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On OTFS Modulation for High-Doppler Fading Channels

K. R. Murali, A. Chockalingam

arXiv:1802.00929v1cs.IT

TL;DR

OTFS targets doubly-dispersive high-Doppler channels where conventional time-frequency signaling is challenged. The paper proposes MCMC-based detection and PN-pilot delay-Doppler channel estimation, reporting robust BER across high Dopplers and small degradation with sufficiently long pilots. Its conclusion is limited by Gibbs-sampling stalling at high SNRs.

  • Problem

    High mobility and operating frequency cause rapid channel variation, motivating a modulation approach suited to high-Doppler fading channels.

  • Method

    The paper proposes low-complexity MCMC sampling for OTFS detection and PN pilot sequence channel estimation in the delay-Doppler domain.

  • Results

    OTFS BER remains robust at 100 Hz, 444 Hz, and 1851 Hz Dopplers; with Np = 1023, estimated-channel degradation is not significant, whereas Np = 127 causes about 1 dB degradation at 10^-2 BER.

  • Takeaways & Limitations

    Delay-Doppler fade invariance and sparse channel structure support robust detection and efficient channel estimation for high-Doppler OTFS links.

  • Takeaways & Limitations

    Gibbs sampling can stall at high SNRs, degrading BER performance and motivating a modified sampling distribution.

Abstract

from arXiv · show

Orthogonal time frequency space (OTFS) modulation is a 2-dimensional (2D) modulation scheme designed in the delay-Doppler domain, unlike traditional modulation schemes which are designed in the time-frequency domain. Through a series of 2D transformations, OTFS converts a doubly-dispersive channel into an almost non-fading channel in the delay-Doppler domain. In this domain, each symbol in a frame experiences an almost constant fade, thus achieving significant performance gains over existing modulation schemes such as OFDM. The sparse delay-Doppler impulse response which reflects the actual physical geometry of the wireless channel enables efficient channel estimation, especially in high-Doppler fading channels. This paper investigates OTFS from a signal detection and channel estimation perspective, and proposes a Markov chain Monte-Carlo sampling based detection scheme and a pseudo-random noise (PN) pilot based channel estimation scheme in the delay-Doppler domain.

I. INTRODUCTION

OTFS addresses the difficulty of supporting high-Doppler doubly-dispersive channels by signaling in the delay-Doppler domain rather than the conventional time-frequency domain. Its sparse channel representation supports efficient estimation, while this paper proposes low-complexity detection and PN-pilot channel estimation.

  • OTFS approach: OTFS transforms the time-varying multipath channel into a 2D delay-Doppler channel and performs modulation and demodulation in that domain.Its 2D basis functions are delocalized in time-frequency but localized in delay-Doppler.
  • Prior results: OTFS has shown lower block error rates than OFDM across vehicle speeds from 30 km/h to 500 km/h, with especially notable robustness at 500 km/h.OFDM performance breaks down in the cited high-Doppler scenario.
  • Contributions: The paper proposes MCMC-based low-complexity OTFS detection and PN-sequence pilot channel estimation in the delay-Doppler domain.The detection proposal assumes perfect equivalent-channel knowledge, while the estimation scheme relaxes that assumption.
  • Motivation: High mobility and high operating frequency make conventional time-frequency channel estimation and adaptation difficult because channel coefficients vary rapidly.Doubly-dispersive channels combine time dispersion from multipath propagation with frequency dispersion from Doppler shifts.
  • Channel representation: Delay-Doppler taps correspond to reflector groups with particular delays and Dopplers, making the representation compact and sparse because only a small number of such groups exist.The representation reflects reflector geometry, with delay related to relative distance and Doppler to relative velocity.

III. OTFS MODULATION

OTFS adds transmitter and receiver transformations around familiar multicarrier time-frequency modulation. These transformations map symbols between delay-Doppler and time-frequency domains, while OTFS demodulation yields an almost constant fade across a frame under a condition involving channel spreads and windows.

  • OTFS architecture: OTFS uses pre- and post-processing around an inner multicarrier time-frequency modulator to implement delay-Doppler-domain modulation.The outer processing implements OTFS, while the inner box is familiar multicarrier modulation.
  • Transmitter: The OTFS transform maps delay-Doppler symbols x[k, l] to time-frequency symbols X[n, m] using the 2D ISFFT and windowing.The Heisenberg transform then converts the time-frequency symbols into the transmitted time-domain signal.
  • Receiver: At the receiver, the Wigner transform produces Y[n, m], which the symplectic finite Fourier transform maps back to delay-Doppler symbols y[k, l].The Wigner transform is the inverse of the Heisenberg transform, and the SFFT performs delay-Doppler demodulation.
  • OTFS channel effect: In OTFS, each delay-Doppler symbol experiences an almost constant fade hw(0, 0) when the cross-symbol interference condition is satisfied.The condition depends on the channel’s delay and Doppler spreads and on the modulation windows.
  • Channel contrast: In time-frequency modulation, each symbol X[n, m] experiences a different fade H[n, m] within a frame.The input-output relation is Y[n, m] = H[n, m]X[n, m] + V[n, m].

