Source-linked AI summary
Low-complexity Equalization of Zak-OTFS Via Neumann Series
Vineetha Yogesh, Saif Khan Mohammed, Sandesh Rao Mattu, Ronny Hadani, Robert Calderbank
TL;DR
When delay-Doppler taps do not lie on a line, the channel can be nondiagonal in the selected carrier basis and symbols interfere. The paper matches a Zak-OTFS carrier basis to channel geometry and uses a Neumann equalizer when the effective channel is diagonally dominant. For fractional two-tap channels, simulations report that the Neumann equalizer outperforms single-tap equalization and approaches Zero-Forcing performance.
Problem
When channel taps do not lie on a line, the channel matrix in the corresponding basis is nondiagonal and information symbols interfere.
Source: §IV · p. 4Method
The method obtains ZC carrier bases from Zak-OTFS pulsones via GDAFT and uses a Neumann-series approximation to Zero-Forcing when the effective channel is diagonally dominant.
Source: §I · p. 1, §IV · p. 4Results
For a two-tap channel with fractional taps, the Neumann equalizer performs better than single-tap equalization and close to Zero-Forcing equalization for some choice of basis parameter.
Source: §IV · p. 4Takeaways & Limitations
When taps lie on a line, the ZC basis diagonalizes the effective channel for single-tap equalization; more generally, the Neumann equalizer mitigates interference from off-diagonal entries.
Source: §VI · p. 6Neumann equalization is not guaranteed to converge when random path gains make the effective channel insufficiently diagonally dominant.
Source: §V · pp. 5–6
Abstract
from arXiv · showhide
We describe a general method for selecting an orthonormal basis of carrier waveforms that aligns the basis with delay / Doppler characteristics of a wireless channel. We show that our method enables low-complexity equalization for two channels of practical interest. The first is satellite communication and the second is communication from a ground station to an unmanned aerial vehicle (UAV). After Doppler compensation both scenarios are characterized by a first line of sight (LOS) path with zero delay and zero Doppler shift, and a second weaker path. We show that our method is robust to fractional delay and Doppler shifts. We consider orthonormal bases of carrier waveforms that are obtained from the pulsone basis of Zak-OTFS carrier waveforms by applying a generalization of the discrete affine Fourier transform (GDAFT). This family of waveforms includes AFDM and other modulations proposed for 6G. What distinguishes these bases is that the carrier waveforms are common eigenvectors of some maximal commutative subgroup S of a Heisenberg-Weyl group of discrete delay and Doppler shifts. We describe how to choose S to mitigate the damaging effects of interference between carriers. We represent the wireless channel by delay-Doppler taps and observe that a channel tap located within S multiplies every waveform by a complex phase. If all channel taps are located within S, then the channel multiplies every waveform by a complex phase, and a single tap equalizer supports reliable communication. This is the case for a linear time-invariant (LTI) channel, where S is the group of discrete time shifts and the carrier waveforms are discrete tones (OFDM). In general, we choose the subgroup S to maximize the diagonal component of the channel energy.
I. INTRODUCTION
The paper frames Zak-OTFS in a discrete-time model and selects carrier bases to fit channel-tap geometry. It proposes a Neumann equalizer when the channel action is not diagonal and evaluates equalizers in two-path scenarios with fractional delay and Doppler shifts.
I. INTRODUCTION
- The system model is discrete-time and describes Zak-OTFS channels and pulse-shaping filters through twisted convolution in the delay-Doppler domain.The paper derives the input-output relation in both discrete-time and matrix-vector forms.Source: §I · p. 1
- ZC carrier bases derived from Zak-OTFS pulsones by GDAFT make the channel act diagonally when all taps lie on a line modulo MN, enabling single-tap equalization.The ZC root determines the line's slope.Source: §I · p. 1
- When taps do not lie on a line, the channel may still be diagonally dominant, allowing a Neumann-series approximation to the Zero-Forcing equalizer.The paper calls this approximation the Neumann Equalizer.Source: §I · p. 1
- The simulations compare single-tap, Neumann, Zero-Forcing, and MMSE equalizers for a two-path channel with a zero-delay, zero-Doppler LOS path and a weaker second path.They also consider fractional delay and Doppler shifts and mismatch between tap locations and the integer grid.Source: §I · p. 1
II. SYSTEM MODEL AND THE ZAK-OTFS I/O RELATION
The section develops Zak-OTFS signal and channel relations in the delay-Doppler domain, then relates them to discrete-time waveforms and an effective channel matrix. It also describes a ZC carrier basis that diagonalizes the channel under a line-support condition, enabling separate symbol detection.
