# Sampling Distributions of Random Electromagnetic Fields in
Mesoscopic or Dynamical Systems

Luk R. Arnaut
Time, Quantum and Electromagnetics Division
National Physical Laboratory, Teddington, United Kingdom
and
Department of Electrical and Electronic Engineering
Imperial College of Science, Technology and Medicine
South Kensington Campus, London, United Kingdom

###### Abstract

We derive the sampling probability density function (pdf) of an ideal
localized random electromagnetic field, its amplitude and intensity in an
electromagnetic environment that is quasi-statically time-varying
statistically homogeneous or static statistically inhomogeneous. The results
allow for the estimation of field statistics and confidence intervals when a
single spatial or temporal stochastic process produces randomization of the
field. Sampling distributions are particularly significant when the number of
degrees of freedom $\nu$ is relatively small (typically, $\nu<40$), e.g., in
mesoscopic systems when the sample set size $N$ is relatively small by choice
or by force. Results for both coherent and incoherent detection techniques are
derived, for Cartesian, planar and full-vectorial fields. We show that the
functional form of the sampling pdf depends on whether the random variable is
dimensioned (e.g., the sampled electric field proper) or is expressed in
dimensionless standardized or normalized form (e.g., the sampled electric
field divided by its sampled standard deviation). For dimensioned quantities,
the electric field, its amplitude and intensity exhibit different types of
Bessel $K$ sampling pdfs, which differ significantly from the asymptotic Gauss
normal and $\chi^{(2)}_{2p}$ ensemble pdfs when $\nu$ is relatively small. By
contrast, for the corresponding standardized quantities, Student $t$, Fisher-
Snedecor $F$ and root-$F$ sampling pdfs are obtained that exhibit heavier
tails than comparable Bessel $K$ pdfs. Statistical uncertainties obtained from
classical small-sample theory for dimensionless quantities are shown to be
overestimated compared to dimensioned quantities. Differences in the sampling
pdfs arising from de-normalization versus de-standardization are obtained.

###### pacs:

PACS: 02.50.-r, 03.50.De, 05.10.Gg, 06.30.Ka, 41.20.-q, 41.20.Jb, 42.25.Bs

##  I Introduction

In the application of statistical methods to disordered and chaotic systems,
particularly nonergodic and/or mesoscopic systems [1]–[6], the role played by
sampling distributions [7] is of fundamental importance because, in practice,
data sets are necessarily of limited (and often small) size $N$. Strictly, the
Central Limit Theorem (CLT) is inapplicable to finite sample sets, a fortiori
to small sets. Therefore, sampling distributions, rather than their underlying
(parent) ensemble distributions, should always be used in any proper
comparison between theoretical and empirical probability distributions based
on numerical data from practical measurement (experiment) or simulation
(computation), particularly when detecting or validating unexpected phenomena.
The finiteness of the number of degrees of freedom for sample sets, $\nu$, can
have a profound effect not only on the statistical moments, but also on the
functional form and shape of the probability density function (pdf) –
particularly near its extremities – and on second- or higher-order statistics,
such as auto- or cross-correlation functions and associated spectral
densities, etc. As a matter of fact, sampling pdfs draw on properties of both
first- and second-order ensemble statistics: while the sampling distributions
are representations for local instantaneous sampled fields, the value of $\nu$
as a sampling distribution parameter is governed by the correlation distance
between points. This dependence on nonlocal properties of the field applies to
any physically limited realizable sampled region. For sample sets in which all
$N$ values are statistically independent (as we shall further assume
throughout), $\nu=N-1$.

Ensemble pdfs for ideal random 3-D electromagnetic (EM) fields are well known
and have been amply investigated in various physical applications, whether in
unbounded space [8] or in the presence of an impedance boundary [9]. By
contrast, their associated sampling pdfs have received little attention yet
are pertinent and require characterization. Idealized random (stochastic)
classical EM fields offer a paradigm for characterizing wave propagation or
transport governed by dynamic multiple scattering (time-varying configuration
or boundaries). Examples include fields inside acoustic or EM mode-tuned or
mode-stirred reverberation chambers (MT/MSRCs) [8]–[12]; mesoscopic
structures; random or turbulent media; polarization and anisotropy in, e.g.,
the cosmic microwave background radiation [13]–[15]; static or dynamic optical
diffusion using random phase screens; diffusing wave spectroscopy [16]–[21];
etc.

For the estimation of statistics of the field, amplitude, intensity, energy
density and power for such ‘wave chaos’, the ensemble pdf of the field is
usually adequate when the number of statistically independent partial
contributions and, hence, $\nu$ approaches infinity. For example, for
MT/MSRCs, the prerequisites are that the wavelength is very short relative to
the physical dimensions of the cavity and that the observation time is long
compared to the correlation time of the stirring process because, physically,
a random EM field is governed by the spatial or temporal extent of the space-
or time-limited process. On occasion, however, $\nu$ is too small for these
ensemble pdfs to be sufficiently accurate. In this case, the CLT or maximum-
entropy principle may not be applicable. Nevertheless, in such low-dimensional
cases, spatial and/or temporal averaging increases the value of $\nu$ [22],
which may have a substantial effect on the measured or perceived distribution
of the field.

A key issue addressed in this paper is that statistically homogeneous random
fields are characterized by pdfs with distribution parameters (e.g., average,
standard deviation, number of degrees of freedom, etc.) whose values have to
be estimated from the same set of data of the fluctuating field. As a result,
these parameters themselves show sampling fluctuations because of the finite
value of $\nu$. These variations give rise to bi- or multivariate fluctuations
and increased uncertainty (i.e., wider confidence intervals) for the sampled
field, compared to the corresponding ensemble distribution where these
parameters are known exactly.

The physical origin of the limitation of the value of $\nu$ can be twofold.
First, the potential (i.e., maximum attainable) number of statistically
independent realizations, $N_{\rm max}$, may be practically restricted, even
if an unlimited number of different states of the statistical ensemble were
physically realized. For example, in a MT/MSRC, this case corresponds to the
so-called undermoded regime (i.e., multi-mode operation but with less than,
say, ten cavity modes being excited simultaneously); typically occurring at
wavelengths where modal overlap is relatively small. As another example, in a
relatively small sample of a random medium, the limitation on the value of
$\nu$ refers to the case of a relatively small loading fraction of inclusions.
In this case, even the ensemble distribution does not possess Gauss normal
statistics. Secondly, $N_{\rm max}$ may be unlimited but, for economical or
other reasons, the sample size may have to be severely restricted ($N\ll
N_{\rm max}$). In this case, while $\nu_{\rm max}$ is potentially large, the
value of $\nu$ actually realized in a sample set is relatively small.

In this paper, we investigate sampling pdfs of the complex value, amplitude
and intensity of 1-D, 2-D and 3-D random EM fields that have ideal Gaussian
ensemble probability distributions, for both the actual (non-standardized)
observed EM quantities and for their standardized forms. Here, the notions of
standarized and normalized random variables refer to a random field quantity
divided by its own (sample or ensemble) standard deviation or mean value,
respectively. Coherent as well as incoherent detection techniques are
considered in each case. However, specific nonstationary effects associated
with the dynamics of multiple scattering, which are relevant to certain
aspects of diffusing wave spectroscopy [17, 18], are not addressed here: each
realized state of the system is considered in its quasi-stationary
approximation. The scenario investigated here is also relevant, in particular,
to several practical applications in wireless communications (e.g., mobile-to-
fixed and mobile-to-mobile transmission), signal processing, wave propagation
in turbulent media, modal noise in optical fibres under restricted-mode launch
conditions [20], etc.

##  II Electric or magnetic field

###  II.1 Coherent detection

####  II.1.1 Non-standardized field

Consider a modulated local analytic electric field $E=E^{\prime}-{\rm
j}E^{\prime\prime}$ received by an antenna (sensor) or scatterer immersed in a
time-varying multi-scattering environment. A harmonic time dependence
$\exp({\rm j}\omega t)$ is assumed and suppressed. If this field is made up of
an arbitrarily large (theoretically infinite) number of fluctuating partial
fields (i.e., modal or angular plane-wave spectral components) whose
realization forms a random walk in the complex plane [23], then, on account of
the CLT (valid under very general but definite conditions [24]), the
associated conditional probability of $E^{\prime(\prime)}$ coincides with the
ensemble pdf, given by

$\displaystyle
f_{E^{\prime(\prime)}|(S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}})}(e^{\prime(\prime)}|(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}}))$
$\displaystyle=\frac{\exp\left[-\frac{\left(e^{\prime(\prime)}-m_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace
s^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi}~{}s_{E^{\prime(\prime)}}}.$
(1)

Here, $M_{E^{\prime(\prime)}}$ and $S_{E^{\prime(\prime)}}$ represent random
variables induced by the sample mean and sample standard deviation of
$E^{\prime(\prime)}$, respectively, whereas lowercase symbols
$m_{E^{\prime(\prime)}}$ and $s_{E^{\prime(\prime)}}$ represent their
corresponding sample values. If the mean and standard deviation of
$E^{\prime(\prime)}$ are known with certainty, with respective constant
ensemble values $\mu_{E^{\prime(\prime)}}$ and $\sigma_{E^{\prime(\prime)}}$,
i.e.,

