User:Cohere-OTFS/sandbox

= 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’ two-dimensional (2D) modulation scheme modulates information (that is, QAM) symbols onto a set of 2D orthogonal basis function, which spans the bandwidth and time duration of the transmission burst or packet. This 2D modulation scheme transforms data carried in a delay-Doppler coordinate system to the familiar time-frequency domain that is 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.

History
Since the introduction of cellphones, every transition to a new generation of wireless network involves a disruption in the underlying air interface. Starting with the transition from 2G networks based on single carrier GSM to 3G networks based on code division multiplexing (CDMA), then followed by the transition to contemporary 4G networks based on orthogonal frequency division multiplexing (OFDM).

The main motivation to introduce a new air interface is made when the demands of a new generation of wireless devices cannot be met by the current (legacy) technology. This can be because of performance, capabilities, and cost. For example, the demand for higher capacity data services drove the transition from interference-limited CDMA network (limited in flexibility to adapt and its inferior achievable throughput) to a network based on an orthogonal narrowband OFDM, which is optimally fit for opportunistic scheduling to achieve higher spectral efficiency.

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. 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 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.
 * Invariance
 * Separability
 * Orthogonality

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.

Key features
The following are the key features of OTFS:

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).

Signal Processing
The general framework of signal processing consists of three signal representations: 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.Note what happens in the limits of the variable 𝜏𝜏 → ∞ and the variable 𝜐𝜏 → ∞. 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.
 * Time
 * Frequency
 * Delay-Doppler

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.

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.

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:

Delay-Doppler Channel Symbol Coupling
The wireless channel is composed of a collection of specular reflectors, some static and some moving. The transmitted waveform propagates through the medium and bounces off each reflector. The signal that arrives at the receiver is a superposition of the direct signal and reflected echoes. Each of the reflected echoes arrives at the receiver at a delayed time (multipath effect) and (possibly) also the shift in frequency (Doppler effect) due to the relative velocity between the reflector and the transmitter/receiver. The channel physics is mathematically modeled through the delay-Doppler impulse response, where each tap represents a cluster of reflectors with specific delay and Doppler characteristics (as shown in the following figure).

Multi-carrier Interpretations of OTFS
One variant of OTFS is more adaptive to the classical multicarrier formalism of time-frequency grids and filter-banks, which illuminates that is aspects of OTFS that are not apparent from the Zak definition. As a result of this definition, OTFS can be viewed as a time-frequency spreading scheme composed of a collection of two-dimensional basis-functions (or codewords) defined over a reciprocal time-frequency grid. Another result is that OTFS can be architected as a simple pre-processing step over an arbitrary multicarrier modulation, such as OFDM. This definition is based on Fourier duality relation between a grid in the delay-Doppler plane and a reciprocal grid in the time-frequency plane.

The delay-Doppler grid consists of 𝑁 points along delay with spacing 𝛥𝜏 = 𝜏r/𝑁 and 𝑀 points along Doppler with spacing 𝛥𝜐= 𝜐r/𝑀 and the reciprocal time-frequency grid consists of 𝑁 points along frequency with spacing 𝛥𝑓= 1/𝜏r and 𝑀 points along time with spacing 𝛥𝑡= 1/𝜐r (as shown in the figure below). The parameter 𝛥𝑡 is the multicarrier symbol duration, and the parameter 𝛥𝑓 is the subcarrier spacing. The time frequency grid can be interpreted as a sequence of 𝑀 multi-carrier symbols each consisting of 𝑁 tones or subcarriers. Note that the bandwidth of the transmission 𝐵= 𝑀𝛥𝑓 is inversely proportional to the delay resolution 𝛥𝜏, and the duration of the transmission 𝑇= 𝑀𝛥𝑡 is inversely proportional to the Doppler resolution 𝛥𝜏. From a broader perspective, the Fourier duality relation between the delay-Doppler grid and the time-frequency grid establishes a mathematical link between Radar and communication, where the first theory is concerned with maximizing the resolution of separation between reflectors/targets (according to their delay-Doppler characteristics). In addition, the second is concerned with maximizing the amount of information that can be reliably transmitted through the communication channel composed of these reflectors.

Summary
OTFS is a novel family modulation scheme based on multiplexing the QAM information symbols over localized pulses in the delay-Doppler signal representation. The OTFS modulation schemes constitute a far-reaching generalization of traditional time and frequency modulation schemes such as TDMA and OFDM, which can be shown to be limiting cases of the OTFS family.

From a broader perspective, OTFS establishes a conceptual link between Radar and communication. The OTFS waveforms couple with the wireless channel in a way that directly captures the underlying physics, yielding a high-resolution delay-Doppler radar image of the constituent reflectors. Thus, the time-frequency selective channel is converted into an invariant, separable and orthogonal interaction, where all received QAM symbols experience the same localized impairment and all the delay-Doppler diversity branches are coherently combined.

The OTFS channel-symbol coupling allow linear scaling of capacity with the MIMO order while satisfying an optimal performance-complexity tradeoff both at the receiver end (using joint ML detection) and at the transmitter end (using Tomlinson-Harashima precode for MU-MIMO). OTFS enables significant spectral efficiency advantages in high order MIMO under general channel conditions over traditional modulation schemes including multicarrier modulations such as OFDM.

OTFS can be viewed as a special type of a time-frequency spreading technique, where each QAM symbol is carried by a two-dimensional basis function spread over the full time-frequency grid. When viewed as a time-frequency spreading technique, OTFS exhibits architectural compatibility with any type of multicarrier modulation scheme, including conventional OFDM. An OTFS packet can be flexibly designed to populate arbitrary regions of the time-frequency grid and be compatible with any convention for channel reference signaling. As a spread spectrum, OTFS enjoys resilience to narrowband interference and full diversity gain.

OTFS resilience to interference makes it ideal to supporting ultra-reliable low latency communication packets overlaid on regular data packets. OTFS diversity gain makes it ideal for communication under mobility conditions. OTFS supports a small packet allocation method (called Doppler transversal allocation) that maximizes link budget and minimizes number of retransmissions under transmit power and latency constraints by achieving the PAPR of single carrier, extracting full time-frequency diversity and maximizing restricted capacity. OTFS, with Doppler transversal allocation, is superior to conventional multicarrier DFT spread techniques such as SC-FDMA and its more elaborate hopped variant for applications of IoT.

3GPP has identified a variety of eMBB deployment scenarios that focus on MU-MIMO. The advantage of OTFS in scaling capacity with the MIMO order makes it ideal for these kinds of deployments. In addition, the new radio air interface must support high spectral efficiency in high Doppler environments. OTFS is ideally suited for these requirements, providing: high spectral efficiency and reliability under diverse channel conditions.