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Transmitter and Receiver Window Designs for Orthogonal Time Frequency Space Modulation

Zhiqiang Wei, Weijie Yuan, Shuangyang Li, Jinhong Yuan, Derrick Wing Kwan Ng

arXiv:2010.13005v1cs.IT

TL;DR

OTFS window design must address fractional-Doppler effects that limit channel estimation and data detection. The paper analyzes transmitter and receiver windows and proposes CSI-dependent transmitter and Dolph-Chebyshev designs. These designs improve effective-channel sparsity or detection performance and yield gains over rectangular windows.

  • Problem

    Fractional Doppler spreads the effective OTFS channel in the delay-Doppler domain, limiting channel estimation and data detection.

  • Method

    The paper derives windowing effects on OTFS channels, power, noise, and estimation, then designs an optimal transmitter window with CSI and a Dolph-Chebyshev window without transmitter CSI.

  • Results

    The proposed windows improve channel estimation and data detection compared with the conventional rectangular window.

  • Takeaways & Limitations

    Enhancing delay-Doppler channel sparsity with a Dolph-Chebyshev window reduces fractional-Doppler channel spread and improves both channel estimation and data detection.

Abstract

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In this paper, we investigate the impacts of transmitter and receiver windows on the performance of orthogonal time-frequency space (OTFS) modulation and propose window designs to improve the OTFS channel estimation and data detection performance. In particular, assuming ideal pulse shaping filters at the transceiver, we derive the impacts of windowing on the effective channel and its estimation performance in the delay-Doppler (DD) domain, the total average transmit power and the effective noise covariance matrix. When the channel state information (CSI) is available at the transceiver, we analyze the minimum squared error (MSE) of data detection and propose an optimal transmitter window to minimize the detection MSE. The proposed optimal transmitter window is interpreted as a mercury/water-filling power allocation scheme, where the mercury is firstly filled before pouring water to pre-equalize the TF domain channels. When the CSI is not available at the transmitter but can be estimated at the receiver, we propose to apply a Dolph-Chebyshev (DC) window at either the transmitter or the receiver, which can effectively enhance the sparsity of the effective channel in the DD domain. Thanks to the enhanced DD domain channel sparsity, the channel spread due to the fractional Doppler is significantly reduced, which leads to a lower error floor in both channel estimation and data detection compared with that of rectangular window. Simulation results verify the accuracy of the obtained analytical results and confirm the superiority of the proposed window designs in improving the channel estimation and data detection performance over the conventional rectangular window design.

I. INTRODUCTION

OTFS targets high-mobility channels by spreading data across the delay-Doppler and time-frequency domains, while this paper develops window designs for practical fractional-Doppler interference.

  • OFDM orthogonality breaks under Doppler spread, causing inter-carrier interference and sharply degrading channel estimation and data detection in time-varying channels.
  • OTFS maps data from the delay-Doppler domain to the time-frequency domain using two-dimensional basis functions before multicarrier transmission.
  • OTFS transforms time-varying channels into effective two-dimensional channels that are sparse and stable in the delay-Doppler domain.
  • Fractional inter-Doppler interference spreads the effective channel across Doppler indices and limits channel estimation, often requiring guard space between pilots and data.
  • The paper analyzes windowing effects and proposes an optimal CSI-aware transmitter window plus a Dolph-Chebyshev window for cases without transmitter CSI.
  • The optimal transmitter window improves detection over the rectangular window, while the Dolph-Chebyshev window improves channel estimation and data detection with reduced error floors.

A. OTFS Transmitter

The OTFS transmitter spreads delay-Doppler symbols across a time-frequency grid, applies a transmitter window, and converts the result into a time-domain waveform.

  • An OTFS frame occupies N time slots and M subcarriers, forming an N × M time-frequency grid.
  • The inverse symplectic finite Fourier transform maps delay-Doppler symbols x[k,l] to time-frequency symbols X[n,m].
  • Each delay-Doppler symbol is spread across the time-frequency domain by a two-dimensional transformation sequence.
  • A transmitter window U[n,m] is applied by point-wise multiplication with the time-frequency signal X[n,m].
  • Because time-frequency multiplication equals two-dimensional circular convolution in delay-Doppler, the window can act as a filter to improve effective-channel sparsity.
  • Fractional Doppler prevents reliable integer-grid localization in practice because resolving Doppler shifts sufficiently may require impractically large speed separation or longer frame latency.

C. OTFS Receiver

The OTFS receiver demodulates the received waveform, applies a receiver window in the time-frequency domain, and transforms the result back to the delay-Doppler domain.