$\displaystyle f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}})$
$\displaystyle=$
$\displaystyle\delta(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}})$ (2)
$\displaystyle f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}})$
$\displaystyle=$
$\displaystyle\delta(s_{E^{\prime(\prime)}}-\sigma_{E^{\prime(\prime)}})$ (3)

then the conditional pdf (1) coincides with the marginal pdf
$f_{E^{\prime(\prime)}}({e^{\prime(\prime)}})$ because, in general,

$\displaystyle f_{E^{\prime(\prime)}}({e^{\prime(\prime)}})$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{-\infty}{\rm
d}m_{E^{\prime(\prime)}}\int^{+\infty}_{0}{\rm d}s_{E^{\prime(\prime)}}$
$\displaystyle\times
f_{E^{\prime(\prime)},S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(e^{\prime(\prime)},s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})$
$\displaystyle=$ $\displaystyle\int^{+\infty}_{-\infty}{\rm
d}m_{E^{\prime(\prime)}}\int^{+\infty}_{0}{\rm d}s_{E^{\prime(\prime)}}$
$\displaystyle\times
f_{E^{\prime(\prime)}|(S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}})}(e^{\prime(\prime)}|(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}}))$
$\displaystyle\times
f_{S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})$
(5)

and because in (and only in) the case of a Gauss normal ensemble distribution
of $E^{\prime(\prime)}$ are the sample mean and sample standard deviation
independent random variables [7], i.e.,

$\displaystyle
f_{S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})=f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}})f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}}).$
(6)

From (1) and (2)–(6), we obtain

$\displaystyle
f_{E^{\prime(\prime)}}(e^{\prime(\prime)})=\frac{\exp\left[-\frac{\left(e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi}~{}\sigma_{E^{\prime(\prime)}}}.$
(7)

Here,
$\sigma_{E^{\prime(\prime)}}=\sigma_{E}/\sqrt{2}=\sqrt{p}\thinspace\sigma_{E^{(\prime)\prime}_{\alpha}}=\sqrt{p/2}\thinspace\sigma_{E_{\alpha}}$
where $\alpha=x,y$ or $z$ for the three Cartesian components $E_{\alpha}$ of
${\bf E}$ and where the value of $p$ corresponds to the number of spatial
dimensions in which ${\bf E}$ is being considered.

Of the possible values of $p$, viz., $1$, $2$, or $3$, its specific value is
governed by the polarization direction of the electric-field sensor and/or by
any EM excitation or boundary conditions that may enforce a certain fixed
(deterministic) polarization state of the field. Thus, the Cartesian component
of a vector field and the 3-D vector field itself correspond to the cases
$p=1$ and $p=3$, respectively. For a paraxially propagating polarized or
unpolarized optical random field, $p=1$ or $p=2$, respectively. The case $p=2$
refers to an unpolarized electric field that is transverse to the local
wavevector (i.e., randomly elliptically polarized field) and will be denoted
by a subscript ‘t’, whereas $p=1$ corresponds to its polarized detection or to
a randomly modulated linearly polarized field, denoted by a subscript
‘$\alpha$’. These values of $p$ apply to local fields detected by an
electrically small sensor, whose characteristic length is less than the
spatial coherence length of the field, i.e., typically $\lambda/2$ in an
unbounded medium. Larger values of $p$ apply to electrically large detectors
whose receiving cross-section (whether physical or as a synthetic aperture) is
larger than $\lambda/2$ in one or more dimensions. For example, for currents
induced in a linear antenna of length $L$, we have $p={\rm
max}[1,L/(\lambda/2)]$.

If, instead of (3), $S_{E^{\prime(\prime)}}$ exhibits random fluctuations or
if its value is not known precisely, the sampling pdf of
$S_{E^{\prime(\prime)}}$ for Gaussian
$E^{\prime(\prime)}|S_{E^{\prime(\prime)}}$ is a $\chi_{pN-1}$ pdf, i.e.,

$\displaystyle f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}};N)$
$\displaystyle=$
$\displaystyle\frac{C_{S_{E^{\prime(\prime)}}}}{\sigma_{S_{E^{\prime(\prime)}}}}\left(\frac{s_{E^{\prime(\prime)}}}{\sigma_{S_{E^{\prime(\prime)}}}}\right)^{pN-2}$
(8) $\displaystyle\times\exp\left[-{\cal
N}\left(\frac{s_{E^{\prime(\prime)}}}{\sigma_{S_{E^{\prime(\prime)}}}}\right)^{2}\right]$

with

$\displaystyle C_{S_{E^{\prime(\prime)}}}$
$\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{2~{}{\cal
N}^{\frac{pN-1}{2}}}{\Gamma\left(\frac{pN-1}{2}\right)},$ (9)
$\displaystyle{\cal N}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle{\frac{pN-1}{2}-\left[\frac{\Gamma\left(\frac{pN}{2}\right)}{\Gamma\left(\frac{pN-1}{2}\right)}\right]^{2}}.$
(10)

The sampling pdf (8) is in its self-sufficient form [22], i.e., it contains
the standard deviation of $S_{E^{\prime(\prime)}}$ itself as a distribution
parameter. Alternatively, (8) can be re-expressed in terms of the ensemble
standard deviation $\sigma_{E^{\prime(\prime)}}$ of $E^{\prime(\prime)}$,
because both statistics are related via

$\displaystyle\sigma_{S_{E^{\prime(\prime)}}}$ $\displaystyle=$
$\displaystyle\sigma_{E^{\prime(\prime)}}\sqrt{\frac{2~{}{\cal N}}{pN-1}}.$
(11)

With the aid of [25, (3.471.9)], the sampling pdf of $E^{\prime(\prime)}$ for
$\mu_{E^{\prime(\prime)}}=0$ in (5) is obtained as a marginal pdf of the joint
pdf
$f_{E^{\prime(\prime)},S_{E^{\prime(\prime)}}}(e^{\prime(\prime)},s_{E^{\prime(\prime)}};N)$,
viz.,

$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}(e^{\prime(\prime)}|s_{E^{\prime(\prime)}})f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}};N){\rm
d}s_{E^{\prime(\prime)}}$ $\displaystyle=$
$\displaystyle\frac{(pN-1)^{\frac{pN-1}{2}}}{2^{\frac{pN}{2}}\sqrt{\pi}\thinspace\Gamma\left(\frac{pN-1}{2}\right)~{}\sigma^{pN-1}_{E^{\prime(\prime)}}}$
$\displaystyle\times\int^{+\infty}_{0}t^{\frac{pN}{2}-2}\exp\left(-\frac{e^{{\prime(\prime)}^{2}}}{2t}-\frac{pN-1}{2}\frac{t}{\sigma^{2}_{E^{\prime(\prime)}}}\right){\rm
d}t,$

i.e.,

$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)=$ (13)
$\displaystyle\frac{C_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\left(\frac{|e^{\prime(\prime)}|}{\sigma_{E^{\prime(\prime)}}}\right)^{\frac{pN}{2}-1}K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}|}{\sigma_{E^{\prime(\prime)}}}\right)$

with

$\displaystyle
C_{E^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{(pN-1)^{\frac{pN}{4}}}{2^{\frac{pN}{2}}\sqrt{\pi}\thinspace\Gamma\left(\frac{pN-1}{2}\right)}.$
(14)

The pdf (13) belongs to McKay’s class of Bessel $K$ distributions [26]. These
pdfs have also been obtained in a variety of applied statistical problems, in
electromagnetics and elsewhere; cf., e.g., [27], [28] and references in [23].
It is further worth noting that the pdf (13) has also been obtained in a
different case, viz., as an ensemble pdf associated with an ad hoc Bessel $K$
distributed field amplitude (implying non-Gaussian/non-Rayleigh ensemble
statistics) and derived via a Blanc-Lapierre transformation [4]. Results that
are at least qualitatively compliant with those obtained from the pdf (13)
have been observed recently in experiments involving correlated scattering
[6]. The pdf (13) is also included in the class of Meijer $G$ limit pdfs for
complex fields in undermoded MT/MSRCs [23], which in turn is a special case of
more general Fox $H$ distributions [29].

It is emphasized that the result (13) strictly holds for a physically ideal,
viz., Gaussian random field if it is subjected to a finite-sized sampling
(detection) process. Only in this case is a Gaussian marginal pdf
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ compatible with a non-delta-
distributed ${S_{E^{\prime(\prime)}}}$. For the physical field itself, a
Gaussian $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ implies the number of
degrees of freedom approaching infinity, hence resulting in a delta
distribution for $f_{S_{E^{\prime(\prime)}}}$, and vice versa.

Fig. 1a shows the pdf (13) for selected values of $N$. Because of symmetry of
$f_{E^{\prime(\prime)}}$ with respect to its mean value, only the positive
half of the distribution is shown.

If $E^{\prime(\prime)}$ exhibits a deterministic, i.e., constant bias
$\mu_{E^{\prime(\prime)}}$ as in (7), then

$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\frac{C_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\left(\frac{|e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right)^{\frac{pN}{2}-1}$
$\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right).$

For nonnegligible “slow” fluctuations of $m_{E^{\prime(\prime)}}$ in (1),
i.e., for a random bias (trend) of $E^{\prime(\prime)}$, the results are
easily generalized as follows. On account of linearity, the sampling variable
$M_{E^{\prime(\prime)}}$ for Gaussian $E^{\prime(\prime)}$ is also Gaussian
with the same expected value but standard deviation
$\sigma_{E^{\prime(\prime)}}/\sqrt{N}$. Therefore, if the bias is itself
random with Gauss normal distribution, i.e.,

$\displaystyle
f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}};N)=\frac{\exp\left[-\frac{N\left(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi/N}~{}\sigma_{E^{\prime(\prime)}}}$
(16)

instead of (2), then with (5)–(6) we arrive at

$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)=$ (17)
$\displaystyle\frac{C^{\prime}_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\int^{+\infty}_{-\infty}\left|e^{\prime(\prime)}-m_{E^{\prime(\prime)}}\right|^{\frac{pN}{2}-1}$
$\displaystyle\times\exp\left[-\frac{N\left(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]$
$\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}-m_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right){\rm
d}m_{E^{\prime(\prime)}}.$

For practical numerical integration of (17), the double-infinite range of
$m_{E^{\prime(\prime)}}$ can be limited to, say,
$[\mu_{E^{\prime(\prime)}}-5\sigma_{E^{\prime(\prime)}}/\sqrt{N},\mu_{E^{\prime(\prime)}}+5\sigma_{E^{\prime(\prime)}}/\sqrt{N}]$.

