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FFT reconstruction of signals from MIMO sampled data

Dong Cheng, Xiaoxiao Hu, Kit Ian Kou

Abstract

This paper introduces an innovative approach for signal reconstruction using data acquired through multi-input-multi-output (MIMO) sampling. First, we show that it is possible to perfectly reconstruct a set of periodic band-limited signals $\{x_r(t)\}_{r=1}^R$ from the samples of $\{y_m(t)\}_{m=1}^M$, which are the output signals of a MIMO system with inputs $\{x_r(t)\}_{r=1}^R$. Moreover, an FFT-based algorithm is designed to perform the reconstruction efficiently. It is demonstrated that this algorithm encompasses FFT interpolation and multi-channel interpolation as special cases. Then, we investigate the consistency property and the aliasing error of the proposed sampling and reconstruction framework to evaluate its effectiveness in reconstructing non-band-limited signals. The analytical expression for the averaged mean square error (MSE) caused by aliasing is presented. Finally, the theoretical results are validated by numerical simulations, and the performance of the proposed reconstruction method in the presence of noise is also examined.

FFT reconstruction of signals from MIMO sampled data

Abstract

This paper introduces an innovative approach for signal reconstruction using data acquired through multi-input-multi-output (MIMO) sampling. First, we show that it is possible to perfectly reconstruct a set of periodic band-limited signals from the samples of , which are the output signals of a MIMO system with inputs . Moreover, an FFT-based algorithm is designed to perform the reconstruction efficiently. It is demonstrated that this algorithm encompasses FFT interpolation and multi-channel interpolation as special cases. Then, we investigate the consistency property and the aliasing error of the proposed sampling and reconstruction framework to evaluate its effectiveness in reconstructing non-band-limited signals. The analytical expression for the averaged mean square error (MSE) caused by aliasing is presented. Finally, the theoretical results are validated by numerical simulations, and the performance of the proposed reconstruction method in the presence of noise is also examined.
Paper Structure (14 sections, 1 theorem, 69 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 69 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Preliminaries
  4. Problem Statement
  5. The MIMO Sampling Expansion of Periodic Band-limited Signals
  6. Consistency Test and Error analysis
  7. Consistency Test
  8. Theoretical Error analysis
  9. Numerical Examples
  10. Error-free Reconstruction
  11. Consistency Test
  12. Aliasing Error Analysis
  13. Signal Reconstruction from Noisy Samples
  14. Conclusion

Key Result

Theorem 3.3

Let $x_r(t)\in B_{\mathbf{N}}$, $1\leq r\leq R$ and $y_m(t)$, $1\leq m\leq M$ be the input signals and the output signals of the MIMO system $\mathbf{H}(t)$. Let $L$, $\mathbf{B}(n)$ be given above. If $\mathbf{B}(n)$ is of full column rank for every $n\in I_1$, then the input signal $x_r(t)$ ($1\le where the $g_{rm}(t)$ is given by (interp_func).

Figures (9)

  • Figure 1: Diagram of MIMO sampling and reconstruction.
  • Figure 2: Diagram of FFT-based signal reconstruction from MIMO samples.
  • Figure 3: Error-free Reconstruction. The band-limited input signals $x_1(t), x_2(t)$ (bandwidth $51$) are sampled by the sampling scheme S-22t (see bottom right). They can be perfectly reconstructed by the proposed FFT-based algorithm.
  • Figure 4: Consistency test under the sampling scheme S-22t. Row 1--2: the original signals and their MIMO samples. Row 3: the reconstructed signals. Row 4: the resampled samples. Row 5: the original and resampled samples are identical.
  • Figure 5: Consistency test under the sampling scheme S-23t. Row 1: the original signals. Row 3: the reconstructed signals. Row 2: The original samples do not match the resampled ones.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3