# Correlated Non-Coherent Radar Detection for Gamma-Fluctuating Targets in
Compound Clutter

Josef Zuk
Defence Science and Technology Group, Australia E-mail:
<EMAIL_ADDRESS>
###### Abstract

This work studies the problem of radar detection of correlated gamma-
fluctuating targets in the presence of clutter described by compound models
with correlated speckle. If the correlation is not accounted for in a radar
model, the required signal-to-interference ratio for a given probability of
detection will be incorrect, resulting in over-estimated performance. Although
more generally applicable, the is focus on airborne maritime radar systems.
Hence K-distributed sea clutter is used as the main example. Detection via
square-law non-coherent pulse integration is formulated in a way that
accommodates arbitrary partial correlation for both target radar cross-section
(RCS) and clutter speckle. The obstacle to including this degree of generality
in previous work was the fact that Swerling’s original characterization of the
standard RCS fluctuation classes as gamma distributions for the power is not
sufficient for the inclusion of both correlation sources (i.e. target and
clutter speckle) for gamma-fluctuating targets. An extension of the model is
required at the quadrature component (i.e. voltage) level, as phase
relationships can no longer be neglected. This is addressed in the present
work, which not only postulates an extended model, but also demonstrates how
to efficiently compute it, with and without a number of simplifying
approximation schemes within the framework of the saddle-point technique.

###### Index Terms:

Radar detection, non-coherent, sea clutter, radar cross-section correlation,
gamma-fluctuating target, K-distribution.

##  I Introduction

Parametric modelling is a useful tool for predicting the detection performance
of a radar system given a statistical description of the environment.
Statistical models of sea clutter are commonly employed in the parametric
modelling of detection by maritime surveillance radar [1, 2]. These are often
compound models that represent the fast varying speckle component of the
clutter with a local Gaussian process which is modulated by a slowly varying
texture component. If the radar employs frequency agility, it may be possible
to treat the speckle as completely decorrelated from pulse to pulse, however
if the radar uses a fixed frequency or the agile frequencies are separated by
less than the pulse bandwidth, the speckle correlation must be taken into
account.

In this paper, the problem of radar detection using non-coherent pulse
integration is considered, where the fluctuations of target RCS are typically
modelled in terms of the standard Swerling classes. Varying degrees of
temporal correlation of target RCS and clutter speckle fluctuations are
relevant considerations for performance modelling of airborne maritime
surveillance systems. Therefore, we formulate a modelling scheme that
accommodates the simultaneous inclusion of arbitrary partial correlation of
both a gamma-fluctuating target RCS and the speckle component of a compound
clutter model. In line with the maritime focus of the paper, K-distributed sea
clutter is used as the main example.

In the case where the only source of interference is (uncorrelated) thermal
noise, general exact calculations for detection probability resulting from
non-coherent pulse integration and partially correlated Rayleigh fluctuating
target RCS were given by Kanter [3], and later for general gamma-fluctuating
targets by Weiner [4]. Various simplified special cases had been previously
considered, and a review of these can be found in [5]. A discussion of
Kanter’s model in which both the target RCS and clutter returns are present
and partially correlated was given in [6]. Subsequently, efficient
approximations to the exact but numerically unstable results of Kanter and
Weiner have been developed, based on effective number-of-looks concepts [1,
2], and saddle-point techniques [7, 8, 9]. The problem of correlated gamma-
fluctuating targets in uncorrelated clutter has also recently been studied in
[10].

In previous work on the saddle-point approach to the problem [9], attention
was restricted to uncorrelated clutter speckle because the full problem of a
correlated gamma-fluctuating target in correlated clutter speckle requires
considerations beyond those of computational methodology (saddle-point or
otherwise); in particular, an extension of the underlying model must first be
established. The classic Swerling models of target RCS fluctuation are
specified in terms of the power distributions of the target returns [11].
While this suffices generally for exponentially fluctuating targets, when the
target power fluctuations are gamma distributed and when both the target and
clutter returns are correlated, the distribution of total returned power
depends also on the relative phases of target quadrature components [12] due
to a breaking of orthogonal symmetry. A similar issue for clutter is discussed
in [13], where it is pointed out that the relative phase between quadrature
components is correlated even in the Gaussian problem unless it comprises
entirely white noise. A general first-principles model that specifies the
joint probability distribution of the quadrature components in a way that
reduces correctly to all known special cases and satisfies all expectations
has not previously appeared in the literature. A solution to this problem,
along with the demonstration of the applicability of the saddle-point
technique, is the subject of the present paper. While attention here is
focussed on pulse-to-pulse integration, the concepts developed are also
applicable to systems employing scan-to-scan integration [14].

The rest of the paper is organized as follows: Section II introduces the
problem and the general method of solution using the inverse Laplace transform
of the moment generating function. Section III describes the representation of
the moment generating function for correlated clutter speckle. Section IV
develops the exact physical solution to the problem, designated as a ‘first
principles’ model, which is not solvable by analytic means. Section V derives
an effective method to evaluate the first-principles model. The resulting
effective model serves as a proxy for the first-principles model for
computational purposes. A number of approximations are then proposed to
further reduce the computational load while maintaining robust performance for
all realistic conditions. In Section VI, Monte Carlo simulation is employed to
demonstrate the high degree of accuracy of the effective method in
approximating the first-principles model, in all physically realistic
situations. It should be noted that the effective model does not stand on its
own: If the first-principles-model were not developed, then there would be no
basis upon which to judge whether the effective model is a valid
representation of the problem. Section VII then summarizes the key results in
the paper.

##  II Problem Formulation

To be specific, we consider square-law detection followed by $M$-pulse non-
coherent integration, in the presence of multiple partially correlated
signal/noise power sources. The test statistic that determines the detection
threshold is the summed returned power random variable (RV)

$Z=\tfrac{1}{2}\sum_{\eta=\text{i},\text{q}}\Bigl{|}\sum_{\alpha}\mathbf{X}_{\eta\alpha}\Bigr{|}^{2}\,,$
(1)

where $\mathbf{X}_{\eta\alpha}$ is an $M$-dimensional vector with elements
$X_{\eta\alpha m}$, $m=1,2,\ldots,M$. Vector RVs are indicated by a bold
typeface. For $\eta=\text{i},\text{q}$, the $X_{\eta\alpha m}$ are
respectively the in-phase and quadrature components on the $m$-th integrated
pulse for signal power source $\alpha$. The source index $\alpha$ will
typically range over thermal noise (n), target return (s) and various types of
clutter (c). The target RCS fluctuation class indexed by $\kappa=1,2,\ldots$,
refers to all models that interpolate (in degree of correlation) between the
Swerling-($2\kappa-1$) and Swerling-($2\kappa$) limiting cases. Thus, the
$\kappa=1$ class interpolates between the fully correlated Swerling 1 and
fully uncorrelated Swerling 2 models, while the $\kappa=2$ class interpolates
between the fully correlated Swerling 3 and fully uncorrelated Swerling 4
models. A steady target RCS (Swerling 0) is obtained asymptotically as
$\kappa\rightarrow\infty$.

The probability density function (PDF) $P_{Z}(z)$ of the RV $Z$ is the inverse
Laplace transform of the moment generating function (MGF)

$\mathcal{M}_{Z}(s)\;\equiv\;\left\langle
e^{-sZ}\right\rangle_{Z}\;=\;\left[\left\langle
e^{-\frac{s}{2}\left(\sum_{\alpha}\mathbf{X}_{\alpha}\right)^{2}}\right\rangle_{\mathbf{X}_{\alpha}}\right]^{2}\,,$
(2)

where $\mathbf{X}_{\alpha}$, representing a single quadrature component, is
distributed identically with both $\mathbf{X}_{\text{i}\alpha}$ and
$\mathbf{X}_{\text{q}\alpha}$. The square on the RHS of the equation above
accounts for the presence of two quadrature components. The notation
$\langle\cdot\rangle_{Z}$ denotes the expectation with respect to the
distribution of the random variable $Z$. If we initially restrict ourselves to
the $\kappa=1$ target RCS fluctuation class then, for each power source
$\alpha$, the joint PDF of the vector RV $\mathbf{X}_{\alpha}$ is a mean-zero
correlated Gaussian with variance $\sigma^{2}_{\alpha}$ representing the
returned power associated with power source $\alpha$, and correlation matrix
$\mathsf{C}_{\alpha}$, It should be noted that the matrices
$\mathsf{C}_{\alpha}$ are taken to be correlation matrices, not covariance
matrices. Therefore, it is assumed that $\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\alpha}\\}=M$. For the thermal noise component, the
correlation matrix is the identity: $\mathsf{C}_{\text{n}}=\mathbb{I}$.

It is convenient to introduce the notation $Z_{\text{avg}}\equiv Z/M$ for the
pulse-averaged returned power, and $Z^{\prime}_{\text{avg}}\equiv
Z_{\text{avg}}/\sigma^{2}_{\text{I}}$ for the average power normalized by the
mean total interference
$\sigma^{2}_{\text{I}}\equiv\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}$.
Also, $S\equiv\sigma^{2}_{\text{s}}/\sigma^{2}_{\text{I}}$ will denote the
signal-to-interference ratio (SIR), and
$q\equiv\sigma^{2}_{\text{c}}/\sigma^{2}_{\text{I}}$ ($0\leq q\leq 1$) is the
clutter-to-interference ratio.

We shall work with the Gauss-Markov correlation model [3], for which the
correlation matrices have the symmetric Toeplitz form:

$[\mathsf{C}_{\text{s}}]_{mn}=\rho_{\text{s}}^{|m-n|}\,,\quad[\mathsf{C}_{\text{c}}]_{mn}=\rho_{\text{c}}^{|m-n|}\,,$
(3)

where $\rho_{\text{s}}$, $\rho_{\text{c}}$ are the target and clutter
correlation coefficients, respectively, such that
$0\leq\rho_{\text{s}},\rho_{\text{c}}\leq 1$. Nevertheless, the methodology
that we develop works equally with any other Toeplitz correlation structure.

By considering the various known special cases, such as correlated
exponentially fluctuating target power ($\kappa=1$) with correlated clutter
speckle [8, 12], and correlated gamma-fluctuating target power ($\kappa>1$)
with uncorrelated clutter speckle [9], one can infer that the mean and
variance of the total returned power in the general case should be given,
respectively, by

$\left\langle Z_{\text{avg}}/\sigma^{2}_{\text{I}}\right\rangle=1+S\,,$ (4)

and

$\mathop{\rm
var}\nolimits\left(Z_{\text{avg}}/\sigma^{2}_{\text{I}}\right)=\frac{1-q^{2}}{M}+\zeta{\cdot}\frac{q^{2}}{L}+\frac{S^{2}}{\kappa
B}+2S{\cdot}\left(\frac{1-q}{M}+\frac{q}{N}\right)\,,$ (5)

where

$L\equiv M^{2}/\mathop{\rm
Tr}\nolimits\\{\mathsf{C}^{2}_{\text{c}}\\},\,B\equiv M^{2}/\mathop{\rm
Tr}\nolimits\\{\mathsf{C}^{2}_{\text{s}}\\},\,N\equiv M^{2}/\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{c}}\mathsf{C}_{\text{s}}\\}\,.$ (6)

The quantity $\zeta$ represents the effect of compound clutter texture, and is
given by

$\zeta\,=\,1+(L+1)\mathop{\rm var}\nolimits(U)\,=\,1+(L+1)/\nu\,,$ (7)

with the second equality holding for K-distributed compound clutter, where the
texture RV $U$ is described by a unit-mean gamma distribution with shape
parameter equal to the clutter texture shape parameter $\nu$, in which case
$\mathop{\rm var}\nolimits(U)=1/\nu$. Standard Rayleigh clutter corresponds to
$\zeta=1$, obtained in the limit $\nu\to\infty$. It should be noted that the
considerations in this work remain valid for any compound clutter model. In
the general case, $U$ is the random variable that realizes the clutter texture
distribution in any compound clutter model.

The simplest way to understand (7) is to set $S=0$ and $q=1$. Then
$Z_{\text{avg}}/\sigma^{2}_{\text{I}}=UV$, where the clutter speckle RV $V$ is
given by the correlated sum of $M$ unit-mean exponential RVs (and scaled by
$M$). Since the second moment of the compound RV factorizes into texture and
speckle components, we have $\mathop{\rm
var}\nolimits\left(Z_{\text{avg}}/\sigma^{2}_{\text{I}}\right)=m_{2}(U)m_{2}(V)-1$,
given that $\langle U\rangle=\langle V\rangle=1$. The result follows upon
observing that $\mathop{\rm var}\nolimits(V)=1/L$ [3, Eq. 65].