For sufficiently large $N$, the field $E^{\prime(\prime)}$ and its sample mean
$M_{E^{\prime(\prime)}}$ are approximately independent (sharing $N-1$ degrees
of freedom), whence $E^{\prime(\prime)}-M_{E^{\prime(\prime)}}$ is then also
Gaussian with zero mean and standard deviation
$\sigma^{*}_{E^{\prime(\prime)}}\simeq\sigma_{E^{\prime(\prime)}}\sqrt{1+1/N}$.
Thus, (LABEL:eq:pdfE_BesselK_determbias) with $\sigma_{E^{\prime(\prime)}}$
replaced by $\sigma^{*}_{E^{\prime(\prime)}}$ serves as a first-order
approximation to the pdf (17) for large $N$.

####  II.1.2 Standardized field

For comparison of an experimental distribution against a theoretical sampling
distribution, it is necessary to standardize the experimentally obtained
$E^{\prime(\prime)}$. Then, rather than determining the pdf of
$E^{\prime(\prime)}$, we require the pdf of the dimensionless variate

$\displaystyle
X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{E^{\prime(\prime)}-M_{E^{\prime(\prime)}}}{S_{E^{\prime(\prime)}}}$
(18)

i.e., for uncertain sample values of $M_{E^{\prime(\prime)}}$ and
$S_{E^{\prime(\prime)}}$ that are unknown a priori. Their values are often to
be estimated from the same limited sample data set of $E^{\prime(\prime)}$.
For Gaussian $E^{\prime(\prime)}$, the $M_{E^{\prime(\prime)}}$ and
$S_{E^{\prime(\prime)}}$ are statistically independent and their sampling
distributions are Gaussian and $\chi_{pN-1}$, respectively [30]. We further
assume that the fluctuations of $M_{E^{\prime(\prime)}}$ are small compared to
those of $S_{E^{\prime(\prime)}}$ and, a fortiori, $E^{\prime(\prime)}$.

It is shown in Sec. A.1 that $X^{\prime(\prime)}$ for
$M_{E^{\prime(\prime)}}=0$ exhibits a Student $t$ distribution [31] with
$pN-1$ degrees of freedom, i.e.,

$\displaystyle
f_{X^{\prime(\prime)}}(x^{\prime(\prime)};N)=C_{X^{\prime(\prime)}}\left(1+\frac{{x^{\prime(\prime)}}^{2}}{pN-1}\right)^{-{pN}/{2}}$
(19)

with
$X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}E^{\prime(\prime)}/S_{E^{\prime(\prime)}}$
for $\langle E^{\prime(\prime)}\rangle=0$, where the sample value
$x^{\prime(\prime)}=e^{\prime(\prime)}/s_{E^{\prime(\prime)}}$ is estimated as
a (dimensionless) ratio of the two sample values $e^{\prime(\prime)}$ and
$s_{E^{\prime(\prime)}}$, and where

$\displaystyle
C_{X^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\Gamma\left(\frac{pN}{2}\right)}{\Gamma\left(\frac{pN-1}{2}\right)\sqrt{(pN-1)\pi}}.$
(20)

The pdf (19) can be used to compare the empirical distribution of the measured
standardized complex field against its ideal theoretical sampling
distribution, particularly when $N$ is relatively small ($N\sim 40$ or less).
For multi-scattering contributions, the value of $N$ can be associated with
the number of scattering centers surrounding the receiver and/or the states
during the motion of the environment around a momentarily stationary receiver.

Without loss of generality, we further consider a deterministically unbiased
field ($\mu_{E^{\prime(\prime)}}=0$). Fig. 1b shows the pdf of
$X^{\prime(\prime)}$ for selected values of $N$. Again, only the positive half
of the distribution is shown. It can be seen that, compared to the asymptotic
Gauss normal pdf, differences with $f_{E^{\prime(\prime)}}$ in Fig. 1a occur
mainly near the origin and in the tails.

###  II.2 Incoherent detection

Of significant practical importance, particular for measurements at optical
wavelengths, is the case where the random field is sampled incoherently, i.e.,
by square-law detection of the electric or magnetic power, energy or intensity
devoid of phase information. Yet one may wish to extract the sampling pdf of
the complex-valued electric or magnetic field from such a scalar measurement.
To this end, we borrow results derived in Sec. III.1, and restrict further
analysis to the case of a circular $f_{E}(e)$. (For an elliptic $f_{E}(e)$,
data for a second linearly independent quadratic form of $E^{\prime}$ and
$E^{\prime\prime}$ are needed, e.g.,
${E^{\prime}}^{2}-{E^{{\prime\prime}}}^{2}$.) From the derivation in Sec.
III.1, it follows that the sampling pdf of
$U^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}{E^{{\prime(\prime)}}}^{2}$
is (LABEL:eq:samplepdfU_final) after replacing $p$ with $p/2$, because the
ensemble pdf of $U^{\prime(\prime)}|S_{U^{\prime(\prime)}}$ is $\chi^{2}_{p}$
whereas $f_{U^{\prime(\prime)}}$ has a $\chi^{2}_{pN-1}$ sampling pdf. With
the variate transformation
$f_{E^{\prime(\prime)}}(e^{\prime(\prime)})=2|e^{\prime(\prime)}|f_{U^{\prime(\prime)}}(u^{\prime(\prime)}=e^{{\prime(\prime)}^{2}})$,
we arrive at

$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle
C^{\prime\prime}_{E^{\prime(\prime)}}\left(\frac{|e^{\prime(\prime)}|}{\sqrt{\sigma_{U}}}\right)^{\frac{1}{2}[p(N+1)-3]}$
$\displaystyle\times
K_{\frac{1}{2}[p(N-1)-1]}\left(\sqrt{\sqrt{{2p}}\left({p}N-1\right)}\frac{|e^{\prime(\prime)}|}{\sqrt{\sigma_{U}}}\right).$

For detection of a Cartesian component of intensity or power, i.e., $p=1$,
(LABEL:eq:samplepdfE_incoh) is equivalent with (13).

##  III Field intensity (energy density, power)

For the field intensity $U=U^{\prime}+U^{\prime\prime}=|E|^{2}$ – as well as
for the energy density or power, which are proportional to $U$ – we can
determine its pdf either on the basis of measured in-phase and/or quadrature
components of the field (i.e., coherent detection, e.g., using a vector
network analyzer), or measured directly using a square-law detector (i.e.,
incoherent detection, e.g., using a power meter, spectrum analyzer, field
probe, etc.). We determine sampling pdfs for both cases.

###  III.1 Incoherent detection

####  III.1.1 Non-standardized intensity

In many practical cases, the intensity or energy density is measured or
perceived through a square-law detector or perception process. Compared to
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ for coherent detection, the
pertinent conditional pdf (cpdf) is now $f_{U|S_{U}}$. On account of the CLT,
the sampling cpdf for $U|S_{U}$ is $\chi^{2}_{2p}$, i.e.,

$\displaystyle
f_{U|S_{U}}(u|s_{U})=\frac{p^{p/2}}{\Gamma(p)~{}s_{U}}\left(\frac{u}{s_{U}}\right)^{p-1}\exp\left(-\sqrt{p}\frac{u}{s_{U}}\right).$
(22)

The sampling pdf of $S_{U}$, associated with an underlying circular Gauss
normal $E$, is a $\chi^{2}_{2pN-1}$ pdf, i.e.,

$\displaystyle f_{S_{U}}(s_{U};N)$ $\displaystyle=$
$\displaystyle\frac{\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left(pN-\frac{1}{2}\right)}}{\Gamma\left(pN-\frac{1}{2}\right)\thinspace\sigma_{S_{U}}}\left(\frac{s_{U}}{\sigma_{S_{U}}}\right)^{pN-\frac{3}{2}}$
(23)
$\displaystyle\times\exp\left(-\sqrt{pN-\frac{1}{2}}~{}\frac{s_{U}}{\sigma_{S_{U}}}\,\right)$

with

$\displaystyle\sigma_{S_{U}}=\frac{\sigma_{U}}{\sqrt{pN-\frac{1}{2}}}.$ (24)

Performing a similar calculation as in Sec. II.1.1 and again using [25,
(3.471.9)] yields the sampling pdf of $U$ as a marginal pdf of
$f_{U,S_{U}}(u,s_{U};N)$. A Bessel $K$ distribution is again obtained, but
compared to (13) it is now of a different type (i.e., the exponent of the
power and the order of the Bessel function differ by a different amount),
viz.,

