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

Luk R. Arnaut and John M. Ladbury

###### 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.

Index Terms–spectral moments, stir noise, stir spectral density.

##  I Introduction

In part I [1], analytical models of correlation functions (CFs) and spectral
density functions (SDFs) for continuously stirred fields and power inside a
mode-stirred reverberation chamber (MSRC) were developed. The normalized
second spectral moment (spectral variance) and the scaled spectral kurtosis
were found to be their key parameters that require estimation and evaluation.
In this part II, the influence of various properties of the physical stirred
field (noise level and bandwidth, understirred field components, external EMI)
and several features in the data acquisition (sample size, sampling rate, stir
speed, IF bandwidth (IFBW)) on the estimation of spectral moments and kurtosis
is analyzed. The primary aim is to quantify and compensate for bias resulting
from these effects in experimental results for CFs and SDFs presented in part
III [2]. The analysis is confined to linear effects of nonlocal temporal
second order (two-point stir correlation) and local spectral fourth order
(stir spectral kurtosis). Sampling effects on local temporal first-order
statistics and probability density functions (PDFs) have been widely studied
previously, e.g., [3], [4]. As shown in [2], discrete autocorrelation
functions (ACFs) and autospectral density functions (ASDFs) of MSRC data can
be applied in performing stir diagnostics.

Specifically, the effect of finite differencing (FD) on the bias and variance
of spectral moments is analyzed in detail. Their evaluation based on the sweep
data is compared with that from the autocovariance (ACV) method, showing
similar levels of performance while being conceptually simpler. Whereas the
methodology applies to general ACFs and SDFs, explicit expressions are derived
for the exponential ASDF in [1].

Theoretical results are evaluated and illustrated with measured data obtained
using a vector network analyzer (VNA). Notwithstanding the high sampling rate
of modern VNAs enabling near-continuous acquisition of stir process data
(i.e., stir traces as sample paths), the implication of discretization remains
an inherent issue for continuous random fields of unknown bandwidth.
Conversely, sampling allows for approaching and investigating fundamental
limits and trade-offs between accuracy, uncertainty and duration of
measurement, as will be shown.

The following conventions and notations are adopted. $m$ overdots indicate
$m$th-order continuous differentiation or discrete differencing. Single- and
double-primed quantities relate to the real (in-phase, I-) and imaginary
(quadrature, Q-) parts of an assumed wide-sense stationary (WSS) complex
random electric field $E(\tau)=E^{\prime}(\tau)+{\rm
j}E^{\prime\prime}(\tau)$, to its complex CF
$\rho_{E}(\tau)=\rho^{\prime}_{E}(\varpi)+{\rm
j}\rho^{\prime\prime}_{E}(\varpi)$ at stir lag $\tau$, or to its single-sided
SDF $g_{E}(\varpi)=g^{\prime}_{E}(\varpi)+{\rm j}g^{\prime\prime}_{E}(\varpi)$
at stir frequency $\varpi=2\pi/\tau$. Spectral moments are denoted as
$\lambda_{m}$ or $\lambda^{\prime(\prime)}_{m}$, the latter implying a
normalization by $\lambda_{0}=\sigma^{2}_{E}$ and association with
$g^{\prime(\prime)}_{E}(\varpi)$. For other notations, cf. [1].

##  II Spectral Moments of Sampled Fields

Spectral-statical properties of $E(\tau)$ can be established by exploiting the
duality between the PDF and the characteristic function of $E(\tau)$ as a pair
of Fourier transforms on one hand, versus its SDF and CF as another such pair
on the other hand. A characterization of CFs and SDFs can be based on spectral
moments. The spectral moments $\lambda_{m}$ of the normalized single-sided
analytic SDF $g_{E}(\varpi\geq 0)$ are related to the averaged product of
$n$th- and $p$th-order stir time derivatives of $E^{(*)}(\tau)$ as [5]–[7]

$\displaystyle\lambda_{m}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma^{2}_{E}\int^{+\infty}_{0}\varpi^{m}g_{E}(\varpi){\rm
d}\varpi=\sigma^{2}_{E}\int^{+\infty}_{-\infty}|\varpi|^{m}s_{E}(\varpi){\rm
d}\varpi$ (1) $\displaystyle={\rm j}^{-n}(-{\rm
j})^{-p}\overline{E^{(n)}(\tau){E^{*}}^{(p)}(\tau)},\hskip
5.69054pt~{}n+p=m=0,1,2,\ldots$ (2)

where $s_{E}(|\varpi|)=g_{E}(\varpi\geq 0)/2$ represents the normalized
double-sided SDF. In the current framework, the overline in (2) denotes sample
path averaging with respect to $\tau$, i.e., taken across a single sweep of
the primary mode stirrer, for an arbitrary state of the secondary tuner,
$\tau_{2,\ell}$. For arbitrary order $m$, the $\lambda_{m}$ are sample values
of random $\Lambda_{m}$ with PDF $f_{\Lambda_{m}}(\lambda_{m})$.
Correspondingly, $g_{E}(\varpi)$ is a sample SDF from an ensemble of SDFs
generated by secondary tuning. The focus is on effective ‘macroscopic’ SDFs of
the observable interior cavity field. In principle, a microscopic
characterization [8] based on moments of SDFs for individual cavity eigenmodes
and their stir evolution is also possible.

Note that (2) is ambiguous, in that different combinations of $n$ and $p$
values may yield the same $m$ but not necessarily the same value of
$\lambda_{m}$. This ambiguity increases with increasing $m$, as more
combinations of $n$ and $p$ become valid. This could be resolved by redefining
$\lambda_{m}$ as the arithmetic mean

$\displaystyle\frac{{\rm
j}^{m}}{m+1}\sum^{m}_{n=0}(-1)^{n}\overline{E^{(n)}(\tau){E^{*}}^{(m-n)}(\tau)}.$
(3)

On the other hand, FD of sampled data provides approximations to the sought
continuous derivatives in (2), a fortiori for large $m$. As clarified in sec.
V-D1, only combination(s) of $n$ and $p$ that minimize(s) $|n-p|$ will be
retained, i.e., $n=p=m/2$ for $m$ even and $n=p-1=\lfloor m/2\rfloor$ for $m$
odd. This choice yields close numerical agreement with ACV-based estimates of
$\lambda_{m}$. Thus, (2) will be evaluated as

$\displaystyle\lambda_{2i}$
$\displaystyle=\overline{\left[{E^{\prime}}^{(i)}(\tau)\right]^{2}}+\overline{\left[{E^{\prime\prime}}^{(i)}(\tau)\right]^{2}}\rightarrow
2\overline{\left[{E^{\prime(\prime)}}^{(i)}(\tau)\right]^{2}}$ (4)
$\displaystyle\lambda_{2i+1}$
$\displaystyle=\overline{{E^{\prime}}^{(i)}(\tau){E^{\prime\prime}}^{(i+1)}(\tau)}-\overline{{E^{\prime}}^{(i+1)}(\tau){E^{\prime\prime}}^{(i)}(\tau)}$
$\displaystyle\hskip 28.45274pt\rightarrow
2\overline{{E^{\prime}}^{(i)}(\tau){E^{\prime\prime}}^{(i+1)}(\tau)}$ (5)

for $m$ even or odd , respectively. In (4) and (5) the limit expressions on
the right hold for ideal circular $E(\tau)$, for which
$\overline{{E^{\prime}}^{(m-n)}(\tau){E^{\prime}}^{(n)}(\tau)}=\overline{{E^{\prime\prime}}^{(n)}(\tau){E^{\prime\prime}}^{(m-n)}(\tau)}$
and
$\overline{{E^{\prime}}^{(m-n)}(\tau){E^{\prime\prime}}^{(n)}(\tau)}=-\overline{{E^{\prime}}^{(n)}(\tau){E^{\prime\prime}}^{(m-n)}(\tau)}$.

For consistency of notation, we shall further replace the overbar for temporal
averaging across a stir sweep with $\langle\cdot\rangle_{N_{s}}$ (or simply
$\langle\cdot\rangle$, at an arbitrary tune state, if there is no confusion),
while reserving $\langle\cdot\rangle_{N_{t}}$ to represent ensemble averaging
over secondary tune states at an arbitrary stir state.

In the literature on spectral moments, e.g., [9]–[15], the main focus has been
on the effect of finite sample size on the bias and variance in the estimation
of $\lambda_{m}$, and on comparing temporal vs. spectral methods of
estimation, restricted almost exclusively to low orders ($m=0,1,2$) and
Gaussian SDFs. Results on higher-order moments and non-Gaussian SDFs are
relatively scant [14]. Here, we compare the bias caused by FD of continuous
vs. sampled stir data, for arbitrary $m$. General results are then applied to
the $[0/1]$-order exponential SDF [1, eq. (34)], for which

$\displaystyle\lambda^{\prime(\prime)}_{m}$
$\displaystyle={m!}/{(\beta^{\prime(\prime)})^{m}},\hskip
14.22636ptm=0,1,2\ldots$ (6)

##  III Bias Caused by Finite Differencing

###  III-A Estimation Based on Stir Sweep Data

####  III-A1 Continuous Stir Sweep

The approximation of an $m$th-order continuous-time derivative $E^{(m)}(\tau)$
through FD can be represented using a scaled $m$-fold convolution of $E(\tau)$
with the anti-symmetric impulse pair function [16]

$\displaystyle\Pi_{\Delta\tau}(\tau)\stackrel{{\scriptstyle\Delta}}{{=}}\delta(\tau+\Delta\tau/2)-\delta(\tau-\Delta\tau/2).$
(7)

In particular, the first-order FD of $E(\tau)$ is

$\displaystyle{{\cal D}}^{1}_{\Delta\tau}[E(\tau)]$
$\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}\frac{E(\tau+\Delta\tau/2)-E(\tau-\Delta\tau/2)}{\Delta\tau}$
(8) $\displaystyle=|\Delta\tau|^{-1}\Pi_{\Delta\tau}(\tau)*E(\tau)$ (9)

where $*$ denotes linear convolution. Since

$\displaystyle{\cal F}[|\Delta\tau|^{-1}\Pi_{\Delta\tau}(\tau)](\varpi)={\rm
j}(2/\Delta\tau)\sin(\varpi\Delta\tau/2)$ (10)

it follows that the first-order FD corresponds to the multiplication of
${{\cal E}}(\varpi)\stackrel{{\scriptstyle\Delta}}{{=}}{\cal
F}[E(\tau)](\varpi)$ by ${\rm j}\varpi\,{\rm sinc}(\varpi\Delta\tau/2)$. More
generally, for $m$th-order FD involving $m$-fold convolution

$\displaystyle{\cal F}[{{\cal D}}^{m}_{\Delta\tau}[E(\tau)]](\varpi)=[{\rm
j}\varpi\,{{\rm sinc}(\varpi\Delta\tau/2)}]^{m}\,{{\cal E}}(\varpi).$ (11)

Parenthetically, a formal correspondence exists between discrete FD (8)–(11)
and continuous local averaging, i.e.,

$\displaystyle{{\cal A}}^{1}_{\Delta\tau}[E(\tau)]$
$\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}|{\Delta\tau}|^{-1}\int^{\Delta\tau/2}_{-\Delta\tau/2}E(\tau+s){\rm
d}s$ (12) $\displaystyle{\cal F}[{{\cal
A}}^{m}_{\Delta\tau}[E(\tau)]](\varpi)$ $\displaystyle=[{\rm
sinc}(\varpi\Delta\tau/2)]^{m}{\cal E}(\varpi).$ (13)

In particular, for uniform first-order averaging ($m=1$), several of the
following methods and results for FD of $E(\tau)$ apply mutatis mutandis to
local averaging of $E(\tau)$ [18].