The challenge is to derive a general model that reproduces this mean and
variance. It is worth mentioning at this point that the related problem of
including partial correlation of clutter speckle in cases where a non-Rayleigh
overall clutter amplitude is being modelled was successfully addressed by the
introduction of the spherically invariant process (SIRP) [13], of which the
compound clutter paradigm is a special case. However, this approach is not
applicable to the correlation of non-Rayleigh target amplitudes because
introducing a SIRP leads to an increase in power variance, as can be seen for
the clutter contribution in (5) and (7) by comparing a finite shape parameter
$\nu$ with $\nu\to\infty$; On the other hand, moving the RCS fluctuation class
index $\kappa$ to values greater than unity will necessarily decrease the
variance.

Compound clutter is incorporated in the usual way [8, 12] by promoting the
mean clutter power to a random variable according to the mapping
$\sigma_{\text{c}}^{2}\mapsto\bar{\sigma}_{\text{c}}^{2}{\cdot}U$ where the RV
$U$ generates the clutter texture distribution with unit mean. The overall
mean interference power becomes

$\sigma_{\text{I}}^{2}\;=\;\sigma_{\text{n}}^{2}+\left\langle\sigma_{\text{c}}^{2}(U)\right\rangle_{U}\;=\;\sigma_{\text{n}}^{2}+\bar{\sigma}_{\text{c}}^{2}\,.$
(8)

The texture expectation is best left to the end, after recovering the survival
function from the MGF. Thus, if $\bar{F}_{\text{cc}}(v;S)$ denotes the
survival function for the total returned power $v$ in the presence of both a
target and compound clutter, and we also use the notation
$\bar{F}_{\text{spk}}(v;S,q)$, where
$q\equiv\bar{\sigma}_{\text{c}}^{2}/\sigma_{\text{I}}^{2}$ is the clutter-to-
interference ratio, to denote the survival function incorporating clutter
speckle but without the texture, then $\bar{F}_{\text{cc}}(v;S)$ is given by

$\displaystyle\begin{aligned}
\bar{F}_{\text{cc}}(v;S)&=\left\langle\bar{F}_{\text{spk}}(v;S,qU)\right\rangle_{U}\\\
&=\int_{0}^{\infty}du\,f_{\nu}(u)\bar{F}_{\text{spk}}(v;S,qu)\,,\end{aligned}$
(9)

where $f_{\nu}(u)$ denotes the clutter texture PDF. For K-distributed clutter,
this is a unit-mean gamma distribution with shape parameter $\nu$ [1]. The
texture integration is most efficiently performed by means of Gaussian
quadrature.

The probability of detection of a target on a single scan $P_{\text{D}}(S)$
when its SIR is known to be $S$, is given by the value of the survival
function (also known as the complementary cumulative distribution function)
for the total returned (signal plus interference) power at the selected
detection threshold $v=v_{\text{b}}$. The threshold is determined as the point
at which the survival function for the total returned interference power
assumes the value that corresponds to a pre-determined desired probability of
false alarm $P_{\text{FA}}$. Thus, we eliminate $v_{\text{b}}$ in the
following pair of equations:

$P_{\text{D}}(S)=\bar{F}_{\text{cc}}(v_{\text{b}};S)\,,\quad
P_{\text{FA}}=\bar{F}_{\text{cc}}(v_{\text{b}};0)\,.$ (10)

In the saddle-point method, just as for the PDF, one computes the survival
function in terms of the inverse Laplace transform of the MGF for its
underlying distribution, by integrating along a suitably constructed contour
in the complex plane [7].

Previous work [9] applied saddle-point techniques to the problem of correlated
targets in uncorrelated clutter speckle, using an existing model for the
problem. The present work, in part, also applies saddle-point techniques, but
to a model that must first be developed, as there is no extant model that
addresses the problem of correlated gamma-fluctuating target in correlated
clutter speckle.

##  III Correlated Clutter

Weiner’s model [4] is applicable to correlated gamma-fluctuating targets
embedded in uncorrelated clutter speckle. In this section, we generalize this
by establishing the representation of the MGF that incorporates speckle
correlation, and discuss the complications encountered when trying to evaluate
it according to the usual specification of the Swerling RCS models.

We assume that there are three sources of returned power
$\alpha=\text{n},\text{c},\text{s}$, corresponding to thermal noise, surface
clutter and target RCS, respectively, and that the correlated gamma-
fluctuating target is embedded in correlated clutter (belonging to the
$\kappa=1$ speckle fluctuation class). The $M$-dimensional vector RV
$\mathbf{X}_{\alpha}$, composed of one element for each of the
$m=1,2,\ldots,M$ pulses, will denote the in-phase quadrature component for the
associated power sources.

The MGF $\mathcal{M}_{Z}(s)$ of the returned-power RV $Z$ is given by [12]

$\displaystyle\begin{aligned} \sqrt{\mathcal{M}_{Z}(s)}&=\left\langle
e^{-\frac{s}{2}\left(\sum_{\alpha}\mathbf{X}_{\alpha}\right)^{2}}\right\rangle_{\mathbf{X}_{\alpha}}\\\
&=\int_{-\infty}^{+\infty}\frac{d^{M}u}{\left(2\pi
s\right)^{M/2}}\,e^{-\mathbf{u}\cdot\mathbf{u}/(2s)}\prod_{\alpha=\text{n},\text{c},\text{s}}\left\langle
e^{-i\mathbf{u}\cdot\mathbf{X}_{\alpha}}\right\rangle_{\mathbf{X}_{\alpha}}\\\
&=\int_{-\infty}^{+\infty}\frac{d^{M}u}{\left(2\pi
s\right)^{M/2}}\,e^{-\mathbf{u}^{\sf T}\mathsf{Q}(s)\mathbf{u}/2}\left\langle
e^{-i\mathbf{u}\cdot\mathbf{X}_{\text{s}}}\right\rangle_{\mathbf{X}_{\text{s}}}\,,\end{aligned}$
(11)

where we have split off the non-Gaussian target component
$\mathbf{X}_{\text{s}}$, evaluated the expectations over the clutter and noise
components, and set

$\mathsf{Q}(s)\equiv(1/s+\sigma^{2}_{\text{n}})\mathbb{I}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}}\,.$
(12)

The square-root on the LHS of the equation arises from the square on the RHS
in (2).

Now, since $\mathsf{Q}$ is a symmetric matrix, we may write
$\mathsf{Q}=\mathsf{R}^{\sf T}\mathsf{\Lambda}\mathsf{R}$, where
$\mathsf{\Lambda}=\mathop{\rm diag}\nolimits(\lambda_{1},\ldots,\lambda_{M})$
and $\mathsf{R}^{\sf T}\mathsf{R}=\mathbb{I}$. Then we have, for any vector
$\mathbf{J}$, the representation

$\displaystyle\begin{aligned}
&\int_{-\infty}^{+\infty}\frac{d^{M}u}{\left(2\pi\right)^{M/2}}\,e^{-\mathbf{u}^{\sf
T}\mathsf{Q}\mathbf{u}/2}{\cdot}e^{-i\mathbf{u}^{\sf T}\mathbf{J}}\\\
&=\int_{-\infty}^{+\infty}\frac{d^{M}u}{\left(2\pi\right)^{M/2}}\,e^{-\mathbf{u}^{\sf
T}\mathsf{\Lambda}\mathbf{u}/2}{\cdot}e^{-i\mathbf{u}^{\sf
T}\mathsf{R}\mathbf{J}}\\\
&=\frac{1}{\sqrt{\det\mathsf{Q}}}\exp\left[-\sum_{m=1}^{M}\left(RJ\right)^{2}_{m}/(2\lambda_{m})\right]\,,\end{aligned}$
(13)

where the first equality is due to a rotational change of integration
variables $\mathbf{u}^{\prime}=\mathsf{R}\mathbf{u}$, and we note that

$\displaystyle\begin{aligned}
\sum_{m=1}^{M}\left(\mathsf{R}J\right)^{2}_{m}/\lambda_{m}&=\sum_{m,a,b=1}^{M}J_{a}\left(R^{\sf
T}_{am}\lambda^{-1}_{m}\mathsf{R}_{mb}\right)J_{b}\\\ &=\mathbf{J}^{\sf
T}\mathsf{Q}^{-1}\mathbf{J}\,,\end{aligned}$ (14)

which leads to the result

$\sqrt{\mathcal{M}_{Z}(s)}=\frac{1}{s^{M/2}\sqrt{\det\mathsf{Q}(s)}}\left\langle
e^{-\frac{1}{2}\mathbf{X}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}(s)\mathbf{X}_{\text{s}}}\right\rangle_{\mathbf{X}_{\text{s}}}\,,$
(15)

upon making the identification $\mathbf{J}\equiv\mathbf{X}_{\text{s}}$.

It is convenient to introduce a scaled target RV $\hat{\mathbf{X}}_{\text{s}}$
according to
$\mathbf{X}_{\text{s}}=\sigma_{\text{s}}\hat{\mathbf{X}}_{\text{s}}$, so that
each power component $\hat{X}^{2}_{\text{s}m}$ has unit mean. For RCS
fluctuation class $\kappa$, each $\hat{X}^{2}_{\text{s}m}$ has a marginal
distribution given by the unit-mean gamma distribution with shape parameter
$\kappa/2$, i.e. $\hat{X}^{2}_{\text{s}m}\sim\Gamma(\kappa/2,2/\kappa)$ for
all $m=1,2,\ldots,M$ [11].

For uncorrelated clutter speckle, $\mathsf{Q}$ becomes proportional to the
unit matrix $\mathsf{Q}=\xi^{2}\mathbb{I}$, with

$\xi^{2}\;\equiv\;1/s+\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\;=\;1/s+\sigma^{2}_{\text{I}}\,.$
(16)

This yields the representation

$\sqrt{\mathcal{M}_{Z}(s)}=\frac{1}{(s\xi^{2})^{M/2}}\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2\xi^{2}}\hat{\mathbf{X}}^{2}_{\text{s}}\right)\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\,,$
(17)

which reproduces the problem considered in the previous work [9] when combined
with the expression [4]

$\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2\xi^{2}}\hat{\mathbf{X}}^{2}_{\text{s}}\right)\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}=\left[\det\left(1+\frac{\sigma^{2}_{\text{s}}}{\kappa\xi^{2}}C_{\text{s}}\right)\right]^{-\kappa/2}\,.$
(18)

The higher-order Swerling models ($\kappa>1$) are defined by giving the common
distribution of the squares of the quadrature components
$\hat{X}_{\text{s}m}^{2}$ [11], but the signs are left unspecified. In the
presence of partial correlation, however, the signs of the target quadrature
components do not drop out and, therefore, the specification is incomplete.
Equivalently, one may note that while $\hat{\mathbf{X}}^{2}_{\text{s}}$ is
rotationally invariant, the quadratic form $\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}\hat{\mathbf{X}}_{\text{s}}$ is not.

##  IV A First-Principles Model

In this section, we review the derivation of Weiner’s model for gamma-
fluctuating targets in uncorrelated interference, and use it to motivate a
generalization to accommodate the presence of clutter with a arbitrarily
correlated speckle component. The aim is to derive a first-principles model,
i.e. one whose degrees of freedom correspond to basic observable physical
quantities. In the present context, these are provided by the electromagnetic
field impinging upon the antenna aperture or, equivalently, by the voltage
variations induced because of it. Due to the stochastic nature of the
detection problem, the relevant physics inputs to the model are the
probability distributions of the two orthogonal quadrature components.

