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MIMO-OTFS in High-Doppler Fading Channels: Signal Detection and Channel Estimation

M. Kollengode Ramachandran, A. Chockalingam

arXiv:1805.02209v1cs.IT

TL;DR

High-Doppler doubly-dispersive channels challenge conventional OFDM, motivating MIMO-OTFS for combined MIMO efficiency and OTFS robustness. The paper develops message-passing detection and delay-Doppler pilot-based channel estimation, achieving strong BER performance and only small degradation from channel estimation.

  • Problem

    High-Doppler MIMO channels create difficult detection and channel-estimation conditions because rapid channel variation degrades OFDM performance and time-frequency estimation.

  • Method

    The paper uses a vectorized MIMO-OTFS input-output model, low-complexity message-passing detection, and delay-Doppler impulses as channel-estimation pilots.

  • Results

    MIMO-OTFS achieves 10−5 BER at about 14 dB SNR for a 2 × 2 system at 1880 Hz Doppler, while estimated-channel BER is close to perfect-channel performance.

  • Takeaways & Limitations

    The reported results support MIMO-OTFS as a robust signaling approach for rapidly varying channels, with efficient delay-Doppler channel estimation.

Abstract

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Orthogonal time frequency space (OTFS) modulation is a recently introduced multiplexing technique designed in the 2-dimensional (2D) delay-Doppler domain suited for high-Doppler fading channels. OTFS converts a doubly-dispersive channel into an almost non-fading channel in the delay-Doppler domain through a series of 2D transformations. In this paper, we focus on MIMO-OTFS which brings in the high spectral and energy efficiency benefits of MIMO and the robustness of OTFS in high-Doppler fading channels. The OTFS channel-symbol coupling and the sparse delay-Doppler channel impulse response enable efficient MIMO channel estimation in high Doppler environments. We present an iterative algorithm for signal detection based on message passing and a channel estimation scheme in the delay-Doppler domain suited for MIMO-OTFS. The proposed channel estimation scheme uses impulses in the delay-Doppler domain as pilots for estimation. We also compare the performance of MIMO-OTFS with that of MIMO-OFDM under high Doppler scenarios.

I. INTRODUCTION

High-Doppler channels create both time and frequency dispersion, challenging OFDM, while OTFS multiplexes symbols in the delay-Doppler domain and supports robust MIMO signaling. The paper develops MIMO-OTFS detection and delay-Doppler channel estimation, reporting strong high-Doppler BER performance and near-perfect-channel performance.

  • I. INTRODUCTION: High mobility and millimeter-wave operation produce doubly-dispersive channels with time dispersion from multipath and frequency dispersion from Doppler shifts.OFDM mitigates ISI but Doppler-induced ICI degrades its performance.
  • I. INTRODUCTION: OTFS multiplexes modulation symbols in the delay-Doppler domain rather than the time-frequency domain used by conventional schemes such as OFDM.Its waveform has been reported as resilient to delay-Doppler shifts and especially robust at high vehicle speeds.
  • I. INTRODUCTION: MIMO-OTFS combines MIMO spectral and energy efficiency with OTFS robustness in rapidly varying channels.The paper addresses vectorized modeling, message-passing detection, and delay-Doppler channel estimation.
  • I. INTRODUCTION: 10−5 BER is achieved at about 14 dB SNR in 2 × 2 MIMO-OTFS at 1880 Hz Doppler, while same-system MIMO-OFDM performance floors at BER 0.02.The reported Doppler corresponds to 500 km/hr at 4 GHz.
  • I. INTRODUCTION: The proposed delay-Doppler pilot-based channel estimation loses less than a fraction of a dB relative to perfect channel knowledge.The paper presents this scheme as simple and effective for high-Doppler MIMO channels.
  • I. INTRODUCTION: The paper proceeds from OTFS modulation and MIMO-OTFS modeling to message-passing detection, channel estimation, performance evaluation, and conclusions.These topics are organized across Sections II–VI.

B. OTFS modulation •

OTFS maps delay-Doppler information symbols into a time-frequency waveform for transmission and reverses the transformations at reception. Its input-output relation incorporates the channel response and windowing operations.

  • B. OTFS modulation •: The SFFT maps periodized time-frequency symbols into delay-Doppler symbols, while the ISFFT provides the inverse mapping.The information symbols x[k, l] are transmitted over a packet burst.
  • B. OTFS modulation •: OTFS preprocessing maps delay-Doppler symbols x[k, l] to time-frequency symbols X[n, m] using a 2D transform and transmit windowing.The resulting X[n, m] is then time-frequency modulated.
  • B. OTFS modulation •: At reception, a receive window is applied and the periodized signal is transformed from the time-frequency domain back to delay-Doppler symbols by the SFFT.This is the OTFS demodulation/post-processing stage.
  • B. OTFS modulation •: The OTFS input-output relation is derived after the transmit and receive transformations and their associated windowing operations.The channel response is incorporated through the derived relation.

