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Generalized Step-Chirp Sequences With Flexible Bandwidth

Cheng Du, Yi Jiang

TL;DR

A family of sequences termed the generalized step-chirp (GSC) sequence with a closed-form expression, where some parameters can be tuned to adjust the bandwidth flexibly for beam sweeping in MIMO broadcast channels.

Abstract

Sequences with low aperiodic autocorrelation sidelobes have been extensively researched in literatures. With sufficiently low integrated sidelobe level (ISL), their power spectrums are asymptotically flat over the whole frequency domain. However, for the beam sweeping in the massive multi-input multi-output (MIMO) broadcast channels, the flat spectrum should be constrained in a passband with tunable bandwidth to achieve the flexible tradeoffs between the beamforming gain and the beam sweeping time. Motivated by this application, we construct a family of sequences termed the generalized step-chirp (GSC) sequence with a closed-form expression, where some parameters can be tuned to adjust the bandwidth flexibly. In addition to the application in beam sweeping, some GSC sequences are closely connected with Mow's unified construction of sequences with perfect periodic autocorrelations, and may have a coarser phase resolution than the Mow sequence while their ISLs are comparable.

Generalized Step-Chirp Sequences With Flexible Bandwidth

TL;DR

A family of sequences termed the generalized step-chirp (GSC) sequence with a closed-form expression, where some parameters can be tuned to adjust the bandwidth flexibly for beam sweeping in MIMO broadcast channels.

Abstract

Sequences with low aperiodic autocorrelation sidelobes have been extensively researched in literatures. With sufficiently low integrated sidelobe level (ISL), their power spectrums are asymptotically flat over the whole frequency domain. However, for the beam sweeping in the massive multi-input multi-output (MIMO) broadcast channels, the flat spectrum should be constrained in a passband with tunable bandwidth to achieve the flexible tradeoffs between the beamforming gain and the beam sweeping time. Motivated by this application, we construct a family of sequences termed the generalized step-chirp (GSC) sequence with a closed-form expression, where some parameters can be tuned to adjust the bandwidth flexibly. In addition to the application in beam sweeping, some GSC sequences are closely connected with Mow's unified construction of sequences with perfect periodic autocorrelations, and may have a coarser phase resolution than the Mow sequence while their ISLs are comparable.
Paper Structure (16 sections, 2 theorems, 30 equations, 6 figures)

This paper contains 16 sections, 2 theorems, 30 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. DFT Codebook-based Beam Sweeping
  4. Chirp-like Sequence-based Omnidirectional Beamforming
  5. Generalized Step-chirp Sequence
  6. Construction of Generalized Step-chirp Sequence
  7. Relationships Between the GSC sequence and Other Sequences
  8. GSC Sequence and DFT Codebook
  9. GSC Sequence and GC Sequence
  10. GSC Sequence and Mow Sequence
  11. Simulations
  12. Tradeoffs Between the Beamforming Gain and the Beam Sweeping Time
  13. Phase Resolution and Spectrum
  14. GSC Sequence versus GC sequence
  15. GSC sequence versus Mow sequence
  16. ...and 1 more sections

Key Result

Theorem 1

The GSC sequence is a family of polyphase sequences with entries $\frac{1}{\sqrt{N}}\omega_N^{m\zeta_n}, n \in {\mathbb Z}_N$, where with parameter set $\{N, \gamma, m, b\ \vert\ N \in {\mathbb Z}^+, m|N, \frac{1}{N}\leq\gamma\leq1, b\in {\mathbb R}\}$. The passband of the power spectrum of the GSC sequence is where $\omega_0 = \frac{2\pi}{N}m\gamma\left(b-\frac{1}{2}\right)$. For beam sweeping

Figures (6)

  • Figure 1: The power spectrums of two kinds of beamformings. (a): the DFT codebook-based beam sweeping; (b) the chirp-like sequence-based omnidirectional beamforming.
  • Figure 2: Step approximation of LFM, $a=1, b=\frac{1}{2}$, and $T=10$.
  • Figure 3: The relationships between the Mow sequence, the GC sequence, the DFT codebook and the GSC sequence.
  • Figure 4: Flexible tradeoffs between the beamforming gain and the beam sweeping time. (a): $\gamma = \frac{1}{2}$; (b): $\gamma = \frac{1}{5}$; (c): $\gamma = \frac{1}{7}$; (d): $\gamma = \frac{1}{13}$.
  • Figure 5: The improvement of phase resolution and spectrum of the GSC sequence against the GC sequence. (a): the passband NRMSE and the stopband leakage ratio of a length-$50$ GSC sequence for different $m$; (b): the phases of the GC sequence corresponding to $m=1$ in (a); (c): the phases of the GSC sequence corresponding to $m=10$ in (a).
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Proposition 1