Deterministic Pilot Design and Channel Estimation for Downlink Massive MIMO-OTFS Systems in Presence of the Fractional Doppler

I. INTRODUCTION

• OTFS maps symbols from the delay-Doppler domain to the time-frequency domain and is described as providing full time-frequency diversity and roughly constant experienced channels at high Doppler.The paper notes that an OFDM modem can implement OTFS as an overlay of OFDM systems.Source: §I · p. 1
• Downlink CSI estimation is needed for FDD massive MIMO-OTFS, but earlier models assume integer Doppler taps and some methods require random pilots or prior uplink estimation.The paper describes the integer-tap assumption as impractical because Doppler-axis resolution is insufficient.Source: §I · p. 2
• The proposed scheme considers fractional Doppler and combines deterministic pilot design with an MSMCE algorithm for downlink CSI acquisition.The algorithm uses previously estimated path delays and Doppler frequencies to modify the sensing matrix, then updates those parameters for the next estimation.Source: §I · p. 2

II. SYSTEM MODEL

• The paper analyzes SISO OTFS input-output behavior using an OFDM modem, then extends the analysis to massive MIMO-OTFS.Its system-model section reviews OTFS concepts before this analysis and extension.Source: §I · p. 2, §II · p. 2

A. SISO OTFS System

• QAM symbols are placed on the delay-Doppler lattice, transformed to the time-frequency domain, and passed to an OFDM modulator that adds a cyclic prefix before transmission.The receiver removes the cyclic prefix, samples the cross-ambiguity function, and applies the SFFT to map received time-frequency data to the delay-Doppler domain.Source: A. SISO OTFS System · p. 3
• The channel is modeled with P dominant paths, each containing S subpaths; subpaths within a dominant path share a delay, and the model includes fractional Doppler.The paper assumes a sparse channel because transmission environments usually contain only a few scatterers.Source: A. SISO OTFS System · p. 3
• The analysis uses OFDM-based OTFS with a cyclic prefix long enough to exceed the maximum dominant-path delay, preventing inter-symbol interference between OFDM symbols within an OTFS symbol.The paper notes that fewer-CP spectral-efficient OTFS models would require an additional case for this inter-symbol interference.Source: A. SISO OTFS System · pp. 3–4
• Fractional Doppler makes each dominant path's received channel response depend on all its subpaths, complicating the input-output model.The authors approximate subpath Doppler frequencies within a dominant path as equal because the subpaths usually originate from one scattering cluster and have slight arrival-direction deviations.Source: A. SISO OTFS System · p. 4
• The simplified relationship is exact when each dominant path has one subpath; the paper identifies prior models as a special case when N tends to infinity.The model also uses dominant-path information in phase compensation, which the authors say makes it more accurate for practical systems.Source: A. SISO OTFS System · p. 4

B. Massive MIMO-OTFS System

• The model considers a centralized base station with a ULA of half-wavelength antenna spacing serving single-antenna users, and focuses on one user terminal.The system has Nt base-station antennas and U user terminals.Source: B. Massive MIMO-OTFS System · p. 4
• The received delay-Doppler symbols are expressed through the channel and transmitted symbols in the delay-Doppler-space domain.The channel representation uses the corresponding received-symbol relationship and channel definition.Source: B. Massive MIMO-OTFS System · p. 4
• A normalized DFT along the space dimension maps the delay-Doppler-space channel into the delay-Doppler-beam domain.The beam-domain channel's dominant elements occur near indices set by subpath Doppler, delay, and angle of departure, yielding three-dimensional sparsity.Source: B. Massive MIMO-OTFS System · p. 4
• The paper identifies downlink CSI acquisition as sparse signal reconstruction, and the received-symbol equality holds precisely when each dominant path has one subpath.The channel is sparse over the delay-Doppler-beam domain.Source: B. Massive MIMO-OTFS System · p. 5

