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Performance Analysis of Coded OTFS Systems over High-Mobility Channels
Shuangyang Li, Jinhong Yuan, Weijie Yuan, Zhiqiang Wei, Baoming Bai, Derrick Wing Kwan Ng
TL;DR
The paper investigates coded OTFS error performance in high-mobility channels, addressing unresolved questions about coding gain. It derives PEP-based bounds and reveals a coding–diversity trade-off, while simulations compare OTFS with coded OFDM and assess the proposed code design.
Problem
The key coding factors governing OTFS performance in high-mobility environments remain unresolved, where receiver implementation can be impractical.
Method
The paper derives coded-OTFS performance bounds from conditional and unconditional pairwise-error probabilities using bounding techniques.
Results
Coding improvement depends on squared Euclidean distance between codewords and independent resolvable paths, with a fundamental trade-off between diversity gain and coding gain.
Takeaways & Limitations
The derived unconditional bound supports a code design criterion, and simulations evaluate OTFS against coded OFDM over high-mobility channels.
Abstract
from arXiv · showhide
Orthogonal time frequency space (OTFS) modulation is a recently developed multi-carrier multi-slot transmission scheme for wireless communications in high-mobility environments. In this paper, the error performance of coded OTFS modulation over high-mobility channels is investigated. We start from the study of conditional pairwise-error probability (PEP) of the OTFS scheme, based on which its performance upper bound of the coded OTFS system is derived. Then, we show that the coding improvement for OTFS systems depends on the squared Euclidean distance among codeword pairs and the number of independent resolvable paths of the channel. More importantly, we show that there exists a fundamental trade-off between the coding gain and the diversity gain for OTFS systems, i.e., the diversity gain of OTFS systems improves with the number of resolvable paths, while the coding gain declines. Furthermore, based on our analysis, the impact of channel coding parameters on the performance of the coded OTFS systems is unveiled. The error performance of various coded OTFS systems over high-mobility channels is then evaluated. Simulation results demonstrate a significant performance improvement for OTFS modulation over the conventional orthogonal frequency division multiplexing (OFDM) modulation over high-mobility channels. Analytical results and the effectiveness of the proposed code design are also verified by simulations with the application of both classical and modern codes for OTFS systems.
I. INTRODUCTION
The paper addresses the missing theoretical analysis of coded OTFS error performance in high-mobility channels. It derives performance bounds, identifies coding and diversity trade-offs, proposes a distance-based code criterion, and reports improved performance over coded OFDM.
- Motivation: High-mobility channels make perfect OFDM subcarrier orthogonality impractical, producing unsatisfactory conventional OFDM performance.Time and frequency dispersion arise from multipath and Doppler shifts, respectively.
- OTFS approach: OTFS modulates symbols in the delay-Doppler domain, where channel parameters are relatively stable and symbols experience the whole time-frequency channel fluctuations.The two-dimensional SFFT performs the domain transformation underlying this modulation.
- OTFS approach: OTFS can potentially exploit full channel diversity, while its sparse delay-Doppler representation supports low-overhead CSI acquisition and low-complexity detection.The paper cites low pilot signaling overhead and low-complexity symbol detection as practical implications.
- Contributions: The paper fills a gap in coded OTFS analysis by deriving conditional PEPs and unconditional performance upper bounds for cases distinguished by independent resolvable channel paths.The analysis begins with a given channel realization and uses bounding techniques because the exact unconditional PEP is generally intractable.
- Contributions: For a given number of independent resolvable paths, coding gain depends on squared Euclidean distance between codewords, motivating maximization of the minimum such distance.The derived criterion is intended to optimize coding gain across all codeword pairs.
- Contributions: OTFS exhibits a fundamental coding-diversity trade-off: diversity gain improves with more independent resolvable paths, whereas coding gain declines.Simulations demonstrate significant performance improvement for coded OTFS over coded OFDM in high-mobility channels.
II. OTFS SYSTEM MODEL
This section reviews the OTFS concept and introduces the coded OTFS system considered in the paper.
- The section begins by reviewing the OTFS concept.
- It then introduces the system model used for analysis.
- The considered system is a coded OTFS system.
A. Coded OTFS System Model
The coded OTFS model maps encoded information through delay-Doppler and time-frequency domains before transmission over a time-varying multipath channel.
- Coded OTFS System Model: An information sequence is channel-encoded and modulated into an MN-length transmitted vector.
- Coded OTFS System Model: The transmitted delay-Doppler representation is transformed into time-frequency symbols using the ISFFT.
- Coded OTFS System Model: OTFS modulation can be viewed as an ISFFT precoder concatenated with a conventional OFDM modulator.
- Coded OTFS System Model: The time-varying channel is represented in the delay-Doppler domain and includes independent resolvable paths with path coefficients, delays, and Doppler shifts.
