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Orthogonal Time Sequency Multiplexing Modulation: Analysis and Low-Complexity Receiver Design

Tharaj Thaj, Emanuele Viterbo, Yi Hong

arXiv:2104.05939v1eess.SPcs.ITmath.IT

TL;DR

The paper addresses the need for modulation robust to fractional Doppler shifts, which are common in channels. It proposes OTSM and reports performance better than OFDM and similar to OTFS, with lower transmission and detection complexity.

  • Problem

    The paper targets modulation robust to fractional Doppler shifts, which are commonly present in channels.

    Source: §I · pp. 2, 4

  • Method

    The paper proposes OTSM as a new modulation scheme.

    Source: §IX · p. 29

  • Results

    OTSM offers better performance than OFDM and similar performance to OTFS, with lower transmission and detection complexity.

    Source: §VIII · p. 26

  • Takeaways & Limitations

    Coded OTSM outperforms coded SC, while coding provides slightly smaller returns.

    Source: §VIII · p. 27

Abstract

from arXiv · show

This paper proposes orthogonal time sequency multiplexing (OTSM), a novel single carrier modulation scheme that places information symbols in the delay-sequency domain followed by a cascade of time-division multiplexing (TDM) and Walsh-Hadamard sequence multiplexing. Thanks to the Walsh Hadamard transform (WHT), the modulation and demodulation do not require complex domain multiplications. For the proposed OTSM, we first derive the input-output relation in the delay-sequency domain and present a low complexity detection method taking advantage of zero-padding. We demonstrate via simulations that OTSM offers high performance gains over orthogonal frequency division multiplexing (OFDM) and similar performance to orthogonal time frequency space (OTFS), but at lower complexity owing to WHT. Then we propose a low complexity time-domain channel estimation method. Finally, we show how to include an outer error control code and a turbo decoder to improve error performance of the coded system.

I. INTRODUCTION

OTSM places information symbols in the delay-sequency domain and uses a Walsh-Hadamard transform (WHT), aiming to support channels with mobility and fractional Doppler shifts. The paper reports performance similar to OTFS at lower complexity, and introduces detection, channel-estimation, and coding methods.

I. INTRODUCTION

  • OTSM places symbols in the delay-sequency domain and applies an inverse WHT to transform them into the delay-time domain for transmission.The paper describes sequency as the number of zero-crossings per unit time interval.Source: §I · p. 2
  • OTSM separates delay-spread and Doppler-spread interference along delay and sequency dimensions, unlike OFDM, where both jointly interfere along frequency.The introduction links OFDM's joint interference to loss of subcarrier orthogonality and failure of single-tap equalization.Source: §I · pp. 2–3
  • OTSM is presented as a low-complexity single-carrier scheme using a one-dimensional orthogonal WHT, in contrast to the cited multi-carrier OTFS schemes using arbitrary two-dimensional unitary transforms.The paper says OTSM can be overlaid on OFDM-based systems using IWHT and FFT relations.Source: §I · p. 3
  • OTSM achieves performance similar to OTFS in static and high-mobility channels, with lower complexity because WHT requires only addition and subtraction.The paper also reports excellent performance with multiple Doppler paths.Source: §I · p. 3
  • Zero-padding between time-domain blocks reduces detection complexity and allows pilot insertion, while the proposed channel estimator reconstructs the delay-time channel from time-domain pilots.The paper also describes an outer error-correcting code as improving error performance and reducing pilot power required for accurate detection.Source: §I · p. 3
  • With integer information symbols, WHT produces integer-valued time-domain samples, allowing fewer transmit-signal bits without quantization error and supporting lower-complexity integer-forcing receivers.The paper also identifies OTSM as suitable for high-mobility and underwater acoustic communications.Source: §I · pp. 3–4
  • The introduction motivates OTSM with fractional Doppler shifts, which are common in channels, and notes that the sequency domain may offer OTFS-like sparsity in all cases.The cited OTFS delay-Doppler sparsity is described as reducing significantly when Doppler shifts are not integer multiples of receiver Doppler resolution.Source: §I · p. 4

II. BACKGROUND

The background introduces Walsh functions as an orthogonal basis and contrasts sequency indexing with frequency indexing. It describes how Walsh and Fourier basis functions differ in their discrete spectra.