$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{U|S_{U}}(u|s_{U})f_{S_{U}}(s_{u};N){\rm
d}s_{U}$ $\displaystyle=$
$\displaystyle\frac{C_{U}}{\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{\frac{1}{2}\left[p(N+1)-\frac{5}{2}\right]}$
$\displaystyle\times
K_{p(N-1)-\frac{1}{2}}\left(2\sqrt{\sqrt{p}\left(pN-\frac{1}{2}\right)}\sqrt{\frac{u}{\sigma_{U}}}\right)$

with normalization constant $C_{U}$ given by

$\displaystyle C_{U}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{2}{\Gamma(p)\Gamma\left(pN-\frac{1}{2}\right)}p^{\frac{1}{4}\left[p(N+1)-\frac{1}{2}\right]}$
(27)
$\displaystyle\times\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left[p(N+1)-\frac{1}{2}\right]}.$

Again, (LABEL:eq:samplepdfU_incoh_BesselK) is one of McKay’s Bessel $K$
distributions. It is remarkable that for the case $p=1$, the same type of
Bessel $K$ distribution (LABEL:eq:samplepdfU_incoh_BesselK) has been obtained
in the context of sea echo for microwave radar [2], but as an ensemble pdf of
$U_{\alpha}$ based on different starting assumptions, viz., for a finite
random walk in the complex plane that assumes a randomly fluctuating and large
but finite number of steps (independent scattering contributions) that is
distributed according to a negative binomial distribution, as a discretization
of a gamma distribution for continuous stepping. For general values of $p$,
the pdf (LABEL:eq:samplepdfU_incoh_BesselK) was also obtained in [23] as a
limit distribution for imperfect reverberation based on a Bayesian model for a
physical process of omnidirectional scattering. Hence, the present derivation
of (LABEL:eq:samplepdfU_incoh_BesselK) shows that Bessel $K$ distributions are
far more universal than previously thought, as they can arise under the much
less restrictive condition of a mere small-sample effect for an underlying
Gaussian field, as opposed to a need for any functional form of a priori
distributions for a fluctuating $\nu$, whether chosen ad hoc or otherwise.
However, any apparent departure from an underlying Gauss normal field
distribution must be interpreted with due care and does not necessarily point
to physical nonlinearity. The identification of Bessel $K$ distributions as
sampling distributions for small sample sets of received power is supported by
measurements of the evolution of the distribution function of pulsed energy in
a reverberant cavity [32]. Such evolution is characterized by a steady growth
in the number of multipath components and, hence, $\nu$.

Figs. 2a, 3a, and 4a show the pdf (LABEL:eq:pdfU_BesselK) of incoherently
detected Cartesian ($p=1$), planar ($p=2$) and total ($p=3$) field
intensities, respectively, for selected values of $N$. The heavier tail and
the sharper peak (mode) of the distribution at smaller values of $N$ are
characteristic features. Fig. 5a shows the standard deviation of
(LABEL:eq:pdfU_BesselK) for these three cases.

####  III.1.2 Standardized intensity

For comparison with the empirical pdf, we consider the sampling pdf of the
ratio of the two random variables $U$ and $S_{U}$, viz.,

$\displaystyle W\stackrel{{\scriptstyle\Delta}}{{=}}\frac{U}{S_{U}}.$ (28)

The calculation of $f_{W}(w;N)$ is detailed in Sec. A.2. The result is a
Fisher-Snedecor $F$ distribution

$\displaystyle f_{W}(w;N)$ $\displaystyle=$
$\displaystyle\frac{C_{W}}{s_{W}}\left(\frac{w}{s_{W}}\right)^{p-1}$ (29)
$\displaystyle\times\left(1+\frac{\sqrt{p}}{pN-\frac{1}{2}}\thinspace
w\right)^{-[p(N+1)-\frac{1}{2}]}$

as a counterpart of (19), where

$\displaystyle
C_{W}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{p^{p/2}}{\left(pN-\frac{1}{2}\right)^{p}}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$
(30)

The pdf (29) for the incoherently detected and sample-standardized Cartesian
($p=1$), planar ($p=2$) and total ($p=3$) field intensities $W_{\alpha}$,
$W_{\rm t}$, and $W$ is shown in Figs. 2b, 3b, and 4b, respectively, at
selected values of $N$. Comparison between the $K$ distributed intensities
with their $F$ distributed sample-standarized values in Figs. 2–4 shows that,
for a given value of $N$, the sampling pdf of $W$ exhibits a more pronounced
spread than for $U$, owing to the larger uncertainty of $W$ caused by
fluctuations of $S_{U}$. Fig. 5b shows the standard deviation of (29) for the
three cases, indicating much larger standard deviations for small $N$ compared
to those in Fig. 5a for nonstandardized intensities.

####  III.1.3 Non-standardized vs. non-normalized intensity

In Sec. III.1.1, $f_{U}(u;N)$ was obtained via the sample-standardized variate
$U/S_{U}$ in the cpdf (22). However, since $\mu_{U}\not=0$, we could have
equally used the sample-normalized variate $U/M_{U}$ for the cpdf to reference
the data in this case. To this end, instead of (III.1.1), we now use

$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{U|M_{U}}(u|m_{U})f_{M_{U}}(m_{u};N){\rm
d}m_{U}~{}~{}~{}~{}$ (31)

with [30]

$\displaystyle f_{U|M_{U}}(u|m_{U})$ $\displaystyle=$
$\displaystyle\frac{p^{p}}{\Gamma(p)~{}m_{U}}\left(\frac{u}{m_{U}}\right)^{p-1}\exp\left(-{p}\frac{u}{m_{U}}\right)~{}~{}~{}~{}~{}$
(32) $\displaystyle f_{M_{U}}(m_{U};N)$ $\displaystyle=$
$\displaystyle\frac{\left(pN\right)^{\frac{pN}{2}-1}}{\Gamma\left(pN\right)\thinspace\sigma_{M_{U}}}\left(\frac{m_{U}}{\sigma_{M_{U}}}\right)^{pN-1}$
(33)
$\displaystyle\times\exp\left(-\sqrt{pN}~{}\frac{m_{U}}{\sigma_{M_{U}}}\,\right)$

and $\mu_{U}=m_{U}$, $\sigma_{U}=\sqrt{N}\sigma_{M_{U}}$, whence (31) becomes

$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\frac{C^{\prime}_{U}}{\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{\frac{1}{2}p(N+1)-1}$
(34) $\displaystyle\times
K_{p(N-1)}\left(2p^{\frac{3}{4}}\sqrt{N}\sqrt{\frac{u}{\sigma_{U}}}\right)~{}~{}~{}~{}$

with normalization constant $C^{\prime}_{U}$ given by

$\displaystyle
C^{\prime}_{U}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2~{}p^{\frac{3}{4}p(N+1)}~{}N^{\frac{1}{2}p(N+1)}}{\Gamma(p)\Gamma\left(pN\right)}.$
(35)

The form (34) is readily re-expressed in terms of $\mu_{U}$ by replacing
$\sigma_{U}$ with $\mu_{U}/\sqrt{p}$. Comparing (34)–(35) to
(LABEL:eq:pdfU_BesselK)–(27) shows that the non-normalized cpdf (34) is
retrieved by replacing $N-(2p)^{-1}$ in the non-standardized cpdf
(LABEL:eq:pdfU_BesselK) by $N$, i.e., resulting in a marginal increase of the
number of independent samples. In other words, the non-normalized cpdf is
marginally closer to the asymptotic ensemble pdf ($N\rightarrow+\infty$) than
the non-standardized cpdf, resulting in slightly smaller uncertainties. This
result is made plausible by the fact that the uncertainty of the sampling mean
value is smaller than for the sampling standard deviation. It is worth
emphasizing that, even though $f_{U}(u;N)$ is for the (dimensioned) energy
density $U$, the chosen route for arriving at this pdf – i.e., whether via
intermediary standardization or normalization in the cpdf of $U$ – has an
effect on the number of degrees of freedom of the end result but not on the
functional form. In summary, whenever $pN\gg 1$ (i.e., for most practical
cases), standardization and normalization yield indistinguishable final
results.

####  III.1.4 Intensity of biased field

Instead of (22), the sampling cpdf of $U|S_{U}$ for the incoherently detected
intensity of a biased field is a generalization of the so-called modified
Nakagami–Rice $m$ distribution, given in self-sufficient form by

$\displaystyle f_{U_{\alpha}|S_{U_{\alpha}}}(u_{\alpha}|s_{U_{\alpha}})$
$\displaystyle=$
$\displaystyle\frac{\sqrt{1+2k_{U_{\alpha}}}}{s_{U_{\alpha}}}$ (36)
$\displaystyle\times\exp\left(-\sqrt{1+2k_{U_{\alpha}}}\frac{u_{\alpha}+u_{\alpha
0}}{s_{U_{\alpha}}}\right)$ $\displaystyle\times
I_{0}\left(2\sqrt{1+2k_{U_{\alpha}}}\frac{\sqrt{u_{\alpha
0}u_{\alpha}}}{s_{U_{\alpha}}}\right)$

with

$\displaystyle
k_{U_{\alpha}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{|E_{\alpha
0}|^{2}}{\langle|E_{\alpha}-E_{\alpha
0}|^{2}\rangle}=\frac{1-n^{2}_{U_{\alpha}}+\sqrt{1-n^{2}_{U_{\alpha}}}}{n^{2}_{U_{\alpha}}}$
(37)

where $n_{U_{\alpha}}$ is a sample value of
$\nu_{U_{\alpha}}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma_{U_{\alpha}}/\mu_{U_{\alpha}}$.
Since $\sigma_{U_{\alpha}}=\sigma_{U_{\alpha}-u_{\alpha 0}}$ for any constant
(deterministic) value $u_{\alpha 0}$, the pdf
$f_{S_{U_{\alpha}}}(s_{U_{\alpha}};N)$ is still given by (23) with $p=1$.
Thus,

