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On the Spectrum of OTFS/VOFDM Signals: PSD Analysis and Bandwidth Allocation

Wei Wang, Xiang-Gen Xia

TL;DR

A null-space-based linear precoding method for OTFS signals to enable flexible bandwidth allocation is proposed and the effectiveness of the proposed NSLP method in tailoring the spectrum of OTFS signals is shown.

Abstract

Orthogonal time frequency space (OTFS)/vector OFDM (VOFDM) is widely regarded as a promising waveform for next-generation mobile communications. However, its spectral characteristics are not yet fully understood. The bandwidth allocation scheme, which is crucial for OTFS's integration into practical wireless standards, also remains unexplored. In this paper, we investigate the spectral characteristics of OTFS signals by analyzing their power spectral density (PSD). We demonstrate that the PSD of discrete-time OTFS signals is periodic with a period of $\frac{1}{MT_s}$, where $M$ is the size of the time/Doppler domain in OTFS, a.k.a., the vector size in VOFDM, and $T_s$ is the sampling interval length of digital to analog converter (DAC), resulting in $M$ identical spectral components within the spectral range $[-\frac{1}{2T_s}, \frac{1}{2T_s})$ of the continuous-time OTFS signal. The periodicity makes bandwidth allocation for OTFS/VOFDM signaling substantially challenging. Furthermore, we establish a relationship between the PSD of OFDM signals and that of OTFS signals, revealing that, when the information symbols are independent, the PSD of OTFS signals is equal to the sum of the PSDs of the component-expanded OFDM (CEP-OFDM) signals. Lastly, we derive a relationship between the information symbols and the corresponding OTFS spectrum, and based on which, we propose a null-space-based linear precoding (NSLP) method for OTFS signals to enable flexible bandwidth allocation. Numerical results validate our analytical results regarding the PSD of OTFS signals and show the effectiveness of our proposed NSLP method in tailoring the spectrum of OTFS signals.

On the Spectrum of OTFS/VOFDM Signals: PSD Analysis and Bandwidth Allocation

TL;DR

A null-space-based linear precoding method for OTFS signals to enable flexible bandwidth allocation is proposed and the effectiveness of the proposed NSLP method in tailoring the spectrum of OTFS signals is shown.

Abstract

Orthogonal time frequency space (OTFS)/vector OFDM (VOFDM) is widely regarded as a promising waveform for next-generation mobile communications. However, its spectral characteristics are not yet fully understood. The bandwidth allocation scheme, which is crucial for OTFS's integration into practical wireless standards, also remains unexplored. In this paper, we investigate the spectral characteristics of OTFS signals by analyzing their power spectral density (PSD). We demonstrate that the PSD of discrete-time OTFS signals is periodic with a period of , where is the size of the time/Doppler domain in OTFS, a.k.a., the vector size in VOFDM, and is the sampling interval length of digital to analog converter (DAC), resulting in identical spectral components within the spectral range of the continuous-time OTFS signal. The periodicity makes bandwidth allocation for OTFS/VOFDM signaling substantially challenging. Furthermore, we establish a relationship between the PSD of OFDM signals and that of OTFS signals, revealing that, when the information symbols are independent, the PSD of OTFS signals is equal to the sum of the PSDs of the component-expanded OFDM (CEP-OFDM) signals. Lastly, we derive a relationship between the information symbols and the corresponding OTFS spectrum, and based on which, we propose a null-space-based linear precoding (NSLP) method for OTFS signals to enable flexible bandwidth allocation. Numerical results validate our analytical results regarding the PSD of OTFS signals and show the effectiveness of our proposed NSLP method in tailoring the spectrum of OTFS signals.
Paper Structure (16 sections, 4 theorems, 39 equations, 8 figures, 1 table)

This paper contains 16 sections, 4 theorems, 39 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. PSD Analysis of OTFS Signals Generated by DAC
  3. The Waveform of OTFS Signals Generated by DAC
  4. The PSD of OTFS Baseband Signal Generated by DAC
  5. On the Impacts of Interpolation Filter
  6. PSD Comparison of OTFS and OFDM Signals
  7. The Waveform of OFDM Baseband Signal Generated by DAC
  8. The PSD of OFDM Signal Generated by DAC
  9. A Study on the Connection Between OTFS and OFDM
  10. Bandwidth Allocation for OTFS Signals
  11. A Review of Bandwidth Allocation for OFDM Signals
  12. The Relationship Between Discrete-Time OTFS Spectrum and the Information Symbols
  13. Null-space Based Linear Precoding (NSLP) for Flexible Bandwidth Allocation Scheme of OTFS Signals
  14. Systematic-Form Linear Precoder
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Proposition 1

When the information symbols $x_{i,l,k}$ are i.i.d. with zero mean and variance $\mathbb{E}\left( {x}^*_{i,l,k} {x}_{{i}, {l}, {k}} \right) = \sigma^2_{l,k}$ with respect to the index $i$, and are independent across the indices $(l,k)$, the resulting discrete-time OTFS signal ${s}_{\eta}$ in Stime i

Figures (8)

  • Figure 1: The procedures of generating the continuous-time OTFS signal
  • Figure 3: The PSDs of OTFS signals with the different interpolation filters, where the Dirac delta interpolation filter corresponds to discrete OTFS signal, the sinc interpolation filter corresponds to ideal digital-to-analog conversion, and the rectangular filter models the sample-and-hold operation of a DAC.
  • Figure 4: The procedures of generating the continuous-time OFDM signal
  • Figure 5: The connection between the OTFS signal and its CEP-OFDM in time and frequency domain
  • Figure 6: The PSDs of OFDM signals with different interpolation filters, where the Dirac delta interpolation filter corresponds to discrete OFDM signal, the sinc interpolation filter corresponds to ideal digital-to-analog conversion, and the rectangular filter models the sample-and-hold operation of a DAC.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Corollary 1
  • Remark 3
  • ...and 4 more