II. SYSTEM MODEL AND THE ZAK-OTFS I/O RELATION
- The channel is represented by paths with complex gains, delays, and Doppler shifts; each discrete tap acts through a delay permutation and Doppler modulation in the effective channel matrix.The channel matrix is decomposed into contributions from its significant delay-Doppler taps.Source: §II · pp. 1, 3
- Zak-OTFS information symbols are embedded in a quasi-periodic discrete delay-Doppler signal, lifted to the information lattice, and represented by pulsone carrier waveforms in discrete time.A pulsone is described as a pulse train modulated by a sinusoid.Source: §II · pp. 2–3
- Transmit and receive pulse shaping and the channel response form an effective delay-Doppler response through twisted convolution, while sampling yields the received discrete signal.The paper also obtains a discrete-time input-output relation by applying the inverse discrete Zak transform.Source: §II · p. 2
III. SINGLE-TAP EQUALIZATION FOR CHANNEL TAPS ON A LINE
- When all channel taps lie on a line modulo MN, the corresponding ZC waveforms are channel eigenvectors and enable single-tap equalization with O(MN) symbol-detection complexity.The ZC basis is obtained from the Zak-OTFS pulsone basis by the unitary GDAFT.Source: §III · pp. 3–4
IV. PROPOSED NEUMANN EQUALIZATION (NE)
When channel taps do not lie on a line, the effective channel is nondiagonal, but it may be diagonally dominant for a suitable basis parameter. The Neumann Equalizer approximates zero-forcing equalization using a truncated series, with lower complexity than exact ZF and reported performance close to ZF for fractional two-tap channels.
IV. PROPOSED NEUMANN EQUALIZATION (NE)
- When channel taps do not lie on a line, the channel matrix is nondiagonal, but a suitable parameter may make it diagonally dominant.The method uses strict row-wise diagonal dominance, which implies that the off-diagonal matrix has spectral radius below one.Source: §IV · p. 4
- The Neumann equalizer approximates the zero-forcing equalizer by restricting the Neumann series to its first R terms.The series converges when the off-diagonal matrix has spectral radius below one.Source: §IV · p. 4
- The Neumann equalizer has complexity O(RM^2N^2), compared with O(M^3N^3) for exact zero-forcing equalization.Its output is formed by applying the truncated series approximation to the received signal.Source: §IV · p. 4
- For fractional two-tap channels, the Neumann equalizer can outperform single-tap equalization and approach zero-forcing performance for a suitable basis parameter.The reported comparison applies even when fractional taps do not lie exactly on any line.Source: §IV · p. 4
V. SIMULATION RESULTS
Simulations evaluate a preconditioned two-path LOS/reflection model across on-grid, off-grid, and near-grid shifts, then test a 3GPP TDL-D channel. Equalizer behavior tracks how well the selected waveforms diagonalize channel energy.
V. SIMULATION RESULTS
- On-grid, u = 285 makes STE nearly match the other equalizers, with slightly worse BER because some channel energy leaks off the selected line.The line is chosen to pass through the two dominant taps at (0, 0) and (4, 7).Source: §V · pp. 4–5, Figure 2 · p. 5
- Off-grid, STE performs poorly for both tested u values, while u = 981 yields better NE performance than u = 285 because its line passes through the continuous-domain tap.The four dominant discrete taps from the fractional shift cannot all lie on one line; NE remains inferior to ZF and MMSE because H_u is not always diagonally dominant.Source: §V · p. 5, Figure 4 · p. 5
- NE is close to ZF/MMSE near-grid and in TDL-D, but falls short off-grid when H_u is not always diagonally dominant and convergence is not guaranteed.Randomness in the second-path gain can leave a small fraction of near-grid realizations where NE does not converge.Source: §V · pp. 5–6
- Near-grid, most energy remains at (0, 0) and (4, 7), so u = 285 gives almost-eigenvectors, less interference than in the off-grid case, and NE performance close to ZF/MMSE.A small fraction of channel realizations still fails to meet NE's diagonal-dominance condition.Source: §V · pp. 5–6, Figure 6 · p. 6
- For TDL-D, most channel energy lies on the zero-delay line; with Z_u = I, NE performs similarly to ZF/MMSE while STE is inferior because of off-line energy.The identity basis is eigen for zero-delay taps, but not for the small fraction of channel energy outside that line.Source: §V · p. 6, Figure 8 · p. 6, Figure 7 · p. 6
VI. CONCLUSION
The paper presents low-complexity Zak-OTFS equalization by matching an orthonormal carrier-waveform basis to the channel’s delay-Doppler geometry. Line-aligned taps enable single-tap equalization, while the Neumann equalizer addresses diagonally dominant channels with off-diagonal interference.
VI. CONCLUSION
- When taps lie on l = −uk (mod MN), ZC carrier waveforms with root u are effective-channel eigenvectors, enabling single-tap equalization.The basis is obtained from the Zak-OTFS pulsone basis using unitary GDAFT.Source: §VI · p. 6
- For diagonally dominant effective channels, the Neumann equalizer mitigates interference from off-diagonal entries.The conclusion positions it as the more general case beyond exact line alignment.Source: §VI · p. 6
- Simulations across two-path and 3GPP TDL-D LOS channels demonstrate the framework for on-grid and fractional delay-Doppler shifts.Source: §VI · p. 6
- In scenarios where the effective channel is mostly diagonally dominant, the Neumann equalizer performs similarly to ZF and MMSE and improves upon single-tap equalization.Source: §VI · p. 6