# Outage Performance Analysis of Widely Linear Receivers in Uplink Multi-user
MIMO Systems

Ronghua Gui, Naveen Mysore Balasubramanya, Lutz Lampe

###### Abstract

This paper considers the application of widely linear (WL) receivers in an
uplink multi-user system using real-valued modulation schemes, where the
cellular base station (BS) with multiple antennas provides connectivity for
randomly deployed single-antenna users. The targeted use case is massive
machine type communication (mMTC) with grant-free access in the uplink, where
the network is required to host a large number of low data rate devices
transmitting in an uncoordinated fashion. Four types of WL receivers are
investigated, namely the WL zero-forcing (ZF) and the WL minimum mean-squared
error (MMSE) receivers, along with their enhanced versions employing
successive interference cancellation (SIC) with channel-dependent ordering,
i.e., the WL-ZF-SIC and WL-MMSE-SIC receivers. The outage performances of
these receivers are analytically characterized in the high signal-to-noise
ratio (SNR) regime and compared to those of conventional linear (CL) receivers
using complex-valued modulation schemes. For the non-SIC receivers, we show
that, when compared to the CL counterparts, the WL receivers yield a higher
diversity gain when decoding the same number of users and have the same
diversity gain but a decreased coding gain when the number of users is nearly
doubled. The outage performance analysis of WL-SIC receivers is facilitated by
the marginal distribution of ordered eigenvalues of a real-valued Wishart
matrix. It is shown that the SIC operation with channel-dependent ordering
brings no additional diversity gain to the WL receivers but instead increases
the coding gain. Moreover, the coding gain of WL-SIC receivers grows as the
number of users increases and even exceeds that of CL-SIC receivers under
suitable conditions. For the mMTC scenario with grant-free transmission, it is
demonstrated that the WL receivers outperform their CL counterparts in terms
of offering a lower outage (and packet drop) probability and a higher system
throughput for a given packet drop probability.

###### Index Terms:

Multi-user communication, grant-free access, massive machine type
communication (mMTC), multi-user detection, widely linear (WL) receiver,
multi-input multi-output (MIMO), zero-forcing (ZF), minimum mean-squared error
(MMSE), successive interference cancellation (SIC), outage probability, real
Wishart matrix, ordered eigenvalues.

##  I Introduction

Uplink multi-user communication to a base station (BS) with multiple receiver
antennas forms a multi-input multi-output (MIMO) system, whose performance is
dominated by the coordination and processing of multi-user signals. A key
innovation has been the conception of multi-user detection techniques, which
permit the simultaneous reliable transmission of multiple user streams [1, 2].
Among various multi-user detection techniques, linear multi-user and
successive interference cancellation (SIC) detection are of particular
interest due to their favorable complexity-performance trade-offs.

The performance of conventional linear (CL) multi-user detection can, in
certain scenarios, be enhanced by widely linear (WL) processing [3]. In the
context of data communication, while CL processing refers to the use of
complex-valued signal constellations for signal transmission and reception, WL
processing refers to the use of one-dimensional constellations transmitted
over a complex-valued channel. This specifically includes real-valued
transmission with pulse amplitude modulation (PAM), but also complex-valued
modulations such as offset quadrature amplitude modulation (offset QAM),
minimum shift keying (MSK) and Gaussian minimum shift keying (GMSK), that can
be interpreted as PAM transmission after a derotation operation at the
receiver [4, 5]. As an important instance of this, WL processing has been
adopted at the receiver to perform single antenna interference cancellation
(SAIC) in cellular networks involving one-dimensional modulated signals, such
as, binary phase shift keying (BPSK), MSK, and GMSK [4, 5]. In this scenario,
it has been demonstrated that the optimal WL receiver with an array of $N$
antennas can process up to $2N-1$ interference signals, while the conventional
receiver requires $2N$ antennas to do the same [5].

More generally, it has been shown that WL processing can improve the receiver
performance in MIMO systems if the received signal is second-order noncircular
or improper [6], such as in the case of PAM or GMSK transmission. For example,
the receiver performance improvement obtained using WL detection for single
user MIMO systems is demonstrated in [7, 8, 9, 10]. Particularly, in [9, 10],
the transmitted codes are constructed using a linear combination of input
symbols and their complex conjugate, which aids in WL processing. The
applications of WL processing in various multi-user MIMO systems with real-
valued constellations were investigated in [11, 12, 13, 14].

In the present age, numerous Internet of things (IoT) applications, such as,
smart metering systems and smart buildings, are envisioned under the massive
machine type communication (mMTC) framework, which hosts a large number of
users demanding low data rates while being delay tolerant. In such scenarios,
grant-free transmission is the preferred multiple access strategy [15, 16,
17]. Our previous works [18, 19] demonstrated that the user density in mMTC
systems with grant-free access can be further enhanced by adopting real-valued
modulation along with WL detection at the base station. We presented an
analytical outage performance characterization of the WL zero-forcing (WL-ZF)
and WL minimum mean-squared error (WL-MMSE) receivers in these works.
Specifically, in [19], we considered the packet drop probability for a fixed
target rate for both CL and WL users and grant-free access in an mMTC scenario
as a performance indicator. Based on this metric, we computed the supported
user density as the maximum number of supported users per cell such that the
packet drop probability does not exceed a given threshold (please see [19, eq.
(39)], and the system throughput as the number of correctly decoded packets
per second per Hz (please see [19, eq. (40)]). Further, we demonstrated that
the packet drop probability was dominated by the collision probability of the
users. We showed that, for a fixed target data rate and transmission time
interval (TTI) of the mMTC user application, (i) WL receivers are capable of
resolving more colliding packets, thereby resulting in a lower packet drop
probability than the CL receivers for a given number of users, or
alternatively, (ii) WL processing supports a higher user density and offers a
larger system throughput than CL processing for a given packet drop
probability.

In this work, we further analyze the outage performance of WL receivers for
uplink multi-user MIMO systems. While the analysis is agnostic with regards to
specific applications, the targeted use-case is mMTC using grant-free access
mechanisms as considered in our previous works [18, 19]. We derive the
analytical expressions for diversity and coding gains of the WL-ZF and WL-MMSE
receivers, based on their outage probability in the high signal-to-noise ratio
(SNR) regime. These expressions enable us to analytically characterize the
performance of these WL receivers in high and low data rate scenarios, unlike
the characterization based on heuristic arguments presented in our previous
works [18, 19]. We also introduce the SIC counterparts of these WL receivers,
i.e., WL-ZF-SIC and WL-MMSE-SIC receivers, analyze their outage performance
and demonstrate the system throughput obtained by the use of the SIC-based WL
receivers in an mMTC scenario with grant-free access. Moreover, the outage
performance analysis of WL-SIC receivers is facilitated by the marginal
distribution of the ordered eigenvalues of real-valued Wishart matrix, which
is constructed from the complex channel matrix. However, for real-valued
Wishart matrices, there are no closed-form expressions for the marginal
distribution of their ordered eigenvalues, that can directly be employed or
derived from the existing results for complex-valued Wishart matrices [20, 21,
22, 23, 24, 25, 26]. Therefore, we derive the polynomial approximation of the
marginal cumulative distribution function (CDF) of those ordered eigenvalues
around zero, which is shown to be accurate enough for the outage performance
analysis of WL-SIC receivers in the high SNR regime. It is worth pointing out
that our results of the polynomial approximation for the ordered-eigenvalue
distribution of real Wishart matrix are also novel.