  • The receiver first performs multicarrier demodulation to obtain the received time-frequency-domain samples.
  • With ideal bi-orthogonal pulse-shaping filters, the time-frequency input-output relation is free of inter-symbol and inter-carrier interference.
  • A receiver window V[n,m] multiplies the received time-frequency signal before OTFS demodulation.
  • The proposed window analysis extends from ideal pulse shaping to commonly used rectangular pulses because the effective channel locations and inter-Doppler interference patterns remain the same up to phase.
  • The symplectic finite Fourier transform converts windowed time-frequency signals into delay-Doppler-domain signals.
  • The transmitter and receiver windows become diagonal matrices in the vectorized time-frequency model and determine the equivalent delay-Doppler channel representation.

III. THE IMPACT OF WINDOWING FOR OTFS MODULATION

The section characterizes how transmitter and receiver windows shape OTFS’s effective delay-Doppler channel, noise, estimation, and detection properties. Fractional Doppler spreads energy under rectangular windows, while window design can control effective-channel sparsity and suppress leakage.

  • Windowing affects OTFS’s effective DD-domain channel, channel-estimation performance, average transmit power, and receiver noise covariance.These analyses provide the basis for practical window designs.
  • Effective Channel in the DD Domain: The received DD-domain signal is a 2D circular convolution between data symbols and the window-dependent effective channel.The effective channel captures the joint transmitter–receiver window effect, while the receiver window also induces a separate filter affecting receiver-side noise properties.
  • Effective Channel Sparsity with Rectangular Window: The DD-domain spreading pattern can be manipulated through the window-induced filter, enabling transmitter or receiver windows to control effective-channel sparsity and suppress fractional-Doppler leakage.The section identifies leakage suppression and improved effective-channel sparsity as desired window-design objectives.
  • Effective Channel in the DD Domain: Transmitter and receiver TF windows have the same effect on the DD-domain channel filter, whereas only the receiver window changes the additional noise-related DD filter.Thus, transmitter and receiver windows jointly control channel spreading, but receiver-window choice additionally alters noise properties.
  • Effective Channel Sparsity with Rectangular Window: With rectangular windows and integer Doppler, the effective channel preserves the original DD-domain sparsity, with nonzero response only at the path’s delay-Doppler indices.The resulting response is a phase-rotated version of the original channel response.
  • Effective Channel Sparsity with Rectangular Window: Fractional Doppler spreads each path across all Doppler indices under a rectangular window, reducing sparsity and introducing power leakage away from the nominal Doppler index.This spreading can degrade channel estimation, increase detection complexity, and impair reliable detection; suppressing leakage therefore motivates appropriate window design.

2) The Total Power of the Effective Channel Gain:

The effective DD-domain channel preserves average total power under independent path coefficients, but inter-spread increases power variance and can harm detection. Window design and pilot-guard choices trade channel-estimation error against signaling overhead.

  • The SFFT’s orthogonality supports deriving the total power of the effective DD-domain channel from the original channel response.
  • For rectangular windows, distinct paths generally produce no inter-spread, except paths sharing delay but having different Doppler shifts.
  • The average total effective-channel power equals the original DD-domain channel power under independent channel coefficients.
  • Inter-spread preserves mean power but increases its variance, with destructive spreading potentially degrading reliable detection.
  • Fractional-Doppler interference creates channel-estimation and data-detection error floors, while a full guard removes interference at higher signaling overhead.
  • Lower window sidelobes reduce estimation error floors and can achieve this with a relatively small additional guard; placing the window at either side yields the same estimation floor.

C. The Impact of TX Window on Average Transmit Power

The TX window controls average OTFS transmit power through its TF-domain squared magnitude, which acts as per-symbol power allocation.

  • The average power of the time-domain transmitted signal is determined by the sum of squared TX-window magnitudes in the TF domain.
  • The squared TX-window magnitude can be interpreted as the power allocated to each TF-domain symbol in an OTFS frame.

D. The Impact of RX Window on Noise Covariance Matrix

RX windows shape the effective DD-domain noise covariance: constant-modulus windows preserve white noise, whereas non-unitary windows produce colored noise. With CSI, the paper designs TX windows for detection and uses DC windows when sparsity is the priority.

  • D. The Impact of RX Window on Noise Covariance Matrix: A constant-modulus RX window satisfying VV^H = I_MN leaves the effective DD-domain noise white.
  • D. The Impact of RX Window on Noise Covariance Matrix: When VV^H ≠ I_MN, the effective DD-domain noise becomes colored, complicating receiver processing.
  • A. With CSI at OTFS Transceiver: With CSI at both transceiver sides, the paper proposes an optimal TX window to minimize data-detection MSE.
  • D. The Impact of RX Window on Noise Covariance Matrix: Whitening colored noise can reverse the RX-window effect and produce a non-sparse DD-domain channel, degrading channel-estimation performance.
  • A. With CSI at OTFS Transceiver: The RX window does not change detection MSE because it simultaneously transforms the effective channel and noise covariance, whereas the TX window can change performance.
  • A. With CSI at OTFS Transceiver: The optimal TX window is interpreted as mercury/water filling: mercury pre-equalizes TF-domain fading before water allocation, while weak channels may receive no allocation.

