Correlation and Spectral Density Functions in Mode-Stirred Reverberation -- II. Spectral Moments, Sampling, Noise, EMI and Understirring

TL;DR

This work quantifies practical biases in estimating spectral moments and kurtosis for mode-stirred reverberation fields, focusing on finite-difference effects, sampling strategies, aliasing, noise, EMI, and understirring. It develops analytic debiasing approaches for continuous and discretely sampled stir data and compares stir-sweep FD with complex autocovariance and spectral-density methods, validating results against high-SNR VNA measurements in a NIST MSRC. Key findings show a leading-order negative quadratic FD bias, aliasing effects are small up to 18 GHz, and noise/EMI can significantly alter kurtosis, underscoring the need for debiasing and careful experimental design. The results provide practical guidelines for accurately extracting spectral moments and kurtosis in real-world MSRC measurements, with implications for EMI diagnostics and reverberation modeling.

Abstract

In part I, spectral moments and kurtosis were established as parameters in analytic models of correlation and spectral density functions for dynamic reverberation fields. In this part II, several practical limitations affecting the accuracy of estimating these parameters from measured stir sweep data are investigated. For sampled fields, the contributions of finite differencing and aliasing are evaluated. Finite differencing results in a negative bias that depends, to leading order, quadratically on the product of the sampling time interval and the stir bandwidth. Numerical estimates of moments extracted directly from sampled stir sweeps show good agreement with values obtained by an autocovariance method. The effects of data decimation and noise-to-stir ratios of RMS amplitudes are determined and experimentally verified. In addition, the dependencies on the noise-to-stir-bandwidth ratio, EMI, and unstirred energy are characterized.

Figures (11)

• Figure 1: (a) Spectral moments $\lambda^{\prime(\prime)}_{m,fd,c/d}$ for $m=0,\ldots 5$, in units (rad/s)$^m$; (b) stir parameters $\kappa^{\prime(\prime)}_{fd,c/d}$, based on FD of ideal continuous (dashed; subscript $c$) or actual sampled (solid; subscript $d$) stir sweeps. Asterisks indicate nominal values of $\Delta\tau/\beta^\prime=0.1128$ and $\Delta\tau/\beta^{\prime\prime}=0.1105$ at $f=18$ GHz.
• Figure 2: Ratios of aliased to non-aliased spectral parameters: (a) relative $\lambda^{\prime(\prime)}_{m,fd,a}$ for $m=0,\ldots,5$; (b) relative $\kappa^{\prime(\prime)}_{fd,a}$, as a function of $\Delta\tau/\beta^{\prime(\prime)}$. Asterisks indicate nominal values of $\Delta\tau/\beta^\prime=0.1128$ and $\Delta\tau/\beta^{\prime\prime}=0.1105$ at $f=18$ GHz.
• Figure 3: (a) ${\rm log}_{10}({\cal B}^\prime_E/{\cal B}^\prime_s)$ and (b) ${\rm log}_{10}(|\kappa^\prime_E|)$ as functions of NSRs for RMS levels and bandwidths, ${\rm log}_{10}(\gamma_N)$ and ${\rm log}_{10}(\gamma_B)$; (c) $\kappa^\prime_E(\gamma_N)$ and (d) $\kappa^\prime_E(\gamma_B)$ for selected $\gamma_B$ and $\gamma_N$, respectively, indicating positive (blue) and negative (red) values.
• Figure 4: Measurement set-up in the chamber B3 at NIST.
• Figure 5: Manufacturer quoted (theoretical) and measured bandwidth--time products (in units rad), its lower limit (\ref{['eq:IFBW_uncertainty']}) for $[0/1]$-order model, and standardized random uncertainty for FFT-based SDF as a function of IFBW, $B$ (in units krad/s).