The remainder of this paper is organized as follows. Section II describes the
system model for the uplink multi-user MIMO system with real-valued modulation
schemes and presents the WL versions of conventional ZF and MMSE receivers.
Section III analyzes the outage performance of WL-ZF and WL-MMSE receivers.
Section IV covers the outage performance analysis of the corresponding WL-SIC
receivers. Numerical results are presented in Section V, followed by
conclusions in Section VI.

##  II System Model and Widely Linear Receivers

###  II-A System Model

We consider a cellular uplink transmission scenario, in which the BS with $M$
receive antennas supports a population of single-antenna users. Assuming that
$N$ users are active in the current transmission interval, the complex
baseband model for the multi-user uplink channel is given by

${\mathchoice{\mbox{\boldmath$\displaystyle\bar{y}$}}{\mbox{\boldmath$\textstyle\bar{y}$}}{\mbox{\boldmath$\scriptstyle\bar{y}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{y}$}}}=\sqrt{p_{\mathrm{T,av}}}{\mathchoice{\mbox{\boldmath$\displaystyle\bar{H}$}}{\mbox{\boldmath$\textstyle\bar{H}$}}{\mbox{\boldmath$\scriptstyle\bar{H}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{H}$}}}{\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}^{\frac{1}{2}}{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}+{\mathchoice{\mbox{\boldmath$\displaystyle\bar{n}$}}{\mbox{\boldmath$\textstyle\bar{n}$}}{\mbox{\boldmath$\scriptstyle\bar{n}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{n}$}}},$
(1)

where
${\mathchoice{\mbox{\boldmath$\displaystyle\bar{y}$}}{\mbox{\boldmath$\textstyle\bar{y}$}}{\mbox{\boldmath$\scriptstyle\bar{y}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{y}$}}}\in\mathbb{C}^{M\times
1}$ is the vector of complex received samples, $p_{\mathrm{T,av}}$ denotes the
average transmit power, and
${\mathchoice{\mbox{\boldmath$\displaystyle\bar{H}$}}{\mbox{\boldmath$\textstyle\bar{H}$}}{\mbox{\boldmath$\scriptstyle\bar{H}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{H}$}}}\in\mathbb{C}^{M\times
N}$ is the matrix of $M\times N$ small-scale fading coefficients. For
concreteness, we assume independent and circularly symmetric standard complex
Gaussian distributed entries of $\textstyle\bar{H}$, i.e.,
$\bar{h}_{i,j}\sim\mathcal{C}\mathcal{N}\left(0,1\right)$ for $1\leq i\leq M$,
$1\leq j\leq N$. The diagonal matrix
$\boldsymbol{\varPsi}\triangleq\text{diag}([\xi_{1},\xi_{2},\ldots,\xi_{N}])$
accounts for the received power variation due to power control and large-scale
fading. Its $i^{\mathrm{th}}$ diagonal entry can be expressed as

$\xi_{i}=\frac{p_{\text{T,}i}\beta_{i}\psi_{i}}{p_{\text{T,av}}},$ (2)

where $p_{\text{T,}i}$, $\beta_{i}$ and $\psi_{i}$ denote the instantaneous
transmit power, the pathloss and the shadowing fading factor for user $i$,
respectively. We assume that $\xi_{i}$, $1\leq i\leq N$, are independently and
identically distributed (i.i.d.) random variables, whose distribution is
determined by the user deployment, large-scale fading model and power control
strategies [27]. Without power control, we have
$p_{\text{T,}i}=p_{\text{T,av}}$, $1\leq i\leq N$, and the power variation
term $\xi_{i}=\beta_{i}\psi_{i}$ is the large-scale fading. With perfect power
compensation (PPC),
$p_{\text{T,}i}=p_{\mathrm{T,av}}\frac{\left(\beta_{i}\psi_{i}\right)^{-1}}{\mathbb{E}_{\beta_{i}\psi_{i}}\left\\{(\beta_{i}\psi_{i})^{-1}\right\\}}$,
where $\mathbb{E}_{x}\\{\cdot\\}$ denotes the statistical expectation with
respect to $x$, and the large-scale fading is fully compensated so that
$\xi_{i}=\frac{1}{\mathbb{E}_{\beta_{i}\psi_{i}}\left\\{(\beta_{i}\psi_{i})^{-1}\right\\}}\triangleq\xi_{\text{PPC}}$,
$1\leq i\leq N$. Other cases lie in between those two extremes. The vector
$\textstyle x$ contains the data symbols simultaneously transmitted by the $N$
users, which are generated from one-dimensional constellations, i.e.,
${\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle
x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}\in\mathbb{R}^{N}$. As the $N$ data streams come from multiple
independent users, we assume that
$\mathbb{E}_{{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}}\left\\{{\mathchoice{\mbox{\boldmath$\displaystyle
xx$}}{\mbox{\boldmath$\textstyle xx$}}{\mbox{\boldmath$\scriptstyle
xx$}}{\mbox{\boldmath$\scriptscriptstyle
xx$}}}^{T}\right\\}={\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle I$}}}_{N}$, where $(\cdot)^{T}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle I$}}{\mbox{\boldmath$\textstyle
I$}}{\mbox{\boldmath$\scriptstyle I$}}{\mbox{\boldmath$\scriptscriptstyle
I$}}}_{N}$ denote the transpose operator and the identity matrix of size
$N\times N$, respectively. The complex noise
${\mathchoice{\mbox{\boldmath$\displaystyle\bar{n}$}}{\mbox{\boldmath$\textstyle\bar{n}$}}{\mbox{\boldmath$\scriptstyle\bar{n}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{n}$}}}\in\mathbb{C}^{N}$
is modelled to be white Gaussian with
${\mathchoice{\mbox{\boldmath$\displaystyle\bar{n}$}}{\mbox{\boldmath$\textstyle\bar{n}$}}{\mbox{\boldmath$\scriptstyle\bar{n}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{n}$}}}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{\mathrm{c}}^{2}\right)$,
where $\sigma_{\mathrm{c}}^{2}$ is the variance. For further use, we define
the transmit SNR as

$\mathsf{snr}\triangleq\frac{p_{\mathrm{T,av}}}{\sigma_{\mathrm{c}}^{2}}.$ (3)

In addition to typical real-valued modulations, i.e., PAM, the MIMO model
considered in (1) is also applicable to complex-valued modulation schemes,
such as Gaussian minimum-shift keying (GMSK), whose complex amplitude can be
considered as a filtered version of a real-valued modulation after a
derotation operation [28].

Considering the real-valued transmission, we apply the WL transform111The WL
transform herein augments the $M$-receiver antenna into an equivalent
$2M$-receiver antenna MIMO system, which allows us to apply the methodology
for the analysis of conventional MIMO systems from e.g. [29, 2]. It is
different from the transform that stacks the complex-valued signal and its
complex conjugate, which has widely been used in the literature [3, 6, 7, 30,
5, 31] for the sake of convenience in differentiating a real-valued function
with respect to a complex argument. $\mathcal{T}:\
{\mathchoice{\mbox{\boldmath$\displaystyle\bar{x}$}}{\mbox{\boldmath$\textstyle\bar{x}$}}{\mbox{\boldmath$\scriptstyle\bar{x}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{x}$}}}\in\mathbb{C}^{M}\,\rightarrow\,{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}}\in\mathbb{R}^{2M}$ with
${\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle
x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}=\left[\text{Re}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{x}$}}{\mbox{\boldmath$\textstyle\bar{x}$}}{\mbox{\boldmath$\scriptstyle\bar{x}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{x}$}}}^{T}\right)\
\text{Im}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{x}$}}{\mbox{\boldmath$\textstyle\bar{x}$}}{\mbox{\boldmath$\scriptstyle\bar{x}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{x}$}}}^{T}\right)\right]^{T}$,
where $\text{Re}\left(\cdot\right)$ and $\text{Im}\left(\cdot\right)$ denote
the real and imaginary parts, respectively, of the complex-valued vector
$\textstyle\bar{y}$ defined above. Then we can rewrite the signal model in (1)
as