$\displaystyle f_{U_{\alpha}}(u_{\alpha};N)$ $\displaystyle=$
$\displaystyle\frac{C^{\prime}_{U_{\alpha}}}{\sigma_{U_{\alpha}}}\int^{+\infty}_{0}\sqrt{1+2k_{U_{\alpha}}}\left(\frac{s_{U_{\alpha}}}{\sigma_{{U_{\alpha}}}}\right)^{N-\frac{5}{2}}$
(38)
$\displaystyle\times\exp\left[-\left({N-\frac{1}{2}}\right)~{}\frac{s_{U_{\alpha}}}{\sigma_{{U_{\alpha}}}}\,\right]$
$\displaystyle\times\exp\left(-\sqrt{1+2k_{U_{\alpha}}}\frac{u_{\alpha}+u_{\alpha
0}}{s_{U_{\alpha}}}\right)$ $\displaystyle\times
I_{0}\left(2\sqrt{1+2k_{U_{\alpha}}}\frac{\sqrt{u_{\alpha
0}u_{\alpha}}}{s_{U_{\alpha}}}\right){\rm d}s_{U_{\alpha}}.~{}~{}~{}~{}~{}$

Note that $k_{U_{\alpha}}$ depends implicitly on $s_{U_{\alpha}}$: to first
approximation, $k_{U_{\alpha}}\simeq s_{U_{\alpha}}/\mu_{U_{\alpha}}$.

###  III.2 Coherent detection

In the case of estimating $f_{U}(u;N)$ from measurements of $E$, the sampling
pdf takes a different form from that for incoherent detection, as we show
next. Knowing the pdf of $E_{\alpha}=E^{\prime}_{\alpha}-{\rm
j}E^{\prime\prime}_{\alpha}$, we can derive the pdf of
$U=|E|^{2}=p|E_{\alpha}|^{2}$, where
$|E_{\alpha}|^{2}\equiv{E^{\prime}_{\alpha}}^{2}+{E^{\prime\prime}_{\alpha}}^{2}=U_{\alpha}$
is the intensity of a Cartesian field component, and similarly for the
electric or magnetic power $P\propto U$. Since
$f_{U^{\prime(\prime)}_{\alpha}}(u^{\prime(\prime)}_{\alpha})\propto
f_{E^{\prime(\prime)}_{\alpha}}(e^{\prime(\prime)}_{\alpha}=\sqrt{u^{\prime(\prime)}_{\alpha}})/\sqrt{u^{\prime(\prime)}_{\alpha}}$,
we obtain

$\displaystyle f_{U_{\alpha}}(u_{\alpha};N)$ $\displaystyle=$ $\displaystyle
C_{U_{\alpha}}\int^{u_{\alpha}}_{0}f_{U^{\prime}_{\alpha}}(x;N)f_{U^{\prime\prime}_{\alpha}}(u_{\alpha}-x;N){\rm
d}x$ $\displaystyle=$ $\displaystyle
C_{U_{\alpha}}\int^{u_{\alpha}}_{0}\frac{f_{E^{\prime}_{\alpha}}(\sqrt{x};N)f_{E^{\prime\prime}_{\alpha}}(\sqrt{u_{\alpha}-x};N)}{\sqrt{x}\sqrt{u_{\alpha}-x}}{\rm
d}x$

where $C_{U_{\alpha}}$ is a normalization constant. The general expression
(LABEL:eq:fUCart) makes allowance for the fact that $E^{\prime}_{\alpha}$ and
$E^{\prime\prime}_{\alpha}$ may, in principle, have different pdfs
(functionally and/or parametrically), although in most cases the pdf of
$E_{\alpha}$ is circular, i.e.,
$f_{E^{\prime}_{\alpha}}=f_{E^{\prime\prime}_{\alpha}}$. The sample pdfs of
$U_{\rm t}\equiv U_{x}+U_{y}$ and $U\equiv U_{x}+U_{y}+U_{z}$ follow similarly
from two- and threefold convolutions of (LABEL:eq:fUCart), respectively. Fig.
6 shows the pdf (LABEL:eq:fUCart) for selected values of $N$.

If ${U_{\alpha}}$ is to be compared with measured data, then the sampling pdf
can be similarly calculated from $f_{X^{\prime}_{\alpha}}$ and
$f_{X^{\prime\prime}_{\alpha}}$ as

$\displaystyle f_{W_{\alpha}}(w_{\alpha})$ $\displaystyle=$ $\displaystyle
C_{W_{\alpha}}\int^{w_{\alpha}}_{0}\frac{f_{X^{\prime}_{\alpha}}(\sqrt{x};N)f_{X^{\prime\prime}_{\alpha}}(\sqrt{w_{\alpha}-x};N)}{\sqrt{x}\sqrt{w_{\alpha}-x}}{\rm
d}x$

where
$W_{\alpha}\stackrel{{\scriptstyle\Delta}}{{=}}U_{\alpha}/S_{U_{\alpha}}$.

##  IV Field amplitude

###  IV.1 Incoherent detection

####  IV.1.1 Non-standardized field amplitude

The sampling pdf of
$A\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{{E^{\prime}}^{2}+{E^{\prime\prime}}^{2}}$
follows in a manner similar to that for $U$. In this case, $A|S_{A}$ has a
$\chi_{2p}$ cpdf, given in self-sufficient form as

$\displaystyle f_{A|S_{A}}(a|s_{A})$ $\displaystyle=$
$\displaystyle\frac{2\left[p-\left(\frac{\Gamma(p+\frac{1}{2})}{\Gamma(p)}\right)^{2}\right]^{p}\thinspace}{\Gamma(p)\thinspace
s_{A}}\left(\frac{a}{s_{A}}\right)^{2p-1}$
$\displaystyle\times\exp\left\\{-\left[p-\left(\frac{\Gamma(p+\frac{1}{2})}{\Gamma(p)}\right)^{2}\right]\left(\frac{a}{s_{A}}\right)^{2}\right\\},$

which has the Rayleigh distribution as a special case for $p=1$, while $S_{A}$
has the $\chi_{2pN-1}$ distribution

$\displaystyle f_{S_{A}}(s_{A};N)$ $\displaystyle=$
$\displaystyle\frac{2\thinspace\left[pN-\frac{1}{2}-\left(\frac{\Gamma(pN)}{\Gamma(pN-\frac{1}{2})}\right)^{2}\right]^{pN-\frac{1}{2}}}{(pN-\frac{1}{2})~{}\Gamma(pN-\frac{1}{2})\thinspace\sigma_{S_{A}}}\left(\frac{s_{A}}{\sigma_{S_{A}}}\right)^{2(pN-1)}$
$\displaystyle\times\exp\left\\{-\left[pN-\frac{1}{2}-\left(\frac{\Gamma(pN)}{\Gamma(pN-\frac{1}{2})}\right)^{2}\right]\left(\frac{s_{A}}{\sigma_{S_{A}}}\right)^{2}\right\\}$

with $\sigma_{S_{A}}$ and $\sigma_{A}$ related via

$\displaystyle\sigma_{S_{A}}=\sigma_{A}\sqrt{1-\frac{1}{pN-\frac{1}{2}}\left(\frac{\Gamma(pN)}{\Gamma\left(pN-\frac{1}{2}\right)}\right)^{2}}.$
(44)

The sampling pdf of $A$ is obtained as a marginal pdf of
$f_{A,S_{A}}(a,s_{A};N)$ and is again a Bessel $K$ distribution, but of yet
another type compared to (13) and (LABEL:eq:samplepdfU_incoh_BesselK), viz.,

$\displaystyle
f_{A}(a;N)=\int^{+\infty}_{0}f_{A|S_{A}}(a|s_{A})f_{S_{A}}(s_{A};N){\rm
d}s_{A}$
$\displaystyle=\frac{C_{A}}{\sigma_{A}}\left(\frac{a}{\sigma_{A}}\right)^{p(N+1)-\frac{3}{2}}$
$\displaystyle\times
K_{p(N-1)-\frac{1}{2}}\left(2\sqrt{\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]\left(pN-\frac{1}{2}\right)}\frac{a}{\sigma_{A}}\right)$

where $C_{A}$ is obtained, with the aid of [25, (6.561.16)], as

$\displaystyle C_{A}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{4}{\Gamma(p)\Gamma\left(pN-\frac{1}{2}\right)}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{\frac{1}{2}\left[p(N+2)-1\right]}$
(46)
$\displaystyle\times\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left[p(N+1)-1\right]}.$

The pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $p=1$ has been obtained in [3]
for the abovementioned scenario of a random walk with fluctuating number of
steps with negative binomial distribution. For general $p$, this pdf has been
retrieved as a limit distribution for imperfect reverberation in [23].

The pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $p=1,2$ and $3$ is shown in
Figs. 7a, 8a, and 9a, respectively, for selected values of $N$. Fig. 10a shows
the corresponding standard deviations of (29). Compared to Fig. 5a, the
increase of the standard deviations with decreasing $N$ is generally smaller
and less dependent on dimensionality.