According to Weiner’s formulation [4], in fluctuation class $\kappa$, the
scaled target power RV may be characterized as

$\hat{\mathbf{X}}_{\text{s}}^{2}=\frac{1}{\kappa}\sum_{k=1}^{\kappa}\mathbf{H}_{k}^{\sf
T}\mathbf{H}_{k}\,,$ (19)

where, for each $k=1,2\ldots,\kappa$, the vector RV $\mathbf{H}_{k}$ is a
zero-mean unit-variance Gaussian with common correlation matrix
$\mathsf{C}_{\text{s}}$. The associated probability measure
$F_{\mathbf{H}}(d\mathbf{h})=P_{\mathbf{H}}(\mathbf{h})d\mathbf{h}$ is thus
given by

$F_{\mathbf{H}}(d\mathbf{h})=\frac{d\mathbf{h}}{\left[(2\pi)^{M}\det\mathsf{C}_{\text{s}}\right]^{\kappa/2}}\exp\left[-\frac{1}{2}\sum_{k=1}^{\kappa}\mathbf{h}_{k}^{\sf
T}\mathsf{C}^{-1}_{\text{s}}\mathbf{h}_{k}\right]\,,$ (20)

where $d\mathbf{h}$ is the product measure in $\kappa M$ dimensions

$d\mathbf{h}\equiv\prod_{k=1}^{\kappa}\prod_{m=1}^{M}dh_{k,m}\;,$ (21)

and where $h_{k,m}$ denotes the $m$-th component of the vector
$\mathbf{h}_{k}$. We observe the explicit orthogonal invariance
$F(d(\mathsf{R}\mathbf{h});\mathsf{R}\mathsf{C}_{\text{s}}\mathsf{R}^{\sf
T})=F(d\mathbf{h};\mathsf{C}_{\text{s}})$, where
$F(d\mathbf{h};\mathsf{C}_{\text{s}})\equiv F_{\mathbf{H}}(d\mathbf{h})$ and
$\mathsf{R}\mathbf{h}\equiv(\mathsf{R}\mathbf{h}_{1},\mathsf{R}\mathbf{h}_{2},\ldots,\mathsf{R}\mathbf{h}_{\kappa})$.
If we write $T\equiv\hat{\mathbf{X}}_{\text{s}}^{2}$, then the induced
probability measure $F_{T}(dt)=P_{T}(t)dt$ on the space of functions of the
scalar RV $T$ is given by

$F_{T}(dt)=\int_{\mathbb{R}^{\kappa
M}}d\mathbf{h}\,P_{\mathbf{H}}(\mathbf{h})\delta\Bigl{(}t-\frac{1}{\kappa}\sum_{k=1}^{\kappa}\mathbf{h}_{k}^{\sf
T}\mathbf{h}_{k}\Bigr{)}{\cdot}dt\,,$ (22)

where $\delta(t)$ is the Dirac delta function. Appealing to the Fourier
transform of the delta function

$\delta(t)=\int^{+\infty}_{-\infty}\frac{d\omega}{2\pi}\,e^{i\omega t}\;,$
(23)

it follows immediately that

$P_{\text{T}}(t)=\int^{+\infty}_{-\infty}\frac{d\omega}{2\pi}\,e^{i\omega
t}\left[\det\left(\mathbb{I}-\frac{i\omega}{\kappa/2}C_{\text{s}}\right)\right]^{-\kappa/2}\,,$
(24)

which depends only on the eigenvalues of $\mathsf{C}_{\text{s}}$, as expected
due to the invariance under the mapping
$\mathsf{C}_{\text{s}}\mapsto\mathsf{R}\mathsf{C}_{\text{s}}\mathsf{R}^{\sf
T}$. When $\mathsf{C}_{\text{s}}=\mathbb{I}$, we recognize, in the integrand,
the characteristic function for the distribution
$T\sim\Gamma(M\kappa/2,2/\kappa)$, which corresponds to shape parameter
$M\kappa/2$ and mean $M$. And if we write

$T_{m}\equiv\frac{1}{\kappa}\sum_{k=1}^{\kappa}H_{mk}^{2}\,,$ (25)

so that $T=\sum_{m=1}^{M}T_{m}$, then $T_{m}\sim\Gamma(\kappa/2,2/\kappa)$ for
each $m=1,2,\ldots,M$. The relevant question now is how to extend this
discussion in order to construct a compatible probability measure on the space
of functions of the vector RV $\hat{\mathbf{X}}_{\text{s}}$.

At this point, we can see that (18) may be established by appealing to the
orthogonal invariance of the expectation on the LHS. Specifically, the
expectation

$\begin{split}\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2\xi^{2}}\hat{\mathbf{X}}^{2}_{\text{s}}\right)\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}=\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2\kappa\xi^{2}}\sum_{k=1}^{\kappa}\mathbf{H}_{k}^{\sf
T}\mathbf{H}_{k}\right)\right\rangle_{\mathbf{H}}\end{split}$ (26)

is unchanged under the mapping $\mathbf{H}_{k}\mapsto\mathsf{R}\mathbf{H}_{k}$
for any orthogonal matrix $\mathsf{R}$, provided that it is accompanied by the
mapping
$\mathsf{C}_{\text{s}}\mapsto\mathsf{R}\mathsf{C}_{\text{s}}\mathsf{R}^{\sf
T}$. To take advantage of this, we write the target correlation matrix in
diagonalized form as $\mathsf{C}_{\text{s}}=\mathsf{R}_{\text{s}}^{\sf
T}\mathsf{\Lambda}_{\text{s}}\mathsf{R}_{\text{s}}$, where
$\mathsf{\Lambda}_{\text{s}}\equiv\mathop{\rm
diag}\nolimits(\gamma^{\text{s}}_{1},\gamma^{\text{s}}_{2},\ldots,\gamma^{\text{s}}_{M})$
and $\mathsf{R}_{\text{s}}^{\sf T}\mathsf{R}_{\text{s}}=\mathbb{I}$, and
introduce the rotated vector RVs
$\mathbf{J}_{k}\equiv\mathsf{R}_{\text{s}}\mathbf{H}_{k}$ whose elements are
independent zero-mean Gaussians such that the $J_{mk}$ have common variance
$\gamma^{\text{s}}_{m}$ for $k=1,2,\ldots,\kappa$. It follows that the sums

$V_{m}\equiv\frac{1}{\kappa}\sum_{k=1}^{\kappa}J^{2}_{mk}$ (27)

are independent and gamma-distributed with respective means
$\gamma^{\text{s}}_{m}$. Then, we explicitly have

$\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2\xi^{2}}\sum_{m=1}^{M}V_{m}\right)\right\rangle_{\mathbf{V}}=\prod_{m=1}^{M}\left(1+\frac{\sigma^{2}_{\text{s}}}{\kappa\xi^{2}}\gamma^{\text{s}}_{m}\right)^{-\kappa/2}\,,$
(28)

and this is equivalent to (18).

We may apply the same rotational change of variable scheme to the problem with
correlated clutter, where $\mathsf{Q}$ is a general matrix, provided we can
assume that the, yet unspecified, mapping
$(\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{\kappa})\mapsto\hat{\mathbf{X}}_{\text{s}}$
satisfies$(\mathsf{R}\mathbf{H}_{1},\mathsf{R}\mathbf{H}_{2},\ldots,\mathsf{R}\mathbf{H}_{\kappa})\mapsto\mathsf{R}\hat{\mathbf{X}}_{\text{s}}$
for any orthogonal matrix $\mathsf{R}$. This requirement is clearly consistent
with (19) but its existence for $\kappa>1$ is not clear. In this case, the
argument of the expectation in (15) is not rotationally invariant, but the
induced probability measure on $\hat{\mathbf{X}}_{\text{s}}$ will be invariant
under the simultaneous mappings
$\hat{\mathbf{X}}_{\text{s}}\mapsto\mathsf{R}_{\text{s}}\hat{\mathbf{X}}_{\text{s}}$,
$C_{\text{s}}\mapsto\mathsf{R}_{\text{s}}\mathsf{C}_{\text{s}}\mathsf{R}_{\text{s}}^{\sf
T}$. Thus, setting
$\mathbf{Y}\equiv\mathsf{R}_{\text{s}}\hat{\mathbf{X}}_{\text{s}}$, we obtain

$\displaystyle\begin{aligned}
&\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}\hat{\mathbf{X}}_{\text{s}}\right)\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\\\
&\quad=\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}\hat{\mathbf{X}}_{\text{s}}\right)\right\rangle_{\mathsf{R}_{\text{s}}\hat{\mathbf{X}}_{\text{s}}}\\\
&\quad=\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\mathbf{Y}^{\sf
T}(\mathsf{R}_{\text{s}}\mathsf{Q}^{-1}\mathsf{R}_{\text{s}}^{\sf
T})\mathbf{Y}\right)\right\rangle_{\mathbf{Y}}\\\
&\quad=\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\hat{\mathbf{Y}}^{\sf
T}(\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}\mathsf{L}_{\text{s}}^{\sf
T})\hat{\mathbf{Y}}\right)\right\rangle_{\hat{\mathbf{Y}}}\,,\end{aligned}$
(29)

where $\mathbf{Y}\equiv\sqrt{\mathsf{\Lambda}_{\text{s}}}\hat{\mathbf{Y}}$,
and
$\mathsf{L}_{\text{s}}\equiv\sqrt{\mathsf{\Lambda}_{\text{s}}}\mathsf{R}_{\text{s}}$
so that $\mathsf{C}_{\text{s}}=\mathsf{L}_{\text{s}}^{\sf
T}\mathsf{L}_{\text{s}}$. We note that elements of the vector RV
$\hat{\mathbf{Y}}$ are independent and identically distributed (iid) such that
$\hat{Y}^{2}_{m}\sim\Gamma(\kappa/2,2/\kappa)$ for all $m=1,2\ldots,M$.

The problem of an uncorrelated gamma-fluctuating target embedded in correlated
clutter is obtained when $\mathsf{L}_{\text{s}}=\mathbb{I}$. Therefore, we see
that introducing target correlation in the presence of correlated clutter
simply amounts to making the mapping
$\mathsf{Q}^{-1}\mapsto\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}\mathsf{L}_{\text{s}}^{\sf
T}$ in the calculation of the target expectation. When
$\mathsf{C}_{\text{s}}=\mathbb{I}$, the only sensible option is for each of
iid RVs $\hat{X}_{m}$ to be described by a two-sided Nakagami distribution
whose PDF is given by

$\begin{split}&f(x;\nu,\Omega)dx={}\\\
&\frac{1}{\Gamma(\nu)}e^{-\left(x\sqrt{\nu/\Omega}\right)^{2}}\left(|x|\sqrt{\nu/\Omega}\right)^{2\nu-1}d(x\sqrt{\nu/\Omega})\,,\end{split}$
(30)

with fading parameter $\nu=\kappa/2$ and scale parameter $\Omega=1$, and this
becomes Gaussian for $\kappa=1$. The characteristic function is given by [15]

$\displaystyle\begin{aligned} \tilde{f}(u;\nu,\Omega)&\equiv\left\langle
e^{-iu\hat{X}}\right\rangle_{\hat{X}}=\frac{\Gamma(2\nu)}{2^{\nu}\Gamma(\nu)}\exp\left(-\frac{\Omega}{8\nu}u^{2}\right)\\\
&{}\times\left[D_{-2\nu}\left(iu\sqrt{\frac{\Omega}{2\nu}}\right)+D_{-2\nu}\left(-iu\sqrt{\frac{\Omega}{2\nu}}\right)\right]\,,\end{aligned}$
(31)

where $D_{-\nu}(z)$ is a parabolic cylinder function [16]. Regardless of
whether it is possible to find a suitable rotationally covariant mapping
$\\{\mathbf{H}_{\kappa}\\}\mapsto\mathbf{X}_{\text{s}}$, we shall adopt (29)
as the definition of a candidate first-principles model: that is, we set
$\hat{\mathbf{X}}_{\text{s}}\equiv\mathsf{L}^{\sf
T}_{\text{s}}\hat{\mathbf{Y}}$, and furthermore postulate that, even for a
general target correlation matrix $\mathsf{C}_{\text{s}}$, the components of
the vector RV $\hat{\mathbf{Y}}$ are iid and generated by the two-sided
Nakagami distribution.

###  IV-A Statistical Properties of the First-Principles Model

The aim of this section is to demonstrate that the model proposed in the
foregoing section yields the required mean and variance as given by (4)–(6).
Recalling that the mean and variance are respectively given by the first and
second derivatives of the cumulant generating function (CGF) evaluated at the
origin, we proceed by writing down the CGF $\mathcal{K}(s)$ for the model and
making a Taylor expansion about $s=0$ up to second order.