C. Vectorized formulation of the input-output relation

The delay-Doppler channel is modeled as a sparse collection of paths, each characterized by delay, Doppler, and complex gain. Vectorization produces a sparse equivalent channel matrix for OTFS detection.

  • C. Vectorized formulation of the input-output relation: Each of P propagation paths is represented by a delay τ_i, Doppler ν_i, and fade coefficient h_i in the delay-Doppler channel impulse response.The channel is therefore described through discrete path parameters.
  • C. Vectorized formulation of the input-output relation: Rectangular windows and integer delay-Doppler tap assumptions yield a discrete input-output relation, with noninteger values approximated by a few taps.The tap indices represent delay and Doppler locations.
  • C. Vectorized formulation of the input-output relation: Vectorization represents transmitted symbols, received symbols, noise, and the channel as x, y, v ∈ C^NM×1 and H ∈ C^NM×NM.Modulo operations produce only P nonzero elements in each row and column of H.

III. MIMO-OTFS MODULATION

MIMO-OTFS applies the sparse OTFS channel model independently across transmit–receive antenna pairs and combines these relations into a block channel system. The resulting equivalent MIMO matrix remains sparse.

  • III. MIMO-OTFS MODULATION: In an n_a-antenna MIMO-OTFS system, each antenna independently transmits OTFS-modulated information symbols through channels with P taps.The transmit and receive antenna counts are assumed equal.
  • III. MIMO-OTFS MODULATION: The vectorized SISO-OTFS formulation is applied to every transmit–receive antenna pair to describe the MIMO-OTFS input-output relation.Each pair has an equivalent channel matrix H_qp.
  • III. MIMO-OTFS MODULATION: Each receive vector is the sum of the channel-matrix contributions from all transmit antennas plus receiver noise.The relations are written explicitly for y_1, y_2, through y_n_a.
  • III. MIMO-OTFS MODULATION: The antenna-pair channel matrices are arranged into a block equivalent matrix H_MIMO for the stacked transmit and receive vectors.The displayed block structure groups H_qp by receive and transmit antenna indices.
  • III. MIMO-OTFS MODULATION: Each row and column of H_MIMO has only n_aP nonzero elements because of the modulo operations.This extends the sparse structure of the single-antenna equivalent channel.

IV. MIMO-OTFS SIGNAL DETECTION

This section presents a message-passing detection scheme for MIMO-OTFS, exploiting the sparsity of the equivalent channel to avoid exponential-complexity MAP detection. The iterative algorithm passes probabilistic messages between variable and observation nodes until convergence or an iteration limit.

  • The section compares MIMO-OTFS and MIMO-OFDM signal detection in high-Doppler scenarios.
  • Exact MAP detection has exponential complexity, so detection uses a symbol-by-symbol MAP rule over the modulation alphabet.
  • Sparsity makes the factor graph locally connected, with each observation and variable node linked only through non-zero channel positions.The connected positions are represented by ζb and ζa, respectively.
  • The algorithm initializes all symbol probabilities uniformly and exchanges interference mean and variance messages between observation and variable nodes.The interference term is approximated as Gaussian, while variable-to-observation messages are probability mass functions over the constellation.
  • Iterations stop when message changes fall below ε or when the maximum iteration count Niter is reached, after which detected symbols are output.

B. Vectorized formulation of the input-output relation for MIMO-OFDM

This subsection develops the vectorized SISO-OFDM input-output formulation used as the basis for the MIMO-OFDM comparison. It models N consecutive OFDM blocks as one frame and explicitly represents cyclic-prefix insertion, removal, and Fourier transforms.

  • For a fair OTFS comparison, N consecutive OFDM blocks of size M are treated as one frame, with joint detection over an NM × 1 transmit vector.
  • Each OFDM block uses a cyclic prefix, making the frame length NL after prefix insertion, where L = M + CP.
  • The formulation defines matrices that insert and remove cyclic prefixes for all N consecutive OFDM blocks.TCP inserts the prefix and RCP removes it for one block; block-diagonal matrices extend these operations across the frame.
  • DFT and IDFT matrices are applied blockwise, and the time-delay channel for a frame is represented by an NL × NL matrix Htd.