####  IV.1.2 Standarized field amplitude

For comparison with the empirical pdf of a field amplitude measured by an
electric or magnetic field probe, we consider the sampling pdf of the ratio of
the variates $A$ and $S_{A}$, viz.,

$\displaystyle V\stackrel{{\scriptstyle\Delta}}{{=}}\frac{A}{S_{A}}.$ (47)

The calculation of $f_{V}(v;N)$ is detailed in Sec. A.3, where the final
result is shown to be

$\displaystyle f_{V}(v;N)$ $\displaystyle=$ $\displaystyle C_{V}~{}{v^{2p-1}}$
(48)
$\displaystyle\times{\left[1+\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{pN-\frac{1}{2}}\thinspace
v^{2}\right]^{-[p(N+1)-\frac{1}{2}]}}$

with

$\displaystyle
C_{V}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2}{\left(pN-\frac{1}{2}\right)^{p}}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{p}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$
(49)

The sampling pdf (48) can be referred to as a root-$F$ distribution and
constitutes a counterpart of (19) and (29). Fig. 10b shows its standard
deviation.

####  IV.1.3 Amplitude of biased field

Unfortunately, unlike (36), the cpdf
$f_{A_{\alpha}|S_{A_{\alpha}}}(a_{\alpha}|s_{A_{\alpha}})$ cannot be expressed
in self-sufficient closed form, because $\sigma_{A_{\alpha}}$ for the
Nakagami–Rice cpdf of $A$ depends in a complicated manner on
$\sigma_{E^{\prime(\prime)}}$. Therefore, we do not further pursue this case.
A simpler approach is to derive $f_{A}(a;N)$ via variate transformation of
(38).

###  IV.2 Coherent detection

Using the variate transformation $A=\sqrt{U}$, the pdf of the local field
magnitude $A=\sqrt{A^{2}_{x}+A^{2}_{y}+A^{2}_{z}}\equiv|E|\propto\sqrt{U}$
follows. With
$A_{\alpha}\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{{E^{\prime}_{\alpha}}^{2}+{E^{\prime\prime}_{\alpha}}^{2}}$,
(LABEL:eq:fUCart) yields

$\displaystyle f_{A_{\alpha}}(a_{\alpha};N)$ (50) $\displaystyle=$
$\displaystyle a_{\alpha}\thinspace f_{U_{\alpha}}(u_{\alpha}=a^{2}_{\alpha})$
$\displaystyle=$ $\displaystyle C_{A_{\alpha}}\thinspace
a_{\alpha}\int^{a^{2}_{\alpha}}_{0}\frac{f_{E^{\prime}_{\alpha}}(\sqrt{x};N)f_{E^{\prime\prime}_{\alpha}}(\sqrt{a^{2}_{\alpha}-x};N)}{\sqrt{x}\sqrt{a^{2}_{\alpha}-x}}{\rm
d}x~{}~{}~{}~{}~{}$

where $C_{A_{\alpha}}$ is a normalization constant. The pdf for $A_{\alpha}$
is shown for selected values of $N$ in Fig. 11 and for $A$ in Fig. 12.

In fact, one may consider the amplitude of the sampling field itself as a
marginal of the joint sampling pdf
$f_{E^{\prime},E^{\prime\prime}}(e^{\prime},e^{\prime\prime};N)$, followed by
variate transformation to the amplitude
$A=\sqrt{{E^{\prime}}^{2}+{E^{\prime\prime}}^{2}}$ and phase
$\Phi=\tan^{-1}(E^{\prime\prime}/E^{\prime})$. Assuming that $E^{\prime}$ and
$E^{\prime\prime}$ are statistically independent (which, strictly, requires
them to be normally distributed) so that
$f_{E^{\prime},E^{\prime\prime}}(e^{\prime},e^{\prime\prime};N)=f_{E^{\prime}}(e^{\prime};N)f_{E^{\prime\prime}}(e^{\prime\prime};N)$,
then

$\displaystyle f_{A}(a;N)$ $\displaystyle=$
$\displaystyle\frac{C_{A}}{\sigma_{A}}a^{{pN}-{1}}\int^{\pi}_{-\pi}\left({|\cos\phi-(a_{0}/a)|}{|\sin\phi|}\right)^{\frac{pN}{2}-1}$
(51) $\displaystyle\times K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|a\cos\phi-
a_{0}|}{\sigma_{E^{\prime}}}\right)$ $\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|a\sin\phi|}{\sigma_{E^{\prime\prime}}}\right){\rm
d}\phi$

where $a_{0}\stackrel{{\scriptstyle\Delta}}{{=}}\mu_{E^{\prime}}$ with
$\mu_{E^{\prime\prime}}=0$.

##  V Extension to non-Gaussian ensemble distributions of the field: Iterative
Bayesian scheme

From the theorem of total probability, it follows that

$\displaystyle f_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}\propto
f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}~{}f_{S_{E^{\prime(\prime)}}}.$
(52)

For non-Gaussian $E^{\prime(\prime)}$, for which the prior pdf
$f_{S_{E^{\prime(\prime)}}}$ is more difficult to determine than in the
exposition given before, (52) can be used in an iterative process by means of
an update equation for $f_{S_{E^{\prime(\prime)}}}$, where the latter can be
assigned an initial $\chi_{N-1}$ distribution
$f^{(0)}_{S_{E^{\prime(\prime)}}}$. This prior distribution, together with the
non-Gaussian $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$, then allows for
calculating the posterior $f_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}$ that
serves as the “prior” pdf for the next iteration. Explicitly, denoting the
$i$th iteration in this scheme by the superscript ‘$(i)$’, the iteration
process is specified by

$\displaystyle
f^{(i+1)}_{S_{E^{\prime(\prime)}}}=f^{(i)}_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}$
(53)

for $i=0,1,\ldots$, thus yielding
$f^{(n)}_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ after $n$ iterations,
with similar relations for the intensity and amplitude.

In general, (52)–(53) can be used even with empirical pdfs
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$. A similar equation and
procedure can be used for determining the intensity or magnitude for
incoherent detection. This outline of a procedure is sketched here for sake of
completeness and guidance. However, no explicit results are included because
they require knowledge of $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$
which must follow from a separate investigation, e.g., from experiment.
Further study is needed with respect to the demonstration of convergence of
(53) to a stable pdf.

##  VI Confidence intervals

$95\%$-confidence interval boundaries from sampling pdfs of $E$, $U$, $A$ as
well as $X$, $W$, $V$ and their components, are compared to corresponding
boundaries for the ensemble pdfs in Figs. 13, 14 and 15, respectively. The
Figures show that $N\sim 10$ or larger is needed in order for the sampling
confidence interval boundaries to be close to the ensemble boundaries. For
example, for an overmoded Fabry-Pérot resonator, a resonator length of the
order of five wavelengths or larger is needed to achieve this.

##  VII Conclusion

In this paper, we studied sampling distributions for the analytic complex-
valued field, the intensity (energy density, power) and the magnitude for
Gaussian statistically homogeneous random electromagnetic waves. The main
results are (13)–(LABEL:eq:samplepdfE_coh_BesselK_determbias), (17), (19),
(LABEL:eq:samplepdfU_incoh_BesselK), (29), (LABEL:eq:samplepdfA_incoh_BesselK)
and (48). A common feature is that lowering the number of degrees of freedom
characterizing sampling distributions results in their tails becoming heavier
and a consequent associated increase of widths of confidence intervals
compared to the Gauss normal, $\chi^{2}_{2p}$ and $\chi_{2p}$ ensemble
distributions, respectively. Furthermore, it was shown that standardized
quantities (i.e., the sampled random field, magnitude or intensity divided by
its own sample standard deviation, whereby the latter is itself considered as
a random variable whose sample value is calculated from the sample data set
itself) exhibit sampling pdfs that are characterized by a wider spread than
for the case where that quantity has an a priori known (ensemble) standard
deviation. The reason is that in the former case the ratio of random variables
defines a bivariate sampling distribution whose (univariate) marginal pdf is
sought, whereas in the latter case the sampling pdf is univariate from the
outset. The differences in sampling distributions arising from choosing
normalization rather than standardization were shown to be negligible in all
practical cases.

While the focus in this paper was on the simplest of a priori chosen
conditional pdfs for the local instantaneous field in unbounded media,
extension to some more general cases and boundary-value problems is
straightforward. For example, compound exponential distributions for
anisotropic ideal random fields near a conducting or dielectric boundary [9,
33] can be used instead, leading to the total (vectorial) intensity and
amplitude of 3-D random fields near a planar isotropic surface by simple
superposition.

##  VIII Acknowledgements

This work was supported in part by the Physical Programme of the U.K. National
Measurement System Policy Unit 2006–2009. I wish to thank Dr. P. Harris (NPL)
for comments to the manuscript, Mr. S. Burbridge (Imperial College) for
assistance with SGI Altix high-performance computation relating to Fig. 6, and
Dr. M. Little (University of Oxford, U.K.) for discussions relating to ref.
[29].

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##  Appendix A Sampling distributions of field, intensity and amplitude

###  A.1 Field

Here we derive the one-dimensional sampling distribution for the intensity
$|E|^{2}$ or energy density $U\propto|E|^{2}$ of a random statistically
homogeneous Cartesian or vector electric (or, by extension, magnetic) field
$E$.