II. BACKGROUND

  • Walsh functions form a complete orthogonal set with values +1 and -1, and waveforms can be synthesized from linear combinations of them,.The paper focuses on sequency-ordered Walsh functions.Source: §II · p. 5

A. Sequency vs Frequency

  • Walsh basis functions have 1, 2, or 4 non-zero spectral lines, whereas Fourier basis functions have only 1 or 2; Walsh spectra can spread across at most N/2 frequency indices.Source: A. Sequency vs Frequency · p. 6

B. Dyadic vs Cyclic convolution

The paper distinguishes the WHT's dyadic-convolution property from the DFT's cyclic-convolution property, then describes OTSM's transmitter mapping from delay-sequency symbols to transmitted time samples.

B. Dyadic vs Cyclic convolution

  • The DFT maps products to cyclic convolution, whereas the WHT maps products to dyadic convolution, with XOR of binary indices defining the dyadic operation.The paper uses this distinction when deriving OTSM's delay-sequency input-output relations and comparing OTSM with OTFS.Source: B. Dyadic vs Cyclic convolution · p. 7

A. Transmitter and Receiver Operation

  • OTSM frames have duration NT and bandwidth MΔf, with Δf = 1/T, critical sampling, and N chosen as a power of 2.The transmitter divides information symbols into vectors, applies an N-point WHT across rows, then vectorizes the delay-time matrix column-wise into time-domain samples.Source: A. Transmitter and Receiver Operation · p. 7, 1) Transmitter · p. 9

1) Transmitter:

  • The transmitter applies an N-point WHT to each row of the delay-sequency symbol matrix, then column-wise vectorizes the transformed matrix to form the transmitted time-domain signal.A cyclic prefix is added before pulse shaping and transmission.Source: 1) Transmitter · p. 9

2) Receiver:

At the receiver, OTSM folds received time samples into a delay-time matrix and applies a row-wise WHT to recover delay-sequency symbols. The block diagram also shows zero-padding regions used for pilots and guards.

2) Receiver:

  • The receiver folds received time-domain samples column-wise into a delay-time matrix and applies an N-point WHT to its rows to obtain delay-sequency information symbols.The transceiver description places channel estimation and detection in the delay-time domain before transforming samples back to delay-sequency symbols.Source: 2) Receiver · pp. 9–10, 3) Transceiver system block diagram · p. 11

3) Transceiver system block diagram:

  • Setting the last 2l_max + 1 rows of X to zero enables insertion of pilot and guard samples; these zeros form interleaved zero padding between time-domain blocks, simplifying detection and channel estimation by removing inter-block interference.Source: 3) Transceiver system block diagram · p. 10

B. Continuous-Time Baseband Channel Model

The channel is modeled through paths with gains, delays, and Doppler shifts under an underspread assumption. The discrete model samples the received waveform and commonly approximates fractional path delays by integer sampling points.

B. Continuous-Time Baseband Channel Model

  • The continuous-time channel has P paths with complex gains, delays, and Doppler shifts, and assumes maximum delay times maximum Doppler is much less than one.The delay-Doppler response is typically sparse because the number of channel coefficients is limited.Source: B. Continuous-Time Baseband Channel… · pp. 11–12

C. Discrete Time Baseband Channel Model

  • The discrete-time model samples the received waveform at MΔf and represents delay shifts as integer multiples of the sampling period.The receiver samples NM complex values and uses discrete delay taps from 0 through l_max.Source: C. Discrete Time Baseband Channel Model · p. 12
  • Fractional path delays introduce interference between different delay responses, but typical wideband systems round delays to the nearest sampling point.This approximation uses the sampling interval 1/(MΔf).Source: C. Discrete Time Baseband Channel Model · pp. 12–13

D. Input-Output Relation in vector form

The input-output relations express OTSM channel effects in delay-time and delay-sequency forms. Zero padding suppresses inter-block interference, while the Walsh-Hadamard transform yields a structured delay-sequency relation.

D. Input-Output Relation in vector form

E. Input-output relation in matrix form

IV. RELATION WITH ZP-OTFS

OTSM and OTFS have equal received information-symbol energy and SNR in the stated analysis, supporting similar error performance. OTSM replaces the DFT-based transform with WHT-based precoding, reducing modulation and demodulation complexity.