${\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle
y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle
y$}}}=\sqrt{p_{\mathrm{T,av}}}{\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle
H\varPsi$}}}^{\frac{1}{2}}{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}+{\mathchoice{\mbox{\boldmath$\displaystyle
n$}}{\mbox{\boldmath$\textstyle n$}}{\mbox{\boldmath$\scriptstyle
n$}}{\mbox{\boldmath$\scriptscriptstyle n$}}}$ (4)

where ${\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}=\left[\text{Re}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{H}$}}{\mbox{\boldmath$\textstyle\bar{H}$}}{\mbox{\boldmath$\scriptstyle\bar{H}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{H}$}}}^{T}\right),\text{Im}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{H}$}}{\mbox{\boldmath$\textstyle\bar{H}$}}{\mbox{\boldmath$\scriptstyle\bar{H}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{H}$}}}^{T}\right)\right]^{T}$
is the $2M\times N$ real-valued channel matrix with i.i.d. entries
$h_{i,j}\sim\mathcal{N}\left(0,0.5\right)$ for $1\leq i\leq 2M$, and
${\mathchoice{\mbox{\boldmath$\displaystyle n$}}{\mbox{\boldmath$\textstyle
n$}}{\mbox{\boldmath$\scriptstyle n$}}{\mbox{\boldmath$\scriptscriptstyle
n$}}}=\left[\text{Re}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{n}$}}{\mbox{\boldmath$\textstyle\bar{n}$}}{\mbox{\boldmath$\scriptstyle\bar{n}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{n}$}}}^{T}\right),\text{Im}\left({\mathchoice{\mbox{\boldmath$\displaystyle\bar{n}$}}{\mbox{\boldmath$\textstyle\bar{n}$}}{\mbox{\boldmath$\scriptstyle\bar{n}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{n}$}}}^{T}\right)\right]^{T}$
is the $2M\times 1$ real-valued noise vector with
${\mathchoice{\mbox{\boldmath$\displaystyle n$}}{\mbox{\boldmath$\textstyle
n$}}{\mbox{\boldmath$\scriptstyle n$}}{\mbox{\boldmath$\scriptscriptstyle
n$}}}\sim\mathcal{N}\left(0,0.5\sigma_{\mathrm{c}}^{2}{\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle I$}}}_{N}\right)$. The WL transform
virtually extends the complex data vector
${\mathchoice{\mbox{\boldmath$\displaystyle\bar{y}$}}{\mbox{\boldmath$\textstyle\bar{y}$}}{\mbox{\boldmath$\scriptstyle\bar{y}$}}{\mbox{\boldmath$\scriptscriptstyle\bar{y}$}}}\in\mathbb{C}^{M}$
to the real vector222The log-likelihood ratio (LLR) computation in WL
detection will be same as that adopted in CL processing. That is, after linear
or SIC processing, an effective additive Gaussian noise (AWGN) channel is
assumed and LLRs are computed based on, for example, the max-log
approximation. ${\mathchoice{\mbox{\boldmath$\displaystyle
y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle
y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}}\in\mathbb{R}^{2M}$.

One could employ the CL-ZF and CL-MMSE receivers to the original complex-
valued data vector $\textstyle\bar{y}$ in (1) for decoding the $N$ user data
streams, regardless of the prior knowledge about real-valued transmission.
Consequently, the maximum number of detectable users for the CL receivers
would be limited to be $M$. However, with the WL transform, the number of
receive antennas will be virtually doubled and thus up-to-$2M$ user data
streams can simultaneously be decoded by applying the WL-ZF or WL-MMSE
detector to the real-valued data vector $\textstyle y$ in (4), as briefly
discussed next.

###  II-B Widely Linear Receivers

We assume that the maximum number of detectable users is restricted to be
$N\leq 2M$, and the channel state information (CSI) can be perfectly tracked
by the BS receiver, i.e., ${\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle H\varPsi$}}}^{\frac{1}{2}}$ is
known at the receiver. In analogy to the CL multi-user detection, we consider
the ZF and MMSE criteria for the WL receiver model (4). One can obtain the
detection matrix ${\mathchoice{\mbox{\boldmath$\displaystyle
W$}}{\mbox{\boldmath$\textstyle W$}}{\mbox{\boldmath$\scriptstyle
W$}}{\mbox{\boldmath$\scriptscriptstyle W$}}}\in\mathbb{R}^{2M\times N}$ using
the expressions for CL multi-user detection [2] as

${\mathchoice{\mbox{\boldmath$\displaystyle W$}}{\mbox{\boldmath$\textstyle
W$}}{\mbox{\boldmath$\scriptstyle W$}}{\mbox{\boldmath$\scriptscriptstyle
W$}}}=\begin{cases}{\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle
H\varPsi$}}}^{\frac{1}{2}}\left({\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}^{\frac{1}{2}}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle
H\varPsi$}}}^{\frac{1}{2}}\right)^{-1}\,\,\,\,\text{for\,\,WL~{}ZF}\\\
{\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle
H\varPsi$}}}^{\frac{1}{2}}\left({\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}^{\frac{1}{2}}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H\varPsi$}}{\mbox{\boldmath$\textstyle
H\varPsi$}}{\mbox{\boldmath$\scriptstyle
H\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle
H\varPsi$}}}^{\frac{1}{2}}+\frac{1}{2\mathsf{snr}}{\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle
I$}}}_{N}\right)^{-1}\,\,\,\,\text{for\,\,WL~{}MMSE}.\\\ \end{cases}$ (5)

Applying $\textstyle W$ to the real-valued data model (4), we obtain the
output signal-to-interference-plus-noise ratios (SINRs) of WL-ZF and WL-MMSE
detectors for the $n^{\mathrm{th}}$ data stream as [29, 19]

$\displaystyle\gamma_{\text{WL-ZF},n}$
$\displaystyle=\frac{2\mathsf{snr}\xi_{n}}{\left[\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}\right)^{-1}\right]_{n,n}}$ (6a)
$\displaystyle=2\mathsf{snr}\xi_{n}{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle
h$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
P$}}{\mbox{\boldmath$\textstyle P$}}{\mbox{\boldmath$\scriptstyle
P$}}{\mbox{\boldmath$\scriptscriptstyle
P$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}}^{\bot}{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle h$}}}_{n}$ (6b)

and

$\displaystyle\gamma_{\text{WL-MMSE},n}$
$\displaystyle=\frac{2\mathsf{snr}\xi_{n}}{\left[\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}+\frac{1}{2\mathsf{snr}}{\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}^{-1}\right)^{-1}\right]_{n,n}}-1$
(7a)
$\displaystyle=2\mathsf{snr}\xi_{n}{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle
h$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle\tilde{P}$}}{\mbox{\boldmath$\textstyle\tilde{P}$}}{\mbox{\boldmath$\scriptstyle\tilde{P}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{P}$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}}^{\bot}{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle h$}}}_{n}$ (7b)

where ${\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle h$}}}_{n}$ is the $n^{\mathrm{th}}$
column of $\textstyle H$ and