The pdf of an ideal random field $E$ is circular with independent and
identically distributed (i.i.d.) real (in-phase) and imaginary (quadrature)
components $E^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}{\rm Re}(E)$ and
$E^{\prime\prime}\stackrel{{\scriptstyle\Delta}}{{=}}{\rm Im}(E)$. As a
result, these components can be studied in isolation from each other. The
parent (ensemble) distribution of $E$ is a central Gauss normal distribution
($\langle E\rangle=0$). Therefore, the standardized $N$-point sample variance
$D_{N-1}\stackrel{{\scriptstyle\Delta}}{{=}}(N{-}1)\thinspace
S^{2}_{E^{\prime(\prime)}}/\sigma^{2}_{E^{\prime(\prime)}}=\sum^{N}_{i=1}(E^{\prime(\prime)}_{i}{-}M_{E^{\prime(\prime)}})^{2}/\sigma^{2}_{E^{\prime(\prime)}}$
exhibits a $\chi^{2}_{N{-}1}$ pdf, while
$D_{1}\stackrel{{\scriptstyle\Delta}}{{=}}(E^{\prime(\prime)}-M_{E^{\prime(\prime)}})^{2}/\sigma^{2}_{E^{\prime(\prime)}}$
has a $\chi^{2}_{1}$ pdf. Consequently, the ratio

$\displaystyle
X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{E^{\prime(\prime)}-M_{E^{\prime(\prime)}}}{S_{E^{\prime(\prime)}}}=\sqrt{\frac{D_{1}}{D_{N-1}/(N{-}1)}}$
(54)

has a Student $t$ sampling distribution with $N{-}1$ degrees of freedom,
whence for $E^{\prime(\prime)}$ itself,

$\displaystyle f_{E^{\prime(\prime)}}\left(e^{\prime(\prime)};N\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{N}{2}\right)}{\sqrt{\pi}~{}\Gamma\left(\frac{N-1}{2}\right)~{}\sqrt{N-1}}$
(55) $\displaystyle\times$
$\displaystyle\left[1+\frac{1}{N-1}\left(\frac{e^{\prime(\prime)}-m_{E^{\prime(\prime)}}}{s_{E^{\prime(\prime)}}}\right)^{2}\right]^{-N/2}$

for $N>1$, where $e^{\prime(\prime)}$, $m_{E^{\prime(\prime)}}$ and
$s_{E^{\prime(\prime)}}$ are sample values. Compared to the sample value
$s_{E^{\prime(\prime)}}$ of the parent pdf, the sampling pdf exhibits an
increased sampling standard deviation, viz.,
$\sqrt{(N{-}1)/(N{-}3)}s_{E^{\prime(\prime)}}$ for $N>3$.

Typically, the Student $t$ distribution arises in the characterization of the
sample mean, i.e., for $[(M_{E^{\prime(\prime)}}-\langle
E^{\prime(\prime)}\rangle)/(\sigma_{E^{\prime(\prime)}}/\sqrt{N})]/(S_{E^{\prime(\prime)}}/\sigma_{E^{\prime(\prime)}})$,
instead of
$[({E^{\prime(\prime)}}-M_{E^{\prime(\prime)}})/\sigma_{E^{\prime(\prime)}}]/(S_{E^{\prime(\prime)}}/\sigma_{E^{\prime(\prime)}})$
as in (54). Without lack of generality, however, we further use the simplified
definition
$X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}{E^{\prime(\prime)}}/{S_{E^{\prime(\prime)}}}$,
i.e., $M_{E^{\prime(\prime)}}=0$, because each sample set can always be
centralized by its own sample mean value while maintaining its sampling pdf.

The square of $X^{\prime(\prime)}$, represented as
$Y^{\prime(\prime)}/S_{Y^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}X^{{\prime(\prime)}^{2}}={E^{\prime(\prime)}}^{2}/S^{2}_{E^{\prime(\prime)}}$,
exhibits a $t^{2}$ sampling distribution that follows from (55) as

$\displaystyle f_{Y^{\prime(\prime)}}(y^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{N}{2}\right)}{2\sqrt{\pi}~{}\Gamma\left(\frac{N-1}{2}\right)~{}\sqrt{N-1}}$
(56)
$\displaystyle\times\left[\frac{y^{\prime(\prime)}}{s_{Y^{\prime(\prime)}}}\left(1+\frac{y^{\prime(\prime)}}{(N-1)\thinspace
s_{Y^{\prime(\prime)}}}\right)^{N}\right]^{-1/2}.$

which is a Fisher-Snedecor $F_{1,N-1}$ pdf of the ratio of the standard
$\chi^{2}_{1}$ variate $D_{1}$ and the standard $\chi^{2}_{N-1}$ variate
$D_{N-1}$ (both being statistically independent, because
$M_{E^{\prime(\prime)}}$ and $S_{E^{\prime(\prime)}}$ are independent for
Gauss normal $E^{\prime(\prime)}$), whereby each variate is divided by its
corresponding number of degrees of freedom, i.e.,

$\displaystyle t^{2}=F_{1,N-1}=\frac{\chi^{2}_{1}/1}{\chi^{2}_{N-1}/(N-1)}.$
(57)

###  A.2 Field intensity

To find the sampling distribution of a squared Cartesian or vectorial EM field
with $2p$ i.i.d. components $Y^{\prime(\prime)}_{i}$ (corresponding to a
$p$-dimensional analytic complex-valued field), i.e.,
$\sigma_{Y^{\prime(\prime)}_{i}}=\sigma^{2}_{X}$, we consider the ratio
$Z/D_{Z}$, where

$\displaystyle
Z\stackrel{{\scriptstyle\Delta}}{{=}}\frac{Y^{\prime}+Y^{\prime\prime}}{\sigma_{Y^{\prime(\prime)}}}=\sum^{p}_{i=1}\left(\frac{Y^{\prime}_{i}}{\sigma_{Y^{\prime}}}+\frac{Y^{\prime\prime}_{i}}{\sigma_{Y^{\prime\prime}}}\right)\sum^{2p}_{i=1}\frac{(X_{i}-m_{X})^{2}}{\sigma^{2}_{X}}~{}~{}~{}$
(58)

has a standard $\chi^{2}_{2p}$ distribution, on account of the addition
theorem for $2p$ i.i.d. standard $\chi^{2}_{1}$ variates, and

$\displaystyle
D_{Z}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{(2pN-1)~{}S^{2}_{X^{\prime(\prime)}}}{\sigma^{2}_{X^{\prime(\prime)}}}$
(59)

exhibits a standard $\chi^{2}_{2pN{-}1}$ distribution, with
$S^{2}_{X^{\prime(\prime)}}=$$\sum^{2pN}_{i=1}(X^{\prime(\prime)}_{i}{-}M_{X^{\prime(\prime)}})^{2}/(2pN{-}1)$.
Hence, their ratio $G$ is a scaled Fisher-Snedecor $F$ variate with
($2p,2pN-1)$ degrees of freedom:

$\displaystyle G\stackrel{{\scriptstyle\Delta}}{{=}}\frac{Z}{D_{Z}}$
$\displaystyle\equiv$
$\displaystyle\frac{2p}{2pN-1}\frac{Z/(2p)}{D_{Z}/(2pN-1)}$ (60)
$\displaystyle=$ $\displaystyle\frac{2p}{2pN-1}F_{2p,2pN-1}.$

With $\sigma_{|E|^{2}}=2\sqrt{p}\thinspace\sigma^{2}_{E^{\prime(\prime)}}$,
the ratio $Z/D_{Z}$ can be related to the standardized power or field
intensity as

$\displaystyle\frac{U}{S_{U}}=\frac{\sum^{2p}_{i=1}{E^{\prime}_{i}}^{2}}{2\sqrt{p}~{}S^{2}_{E^{\prime}}}$
$\displaystyle=$
$\displaystyle\frac{2pN-1}{2\sqrt{p}}\frac{\sum^{2p}_{i=1}(E^{\prime}_{i}/\sigma_{E^{\prime}})^{2}}{(2pN-1)~{}S^{2}_{E^{\prime}}/\sigma^{2}_{E^{\prime}}}$
(61) $\displaystyle=$ $\displaystyle\frac{2pN-1}{2\sqrt{p}}\frac{Z}{D_{Z}}.$

Thus, combining (60) and (61) yields

$\displaystyle
W\stackrel{{\scriptstyle\Delta}}{{=}}\frac{U}{S_{U}}=\sqrt{p}~{}F_{2p,2pN-1}.$
(62)

With (60), upon scaling the standard $F_{2p,2pN-1}$ distribution, the sampling
pdf of $G$ is

$\displaystyle f_{G}(g;N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}~{}\frac{g^{p-1}}{\left(1+g\right)^{pN-\frac{1}{2}+p}}~{}~{}~{}.$
(63)

For the square of the Cartesian in-phase field component $[{\rm Re}(E)]^{2}$
or the quadrature component $[{\rm Im}(E)]^{2}$ (i.e., for $p=1/2$), (63)
reduces to (56) as expected. Finally, from (62), the sampling distribution of
$W=U/S_{U}$ follows as

$\displaystyle f_{W}(w;N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}~{}\frac{p^{p/2}}{\left(pN-\frac{1}{2}\right)^{p}}$
(64)
$\displaystyle\times\frac{w^{p-1}}{\left(1+\frac{\sqrt{p}}{pN-\frac{1}{2}}\thinspace
w\right)^{pN-\frac{1}{2}+p}}.$

This represents the pdf of the standardized received sampled power $W$ at $w$,
i.e., for the ratio of sampled values $u$ and $s_{U}$.