IV. RELATION WITH ZP-OTFS

  • OTSM can be interpreted as one-dimensional orthogonal precoding using an IFFT and WHT cascade along the Doppler domain.Its inverse Walsh Zak interpretation replaces the traditional Zak transform's DFT with WHT.Source: §IV · pp. 17–18
  • OTFS circular convolution along Doppler becomes dyadic convolution in sequency, where channel-coefficient ordering depends on the information symbol's sequency index.In OTFS, delay-Doppler coefficient ordering is independent of the information-symbol location.Source: §IV · pp. 17–18
  • OTSM retains OTFS's property that all information symbols experience approximately the same channel gain.Source: §IV · p. 18
  • OTSM and OTFS have equal received information-symbol energy and SNR, supporting similar error performance with lower OTSM modulation and demodulation complexity.The analysis attributes the equal SNR to unitary DFT and WHT transformations preserving noise power.Source: §IV · p. 18

V. LOW-COMPLEXITY DETECTION

The detector uses zero padding to prevent interference between time-domain blocks, then estimates symbols block by block. A matched-filtered channel system is solved iteratively with Gauss-Seidel updates and hard QAM decisions.

V. LOW-COMPLEXITY DETECTION

  • Time-domain zero padding prevents inter-block interference, allowing the received relation to be split and processed independently by block.Each block follows r_n = G_n · s_n + w_n.Source: §V · pp. 18–19
  • The detector applies Gauss-Seidel iteration to matched-filtered channel-matrix blocks to find the least-squares solution of an M-dimensional system.The matched-filter quantities are R_n = G_n† · G_n and z_n = G_n† · r_n.Source: §V · p. 19
  • Each iteration makes nearest-Euclidean-distance QAM decisions in the delay-sequency domain and transforms hard decisions back to time domain for the next update.The relaxation parameter is used to improve convergence for higher modulation schemes such as 64-QAM.Source: §V · pp. 19–20
  • The iterative detector can start from zero or an MMSE solution, with the MMSE initialization yielding faster convergence.Source: §V · p. 20

VI. EMBEDDED PILOT-AIDED CHANNEL ESTIMATION

OTSM uses an embedded delay-sequency pilot to estimate the channel in the delay-time domain, then reconstructs channel coefficients between pilot observations. Pilot-power allocation trades estimation quality against energy efficiency.

VI. EMBEDDED PILOT-AIDED CHANNEL ESTIMATION

  • For OTSM, linear interpolation is sufficient; unlike the single-pilot OTFS scheme, proposed channel estimation is performed in the delay-time domain rather than the DD domain.Source: §VI · p. 20

A. Pilot placement

  • A single delay-sequency pilot vector is flanked by guard vectors to limit data-pilot interference from delay spread; a cyclic prefix supports interpolation before the pilot time.The prefix copies the last lmax+1 time-domain samples, including the pilot sample.Source: A. Pilot placement · pp. 20–21

VI. EMBEDDED PILOT-AIDED CHANNEL ESTIMATION

  • After conversion to the time domain, interleaved pilot samples provide parallel, factor-M-subsampled observations of the channel, estimated directly in the delay-time domain.Interpolation reconstructs the time-domain channel coefficients across the frame for each delay tap.Source: §VI · p. 20, A. Pilot placement · p. 21

B. Pilot power allocation

  • With total transmit power fixed, baseline pilot power scales with N and lzp, and lzp is generally less than 10% of M under the underspread-channel assumption.The paper notes this allocation benefits frames with large N and M because pilot power, and thus pilot-sample SNR, increases proportionately.Source: B. Pilot power allocation · pp. 21–22
  • In the EVA example, the baseline pilot power ratio is approximately 10%; increasing excess pilot power improves channel estimation but reduces energy efficiency.The stated trade-off is that more energy is spent per information bit as the pilot-power ratio increases.Source: B. Pilot power allocation · p. 22

C. Delay-time channel estimation

The estimator obtains subsampled delay-time channel values from pilot observations and interpolates between them to reconstruct the channel across the frame. Its reconstruction condition depends on Doppler, and linear interpolation offers a lower-complexity alternative to spline interpolation in the considered setting.

