak transform’s frequency representation is given by the Fourier transform along a delay period. In addition, the quasi-periodicity condition is required for the Zak transform to be a one-to-one equivalence between functions on the one-dimensional line and on the two-dimensional plane. Without it, a signal on the line will admit (infinitely) many delay-Doppler representations.^[2]

Signal Processing

The general framework of signal processing consists of three signal representations:

Time
Frequency
Delay-Doppler

All three are interchangeable by means of canonical transforms. The setting can be organized in a form of a triangle, as shown in the following figure. The nodes of the triangle represent the three representations (time, frequency and delay-Doppler), and the edges represent the canonical transformation rules that converts them.

It is important to note that in this diagram, any pair of transform is equal in that it traverses along the edges of the triangle and results in the same answer no matter of which path is chosen. As a result, one can write the Fourier transform as a composition of two Zak transforms:

FT = Z_t ° Z_f^–1

This means that instead of transforming from frequency to time using the Fourier transform, OTFS transforms from frequency to delay-Doppler using the inverse Zak transform (Z_f^–1), as well as from delay-Doppler to time using the Zak transform (Z_t). The above decomposition yields an alternative algorithm for computing the Fourier transform, which turns out to coincide with the fast Fourier transform algorithm discovered by Cooley-Tukey^[3]. This fact is an evidence that the delay-Doppler representation silently plays an important role in classical signal processing. Note that the delay-Doppler representation is not unique, but depends on a choice of a pair of periods (𝜏_r, 𝜐_r), which satisfies the relation: 𝜏_r∙ 𝜐_r = 1. This implies that there is a continuous family of delay-Doppler representations that correspond to points on the hyperbola 𝜐_r = 1/𝜏_r (shown in the following figure).

Note what happens in the limits of the variable 𝜏_𝜏 → ∞ and the variable 𝜐_𝜏 → ∞. In the first limit, the delay period is extended at the expense of the Doppler period contracting; thus, converging in the limit to a one-dimensional representation coinciding with the time representation. Equally, in the second limit, the Doppler period is extended at the expense of the delay period contracting; thus, converging in the limit to a one-dimensional representation coinciding with the frequency representation.

Therefore, the time and frequency representations is a limiting case of the more general family of delay-Doppler representations. That is, all delay-Doppler representations are interchangeable by means of appropriately defined Zak transforms, which satisfies the commutativity relations generalizing the triangle relation. This means that the conversion between any pair of representations along the curve is independent of the polygonal path that connects them. Furthermore, the delay-Doppler representations and the associated Zak transforms constitute the building blocks of signal processing; in particular, to the classical notions of time and frequency and the associated Fourier transformation rule.

OTFS Modulation Scheme

Basically, communication theory is the transfer of information through two main physical media: wired and wireless. The method that couples a sequence of information-carrying QAM symbols with the communication channel is referred to as a modulation scheme. Thus, the channel-symbol coupling depends both on the physics of the channel and on the modulation carrier waveform. Consequently, every modulation scheme gives rise to a unique coupling pattern, which reflects the way the modulation waveforms interact with the channel.

The classical communication theory revolves around two basic modulation schemes, which are associated with the time and frequency signal representations. The first scheme multiplexes QAM symbols over localized pulses in the time representation called TDM (Time Division Multiplexing). The second scheme multiplexes QAM symbols over localized pulses in the frequency representation (and transmits them using the Fourier transform) called FDM (Frequency Division Multiplexing).

When converting the TDM and FDM carrier pulses to the delay-Doppler representation using the respective inverse Zak transforms, the TDM pulse reveals a quasi-periodic function that is localized in delay, but non-localized in Doppler. Conversely, converting the FDM pulse reveals a quasi-periodic signal that is localized in Doppler, but non-localized in delay. The polarized, non-symmetric delay-Doppler representation of the TDM and FDM pulses suggests a superior modulation based on symmetrically localized signals in the delay-Doppler representation.

OTFS allows for an infinite number of corresponding modulation schemes to the different delay-Doppler representation, which are parameterized by points of the delay-Doppler curve. However, the traditional time and frequency modulation schemes (that is, TDMA and OFDM) appear as limiting cases of the OTFS family, when the delay-Doppler periods approach infinity. The OTFS family of modulation schemes smoothly interpolates between time and frequency division multiplexing.

OTFS Carrier Waveform

An explicit description of the OTFS carrier waveform as a function of time, consider a two-dimensional grid in the delay-Doppler plane as specified by the following parameters:

This defined grid consists of an 𝑁 point along the delay period, with spacing Δ𝜏 and 𝑀 points along the Doppler period, with spacing 𝛥𝜐, resulting with a total of 𝑁𝑀 grid points inside the fundamental rectangular domain. Next, there is a localized pulse (𝑤_n,m) in the delay-Doppler representation at a specific grid point 𝑛Δ𝜏,𝑚Δ𝜐. In addition, a pulse is only localized inside the boundaries of the fundamental domain (enclosed by the delay-Doppler period), and will repeat itself quasi-periodically over the whole delay-Doppler plane (shown in the following figure) with 𝑛 = 3 and 𝑚 = 2. It is assumed that 𝑤_n,m is a product of two one-dimensional pulses:

^ After Joshua Zak, Department of Physics, Technion – Israel Institute of Technology
^ In a sense, the situation resembles the Fourier equivalence between sampled functions on the line and periodic functions on the line. Without imposing periodicity, a sampled function will have infinitely many representations in the Fourier domain.
^ More accurately, the FFT algorithm amounts to a decomposition of the Fourier transform into a sequence of intermediate Zak transforms converting between the points of a polygonal decomposition of the delay- Doppler curve.

[1] After Joshua Zak, Department of Physics, Technion – Israel Institute of Technology

[2] In a sense, the situation resembles the Fourier equivalence between sampled functions on the line and periodic functions on the line. Without imposing periodicity, a sampled function will have infinitely many representations in the Fourier domain.

[3] More accurately, the FFT algorithm amounts to a decomposition of the Fourier transform into a sequence of intermediate Zak transforms converting between the points of a polygonal decomposition of the delay- Doppler curve.

[1]

[2]

[3]