1) MIMO-OFDM:

The MIMO-OFDM formulation extends the vectorized SISO-OFDM input-output relation to multiple transmit and receive antennas. Each antenna pair is represented by an equivalent channel matrix within the stacked system model.

  • The SISO-OFDM vectorized formulation is extended to MIMO-OFDM in the same manner as the MIMO-OTFS formulation.
  • HOFDMqp denotes the equivalent channel matrix from transmit antenna p to receive antenna q.
  • The transmit and received vectors are indexed by antenna, and the channel matrices are arranged in a block matrix spanning all antenna pairs.

T , xOFDM2

The supplied passage contains a fragment of the MIMO-OFDM vectorized notation involving the transmit-vector variables.

  • The notation includes transmit-vector terms for multiple OFDM transmit antennas.
  • The displayed fragment places xOFDM2 among the antenna-indexed transmit variables.
  • The passage does not state an additional operation or result for these variables.

T , yOFDM2

Under high-Doppler conditions, the proposed message-passing detector gives MIMO-OTFS strong BER performance, while MIMO-OFDM degrades substantially in the 2 × 2 comparison.

  • Performance setup: The evaluation assumes perfect receiver channel knowledge and applies message passing to both MIMO-OTFS and MIMO-OFDM.The simulations use a five-tap channel model and specified iterative-detector settings.
  • MIMO-OTFS performance: 10^-5 BER requires about 14 dB SNR for 2 × 2 MIMO-OTFS at 1880 Hz Doppler.The Doppler corresponds to approximately 507.6 kmph at a 4 GHz carrier frequency.
  • MIMO-OTFS performance: The same 10^-5 BER requires about 2 dB less SNR in the 3 × 3 configuration than in the 2 × 2 configuration.Figure 3 compares SISO, 2 × 2, and 3 × 3 MIMO-OTFS systems.
  • MIMO-OTFS versus MIMO-OFDM: MIMO-OFDM performance floors at about 2 × 10^-2 BER in the 2 × 2 system under 1880 Hz Doppler.The passage attributes this degradation to severe inter-carrier interference in the time-frequency domain.
  • MIMO-OTFS versus MIMO-OFDM: MIMO-OTFS reaches 10^-5 BER at about 14 dB SNR in the 2 × 2 comparison, whereas MIMO-OFDM breaks down and floors near 2 × 10^-2 BER.The comparison is made under rapidly varying channel conditions with high Doppler spread.

V. CHANNEL ESTIMATION FOR MIMO-OTFS

The channel-estimation scheme uses delay-Doppler impulse pilots whose spacing prevents channel-response overlap, enabling simultaneous estimation across MIMO antenna pairs from one frame.

  • Pilot-based estimation: The proposed scheme relaxes perfect channel knowledge by using impulses in the delay-Doppler domain as pilots.Each transmit-receive antenna pair has a finite delay-Doppler support determined by channel delay and Doppler spreads.
  • Pilot-based estimation: Known pilot coordinates allow the receiver to recover each equivalent channel response from the received signal.The vectorized formulation is used to obtain the equivalent channel matrix for each transmit-receive pair.
  • Pilot placement: Sufficiently spaced pilot impulses remain nonoverlapping after 2D-convolution spreading, so all antenna-pair responses can be estimated simultaneously.The resulting estimate of the equivalent MIMO-OTFS channel matrix uses a single frame.
  • Pilot placement: In the 2×1 illustration, pilots at (0, 0) and (16, 16) produce nonoverlapping impulse responses at the receiver.The example uses a (32, 32) frame at 4 dB SNR.

A. Performance results and discussions

The proposed delay-Doppler channel estimation reduces estimation error as pilot SNR increases and yields BER close to perfect-channel knowledge in a 2×2 MIMO-OTFS system. At 1880 Hz Doppler, its sparse, time-invariant channel representation supports simple and efficient estimation.

  • The Frobenius-norm estimation error decreases with pilot SNR in the 2×2 MIMO-OTFS system.The metric is the norm of the difference between HMIMO and the estimated equivalent channel matrix.
  • The estimated-channel BER is quite close to the perfect-channel BER for the 2×2 MIMO-OTFS system.The proposed channel estimation scheme is evaluated with message-passing detection.
  • 2 × 10^-5 BER requires about 12.5 dB with perfect channel knowledge and 13 dB with estimated channel knowledge.
  • At 1880 Hz maximum Doppler, time-frequency estimation is inaccurate, whereas sparse delay-Doppler channel representation remains time-invariant over a larger observation time.This property, together with 2D periodic convolution, enables simple and efficient MIMO-OTFS channel estimation.
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