B. OTFS modulation and the delay-Doppler lattice

The delay-Doppler lattice samples a unit-area rectangle, and OTFS uses symplectic Fourier transforms to move between this domain and the time-frequency domain. Under the stated condition, symbols share an almost constant fade.

  • Delay-Doppler lattice: The delay-Doppler representation uses a quasi-periodic lattice whose delay and Doppler periods satisfy τrνr = 1.The representation is non-unique because the periods can vary while preserving their unit-area product.
  • Delay-Doppler lattice: The delay-Doppler lattice samples delay at 1/(M∆f) and Doppler at 1/(NT) over indices k = 0, …, N−1 and l = 0, …, M−1.These samples define the finite delay-Doppler grid used by OTFS.
  • Domain transformations: The SFFT maps periodized time-frequency symbols to the delay-Doppler domain, while the inverse mapping uses the OTFS transform and transmit window.The transmit symbols x[k, l] are mapped to X[n, m] before time-frequency modulation.
  • Receiver processing: Receiver processing applies a receive window, periodizes Y[n, m], and then uses the symplectic Fourier transform to return to the delay-Doppler domain.This processing produces the demodulated delay-Doppler signal.
  • Resulting channel: Each demodulated symbol experiences the same fade hw(0, 0), and each symbol in a frame therefore experiences an almost constant fade.Cross-symbol interference vanishes when the stated condition involving channel spreads and windows holds.

C. Vectorized formulation of the input-output relation

The paper models delay-Doppler OTFS channels through sparse paths with integer and fractional Doppler components, then derives a vectorized input-output relation. Fractional Doppler spreads interference across neighboring Doppler taps.

  • Channel model: The channel is modeled with P propagation paths, each specified by delay τi, Doppler νi, and fade coefficient hi.P is also called the channel sparsity.
  • Tap indexing: Delay and Doppler taps are indexed by integers αi and βi, while fractional Doppler γi accounts for shifts that fall between sampled Doppler points.The channel parameters satisfy 0 ≤ γi < 1.
  • Sparse approximation: Because significant Doppler response is concentrated near each path, only 2Ei + 1 Doppler values need consideration for each path.The retained indices correspond to q from −Ei to Ei, with Ei much smaller than N.
  • Input-output relation: The resulting OTFS input-output relation uses modulo-N indexing and additive Gaussian noise, with fractional Doppler producing inter-Doppler interference across neighboring taps.The neighboring range is −Ei to Ei.
  • Vectorized formulation: Vectorization stacks the delay-Doppler symbols into x and y and forms an NM × NM channel matrix H with P(2Ei + 1) nonzero elements per row and column.This sparsity follows from the modulo operations in the vectorized representation.

IV. OTFS SIGNAL DETECTION

This section presents OTFS signal detection algorithms using MCMC sampling-based techniques.

  • The section presents OTFS signal detection algorithms based on MCMC sampling techniques.
  • MCMC sampling is applied specifically to OTFS signal detection.
  • The detection methods use computational sampling techniques rather than a single explicitly specified detection rule.

A. OTFS signal detection using MCMC sampling

The paper frames OTFS detection as an exponentially complex ML problem and uses Gibbs-sampling MCMC updates to obtain approximate solutions, while addressing stalling through modified sampling distributions.

  • MCMC generates samples through transitions whose probabilities depend only on the previous sample.
  • The ML detector minimizes ∥y − Hx∥2 over the modulation alphabet, but its complexity is exponential in NM.
  • Gibbs sampling approximates the ML solution by randomly initializing x and sequentially sampling coordinate updates from conditional distributions.
  • After burn-in, the detected vector is the sampled vector with the lowest ML cost across iterations.
  • Gibbs sampling can stall and degrade BER at high SNRs, motivating modified distributions intended to reduce stalling and iterations.
  • A temperature parameter α can reduce the expected number of iterations for finding the solution.

2) Gibbs sampling with temperature parameter

The paper modifies Gibbs sampling with randomized updates and notes that temperature selection depends on operating SNR.

  • α depends on the operating SNR, so its value is not fixed across operating conditions.
  • Randomized Gibbs sampling combines conventional Gibbs updates with uniform-distribution sampling using probability r = 1/NM.
  • The randomized Gibbs sampling detector is specified in Algorithm 1.