$\displaystyle{\mathchoice{\mbox{\boldmath$\displaystyle
P$}}{\mbox{\boldmath$\textstyle P$}}{\mbox{\boldmath$\scriptstyle
P$}}{\mbox{\boldmath$\scriptscriptstyle
P$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}}^{\bot}$
$\displaystyle={\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle
I$}}}_{2M}-{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\right)^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}^{T}$ (8a)
$\displaystyle{\mathchoice{\mbox{\boldmath$\displaystyle\tilde{P}$}}{\mbox{\boldmath$\textstyle\tilde{P}$}}{\mbox{\boldmath$\scriptstyle\tilde{P}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{P}$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}}^{\bot}$
$\displaystyle={\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle
I$}}}_{2M}-{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}+\frac{1}{2\mathsf{snr}}{\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}_{n}^{-1}\right)^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}^{T}.$ (8b)

In ${\mathchoice{\mbox{\boldmath$\displaystyle P$}}{\mbox{\boldmath$\textstyle
P$}}{\mbox{\boldmath$\scriptstyle P$}}{\mbox{\boldmath$\scriptscriptstyle
P$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}}^{\bot}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle\tilde{P}$}}{\mbox{\boldmath$\textstyle\tilde{P}$}}{\mbox{\boldmath$\scriptstyle\tilde{P}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{P}$}}}_{{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}_{n}}^{\bot}$,
${\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle
H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\in\mathbb{R}^{2M\times\left(N-1\right)}$ is $\textstyle H$ with the
$n^{\mathrm{th}}$ column removed and
${\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}_{n}\in\mathbb{R}^{\left(N-1\right)\times\left(N-1\right)}$
is $\textstyle\varPsi$ with the $n^{\mathrm{th}}$ column and $n^{\mathrm{th}}$
row removed.

Similar to their CL counterparts, the WL receivers discussed above enable
relatively low-complexity detection at the price of sub-optimal error-rate
performance. Considering that the performance of CL receivers can be enhanced
using SIC with channel-dependent ordering [2], we attempt to improve the
performance of WL receivers using the same methodology. Specifically, for CL-
SIC receivers, the optimal ordering is to choose the user to decode such that
the output SINR is maximized at each decoding stage [32]. We apply this SINR-
maximization ordering rule to the WL-SIC receivers, and then obtain the
$n^{\mathrm{th}}$-stage output SINRs of the WL-ZF-SIC and WL-MMSE-SIC
receivers as

$\displaystyle\gamma_{\text{WL}-\text{ZSIC}}^{\left(n\right)}$
$\displaystyle=\max_{n}\left\\{\gamma_{\text{WL}-\text{ZF,}1}^{\left(n\right)},\gamma_{\text{WL}-\text{ZF,}2}^{\left(n\right)},\cdots,\gamma_{\text{WL}-\text{ZF,}N-n+1}^{\left(n\right)}\right\\}$
(9a) $\displaystyle\gamma_{\text{WL}-\text{MSIC}}^{\left(n\right)}$
$\displaystyle=\max_{n}\left\\{\gamma_{\text{WL}-\text{MMSE,}1}^{\left(n\right)},\gamma_{\text{WL}-\text{MMSE,}2}^{\left(n\right)},\cdots,\gamma_{\text{WL}-\text{MMSE,}N-n+1}^{\left(n\right)}\right\\}$
(9b)

where $\gamma_{\text{WL}-\text{ZF,}i}^{\left(n\right)}$ and
$\gamma_{\text{WL}-\text{MMSE,}i}^{\left(n\right)}$ represent the
$n^{\mathrm{th}}$-stage output SINRs of the WL-ZF and WL-MMSE detectors,
respectively, for the $i^{\mathrm{th}}$ data stream that remains after
previous $(n-1)$ SIC stages, i.e., $1\leq i\leq N-n+1$.

##  III Outage Performance of Widely Linear Receivers

In this section, we consider the WL-ZF and WL-MMSE receivers using $\textstyle
W$ from (5). We first briefly state the SINR and outage probability results
from our previous work [19], which are then used to establish the diversity
and coding gains.

The outage probability for WL receivers can be expressed as [2]

$\mathcal{P}=\mathrm{Pr}\left(\frac{1}{2}\log\left(1+\text{SINR}\right)\leq
R\right)$ (10)

where the factor $1/2$ is due to the use of real-valued transmission and $R$
is the target data rate in bits/sec/Hz. Based on the behaviour of
$\mathcal{P}$ in the high SNR regime, the outage probability of a
communication system as defined in (10) can be expressed as [2]

$\mathcal{P}\left(\mathsf{snr}\right)\simeq\left(C\cdot\mathsf{snr}\right)^{-d}$
(11)

where $C$ is the outage coding gain, $d$ is the outage diversity gain, and
$\simeq$ denotes the asymptotic equality in the high SNR regime. Using the
diversity and coding gains, we can make an analytical comparison between the
asymptotic outage behaviours of WL and CL receivers.

###  III-A Previous Results[19]

####  III-A1 SINR Distribution

The SINR of the WL-ZF receiver is distributed according to

$\frac{\gamma_{\text{WL}-\text{ZF},n}}{\xi_{n}}\sim\mathsf{snr}\cdot\chi_{2M-N+1}^{2}\;,$
(12)

where $\chi_{2M-N+1}^{2}$ represents the standard Chi-squared distribution
with $2M-N+1$ degrees of freedom. As it is difficult to directly characterize
the SINR distribution of the WL-MMSE receiver, we decompose the scaled SINR of
WL-MMSE receiver, i.e., $\frac{\gamma_{\text{WL}-\text{MMSE,}n}}{\xi_{n}}$ as
the sum of the scaled WL-ZF SINR, i.e.,
$\frac{\gamma_{\text{WL}-\text{ZF,}n}}{\xi_{n}}$ and the residual term
$\eta_{\text{WL,}n}$ given by

$\frac{\gamma_{\text{WL}-\text{MMSE,}n}}{\xi_{n}}=\frac{\gamma_{\text{WL}-\text{ZF,}n}}{\xi_{n}}+\eta_{\text{WL,}n}\;,$
(13)

where the residual term
$\eta_{\text{WL},n}\triangleq\frac{\gamma_{\text{WL}-\text{MMSE},n}}{\xi_{n}}-\frac{\gamma_{\text{WL}-\text{ZF},n}}{\xi_{n}}=2\mathsf{snr}\boldsymbol{h}_{n}^{T}\left(\boldsymbol{P}_{\boldsymbol{H}_{n}}^{\bot}-\boldsymbol{\tilde{P}}_{\boldsymbol{H}_{n}}^{\bot}\right)\boldsymbol{h}_{n}$
can be shown to be independent of $\gamma_{\text{WL-ZF},n}/\xi_{n}$ and
approximated as

$\eta_{\text{WL},n}\simeq{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle
h$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\right)^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}_{n}^{-1}\left({\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}\right)^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
h$}}{\mbox{\boldmath$\textstyle h$}}{\mbox{\boldmath$\scriptstyle
h$}}{\mbox{\boldmath$\scriptscriptstyle h$}}}_{n}$ (14)

in the high SNR regime, i.e., $\mathsf{snr}\rightarrow\infty$. Intuitively,
the term $\eta_{\text{WL},n}$ represents the power of the signal component
“hidden” in the range of $\boldsymbol{H}_{n}$ that is recovered by the WL-MMSE
receiver, but nulled out by the WL-ZF receiver.

