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Derivation of OTFS Modulation from First Principles

Saif Khan Mohammed

arXiv:2007.14357v1cs.IT

TL;DR

The paper addresses the missing rigorous first-principles derivation of OTFS and develops one using the ZAK representation of time-domain signals. It constructs an approximately time- and bandwidth-limited orthonormal DD-domain basis, derives OTFS modulation from it, and shows that increased time duration localizes received basis signals and reduces interference, while increasing latency.

  • Problem

    Prior work had not rigorously derived OTFS modulation from first principles, making its robustness to channel-induced Doppler shift difficult to understand.

  • Method

    Using the ZAK representation, the paper derives an orthonormal basis of approximately time- and bandwidth-limited signals localized in the delay-Doppler domain, then derives DD modulation and OTFS.

  • Results

    Increasing time duration increasingly localizes DD-domain basis signals irrespective of Doppler shift, reducing interference and enabling joint DD-domain equalization.

  • Takeaways & Limitations

    DD-domain modulation is robust to channel-induced Doppler shift, but achieving greater localization and robustness increases transmit-signal duration and latency.

Abstract

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Orthogonal Time Frequency Space (OTFS) modulation has been recently proposed to be robust to channel induced Doppler shift in high mobility wireless communication systems. However, to the best of our knowledge, none of the prior works on OTFS have derived it from first principles. In this paper, using the ZAK representation of time-domain (TD) signals, we rigorously derive an orthonormal basis of approximately time and bandwidth limited signals which are also localized in the delay-Doppler (DD) domain. We then consider DD domain modulation based on this orthonormal basis, and derive OTFS modulation. To the best of our knowledge, this is the first paper to rigorously derive OTFS modulation from first principles. We show that irrespective of the amount of Doppler shift, the received DD domain basis signals are localized in a small interval of size roughly equal to the inverse time duration along the Doppler domain and of size roughly equal to the inverse bandwidth along the delay domain (time duration refers to the length of the time-interval where the TD transmit signal has been limited). With sufficiently large time duration and bandwidth, there is little interference between information symbols modulated on different basis signals, which allows for joint DD domain equalization of all information symbols. This explains the inherent robustness of DD domain modulation to channel induced Doppler shift when compared with Orthogonal Frequency Division Multiplexing (OFDM). The degree of localization of the DD domain basis signals is inversely related to the time duration of the transmit signal, which explains the trade-off between robustness to Doppler shift and latency.

I. INTRODUCTION

The paper addresses the lack of a rigorous first-principles derivation of OTFS by using the ZAK representation to construct localized orthonormal DD-domain basis signals and derive DD modulation. It links DD localization to reduced Doppler-induced interference, joint equalization, and a robustness–latency trade-off.

  • Motivation: Prior OTFS work had not rigorously derived its modulation and basis waveforms from first principles, limiting understanding of its Doppler robustness.The paper motivates deeper analysis for high-mobility scenarios such as high-speed trains and air-to-ground communication.
  • Approach: Using the ZAK representation of TD signals, the paper derives OTFS modulation from DD-domain modulation based on an orthonormal basis.DD information symbols linearly modulate the derived DD-domain basis signals before conversion to a TD transmit signal.
  • Basis construction: The derived basis is approximately time-limited to NT seconds, bandwidth-limited to M∆f Hz, and localized over inverse-bandwidth and inverse-time-duration DD intervals.The basis dimensionality is the time-bandwidth product NT × M∆f = MN.
  • Robustness: DD-domain basis signals experience a smaller fraction of interference than OFDM sub-carriers, with less variation as Doppler shift increases.The paper identifies this localization behavior as explaining DD modulation’s robustness to channel-induced Doppler shift.
  • Trade-off: For fixed bandwidth M∆f, increasing N decreases the fraction of interfered DD basis signals and facilitates joint equalization of all MN information symbols.The corresponding time duration NT increases with N, increasing latency.