In the limit $N\rightarrow+\infty$, the exponent $p(N+1)-\frac{1}{2}$ in (64)
reduces to $pN-\frac{1}{2}$; the prefactor
$\Gamma(pN-\frac{1}{2}+p)/\Gamma(pN-\frac{1}{2})=(pN-\frac{1}{2})\cdot\ldots\cdot(pN-\frac{3}{2}+p)$
for $p\geq 1$ becomes $(pN-\frac{1}{2})^{p}$; and $s_{U}$ approaches the
ensemble statistic $\sigma_{U}$, whence

$\displaystyle f_{W}(w;N)\rightarrow
f_{U}(u)=\frac{p^{p/2}}{\Gamma(p)\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{p-1}\exp\left(-\sqrt{p}\thinspace\frac{u}{\sigma_{U}}\right)$

which is the ensemble $\chi^{2}_{2p}$ limit pdf, as expected. The result
agrees with a well-known limit theorem from probability theory [7] stating
that $F_{m,n}(x)$ converges to $\chi^{2}_{m}(mx)$ when
$n/m{\rightarrow}+\infty$, i.e., $F_{q}(m,n)\rightarrow\chi^{2}_{q}(m)/m$ for
the corresponding quantiles. In the same limit $N{\rightarrow}{+}\infty$, the
asymptotic mean value of $W$ is

$\displaystyle\langle
W\rangle=\frac{2pN{-}1}{2pN{-}3}\sqrt{p}\rightarrow\sqrt{p}$ (66)

and the standard deviation for $pN>5/2$ is

$\displaystyle\sigma_{W}=\sqrt{\frac{2(2pN{-}1)^{2}(2pN{+}2p{-}3)}{2p(2pN{-}3)^{2}(2pN{-}5)}{p}}\rightarrow
1.$ (67)

Thus, we find that the coefficient of variation, i.e.,

$\displaystyle\frac{\sigma_{W}}{\langle{W}\rangle}=\sqrt{\frac{2pN{+}2p{-}3}{p(2pN{-}5)}}~{}\rightarrow\frac{1}{\sqrt{p}}$
(68)

has its value reduced to that for the $\chi^{2}_{2p}$ ensemble distribution
when $N{\rightarrow}+\infty$, whereas for small $N$ its value is substantially
larger, indicating larger relative uncertainty.

###  A.3 Field amplitude

For the standardized field magnitude
$V\stackrel{{\scriptstyle\Delta}}{{=}}A/S_{A}=\sqrt{(S^{2}_{V}/S_{W})~{}W}$,
with

$\displaystyle\frac{\sigma^{2}_{Z}}{\sigma_{W}}=\frac{\sigma^{2}_{A}}{\sigma_{A^{2}}}=\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{\sqrt{p}}$
(69)

valid for $\chi^{(2)}_{2p}$ distributions, from variate transformation of (64)
and noting that

$\displaystyle\frac{w}{s_{W}}=\left(\frac{v}{s_{V}}\right)^{2}=\frac{v^{2}}{s_{W}}\cdot\frac{s_{W}}{s^{2}_{V}},~{}~{}{\rm
d}w=2v\frac{s_{W}}{s^{2}_{V}}{\rm d}v,~{}~{}~{}$ (70)

we obtain the sampling pdf of $V$ at $v=a/s_{A}$ as

$\displaystyle f_{V}(v;N)$ $\displaystyle=$ $\displaystyle C_{V}{v^{2p-1}}$
(71)
$\displaystyle\times{\left[1+\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{pN-\frac{1}{2}}\thinspace
v^{2}\right]^{-(pN+p-\frac{1}{2})}}$

which can be referred to as a root-$F$ distribution, where

$\displaystyle
C_{V}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2}{\left(pN-\frac{1}{2}\right)^{p}}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{p}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$

## Figures

|
---|---
Figure 1:  (color on-line) Sampling probability density function for real or
imaginary Cartesian component of electric field ($p{=}1$) at selected values
of $N$: (a) Bessel $K$ sampling pdfs of $E^{\prime(\prime)}_{\alpha}$ [eq.
(13)]; (b) Student $t$ sampling pdfs of
$X^{\prime(\prime)}=E^{\prime(\prime)}_{\alpha}/S_{{E^{\prime(\prime)}_{\alpha}}}$
[eq. (19)]. In both plots, the right tail becomes thinner for increasing
values of $N$.

|
---|---
Figure 2:  (color on-line) Sampling probability density function for intensity
or energy density of Cartesian component of field ($p{=}1$) at selected values
of $N$ based on incoherent detection: (a) Bessel $K$ sampling pdfs of
$U_{\alpha}$ [eq. (LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor
$F$ sampling pdfs of $W_{\alpha}=U_{\alpha}/S_{U_{\alpha}}$ [eq. (29)]. In
both plots, the right tail becomes thinner for increasing values of $N$.

|
---|---
Figure 3:  (color on-line) Probability density function of intensity or energy
density of planar field ($p{=}2$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdfs of $U_{\rm t}$ [eq.
(LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor $F$ sampling pdfs of
$W_{\rm t}=U_{\rm t}/S_{U_{\rm t}}$ [eq. (29)]. In both plots, the right tail
becomes thinner for increasing values of $N$.

|
---|---
Figure 4:  (color on-line) Probability density function of intensity or energy
density of total (vector) field ($p{=}3$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdfs of $U$ [eq.
(LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor $F$ sampling pdfs of
$W=U/S_{U}$ [eq. (29)]. In both plots, the right tail becomes thinner for
increasing values of $N$.

|
---|---
(a) | (b)
Figure 5:  (color on-line) Sampling standard deviations as a function of $N$
(a) for Bessel $K$ sampling distributions (LABEL:eq:samplepdfU_incoh_BesselK)
of $U_{\alpha}$, $U_{\rm t}$, and $U$; (b) for Fisher-Snedecor $F$ sampling
distributions (29) of $U_{\alpha}/S_{U_{\alpha}}$, and $U_{\rm t}/S_{U_{\rm
t}}$, $U/S_{U}$.

---
Figure 6:  (color on-line) Probability density function of intensity or energy
density of Cartesian component of field ($p{=}1$) derived from coherent
detection at selected values of $N$: Bessel $K$ sampling pdf of $U_{\alpha}$
[eqn. (LABEL:eq:samplepdfU_coh_BesselK)].

|
---|---
Figure 7:  (color on-line) Sampling probability density function of amplitude
of Cartesian component of field ($p=1$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdf
(LABEL:eq:samplepdfA_incoh_BesselK) for $A_{\alpha}$; (b) root-$F$ pdf (48)
for $V_{\alpha}=A_{\alpha}/S_{A_{\alpha}}$. In both plots, the right tail
becomes thinner for increasing values of $N$.

|
---|---
Figure 8:  (color on-line) Sampling probability density function of amplitude
of planar field ($p{=}2$) at selected values of $N$ based on incoherent
detection: (a) Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $A_{\rm
t}$; (b) root-$F$ pdf (48) for $V_{\rm t}=A_{\rm t}/S_{A_{\rm t}}$. In both
plots, the right tail becomes thinner for increasing values of $N$.

|
---|---
Figure 9:  (color on-line) Sampling probability density function of amplitude
of total (vector) field ($p{=}3$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK)
for $A$; (b) root-$F$ pdf (48) for $V=A/S_{A}$. In both plots, the right tail
becomes thinner for increasing values of $N$.

|
---|---
(a) | (b)
Figure 10:  (color on-line) Sampling standard deviation as a function of $N$
(a) for Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK) of $A_{\alpha}$,
$A_{\rm t}$, and $A$; (b) for Fisher-Snedecor $F$ pdf (48) of
$A_{\alpha}/S_{A_{\alpha}}$, $A_{\rm t}/S_{A_{\rm t}}$, and $A/S_{A}$.

---
Figure 11:  (color on-line) Sampling probability density function of magnitude
of Cartesian component of field ($p{=}1$) at selected values of $N$, derived
for coherent detection. The right tail becomes thinner for increasing values
of $N$.

---
Figure 12:  (color on-line) Sampling probability density function of magnitude
of total (vector) field ($p{=}3$) at selected values of $N$, derived for
coherent detection. In both plots, the right tail becomes thinner for
increasing values of $N$.

---
Figure 13:  (color on-line) Upper boundaries of $95\%$-confidence intervals
for real or imaginary parts of $E_{\alpha}$, $E_{\rm t}$, or $E$. The lower
boundary $F^{-1}(0.025)\equiv-F^{-1}(0.975)$ is symmetric with respect to
$e=0$.

---
(a)
(b)
(c)
Figure 14:  (color on-line) Lower and upper boundaries of $95\%$-confidence
intervals (a) for a 1D Cartesian component $U_{\alpha}$ ($p{=}1$), (b) for a
2D planar field $U_{\rm t}$ ($p{=}2$), and (c) for the 3D vectorial $U$
($p{=}3$), normalized by the respective sampling standard deviations. For each
line type, the lower and upper curves represent $2.5\%$ and $97.5\%$
percentiles, respectively.

---
(a)
(b)
(c)
Figure 15:  (color on-line) Lower and upper boundaries of $95\%$-confidence
intervals (a) for a 1D Cartesian component $A_{\alpha}$ ($p{=}1$), (b) for a
2D planar field $A_{\rm t}$ ($p{=}2$), and (c) for the 3D vectorial $A$
($p{=}3$), normalized by the respective sampling standard deviations. For each
line type, the lower and upper curves represent $2.5\%$ and $97.5\%$
percentiles, respectively.