- Coded OTFS System Model: The receiver filters the received signal, obtains time-frequency symbols, and transforms them into delay-Doppler symbols using the SFFT.
B. Vector Form Representation of OTFS
The vector representation organizes OTFS symbols and channel paths into an effective matrix model, while specifying path-index and channel-coefficient assumptions.
- Vector Form Representation of OTFS: The effective channel matrix is constructed using permutation and diagonal matrices associated with path delays and Doppler shifts.
- Vector Form Representation of OTFS: Fractional Doppler is represented by κ_i, while the analysis thereafter considers only integer Doppler indices.
- Vector Form Representation of OTFS: The delay and Doppler indices are assumed to follow discrete uniform distributions over bounded index ranges.
- Vector Form Representation of OTFS: The equivalent codeword matrix concatenates path-dependent column vectors and has dimensions MN × P.
- Vector Form Representation of OTFS: Path coefficients are modeled as independent identically distributed complex Gaussian variables with mean μ and variance 1/(2P) per real dimension.
III. ERROR PERFORMANCE ANALYSIS
The error analysis derives conditional and unconditional PEP expressions from codeword-distance matrices, eigenvalues, and channel-path statistics, with diversity linked to matrix rank.
- ERROR PERFORMANCE ANALYSIS: The analysis assumes ideal channel state information at the receiver, including path coefficients and delay-Doppler indices.
- ERROR PERFORMANCE ANALYSIS: Conditional PEP is bounded using the codeword difference matrix and average symbol energy.
- ERROR PERFORMANCE ANALYSIS: The codeword difference matrix is Hermitian positive semidefinite and is analyzed through its nonnegative eigenvalues and rank.
- ERROR PERFORMANCE ANALYSIS: Unconditional PEP is obtained by averaging conditional PEP over channel distributions and considering delay-Doppler index distributions.
- ERROR PERFORMANCE ANALYSIS: The rank of the codeword difference matrix is defined as the diversity gain of OTFS modulation.
- ERROR PERFORMANCE ANALYSIS: The coding-gain analysis focuses on the full-diversity case in which the matrix rank equals the number of independent resolvable paths.
A. Error Performance Analysis for Coded OTFS systems
The analysis derives unconditional error-performance bounds for coded OTFS from conditional PEP using properties of the codeword difference matrix. It identifies how squared Euclidean distance and independent resolvable paths shape coding and diversity gains.
- Unconditional PEP bounds: Bounding techniques replace the generally intractable averaging over channel parameters with unconditional PEP and coding-gain bounds.The derivation uses properties of the codeword difference matrix, including its trace, inverse trace, eigenvalue squares, and determinant.
- Unconditional PEP bounds: The unconditional PEP depends on the squared Euclidean distance d2_E(e) and the number of independent resolvable paths P.The factor P represents energy averaging across independent paths, while d2_E(e)/P characterizes coding gain.
- Diversity–coding trade-off: The diversity gain improves with P, whereas the coding gain declines for a given channel code.This establishes a fundamental trade-off between diversity gain and coding gain in coded OTFS systems.
- Diversity–coding trade-off: For small P, squared Euclidean distance is especially important because optimized coding can substantially improve error performance.When P is large, diversity gain is already high and code design is expected to provide only limited additional improvement.
- Code-design implication: The code-design guideline is to maximize the minimum d2_E(e) among all pairs of codewords.The analysis also states that coding gain increases with d2_E(e), regardless of P.
- Numerical verification: The derived coding-gain bounds closely match average coding gains, especially for small P, but diverge slightly when P is large.This discrepancy motivates a more suitable approximation for large path counts.
B. Error Performance Analysis for Large Values of P
For many independent resolvable paths, the analysis approximates the unconditional PEP using Gaussian behavior and derives a large-P upper bound. The approximation is sufficiently accurate for P ≥ 4 and the channel approaches an AWGN model as diversity paths increase.
- Large-P approximation: For large P and reasonably high SNR, the unconditional PEP is approximately upper-bounded using a Gaussian approximation.The approximation follows from the aggregate channel terms approaching Gaussian behavior through the central limit theorem and strong law of large numbers.
- Large-P approximation: For P ≥ 4, the approximation in Theorem 3 is sufficiently accurate.The stated accuracy is attributed to the strong law of large numbers.
- Interpretation: The large-P unconditional PEP depends on squared Euclidean distance d2_E(e) rather than delay and Doppler indices.This form is similar to error performance over AWGN channels because fading impact is mitigated by many diversity branches.
- Interpretation: A channel with many diversity paths approaches an AWGN model as the impact of fading is mitigated.The paper connects this behavior to averaging across a large number of diversity branches.
- Scope: The analysis considers only the integer Doppler case, although extending it to fractional Doppler is described as straightforward.This is the stated scope boundary of the error-performance analysis.