Returning to (11), since $g^{\prime}_{E}(\varpi)=\langle{\cal E}(\varpi){\cal
E}^{*}(\varpi)\rangle/\sigma^{2}_{E}$, the FD of $E(\tau)$ corresponds to a
multiplication of $g_{E}(\varpi)$ by ${\rm sinc}^{m}(\varpi\Delta\tau/2)$. Its
effect on the spectral moments is

$\displaystyle\lambda^{\prime(\prime)}_{m,fd,c}$
$\displaystyle=\int^{+\infty}_{0}\varpi^{m}{\rm
sinc}^{m}\left({\varpi\Delta\tau}/{2}\right)g^{\prime(\prime)}_{E}(\varpi){\rm
d}\varpi$ (14)

where the subscript $fd,c$ denotes FD performed on a continuous function
(analog stir sweep).

For a continuous and mean-square differentiable $E(\tau)$ [17], its ASDF can
be represented by the $[0/1]$ Padé model
$\tilde{g}^{\prime}_{E}(\varpi)=\beta^{\prime}\exp(-\beta^{\prime}\varpi)$
with $\beta^{\prime}=\sqrt{2/\lambda^{\prime}_{2}}$. An independent or
iterative estimate of $\beta^{\prime}$ is desirable. For even-order moments
($m=2i$), substituting this $\tilde{g}^{\prime}_{E}(\varpi)$ into (14) yields

$\displaystyle\lambda^{\prime}_{2i,fd,c}$
$\displaystyle=\overline{\left|\frac{\Delta^{i}E(\tau)}{\Delta\tau^{i}}\right|^{2}}=\frac{(2i)!}{(\beta^{\prime})^{2i}}\prod^{i}_{k=1}\left[1+\left(\frac{k\Delta\tau}{\beta^{\prime}}\right)^{2}\right]^{-1}.$
(15)

If $\Delta\tau/\beta^{\prime}\ll 1$ then, to leading order, (15) is
approximated as

$\displaystyle\lambda^{\prime}_{2i,fd,c}$
$\displaystyle\simeq\frac{(2i)!}{(\beta^{\prime})^{2i}}\left[1-\frac{i(i+1)(2i+1)}{6}\left(\frac{\Delta\tau}{\beta^{\prime}}\right)^{2}\right].$
(16)

In particular, the estimated RMS power $\lambda_{0}$ is not affected by this
FD on continous $E(\tau)$, as expected, while the relative bias of
$\lambda^{\prime}_{2i}$ for $i>0$ increases rapidly with $i$. Compensation for
FD bias yields debiased moments as

$\displaystyle\lambda^{\prime}_{2i}$
$\displaystyle=\lambda^{\prime}_{2i,fd,c}\prod^{i}_{k=1}\left[1+\left(\frac{k\Delta\tau}{\beta^{\prime}}\right)^{2}\right]$
(17)
$\displaystyle\simeq\lambda^{\prime}_{2i,fd,c}\left[1+\frac{i(i+1)(2i+1)}{6}\left(\frac{\Delta\tau}{\beta^{\prime}}\right)^{2}\right].$
(18)

The FD bias of
$\kappa^{\prime}\equiv\lambda^{\prime}_{4}/]6(\lambda^{\prime}_{2})^{2}]-1$
follows as

$\displaystyle\Delta\kappa^{\prime}_{fd,c}$
$\displaystyle=\frac{1+\left(\Delta\tau/\beta^{\prime}\right)^{2}}{1+4\left(\Delta\tau/\beta^{\prime}\right)^{2}}-1$
(19)
$\displaystyle\simeq\left\\{\begin{array}[]{ll}-3\left(\Delta\tau/\beta^{\prime}\right)^{2},&\Delta\tau/\beta^{\prime}\ll
1\\\ -3/4,&\Delta\tau/\beta^{\prime}\gg 1\end{array}\right.$ (22)

which must be similarly compensated for when estimating $\kappa^{\prime}$.

For odd-order moments ($m=2i+1$), the $[1/2]$-order model
$\tilde{g}^{\prime\prime}_{E}(\varpi)=\beta^{\prime\prime}\exp(-\beta^{\prime\prime}\varpi)$
with
$\beta^{\prime\prime}=\sqrt{6\lambda^{\prime\prime}_{1}/\lambda^{\prime\prime}_{3}}$
in (14) yields

$\displaystyle\lambda^{\prime\prime}_{2i+1,fd,c}=\frac{(2i+1)!}{(\beta^{\prime\prime})^{2i+1}}\prod^{i}_{k=0}\left[1+\left(\frac{\left(k+\frac{1}{2}\right)\Delta\tau}{\beta^{\prime\prime}}\right)^{2}\right]^{-1}$
(23)
$\displaystyle\simeq\frac{(2i+1)!}{(\beta^{\prime\prime})^{2i+1}}\left[1-\frac{4i^{3}+12i^{2}+11i+3}{12}\left(\frac{\Delta\tau}{\beta^{\prime\prime}}\right)^{2}\right]$
(24)

where the latter approximation again holds for
$\Delta\tau/\beta^{\prime\prime}\ll 1$. After compensation for FD bias, these
become

$\displaystyle\lambda^{\prime\prime}_{2i+1}=\lambda^{\prime\prime}_{2i+1,fd,c}\prod^{i}_{k=0}\left[1+\left(\frac{\left(k+\frac{1}{2}\right)\Delta\tau}{\beta^{\prime\prime}}\right)^{2}\right]$
(25)
$\displaystyle\simeq\lambda^{\prime\prime}_{2i+1,fd,c}\left[1+\frac{4i^{3}+12i^{2}+11i+3}{12}\left(\frac{\Delta\tau}{\beta^{\prime\prime}}\right)^{2}\right]$
(26)

resulting in the FD debiased squared CSDF bandwidth

$\displaystyle\frac{\lambda^{\prime\prime}_{3}}{6\lambda^{\prime\prime}_{1}}=\frac{\lambda^{\prime\prime}_{3,fd,c}}{6\lambda^{\prime\prime}_{1,fd,c}}\left[1+\frac{9}{4}\left(\frac{\Delta\tau}{\beta^{\prime\prime}}\right)^{2}\right].$
(27)

The FD bias of $\kappa^{\prime\prime}\equiv
3\lambda^{\prime\prime}_{1}\lambda^{\prime\prime}_{5}/[10(\lambda^{\prime\prime}_{3})^{2}]-1$
follows as

$\displaystyle\Delta\kappa^{\prime\prime}_{fd,c}$
$\displaystyle=\frac{1+9\left({\Delta\tau}/{\beta^{\prime\prime}}\right)^{2}/4}{1+25\left({\Delta\tau}/{\beta^{\prime\prime}}\right)^{2}/4}-1$
(28)
$\displaystyle\simeq\left\\{\begin{array}[]{ll}-4\left(\Delta\tau/\beta^{\prime\prime}\right)^{2},&\Delta\tau/\beta^{\prime\prime}\ll
1\\\ -16/25,&\Delta\tau/\beta^{\prime\prime}\gg 1.\end{array}\right.$ (31)

In summary, FD of a continuous stir trace results in a negative quadratic bias
proportional to $-(\Delta\tau/\beta^{\prime(\prime)})^{2}$ when
$\Delta\tau/\beta^{\prime(\prime)}\ll 1$, to leading order. This bias
underestimates all spectral moments and $\kappa^{\prime(\prime)}$. At higher
CW frequencies (implying larger stir bandwidths $1/\beta^{\prime(\prime)}$) or
for slower sampling, this bias increases.

####  III-A2 Sub-Nyquist Sampling

When sampling a continuous stir sweep at $N_{s}$ uniformly spaced points
$\tau_{n}\equiv n\Delta\tau$, the spectral expansion of $E(\tau)$ [1, eq. (3)]
in based on the stir discrete-time Fourier transform (DTFT)

$\displaystyle{E}(\tau_{n})$
$\displaystyle=\frac{\Delta\tau}{2\pi}\int^{+\pi/\Delta\tau}_{-\pi/\Delta\tau}{{\cal
E}}(\varpi)\exp({\rm j}\varpi n\Delta\tau){\rm d}\varpi.$ (32)

The corresponding stir IDFT at $\tau_{n}$ is a discrete sum over $N_{s}$ stir
frequencies $\varpi_{k}=k\Delta\varpi$ ($k=0,1,\ldots,N_{s}-1)$ with stir-
spectral resolution $\Delta\varpi=(2\pi/\Delta\tau)/N_{s}$, i.e.,

$\displaystyle{E}(n)$
$\displaystyle=\frac{1}{N_{s}}\sum^{N_{s}-1}_{k=0}\underline{{\cal
E}}(\varpi_{k})\exp({\rm j}2\pi kn/N_{s}).$ (33)

Therefore, for discretized stir sweeps, [1, eq. (33)] becomes

$\displaystyle\\{E(\tau_{n})\\}\stackrel{{\scriptstyle{\rm
DFT}}}{{\longrightarrow}}\\{{\cal
E}(\varpi_{k})\\}\stackrel{{\scriptstyle[1,(32)]}}{{\longrightarrow}}\\{g_{E}(\varpi_{k})\\}\stackrel{{\scriptstyle{\rm
IDFT}}}{{\longrightarrow}}\\{\rho_{E}(\tau_{n})\\}.$ (34)

The discrete SDF $g_{E}(\varpi_{{\color[rgb]{0,0,1}k}})$ is obtained from
either the DFT of the sampled field (periodogram based SDF) or its correlation
matrix, with the integration limits $\pm\infty$ replaced by the Nyquist
limits, $\pm\varpi_{\rm max}=\pm\pi/\Delta\tau$.

The estimated spectral moments are now

$\displaystyle\lambda^{\prime(\prime)}_{m,fd,d}$
$\displaystyle=\alpha^{\prime(\prime)}_{m,fd,d}\lambda^{\prime(\prime)}_{m,fd,c}$
(35)

where the subscript $fd,d$ refers to FD performed on discrete stir data, with
[19, eq. (3.895.3)]

$\displaystyle\alpha^{\prime}_{2i,fd,d}=1-\exp\left(-\frac{\pi
q^{\prime}}{2}\right)\left[1+\frac{{q^{\prime}}^{2}}{2!}+\frac{{q^{\prime}}^{2}({q^{\prime}}^{2}+2^{2})}{4!}\right.$
$\displaystyle\hskip
28.45274pt\left.+\ldots+\frac{{q^{\prime}}^{2}({q^{\prime}}^{2}+2^{2})\ldots({q^{\prime}}^{2}+(2i-2)^{2})}{(2i)!}\right]$
(36)
$\displaystyle\alpha^{\prime\prime}_{2i+1,fd,d}=1-q^{\prime\prime}\exp\left(-\frac{\pi
q^{\prime\prime}}{2}\right)\left[1+\frac{{q^{\prime\prime}}^{2}+1^{2}}{3!}+\ldots\right.$
$\displaystyle\left.\hskip
28.45274pt+\frac{({q^{\prime\prime}}^{2}+1^{2})({q^{\prime\prime}}^{2}+3^{2})\ldots({q^{\prime\prime}}^{2}+(2i-1)^{2})}{(2i+1)!}\right]$
(37)

in which
$q^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}2\beta^{\prime(\prime)}/\Delta\tau\gg
1$ is double the normalized stir correlation time. Since
$\alpha^{\prime(\prime)}_{m,fd,d}<1$, it follows that imposing the Nyquist
criterion further reduces $\lambda^{\prime(\prime)}_{m}$. In particular, from
(36),

$\displaystyle\alpha^{\prime}_{0,fd,d}$
$\displaystyle=1-\exp(-\pi\beta^{\prime}/\Delta\tau)<1$ (38)

i.e., the normalized RMS power $\lambda^{\prime}_{0,fd,d}$ is negatively
biased, unlike $\lambda^{\prime}_{0,fd,c}\equiv 1$, prompting a
renormalization of $g_{E}(\varpi)$.