In general, it is difficult to derive the analytical distribution of SINR gap
$\eta_{\text{WL},n}$. However, for the case of perfect compensation of large-
scale fading we have $\xi_{n}=\xi_{\text{PPC}}$, $1\leq n\leq N$, and thus
${\mathchoice{\mbox{\boldmath$\displaystyle\varPsi$}}{\mbox{\boldmath$\textstyle\varPsi$}}{\mbox{\boldmath$\scriptstyle\varPsi$}}{\mbox{\boldmath$\scriptscriptstyle\varPsi$}}}=\xi_{\text{PPC}}{\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle I$}}}_{N}$. Then, we can provide the
closed-form distribution for the scaled version of $\eta_{\text{WL},n}$ as

$\frac{2M-N+2}{N-1}\xi_{\text{PPC}}\cdot\eta_{\text{WL},n}\sim\mathcal{F}_{N-\text{1},2M-N+2},$
(15)

where $\mathcal{F}_{N-\text{1},2M-N+2}$ denotes the standard $F$-distribution
with degree-of-freedom (DoF) parameters $d_{1}=N-\text{1}$ and $d_{2}=2M-N+2$
as defined in [29, Footnote 2]. Note that the F distribution is interpreted as
the ratio of two chi-squared distributions with degrees of freedom $d_{1}$ and
$d_{2}$ respectively.

####  III-A2 Outage Probability

Applying the SINR distributions from above to the outage probability (10), we
obtain

$\displaystyle\begin{split}\mathcal{P}_{\text{WL}-\text{ZF}}&=\text{Pr}\left(\frac{1}{2}\log\left(1+\gamma_{\text{WL}-\text{ZF},n}\right)\leq
R\right)\\\
&=\text{Pr}\left(\frac{\gamma_{\text{WL}-\text{ZF},n}}{\mathsf{snr}\cdot\xi_{n}}\leq\frac{2^{2R}-1}{\mathsf{snr}\cdot\xi_{n}}\right)\\\
&=\mathbb{E}_{\xi_{n}}\left\\{F_{\chi_{2M-N+1}^{2}}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}\cdot\xi_{n}}\right)\right\\}\end{split}$
(16a)
$\displaystyle\begin{split}\mathcal{P}_{\text{WL}-\text{MMSE}}&=\text{Pr}\left(\frac{1}{2}\log\left(1+\gamma_{\text{WL}-\text{MMSE},n}\right)\leq
R\right)\\\
&\simeq\text{Pr}\left(\frac{\gamma_{\text{WL}-\text{ZF},n}}{\mathsf{snr}\cdot\xi_{n}}+\frac{\eta_{\text{WL},n}}{\mathsf{snr}}\leq\frac{2^{2R}-1}{\mathsf{snr}\cdot\xi_{n}}\right)\\\
&=\mathbb{E}_{\xi_{n},\eta_{\text{WL},n}}\left\\{F_{\chi_{2M-N+1}^{2}}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\left[\frac{1}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}\right]^{+}\right)\right\\}\end{split}$
(16b)

where $\gamma_{\text{WLT}}=2^{2R}-1$ is the SINR threshold of WL receivers,
$F_{\chi_{2M-N+1}^{2}}$ is the CDF of $\chi_{2M-N+1}^{2}$ and
$[x]^{+}=\max(x,0)$. In the PPC case, the outage probability expressions can
be simplified to

$\displaystyle\mathcal{P}_{\text{WL}-\text{ZF}}$
$\displaystyle=F_{\chi_{2M-N+1}^{2}}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}\cdot\xi_{\text{PPC}}}\right)$
(17a) $\displaystyle\mathcal{P}_{\text{WL}-\text{MMSE}}$
$\displaystyle\simeq\mathbb{E}_{\eta_{\text{WL},n}}\left\\{F_{\chi_{2M-N+1}^{2}}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}\cdot\xi_{\text{PPC}}}\left[1-\frac{\xi_{\text{PPC}}}{\gamma_{\text{WLT}}}\eta_{\text{WL},n}\right]^{+}\right)\right\\}$
(17b)

where the probability density function (PDF) of $\eta_{\text{WL},n}$ can be
obtained according to (15).

###  III-B Comparative Analysis

We now use the expressions in (16) to study the diversity and coding gains for
WL detection. For this, we note that the polynomial approximation of
$F_{\chi_{k}^{2}}\left(x\right)$ around $x=0$ is given by

$F_{\chi_{k}^{2}}\left(x\right)=\frac{1}{\left(k/2\right)2^{k/2}\Gamma\left(k/2\right)}x^{k/2}+o\left(x^{k/2}\right),$
(18)

where $\Gamma(a)\triangleq\int_{0}^{\infty}{e^{-t}t^{a-1}\mathrm{d}t}$ is the
Gamma function. Considering the high SNR regime such that
$\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\rightarrow 0$ and letting
$d_{\text{WL}}=2M-\left(N-1\right)/2$, we can expand the outage probability
expressions of WL-ZF and WL-MMSE receivers using (18) in (16a) and (16b), as
follows

$\displaystyle\begin{split}\mathcal{P}_{\text{WL}-\text{ZF}}&\simeq\mathbb{E}_{\xi_{n}}\left\\{\frac{1}{d_{\text{WL}}2^{d_{\text{WL}}}\Gamma\left(d_{\text{WL}}\right)}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}\cdot\xi_{n}}\right)^{d_{\text{WL}}}\right\\}\\\
&=\left(\left[\mathbb{E}_{\xi_{n}}\left\\{\frac{1}{d_{\text{WL}}2^{d_{\text{WL}}}\Gamma\left(d_{\text{WL}}\right)}\left(\frac{\gamma_{\text{WLT}}}{\xi_{n}}\right)^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}}\cdot\mathsf{snr}\right)^{-d_{\text{WL}}}\\\
\end{split}$
$\displaystyle\begin{split}\mathcal{P}_{\text{WL}-\text{MMSE}}&\simeq\mathbb{E}_{\xi_{n},\eta_{\text{WL},n}}\left\\{\frac{1}{d_{\text{WL}}2^{d_{\text{WL}}}\Gamma\left(d_{\text{WL}}\right)}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\left[\frac{1}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\\\
&=\left(\left[\mathbb{E}_{\xi_{n},\eta_{\text{WL},n}}\left\\{\frac{1}{d_{\text{WL}}2^{d_{\text{WL}}}\Gamma\left(d_{\text{WL}}\right)}\left(\gamma_{\text{WLT}}\left[\frac{1}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}}\cdot\mathsf{snr}\right)^{-d_{\text{WL}}}\end{split}$

It is clear that the outage diversity and coding gains of WL-ZF and WL-MMSE
receivers are expressed as

$\displaystyle d_{\text{WL}-\text{ZF}}$
$\displaystyle=d_{\text{WL}-\text{MMSE}}=d_{\text{WL}}$ (19a) $\displaystyle
C_{\text{WL}-\text{ZF}}$
$\displaystyle=\frac{2\left(d_{\text{WL}}\Gamma(d_{\text{WL}})\right)^{1/d_{\text{WL}}}}{\gamma_{\text{WLT}}}T_{\text{WL}-\text{ZF}}$
(19b) $\displaystyle C_{\text{WL}-\text{MMSE}}$
$\displaystyle=\frac{2\left(d_{\text{WL}}\Gamma(d_{\text{WL}})\right)^{1/d_{\text{WL}}}}{\gamma_{\text{WLT}}}T_{\text{WL}-\text{MMSE}}\;,$
(19c)

where

$\displaystyle T_{\text{WL}-\text{ZF}}$
$\displaystyle=\left[\mathbb{E}_{\xi_{n}}\left\\{\xi_{n}^{-d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}}$
(20a) $\displaystyle T_{\text{WL}-\text{MMSE}}$
$\displaystyle=\left[\mathbb{E}_{\xi_{n},\eta_{\text{WL},n}}\left\\{\left(\left[\frac{1}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}}$
(20b)

We observe that the WL-MMSE receiver does not provide an extra diversity gain
over the WL-ZF receiver but rather yields an increase in coding gain since
$T_{\text{WL}-\text{MMSE}}>T_{\text{WL}-\text{ZF}}$. As
$\gamma_{\text{WLT}}\rightarrow\infty$, we have
$T_{\text{WL}-\text{MMSE}}\rightarrow T_{\text{WL}-\text{ZF}}$, i.e., in a
high data rate scenario, the WL-MMSE receiver has no significant outage
performance superiority over the WL-ZF receiver. In contrast, in a low rate
scenario where $\gamma_{\text{WLT}}\rightarrow 0$, the WL-MMSE receiver is
expected to have a notably increased coding gain when compared to the WL-ZF
receiver. It should be noted that these insights were presented using
heuristic arguments in our previous work [19].