Thus, in order to compute the mean and variance arising from (15), let us
introduce the functions

$\displaystyle\begin{aligned}
\mathcal{K}_{0}(s)&\equiv-\ln\det(s{\cdot}\mathsf{Q}(s))\,,\\\
\mathcal{K}_{1}(s)&\equiv
2\ln\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}(s)\hat{\mathbf{X}}_{\text{s}}\right)\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\,.\end{aligned}$
(32)

Then, the cumulant generating function for the model defined by (15) is given
by

$\mathcal{K}(s)\;\equiv\;\ln\left\langle
e^{-sZ_{\text{avg}}}\right\rangle_{Z_{\text{avg}}}\;=\;\mathcal{K}_{0}(s)+\mathcal{K}_{1}(s)\,.$
(33)

The function $\mathcal{K}_{0}(s)$ generates the contribution to the CGF that
is independent of the SIR. Thus, performing a Taylor expansion in $s$,

$\begin{split}\mathcal{K}_{0}(s)=&-s(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}})+\frac{s^{2}}{2}\biggl{[}\frac{\sigma^{4}_{\text{n}}}{M}+\frac{\sigma^{4}_{\text{c}}}{M^{2}}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{c}}^{2}\\}+\frac{2\sigma^{2}_{\text{n}}\sigma^{2}_{\text{c}}}{M}\biggr{]}\\\
&{}+O(s^{3}).\end{split}$ (34)

Also performing a Taylor expansion in $s$ for the SIR-dependent term yields

$\begin{split}&\mathcal{K}_{1}(s)=-\sigma^{2}_{\text{s}}\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}(s)\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\\\
&{}+\frac{\sigma^{4}_{\text{s}}}{4}\left[\left\langle\left(\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}(s)\hat{\mathbf{X}}_{\text{s}}\right)^{2}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}-\left(\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{Q}^{-1}(s)\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\right)^{2}\right]\\\
&{}+O(s^{3})\,,\end{split}$ (35)

and we note that

$\displaystyle\begin{aligned}
\mathsf{Q}^{-1}(s)&=\frac{s}{1+s(\sigma^{2}_{\text{n}}\mathbb{I}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}})}\\\
&=s\mathbb{I}-s^{2}(\sigma^{2}_{\text{n}}\mathbb{I}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}})+O(s^{3})\,,\end{aligned}$
(36)

which yields

$\begin{split}&\mathcal{K}_{1}(s)=-s\sigma^{2}_{\text{s}}\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}+s^{2}\biggl{[}\sigma^{2}_{\text{n}}\sigma^{2}_{\text{s}}\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\\\
&{}+\sigma^{2}_{\text{c}}\sigma^{2}_{\text{s}}\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{C}_{\text{c}}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}+\frac{\sigma^{2}_{\text{s}}}{4}\mathop{\rm
var}\nolimits(\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\hat{\mathbf{X}}_{\text{s}})\biggr{]}+O(s^{3})\,.\end{split}$ (37)

A matrix in the denominator, as in (36), denotes the inverse.

By construction, (19) holds in distribution for the first-principles model.
This is easily confirmed, based on the observation that

$H\sim\mathcal{N}(0,1)\Rightarrow H^{2}\sim\Gamma(1/2,2)\,.$ (38)

It follows immediately that

$\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}=\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\\}\,,\quad\mathop{\rm
var}\nolimits(\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\hat{\mathbf{X}}_{\text{s}})=\frac{2}{\kappa}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}^{2}\\}\,.$ (39)

From this, we see that the mean and variance are given as in (4)–(6), but with

$N=M^{2}/\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{C}_{\text{c}}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}\,.$
(40)

To compute this expectation, we recall that the RVs $\hat{Y}_{m}$,
$m=1,2,\ldots,M$ are iid such that
$\hat{Y}_{m}^{2}\sim\Gamma(\kappa/2,2/\kappa)$ and with random signs (i.e.
randomly positive or negative, so that $\mathop{\rm
sgn}\nolimits(\hat{Y}_{m})=\pm 1$ is by construction a Rademacher RV), and set
$\hat{\mathbf{X}}_{\text{s}}\equiv\mathsf{L}^{\sf
T}_{\text{s}}\hat{\mathbf{Y}}$, where $\mathsf{C}_{\text{s}}=\mathsf{L}^{\sf
T}_{\text{s}}\mathsf{L}_{\text{s}}$. Accordingly,

$\displaystyle\begin{aligned}
\left\langle\hat{X}_{m}\hat{X}_{n}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}&=\sum_{a,b=1}^{M}\left[\mathsf{L}_{\text{s}}\right]_{am}\left[\mathsf{L}_{\text{s}}\right]_{bn}\left\langle\hat{Y}_{a}\hat{Y}_{b}\right\rangle_{\mathbf{Y}}\\\
&=\sum_{a,b=1}^{M}\left[\mathsf{L}_{\text{s}}\right]_{am}\left[\mathsf{L}_{\text{s}}\right]_{bn}\delta_{ab}\\\
&=\left[\mathsf{L}^{\sf T}_{\text{s}}\mathsf{L}_{\text{s}}\right]_{mn}\\\
&=\left[\mathsf{C}_{\text{s}}\right]_{mn}\,.\end{aligned}$ (41)

This means that we obtain $\left\langle\hat{\mathbf{X}}_{\text{s}}^{\sf
T}\mathsf{C}_{\text{c}}\hat{\mathbf{X}}_{\text{s}}\right\rangle_{\hat{\mathbf{X}}_{\text{s}}}=\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{c}}\mathsf{C}_{\text{s}}\\}$ as desired. The
upshot of all this is that the candidate first-principles model reproduces the
expected mean and variance for the returned power, and (29) holds, but the
rotational covariance assumption that initially led to it is redundant.

###  IV-B Model Summary

In summary, the first-principles model is defined by specifying that the
(normalized) target quadrature component is given by
$\hat{\mathbf{X}}_{\text{s}}\equiv\mathsf{L}^{\sf
T}_{\text{s}}\hat{\mathbf{Y}}$, where the iid vector RV $\hat{\mathbf{Y}}$ has
elements distributed according to the two-sided Nakagami distribution of (30),
and the matrix $\mathsf{L}_{\text{s}}$ is obtained from the eigenvalue
decomposition of the target correlation matrix according to

$\mathsf{C}_{\text{s}}=\mathsf{R}_{\text{s}}^{\sf
T}\mathsf{\Lambda}_{\text{s}}\mathsf{R}_{\text{s}}\,,\quad\mathsf{L}_{\text{s}}\equiv\sqrt{\mathsf{\Lambda}_{\text{s}}}\mathsf{R}_{\text{s}}\;.$
(42)

Hence, the MGF of the first-principles model is given by

$\begin{split}&\mathcal{M}_{Z}(s)={}\\\
&\frac{1}{s^{M}\det\mathsf{Q}(s)}{\cdot}\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\hat{\mathbf{Y}}^{\sf
T}(\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T})\hat{\mathbf{Y}}\right)\right\rangle^{2}_{\hat{\mathbf{Y}}}\,.\end{split}$
(43)

In the Appendix, we discuss some minor shortcomings of the first-principles
model. We also show that it may be computed exactly for the special case of a
fully correlated target.

Compound clutter may be incorporated as described at the end of Section II by
making the Q-matrix of (12) dependent on the clutter texture integration
variable $u$, according to

$\mathsf{Q}(s;u)\equiv(1/s+\sigma^{2}_{\text{n}})\mathbb{I}+u\bar{\sigma}^{2}_{\text{c}}\mathsf{C}_{\text{c}}\,,$
(44)

or, equivalently, the normalized form

$\mathsf{Q}(s;u)/\sigma_{\text{I}}^{2}=\left(\frac{1}{s\sigma_{\text{I}}^{2}}+1-q\right)\mathbb{I}+qu\mathsf{C}_{\text{c}}\,.$
(45)

The calculations proceed for an arbitrary value of $u$, until the texture
integration is performed at the end, as indicated in (9).

##  V Effective Model

A ‘first-principles’ model for the problem of a correlated gamma-fluctuating
target in correlated clutter would be characterized by the specification of
the joint probability distribution for the quadrature RVs
$\hat{\mathbf{X}}_{\text{s}}$ that appear in (15) such that (29) holds.
However, having such a joint probability distribution does not necessarily, or
even likely, lead to a convenient closed-form MGF to which saddle-point
techniques could be easily applied. This is the case with the proposal of the
preceding section, as defined by (43), which appears to be analytically
intractable. So in this section, we propose an alternative ‘effective’ model —
one whose outputs are expected to be that same as those of the first-
principles model, though not specified in terms of physical degrees of freedom
— that is amenable to analytical treatment. Specifically, our approach is to
postulate an explicit MGF that reduces to the expected results in all known
computable special cases. This MGF would then define an ‘effective model’ for
the system. We examine its relationship with the previously defined first-
principles model, show how to compute the survival function from the MGF using
saddle-point techniques and, finally, present two approximation schemes that
reduce computational load significantly while maintaining a high degree of
accuracy.

Let us consider the following ‘effective’ MGF for the problem with correlated
clutter:

$\mathcal{M}_{Z_{\text{avg}}}(s)=\frac{\left[\det\left(\mathbb{I}+(s/M)\left(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}}\right)\right)\right]^{\kappa-1}}{\left[\det\left(\mathbb{I}+(s/M)\left(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}}+\sigma^{2}_{\text{s}}\mathsf{C}_{\text{s}}/\kappa\right)\right)\right]^{\kappa}}\,,$
(46)

which correctly reproduces the special cases (i) $\sigma^{2}_{\text{c}}=0$,
(ii) $\sigma^{2}_{\text{s}}=0$, (iii) $\rho_{\text{c}}=0$, and (iv)
$\kappa=1$. It also satisfies the essential requirement of yielding the
expected mean and variance, as given by (4)–(6). To see that, we construct the
CGF

$\mathcal{K}(t)\;\equiv\;\ln\mathcal{M}_{Z_{\text{avg}}}(t)\;=\;\ln\left\langle
e^{-tZ_{\text{avg}}}\right\rangle_{Z_{\text{avg}}}\,,$ (47)

which, for the present problem, reads

$\displaystyle\begin{aligned} \mathcal{K}(t)&=(\kappa-1)\mathop{\rm
Tr}\nolimits\left\\{\ln\left[\mathbb{I}+(t/M)(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}})\right]\right\\}\\\
&\quad{}-\kappa\mathop{\rm
Tr}\nolimits\left\\{\ln\left[\mathbb{I}+(t/M)(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}}+\sigma^{2}_{\text{s}}\mathsf{C}_{\text{s}}/\kappa)\right]\right\\}\\\
&=-t(\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}+\sigma^{2}_{\text{s}})\\\
&\quad{}+\frac{t^{2}}{2}\biggl{[}\frac{\sigma^{4}_{\text{n}}}{M}+\frac{\sigma^{4}_{\text{c}}}{M^{2}}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{c}}^{2}\\}+\frac{\sigma^{4}_{\text{s}}}{\kappa
M^{2}}\mathop{\rm Tr}\nolimits\\{\mathsf{C}_{\text{s}}^{2}\\}\\\
&\quad{}+2\left(\frac{\sigma^{2}_{\text{n}}\sigma^{2}_{\text{c}}}{M}+\frac{\sigma^{2}_{\text{n}}\sigma^{2}_{\text{s}}}{M}+\frac{\sigma^{2}_{\text{c}}\sigma^{2}_{\text{s}}}{M}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{c}}\mathsf{C}_{\text{s}}\\}\right)\biggr{]}+O(t^{3})\,,\end{aligned}$
(48)

and recall that the mean and variance are given by the first and second
derivatives of the CGF evaluated at the origin. This expression leads directly
to the results given in (4) and (5) for the returned power mean and variance,
respectively, noting that we must set $\nu=\infty$ for the clutter shape
parameter as we have not yet introduced the compound clutter texture. The form
of the MGF in (46) is an ansatz (i.e. trial form) motivated by previous work
on the heuristic generalized Dalle Mese Giuli (DMG) approximation [12].

In order to make contact with the previous approach, we can write

$\displaystyle\begin{aligned}
\mathcal{M}_{Z}(s)&=\frac{1}{s^{M}\det\mathsf{Q}(s)}\left[\frac{\det\mathsf{Q}(s)}{\det\left(\mathsf{Q}(s)+\sigma^{2}_{\text{s}}\mathsf{C}_{\text{s}}/\kappa\right)}\right]^{\kappa}\\\
&=\frac{1}{s^{M}\det\mathsf{Q}(s)}\left[\det\left(\mathbb{I}+\frac{\sigma^{2}_{\text{s}}}{\kappa}\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T}\right)\right]^{-\kappa}\,.\end{aligned}$ (49)

We see that reconciling the first-principles approach of (15) and (29) with
the effective model of (46) would require that

$\left\langle\exp\left(-\tfrac{1}{2}\hat{\mathbf{Y}}^{\sf
T}\mathsf{A}\hat{\mathbf{Y}}\right)\right\rangle_{\hat{\mathbf{Y}}}=\left[\det\left(\mathbb{I}+\mathsf{A}/\kappa\right)\right]^{-\kappa/2}\,,$
(50)

for any positive semi-definite symmetric matrix $\mathsf{A}$. We should note
that this statement does indeed hold for any such diagonal matrix
$\mathsf{A}$, or when $\kappa=1$ in which case $\hat{\mathbf{Y}}$ is a
Gaussian RV. However, this relationship will not hold in general: The RHS
manifestly only depends on the eigenvalues of $\mathsf{A}$. The LHS will
depend only on the eigenvalues if the probability measure on
$\hat{\mathbf{Y}}$ is rotationally invariant, which is not expected to be that
case for $\kappa>1$. Nevertheless, it is useful to consider the special case
where, as is true in the generalized DMG model [12] (discussed in more detail
in a later section), the matrices $\mathsf{C}_{\text{c}}$ and
$\mathsf{C}_{\text{s}}$ commute, and are thus simultaneously diagonalizable.
Then, upon writing
$\mathsf{C}_{\text{c}}=\mathsf{R}^{-1}\mathsf{\Lambda}_{\text{c}}\mathsf{R}$
and
$\mathsf{C}_{\text{s}}=\mathsf{R}^{-1}\mathsf{\Lambda}_{\text{s}}\mathsf{R}$
so that $\mathsf{L}_{\text{s}}=\sqrt{\mathsf{\Lambda}_{\text{s}}}\mathsf{R}$,
we have

$\mathsf{A}\;=\;\sigma^{2}_{\text{s}}\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T}\;=\;\frac{\sigma^{2}_{\text{s}}\mathsf{\Lambda}_{\text{s}}}{1/s+\sigma^{2}_{\text{n}}\mathbb{I}+\sigma^{2}_{\text{c}}\mathsf{\Lambda}_{\text{c}}}\,,$
(51)

which is diagonal, and so the equivalence holds. It follows that the DMG model
is simultaneously an approximation to the effective model and any valid first-
principles model.