Fig. 1 compares $\lambda^{\prime(\prime)}_{m,fd,c/d}$ and
$\kappa^{\prime(\prime)}_{fd,c/d}$ for $0\leq m\leq 5$ as a function of
$2/q^{\prime(\prime)}\equiv\Delta\tau/\beta^{\prime(\prime)}$ for FD on analog
(i.e., continuous) vs. sampled (i.e., discretized) stir traces. The moment
characteristics
$\lambda^{\prime(\prime)}_{m,fd,c/d}(\Delta\tau/\beta^{\prime(\prime)})$ are
insensitive when $\Delta\tau/\beta^{\prime(\prime)}\ll 1$, in particular for
$\Delta\tau/\beta^{\prime(\prime)}\leq 0.1128$ as in our experiments for
$1\leq f\leq 18$ GHz. By contrast, $\kappa^{\prime(\prime)}_{fd,c/d}$ are
negative, significantly biased from zero, and sensitive to small changes of
$\Delta\tau/\beta^{\prime(\prime)}$ when $\Delta\tau/\beta^{\prime(\prime)}\ll
1$. Recall that decreasing $f$ increases $\beta^{\prime(\prime)}$, which
lowers the FD bias of $\lambda^{\prime(\prime)}_{m}$ and
$\kappa^{\prime(\prime)}$ for arbitrary $\Delta\tau$. Hence, the consideration
of FD bias at high $f$ offers a conservative estimate.

---
(a)
(b)
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.

####  III-A3 Super-Nyquist Sampling

The prevention of aliasing in stir data via low-pass pre-filtering in the
spectral stir domain requires prior knowledge of the maximum rate of stir
fluctuation [20] to decide on the appropriate sampling rate. Aliasing has
previously been analyzed in the context of late time decay [21]. Although stir
aliasing cannot be eradicated after measurement, its influence on the spectral
moments and SDF can be gleaned from a adaptation of the previous FD analysis.
To this end, extending the integration limits in (32) to $\pm\varpi_{s}=\pm
2\pi/{\Delta\tau}$ yields the SDF and moments (with subscript $s$) including
the contributions by the aliased power between $\pm\varpi_{\rm max}$ and
$\pm\varpi_{\rm s}$.111Note that extending the spectral range also captures
out-of-band stir noise. With

$\displaystyle\lambda^{\prime(\prime)}_{m,fd,s}$
$\displaystyle=\alpha^{\prime(\prime)}_{m,fd,s}\lambda^{\prime(\prime)}_{m,fd,c}$
(39)

where [19, eq. (3.895.6)]

$\displaystyle\alpha^{\prime}_{2i,fd,s}\stackrel{{\scriptstyle\Delta}}{{=}}1-\exp\left(-{\pi
q^{\prime}}\right),\hskip
5.69046pt\alpha^{\prime\prime}_{2i+1,fd,s}\stackrel{{\scriptstyle\Delta}}{{=}}1+\exp\left(-{\pi
q^{\prime\prime}}\right)$ (40)

this yields the residual aliased contribution (subscript $a$) as a fraction of
the sub-Nyquist contribution to $\lambda^{\prime(\prime)}_{m,fd,c}$, i.e.,

$\displaystyle\lambda^{\prime(\prime)}_{m,fd,a}$
$\displaystyle=\alpha^{\prime(\prime)}_{m,fd,a}\lambda^{\prime(\prime)}_{m,fd,c}$
(41)

where

$\displaystyle\alpha^{\prime}_{2i,fd,a}\stackrel{{\scriptstyle\Delta}}{{=}}\exp\left(-\frac{\pi
q^{\prime}}{2}\right)\left[1+\frac{{q^{\prime}}^{2}}{2!}+\frac{{q^{\prime}}^{2}({q^{\prime}}^{2}+2^{2})}{4!}+\ldots~{}\right.$
$\displaystyle\hskip
8.5359pt\left.+\frac{{q^{\prime}}^{2}({q^{\prime}}^{2}+2^{2})\ldots({q^{\prime}}^{2}+(2i-2)^{2})}{(2i)!}\right]-\exp\left(-{\pi
q^{\prime}}\right)$ (42)
$\displaystyle\alpha^{\prime\prime}_{2i+1,fd,a}\stackrel{{\scriptstyle\Delta}}{{=}}q^{\prime\prime}\exp\left(-\frac{\pi
q^{\prime\prime}}{2}\right)\left[1+\frac{{q^{\prime\prime}}^{2}+1^{2}}{3!}+\ldots+\right.$
$\displaystyle\left.\frac{({q^{\prime\prime}}^{2}+1^{2})({q^{\prime\prime}}^{2}+3^{2})\ldots({q^{\prime\prime}}^{2}+(2i-1)^{2})}{(2i+1)!}\right]+\exp\left(-{\pi
q^{\prime\prime}}\right).$ (43)

For sampled data,
$\lambda^{\prime(\prime)}_{m,fd,a}=(\alpha^{\prime(\prime)}_{m,fd,a}/\alpha^{\prime(\prime)}_{m,fd,d})\lambda^{\prime(\prime)}_{m,fd,d}$.

Fig. 2 shows
$\lambda^{\prime(\prime)}_{m,fd,a}/\lambda^{\prime(\prime)}_{m,fd,d}$ and
$\kappa^{\prime(\prime)}_{fd,a}/\kappa^{\prime(\prime)}_{fd,d}$ for $0\leq
m\leq 5$ as a function of $\Delta\tau/\beta^{\prime(\prime)}$ at $f=18$ GHz.
The results indicate a finite transition region, approximately
$0.5<\Delta\tau/\beta^{\prime(\prime)}<500$. Higher-order moments exhibit an
earlier onset of increasing
$\lambda^{\prime(\prime)}_{m,fd,a}/\lambda^{\prime(\prime)}_{m,fd,d}$. The
deviation of $\kappa^{\prime(\prime)}_{fd,a}/\kappa^{\prime(\prime)}_{fd,d}$
from unity is less than $4\%$ across the transition region, and less than
$0.1\%$ in our measurements. Since stir spectra are in general flatter for
higher $f$ (larger stir bandwidth), the fraction of aliased power for lower
$f$ is expected to be smaller, although the stir LF-to-HF transition of stir
power density is then also less abrupt.

In summary, the estimated effect of stir aliasing on the FD bias of
$\kappa^{\prime(\prime)}$ is negligible for $f\leq 18$ GHz, being just a few
percent or less at the highest frequency, for arbitrary
$\Delta\tau/\beta^{\prime(\prime)}$.

---
(a) (b)
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.

The foregoing analysis was performed in the stir-spectral DTFT domain for
continuous $\varpi$. It is easily extended to DFTs for discrete $\varpi_{k}$
by replacing integrations with sums over $\varpi_{k}=k\Delta\varpi=2\pi
k/(N_{s}\Delta\tau)$, where $k=0,1,\ldots,N_{s}-1$.

###  III-B Estimation Based on Complex Autocorrelation

In sec. III-A1, independent estimates of $\beta^{\prime(\prime)}$ were sought
for estimating FD bias. These can be provided by the ACV (or ACF) method [10].
This alternative technique for estimating any $\lambda_{m}$ still operates
within the temporal stir domain, but uses the $m$th-order FD of the complex
ACF instead, i.e.,

$\displaystyle\rho_{E}(\tau)\equiv a_{\rho_{E}}(\tau)\exp[{\rm
j}\phi_{\rho_{E}}(\tau)]$ (44)

for complex $E(\tau)$, where
$a_{\rho_{E}}(\tau)=[\rho^{\prime^{2}}_{E}(\tau)+\rho^{\prime\prime^{2}}_{E}(\tau)]^{1/2}$
and
$\phi_{\rho_{E}}(\tau)=\tan^{-1}(\rho^{\prime\prime}_{E}(\tau)/\rho^{\prime}_{E}(\tau))$
are both to be evaluated at zero lag, or alternatively using the ACV function
$\gamma_{E}(\tau)$. Thus,

$\displaystyle{\lambda_{m}}=(-{\rm
j})^{m}{\rho^{(m)}_{E}}(0)\sigma^{2}_{E}=(-{\rm
j})^{m}{{\gamma^{(m)}_{E}}(0)}.$ (45)

Extending the results for $m\leq 2$ obtainable from [10] to the first six
moments that govern $\kappa^{\prime(\prime)}$ yields

$\displaystyle\lambda^{\prime}_{0}$ $\displaystyle=a_{\rho_{E}}(0)=1$ (46)
$\displaystyle\lambda^{\prime\prime}_{1}$
$\displaystyle=\dot{\phi}_{\rho_{E}}(0)$ (47)
$\displaystyle\lambda^{\prime}_{2}$
$\displaystyle=-\ddot{a}_{\rho_{E}}(0)+(\lambda^{\prime\prime}_{1})^{2}$ (48)
$\displaystyle\lambda^{\prime\prime}_{3}$
$\displaystyle=-\dddot{\phi}_{\rho_{E}}(0)-2(\lambda^{\prime\prime}_{1})^{3}+3\lambda^{\prime\prime}_{1}\lambda^{\prime}_{2}$
(49) $\displaystyle\lambda^{\prime}_{4}$
$\displaystyle={a}^{(4)}_{\rho_{E}}(0)+3(\lambda^{\prime\prime}_{1})^{4}-6(\lambda^{\prime\prime}_{1})^{2}\lambda^{\prime}_{2}+4\lambda^{\prime\prime}_{1}\lambda^{\prime\prime}_{3}$
(50) $\displaystyle\lambda^{\prime\prime}_{5}$
$\displaystyle={\phi}^{(5)}_{\rho_{E}}(0)-24(\lambda^{\prime\prime}_{1})^{5}+60(\lambda^{\prime\prime}_{1})^{3}\lambda^{\prime}_{2}-20(\lambda^{\prime\prime}_{1})^{2}\lambda^{\prime\prime}_{3}$
$\displaystyle~{}~{}~{}~{}+5\lambda^{\prime\prime}_{1}\lambda^{\prime}_{4}-30\lambda^{\prime\prime}_{1}(\lambda^{\prime}_{2})^{2}+10\lambda^{\prime}_{2}\lambda^{\prime\prime}_{3}.$
(51)