For the purpose of a WL-vs.-CL comparison, we present the diversity and coding
gains of CL-ZF and CL-MMSE receivers for decoding data streams generated by
complex-valued modulations333Since the diversity and coding gains of CL-
ZF/-MMSE receivers with complex-valued modulation can be derived using the
same approach as that for deriving (19a)-(19c), their intermediate steps are
omitted due to the space limit. Note that the outage probability expression
for complex-valued transmission should be applied to the output SINR of CL-
ZF/MMSE receivers derived from the complex-valued model (1).

$\displaystyle d_{\text{CL}-\text{ZF}}$
$\displaystyle=d_{\text{CL}-\text{MMSE}}=d_{\text{CL}}=M-(N-1)$ (21a)
$\displaystyle C_{\text{CL}-\text{ZF}}$
$\displaystyle=\frac{\left(d_{\text{CL}}\Gamma(d_{\text{CL}})\right)^{1/d_{\text{CL}}}}{\gamma_{\text{CLT}}}T_{\text{CL}-\text{ZF}}$
(21b) $\displaystyle C_{\text{CL}-\text{MMSE}}$
$\displaystyle=\frac{\left(d_{\text{CL}}\Gamma(d_{\text{CL}})\right)^{1/d_{\text{CL}}}}{\gamma_{\text{CLT}}}T_{\text{CL}-\text{MMSE}}\;,$
(21c)

where $\gamma_{\text{CLT}}=2^{R}-1$ and

$\displaystyle T_{\text{CL}-\text{ZF}}$
$\displaystyle=\left[\mathbb{E}_{\xi_{n}}\left\\{\xi_{n}^{-d_{\text{CL}}}\right\\}\right]^{-1/d_{\text{CL}}}$
(22a) $\displaystyle T_{\text{CL}-\text{MMSE}}$
$\displaystyle=\left[\mathbb{E}_{\xi_{n},\eta_{\text{CL},n}}\left\\{\left(\left[\frac{1}{\xi_{n}}-\frac{\eta_{\text{CL},n}}{\gamma_{\text{CLT}}}\right]^{+}\right)^{d_{\text{CL}}}\right\\}\right]^{-1/d_{\text{CL}}}$
(22b)

with $\eta_{\text{CL},n}$ being the SINR difference between the CL-ZF and CL-
MMSE receivers. In the PPC case, the additional term $\eta_{\text{CL},n}$ is a
scaled $F$-distributed random variable with DoF parameters $d_{1}=2(N-1)$ and
$d_{2}=2(M-N+2)$ [29].

For a fair comparison, we assume that the number of receive antennas $M$ at
the BS and the target rate $R$ for each user are the same for the WL and CL
systems. Denoting the number of users by $N_{\text{WL}}$ and $N_{\text{CL}}$
for the two system paradigms, we can make the following observations.

  * •

For $N_{\text{WL}}=N_{\text{CL}}$, we have $d_{\text{WL}}>d_{\text{CL}}$,
i.e., for detection of the same number of users, the diversity gain of WL
receivers is greater than that of CL receivers. Therefore, in the high SNR
regime, the WL system will have a better error-rate performance than the CL
system.

  * •

For $N_{\text{WL}}<2N_{\text{CL}}-1$, we have $d_{\text{WL}}>d_{\text{CL}}$.
This means that the number of users can be nearly doubled with WL detection,
while having an improved diversity gain.

  * •

For $N_{\text{WL}}=2N_{\text{CL}}-1$, we have $d_{\text{WL}}=d_{\text{CL}}$
and

$\frac{C_{\text{WL}-\text{ZF}}}{C_{\text{CL}-\text{ZF}}}=\frac{2\gamma_{\text{CLT}}}{\gamma_{\text{WLT}}}=\frac{2\left(2^{R}-1\right)}{2^{2R}-1}\triangleq\mathcal{L}(R).$
(23)

Since $\mathcal{L}(R)<1$, the WL-ZF receiver suffers from a decrease in coding
gain in this case. However, as $\mathcal{L}(R)\rightarrow 1$ for $R\rightarrow
0$, the decrease in coding gain of the WL-ZF receiver becomes negligible for
lower target rates.

Since the MMSE receivers converge to the ZF receivers in the high data rate
regime, it follows that

$\frac{C_{\text{WL}-\text{MMSE}}}{C_{\text{CL}-\text{MMSE}}}\approx\frac{C_{\text{WL}-\text{ZF}}}{C_{\text{CL}-\text{ZF}}}=\mathcal{L}(R),\
\ \text{for~{}large}\ R.$ (24)

Hence, the WL-MMSE receiver also has a decreased coding gain compared to the
CL-MMSE counterpart. When $R$ is small, i.e., in the low rate regime, it is
difficult to analytically compare $C_{\text{WL}-\text{MMSE}}$ and
$C_{\text{CL}-\text{MMSE}}$. Through the numerical results, it will be shown
that the decrease in coding gain for the WL-MMSE receiver also becomes
negligible in this case.

  * •

As stipulated from the WL MIMO channel model earlier, the largest number of
users supported by the WL receivers with a positive diversity gain is
$N_{\text{WL}}=2M$.

##  IV Outage Performance of WL-SIC Receivers

In this section, we enhance the WL-ZF and WL-MMSE receivers through the
application of SIC and analyze their outage performances in the high SNR
regime. In order to facilitate the outage probability analysis for WL-SIC
receivers, we first derive the polynomial approximation of the marginal
distribution of ordered eigenvalues for real Wishart matrices .

###  IV-A Ordered-Eigenvalue Distribution of Real Wishart Matrix

Let ${\mathchoice{\mbox{\boldmath$\displaystyle
X$}}{\mbox{\boldmath$\textstyle X$}}{\mbox{\boldmath$\scriptstyle
X$}}{\mbox{\boldmath$\scriptscriptstyle X$}}}\in\mathbb{R}^{n\times m}$ be a
matrix with its entries drawn i.i.d. from a real-valued Gaussian distribution
with zero mean and unit variance and assume $n\leq m$. Furthermore, let
$\lambda_{k}$ for $1\leq k\leq n$ denote the sorted eigenvalues of the real
central Wishart matrix ${\mathchoice{\mbox{\boldmath$\displaystyle
XX$}}{\mbox{\boldmath$\textstyle XX$}}{\mbox{\boldmath$\scriptstyle
XX$}}{\mbox{\boldmath$\scriptscriptstyle XX$}}}^{T}$ with
$0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}<\infty$.