It is also true, in the general case, that the correlation matrices
$\mathsf{C}_{\text{c}}$ and $\mathsf{C}_{\text{s}}$ commute asymptotically in
the limit as $M\to\infty$, in the sense that the weak matrix norm of the
commutator vanishes in this limit, provided that they are symmetric Toeplitz
and satisfy the summability conditions [17]

$\sum_{m=1}^{\infty}\left|[\mathsf{C}_{\text{c}}]_{m1}\right|<\infty\,,\quad\sum_{m=1}^{\infty}\left|[\mathsf{C}_{\text{s}}]_{m1}\right|<\infty\,.$
(52)

Thus, (51) holds asymptotically, from which it follows that the effective
model is (at least) asymptotically equivalent to the first-principles model.
Finally, we remark that, when $M=2$, all correlation matrices will have the
circulant form

$\mathsf{C}=\begin{pmatrix}1&\rho\\\ \rho&1\end{pmatrix}\,,$ (53)

and therefore all such matrices commute. We conclude that the first-principles
and effective models must be identical for $M=2$.

###  V-A Saddle-point Method for the Effective Model

A robust and accurate way of computing the survival function for a
distribution from its MGF is provided by saddle-point techniques, which are
especially effective when the MGF is a rational function. This approach has
previously been developed for the case of a correlated gamma-fluctuating
target embedded in uncorrelated clutter speckle [9], and is easily extended to
the general problem.

In the presence of clutter texture, the speckle MGF for the effective model
can be expressed as

$\begin{split}\mathcal{M}^{\text{spk}}_{Z^{\prime}_{\text{avg}}}(s;u)=\frac{\left[\det\left(\mathbb{I}+(s/M)(1-q+qu\mathsf{C}_{\text{c}})\right)\right]^{\kappa-1}}{\left[\det\left(\mathbb{I}+(s/M)(1-q+qu\mathsf{C}_{\text{c}}+S\mathsf{C}_{\text{s}}/\kappa)\right)\right]^{\kappa}}\,.\end{split}$
(54)

where $u$ denotes the normalized texture integration variable. It is
convenient to introduce an aggregated target/clutter correlation matrix
according to

$\mathsf{C}_{\text{sc}}(u)\equiv\frac{1}{qu+S/\kappa}\left[qu\mathsf{C}_{\text{c}}+(S/\kappa)\mathsf{C}_{\text{s}}\right]\,,$
(55)

such that $\mathop{\rm Tr}\nolimits\mathsf{C}_{\text{sc}}=M$. Then, we can
write

$\mathcal{M}^{\text{spk}}_{Z^{\prime}_{\text{avg}}}(s;u)=\frac{\prod_{m=1}^{M}\left[1+a_{qm}(u)s\right]^{\kappa-1}}{\prod_{m=1}^{M}\left[1+a_{m}(u)s\right]^{\kappa}}\,,$
(56)

with

$\displaystyle\begin{aligned}
a_{qm}(u)&=\left[1-q+qu\gamma^{\text{c}}_{m}\right]/M\,,\\\
a_{m}(u)&=\left[1-q+(qu+S/\kappa)\gamma^{\text{sc}}_{m}(u)\right]/M\,,\end{aligned}$
(57)

where

$\displaystyle\begin{aligned}
\\{\gamma^{\text{c}}_{m}:m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{c}})\,,\\\
\\{\gamma^{\text{sc}}_{m}(u):m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{sc}}(u))\,.\end{aligned}$ (58)

It is useful to note that

$\sum_{m=1}^{M}(a_{m}-a_{qm})=S/\kappa\,.$ (59)

The saddle-point equation, for the saddle-point (SP) $s=s_{0}$, is obtained as
the extremum ($\Phi^{\prime}(s)=0$) of Helstrom’s phase function [7]

$\Phi(s;v,u)\equiv\ln\mathcal{M}^{\text{spk}}_{Z^{\prime}_{\text{avg}}}(s;u)-\ln(-s)+sv\,,$
(60)

and reads

$v=\frac{1}{s_{0}}+\kappa\sum_{m=1}^{M}\frac{a_{m}(u)}{1+a_{m}(u)s_{0}}-(\kappa-1)\frac{a_{qm}(u)}{1+a_{qm}(u)s_{0}}\,.$
(61)

The integration contour for the inverse Laplace transform of the MGF that
yields the survival function is taken to be the steepest descent path (SDP),
which is defined by $\mathop{\rm Im}\nolimits\Phi(s)=0$ and passes through the
saddle-point $s_{0}$. Next, for convenience, we set $a_{0}\equiv
a_{q0}\equiv+\infty$ and define

$c_{m}\equiv\frac{1}{s_{0}v+v/a_{m}(u)}\,,\quad
c_{qm}\equiv\frac{1}{s_{0}v+v/a_{qm}(u)}\,,$ (62)

for $m=0,1,\ldots,M$. Then, with $z\equiv(s_{0}-s)v$, we have the associated
$\tau$-phase function [9] defined by

$\tau(z)\equiv\Phi(s_{0};v)-\Phi(s_{0}-z/v;v)\,,$ (63)

and evaluated as

$\displaystyle\begin{aligned}
\tau(z)&=z+\kappa\sum_{m=0}^{M}\ln(1-c_{m}z)-(\kappa-1)\sum_{m=0}^{M}\ln(1-c_{qm}z)\\\
&=-\sum_{n=2}^{\infty}\frac{r_{n}}{n}z^{n}\,,\end{aligned}$ (64)

with coefficients

$r_{n}\equiv\sum_{m=0}^{M}\left[\kappa
c^{n}_{m}-(\kappa-1)c^{n}_{qm}\right]\,,$ (65)

for $n=1,2,\ldots$, arising in its Taylor expansion. Inversion of this
equation (to obtain $z(\tau)$), followed by integration over $\tau$ leads to
the exact representation of the speckle survival function

$\bar{F}_{\text{spk}}(v;qu)=\bar{F}_{\text{sp}}(v;qu){\cdot}\sqrt{\frac{2r_{2}}{\pi}}\int_{0}^{\infty}d\tau\,e^{-\tau}\mathop{\rm
Im}\nolimits z(\tau)\,,$ (66)

assuming that $s_{0}<0$111 The adjustments necessary for positive saddle-
points $s_{0}>0$ and the choice of saddle-point sign are described in [7, 9].,
with the factor

$\bar{F}_{\text{sp}}(v;qu)=\frac{1}{v\sqrt{2\pi
r_{2}}}{\cdot}e^{\Phi(s_{0};v,u)}$ (67)

constituting the basic saddle-point approximation. It only remains to perform
the clutter texture integration to obtain the full survival function

$\bar{F}_{\text{cc}}(v)\simeq\sum_{\ell=1}^{L}w_{\ell}\bar{F}_{\text{spk}}(v;qu_{\ell})\,,$
(68)

where the $w_{\ell}$, $u_{\ell}$ are the weights and nodes, respectively, of
an appropriate Gaussian quadrature of order $L$.

The Padé-adapted version of the $\tau$-phase function reads

$\begin{split}\tau(z)&=z+\ln(1-c_{0}z)\\\
&{}+\kappa\left[(M-1)\ln(1-\bar{c}z)+\ln(1-c_{M}z)\right]\\\
&{}-(\kappa-1)\left[(M-1)\ln(1-\bar{c}_{q}z)+\ln(1-c_{qM}z)\right]\\\
&{}-\sum_{n=2}^{\infty}\frac{\bar{r}_{n}}{n}z^{n}\,,\end{split}$ (69)

with

$\bar{c}\equiv\frac{1}{M-1}\sum_{m=1}^{M-1}c_{m}\,,\quad\bar{c}_{q}\equiv\frac{1}{M-1}\sum_{m=1}^{M-1}c_{qm}\,,$
(70)

and

$\bar{r}_{n}\equiv\sum_{m=1}^{M-1}\left[\kappa(c_{m}^{n}-\bar{c}^{n})-(\kappa-1)(c_{qm}^{n}-\bar{c}_{q}^{n})\right]\,.$
(71)

In the special case of uncorrelated clutter speckle (i.e. the problem
previously studied in [9]), we have $c_{qm}=\bar{c}_{q}$ for all
$m=1,2,\ldots,M$. In this case, (69) reduces to equation (50) of [9]. Also,
the SP equation implies that $r_{1}=1$, which expands to the relationship

$c_{0}+\kappa\left[c_{m}+(M-1)\bar{c}\right]-(\kappa-1)\left[c_{qM}+(M-1)\bar{c}_{q}\right]=1\,.$
(72)

Using this, we can confirm that the contribution to the RHS of (69) linear in
$z$ vanishes, as expected.

The Padé approximation comprises approximating the infinite series in (69) by
a low order Padé approximant, as described in [9]. This serves to greatly
reduce the computational effort required to perform the inversion $z(\tau)$
implicit in the integral of (66) for very little loss in accuracy. It obviates
the need for the large summations that appear in the first equality of (64) to
be performed at every iteration of the complex-plane Newton-Raphson root
finding.

The integration in (66) is equivalent to integration over the steepest descent
path (SDP) of the phase function (60) and leads to a numerically exact result.
On the other hand, depending on the accuracy required, the coefficients
$r_{n}$ of (65) may be used to construct the first few terms of the residuum
series, of which the basic saddle-point (SP) approximation, given by (67), is
the leading contribution.

One should note that, when this technique is used in a compound clutter
problem, because the eigenvalues $\gamma^{\text{sc}}_{m}$ in (57) depend on
the texture integration variable $u$, a separate eigenvalue problem must be
solved at each step of the numerical integration. We proceed to describe some
approximation schemes that avoid this complication.

###  V-B Dalle Mese Giuli (DMG) Approximation

The generalized DMG approximation, as described in [12], is based on the use
of simplified correlation matrices, chosen to reproduce the effective number
of looks corresponding to the given correlation coefficients [2]. It has
already been found to work well in the case of correlated gamma-fluctuating
targets embedded in uncorrelated clutter speckle [12]. In that application,
its adoption reduced computation time relative to the steepest-descent-path
integration of the full effective model by a significant factor (primarily as
all the $M$-fold summations over pulses are eliminated). It also provided the
basis for proposing the MGF of the effective model. Furthermore, as discussed
above, in addition to being an approximation to the effective model, it is
also a bone fide approximation to the first-principles model.