Since the complex ACF exhibits Hermitean symmetry, i.e.,

$\displaystyle
a_{\rho_{E}}(\tau)=a_{\rho_{E}}(-\tau),~{}~{}~{}\phi_{\rho_{E}}(\tau)=-\phi_{\rho_{E}}(-\tau)$
(52)

this can be exploited to halve the number of lags $\tau_{n}$ required for
estimating $\lambda^{\prime(\prime)}_{m}$. Specifically, for FD of sampled
data

$\displaystyle\dot{\phi}_{\rho_{E}}(0)$
$\displaystyle\simeq\frac{2\left[{\phi}_{\rho_{E}}(\Delta\tau)-{\phi}_{\rho_{E}}(0)\right]}{2\Delta\tau}\equiv\frac{{\phi}_{\rho_{E}}(\Delta\tau)}{\Delta\tau}$
(53) $\displaystyle\ddot{a}_{\rho_{E}}(0)$
$\displaystyle\simeq\frac{2\left[{a}_{\rho_{E}}(\Delta\tau)-{a}_{\rho_{E}}(0)\right]}{(\Delta\tau)^{2}}\equiv\frac{2\left[{a}_{\rho_{E}}(\Delta\tau)-1\right]}{(\Delta\tau)^{2}}$
(54) $\displaystyle\dddot{\phi}_{\rho_{E}}(0)$
$\displaystyle\simeq\frac{2\left[{\phi}_{\rho_{E}}(3\Delta\tau)-3{\phi}_{\rho_{E}}(\Delta\tau)\right]}{(2\Delta\tau)^{3}}$
(55) $\displaystyle{a}^{(4)}_{\rho_{E}}(0)$
$\displaystyle\simeq\frac{2\left[{a}_{\rho_{E}}(2\Delta\tau)-4{a}_{\rho_{E}}(\Delta\tau)+3{a}_{\rho_{E}}(0)\right]}{(\Delta\tau)^{4}}$
(56) $\displaystyle{\phi}^{(5)}_{\rho_{E}}(0)$
$\displaystyle\simeq\frac{2\left[{\phi}_{\rho_{E}}(5\Delta\tau)-5{\phi}_{\rho_{E}}(3\Delta\tau)+10{\phi}_{\rho_{E}}(\Delta\tau)\right]}{(2\Delta\tau)^{5}}$
(57)

etc. Because of the high oversampling rate offered by a VNA, multi-point ACF
estimation schemes [13] for complementing (53)–(57) offer no further
advantage, even for large stir bandwidths. Since the order of differentiation
for $\rho_{E}(0)$ in (45) equals the moment order $m$, an FD bias compensation
similar to that for stir sweeps in sec. III-A can be applied.

###  III-C Estimation Based on Spectral Density Function

Naturally, spectral moments can also be estimated in the spectral stir domain,
from their definition (1) [9]. Estimating ${\cal E}(\varpi){\cal
E}^{*}(\varpi)$ based on a DFT of $E(\tau)$, applying the forward
transformation in (34), with $N_{s}\Delta\varpi/2=\pi/\Delta\tau$, and
following (14), this yields

$\displaystyle\lambda_{m}\simeq\sigma^{2}_{E}\sum^{{N_{s}}/{2}-1}_{k=0}[{\varpi_{k}}{\rm
sinc}(\pi k/N_{s})]^{m}\hat{g}_{E}(\varpi_{k})\Delta\varpi.$ (58)

However, it is well known that this spectral method is in general ill-
conditioned. An FFT performed on a finite record of sampled stir data or
correlation series causes its sample SDF to cover only a finite stir band
$N_{s}\Delta\varpi$, resulting in a biased estimate of the sought continuous
unrestricted SDF. Sample spectrum estimates are in general neither unbiased
(except for ideal white noise) nor consistent. On the other hand, by the
Nyquist theorem, a periodogram-based SDF estimate is aliased when it is not
bandlimited. For the moments, a bias in the estimated SDF causes in turn a
bias of $\lambda_{m}$ when derived from (58), a fortiori for large spectral
bandwidths approaching $\pi/\Delta\tau$ [12], [15]. Especially higher-order
$\lambda_{m}$ increasingly depend on the SDF tail and are, hence, increasingly
affected by aliasing or truncation. Furthermore, the standard uncertainty of
$\Lambda_{m}$ rapidly increases with $m$ and is larger than in temporal
estimation [12]. Finally, in spectral estimation, $\tilde{\lambda}_{m}$ is
merely asymptotically unbiased, i.e., for $T\rightarrow+\infty$ in a single
stir sweep, which is not applicable to rotational periodic stirring.

In general, widening the IFBW $B$ and stir bandwidth
$\sqrt{\lambda^{\prime}_{2}/2}$ (both in units rad/s) results increasingly in
spectral nonuniformity for the bias of the ASDF, leading to the peaks and
troughs in the ASDF becoming under- and overestimated, respectively. The bias
error of the ASDF, $b_{g_{E}}$, is strongly nonuniform when
$B\ll\sqrt{\lambda^{\prime}_{2}/2}$: for the $[0/1]$ model [22]

$\displaystyle
b_{g_{E}}(\varpi)\simeq\frac{B^{2}\ddot{g}_{E}(\varpi)}{96\pi^{2}}\propto\frac{B^{2}}{(\lambda^{\prime}_{2})^{3/2}}\exp\left(-\sqrt{{2}/{\lambda^{\prime}_{2}}}\varpi\right)$
(59)

i.e., decreasing exponentially. This strong nonuniformity also leads to
unequal bias of estimated $\lambda_{m}$ for different $m$. Note, however, that
the relative bias $\varepsilon_{b}={b_{g_{E}}(\varpi)}/{g_{E}(\varpi)}\propto
B^{2}/\lambda^{\prime}_{2}$ is independent of $\varpi$ in this model.

##  IV Bias Caused by Noise, Interference, or Understirring

###  IV-A Stir Noise

The FD bias analyzed in sec. III is a measurement artefact and by-product of
sampling by the VNA. Physical imperfections to ideal randomization of the
continuous field itself inside the MSRC, prior to sampling, are now
investigated for their contribution to spectral moments. Unlike for FD bias,
physical imperfections are more difficult to compensate in spectral moments
and kurtosis, because they require more detailed information about field
contributions, as shown below. For brevity, only the real ASDF
$g^{\prime}_{E}(\varpi)$ is analyzed here.

####  IV-A1 Continuous Ideal Stirred Field and Baseband Stir Noise

Consider a random noise field, $N(\tau)$, independent of but additive to the
stirred field $E_{s}(\tau)$ [1, eq. (81)], i.e.,

$\displaystyle E(\tau)=E_{s}(\tau)+N(\tau).$ (60)

Such $N(\tau)$ may have an electrical, electromechanical, thermoelectric, or
other origin. It is modelled here as bandlimited baseband white noise with
$g^{\prime}_{N}(\varpi)=1/{\cal B}^{\prime}_{N}$ for $0\leq\varpi\leq{\cal
B}^{\prime}_{N}$. On account of the additivity of ACV functions and Bochner’s
theorem [17], the non-normalized ASDF of $E(\tau)$ is

$\displaystyle\sigma^{2}_{E}g^{\prime}_{E}(\varpi)=\sigma^{2}_{E_{s}}g^{\prime}_{E_{s}}(\varpi)+\sigma^{2}_{N}g^{\prime}_{N}(\varpi)$
(61)

and $\sigma^{2}_{E}=\sigma^{2}_{E_{s}}+\sigma^{2}_{N}$. Denote ${\cal
B}^{\prime}_{s}$ as the stir bandwidth of $E_{s}(\tau)$, where ${\cal
B}^{\prime}_{s}\equiv
1/\beta^{\prime}_{s}=(\lambda^{\prime}_{2,E_{s}}/2)^{1/2}$ with
$g^{\prime}_{E_{s}}(\varpi)=\exp(-\varpi/{\cal B}^{\prime}_{s})/{\cal
B}^{\prime}_{s}$ for ideal stirred $E_{s}(\tau)$ and $0<\varpi<+\infty$. With
(6), the spectral moments of $E(\tau)$ are then

$\displaystyle\lambda^{\prime}_{m,E}$ $\displaystyle={m!}{({\cal
B}^{\prime}_{s})^{m}}\frac{\sigma^{2}_{E_{s}}}{\sigma^{2}_{E}}+\frac{({\cal
B}^{\prime}_{N})^{m}}{m+1}\frac{\sigma^{2}_{N}}{\sigma^{2}_{E}}$ (62)
$\displaystyle={m!}({\cal
B}^{\prime}_{s})^{m}\frac{1+\gamma^{2}_{N}\gamma^{m}_{B}/(m+1)!}{1+\gamma^{2}_{N}}$
(63)

where

$\displaystyle\gamma_{B}\stackrel{{\scriptstyle\Delta}}{{=}}{\cal
B}^{\prime}_{N}/{\cal B}^{\prime}_{s},\hskip
8.5359pt\gamma_{N}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma_{N}/\sigma_{E_{s}}$
(64)

denote the noise-to-stir ratios (NSRs) of the bandwidths and RMS amplitudes,
respectively, between $N(\tau)$ and $E_{s}(\tau)$. The overall bandwidth
${\cal B}^{\prime}_{E}=({\lambda^{\prime}_{2,E}}/{2})^{1/2}$ and kurtosis
$\kappa^{\prime}_{E}={\lambda^{\prime}_{4,E}}/[{6(\lambda^{\prime}_{2,E})^{2}}]-1$
for $E(\tau)$ then follow as

$\displaystyle{\cal B}^{\prime}_{E}$ $\displaystyle={\cal
B}^{\prime}_{s}\sqrt{\frac{1+\gamma^{2}_{N}\gamma^{2}_{B}/6}{1+\gamma^{2}_{N}}}$
(65) $\displaystyle\kappa^{\prime}_{E}$
$\displaystyle=\frac{\gamma^{2}_{N}}{\left(1+{\gamma^{2}_{N}\gamma^{2}_{B}}/{6}\right)^{2}}\left(1-\frac{\gamma^{2}_{B}}{3}+\frac{\gamma^{4}_{B}}{120}-\frac{7\gamma^{2}_{N}\gamma^{4}_{B}}{360}\right).$
(66)

Fig. 3(b) indicates that $\kappa^{\prime}_{E}(\gamma_{N},\gamma_{B})$ exhibits
loci of matching pairs $(\gamma_{N},\gamma_{B})$ where
$\kappa^{\prime}_{E}=0$, which are related via

$\displaystyle 1-40\gamma^{-2}_{B}+120\gamma^{-4}_{B}=7\gamma^{2}_{N}/3.$ (67)

If the RMS noise level overwhelms the RMS magnitude of the stirred field
($\gamma_{N}\gg 1$), then

$\displaystyle\kappa^{\prime}_{E}\simeq-7/10+[3/10-(18/5)\gamma^{-2}_{B}+36\gamma^{-4}_{B}]\gamma^{-2}_{N}.$
(68)

Conversely, for vanishing RMS noise levels ($\gamma_{N}\rightarrow 0$) and
bandwidths ($\gamma_{B}\rightarrow 0$), $\kappa^{\prime}_{E}$ represents
asymptotically the NSR of power levels, viz.,

$\displaystyle\kappa^{\prime}_{E}\simeq\left(1-\gamma^{2}_{B}/3+\gamma^{4}_{B}/120\right)\gamma^{2}_{N}\rightarrow\gamma^{2}_{N}.$
(69)

For typical VNA measurements inside efficiently stirred MSRCs (${\cal
B}^{\prime}_{N}\gg{\cal B}^{\prime}_{s}$, $\sigma_{N}\ll\sigma_{E_{s}}$), we
have $\gamma_{N}\ll 1$ and222While the bandwidth is manually selected on the
analyzer, the bandwidth of a mechanical stir process is typically much lower
than that of $N_{e}(\tau)$. $\gamma_{B}\gg 1$. In this regime, Fig. 3(c)
indicates that $\kappa^{\prime}_{E}(\gamma_{N})$ is predominantly positive but
rapidly decreasing with increasing $\gamma_{N}$. Conversely,
$\kappa^{\prime}_{E}(\gamma_{B})$ also increases by several orders of
magnitude to high positive values, possibly with one or two zero crossings, as
is apparent from Fig. 3(d). Thus, typically, $\kappa^{\prime}_{E}$ is positive
but highly sensitive to variations of $\gamma_{N}$ and $\gamma_{B}$.