######  Theorem 1.

The polynomial expansion of the marginal CDF of $\lambda_{k}$ around
$\lambda_{k}=0$ can be expressed as

$\mathrm{Pr}(\lambda_{k}<\epsilon)=\beta_{k}\epsilon^{d_{k}}+o(\epsilon^{d_{k}}),$
(25)

where $\beta_{k}$ is a coefficient independent of $\epsilon$ and

$d_{k}=\frac{1}{2}k\left(m-n+k\right).$ (26)

Hence, the marginal CDF of $\lambda_{k}$ satisfies the following asymptotical
property:

$\lim_{\epsilon\rightarrow
0^{+}}\frac{\log\mathrm{Pr}(\lambda_{k}<\epsilon)}{\log\epsilon}=d_{k},\,\,1\leq
k\leq n.$ (27)

The proof is provided in Appendix A. Theorem 1 has complemented the polynomial
approximation of the marginal CDF of the ordered eigenvalues for complex-
valued Wishart matrices provided in [33].

######  Theorem 2.

The polynomial expansion (25) for the marginal CDF of the smallest eigenvalue
$\lambda_{1}$ has the parameters

$\displaystyle d_{1}$ $\displaystyle=\frac{1}{2}(m-n+1)$ (28a)
$\displaystyle\beta_{1}$
$\displaystyle=K_{nm}^{-1}d_{1}^{-1}\sqrt{|{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}|}$ (28b)

with
$K_{nm}=\left(\frac{2^{m}}{\pi}\right)^{\frac{n}{2}}\prod_{i=1}^{n}{\Gamma\left(\frac{m-i+1}{2}\right)\Gamma\left(\frac{n-i+1}{2}\right)}$.
For $n$ odd, the skew-symmetric matrix $\textstyle J$ is $(n-1)\times(n-1)$
with entries

$[{\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}]_{i,j}=-[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}]_{j,i}=2^{b_{i}+b_{j}+1}\Gamma(b_{i})\Gamma(b_{j})\sum_{k=1}^{j-i}{\frac{2^{-(b_{i}+b_{j}-k)}\Gamma(b_{i}+b_{j}-k)}{\Gamma(b_{i})\Gamma(b_{j-k+1})}}$
(29)

where $1\leq i<j\leq n-1$ and $b_{i}=\frac{1}{2}(m-n+1)+i$. For $n$ even, the
skew-symmetric matrix $\textstyle J$ is $n\times n$ with
$[{\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}]_{i,j}$ given by (29) for $1\leq i<j\leq n-1$ and the additional entries
given by $[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}]_{i,n}=-[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}]_{n,i}=2^{b_{i}}\Gamma(b_{i})$
for $1\leq i\leq n-1$ and $[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}]_{n,n}=0$.

The proof is provided in Appendix B.

###  IV-B Asymptotic Outage Probability

Following the analysis of diversity-multiplexing tradeoff for CL-SIC receivers
in [29], one can show that, for any ordering rule, the outage probability of
WL-SIC receivers for the $n^{\mathrm{th}}$ user is asymptotically equal to
that for the first-layer detection. Hence, for the $n^{\mathrm{th}}$ user, the
outage probability of WL-ZF-SIC and WL-MMSE-SIC receivers with SINR-
maximization ordering is given by

$\displaystyle\mathcal{P}_{\text{WL}-\text{ZSIC},n}$
$\displaystyle\simeq\Pr\left(\max_{1\leq i\leq
N}\left\\{\gamma_{\text{WL}-\text{ZF},i}\right\\}\leq\gamma_{\text{WLT}}\right)$
(30a) $\displaystyle\mathcal{P}_{\text{WL}-\text{MSIC},n}$
$\displaystyle\simeq\Pr\left(\max_{1\leq i\leq
N}\left\\{\gamma_{\text{WL}-\text{MMSE},i}\right\\}\leq\gamma_{\text{WLT}}\right).$
(30b)

If the output SINRs were independent, we could readily calculate
$\mathcal{P}_{\text{WL}-\text{ZSIC,}n}$ and
$\mathcal{P}_{\text{WL}-\text{MSIC,}n}$ using the simple order statistics for
independent random variables as in [34]. However, from (6b) and (7b), we
observe that the output SINRs are highly correlated, and thus a different
approach is needed.

####  IV-B1 WL-ZF-SIC Receiver

For convenience, we introduce

$\omega_{n}\triangleq\left[\left(2{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}\right)^{-1}\right]_{n,n},$ (31)

such that (6a) becomes

$\gamma_{\text{WL}-\text{ZF,}n}=\mathsf{snr}\frac{\xi_{n}}{\omega_{n}}.$ (32)

Let $2{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}={\mathchoice{\mbox{\boldmath$\displaystyle
V$}}{\mbox{\boldmath$\textstyle V$}}{\mbox{\boldmath$\scriptstyle
V$}}{\mbox{\boldmath$\scriptscriptstyle
V$}}}{\mathchoice{\mbox{\boldmath$\displaystyle\varLambda
V$}}{\mbox{\boldmath$\textstyle\varLambda
V$}}{\mbox{\boldmath$\scriptstyle\varLambda
V$}}{\mbox{\boldmath$\scriptscriptstyle\varLambda V$}}}^{T}$ be the eigenvalue
decomposition, where ${\mathchoice{\mbox{\boldmath$\displaystyle
V$}}{\mbox{\boldmath$\textstyle V$}}{\mbox{\boldmath$\scriptstyle
V$}}{\mbox{\boldmath$\scriptscriptstyle V$}}}\in\mathbb{R}^{N\times N}$ is an
orthonormal matrix and
${\mathchoice{\mbox{\boldmath$\displaystyle\varLambda$}}{\mbox{\boldmath$\textstyle\varLambda$}}{\mbox{\boldmath$\scriptstyle\varLambda$}}{\mbox{\boldmath$\scriptscriptstyle\varLambda$}}}=\text{diag}\left\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\right\\}$
is diagonal with the ordered eigenvalues
$\lambda_{1}\leq\cdots\leq\lambda_{N}$. Since

$\omega_{n}=\left[{\mathchoice{\mbox{\boldmath$\displaystyle
V\varLambda$}}{\mbox{\boldmath$\textstyle
V\varLambda$}}{\mbox{\boldmath$\scriptstyle
V\varLambda$}}{\mbox{\boldmath$\scriptscriptstyle
V\varLambda$}}}^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle
V$}}{\mbox{\boldmath$\textstyle V$}}{\mbox{\boldmath$\scriptstyle
V$}}{\mbox{\boldmath$\scriptscriptstyle
V$}}}^{T}\right]_{n,n}={\mathchoice{\mbox{\boldmath$\displaystyle
v$}}{\mbox{\boldmath$\textstyle v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle
v$}}}_{n}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle\varLambda$}}{\mbox{\boldmath$\textstyle\varLambda$}}{\mbox{\boldmath$\scriptstyle\varLambda$}}{\mbox{\boldmath$\scriptscriptstyle\varLambda$}}}^{-1}{\mathchoice{\mbox{\boldmath$\displaystyle
v$}}{\mbox{\boldmath$\textstyle v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle
v$}}}_{n}=\sum_{i=1}^{N}{v_{n,i}^{2}\lambda_{i}^{-1}}\geq
v_{n,1}^{2}\lambda_{1}^{-1},$ (33)

where ${\mathchoice{\mbox{\boldmath$\displaystyle
v$}}{\mbox{\boldmath$\textstyle v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle v$}}}_{n}$ is the $n^{\mathrm{th}}$
column of ${\mathchoice{\mbox{\boldmath$\displaystyle
V$}}{\mbox{\boldmath$\textstyle V$}}{\mbox{\boldmath$\scriptstyle
V$}}{\mbox{\boldmath$\scriptscriptstyle V$}}}^{T}$ and $v_{n,i}$ is the
$i^{\mathrm{th}}$ entry of ${\mathchoice{\mbox{\boldmath$\displaystyle
v$}}{\mbox{\boldmath$\textstyle v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle v$}}}_{n}$. Then, the output SINR of
the WL-ZF receiver is upper bounded as