III. DOWNLINK CSI ACQUISITION FOR MASSIVE MIMO-OTFS SYSTEMS

• Downlink CSI acquisition is framed as a sparse signal reconstruction problem.The section grounds this formulation in the system models and delay-Doppler-beam channel sparsity.Source: §III · p. 5
• Deterministic pilots are designed to improve sensing performance, and pilot overhead is discussed.The section proceeds from the acquisition model to pilot design.Source: §III · p. 5
• The MSMCE algorithm is proposed to reconstruct the sparse delay-Doppler-beam channel, and its performance is analyzed.Source: §III · p. 5

• Pilot positions are shared across transmit antennas, and received symbols for estimation occupy the same delay-Doppler region as the pilots.Source: A. Downlink CSI Acquisition Model · p. 5
• Guard intervals use zero symbols to separate pilots from data, with delay guards sized for maximum delay and Doppler guards accounting for fractional-Doppler leakage.Source: A. Downlink CSI Acquisition Model · p. 5
• Downlink CSI acquisition becomes sparse signal reconstruction, with the delay-Doppler-beam channel as the sparse vector and Θ as the sensing matrix.Source: A. Downlink CSI Acquisition Model · p. 6
• The channel-dependent phase matrix Φ is more accurate but requires path delays and Doppler frequencies, whereas channel-independent ˜Φ is inaccurate; MSMCE combines their advantages.Source: A. Downlink CSI Acquisition Model · p. 6

• Random sensing matrices can be memory-intensive and difficult to implement, motivating deterministic pilots with low column coherence for reliable sparse recovery.Source: B. Deterministic Pilot Design · p. 6
• For one beam, each dashed pilot-position box contributes MpNp symbols to a pilot-matrix column, with MgNg columns constructed in total.Source: B. Deterministic Pilot Design · p. 6
• The revised placement wraps Doppler-edge pilots and adds delay-edge pilots, avoiding zero entries in the pilot matrix that would complicate coherence design.Source: B. Deterministic Pilot Design · pp. 6–7
• Pilot sequences must be orthogonal to their cyclic shifts within a beam, while columns from additional beams must maintain low coherence across beam groups.Source: B. Deterministic Pilot Design · p. 7
• Zadoff-Chu sequences provide cyclic orthogonality and low cross-correlation, and the proposed design arranges pilots across beams into a matrix of size MpNp × Nt.Source: B. Deterministic Pilot Design · p. 7
• The designed pilot matrix has maximal coherence close to the Welch bound and can be generated for system configurations, unlike random pilots that require substantial memory.Source: B. Deterministic Pilot Design · p. 8

• With the same MpNp pilots, the design occupies Ng(Mp + 2Mg) more delay-Doppler grids than conventional random pilots, yet simulations show about 5–10 percent fewer pilots are needed.Source: C. Pilot Overhead · p. 8
• When practical fractional Doppler requires Np = N, Doppler guard intervals are unnecessary and the designed placement occupies the same grids as the conventional placement, further increasing saved overhead.Source: C. Pilot Overhead · p. 8

• The channel-independent ˜Θ avoids needing path information initially, while Φ incorporates path delays and Doppler frequencies for a more accurate sensing matrix.Source: D. MSMCE Algorithm · p. 8
• The extraction procedure locates delays from dominant values after summing over Doppler and beam dimensions, and estimates fractional Doppler from the channel magnitude’s peak pattern.Source: D. MSMCE Algorithm · pp. 8–9
• The extraction procedure assumes no fractional delay, so dominant paths do not interfere in that step.Source: D. MSMCE Algorithm · p. 9
• MSMCE first estimates the channel with ˜Θ, extracts dominant-path delays and Doppler frequencies, then constructs Φ and refines the estimate using Θ.Source: D. MSMCE Algorithm · p. 9
• Because channel geometry changes slowly relative to communication, later estimates can reuse Del and Dop to form the modified sensing matrix.Source: D. MSMCE Algorithm · p. 9