II. THE ZAK REPRESENTATION OF TIME-DOMAIN (TD) SIGNALS

The ZAK representation maps time-domain signals into a quasi-periodic delay–Doppler representation, where channel delay and Doppler shifts become shifts along the DD axes. The paper develops localized DD signals and their corresponding time-domain basis waveforms.

  • DD interpretation: A channel-induced TD delay and Doppler shift correspond to shifts along the delay and Doppler axes of the signal’s ZAK representation.This motivates calling the two ZAK coordinates the delay and Doppler domains.
  • ZAK properties: The ZAK representation is periodic along Doppler with period ∆f = 1/T and quasi-periodic along delay with period T.These properties support the correspondence between valid DD representations and time-domain signals.
  • DD localization: The ideal DD-localized signals are simultaneously localized in delay and Doppler, unlike signals requiring exact finite time and frequency support.Their DD representation consists of impulses whose locations and complex values are illustrated in Fig. 1.
  • Ideal basis signals: The ideal DD-localized signal corresponds to a time-shifted impulse train multiplied by a complex exponential.Its impulses are spaced T seconds apart, with the impulse in [0,T) located at τ0.
  • Basis representation: For 0 ≤ τ0 < T and 0 ≤ ν0 < ∆f, the TD signals p(τ0,ν0)(t) form a basis for the space of TD signals.The coefficient of each basis signal is the value of the signal’s ZAK representation at the corresponding delay and Doppler coordinates.

III. AN ORTHONORMAL BASIS FOR TIME AND BANDWIDTH LIMITED SIGNALS WHICH ARE LOCALIZED IN DD DOMAIN

The paper constructs an MN-dimensional orthonormal basis of approximately time- and bandwidth-limited signals localized in the delay-Doppler domain. Their localization supports low-interference DD modulation, while time and bandwidth limitation determines finite localization widths.

  • Signal construction: The construction targets signals approximately limited to [0, NT) in time and [0, M∆f) in frequency for DD-domain modulation.The signals are sought because DD localization can reduce inter-symbol interference and is expected to support Doppler robustness.
  • Signal construction: Time and bandwidth limitation converts ideally point-localized DD impulses into signals spread over finite delay-Doppler intervals.The construction multiplies the ideal basis by a time-limited signal and convolves it with an approximately band-limited signal.
  • DD localization: Most energy is localized around (τ0, ν0) within widths 1/(M∆f) along delay and ∆f/N along Doppler.The resulting sinc pulses have width roughly twice the inverse bandwidth and are restricted to [0, NT) by time limitation.
  • DD localization: Two DD-domain basis signals will not interfere significantly when their locations are separated by roughly 1/(M∆f) in delay and ∆f/N in Doppler.This separation motivates sampling the DD locations on corresponding delay and Doppler grids.
  • Orthonormal basis: The sampled signals form an MN-dimensional orthonormal basis, with indices k = 0, …, N−1 and l = 0, …, M−1.The basis dimension equals the time-bandwidth product M∆f × NT = MN, so DD localization does not reduce the dimensionality.

IV. DELAY-DOPPLER (DD) DOMAIN MODULATION

DD-domain modulation linearly maps complex information symbols onto the derived orthonormal basis signals, producing the time-domain transmit signal through a two-stage transformation.

  • DD-domain modulation: Complex information symbols x[k, l] linearly modulate the basis signals α^(k,l)(t) for k = 0, …, N−1 and l = 0, …, M−1.The resulting time-domain signal is formed by summing the modulated basis signals.
  • Two-stage transformation: The modulation first transforms DD symbols into frequency-domain signals X_n(f), then transforms each X_n(f) into a time-shifted signal x_n(t).The time-shifted signals are added to form the transmit waveform.

V. DERIVATION OF OTFS MODULATION

The paper derives OTFS modulation from DD-domain modulation by showing that, for sufficiently large M, the resulting waveform is equivalent to rectangular-pulse OTFS and can use practical OFDM hardware.