By extension, for alias-free sampled $E_{s}(\tau_{n})$, the first term in (62)
generalizes to

$\displaystyle\left[{m!}{({\cal
B}^{\prime}_{s})^{m}}-\sum^{m}_{i=0}\frac{m!~{}\varpi^{m-i}_{\rm max}({\cal
B}^{\prime}_{s})^{i}\exp(-\varpi_{\rm max}/{\cal
B}^{\prime}_{s})}{(m-i)!}\right]\frac{\sigma^{2}_{E_{s}}}{\sigma^{2}_{E}}.$
(70)

If the stir noise is not baseband (i.e., not coupled to stir DC; cf. sec.
IV-B) but restricted to some finite band $[{\cal B}^{\prime}_{N,\ell},{\cal
B}^{\prime}_{N,u}]$, then $({\cal B}^{\prime}_{N})^{m}$ in (62) is to be
replaced with $[({\cal B}^{\prime}_{N,u})^{m+1}-({\cal
B}^{\prime}_{N,\ell})^{m+1}]/({\cal B}^{\prime}_{N,u}-{\cal
B}^{\prime}_{N,\ell})$, with corresponding changes to (64)–(66).

---
(a)
(b)
(c) (d)
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.

####  IV-A2 Stir Noise vs. Electrical Noise

Stir noise $N_{s}(\tau)$ represents contributing random fluctuations that vary
at a rate that is much faster than that for the stirred field process
$E_{s}(\tau)$ itself. Such noise may have different physical origins, also
including quantization noise. Typically, $N_{s}(\tau)$ varies more slowly than
electrical noise $N_{e}(t)$ that is already present in the absence of
stirring. Observations of typical VNA stir sweeps at high SNR level ratios
indicate that $N_{s}(t)$ resembles shot noise, i.e., appearing as Poissonian,
approximately white and additive, $N(t)=N_{s}(t)+N_{e}(t)$.

Another case arises when signal and noise intensities are of comparable RMS
magnitude, i.e., $\overline{E^{2}_{s}(\tau)}\sim\overline{N^{2}(\tau)}$, a
fortiori when $\overline{E^{2}_{s}}$ is small and $N(t)$ is the thermal noise
field [23]. Then, stirring of this thermal noise must be taken into account
(stirred $N_{e}(t)$). With all interior surfaces of the MSRC then radiating at
nonzero temperature, $g_{N}(\varpi)$ then exhibits a significant unstirred
component.

Pure nonelectrical contributions may be separated from electrical noise using
static or (ideal) mode tuning, provided that the tuner is sufficiently rigid
in order to produce either a mechanically truly static state, or to move at a
rate that is much slower than the rates of the electrical noise and cavity
field relaxation (decay) [20].

###  IV-B Harmonic EMI and Understirred Fields

To evaluate the contribution of permanent periodic interfering fields on the
spectral moments, consider $E(\tau)$ to be the superposition of a perfectly
stirred $E_{s}(\tau)$ and a discrete set of deterministic harmonic EMI
contributions $E_{h}(\tau)$

$\displaystyle E(\tau)=E_{s}(\tau)+\sum_{h}E_{h}(\tau).$ (71)

Whilst $g_{E_{s}}(\varpi)=\exp(-\varpi/{\cal B}^{\prime}_{s})/{\cal
B}^{\prime}_{s}$ remains continuous, each spectral EMI component ${\cal
E}_{h}(\varpi)={\cal E}_{h,0}\exp({\rm j}\phi_{h})\delta(\varpi-\varpi_{h})$
originating from the $\omega$-domain adds a discrete contribution
$g_{E_{h}}(\varpi)=\delta(\varpi-\varpi_{h})$ to the SDF of $E_{s}(\tau)$.
Proceeding as in sec. IV-A, with
$\sigma^{2}_{E}=\sigma^{2}_{E_{s}}+\sum_{h}\sigma^{2}_{E_{h}}$, the ASDF of
$E(\tau)$ and its moments follow from (60), (61) and (71) as

$\displaystyle\sigma^{2}_{E}g^{\prime}_{E}(\varpi)$
$\displaystyle=\sigma^{2}_{E_{s}}g^{\prime}_{E_{s}}(\varpi)+\sum_{h}|{\cal
E}_{h,0}|^{2}g^{\prime}_{E_{h}}(\varpi)$ (72)
$\displaystyle\lambda^{\prime}_{m,E}$ $\displaystyle={m!}({\cal
B}^{\prime}_{s})^{m}\frac{\sigma^{2}_{E_{s}}}{\sigma^{2}_{E}}+\sum_{h}\varpi^{m}_{h}\frac{|{\cal
E}_{h,0}|^{2}}{\sigma^{2}_{E}}.$ (73)

In particular, if EMI is confined to a single field harmonic then

$\displaystyle\lambda^{\prime}_{m,E}$ $\displaystyle={m!}({\cal
B}^{\prime}_{s})^{m}\frac{1+\gamma^{2}_{E_{h}}\gamma^{m}_{\varpi_{h}}/m!}{1+\gamma^{2}_{E_{h}}}$
(74) $\displaystyle{\cal B}^{\prime}_{E}$ $\displaystyle={\cal
B}^{\prime}_{s}\sqrt{\frac{1+\gamma^{2}_{E_{h}}\gamma^{2}_{\varpi_{h}}/2}{1+\gamma^{2}_{E_{h}}}}$
(75) $\displaystyle\kappa^{\prime}_{E}$
$\displaystyle=\frac{\gamma^{2}_{E_{h}}}{\left(1+\gamma^{2}_{E_{h}}\gamma^{2}_{\varpi_{h}}/{2}\right)^{2}}\left(1-\gamma^{2}_{\varpi_{h}}+\frac{\gamma^{4}_{\varpi_{h}}}{24}-\frac{5\gamma^{2}_{E_{h}}\gamma^{4}_{\varpi_{h}}}{24}\right)$
(76)

in which the interference-to-stir ratios (ISRs) for frequencies and RMS
amplitudes are defined by

$\displaystyle\gamma_{\varpi_{h}}\stackrel{{\scriptstyle\Delta}}{{=}}{\varpi_{h}}/{\cal
B}^{\prime}_{s},~{}~{}~{}\gamma_{E_{h}}\stackrel{{\scriptstyle\Delta}}{{=}}|{\cal
E}_{h,0}|/\sigma_{E_{s}}.$ (77)

If the EMI is random or intermittent, then the amplitude ISR can be redefined
as
$\gamma_{E_{h}}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma_{E_{h}}/\sigma_{E_{s}}$
with $\sigma_{E_{h}}=\langle|{\cal E}_{h}|^{2}\rangle^{1/2}$.

Since (76) is formally similar to (66), the plot of
$\kappa^{\prime}_{E}(\gamma_{{E_{h}}},\gamma_{\varpi_{h}})$ [24] is similar to
$\kappa^{\prime}_{E}(\gamma_{N},\gamma_{B})$ in Fig. 3(b), now with its zero
crossings occurring for

$\displaystyle
1-24\gamma^{-2}_{\varpi_{h}}+24\gamma^{-4}_{\varpi_{h}}={5}\gamma^{2}_{E_{h}}.$
(78)

Some special cases of (76) can now be considered. Firstly, an unstirred field
of stir-spectral magnitude ${\cal E}_{u,0}$ represents the stir DC limit of a
stir ‘harmonic’, i.e.,
$\varpi_{h}=0\stackrel{{\scriptstyle\Delta}}{{=}}\varpi_{u}$. Hence ${\cal
E}_{u}(\varpi)={\cal E}_{u,0}\exp({\rm j}\phi_{u})\delta(\varpi)$, in which
${\cal E}_{u,0}$ and/or $\phi_{u}$ are (quasi-)stationary or deterministic
relative to ${\cal E}_{s}(\varpi)$, by the same token as for
$\sigma^{2}_{E_{h}}$ before. After normalization by $|{\cal E}_{u,0}|$ or
$\langle|{\cal E}_{u,0}|^{2}\rangle^{1/2}$, its SDF contribution is
$g_{E_{u}}(\varpi)=\delta(\varpi)$. From (76) with
$\gamma_{E_{u}}\gamma_{\varpi_{u}}\gg 1$, regardless of the magnitude of
$\gamma_{E_{u}}$,

$\displaystyle\kappa^{\prime}_{E}\simeq\left(1-\gamma^{2}_{\varpi_{u}}+\gamma^{4}_{\varpi_{u}}/24\right)\gamma^{2}_{E_{u}}\rightarrow\gamma^{2}_{E_{u}}$
(79)

in which $\gamma^{2}_{E_{u}}$ is the familiar Ricean $K$-factor. Naturally,
(79) parallels (69) because baseband stir noise with a vanishingly small
bandwidth ($\gamma_{B}\rightarrow 0$) is equivalent to an unstirred or quasi-
statically stirred (i.e., weakly stir modulated) field contribution
($\gamma_{\varpi_{u}}\rightarrow 0$) to $E(\tau)$. In this case, from (79),
$\kappa^{\prime}_{E}$ increases for an increasing level ${\cal E}_{u,0}$ or
$\sigma_{E_{u}}$. This result agrees with one interpretation of kurtosis as
representing ‘peakedness’ in a density function. Thus, in the absence of stir
noise, understirring gives rise to a positive contribution to
$\kappa^{\prime}_{E}(\gamma_{E_{u}})$.

As a second case, for a dominant non-DC quasi-harmonic contribution at a high
stir frequency ($\gamma_{E_{h}}\gg 1$, $\gamma_{\varpi_{h}}\not\ll 1$,
$\gamma_{E_{h}}\gamma_{\varpi_{h}}\gg 1$), it follows from (76) that

$\displaystyle\kappa^{\prime}_{E}\simeq-5/6+[1/6-(2/3)\gamma^{-2}_{\varpi_{h}}+4\gamma^{-4}_{\varpi_{h}}]\gamma^{-2}_{E_{h}}$
(80)

reducing to the lower bound $-5/6$ for a purely deterministic $E_{s}(\tau)$
[1, sec. III-C]. Dependencies for the cases of quasi-DC
($\gamma_{\varpi_{h}}\ll 1$) or weak harmonics ($\gamma_{E_{h}}\ll 1$) follow
as in sec. IV-A.

