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Low Complexity LMMSE Receiver for OTFS
Shashank Tiwari, Suvra Sekhar das, Vivek Rangamgari
TL;DR
OTFS requires advanced receivers because its time-frequency spreading encounters interference in time-varying channels while enabling diversity gain. This paper develops a low-complexity LMMSE receiver that exploits sparse, quasi-banded matrices through two-stage equalization and LU factorization. The receiver achieves up to 107 times lower complexity without BER degradation, including a 13 dB SNR gain over OFDM-MMSE at BER 5 × 10^-4.
Problem
OTFS receivers must manage time-varying-channel interference, while direct LMMSE processing has O(M^3N^3) complexity for large grids.
Method
The receiver exploits sparsity and quasi-banded structure in LMMSE matrices, using two-stage equalization and low-complexity LU factorization.
Results
Up to 107 times lower complexity is achieved without BER degradation; at BER 5 × 10^-4, OTFS-LMMSE provides a 13 dB SNR gain over OFDM-MMSE.
Takeaways & Limitations
The complexity reduction with linear processing is presented as supporting practical realization of future wireless OTFS-based systems.
Abstract
from arXiv · showhide
Orthogonal time frequency space modulation is a two dimensional (2D) delay-Doppler domain waveform. It uses inverse symplectic Fourier transform (ISFFT) to spread the signal in time-frequency domain. To extract diversity gain from 2D spreaded signal, advanced receivers are required. In this work, we investigate a low complexity linear minimum mean square error receiver which exploits sparsity and quasi-banded structure of matrices involved in the demodulation process which results in a log-linear order of complexity without any performance degradation of BER.
I. INTRODUCTION
OTFS spreads data across time-frequency resources to address interference in challenging time-varying channels and enable diversity gain. The paper motivates a lower-complexity LMMSE receiver for large OTFS grids.
- 5G-NR’s OFDM can experience interference when delay and Doppler spreads are both significant, motivating OTFS for such channel conditions.
- OTFS spreads data symbols across available time-frequency resources, which can be exploited to extract diversity gain.
- Time-varying channels cause inter-symbol and inter-carrier interference, limiting the ability of matched-filter receivers to suppress interference sufficiently.
- Non-linear receivers approach maximum-likelihood performance but have iterative, high-complexity structures, whereas linear receivers are simpler but generally perform worse.
- Direct LMMSE implementation has O(M^3N^3) complexity, while the proposed receiver targets O(MN log2(N)) complexity without BER degradation.
B. Channel
The channel model represents a time-varying wireless channel through multiple delay-Doppler paths with complex attenuation, delay, and Doppler parameters.
- The model contains P channel paths, each characterized by complex attenuation, delay, and Doppler value.
- Each path is assigned a delay and Doppler bin on the Doppler-delay lattice, with sufficiently large M and N assumed to avoid fractional-delay and fractional-Doppler effects.
- The receiver is assumed to have perfect knowledge of each path’s attenuation, delay-bin, and Doppler-bin parameters.
- Channel delay and Doppler lengths are defined from the maximum spreads, with α << MN and β << MN for typical wireless channels.
C. Receiver
After cyclic-prefix removal, the received OTFS signal is modeled with a channel matrix and Gaussian noise, then processed by an LMMSE equalizer to estimate the data vector.
- After CP removal, the received signal is represented using a channel matrix H and a white Gaussian noise vector.
- The channel matrix is built from circulant delay operations and a diagonal Doppler matrix.
- The receiver further processes the received vector through an LMMSE equalizer to obtain estimated data.
III. LOW COMPLEXITY LMMSE RECEIVER FOR OTFS
The proposed receiver reduces LMMSE processing cost by exploiting the unitary OTFS modulation structure and the sparse, quasi-banded form of the equalization matrix. It uses a two-stage equalizer and low-complexity LU factorization.
- With a rectangular pulse, the OTFS modulation matrix A is unitary, simplifying the LMMSE equalization expression.
- LMMSE equalization is split into channel equalization producing rce = Heqr, followed by an OTFS matched-filter operation producing d̃ = A†rce.
- Direct computation of rce requires inversion of Ψ and multiplication by H†, with O(M^3N^3) complexity.
- The matrix Ψ is quasi-banded with bandwidth 2α − 1 and sparse because α << MN, enabling a low-complexity LU-based implementation.
B. Low complexity LU factorization of Ψ
The receiver factorizes the sparse, quasi-banded matrix Ψ using a partition that enables low-complexity triangular solves and LU decomposition.
- The proposed partition separates Ψ using θ = α −1 and Q = MN −θ to structure its LU factorization.
- Banded-matrix algorithms compute the LU decomposition of T, while forward substitution solves the lower-triangular banded system.
- The factorization computes V through V† = (U†)−1S† followed by Hermitian transposition, exploiting the banded structure of U†.
- Direct computation of (18) requires O(θ2MN) computations because θ << MN, while F and G are obtained through LU decomposition.
- The resulting triangular factors support LMMSE processing because L and U must later be inverted.
Note on the non-singularity of L and U:
The positive definiteness of Ψ ensures invertibility, and the LU factors L and U are consequently non-singular.
- For finite SNR ranges with σ2d > 0, Ψ is positive definite and therefore invertible; L and U are non-singular.
C. Computation of r
The receiver computes the equalized vector through triangular solves, channel-dependent circular shifts and diagonal multiplications, then FFT-based reshaping.
- The equalized vector is obtained by first computing r(1) = L−1r and then r(2) = U−1r(1).
- Because L is quasi-banded lower triangular, r(1) is computed using low-complexity forward substitution, followed by an upper-triangular solve for r(2).
- To form rce, each r(2) is circularly shifted by −lp and multiplied by h̄p diag{∆−kp} for path p before summing the resulting vectors.
- The vector rce is reshaped into an M × N matrix before applying FFT operations to compute the final demodulation step.
- The reshape-based implementation uses M number of N-point FFT operations instead of directly computing the matrix operation A†rce.
D. LMMSE receiver for OFDM over TVC
The low-complexity OTFS receiver can be extended to OFDM by selecting A = IN ⊗ WM and evaluating the corresponding transformations with FFTs.
- The OTFS receiver extends to OFDM by setting A = IN ⊗ WM and computing rce = Heq r through the same procedures.
- The final estimate d̂ = (IN ⊗ W†M)rce can be computed using N number of M-point FFTs.
IV. RESULT
The proposed receiver’s complexity is quantified through complex multiplications and scales log-linearly with system dimensions, exploiting structured operations rather than direct matrix processing.
- Complexity is evaluated using complex multiplications for FFTs, matrix products, matrix inversion, and LU decomposition.The receiver’s operation counts are assembled from these constituent computations.
- The proposed receiver has complexity O(MN[log2(N) + α2 + P 2β]).The expression accounts for FFT-related processing and channel-structure terms.
- The evaluation considers EVA and EVB channels with N = 16 and N = 128 across M ∈[2 4096].The scenarios use P = 9 for EVA and P = 6 for EVB.
B. BER Evaluation
BER evaluation compares the proposed receiver with direct implementations and reports no BER degradation, while the conclusion reports a substantial complexity reduction.
- The proposed receiver does not suffer performance degradation compared with direct implementations using (9).The BER comparison is conducted in an EVA channel with 4 QAM modulation.
- At BER 5 × 10−4, OTFS-LMMSE achieves an SNR gain of 13 dB over OFDM-MMSE.This result demonstrates the reported diversity gain in the EVA-channel evaluation.
- The proposed receiver achieves up to 107 times complexity reduction over direct implementation without performance penalty.The reported reduction is presented as supporting practical realization of OTFS-based systems.