B. Without CSI at OTFS Transmitter

Without transmitter CSI, the paper targets effective DD-domain sparsity through practical windowing. Dolph-Chebyshev windows suppress fractional-Doppler spreading, but the ideal zero-spread response is unrealizable with finite time length.

  • B. Without CSI at OTFS Transmitter: Without transmitter CSI, a universal window design that enhances effective-channel sparsity is proposed to improve channel estimation.
  • B. Without CSI at OTFS Transmitter: The ideal Doppler-domain window tolerates fractional Doppler without channel-gain loss or spreading, but requires infinite time-domain length and is impractical.
  • 2) Dolph-Chebyshev Window: A Dolph-Chebyshev window minimizes sidelobes for a specified mainlobe width, thereby reducing spread to other Doppler indices.
  • 2) Dolph-Chebyshev Window: With kmain ≈ 3 and a −40 dB sidelobe level, the effective channel has approximately three considerable Doppler-domain entries.
  • 2) Dolph-Chebyshev Window: Compared with a rectangular window, the DC window significantly suppresses channel spread and improves effective-channel sparsity.

V. NUMERICAL RESULTS

Simulations validate the analytical results and show that DC and optimized transmitter windows improve OTFS channel estimation and detection relative to rectangular windows under the tested settings.

  • Simulation setup: The simulations use QPSK and BPSK over OTFS frames with N = [16, 20] time slots and M = [8, 30] subcarriers.Channels use P = [2, 5] paths, lmax = [2, 4], kmax = [2, 3], and guard space k̂ = [0, 1].
  • Channel estimation: Fractional Doppler causes an effective-channel estimation error floor through interference spreading from data symbols into the guard space.The analytical error-floor level closely matches high-SNR simulation results.
  • Channel estimation: Increasing pilot power lowers channel-estimation MSE, while adding guard space lowers MSE at the cost of higher overhead.This trade-off is observed across the considered channel-estimation experiments.
  • Channel estimation: The designed DC window at the transmitter achieves significantly lower effective-channel estimation MSE than the rectangular window by enhancing DD-domain channel sparsity.Using the DC window at either transmitter or receiver produces identical channel-estimation results in the reported setting.
  • Data detection: With perfect CSI, the optimized transmitter window achieves the best reported FER and minimizes detection MSE, whereas receiver-side DC can match rectangular-window FER under SPA detection.The optimized design is evaluated against the other window and CSI configurations.
  • Data detection: With estimated CSI, DC windows at either side yield lower FER error floors than rectangular windows, while transmitter-side DC gives better FER than receiver-side DC for MMSE detection.Receiver-side DC colors the noise, making MMSE detection more sensitive to CSI imperfections.

VI. CONCLUSIONS

The paper derives how transmitter and receiver windows affect OTFS channels, estimation, power, and noise, then proposes CSI-dependent designs that improve estimation and detection.

  • Contributions: The analysis covers effective-channel behavior, channel-estimation performance, average transmit power, and noise covariance under windowing.These results provide analytical foundations for the proposed designs.
  • Findings: Fractional Doppler can spread the effective channel and create an error floor in effective channel estimation.This identifies a central limitation addressed by the window designs.
  • Findings: Transmitter- and receiver-side windows achieve identical effective-channel estimation performance in the reported analysis.The transmitter window is also interpreted as TF-domain power allocation, whereas the receiver window causes colored noise.
  • CSI-aware design: With CSI at both ends, the paper proposes an optimal transmitter window that minimizes data-detection MSE.The design targets the CSI-aware detection problem.
  • CSI-unavailable transmitter: Without transmitter CSI but with receiver-side estimation, a DC window enhances channel sparsity and improves channel estimation and data detection.Simulations verify the analytical results and demonstrate substantial performance gains from the proposed designs.

APPENDIX A. Sum-product Algorithm (SPA)-based Detector [31]

The SPA detector models dominant DD-domain interference probabilistically and iteratively updates symbol probabilities to produce posterior-based decisions.

  • Interference model: Fractional Doppler increases the number of interfered symbols because the effective DD-domain channel contains more nonzero entries.Integer Doppler instead gives P − 1 interferers for a received sample.
  • Interference model: SPA retains the L largest effective-channel entries and treats the remaining signals as noise to reduce detection complexity.Each data symbol is modeled as interfering with L received samples.
  • Probability updates: The algorithm iteratively applies sum and product updates to obtain each symbol’s posterior probability from the received DD-domain observations.The posterior is then used for data-symbol detection.
  • Complexity: For constellation size Q and L interfered symbols, the sum step has complexity scale O(Q^L).The complexity is dominated by the sum step.
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