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A Simple Variational Bayes Detector for Orthogonal Time Frequency Space (OTFS) Modulation
Weijie Yuan, Zhiqiang Wei, Jinhong Yuan, Derrick Wing Kwan Ng
TL;DR
OTFS detection must retain performance in dynamic, multipath channels without incurring the exponential complexity of MAP detection. This paper develops a variational Bayes approximation that iteratively maximizes the ELBO, with guaranteed convergence of the approximated detector. Simulations show fast convergence and better detection performance than conventional message passing, especially in multipath settings.
Problem
MAP detection handles OTFS inter-symbol interference but has exponentially increasing complexity, while message passing can converge locally in loopy factor graphs.
Method
The receiver approximates the joint posterior by a product of local marginals, minimizing KL divergence through iterative ELBO maximization.
Results
The VB receiver converges rapidly, outperforms MP in practical multipath scenarios, and approaches 4-path MAP performance with significantly lower complexity.
Takeaways & Limitations
The convergence-guaranteed VB receiver provides a lower-complexity alternative to MAP and MP detection for OTFS systems.
Abstract
from arXiv · showhide
The emerging orthogonal time frequency space (OTFS) modulation technique has shown its superiority to the current orthogonal frequency division multiplexing (OFDM) scheme, in terms of its capabilities of exploiting full time-frequency diversity and coping with channel dynamics. The optimal maximum a posteriori (MAP) detection is capable of eliminating the negative impacts of the inter-symbol interference in the delay-Doppler (DD) domain at the expense of a prohibitively high complexity. To reduce the receiver complexity for OTFS scheme, this paper proposes a variational Bayes (VB) approach as an approximation of the optimal MAP detection. Compared to the widely used message passing algorithm, we prove that the proposed iterative algorithm is guaranteed to converge to the global optimum of the approximated MAP detector regardless the resulting factor graph is loopy or not. Simulation results validate the fast convergence of the proposed VB receiver and also show its promising performance gain compared to the conventional message passing algorithm.
I. INTRODUCTION
OTFS is presented as a modulation scheme suited to doubly selective, high-mobility channels, while practical OTFS detection must balance MAP performance against computational complexity and message-passing convergence.
- Motivation: High-mobility channels can cause OFDM performance degradation through Doppler-induced inter-carrier interference.The motivation includes vehicular networks and high-speed railway applications.
- Motivation: OTFS is robust in doubly selective channels and multiplexes information symbols in the delay-Doppler domain.OTFS can be implemented by adding pre- and post-processing to conventional OFDM schemes.
- Detection challenge: Optimal MAP detection can exploit time-frequency diversity, but its complexity grows exponentially with the OTFS frame block size.The MAP optimization requires complexity of order |χ|NM.
- Detection challenge: Gaussian-approximation message-passing receivers reduce complexity but may converge only to a locally optimal point in loopy factor graphs.Loopy graphs are likely for OTFS in multipath scenarios.
- Proposed approach: The proposed VB detector approximates the posterior using KL divergence and iteratively maximizes the ELBO to obtain point-wise MAP decisions.The selected variational family makes ELBO maximization strictly convex, yielding a globally optimal approximate solution with lower complexity than MP.
II. SYSTEM MODEL
The system model describes OTFS transmission and reception through two-dimensional delay-Doppler/time-frequency processing, followed by propagation over a linear time-variant multipath channel and transformation back to the delay-Doppler domain.
- Delay-Doppler representation: Coded bits are mapped to delay-Doppler data symbols d_k,l from constellation χ, indexed over N time slots and M subcarriers.The Doppler-shift index k ranges from 0 to N−1, while the delay index l ranges from 0 to M−1.
- Two-dimensional processing: The delay-Doppler symbols are transformed into time-frequency symbols X_n,m through the inverse symplectic finite Fourier transform.The transformation uses matrix representations of X and D together with the normalized N-dimensional DFT matrix F_N.
- Two-dimensional processing: The time-frequency symbols are converted into a continuous transmitted signal by time-domain conversion and pulse shaping with g_tx(t).The stated process first transforms time-frequency symbols to time-domain symbols and then shapes the pulse using the transmit function g_tx(t).
- Channel model: Propagation is modeled by a linear time-variant channel whose delay-Doppler impulse response contains P resolvable paths with path delays τ_i and Doppler shifts ν_i.The path parameters describe the delay and Doppler shift associated with each path.
- Receiver processing: At the receiver, a receiving filter and DFT produce sampled time-frequency observations, which are then transformed by SFFT to obtain delay-Doppler received samples y_k,l.The samples are taken at t=nT and f=m∆f; under ideal transmit and receive pulse shaping, the time-frequency input-output relation simplifies.