E. The Performance Analysis of the MSMCE Algorithm

• MSMCE achieves a significant performance gain at negligible complexity increase, while most Algorithm 2 operations are comparisons and are low-complexity relative to the selected CS algorithm.The comparison count is about P(Mg + 2Ng − 1).Source: E. The Performance Analysis of the… · pp. 9–10
• With multiple subpaths per dominant path, approximation error increases, reducing the accuracy of the channel model and extracted path information and degrading MSMCE performance.With one subpath per dominant path, the error is zero and MSMCE achieves optimal performance.Source: E. The Performance Analysis of the… · p. 10
• Higher user velocity and larger delay-dimension pilot positions increase approximation error, so the initial delay pilot should be as close to zero as possible; Doppler-dimension pilot position does not affect performance.Source: E. The Performance Analysis of the… · p. 10
• When Doppler resolution maps each path to an integer tap, the model is accurate and phase compensation is exact; rapid geometry changes or long estimation intervals make prior path information less reliable, requiring re-extraction.In the integer-tap case, only steps 2–3 of MSMCE are needed.Source: E. The Performance Analysis of the… · p. 10

IV. SIMULATION RESULTS

• At 35% pilot overhead, deterministic pilots achieve NMSE 0.013 with MSMCE, while random pilots need at least 45% pilot overhead.The comparison uses 16 BS antennas, 5 dB SNR, and 100 m/s user velocity.Source: §IV · p. 10, Figure 4 · p. 11
• At 89% pilot overhead, MSMCE substantially outperforms ISMCE across S=5 and 20, while deterministic pilots outperform random pilots.The setting uses 64 BS antennas, 100 m/s user velocity, and N = 19.Source: Figure 5 · p. 11, §IV · p. 11
• With fewer subpaths per dominant path, MSMCE performance improves and is best at one subpath, where the sensing matrix is exact.The comparison uses 64 BS antennas, 100 m/s user velocity, N = 19, and 89% pilot overhead.Source: Figure 6 · p. 11, §IV · p. 11
• With 20 subpaths, increasing initial pilot position lp degrades MSMCE, especially at high SNR; with one subpath, performance is position-independent.The comparison uses 32 BS antennas, 100 m/s user velocity, N = 19, and 49% pilot overhead.Source: §IV · p. 11, Figure 7 · p. 12
• MSMCE NMSE degrades as user velocity increases, but it retains a significant gain over ISMCE at different velocities.The comparison uses 16 BS antennas, 15 dB SNR, N = 19, 35% pilot overhead, and 20 subpaths per dominant path.Source: §IV · pp. 11–12, Figure 8 · p. 12
• As OTFS symbol size N decreases, ISMCE performance degrades while MSMCE remains consistent, because lower Doppler resolution makes fractional Doppler more consequential.The comparison uses 16 BS antennas, 100 m/s user velocity, 35% pilot overhead, and 20 subpaths per dominant path.Source: Figure 9 · p. 12, §IV · p. 12
• The MSMCE algorithm outperforms ISMCE in BER and comes very close to perfect CSI, with fractional-Doppler consideration and extracted path information.The comparison uses a 4QAM signal, an MMSE detector, 100 m/s user velocity, N = 11, and 35% pilot overhead.Source: Figure 10 · p. 12, §IV · pp. 12–13

• Simulations demonstrated that the proposed scheme could acquire accurate downlink CSI.Source: §V · p. 13
• The scheme designs deterministic pilots from the downlink CSI acquisition model and uses MSMCE to reconstruct the delay-Doppler-beam domain channel.Source: §V · p. 13
• The analysis derives the SISO OTFS input-output relationship based on the OFDM modem and extends it to massive MIMO-OTFS with fractional Doppler.Source: §V · p. 13

• Substituting (9) into (10) expresses Y DD[k, l].Source: APPENDIX PROOF OF PROPOSITION 1 · p. 13
• Substituting (42) into (41) writes Y DD[k, l] in terms that include N + ZDD[k, l].Source: APPENDIX PROOF OF PROPOSITION 1 · p. 13
• The passage states that this completes the proof.Source: APPENDIX PROOF OF PROPOSITION 1 · p. 13