C. Delay-time channel estimation

  • Pilot observations yield subsampled delay-time channel values, which are interpolated to reconstruct intermediate samples and the channel across the OTSM frame.The estimated values are at discrete pilot sample locations separated by M.Source: C. Delay-time channel estimation · pp. 22–23
  • Accurate reconstruction requires maximum Doppler shift below half the subsampled observation frequency, a condition described as reasonable for underspread channels.The estimated channel is sampled at M times lower frequency than the receiver sampling rate.Source: C. Delay-time channel estimation · p. 23
  • Spline interpolation captures channel variations better when delay-time responses have more oscillations at high Doppler, while linear interpolation can fail to trace those responses accurately.The comparison is illustrated for an EVA channel at 500 km/hr.Source: C. Delay-time channel estimation · pp. 23–24, Figure 6 · p. 23
  • For the considered underspread channels, linear and spline interpolation give very similar BER performance at speeds below at least 500 km/hr, supporting linear interpolation for lower-complexity estimation.The BER comparison uses 4-QAM and includes speeds of 500 and 1000 km/hr.Source: C. Delay-time channel estimation · p. 24, Figure 7 · p. 24

D. Channel Estimation Complexity

The section quantifies the cost of reconstructing channel samples, describes iterative detector-decoder operation, and reports OTSM comparisons for uncoded and coded systems. The simulations show gains over OFDM and coded single-carrier transmission, alongside performance close to OTFS or perfect-CSI operation in the stated settings.

D. Channel Estimation Complexity

  • Linear interpolation reconstructs each intermediate channel sample with one scalar multiplication by a complex number, ignoring additions.The initial slope calculation requires 2(N−1)ML scalar multiplications.Source: D. Channel Estimation Complexity · p. 25

VII. TURBO DECODER FOR CODED OTSM

  • Turbo decoding alternates time-domain detection with LDPC decoding, using decoder-improved symbol estimates to update the next detector iteration.Each turbo iteration includes at least one detector and one decoder iteration.Source: §VII · p. 25

VIII. SIMULATION RESULTS AND DISCUSSION

  • Uncoded OTSM outperforms OFDM and performs similarly to OTFS, with lower transmission and detection complexity.The comparison is reported for uncoded BER with perfect CSI.Source: §VIII · p. 26, Figure 9 · p. 27
  • At FER 10^-2, coded OTSM gains 4 dB over OFDM with the short rate-1/2 code and 2 dB with the long rate-1/2 code.The comparison uses the single-tap equalizer for OTSM and OFDM.Source: §VIII · p. 27, Figure 10 · p. 28
  • Coded OTSM outperforms coded single-carrier transmission, while coding reduces the performance gap because it helps the single-carrier scheme recover some channel diversity.The comparison is shown for 4-QAM at rates 1/2 and 3/4 with turbo decoding.Source: §VIII · p. 27, Figure 11 · p. 28
  • Coded OTSM achieves performance similar to perfect CSI for 16-QAM and 64-QAM with less pilot power than the uncoded case, and longer codewords further reduce required pilot power.Source: §VIII · pp. 27, 29

IX. CONCLUSION

OTSM multiplexes information using time- and sequency-shifted rectangular-pulse components and derives their input-output relation in the delay-sequency domain. Simulations report better performance than OFDM and single-carrier transmission, and performance similar to OTFS at lower complexity.

IX. CONCLUSION

  • OTSM multiplexes information using time- and sequency-shifted versions of a basic rectangular pulse and derives the input-output relation in the delay-sequency domain.Source: §IX · p. 29
  • Zero padding between time-domain blocks supports low-complexity channel estimation and detection for OTSM.Source: §IX · p. 29
  • Simulations show OTSM performs significantly better than OFDM and single-carrier transmission, and similarly to OTFS at much lower complexity thanks to the WHT.The authors describe OTSM as promising for future high-mobility systems requiring low-complexity transceivers.Source: §IX · p. 29
  • Further analysis will explore channel representation and low-complexity channel estimation and detection in the delay-sequency domain.Source: §IX · p. 29
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