In the generalized DMG approximation, the target and clutter correlation
matrices $\mathsf{C}_{\text{s}}$, $\mathsf{C}_{\text{c}}$ are constructed such
that they commute, and are therefore simultaneously diagonalizable.
Consequently, the speckle MGF of (56) holds with

$\displaystyle\begin{aligned}
a_{qm}(u)&=\left(1-q+qu\gamma^{\text{c}}_{m}\right)/M\,,\\\
a_{m}(u)&=a_{qm}(u)+S\gamma^{\text{s}}_{m}/(\kappa M)\,,\end{aligned}$ (73)

where

$\displaystyle\begin{aligned}
\\{\gamma^{\text{c}}_{m}:m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{c}})\,,\\\
\\{\gamma^{\text{s}}_{m}:m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{s}})\,.\end{aligned}$ (74)

Specifically,

$\displaystyle\begin{aligned}
\gamma^{\text{c}}_{m}=1-\rho^{\prime}_{\text{c}}+M\rho^{\prime}_{\text{c}}\delta_{mM}\,,\\\
\gamma^{\text{s}}_{m}=1-\rho^{\prime}_{\text{s}}+M\rho^{\prime}_{\text{s}}\delta_{mM}\,,\end{aligned}$
(75)

where $\delta_{mn}$ denotes the Kronecker delta. The eigenvalues are expressed
in terms of effective correlation coefficients, given by

$\displaystyle\begin{aligned}
\rho^{\prime}_{\text{s}}&=\sqrt{(M/B-1)/(M-1)}\,,\\\
\rho^{\prime}_{\text{c}}&=\sqrt{(M/L-1)/(M-1)}\,,\end{aligned}$

where the quantities $B$, $L$ represent the effective numbers of looks on the
target and clutter, respectively. These, in turn, can be obtained from the
true correlation coefficients associated with an assumed correlation
structure. For Kanter’s Gauss-Markov correlation model, one has

$\displaystyle\begin{aligned}
B&=M{\cdot}\left[1+\frac{2\rho^{2}_{\text{s}}}{1-\rho^{2}_{\text{s}}}\left(1-\frac{1}{M}{\cdot}\frac{1-\rho^{2M}_{\text{s}}}{1-\rho^{2}_{\text{s}}}\right)\right]^{-1}\,,\\\
L&=M{\cdot}\left[1+\frac{2\rho^{2}_{\text{c}}}{1-\rho^{2}_{\text{c}}}\left(1-\frac{1}{M}{\cdot}\frac{1-\rho^{2M}_{\text{c}}}{1-\rho^{2}_{\text{c}}}\right)\right]^{-1}\,.\end{aligned}$
(76)

All of this ensures that the DMG approximation reproduces the power variance
of the underlying model to the extent possible within an effective-looks
approach.

From (75), we see that $c_{m}=\bar{c}$ and $c_{qm}=\bar{c}_{q}$ for all
$m=1,2,\ldots,M-1$, so that the $\tau$-phase function reads

$\begin{split}\tau(z)&=z+\ln(1-c_{0}z)\\\
&{}+\kappa\left[(M-1)\ln(1-\bar{c}z)+\ln(1-c_{M}z)\right]\\\
&{}-(\kappa-1)\left[(M-1)\ln(1-\bar{c}_{q}z)+\ln(1-c_{qM}z)\right]\,.\end{split}$
(77)

No Padé approximation is necessary in this case, as $\bar{r}_{n}=0$ for all
$n=1,2,\ldots$. The SP equation reads

$\begin{split}v&=\frac{1}{s_{0}}+\kappa\left[\frac{(M-1)a_{1}(u)}{1+a_{1}(u)s_{0}}+\frac{a_{M}(u)}{1+a_{M}(u)s_{0}}\right]\\\
&{}-(\kappa-1)\left[\frac{(M-1)a_{q1}(u)}{1+a_{q1}(u)s_{0}}+\frac{a_{qM}(u)}{1+a_{qM}(u)s_{0}}\right]\,.\end{split}$
(78)

Its solution is equivalent to finding the roots of a quintic polynomial. We
may also note that, here,

$r_{n}=c_{0}^{n}+\kappa\left[(M-1)\bar{c}^{n}+c_{M}^{n}\right]-(\kappa-1)\left[(M-1)\bar{c}_{q}^{n}+c_{qM}^{n}\right]\,,$
(79)

for $n=1,2,\ldots$, with the identity $r_{1}\equiv 1$ following from the SP
equation.

Figure 1: Detection probability for a $\kappa=2$ correlated target in
correlated K-clutter according to the various computational schemes.

Figure 1 shows the detection probability for a false-alarm probability of
$P_{\text{FA}}=10^{-n_{\text{FA}}}$, $n_{\text{FA}}=6$ as a function of SIR
for both exact effective model and DMG approximation, each computed via
integration along the SDP and with the basic SP approximation. The difference
between the SDP integration and SP approximation (solid versus dashed lines)
is barely perceptible.

###  V-C The Diagonal Approximation

Unlike the DMG approximation of the previous section, which has a theoretical
basis arising from the effective looks concept of dealing with correlation,
the diagonal approximation that we now introduce is an ad-hoc scheme, albeit
one that turns out to be remarkably accurate, while conceptually simple, and
exerting a somewhat reduced computational load relative to the full effective
model. It comprises adoption of the effective model while pretending that the
clutter and target correlation matrices commute, and can therefore be
represented by their diagonalized forms. Then, as with the DMG approximation,
we set

$\displaystyle\begin{aligned}
a_{qm}(u)&=\left(1-q+qu\gamma^{\text{c}}_{m}\right)/M\,,\\\
a_{m}(u)&=a_{qm}(u)+S\gamma^{\text{s}}_{m}/(\kappa M)\,,\end{aligned}$ (80)

where

$\displaystyle\begin{aligned}
\\{\gamma^{\text{c}}_{m}:m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{c}})\,,\\\
\\{\gamma^{\text{s}}_{m}:m=1,2,\ldots,M\\}&=\mathop{\rm
eig}\nolimits(\mathsf{C}_{\text{s}})\,,\end{aligned}$ (81)

but now $\mathsf{C}_{\text{c}}$, $\mathsf{C}_{\text{s}}$ are the actual
correlation matrices rather than their simplified DMG counterparts.

The advantage of this over the full effective model is that, in the presence
of compound clutter, eigenvalues of a target-plus-clutter correlation matrix
do not have to be recomputed separately for each texture integration
quadrature node. The diagonal approximation is manifestly exact for the
following special cases:

  * •

when $\rho_{\text{c}}=0$ or $\rho_{\text{s}}=0$;

  * •

when $\rho_{\text{c}}=\rho_{\text{s}}$, provided both the target and clutter
have the same correlation structure;

  * •

when $M=2$ and in the $M\to\infty$ limit;

  * •

and obviously when $S=0$ or $q=0$.

It exhibits high accuracy across the full range of parameters. The deviation
with respect to the exact result increases with power level $v$, but lies well
within 1% for survival probabilities above $10^{-3}$. This is illustrated in
Figure 2 for a case where the deviation is amplified.

Figure 2: Deviation of the survival function from the exact effective model
result for the diagonal (magenta) and basic saddle-point (green)
approximations. The dashed blue curve is the negative logarithm of the exact
survival probability with scale displayed on the RHS.

The magenta curve shows the difference in base-10 logarithms of the survival
function between the exact effective model and the diagonal model. The green
curve is the corresponding difference between the exact effective model and
its basic saddle-point approximation. It can be seen that the basic SP
approximation is worst around the mean of the distribution $v=\bar{v}$, which
occurs at a value of the power level normalized to mean interference given by
$\bar{v}=1+S=6$. The diagonal approximation slowly degrades with increasing
power level $v$, but is generally superior. The power-axis interval shown in
the graph spans a range of survival probabilities from unity to $10^{-3.5}$,
as can be seen from the dashed blue curve representing the exact negative log-
survival function.

###  V-D Effective Model for Non-Fluctuating Targets

To derive the large-$\kappa$ limit of the effective model, we can look at

$\displaystyle\begin{aligned}
&\left[\frac{\det(1/s+\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}})}{\det(1/s+\sigma^{2}_{\text{n}}+\sigma^{2}_{\text{c}}\mathsf{C}_{\text{c}}+\sigma^{2}_{\text{s}}\mathsf{C}_{\text{s}}/\kappa)}\right]^{\kappa}\\\
&=\frac{1}{\det(\mathbb{I}+\sigma^{2}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{C}_{\text{s}}/\kappa)^{\kappa}}\;\mathrel{\mathop{\sim}_{\kappa\to\infty}}\;e^{-\sigma^{2}_{\text{s}}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}}\,,\end{aligned}$
(82)

from which we infer that

$\mathcal{M}_{Z}(s)\mathrel{\mathop{\sim}_{\kappa\to\infty}}\frac{1}{s^{M}\det\mathsf{Q}(s)}e^{-\sigma^{2}_{\text{s}}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}}\,.$ (83)

Equivalently, including the texture variable,

$\begin{split}\mathcal{M}^{\text{spk}}_{Z^{\prime}_{\text{avg}}}(s,u)&\mathrel{\mathop{\sim}_{\kappa\to\infty}}\exp\Biggl{[}-\mathop{\rm
Tr}\nolimits\Bigl{\\{}\ln\left(\mathbb{I}+(s/M)(1-q+qu\mathsf{C}_{\text{c}})\right)\\\
&{}+(s/M)\frac{S\mathsf{C}_{\text{s}}}{\mathbb{I}+(s/M)(1-q+qu\mathsf{C}_{\text{c}})}\Bigr{\\}}\Biggr{]}\,.\end{split}$
(84)

We may now write
$\mathsf{C}_{\text{c}}=\mathsf{R}^{-1}_{\text{c}}\mathsf{\Lambda}_{\text{c}}\mathsf{R}_{\text{c}}$,
so that

$\begin{split}&\mathop{\rm
Tr}\nolimits\frac{\mathsf{C}_{\text{s}}}{\mathbb{I}+(s/M)(1-q+qu\mathsf{C}_{\text{c}})}\\\
&=\sum_{m=1}^{M}\frac{\left[\mathsf{R}_{\text{c}}\mathsf{C}_{\text{s}}\mathsf{R}^{-1}_{\text{c}}\right]_{mm}}{1+(s/M)(1-q+qu\gamma_{m}^{\text{c}})}\,,\end{split}$
(85)

and introduce the quantities

$b_{m}\,\equiv\,(1/M)\left[\mathsf{R}_{\text{c}}\mathsf{C}_{\text{s}}\mathsf{R}^{-1}_{\text{c}}\right]_{mm}\,=\,(1/M)\sum_{n=1}^{M}\left[\mathsf{R}_{\text{c}}\mathsf{R}_{\text{s}}^{\sf
T}\right]_{mn}^{2}\gamma^{\text{s}}_{n}\,.$ (86)

This leads to the representation

$\begin{split}&\mathcal{M}^{\text{spk}}_{Z^{\prime}_{\text{avg}}}(s,u)\mathrel{\mathop{\sim}_{\kappa\to\infty}}{}\\\
&\exp\left\\{-\sum_{m=1}^{M}\left[\ln(1+a_{m}(u)s)+S{\cdot}\frac{b_{m}s}{1+a_{m}(u)s}\right]\right\\}\,,\end{split}$
(87)

and it follows directly from (86) that $\sum_{m=1}^{M}b_{m}=1$ and that
$b_{m}\geq 0$ for all $m=1,2,\ldots,M$. For a true steady target, in which
case $\mathsf{C}_{\text{s}}=\mathsf{\Omega}$, where
$\mathsf{\Omega}_{ij}\equiv 1$ for all $i,j$, we obtain

$b_{m}=\frac{1}{M}\left(\sum_{n=1}^{M}[\mathsf{R}_{\text{c}}]_{mn}\right)^{2}\,.$
(88)

A Swerling-0 target in the first-principles model (given the same target
correlation matrix) yields, in (11), the expectation value

$\left\langle
e^{-i\mathbf{u}\cdot\mathbf{X}_{\text{s}}}\right\rangle_{\mathbf{X}_{\text{s}}}=e^{-i\mathbf{u}\cdot\mathbf{J}}\,,$
(89)

with $\mathbf{J}\equiv\sigma_{\text{s}}{\cdot}(1,\ldots,1)^{\sf T}$, which
leads to the MGF being given precisely by the RHS of (83), upon substituting
$\mathsf{C}_{\text{s}}=\mathsf{\Omega}$ therein. Therefore, we see that the
effective and first-principles models concur in the case of a Swerling-0
target (which, by definition, is fully correlated) in correlated clutter. In
the opposite extreme, where $\mathsf{C}_{\text{s}}=\mathbb{I}$, we have

$\left\langle
e^{-i\mathbf{u}\cdot\mathbf{X}_{\text{s}}}\right\rangle_{\mathbf{X}_{\text{s}}}=\prod_{m=1}^{M}\cos\left(\sigma_{\text{s}}u_{m}\right)\,,$
(90)

which can be shown to yield the MGF

$\begin{split}\mathcal{M}_{Z}(s)&=\frac{1}{s^{M}\det\mathsf{Q}(s)}e^{-\sigma^{2}_{\text{s}}\mathop{\rm
Tr}\nolimits\\{\mathsf{Q}^{-1}(s)\\}}\\\
&{}\times\prod_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{M}\cosh\left(\sigma^{2}_{\text{s}}\mathsf{Q}^{-1}_{ij}(s)\right)\,.\end{split}$
(91)

This result provides evidence that the effective and first-principles models
are not strictly identical. However, as discussed in the Appendix, the fully
phase-uncorrelated steady target is an exceptional case. In the general
partially correlated case, the result generalizes to

$\begin{split}\mathcal{M}_{Z}(s)&=\frac{1}{s^{M}\det\mathsf{Q}(s)}e^{-\sigma^{2}_{\text{s}}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}}\\\
&{}\times\prod_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{M}\cosh\left(\sigma^{2}_{\text{s}}[\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T}]_{ij}\right)\,,\end{split}$ (92)

which follows from the representation
$\mathbf{X}_{\text{s}}=\sigma_{\text{s}}\mathsf{L}_{\text{s}}^{\sf
T}\mathbf{B}$, where the $B_{m}$ are iid Rademacher RVs.

