# Orthogonal Time Frequency Space

Orthogonal Time Frequency Space (OTFS) is a 2D modulation technique that transforms the information carried in the Delay-Doppler coordinate system. The information is transformed in a similar time-frequency domain as utilized by the traditional schemes of modulation such as TDMA, CDMA, and OFDM. It was first used for fixed wireless, and is now a contending waveform for 6G technology due to its robustness in high-speed vehicular scenarios.

## Overview

OTFS is a modulation scheme where each transmitted symbol experiences a near-constant channel gain even in channels at high carrier frequencies (mm-wave) or with high Doppler. This OTFS signal is well localized in both the time and frequency domain. The transmitted signal is in the delay-doppler domain. OTFS waveform remains invariant under the operation of the time and frequency domains. When we transmit an OTFS waveform in the delay-doppler domain, we use the Zak transform. This OTFS will satisfy the Heisenberg Uncertainty principle (signal is localized in delay-doppler representation).
It effectively transforms the time-varying multipath channel into a 2D channel in the Delay-Doppler domain. Using this transformation, along with equalization within this domain, each symbol experiences similar channel gain throughout the transmission.
The modulation begins with first mapping the information symbols x[k,l] in the Delay–Doppler domain to symbols X [n, m] for creating the time-domain signal s(t) which is transmitted over a wireless channel. At the receiver end, the time-domain signal r(t) is mapped to the domain of time-frequency using the Wigner transform, which is the inverse of the Heisenberg transform and then for symbol demodulation uses the Delay–Doppler domain.
The technology is being considered for 6G networks.
In terms of transmission, the transmit signals of OTFS in either discrete time sequence or continuous time waveform are the same as that of single antenna vector OFDM (VOFDM) systems (Proceedings of ICC 2000, New Orleans, and IEEE Trans. on Communications, Aug. 2001), no matter a channel is stationary or not.

## Channel Equalization and Estimation

Low complexity equalization has been proposed based on Message Passing (MP), Markov Chain Monte Carlo (MCMC), and Linear equalization methods. The diversity of OTFS modulation has been studied in. Channel estimation pilots are transmitted in the delay Doppler domain.
Iterative Rake decision feedback equalization achieves equivalent performance to message passing with a much lower complexity that is independent of the modulation size.
The performance of OTFS modulation in static multi-path channels has also been studied.

## Practical Pulse Shaping Waveforms

It is impossible to transmit an ideal pulse shape due to the time-frequency uncertainty principle. This motivated some works for practical pulse shaped OTFS systems.

## Pulsone

A pulsone (stands for pulse + tone) is the time realization of a quasi-periodic pulse in delay-Doppler and it serves as the carrier waveform of the OTFS modulation format. Of particular interest are pulsones in the crystalline regime (when the periods are greater than the spread of the channel). In this regime, the pulsone remains invariant under the operations of time delay and Doppler shift which results in non-fading and predictable channel interaction, rendering pulsones ideal for mobility and machine learning applications.

## Application

OTFS offers several advantages in particular environments where the dispersion is at high frequency. Environments such as these are encountered in mm-wave systems, due to both larger Doppler spreads and higher phase noise. Application of OTFS waveforms for Radio Detection and Ranging (RADAR) have also been proposed recently.
High mobility scenarios, such as fast-moving vehicles or dynamic wireless networks, introduce severe channel impairments due to rapid time-varying fading, Doppler shift, and time dispersion. OFDM, with its fixed orthogonal subcarriers, struggles to cope with severe channel variations. As a result, the performance of OFDM degrades significantly, leading to reduced data rates and increased error rates.
OTFS addresses the challenges posed by high mobility scenarios by employing time and frequency transformations. OTFS converts the time-varying fading channel into a quasi-static channel, eliminating the need for Doppler compensation. This transformation turns the time-varying channel into stable flat fading, improving signal reception and reducing packet loss significantly.
OTFS achieves better spectral efficiency due to its ability to mitigate inter-symbol interference (ISI) and inter-carrier interference (ICI), which are common in OFDM systems under high mobility.
OTFS also demonstrates improved energy efficiency compared to OFDM in high mobility scenarios. The reduced packet loss and improved spectral efficiency in OTFS lead to fewer retransmissions, resulting in lower power consumption and increased battery life in mobile devices.

## Patents

The idea for OTFS was first patented in 2010 by Ronny Hadani and Shlomo Rakib and transferred to Cohere Technologies Inc in 2011. In December 2022, during the inaugural 6G Evolution Summit event opening keynote, Fierce Wireless moderator referred to Hadani as “The Father of OTFS.”