Finally, the combination of noise plus EMI is analyzed in [24], exhibiting
additional cross-coupling terms between stir noise and EMI in the expressions
for ${\cal B}^{\prime}_{E}$ and $\kappa^{\prime}_{E}$.

##  V Experimental Results

###  V-A Measurement Configuration

Measurements were performed inside the reverberation chamber B3 at NIST,
Boulder, CO, with an interior volume of $2.7\times 3.3\times 2.8$ m3 (Fig. 4)
using a $32\,001$-point sampling VNA (model Keysight PNA-L). The chamber is
furnished with two rotating mode tuners/stirrers: (i) a “large” short wide
five-blade paddle wheel suspended from the ceiling, whose blades are oriented
at different randomly chosen angles, generating a swept volume of
approximately $1.696$ m3; and (ii) a “small” tall slender paddle wheel
stretching from floor to ceiling, with a swept volume of approximately $0.391$
m3. Both paddles can be operated in tuned (stepped) or stirred (continuous)
mode of rotation. For most results reported here, and unless stated otherwise,
the small paddle is used in tuned mode, generating $N_{t}=72$ tune states with
equiangular separation of $5$ deg and observing a one-minute settling time
after each tuner rotation step, whereas the large paddle is operated in
stirred mode, sampled at $N_{s}=29\,869$ time points across one full stirrer
rotation (cf. sec. V-C). Each one of two dual ridged horn antennas is directed
towards one paddle, without any intentional line-of-sight direction of
coupling. An amplifier (rated 2–18 GHz) was connected at the transmitter side,
to mitigate secondary noise effects generated by the amplifier affecting the
received signal. Collected $S_{21}(t)$ data were not normalized by
$1-S_{11}(t)$ because $S_{11}(t)$ cannot be measured simultaneously in a MSRC,
while synchronization with $S_{11}(t)$ data measured during a subsequent
rotation suffers from potential misalignment, jitter and drift. Typical VNA-
based measurement systems exhibit lower sampling rates but higher SNR levels
than vector spectrum analyzer (VSA) systems, so that the effects of level and
bandwidth of noise on the bias and variance of the spectral moments can be
mitigated.

---
Figure 4:  Measurement set-up in the chamber B3 at NIST.

###  V-B ASDF/ACF Uncertainty Relationship

The selected IFBW $B$ and the corresponding minimum time interval $T$ needed
to capture $N$ samples (dwell time $\Delta\tau=T/N$ per sample) are related
via [22]

$\displaystyle B\times(T/N)\geq\pi.$ (81)

For the $[0/1]$-order $g^{\prime}_{E}(\varpi)$ [1, eq. (34)], the lower limit
in (81) is reached for the bandwidth and correlation length [22]

$\displaystyle B$
$\displaystyle=\frac{\int^{\infty}_{0}g^{\prime}_{E}(\varpi){\rm
d}\varpi}{{\rm
max}_{\varpi}\\{g^{\prime}_{E}(\varpi)\\}}=\sqrt{\frac{\lambda^{\prime}_{2}}{2}}$
(82) $\displaystyle\tau_{c}$
$\displaystyle=\frac{\int^{+\infty}_{-\infty}|\rho^{\prime}_{E}(\tau)|{\rm
d}\tau}{\rho^{\prime}_{E}(0)}=\pi\sqrt{\frac{2}{\lambda^{\prime}_{2}}}$ (83)

respectively, such that $\tau_{c}=\Delta\tau$,
$g^{\prime}_{E}(B)=\sqrt{2/\lambda^{\prime}_{2}}/{\rm e}$ and
$\rho^{\prime}_{E}(\tau_{c})=1/(1+\pi^{2})$. For $U=|E|^{2}$, the [0/1] model
[1, eqs. (73) and (76)] produces the same lower limit as in (81).

For an experimental verification of (81), a different VNA was employed,
capable of collecting $100\,003$ samples across one rotation, thus allowing
for slower stirring without increasing $\Delta\tau$. Fig. 5 compares results
quoted by the manufacturer against our measured data inside the MSRC. The
estimated uncertainty for the SDF computed via FFT, from sampled data or ACF,
viz., $\sqrt{2\pi N/(BT)}$ [22, eq. (8.157)], is also plotted. In the
remainder, $B=2\pi\times 10$ krad/s will be selected as the nominal IFBW for
all measurements whence $B\times\Delta\tau\simeq 16.7$, as a compromise
between measurement duration vs. accuracy.

---
Figure 5:  Manufacturer quoted (theoretical) and measured bandwidth–time
products (in units rad), its lower limit (81) for $[0/1]$-order model, and
standardized random uncertainty for FFT-based SDF as a function of IFBW, $B$
(in units krad/s).

The lower limit in (81) is fundamental to the achievable uncertainty in
estimating the SDF or ACF in stir sweeps and, hence, for the spectral moments.
A narrow IFBW achieves a small bias but yields a large variance of spectral
amplitudes, and vice versa. Reducing the IFBW necessitates a longer dwell time
or larger correlation between samples, requiring a longer time $T$ to complete
the stir sweep. Thus, for a preset $T$, both temporal and spectral methods of
estimating $\lambda_{m}$ incur a bias that can be decreased by reducing $B$
and increasing $N$.

###  V-C Quasi-Periodicity and Drift

In an ideal MSRC, the start and end points of one mechanical rotation period
match perfectly. In practice, electrical and mechanical drift cause a mismatch
in the field data after each rotation and raises uncertainty about the true
stir period of $E(\tau)$. Underestimation of the true period causes a
suboptimal exploitation of the stir process, while overestimation results in
overlapping data reducing statistical accuracy. Hence the stir period should
be estimated conservatively. Improved control of start and stop times can be
achieved using triggering.

In the presence of long-term drift, the centering of data is an issue because
averages dependent on the length $T$, such that averaging over one stir period
$T_{s}$ does not necessarily coincide with averaging over longer or shorter
$T$. For lack of a better alternative, the drift is assumed to be constant
during one full rotation.

The stir period $T_{s}$ was estimated by cross-correlating the data with a
contiguous subset of duration $T_{w}\ll T_{s}$ using a rectangular window. The
uncertainty on the extracted number $N_{s}$ was found to be a weak function of
the choice of $T_{w}$ and estimated to be on the order of $0.03\%$. The number
of nonoverlapping samples is, thus, estimated to be $N_{s}=29\,869\pm 9$,
constituting one stir period $T_{s}=7.9340\pm 0.0024$ s, extracted from a stir
sweep of $N=32\,001$ samples over $T=8.5$ s that includes an arc of partial
overlap spanning $\pi/15$ rad. The sampling time resolution is then
$\Delta\tau=T_{s}/N_{s}\simeq 265.625$ $\mu$s; the corresponding sampling
spectral resolution (angular stir speed) is
$\Delta\varpi=2\pi/(N_{s}\Delta\tau)\simeq 0.79194\pm 0.0003$ rad/s.

###  V-D Estimation of Spectral Parameters

####  V-D1 Effect of Field Noncircularity

As noted in sec. II, unequal orders of differentiation $n$ and $p$ for
$E(\tau)$ and $E^{*}(\tau)$ may lead to discrepancies in the estimated value
of $\lambda_{m}$ when $E(\tau)$ is noncircular. Tbl. I illustrates the effect
of different choices on the first four moments extracted from $S_{21}(\tau)$,
denoted here as $E(\tau)$, at $f=18$ GHz. In the remainder, the calculated
$\lambda_{m}$ use the complex field instead of individual I- and/or
Q-components. In other words, $\lambda_{0}=\langle|{E}(\tau)|^{2}\rangle$,
$\lambda_{1}=-{\rm Im}[\langle{E}(\tau)\dot{E}^{*}(\tau)\rangle]$,
$\lambda_{2}=\langle|\dot{E}(\tau)|^{2}\rangle$, $\lambda_{3}=-{\rm
Im}[\langle\dot{E}(\tau)\ddot{E}^{*}(\tau)\rangle]$, etc., are selected.

$\lambda_{0}/2$ | $-$ | $\lambda_{1}/2$ | rad/s
---|---|---|---
$\langle[{E}^{\prime}(\tau)]^{2}\rangle$ | $3.91\times 10^{-5}$ | $\langle{E}^{\prime}(\tau)\dot{E}^{\prime\prime}(\tau)\rangle$ | $6.03\times 10^{-4}$
$\langle[{E}^{\prime\prime}(\tau)]^{2}\rangle$ | $3.77\times 10^{-5}$ | $-\langle\dot{E}^{\prime}(\tau){E}^{\prime\prime}(\tau)\rangle$ | $6.12\times 10^{-4}$
$\langle|{E}(\tau)|^{2}\rangle/2$ | $3.84\times 10^{-5}$ | $-{\rm Im}[\langle{E}(\tau)\dot{E}^{*}(\tau)\rangle]/2$ | $6.08\times 10^{-4}$

$\lambda_{2}/2$ | (rad/s)2 | $\lambda_{3}/2$ | (rad/s)3
---|---|---|---
$\langle[\dot{E}^{\prime}(\tau)]^{2}\rangle$ | $13.75$ | $\langle\dot{E}^{\prime}(\tau)\ddot{E}^{\prime\prime}(\tau)\rangle$ | $615.5$
$\langle[\dot{E}^{\prime\prime}(\tau)]^{2}\rangle$ | $13.48$ | $-\langle\ddot{E}^{\prime}(\tau)\dot{E}^{\prime\prime}(\tau)\rangle$ | $626.9$
$\langle|\dot{E}(\tau)|^{2}\rangle/2$ | $13.61$ | $-{\rm Im}[\langle\dot{E}(\tau)\ddot{E}^{*}(\tau)\rangle]/2$ | $621.2$
$-\langle E^{\prime}(\tau)\ddot{E}^{\prime}(\tau)\rangle$ | $12.73$ | $-\langle{E}^{\prime}(\tau)\dddot{E}^{\prime\prime}(\tau)\rangle$ | $468.1$
$-\langle{E}^{\prime\prime}(\tau)\ddot{E}^{\prime\prime}(\tau)\rangle$ | $12.45$ | $\langle\dddot{E}^{\prime}(\tau){E}^{\prime\prime}(\tau)\rangle$ | $476.5$
$-{\rm Re}[\langle{E}(\tau)\ddot{E}^{*}(\tau)\rangle]/2$ | $12.59$ | ${\rm Im}[\langle{E}(\tau)\dddot{E}^{*}(\tau)\rangle]/2$ | $472.3$

TABLE I:  Alternative discrete-time estimates of non-normalized
$\lambda_{0\ldots 3}/2$ from sampled stir sweeps for large stirrer at $f=18$
GHz. Overdots denote FD; $\tau$ denotes $\tau_{n}=n\Delta\tau$ for
$n=0,1,\ldots,N_{s}-1$.

####  V-D2 Effect of Input Power and Electrical Noise

In [9], [10], effects of additive white noise on the first two spectral
moments of a Gaussian ASDF were analyzed; for the effect on a complete
exponential ASDF, cf. [1, sec. VI]. The effect of noise becomes progressively
more significant when $E(\tau)$ fades at one or more stir states, on
approaching the noise floor of the VNA or when introducing an external noise
source.