##  VI Monte Carlo Simulation

While the first-principles model, as defined in (43), seems analytically
intractable, it is highly amenable to Monte Carlo simulation, from which an
empirical survival function can be derived. This enables a comparison with the
effective model whose survival function can be explicitly computed via saddle-
point techniques. Therefore, we can call upon the Kolomogorov-Smirnov
statistic to test the null hypothesis that the empirical survival function
derived from MC simulation of the first-principles model and the analytically
computed survival function for the effective model represent the same
distribution.

A Monte Carlo (MC) simulation of the first-principles model, defined by (43),
can be easily performed at the quadrature component level. In this case, the
returned power RV is realized according to

$\begin{split}&Z_{\text{avg}}/\sigma_{\text{I}}^{2}={}\\\
&\frac{1}{2M}\sum_{m=1}^{M}\sum_{\eta=\text{i,q}}\left(\sqrt{1-q}H_{\text{n},\eta
m}+\sqrt{qU}\hat{X}_{\text{c},\eta m}+\sqrt{S}\hat{X}_{\text{s},\eta
m}\right)^{2}\,.\end{split}$ (93)

Assuming Gauss-Markov correlated clutter, we can write

$\hat{X}_{\text{c},\eta m}=\rho_{\text{c}}\hat{X}_{\text{c},\eta
m-1}+\sqrt{1-\rho^{2}_{\text{c}}}H_{\text{c},\eta m}\,,$ (94)

for $m=2,3,\ldots,M$, with $\hat{X}_{\text{c},\eta 1}=H_{\text{c},\eta 1}$.
Here, $H_{\alpha,\eta m}\sim\mathcal{N}(0,1)$ for both
$\alpha=\text{n},\text{c}$. For the target RVs, we have

$\displaystyle\begin{aligned} \hat{X}_{\text{s},\eta
m}&=\sum_{m^{\prime}=1}^{M}\hat{Y}_{\text{s},\eta
m^{\prime}}[\mathsf{L}_{\text{s}}]_{m^{\prime}m}\,,\\\ \hat{Y}_{\text{s},\eta
m}&=(2B_{\eta m}-1)\sqrt{G_{\eta m}}\,,\end{aligned}$ (95)

where the $B_{\eta m}$ represent iid Bernoulli trials such that $B_{\eta
m}\sim\text{Bin}(1,1/2)$, and the $G_{\eta m}$ are iid gamma variates such
that $G_{\eta m}\sim\Gamma(\kappa/2,2/\kappa)$. The RV $U$ generates the
clutter texture distribution. For K-distributed surface clutter,
$U\sim\Gamma(\nu,1/\nu)$.

Figure 3: Survival function for a $\kappa=2$ correlated target in correlated
K-clutter according to the various computational schemes. All curves except
for those from the DMG approximation (red) coincide to within the line-width
of the graph. Figure 4: Log-survival function for a $\kappa=2$ correlated
target in correlated K-clutter according to the various computational schemes.
For the effective model (green) and DMG approximation (red), the dashed curve
for the SP approximation coincides with the solid SDP curve to within the
line-width of the graph.

With the aid of MC simulation, we can compare the empirical survival function
for the first-principles model with the computed survival function for the
effective model. Figure 3 compares two MC runs of the first-principles model
for a gamma-fluctuating target with $\kappa=2$ with the effective and DMG
models each computed via integration along the SDP and basic SP approximation.
A high level of coincidence is observed among all the cases. Figure 4 draws
the same survival function on a logarithmic scale where some discrepancies
become apparent for the DMG approximation (red curve) in the tail. The dashed
green and red curves for the effective model and DMG approximation arising
from the SP approximation coincide with their SDP counterparts to within the
line-width of the graph. Also, no difference between the first-principles
(black and blue curves) and effective (green curve) models is evident. The MC
sample size for each run was $N_{\text{mc}}=10^{4}$.

The null hypothesis that the two models generate the same distribution can be
tested by means of the one-sample Kolmogorov-Smirnov (KS) statistic [18].
Since the population of returned powers associated with the first-principles
model is sampled by MC simulations, we are able generate repeated samples, and
thus construct an empirical distribution for the KS statistic. The results for
the empirical survival function of the KS statistic are presented in Figure 5
and Figure 6 for target fluctuation classes $\kappa=1,2$, respectively.

Figure 5: Empirical KS survival function for a $\kappa=1$ correlated target in
correlated K-clutter, comparing the first-principles and effective models.
Figure 6: Empirical KS survival function for a $\kappa=2$ correlated target in
correlated K-clutter, comparing the first-principles and effective models.

We find that we are unable to reject the null hypothesis that the first-
principles and effective models generate the same distribution at the
$\alpha=0.01$ significance level for almost any combination of parameters,
with the exception of $\kappa>6$ whenever $\mathsf{C}_{\text{s}}$ is very
close (but not exactly equal) to the identity matrix, in which case a small
discrepancy is detectable. The green region is generated by empirical survival
functions derived via bootstrap resampling, and roughly delineates the 1%
confidence interval. The red curve is the Dvoretzky-Kiefer-Wolfowitz (DKW)
bound. The appearance of white space between the red curve and the lower edge
of the green bootstrap region indicates that the null hypothesis should be
rejected. The magenta curve is the survival function for the Kolmogorov
distribution that represents the theoretical distribution for the KS statistic
under the null hypothesis, and is generated by the stochastic process known as
the Brownian bridge. We note that the red DKW bound essentially tracks the
Brownian bridge due to the large sample size. We have also explicitly
indicated the upper and lower confidence limits as computed from Greenwood’s
formula for a significance level of $\alpha=0.01$, indicated by the dashed
black curves. These curves are seen to bound the green bootstrap region.

We have also compared the first-principles model with the DMG approximation.
The results for the empirical KS survival function for a $\kappa=2$ target are
presented in Figure 7. Even though the DMG approximation agrees well with both
the effective and first-principles model, as observed in the previous graphs,
our statistical test has more than sufficient power to unambiguously reject
the null hypothesis that it corresponds to the same distribution.

Figure 7: Empirical KS survival function for a $\kappa=2$ correlated target in
correlated K-clutter, comparing the first-principles model with the DMG
approximation.

###  VI-A Statistical Power

False alarms occur in cases where the null hypothesis is rejected when it
should not have been. Missed detections occur in cases where the null
hypothesis is not rejected when it should have been. The power of a
statistical test is formally defined as one minus the probability of a missed
detection. The statistical power generally increases with sample size, and is
a measure of the sensitivity of the statistical test.

In order to quantify the power (i.e. sensitivity) of the KS test employed in
the foregoing section, one may ask how much must one deform the distribution
of the effective model away from its true functional form before the null
hypothesis that it agrees with the empirical distribution of the first-
principles model is rejected at the desired confidence level. In this section,
we show that the required amount of deformation is tiny. In other words, if
the survival function of the effective model were only very slightly
different, then a discrepancy with respect to the first-principles model would
be detected.

Thus, in order to characterize the statistical power of the KS test, we shall
consider, as an alternative hypothesis, the slightly perturbed effective-model
survival function [19, 20]

$G(x)\equiv\left[\bar{F}(x)\right]^{1+\epsilon}\,,$ (96)

where $|\epsilon|\ll 1$. The value of the perturbation parameter $\epsilon$
can be related to the maximum difference $\delta_{\text{max}}$ between
$\bar{F}(x)$ and $G(x)$. Writing

$\delta(x)\equiv\bar{F}(x)-\left[\bar{F}(x)\right]^{1+\epsilon}\,,$ (97)

we obtain the derivative

$\delta^{\prime}(x)=f(x)\left[(1+\epsilon)\bar{F}(x)^{\epsilon}-1\right]\,.$
(98)

where $f(x)$ denotes the PDF associated with $\bar{F}(x)$. The extremum
condition $\delta^{\prime}(x)=0$ is satisfied when

$\bar{F}(x)=\left(\frac{1}{1+\epsilon}\right)^{1/\epsilon}\,.$ (99)

Hence, we find that

$\delta_{\text{max}}\;=\;\bar{F}(x)\left[1-\bar{F}(x)^{\epsilon}\right]\;=\;\epsilon{\cdot}\left(\frac{1}{1+\epsilon}\right)^{1+1/\epsilon}\,.$
(100)

This may be expressed as

$\displaystyle\begin{aligned}
\ln\delta_{\text{max}}&=\ln\epsilon-(1+1/\epsilon)\ln(1+\epsilon)\\\
&=\ln\epsilon-1+O(\epsilon^{2})\,,\end{aligned}$ (101)

in which case we obtain the approximate inversion $\epsilon\simeq
e{\cdot}\delta_{\text{max}}$. We may now choose small values of
$\delta_{\text{max}}$ and study at which point the theoretical distribution of
the KS statistic under the null hypothesis falls outside the confidence
intervals of the empirical distribution. With $N_{\text{mc}}=10^{4}$ MC
trials, a KS sample size of $K=400$, and model parameters as given in Figure
8, we find a statistical power close to unity at a significance level of
$\alpha=0.01$ when $\delta_{\text{max}}=0.003$. The null and alternative
survival functions for this value of $\delta_{\text{max}}$ are plotted in
Figure 9, where they are seen to be barely distinguishable.

Figure 8: Comparison of the empirical survival function relative to the
perturbed effective model with $\delta_{\text{max}}=0.003$, showing the
$\alpha=0.01$ confidence interval, and the theoretical curve under the null
hypothesis. Figure 9: Comparison of the survival function for the effective
model with its perturbed counterpart for $\delta_{\text{max}}=0.003$ with the
same parameters as in the previous figure.

It is clear in Figure 8, that the Brownian bridge, represented by the magenta
curve, lies just outside the $\alpha=0.01$ confidence intervals, indicated by
the dashed black curves, and hence the situation depicted here shows the
smallest deformation of the effective-model survival function that leads to a
rejection of the null hypothesis.

The upshot of this is that while the KS test is able to detect the difference
between the effective model and a deformation of it that is barely
discernible, as shown in Figure 9, it is unable to detect a discrepancy
between the effective and first-principles models at the same significance
level. It follows that the effective model is a sound proxy for the first-
principles model.

##  VII Discussion

The first-principles and effective models always exhibit the same mean and
variance, and are in exact agreement in the following situations:

  * •

Uncorrelated clutter ($\mathsf{C}_{\text{c}}=\mathbb{I}$);

  * •

Gaussian target ($\kappa=1$) with arbitrary target and clutter correlations;

  * •

Swerling-0 target ($\kappa\to\infty,\mathsf{C}_{\text{s}}=\Omega$) with
arbitrary clutter correlation;

  * •

Fully correlated target ($\mathsf{C}_{\text{s}}=\mathsf{\Omega}$) with
arbitrary clutter correlation;

  * •

$M\leq 2$ and $M\to\infty$;

  * •

Commuting target and clutter correlation matrices (provided
$\mathsf{C}_{\text{s}}\neq\mathbb{I}$);

  * •

Uncorrelated target ($\mathsf{C}_{\text{s}}=\mathbb{I}$), (provided one sets
$\mathsf{R}_{\text{s}}=\mathsf{R}_{\text{c}}$ when $\kappa>1$);

  * •

And, trivially, as $S\to 0$ and $S\to\infty$.

The first and final five cases apply to targets in all finite Swerling
fluctuation classes $\kappa=1,2,\ldots$.

In all other cases studied, provided $\kappa\leq 6$ when
$\mathsf{C}_{\text{s}}$ is close to the identity matrix, the first-principles
and effective models are seen to agree for all practical purposes, as
demonstrated by the examining the KS statistic. Thus, the effective model is a
good proxy for the first-principles model, since the former lends itself to
efficient computation via the saddle-point technique, whereas for the latter
one must resort to MC simulation.

By studying the $\kappa\to\infty$ limit, we are able to show that the first-
principles and effective models do not always coincide. The worst-case
scenario occurs when $\rho_{\text{c}}=1$, $\rho_{\text{s}}=0^{+}$ and $q=1$,
with the SIR and integrated pulses around $S=5$ and $M=10$, respectively.
Physically, this represents a highly unlikely scenario. The fluctuation
parameter $\kappa$ must exceed $\kappa=6$ before we are able to reject the
null hypothesis.

