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Discrete Spectrum Analysis of Vector OFDM Signals

Xiang-Gen Xia, Wei Wang

TL;DR

The paper addresses the unclear discrete spectrum of Vector OFDM (VOFDM) signals and proposes a rigorous spectrum analysis. It presents the main finding that the $k$-th discrete spectrum vector ${\bf y}_k$ is the linear image of the $k$-th information vector ${\bf x}_k$ via $y_k(m) = \frac{1}{\sqrt{M}} \sum_{n=0}^{M-1} \left(x_k(n) W_{MN}^{nk}\right) W_M^{nm}$, i.e., the $M$-point DFT of the modulated ${\bf x}_k$, and notes that ${\bf x}_k=0$ implies ${\bf y}_k=0$, enabling spectrum control similar to OFDM. Using this relation, it designs vector-wise precoding to force selected spectrum components to zero by partitioning ${\bf x}_k$ and solving a linear system with the ${\bf W}_M$ matrix; this preserves low complexity when $M$ is small. Simulations for $M=N=4$ and for $M=8$, $N=64$ validate the theory, illustrate zeroing of spectrum portions, and discuss PAPR implications of precoding. Overall, the work enables practical spectrum shaping in VOFDM under time-varying channels and connects VOFDM with OTFS perspectives, while highlighting a trade-off between spectral control and PAPR.

Abstract

Vector OFDM (VOFDM) is equivalent to OTFS and is good for time-varying channels. However, due to its vector form, its signal spectrum is not as clear as that of the conventional OFDM. In this paper, we study the discrete spectrum of discrete VOFDM signals. We obtain a linear relationship between a vector of information symbols and a vector of the same size of components evenly distributed in the discrete VOFDM signal spectrum, and show that if a vector of information symbols is set to 0, then a corresponding vector of the same size of the discrete VOFDM signal spectrum is 0 as well, where the components of the 0 vector are not together but evenly distributed in the spectrum. With the linear relationship, the information symbol vectors can be locally precoded so that any of the discrete spectrum of VOFDM signals can be set to 0, similar to that of the conventional OFDM signals. These results are verified by simulations.

Discrete Spectrum Analysis of Vector OFDM Signals

TL;DR

The paper addresses the unclear discrete spectrum of Vector OFDM (VOFDM) signals and proposes a rigorous spectrum analysis. It presents the main finding that the -th discrete spectrum vector is the linear image of the -th information vector via , i.e., the -point DFT of the modulated , and notes that implies , enabling spectrum control similar to OFDM. Using this relation, it designs vector-wise precoding to force selected spectrum components to zero by partitioning and solving a linear system with the matrix; this preserves low complexity when is small. Simulations for and for , validate the theory, illustrate zeroing of spectrum portions, and discuss PAPR implications of precoding. Overall, the work enables practical spectrum shaping in VOFDM under time-varying channels and connects VOFDM with OTFS perspectives, while highlighting a trade-off between spectral control and PAPR.

Abstract

Vector OFDM (VOFDM) is equivalent to OTFS and is good for time-varying channels. However, due to its vector form, its signal spectrum is not as clear as that of the conventional OFDM. In this paper, we study the discrete spectrum of discrete VOFDM signals. We obtain a linear relationship between a vector of information symbols and a vector of the same size of components evenly distributed in the discrete VOFDM signal spectrum, and show that if a vector of information symbols is set to 0, then a corresponding vector of the same size of the discrete VOFDM signal spectrum is 0 as well, where the components of the 0 vector are not together but evenly distributed in the spectrum. With the linear relationship, the information symbol vectors can be locally precoded so that any of the discrete spectrum of VOFDM signals can be set to 0, similar to that of the conventional OFDM signals. These results are verified by simulations.
Paper Structure (4 sections, 1 theorem, 18 equations, 7 figures)

This paper contains 4 sections, 1 theorem, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Discrete Spectrum and Precoding
  3. Simulations
  4. Conclusion

Key Result

Theorem 1

The $k$th discrete spectrum vector ${\bf y}_k$ in (5) of a discrete VOFDM signal and the $k$th information symbol vector ${\bf x}_k$ in (0) satisfy the linear relationship (6), where $0\leq k\leq N-1$. If vector ${\bf x}_{k_0}=0$ for some $0\leq k_0\leq N-1$, then vector ${\bf y}_{k_0}=0$.

Figures (7)

  • Figure 1: Spectrum of a VOFDM signal of length $16$ with $M=N=4$ and the first information symbol vector ${\bf x}_0$ is set to $0$.
  • Figure 2: Spectrum of a VOFDM signal of length $512$ with $M=8$ and $N=64$, and the first $16$ information symbol vectors ${\bf x}_k$, $0\leq k\leq 15$, are set to $0$.
  • Figure 3: Averaged magnitudes of a precoded signal ${\bf x}_k$ with $M=8$ and $N=64$, where the last $2$ frequency components in ${\bf y}_k$ are set to $0$.
  • Figure 4: Averaged magnitudes of the precoded part ${\bf x}_{k,2}$ for different vector sizes $M$, where $N=64$ and the last $2$ frequency components in ${\bf y}_k$ are set to $0$.
  • Figure 5: Frequency spectrum of a VOFDM signal with the last $32$ frequency components $0$.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1