Figs. 6(a) and (c) show the effect of lowering $P_{\rm in}$ on $S_{21}(\tau)$
at $f=1$ GHz without amplifier. This reduction of the input power also
manifests itself in the corresponding SDFs, shown in Figs. 6(b) and (d). These
plots confirm that lowering the SNR in a stir trace ($P_{\rm in}=-60$ dBm)
results in a whitening of the original SDF at $P_{\rm in}=0$ dBm.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 6:  (a)(c) Scatter plots of stir traces $({\rm Re}[S_{21}(\tau)],{\rm
Im}[S_{21}(\tau)])$; (b)(d) associated stir ASDFs
$g^{\prime}_{S_{21}}(\varpi)$, for (a)(b) $P_{\rm in}=0$ dBm and for (c)(d)
$P_{\rm in}=-60$ dBm, all measured at $f=1$ GHz, without amplifier, using
small paddle as stirrer.

Fig. 7 shows the experimental non-normalized $\lambda_{2i}(P_{\rm in})$ and
$\kappa^{\prime}(P_{\rm in})$ at $f=1$ GHz when reducing $P_{\rm in}$ in steps
of $1$ dBm, together with first-order logarithmic fits, obtained as

$\displaystyle\lambda_{0}(P_{\rm in})$ $\displaystyle\simeq 1.288\times
10^{-3}$ (84) $\displaystyle\lambda_{2}(P_{\rm in})$
$\displaystyle\simeq\left\\{\begin{array}[]{ll}(5.37\times
10^{-5})P^{-0.96}_{\rm in},&P_{\rm in}\ll 1\,{\rm mW}\\\ 4.074\times
10^{-2},&P_{\rm in}\gg 1\,{\rm mW}\end{array}\right.$ (87)
$\displaystyle\lambda_{4}(P_{\rm in})$
$\displaystyle\simeq\left\\{\begin{array}[]{ll}(2.60\times
10^{3})P^{-0.95}_{\rm in},&P_{\rm in}\ll 3\,{\rm mW}\\\ 6.310\times
10^{5},&P_{\rm in}\gg 3\,{\rm mW}\end{array}\right.$ (90)
$\displaystyle\kappa^{\prime}(P_{\rm in})$
$\displaystyle\simeq\left\\{\begin{array}[]{ll}(1.94\times
10^{8})P^{0.97}_{\rm in}-1,&P_{\rm in}\ll 1\,{\rm mW}\\\ 8.161\times
10^{4},&P_{\rm in}\gg 3\,{\rm mW}\end{array}\right.$ (93)

suggesting that, at least to good approximation,

$\displaystyle\lambda^{\prime}_{2},\,\lambda^{\prime}_{4}\propto(P_{\rm
in})^{-1},\hskip 14.22636pt\kappa^{\prime}\propto P_{\rm in}$ (94)

when $P_{\rm in}\ll 0$ dBm. For $\kappa^{\prime}$, this result agrees with the
theoretical relationship
$|\kappa^{\prime}_{E}(\gamma_{N})|\propto\gamma^{-2}_{N}\propto P_{\rm
in}/P_{N}$ for $\gamma_{N}\gg 1$, constant $P_{N}$ and ${\cal
B}^{\prime}_{N}=B$; cf. (68) and Fig. 3(b).

The findings for $\lambda_{0}(P_{\rm in})$ and $\lambda_{2}(P_{\rm in})$
agree333In [9], explicit numerical results apply to the case of Gaussian SDFs.
with the theoretical result that the bias of $\lambda^{\prime}_{2}(P_{\rm
in})$ increases rapidly with decreasing SNR [9, sec. V.A.]. When the spectral
width of the stir process is much larger than the stir spectral resolution,
this increase happens at a rate of approximately $1/P_{\rm in}$. Thus, noise
induces a bias for estimated $\lambda^{\prime}_{2i}$; for $\gamma_{N}\ll 1$
(cf. (63))

$\displaystyle\lambda^{\prime}_{m}\simeq{m!}{({\cal
B}^{\prime}_{s})^{m}}\left[1+\left(\frac{\gamma^{m}_{B}}{(m+1)!}-1\right)\gamma^{2}_{N}\right].$
(95)

---
Figure 7:  Experimental values (solid) and first-order logarithmic fits
(84)–(93) (dashed) for $\lambda_{2i}$ (in units (rad/s)2i) and
$\kappa^{\prime}$ as a function of drive input power $P_{\rm in}$ (in units
dBm) at $f=1$ GHz, without amplifier, using small paddle as stirrer. Asterisks
indicate nominal $P_{\rm in}=0$ dBm.

###  V-E Comparison of Stir-Domain Estimation Methods

Tbl. II compares estimates of $(\lambda^{\prime(\prime)}_{m})^{1/m}$ obtained
from the sweep vs. ACV FD methods, indicating close numerical agreement.
Associated centered moment-based quantities are similarly close; e.g., the
relative difference of
$[\lambda^{\prime}_{2}-(\lambda^{\prime\prime}_{1})^{2}]^{1/2}$ between both
methods is less than $1.4\times 10^{-4}$. As functions of ratios of raw
moments and $\lambda_{0}$, these estimates are still subject to sample
statistical uncertainty [25, sec. 27.7]. Compensation for FD bias yields a
positive and smaller (in magnitude) $\kappa^{\prime}$ for both methods,
whereas $\kappa^{\prime\prime}$ becomes closer to zero but remains negative
for the stir sweep method, unlike for ACV. FD debiasing does not remove the
overall bias, as other contributions including aliasing, finite sample sweep
length, etc., remain [11], but are relatively small ($\sim 1/N_{s}$).

From $(\lambda^{\prime(\prime)}_{m})^{1/m}$, several spectral statistics
follow, including the mean
$\mu_{g^{\prime(\prime)}_{E}}=\lambda^{\prime(\prime)}_{1}$, variance
$\sigma^{2}_{g^{\prime(\prime)}_{E}}=\lambda^{\prime(\prime)}_{2}-(\lambda^{\prime(\prime)}_{1})^{2}$,
and normalized bandwidths [5], [8]

$\displaystyle\delta^{\prime(\prime)}_{E}$
$\displaystyle\equiv\sigma_{g^{\prime(\prime)}_{E}}/\sqrt{\lambda^{\prime(\prime)}_{2}}=\sqrt{1-\left(\lambda^{\prime(\prime)}_{1}\right)^{2}/\lambda^{\prime(\prime)}_{2}}$
(96) $\displaystyle\delta^{\prime}_{E}$
$\displaystyle\simeq\sqrt{\frac{1}{2}}\left[1-\frac{1}{4}\left(\frac{\Delta\tau}{\beta^{\prime}}\right)^{2}\right]$
(97) $\displaystyle\epsilon^{\prime(\prime)}_{E}$
$\displaystyle\equiv\sqrt{1-1/k_{g^{\prime(\prime)}_{E}}}=\sqrt{1-\left(\lambda^{\prime(\prime)}_{2}\right)^{2}/\lambda^{\prime(\prime)}_{4}}$
(98) $\displaystyle\epsilon^{\prime}_{E}$
$\displaystyle\simeq\sqrt{\frac{5}{6}}\left[1-\frac{3}{10}\left(\frac{\Delta\tau}{\beta^{\prime}}\right)^{2}\right].$
(99)

The approximations (97) and (99) account for FD bias, which is again negative
and quadratic to leading order. In Tbl. II, the FD compensated
$\delta^{\prime\prime}_{E}$ at $f=18$ GHz is within $0.2\%$ of its maximum
value $1$, while $\delta^{\prime}_{E}$ and $\epsilon^{\prime}_{E}$ also remain
marginally below their respective theoretical minimum values, $\sqrt{1/2}$ and
$\sqrt{5/6}$, for ideal I/Q isotropy in the complex $E$-plane, i.e., for
circular local fields [18]. This indicates a wide stir (dynamic) bandwidth
despite the narrow excitation (static) bandwidth.

Parameter | Sweep FD | ACV FD
---|---|---
$\lambda^{\prime\prime}_{1}$ | $15.8299$ | $16.0401$
$(\lambda^{\prime}_{2})^{1/2}$ | $595.542$ | $595.630$
$(\lambda^{\prime\prime}_{3})^{1/3}$ | $252.940$ | $257.284$
$(\lambda^{\prime}_{4})^{1/4}$ | $926.161$ | $926.478$
$(\lambda^{\prime\prime}_{5})^{1/5}$ | $550.232$ | $572.812$
$\kappa^{\prime}$ (biased) | $-0.02514$ | $-0.02437$
$\kappa^{\prime}$ (FD compensated) | $+0.01118$ | $+0.01195$
$\kappa^{\prime\prime}$ (biased) | $-0.08541$ | $+0.02307$
$\kappa^{\prime\prime}$ (FD compensated) | $-0.04193$ | $+0.06804$
$\delta^{\prime}_{E}$ (biased) | $0.69542$ | $0.69533$
$\delta^{\prime}_{E}$ (FD compensated) | $0.70494$ | $0.70485$
$\delta^{\prime\prime}_{E}$ (biased) | $0.99965$ | $0.99964$
$\delta^{\prime\prime}_{E}$ (FD compensated) | $0.99805$ | $0.99811$
$\epsilon^{\prime}_{E}$ (biased) | $0.90704$ | $0.90711$
$\epsilon^{\prime}_{E}$ (FD compensated) | $0.91051$ | $0.91059$
TABLE II:  $(\lambda^{\prime(\prime)}_{m})^{1/m}$ for $1\leq m\leq 5$, in
units rad/s; $\kappa^{\prime(\prime)}$; $\delta^{\prime(\prime)}_{E}$;
$\epsilon^{\prime}_{E}$, extracted using stir sweep vs. ACV FD methods, at
$f=18$ GHz, for an arbitrary secondary tune state, with
$\lambda_{0}=7.6774\times 10^{-5}$.

###  V-F SDF Symmetry, Phase Linearity and Doppler Shift

WSS $E^{\prime}(\tau)$ and $E^{\prime\prime}(\tau)$ have Hermitean ASDFs
$g_{E^{\prime}}(\varpi)$ and $g_{E^{\prime\prime}}(\varpi)$. This complex
conjugate symmetry does not necessarily extend to $g_{E}(\varpi)$ for sampled
fields [9], [13]. An asymmetric ASDF corresponds to a nonlinearity of
$\phi_{\rho_{E}}(\tau)$ that contributes to (49)–(51) and, hence, to spectral
skewness. Such phase discrimination can be used, e.g., for characterizing
narrowband interference [26].

The continuous motion of a mechanical stirrer carries an associated Doppler
shift (mean velocity and $\langle\Lambda_{1}\rangle\not=0$), raising the
question of significance of its effect on $g_{E}(\varpi)$,
$\delta^{\prime(\prime)}_{E}$ and $\kappa$. To this end, a perturbation
expansion444Applicability of the perturbation expansion method requires
$\rho^{2}_{E}(\Delta\tau)\gg 1/N_{s}$ and $\Delta\tau/\beta\gg 1/N_{s}$ [12].
of the complex ACF of $E(\tau)$ can be applied [12]. If $N_{e}(\tau)$ is
negligible, then the variance of $\Lambda_{1}/\Lambda_{0}$ for
$\rho_{E}(\Delta\tau)=[1+(\Delta\tau/\beta)^{2}]^{-1}$ is estimated as

$\displaystyle\sigma^{2}_{\Lambda_{1}/\Lambda_{0}}\simeq\frac{\pi}{2(\Delta\tau)\beta
N_{s}}\left[1+\frac{1}{2}\left(\frac{\Delta\tau}{\beta}\right)^{2}\right].$
(100)

At $f=18$ GHz, this yields $\sigma_{\Lambda_{1}/\Lambda_{0}}\simeq 9.2$ rad/s,
i.e., comparable in magnitude to
$\langle\Lambda_{1}/\Lambda_{0}\rangle_{N_{t}}\simeq 21.22$ rad/s, but about
two orders of magnitude smaller than $\sigma_{g_{E}}$. On this basis, the
assumption of a quasi-stationary symmetric ASDF with negligible stir frequency
shift is justifiable.