  • Derivation: For sufficiently large M, each component x_n(t) is approximately time-limited to [0, T), enabling its shifted copy to occupy approximately [nT, (n+1)T).The corresponding X_n(f) is called the n-th time-frequency signal.
  • Derivation: Theorem 4 derives the OTFS modulation expression from the DD-domain modulation equation under the sufficiently-large-M condition.The theorem identifies the derived signal with the OTFS construction used subsequently.
  • Derivation: The resulting equation is exactly the OTFS modulation equation with a rectangular transmit pulse g(t).The rectangular pulse is defined over [0,T) and is zero otherwise.
  • Implementation: For sufficiently large M, DD-domain modulation is approximately equal to a waveform computable by an OFDM modulator with sub-carrier spacing ∆f and M sub-carriers.This provides a practical implementation using existing 4G/5G modem hardware.
  • Performance: For sufficiently large M, the paper expects OTFS spectral efficiency to match DD-domain modulation, and simulations report that they are the same.The claim concerns the spectral-efficiency performance of the two modulation forms.

VI. SPECTRAL EFFICIENCY OF DD DOMAIN MODULATION

This section derives spectral efficiency for DD domain modulation using a ZAK receiver and a discrete sampled received DD-domain signal.

  • Receiver model: A ZAK receiver samples the received signal's ZAK representation at discrete DD-domain points.The sampled signal is denoted Y[k′, l′], from which DD-domain information symbols are decoded.
  • Received-signal expression: Theorem 5 expresses each sampled received DD-domain signal Y[k′, l′] in terms of the information symbols and transformed additive white Gaussian noise.The theorem identifies Zn(τ, ν) as the ZAK representation of receiver noise n(t).
  • Spectral efficiency: Theorem 6 gives the spectral efficiency achieved by DD domain modulation with i.i.d. complex Gaussian information symbols and a ZAK receiver.The expression uses ρ, the ratio of average transmit power to received noise power within the communication bandwidth M∆f.
  • Matrix formulation: The spectral-efficiency expression is parameterized by MN × MN identity and channel-related matrices ˜H and ˜K.The elements of ˜H and ˜K are specified through the channel coefficients in the theorem's matrix formulation.

VII. WHY IS DD DOMAIN MODULATION BETTER THAN OFDM ?

The section analyzes Doppler-induced interference and reports that DD-domain modulation keeps interference substantially more localized than CP-OFDM, while larger N trades lower interference for higher latency.

  • Interference mechanism: For integer-grid delay and Doppler shifts, the received DD-domain basis signal has no inter-symbol interference.The grid spacings are 1/(M∆f)=T/M for delay and ∆f/N=1/(NT) for Doppler.
  • Interference mechanism: Non-integer delay or Doppler shifts spread a symbol's DD-domain energy into other basis signals, creating inter-symbol interference.The received coefficient is analyzed as a product of terms depending separately on Doppler-index and delay-index differences.
  • DD localization: The Doppler-domain main-lobe width is two, independent of both N and ν′.The function governing Doppler leakage is periodic with period N, and its main lobe lies between adjacent zero crossings.
  • DD-domain interference: For N=23, 46, 92, the reported interfered-symbol fractions are 12.67%, 8.55% and 6.5%, respectively.The section states that the upper bound decreases as N increases for fixed M.
  • Comparison with OFDM: The maximum interfered-symbol fraction is 7.6% for DD domain modulation versus about 48% for CP-OFDM.The lower DD-domain interference makes joint equalization of all MN information symbols practically feasible.
  • Trade-off: Increasing N improves DD energy localization and joint equalization but increases signal duration to NT, producing a robustness–latency trade-off.The modulated signal lasts N times longer than an OFDM symbol.

VIII. NUMERICAL SIMULATION

The numerical simulation compares DD domain modulation with OTFS in a two-path unmanned-aircraft communication scenario across aircraft speeds and power ratios.