In order to derive the first-principles MGF for the worst case discrepancy, we
set $q=1$, $\mathsf{C}_{\text{s}}=\mathbb{I}$ and

$\mathsf{C}_{\text{c}}\;=\;\mathsf{\Omega}\;=\;\mathsf{R}_{\text{c}}^{\sf
T}\mathsf{\Lambda}_{\text{c}}\mathsf{R}_{\text{c}}\,,\quad\mathsf{\Lambda}_{\text{c}}=\mathop{\rm
diag}\nolimits(M,0,\ldots,0)\,,$ (102)

so that

$\begin{split}\mathcal{M}_{Z^{\prime}_{\text{avg}}}(s)&=\frac{1}{\det[s\hat{\mathsf{Q}}(s)]}e^{-(S/M)\mathop{\rm
Tr}\nolimits\\{\hat{\mathsf{Q}}^{-1}(s)\\}}{\cdot}\\\
&{}\times\prod_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{M}\cosh\left((S/M)\hat{\mathsf{Q}}^{-1}_{ij}(s)\right)\,,\end{split}$
(103)

with $\hat{\mathsf{Q}}(s)=1/s+\mathsf{\Omega}/M$. Then, we have

$\displaystyle\begin{aligned}
\hat{\mathsf{Q}}^{-1}(s)&=\mathsf{R}_{\text{c}}^{\sf T}\mathop{\rm
diag}\nolimits(s/(1+s),s,\ldots,s)\mathsf{R}_{\text{c}}\\\
&=s\mathbb{I}-\frac{s^{2}}{1+s}\mathsf{\Omega}/M\,,\end{aligned}$ (104)

in which case

$\mathop{\rm Tr}\nolimits\\{\hat{\mathsf{Q}}^{-1}(s)\\}=(M-1)s+s/(1+s)\,,$
(105)

and so

$\begin{split}\mathcal{M}_{Z^{\prime}_{\text{avg}}}(s)&=\frac{1}{1+s}e^{-(1-1/M)Ss}{\cdot}e^{-(S/M)s/(1+s)}{\cdot}\\\
&{}\times\cosh^{M(M-1)}\left(\frac{(S/M^{2})s^{2}}{1+s}\right)\,.\end{split}$
(106)

On comparing this with (83) we see that, by neglecting the cosh term, the
survival function for the corresponding effective model is recovered and it
reduces to

$\bar{F}_{\text{eff}}(v;S,M)=\bar{F}_{\text{SW0}}(v-(1-1/M)S;S/M,1)\,,$ (107)

where $\bar{F}_{\text{SW0}}(v;S,M)$ denotes the $M$-pulse Swerling-0 (i.e.
Rician) survival function for SIR $S$, normalized such that

$\int_{0}^{\infty}dv\,\bar{F}_{\text{SW0}}(v;S,M)=1+S\,.$ (108)

This serves to quantify the extent of the discrepancy. The mean KS statistic
in this worst case is around $0.04$ with the discrepancy highly localized in
the vicinity of the knee of the survival function at $v=(1-1/M)S$ as shown in
Figure 10. The survival function for the effective model was computed from
(107) which holds strictly in the limit $\rho_{\text{s}}\to 0$. If we select a
tiny but finite value, such as $\rho_{\text{s}}=10^{-4}$, the KS statistic
drops to $0.008$, for which the null hypothesis cannot be rejected.

Figure 10: Comparison of the survival functions for the effective model and two MC runs of the first-principles model in the worst case scenario. TABLE I: Run-times and accuracies Method | Run-time | Probability error
---|---|---
| (sec) | Absolute | Relative
Effective – SDP | $5.59$ |  |
Effective – SP | $1.45$ | $3.58\times 10^{-2}$ | $2.39\times 10^{-3}$
DMG – SDP | $0.94$ | $3.43\times 10^{-2}$ | $8.45\times 10^{-4}$
DMG – SP | $0.66$ | $6.75\times 10^{-2}$ | $2.30\times 10^{-3}$
Diagonal – SDP | $5.13$ | $1.92\times 10^{-4}$ | $5.50\times 10^{-5}$
Diagonal – SP | $1.12$ | $3.58\times 10^{-2}$ | $2.44\times 10^{-3}$

In Table I, we compare the run-time and accuracy of each of the methods for
calculating the effective model: These comprise the full effective model, the
DMG approximation and the diagonal approximation, each computed within the
basic saddle-point approximation (SP) and with integration along the path of
steepest descent (SDP). The run-time is that required to compute a grid of 100
points on the survival function from the origin to the point where the
probability has decreased to $10^{-3}$. Each probability error measures the
maximum deviation of the survival function relative to the SDP result for the
full effective model, expressed as both absolute and relative error. Each run-
time and error represent an average of 100 randomized parameter combinations.
The SIR is drawn in the range $1\leq S\leq 10$, the shape parameter is drawn
in the range $1\leq\nu\leq 10$, and the clutter-to-interference ratio in the
range $0.5\leq q\leq 1$. The remaining parameters are kept constant with
values $M=100$, $\rho_{\text{c}}=0.75$, $\rho_{\text{s}}=0.95$, $\kappa=2$.
The calculations were performed in Matlab on a laptop computer with an Intel
Core i5-4210U processor running at 1.70 GHz. We see that the DMG approximation
is very fast but is less accurate than the diagonal approximation, which
maintains a higher degree of accuracy but runs only marginally faster than the
full effective model. The main draw-card of the diagonal model is its
conceptual simplicity.

##  VIII Conclusions

Our main conclusion is that, for all practical purposes, the effective model,
defined in terms of its MGF, solves the problem of non-coherent detection of a
gamma-fluctuating target in the presence of simultaneous arbitrary correlation
of target RCS and clutter speckle, and solves it in a computationally
efficient manner. One open question that arises from this work is whether
there exists an alternative first-principles model whose MGF is exactly that
of the effective model.

## Acknowledgment

I would like to thank Stephen Bocquet and Luke Rosenberg for useful
discussions and encouragement during the course of this research.

A shortcoming of the first-principles model appears when dealing with an
uncorrelated target, for which $\mathsf{C}_{\text{s}}=\mathbb{I}$. Here,
$\mathsf{\Lambda}_{\text{s}}=\mathbb{I}$, and so
$\mathsf{L}_{\text{s}}=\mathsf{R}_{\text{s}}$ can be an arbitrary rotation
matrix. However, the MGF given in Section IV is not independent of
$\mathsf{R}_{\text{s}}$. A practical work-around is to select
$\mathsf{R}_{\text{s}}=\mathsf{R}_{\text{c}}$ in this case. This also serves
to bring it into agreement with the associated effective model, introduced in
Section V, which is explicitly independent of $\mathsf{R}_{\text{s}}$ for an
uncorrelated target.

It may also appear that the first-principles model is not well-defined since
the matrix
$\mathsf{L}_{\text{s}}=\sqrt{\mathsf{\Lambda}_{\text{s}}}\mathsf{R}_{\text{s}}$
is not uniquely constructed from the eigenvalue decomposition
$\mathsf{C}_{\text{s}}=\mathsf{R}_{\text{s}}^{\sf
T}\mathsf{\Lambda}_{\text{s}}\mathsf{R}_{\text{s}}$. While this is true for
$\mathsf{C}_{\text{s}}=\mathbb{I}$, it is not generally so: The uncorrelated
target is an exceptional case. The underlying theory assumes a non-degenerate
eigenvalue spectrum (which is violated for an uncorrelated target). The
eigenvalue decomposition partitions the Euclidean space $\mathbb{R}^{M}$ into
$M$ orthogonal directions, and the columns of the matrix
$\mathsf{R}_{\text{s}}$ are the respective unit vectors associated with these
directions. Each of these is uniquely determined up to an overall sign. Thus,
we also have $\mathsf{C}_{\text{s}}=\mathsf{R}_{\text{s}}^{\prime{\sf
T}}\mathsf{\Lambda}_{\text{s}}\mathsf{R}^{\prime}_{\text{s}}$ where
$\mathsf{R}^{\prime}_{\text{s}}=\mathsf{B}\mathsf{R}_{\text{s}}$ for any
diagonal matrix $\mathsf{B}$ with diagonal elements $\pm 1$. Fortunately, the
first-principles model as given in (43) is invariant under the transformation
$\mathsf{L}_{\text{s}}\mapsto\mathsf{B}\mathsf{L}_{\text{s}}$.

It is worth pointing out that the first-principles MGF can be evaluated
explicitly in the case of a fully correlated target
($\mathsf{C}_{\text{s}}=\mathsf{\Omega}$, where $\mathsf{\Omega}_{ij}\equiv 1$
for all $i,j$) for any $\kappa$. Here, the matrix $\mathsf{L}_{\text{s}}$ is
such that, for any matrix $\mathsf{A}$,

$\mathsf{L}_{\text{s}}\mathsf{A}\mathsf{L}_{\text{s}}^{\sf T}=\mathop{\rm
diag}\nolimits\left(\mathop{\rm
Tr}\nolimits\\{\mathsf{\Omega}\mathsf{A}\\},0,\ldots,0\right)\,.$ (109)

In particular,

$\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T}=\mathop{\rm diag}\nolimits\left(\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\},0,\ldots,0\right)\,,$
(110)

in which case

$\hat{\mathbf{Y}}^{\sf
T}(\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}(s)\mathsf{L}_{\text{s}}^{\sf
T})\hat{\mathbf{Y}}=\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}{\cdot}\hat{Y}_{1}^{2}\,.$
(111)

It follows that

$\begin{split}\mathcal{M}_{Z}(s)&=\frac{1}{s^{M}\det\mathsf{Q}(s)}\\\
&{}\times\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}{\cdot}\hat{Y}_{1}^{2}\right)\right\rangle^{2}_{\hat{Y}_{1}}\,.\end{split}$
(112)

The expectation is given by

$\displaystyle\begin{aligned}
&\left\langle\exp\left(-\frac{\sigma^{2}_{\text{s}}}{2}\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}{\cdot}\hat{Y}_{1}^{2}\right)\right\rangle_{\hat{Y}_{1}}\\\
&\quad=\frac{1}{\Gamma(\kappa/2)}\int_{0}^{\infty}dx\,x^{\kappa/2-1}\\\
&\quad\quad{}\times\exp\left\\{-\left[1+(\sigma^{2}_{\text{s}}/\kappa)\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}\right]x\right\\}\\\
&\quad=\left[1+(\sigma^{2}_{\text{s}}/\kappa)\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\\}\right]^{-\kappa/2}\,.\end{aligned}$
(113)

Next, let us look at

$\displaystyle\begin{aligned}
&\ln\det\left[\mathbb{I}+(\sigma^{2}_{\text{s}}/\kappa)\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\right]\\\
&\quad=\mathop{\rm
Tr}\nolimits\left\\{\ln\left[\mathbb{I}+(\sigma^{2}_{\text{s}}/\kappa)\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\right]\right\\}\\\
&\quad=\sum_{n=1}^{\infty}\frac{1}{n}\left(\sigma^{2}_{\text{s}}/\kappa\right)^{n}\mathop{\rm
Tr}\nolimits\left\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)\right\\}^{n}\,,\end{aligned}$
(114)

and observe that

$\mathop{\rm
Tr}\nolimits\left\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}\right\\}^{n}\;=\;\mathop{\rm
Tr}\nolimits\left\\{\mathsf{L}_{\text{s}}\mathsf{Q}^{-1}\mathsf{L}_{\text{s}}^{\sf
T}\right\\}^{n}\;=\;\left(\mathop{\rm
Tr}\nolimits\\{\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}\\}\right)^{n}\,,$ (115)

from which it follows that

$\mathcal{M}_{Z}(s)=\frac{1}{s^{M}\det\mathsf{Q}(s)}{\cdot}\frac{1}{\left[\det\left(\mathbb{I}+\sigma^{2}_{\text{s}}\mathsf{C}_{\text{s}}\mathsf{Q}^{-1}(s)/\kappa\right)\right]^{\kappa}}\,,$
(116)

which, as one can see by appealing to (49) or (83), is in agreement with the
effective model discussed in Section V.

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|  Josef Zuk was awarded a B.Sc.(Hons) in Physics and Mathematics from the
University of Melbourne in 1982. In 1985, he completed a D.Phil. in
Theoretical Particle Physics at the University of Oxford, where he remained as
a research fellow until 1987, before moving on to the Max Planck Institute for
Nuclear Physics in Heidelberg, Germany to take up a research fellowship in
Theoretical Condensed-Matter Physics. In 1990, he commenced a faculty position
in the Department of Physics at the University of Manitoba, Winnipeg, Canada.
Having joined DST Group (then DSTO) in 1995, he is currently a senior research
scientist in the Joint and Operations Analysis Division (JOAD), where he works
in operations research, and is mainly engaged in operational aspects of radar
performance modelling and simulation, particularly for airborne maritime
surveillance radar systems.
---|---