Physically, a significantly nonzero value of
$\langle\Lambda_{1}/\Lambda_{0}\rangle_{N_{t}}$ can be associated with a net
mean radial flux of directed energy produced by reflections from the stirrer’s
surface elements across the stir process. This flux arises most prominently
when the antenna’s boresight direction is pointing towards an approaching or
receding paddle blade, i.e., directed off-axis with reference to the stirrer’s
shaft. The spectral width
$\langle(\Lambda_{2}-{\Lambda^{2}_{1}})/\Lambda_{0}\rangle^{1/2}_{N_{t}}$ then
measures the RMS spread of the instantaneous radial flux during the stir
process, serving as a measure of stir efficiency. Higher-order spectral
moments can be used to further refine the probability distribution of this
flux.

###  V-G Effect of Sampling Rate and Decimation

The effect of sampling on the spectral moments can also be investigated a
posteriori, by subsampling the original data with a decimation factor
$d=\tau/\Delta\tau\geq 1$ [27]. A continuous stir trace corresponds to the
limit $\Delta\tau\rightarrow 0$ and $d=1$. Fig. 8 shows experimental values of
$\lambda_{0}$, $(\lambda^{\prime(\prime)}_{m})^{1/m}$ and
$\kappa^{\prime(\prime)}$ as functions of $d$ at $f=18$ GHz for the large
stirrer, obtained using the stir sweep method. Expanded confidence intervals
$[\langle\kappa^{\prime(\prime)}(d)\rangle-2\sigma_{\kappa^{\prime(\prime)}},\langle\kappa^{\prime(\prime)}(d)\rangle+2\sigma_{\kappa^{\prime(\prime)}}]$
are shown with black dots.

For even-order moments and coarse subsampling ($30<d<1000$), the slope of the
mean $\langle(\lambda^{\prime}_{m}(d))^{1/m}\rangle_{N_{t}}$ for $m=2$ and $4$
in Fig. 8(a) is $-0.9991$ and $-0.9995$, respectively, in good agreement with
(15) for $\Delta\tau/\beta^{\prime}\gg 1$, to leading order. For
$\Delta\tau/\beta^{\prime}\ll 1$, the quadratic dependence of
$\langle(\lambda^{\prime}_{m}(d))^{1/m}\rangle_{N_{t}}$ in (16) is also
apparent. The asymptotic value of $\kappa^{\prime}(d\rightarrow
N_{s}/2)\simeq-0.584$ is slightly above the theoretical value $-3/4$ in (22),
valid for ideal stirring in the absence of noise, thus indicating good stir
performance. For $d\rightarrow 1$, the slope of $\log(|\kappa^{\prime}|)$ vs.
$\log(d)$ is sensitive to the specific small value of $|\kappa^{\prime}|$ at
$d=1$, as expected, ranging between $-1.76$ and $-2.49$ across the 72
secondary tune states $\tau_{2,\ell}$. This range incorporates the theoretical
value $-2$ for $\Delta\tau/\beta^{\prime}\ll 1$ from (22), applicable to ideal
$\kappa^{\prime}(\Delta\tau/\beta^{\prime}\rightarrow 0)=0$.

For odd-order moments and $\kappa^{\prime\prime}$ shown in Fig. 8(b), their
dependence on $d$ exhibits considerably larger fluctuations across tune states
than for even-order moments. For $100<d<1000$, the slope of
$\langle(\lambda^{\prime\prime}_{m}(d))^{1/m}\rangle_{N_{t}}$ for $m=1$, $3$
and $5$ is $-1.0372$, $-0.9978$ and $-0.9981$, respectively. The asymptotic
value $\kappa^{\prime\prime}(d\rightarrow N_{s}/2)\simeq-0.49$ is again
slightly above the theoretical value, now $-16/25$ in (31), for ideal noise-
free stirring.

The differences in concentration of individual $d$-characteristics between
even- vs. odd-order moments in Figs. 8(a) and (b) can be traced to the unequal
orders of differentiation $n\not=p$ when $m$ is odd. This leads to mismatched
FD grids of sampled data points for extracting odd-order moments, resulting in
different values of $\lambda_{m}$ depending on whether $n=p+1$ or $n=p-1$; cf.
Tbl. I.

---
(a)
(b)
Figure 8:  (a) Even- and (b) odd-order normalized spectral moments
$(\lambda_{m}/\lambda_{0})^{1/m}$ and $\kappa^{\prime(\prime)}$ as a function
of data decimation factor $d$, based on: (i) individual stir sweeps for 72
tune states (solid), (ii) averaging across 72 tune states (black dashed),
(iii) concatenation of all $N_{t}N_{s}=2\,150\,568$ samples (cyan dashed),
including confidence intervals
$[\langle\kappa^{\prime(\prime)}\rangle-2\sigma_{\kappa^{\prime(\prime)}},\langle\kappa^{\prime(\prime)}\rangle+2\sigma_{\kappa^{\prime(\prime)}}]$
(black dotted) at $f=18$ GHz, for stirring by large paddle, secondary tuning
by small paddle, including amplifier.

###  V-H CW Frequency Characteristics

####  V-H1 Even-Order Moments

Fig. 9(a) shows $\lambda_{0}(f)$ and $(\lambda_{m}(f)/\lambda_{0}(f))^{1/m}$
for $m=0,2,4$. The latter characteristics increase proportionally with $f$
when $d\rightarrow 0$. At higher $f$, the curves increasingly sag with
increasing $\Delta\tau$; cf. [27, Fig. 6].

In Fig. 9(b), ${\kappa^{\prime}}(f)$ rapidly converges towards ideal zero with
increasing $f$, on average, as predicted theoretically. Fig. 9(b) and (c) show
that residual fluctuations remain, caused by FD numerical underflow. These can
be reduced by taking the RMS value of ${\kappa^{\prime}}^{2}$ across the
$N_{t}$ tune states. At $f=18$ GHz, this results in
$\langle{\kappa^{\prime}}^{2}\rangle^{1/2}_{N_{t}}\simeq 0.0260$ whereas
$\langle{|\kappa^{\prime}|}\rangle^{1/2}_{N_{t}}\simeq 0.1448$.

---
(a)
(b)
(c)
Figure 9:  (a) Normalized even-order spectral moments as a function of CW
frequency $f$: 72 individual stir traces (colored, solid), tune average (cyan,
dashed) and tune standard deviation (blue, dashed), for large stirrer,
including amplifier; (b) FD debiased ${\kappa^{\prime}}(f)$ based on stir
sweep data; (c) FD debiased ${\kappa^{\prime}}(f)$ based on complex ACV.

####  V-H2 Odd-Order Moments

Fig. 10(a) shows the frequency characteristics
$(\lambda_{m}(f)/\lambda_{0}(f))^{1/m}$ for $m=1,3,5$. Compared to those for
even-order moments, considerably larger variations occur again across
secondary tune states; similarly for ${\kappa^{\prime\prime}}(f)$ in Fig.
10(b). At high $f$, the convergence of ${\kappa^{\prime\prime}}(f)$ to zero is
slower.

Fig. 10(c) indicates that the ACV method predicts relatively large magnitudes
of $\langle\kappa^{\prime\prime}(f)\rangle_{N_{t}}$ at any frequency, and
exhibits a slower decrease with $f$ compared to the stir sweep method. The
difference is attributable to the values of $\lambda_{5}(f)/\lambda_{0}(f)$,
which increase more slowly in the ACV method because of the less accurate
estimation of the ACF phase angle derivative $\phi^{(5)}_{\rho_{E}}(0)$ at
lower frequencies. A similarly slower decrease is found for
$\langle\kappa^{\prime}(f)\rangle_{N_{t}}$ in Fig. 9(c), owing to larger
values and fluctuations of $\lambda_{4}(f)/\lambda_{0}(f)$ traceable to
$a^{(4)}_{\rho_{E}}(0)$ at lower $f$.

###  V-I Dependence on Tune States and Effect of FD Debiasing

Fig. 11 compares fluctuations of $\kappa^{\prime(\prime)}$ across all tune
states before and after FD debiasing at $f=18$ GHz, using the stir sweep FD
method. While fewer isolated tune state maintain a negative $\kappa^{\prime}$,
the vast majority incur a positive shift to small values as a result of FD
debiasing, as expected in view of the observed LF-to-HF level drop in the ASDF
for $\kappa^{\prime}>0$ [1, eq. (43)]. By contrast, FD debiasing has a
significantly smaller effect on $\kappa^{\prime\prime}$ across all secondary
tune states.

---
(a)
(b)
(c)
Figure 10:  Same as Fig. 9 but for odd-order moments.
---
Figure 11:  ${\kappa^{\prime}}$ and ${\kappa^{\prime\prime}}$ across $72$
secondary tune states at $f=18$ GHz: raw data (red), estimated FD bias
(black), FD debiased (blue).

##  VI Conclusion

Sampling, FD, noise, interference, and understirring/line-of-sight coupling in
a continuously and uniformly stirred field all affect its spectral and
correlation characteristics. These were analyzed theoretically and
demonstrated experimentally in this article. Absorption is another major
contributor, which will be investigated in future work.

Direct FD of a sampled stir sweep trace, or its complex ACF using (46)–(57),
were shown to yield comparable numerical estimates of spectral moments at high
excitation frequencies, provided the FD bias is compensated for; cf. Tbl. II.
The former technique can be implemented as an online (real-time) method, while
the latter requires batch processing after all data samples have been
acquired. To leading order, the spectral moments, (16) and (24), as well as
the scaled spectral kurtoses, (22) and (31), exhibit a negative quadratic bias
with respect to the normalized sampling resolution,
$\Delta\tau/\beta^{\prime(\prime)}$, that can be compensated for. Odd-order
spectral moments $\lambda^{\prime\prime}_{2i+1}$ and
$\kappa^{\prime\prime}(f)$ involve unequal orders of FD and exhibit
considerably larger spread for individual stir traces than even-order
$\lambda^{\prime}_{2i}$ and $\kappa^{\prime}(f)$.

Noise, interference and understirring were shown to have a major effect on the
spectral kurtosis of ideal stirred continuous fields ($\kappa^{\prime}=0$) via
(66), (76) and (79). A rich dependence structure on noise-to-stir and noise-
to-interference ratios of levels and bandwidths was found (Figs. 3 and 7).
This demonstrates the high sensitivity of CFs and SDFs on field imperfections,
even after FD bias compensation. The results in Figs. 9–11 illustrate the
challenge of accurately estimating the small magnitude and sign of
$\kappa^{\prime(\prime)}$.

Whereas the analysis for second- and higher-order Padé-based models is more
involved, the simple $[0/1]$ model limits the effects on sampling, noise and
interference to a consideration of the parameter $\lambda^{\prime}_{2}$ only.

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