  • Channel model: The simulation models control and non-payload communication between an Unmanned Aircraft System and a Ground Station using a direct and reflected path.The reflected-path delay is 33 µs, and the model uses a Rician factor typically set to 15 dB.
  • Simulation parameters: The simulated channel uses 90 KHz bandwidth, 23 ms time duration, carrier frequency 5.06 GHz, M=45, N=46, and ∆f=2 KHz.These choices give T=0.5 ms, M∆f=90 KHz, and NT=23 ms.
  • Evaluation: Figure 8 plots average spectral efficiency against aircraft speed for DD domain modulation and OTFS at different values of ρ.The DD-domain result is E[C], while the OTFS result is E[CZak].
  • Results: For a given ρ, spectral-efficiency performance remains constant as aircraft speed and Doppler shift increase for both modulation schemes.The section also reports identical spectral-efficiency performance for DD domain modulation and OTFS.

IX. CONCLUSION

The conclusion presents a first-principles derivation of OTFS from ZAK-represented time-domain signals and connects DD localization with reduced interference and a latency cost.

  • Conclusion: The paper derives an orthonormal basis of approximately time- and bandwidth-limited signals that are localized in the delay-Doppler domain.The basis is constructed using the ZAK representation of time-domain signals.
  • Conclusion: DD-domain modulation is built from this basis, and OTFS modulation is derived from the resulting DD-domain modulation.The paper identifies this as a rigorous first-principles derivation.
  • Conclusion: Increasing time duration increasingly localizes the basis signals in the DD domain regardless of Doppler-shift magnitude.The localized energy reduces interference between symbols carried by different basis signals.
  • Conclusion: Reduced interference enables joint DD-domain equalization of all information symbols and supports robustness to Doppler shift.The conclusion attributes this robustness to the increased DD-domain localization associated with longer time duration.
  • Conclusion: DD-domain robustness to Doppler shift is achieved at the cost of increased latency.The conclusion links the cost to the time duration of the transmit signal.

APPENDIX H PROOF OF THEOREM 3

The proof establishes that the constructed signal family forms an orthonormal basis with MN signals.

  • The sinc-product orthogonality condition is nonzero only when both signal indices match.Because the index differences cannot be nonzero integer multiples of M, distinct basis signals are orthogonal.
  • The basis contains MN orthonormal signals and is therefore MN-dimensional.

APPENDIX I PROOF OF THEOREM 5

The proof uses ZAK representations and linearity to express transmitted and received signals in terms of delay-Doppler information symbols.

  • The ZAK representation of the received signal y(t) is introduced as the starting point for the derivation.
  • The ZAK representations of the transmit signal and noise are combined using the received-signal relation.The derivation explicitly invokes the representations Zx(τ, ν) and Zn(τ, ν).
  • Linearity of the ZAK representation and the basis-signal expression produce the representation of the modulated signal.
  • Substituting the basis representation into the received-signal expression yields the sampled DD-domain signal Y[k′, l′].The derivation proceeds from the information symbols x[k, l] to an expression identified as equation (34).

APPENDIX J PROOF OF THEOREM 6

The proof develops the DD-domain input-output and noise representation, then analyzes OFDM interference under Doppler shift using energy concentration across subcarriers.

  • The received signal is limited to [0, (N + 1)T), and the information symbols are modeled as independent CN(0, ρ) variables.
  • The average transmit power is Mρ/T, while unit-PSD AWGN contributes receiver noise power M∆f.
  • The received DD-domain samples and information symbols are organized into vectors, with an effective DD-domain channel matrix relating them.The matrix entries are defined from the DD-domain channel coefficients, while the noise samples form a separate vector.
  • The DD-domain noise covariance is specified through the entries of the covariance matrix Kz.The indexed covariance expression distinguishes equal and unequal DD-domain indices.
  • OFDM comparison: Under Doppler shift ν′, OFDM energy from subcarrier k spreads according to |Hofdm[m, k]|2 = |h′|2 sinc2(ν′T + k − m).Most energy is received around subcarrier m = ⌊k + ν′T⌋.
  • OFDM comparison: The smallest subcarrier set Gk capturing at least 0.99 of x[k]'s energy defines the interference calculation.The fraction of interfered information symbols is then derived from this concentration set.
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