A. Variational Bayes Approach
The detector approximates the OTFS MAP posterior with independent variational marginals and iteratively maximizes the ELBO. This converts joint MAP detection into lower-complexity point-wise decisions.
- Variational approximation: The proposed detector approximates p(d|y) with a tractable distribution q(d) by minimizing the KL divergence and maximizing the ELBO.The variational family assumes mutually independent symbol variables, known as the mean field approximation.
- Iterative updates: The mean field factorization assigns each DD-domain symbol its own variational factor q_k,l(d_k,l), updated iteratively using the other symbols’ current distributions.The updates are obtained by setting the relevant ELBO partial derivative to zero.
- Signal model: The likelihood is modeled in pairwise form after the orthogonal SFFT preserves Gaussian noise statistics, enabling the local variational updates.The equivalent channel representation supports the compact stacked input-output model used by the detector.
- Point-wise detection: The resulting marginal distributions q*_k,l(d_k,l) replace joint MAP detection with point-wise MAP decisions for each symbol.The detector selects the constellation value maximizing each local marginal distribution.
- Complexity: The detection complexity is reduced to order |χ|NM, substantially below conventional MAP detection.This complexity follows from maximizing each symbol’s marginal over the constellation rather than searching jointly over all symbols.
B. Comparison with MP Receiver
The VB receiver is designed to avoid the local-optimum behavior of message passing on loopy OTFS factor graphs. Its ELBO optimization has a global optimum, while its complexity is lower than MP for multipath channels.
- Convergence motivation: Message passing can converge to a local optimum when OTFS factor graphs contain loops, which commonly arise in multipath or rich-scattering channels.Even channels with only two paths may produce many girth-4 loops.
- Convergence guarantee: The ELBO is a concave functional under the proposed variational formulation, so its local maximum is the global optimum.The diagonal Hessian structure supports this concavity conclusion.
- Convergence guarantee: Iterative updates of each local distribution are therefore guaranteed to converge to the optimal approximate distribution maximizing the ELBO.The guarantee applies to the variational approximation rather than the exact MAP posterior.
- Complexity comparison: For P propagation paths, the proposed algorithm has complexity O(|χ|NMP), whereas MP requires O(|χ|NMP^2).VB reuses marginal means across an iteration, while MP computes messages separately for connected factor vertices.
- Overall comparison: The proposed VB approach is attractive in rich-scattering environments because it combines low complexity with guaranteed convergence.These properties directly address the convergence and computational limitations identified for MP and MAP detection.
IV. NUMERICAL RESULTS
Simulations compare the proposed VB detector with MP across multipath OTFS channels and show faster convergence and stronger BER performance as path count increases.
- Simulation setup: The simulations average results over 2 × 10^4 OTFS frames using M = 512 subcarriers, N = 128 time slots, QPSK, and a 120 km/h mobile user.The receiver is assumed to have perfect channel information, and the number of iterations is set to 10.
- BER performance: In the 4-path scenario, the proposed VB algorithm and conventional MP achieve virtually the same BER performance.BER is evaluated versus Es/σ^2 for OTFS over delay-Doppler channels.
- BER performance: For 9 paths, the proposed VB method significantly outperforms MP because MP converges to a local optimum in the presence of many girth-4 loops.The performance improvement grows with the number of paths, while VB can exploit OTFS time-frequency diversity more efficiently.
- BER performance: The proposed VB receiver approaches the 4-path MAP detector’s performance while substantially reducing complexity.The MAP benchmark is obtained by exhaustive search.
- Convergence: At Es/σ^2 = 15 dB, increasing iterations improves both methods, while VB converges faster than MP in both path scenarios.For 9 paths, VB also achieves better detection performance; for 4 paths, both methods converge to almost the same BER.
V. CONCLUSIONS
The paper develops a variational Bayes receiver that reduces OTFS detection complexity through ELBO optimization and reports fast convergence and improved performance over MP in practical multipath settings.
- Conclusions: The proposed receiver approximates the joint a posteriori distribution by a product of local marginals while maximizing the ELBO.The selected variational family convexifies the ELBO and guarantees convergence of the VB approach.
- Conclusions: The proposed algorithm has lower complexity than both MAP and MP receivers.Its complexity is analyzed as lower than the conventional alternatives.
- Conclusions: Simulation results confirm fast convergence and superior detection performance compared with MP, especially in practical multipath scenarios.The conclusion emphasizes the performance advantage in multipath channels.