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MIMO Array Calibration in Non-stationary Channels with Residual Surfaces and Slepian Spherical Harmonics

Oliver Kirkpatrick, Santiago Ozafrain, Christopher Gilliam, Beth Jelfs

TL;DR

This work addresses calibrating MIMO arrays in non-stationary channels by introducing residual surfaces that capture element-wise hardware perturbations relative to a reference. The authors model these residuals with a Slepian spherical harmonic basis tailored to the radar's field of view and estimate the basis weights from scatterers of opportunity using a least-squares fit. Simulations demonstrate that residual-surface calibration improves beamforming performance, yielding closer-to-ideal main lobes, deeper nulls, and more accurate direction finding, even when calibration observations are moderately noisy. The method supports in-situ calibration for large arrays and bistatic configurations, with practical effectiveness when observation SNRs are at or above 4 dB.

Abstract

The fundamental mechanism driving MIMO beamforming is the relative phases of signals departing the transmit array and arriving at the receive array. If a propagation channel affects all transmitted signals equally, the relative phases are a function of the directions of departure and arrival, as well as the transmit and receive hardware. In a non-stationary channel, the amplitudes and phases of arriving signals may vary significantly over time, making it infeasible to directly measure the influence of hardware. In this paper, we present a calibration method for achieving indirect measurement and compensation of hardware influences in non-stationary channels. Our method characterizes the patterns of array elements relative to a reference element and estimates these relative patterns, termed residual surfaces, using a Slepian spherical harmonic basis. Using simulations, we demonstrate that our calibration method achieves beamforming gains that closely match theoretical optimums. Our results also show a reduction in the error in estimating the target direction, lower side lobes, and improve null-steering capabilities.

MIMO Array Calibration in Non-stationary Channels with Residual Surfaces and Slepian Spherical Harmonics

TL;DR

This work addresses calibrating MIMO arrays in non-stationary channels by introducing residual surfaces that capture element-wise hardware perturbations relative to a reference. The authors model these residuals with a Slepian spherical harmonic basis tailored to the radar's field of view and estimate the basis weights from scatterers of opportunity using a least-squares fit. Simulations demonstrate that residual-surface calibration improves beamforming performance, yielding closer-to-ideal main lobes, deeper nulls, and more accurate direction finding, even when calibration observations are moderately noisy. The method supports in-situ calibration for large arrays and bistatic configurations, with practical effectiveness when observation SNRs are at or above 4 dB.

Abstract

The fundamental mechanism driving MIMO beamforming is the relative phases of signals departing the transmit array and arriving at the receive array. If a propagation channel affects all transmitted signals equally, the relative phases are a function of the directions of departure and arrival, as well as the transmit and receive hardware. In a non-stationary channel, the amplitudes and phases of arriving signals may vary significantly over time, making it infeasible to directly measure the influence of hardware. In this paper, we present a calibration method for achieving indirect measurement and compensation of hardware influences in non-stationary channels. Our method characterizes the patterns of array elements relative to a reference element and estimates these relative patterns, termed residual surfaces, using a Slepian spherical harmonic basis. Using simulations, we demonstrate that our calibration method achieves beamforming gains that closely match theoretical optimums. Our results also show a reduction in the error in estimating the target direction, lower side lobes, and improve null-steering capabilities.
Paper Structure (8 sections, 8 equations, 4 figures, 1 table)

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

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proposed Residual Surface Calibration Method
  4. Orthogonal basis functions tailored to a subregion
  5. Fitting Slepian basis to residual surface measurements
  6. Comments
  7. Analysis of Simulated Calibration Performance
  8. Conclusion

Figures (4)

  • Figure 1: Geometry of orthogonal and co-located uniform linear transmit and receive arrays.
  • Figure 2: Slepian spherical harmonic basis functions of varying rank $\alpha$, all well concentrated into a target subregion (red).
  • Figure 3: Relationship between number of samples and absolute difference between the estimated and ground truth Slepian weights.
  • Figure 4: Beamformer output comparison for an ideal array (identical elements), a perturbed array, uncalibrated, and calibrated with a Slepian residual surface approximation computed from observations with an 8 dB SNR. Dashed black line indicates true azimuth.