C. Code Design Issues
The proposed code-design criterion maximizes minimum squared Euclidean distance, while practical performance can still depend on channel parameters. An interleaver is suggested to reduce this parameter-dependent variation.
- Code-design criterion: The channel code should maximize the minimum squared Euclidean distance among all pairs of possible codewords.This criterion is presented as the rule-of-thumb design principle for coded OTFS systems.
- Channel dependence: Even with the designed code, coded OTFS error performance may vary with channel parameters such as delay and Doppler indices.The paper identifies this variation as a detrimental effect associated with different channel realizations.
- Robustness measure: An interleaver can permute coded symbols before constellation mapping or DD-domain OTFS modulation to whiten the transmitted symbols.The paper states that this can alleviate detrimental error-performance effects due to channel parameters.
- Evaluation: The proposed performance analysis and code-design approach is examined through numerical simulations of OTFS systems over high-mobility channels.The simulations are intended to examine the performance analysis developed in the paper.
IV. NUMERICAL RESULTS
Numerical simulations evaluate coded OTFS under high-mobility Rayleigh fading and compare codes, path counts, coding, and OFDM baselines. The results support the analytical code-design criterion and show OTFS advantages in error performance and diversity.
- Code comparisons: A larger minimum squared Euclidean distance yields a larger coding gain for OTFS at P = 8, while coded systems retain steeper FER behavior than uncoded OTFS.This observation is reported as consistent with the analytical result and code-design criterion.
- Path-count effects: Increasing the number of distinguishable paths improves coded OTFS error performance and increases diversity gain for the same code.The path-count comparison uses code D at a relative UE speed of 250 km/h.
- Diversity–coding-gain trade-off: At FER ≈10^-3, code D provides around 5.7 dB coding gain at P = 3 but around 5.0 dB at P = 8, demonstrating declining coding gain with more paths.The result matches the predicted diversity–coding-gain trade-off.
V. CONCLUSION
The paper derives performance bounds for coded OTFS over high-mobility channels and uses them to relate coding gain, diversity gain, and code design. Simulations verify the analysis and the proposed criterion, while modern OTFS code design remains future work.
- Conclusion: The paper derives conditional and unconditional pairwise-error probabilities by analyzing OTFS codeword distances and applying bounding techniques.The analysis distinguishes transmission cases according to the number of independent resolvable channel paths.
- Conclusion: Coding improvement depends on the squared Euclidean distance between codeword pairs and the number of independent resolvable paths.The conclusion identifies codeword distance as the relevant coding-performance quantity for the analyzed path conditions.
- Conclusion: The analysis demonstrates a fundamental trade-off between diversity gain and coding gain in OTFS systems.The simulations in the numerical-results section verify this reported relationship.
- Conclusion: The proposed code-design criterion is based on the derived unconditional performance bound.Its analytical basis and effectiveness are verified through numerical simulations.
- Future work: Future work will address modern OTFS code design using analytical tools including density evolution and extrinsic information transfer.This scope boundary is stated as planned future work rather than as a completed contribution.
APPENDIX A PROOF OF LEMMA 3
The appendix proves Lemma 3 using eigenvalue inequalities and identifies the equality condition for the determinant-related bound. Equality requires equal eigenvalues, including the diagonal codeword-difference case.
- Proof of Lemma 3: The proof uses the positive-definite Hermitian property of Ω(e), so all eigenvalues {λ_i} are positive.It then applies the arithmetic–geometric mean and Cauchy–Schwarz inequalities.
- Proof of Lemma 3: Equality in the relevant bound holds when the eigenvalues {λ_i} have the same value.The appendix gives a diagonal Ω(e) as an example of this condition.
- Proof of Lemma 3: The determinant of the codeword-difference matrix Ω(e) is expressed through its eigenvalues.The proof completes Lemma 3 after applying the eigenvalue-based inequality.
APPENDIX C PROOF OF THEOREM 2
The appendix justifies a determinant approximation used in the performance analysis through the P-condition number and eigenvalue properties. It also notes that the approximation may be loose for ill-conditioned matrices and relies on an SNR assumption.
- Proof of Theorem 2: The approximation uses slow variation of the exponential term, but its justification additionally depends on the SNR assumption associated with the bound.The appendix explicitly restricts the validity of the relevant result through that SNR condition.
- Proof of Theorem 2: The determinant approximation is exact when Ω(e) is diagonal, while its mathematical accuracy may be loose when Ω(e) is ill-conditioned.The appendix motivates checking conditioning with the P-condition number.
- Proof of Theorem 2: The P-condition number is defined using the spectral radius and is used to distinguish ill-conditioned from well-conditioned matrices.A large P-condition number indicates ill-conditioning, whereas a small value indicates good conditioning.
- Proof of Theorem 2: The proof establishes that the P-condition number of Ω(e) is at least 1 and concludes that Ω(e) is generally well-conditioned.This supports the stated accuracy of the determinant approximation in the analyzed setting.