B. Performance results

The performance study evaluates randomized Gibbs-sampling detection for OTFS under three Doppler frequencies and reports nearly Doppler-invariant BER performance.

  • The evaluation assumes perfect channel knowledge at the receiver and uses a five-tap propagation model.
  • OTFS uses randomized Gibbs-sampling detection with a random initial vector, iterative coordinate updates, and minimum-cost output selection.
  • The simulations represent UE speeds of 27, 100, and 500 kmph as Doppler frequencies of 100, 444.44, and 1851 Hz at 4 GHz.
  • 10^-3 BER is achieved at about 13 dB SNR for all three Doppler frequencies considered.
  • The configuration uses M = 128, N = 32, BPSK, 3.75 kHz subcarrier spacing, and an 8.6 msec frame length.
  • The BER performance is almost invariant to Doppler, illustrating OTFS robustness in high-Doppler fading channels.

V. DELAY-DOPPLER CHANNEL ESTIMATION

The paper relaxes perfect channel knowledge by estimating delay shifts, Doppler shifts, and fade coefficients from a PN pilot in a discrete delay-Doppler model. Sampling and sequence representations simplify the channel-estimation problem.

  • Channel-estimation setup: The PN-pilot estimator targets each path’s delay shift δ_i, Doppler shift ν_i, and fade coefficient α_i.The method operates in the discrete domain and estimates the three channel quantities for each path.
  • Channel-estimation setup: The channel model uses a finite-integer function space H with arithmetic modulo Np.H consists of complex-valued functions on Z_Np = {0, 1, ..., Np−1}.
  • Discrete channel model: Representing waveforms as sequences yields a discrete channel model that simplifies channel estimation.The received signal is sampled to obtain the sequence R[n].
  • Discrete channel model: The pilot duration is extended to M = Np + K, where K = ⌈WT_spread⌉ captures the channel time spread.Here T_spread is the maximum path delay and M must satisfy M ≥ Np.
  • Discrete channel model: Under the stated delay and Doppler grid conditions, Proposition 1 supplies the discrete received-signal model used for channel estimation.The model assumes τ_i ∈ (1/W)Z+ and ν_i ∈ (W/Np)Z.

B. PN pilot based channel estimation

The PN-pilot scheme estimates channel shifts by searching a matched-filter matrix and then recovers path fades at the detected shift locations. Its examples show concentrated peaks and improved separation for longer PN sequences.

  • Shift and fade estimation: The estimator computes the matched-filter matrix M(R,S)[δ,ω] over delay and Doppler shifts to solve the time-frequency shift problem.The matrix is defined as the inner product between the received sequence and a shifted, modulated PN sequence.
  • Shift and fade estimation: It selects each shift pair where the matched-filter value is approximately one, then estimates the corresponding fade as α_i ≈ M(R,S)[δ_i,ω_i].This procedure estimates (α_i, δ_i, ω_i) for each path.
  • Single-path behavior: At SNR = 0 dB with Np = 127, matched-filter magnitudes are near one at the true shift and near zero elsewhere for two single-path settings.The settings are (δ0,ω0) = (40,90) and (80,60).
  • Multi-path behavior: For two-path channels at SNR = 20 dB, the matched-filter matrix exhibits two strong peaks at the two path shifts, whose entries provide the corresponding α_i values.Off-shift magnitudes are very small.
  • Sequence-length effect: For P = 5 at SNR = 20 dB, Np = 1023 produces lower off-shift magnitudes than Np = 127 and can yield more accurate estimates.The comparison uses five specified delay-Doppler path locations.

C. Performance results

The proposed PN-pilot channel estimator achieves lower estimation error with higher pilot SNR and longer sequences, while BER degradation depends on pilot length. Large pilot sequences preserve detection performance more closely but increase estimation complexity.

  • Channel estimation accuracy: Estimation error decreases as pilot SNR increases and as PN sequence length Np grows.The evaluated sequence lengths are Np = 31, 127, and 1023.
  • BER performance: Estimated-channel BER remains close to perfect-channel BER for Np = 1023, whereas degradation increases as Np is reduced.At 10^-2 BER, the degradation is about 1 dB for Np = 127 and becomes severe for Np = 15.
  • BER performance: The pilot-length choice trades estimation accuracy against complexity.Increasing Np improves estimates but also increases estimation complexity.
  • Overall performance: The study reports robust OTFS BER performance at Dopplers of 100 Hz, 444 Hz, and 1851 Hz.The conclusion characterizes OTFS as suited to high-Doppler fading channels.
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