$\mathsf{snr}\frac{\xi_{n}}{\omega_{n}}\leq\mathsf{snr}\cdot\lambda_{1}\frac{\xi_{n}}{v_{n,1}^{2}}.$
(34)

Hence, we have the lower bound

$\mathcal{P}_{\text{WL}-\text{ZF},n}\geq\mathrm{Pr}\left(\mathsf{snr}\cdot\lambda_{1}\frac{\xi_{n}}{v_{n,1}^{2}}\leq\gamma_{\text{WLT}}\right).$
(35)

Using Theorem 1 and a proof similar to that for the CL case [29, Lemma VI.1] ,
we can show that the lower bound above is asymptotically tight, i.e.,

$\mathcal{P}_{\text{WL}-\text{ZF},n}\simeq\mathrm{Pr}\left(\mathsf{snr}\cdot\lambda_{1}\frac{\xi_{n}}{v_{n,1}^{2}}\leq\gamma_{\text{WLT}}\right).$
(36)

Hence, the asymptotic outage performance of the WL-ZF receiver is determined
by the smallest eigenvalue of real Wishart matrix
$2{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle
H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}$. Applying this result to (30a)
and defining $\theta_{n}\triangleq v_{n,1}^{2}/\xi_{n}$ and
$\theta_{\min}\triangleq\min_{n}\\{\theta_{n}\\}$, we obtain the asymptotic
outage probability for the WL-ZF-SIC receiver as

$\mathcal{P}_{\text{WL}-\text{ZSIC},n}\simeq\mathrm{Pr}\left(\max_{n}\left\\{\mathsf{snr}\frac{\lambda_{1}}{\theta_{n}}\right\\}\leq\gamma_{\text{WLT}}\right)=\mathrm{Pr}\left(\lambda_{1}\leq\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\theta_{\min}\right)$
(37)

Since $2{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}$ is a real central Wishart
matrix, the orthonormal matrix $\textstyle V$ is Haar-distributed and
independent of $\textstyle\varLambda$ [35, Sec. 3.2.5], which implies the
statistical independence between $\\{\theta_{1},\ldots,\theta_{N}\\}$ and
$\\{\lambda_{1},\ldots,\lambda_{N}\\}$. From (37) and Theorem 2, it follows
that

$\mathcal{P}_{\text{WL}-\text{ZSIC},n}\simeq\mathbb{E}_{\theta_{\min}}\left\\{\beta_{\text{WL}}\left(\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\theta_{\min}\right)^{d_{\text{WL}}}\right\\},$
(38)

where
$\beta_{\text{WL}}=K_{N,2M}^{-1}d_{\text{WL}}^{-1}\sqrt{\left|{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}_{\text{WL}}\right|}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}_{\text{WL}}$ equals to the matrix $\textstyle J$ in Theorem 2 with $n=N$
and $m=2M$. From this, the diversity and coding gains follow as

$\displaystyle d_{\text{WL}-\text{ZSIC}}$ $\displaystyle=d_{\text{WL}}$ (39a)
$\displaystyle C_{\text{WL}-\text{ZSIC}}$
$\displaystyle=\frac{\beta_{\text{WL}}^{-1/d_{\text{WL}}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{\theta_{\min}}\left\\{(\theta_{\min})^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}}.$
(39b)

From (36) we also obtain an alternative expression of the coding gain of WL-ZF
receiver

$C_{\text{WL}-\text{ZF}}=\frac{\beta_{\text{WL}}^{-1/d_{\text{WL}}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{\theta_{n}}\left\\{(\theta_{n})^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}},$
(40)

which can be shown to be equal to (19b).

Comparing (39b) and (40), we observe that the SIC operation with channel-
dependent ordering, although not improving the diversity gain, increases the
coding gain. Moreover, the coding gain $C_{\text{WL}-\text{ZSIC}}$ increases
with $N$, because the SIC receiver has more candidate users to choose from for
ordering.

####  IV-B2 WL-MMSE-SIC Receiver

It is not tractable to analyze the outage probability of the WL-MMSE-SIC
receiver directly from (7a), since the diagonal entries of $\textstyle\varPsi$
are unequal. Therefore, we turn to the lower and upper bounds of (7a). In
particular, one can easily show that the output SINR (7a) can be bounded by

$\gamma_{n}^{\text{LB}}\leq\gamma_{\text{WL}-\text{MMSE,}n}\leq\gamma_{n}^{\text{UB}},$
(41)

where

$\displaystyle\gamma_{n}^{\text{LB}}$
$\displaystyle=\frac{\mathsf{snr}\cdot\xi_{n}}{\left[\left(2{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}+\frac{\xi_{\max}^{-1}}{\mathsf{snr}}{\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle I$}}}_{N}\right)^{-1}\right]_{n,n}}-1$
(42a) $\displaystyle\gamma_{n}^{\text{UB}}$
$\displaystyle=\frac{\mathsf{snr}\cdot\xi_{n}}{\left[\left(2{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}^{T}{\mathchoice{\mbox{\boldmath$\displaystyle
H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle
H$}}{\mbox{\boldmath$\scriptscriptstyle
H$}}}+\frac{\xi_{\min}^{-1}}{\mathsf{snr}}{\mathchoice{\mbox{\boldmath$\displaystyle
I$}}{\mbox{\boldmath$\textstyle I$}}{\mbox{\boldmath$\scriptstyle
I$}}{\mbox{\boldmath$\scriptscriptstyle I$}}}_{N}\right)^{-1}\right]_{n,n}}-1$
(42b)

with $\xi_{\max}\triangleq\max_{n}\\{\xi_{n}\\}$ and
$\xi_{\min}\triangleq\min_{n}\\{\xi_{n}\\}$. Following the same derivations
for the WL-ZF-SIC receiver above, we arrive at

$\mathcal{P}_{\text{WL}-\text{MSIC},n}\simeq\left(C_{\text{WL}-\text{MSIC},n}\mathsf{snr}\right)^{-d_{\text{WL}}},$
(43)

where $C_{\text{WL}-\text{MSIC,}n}$ is the coding gain bounded as
$C_{n}^{\text{LB}}\leq C_{\text{WL}-\text{MSIC,}n}\leq C_{n}^{\text{UB}}$ with

$\displaystyle C_{n}^{\text{LB}}$
$\displaystyle=\frac{\beta_{\text{WL}}^{-\text{1/}d_{\text{WL}}}}{\gamma_{\text{WLT}}+1}\left[\mathbb{E}_{\theta_{\min},\xi_{\max}}\left\\{\left(\left[\theta_{\min}-\frac{\xi_{\max}^{-1}}{\gamma_{\text{WLT}}+1}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-\text{1/}d_{\text{WL}}}$
(44) $\displaystyle C_{n}^{\text{UB}}$
$\displaystyle=\frac{\beta_{\text{WL}}^{-\text{1/}d_{\text{WL}}}}{\gamma_{\text{WLT}}+1}\left[\mathbb{E}_{\theta_{\min},\xi_{\min}}\left\\{\left(\left[\theta_{\min}-\frac{\xi_{\min}^{-1}}{\gamma_{\text{WLT}}+1}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-\text{1/}d_{\text{WL}}}$
(45)

obtained from $\gamma_{n}^{\text{LB}}$ and $\gamma_{n}^{\text{UB}}$,
respectively.

Since it is difficult to gain insights into the effect of SIC on the coding
gain for WL-MMSE, we consider an alternative path to bound
$C_{\text{WL}-\text{MSIC},n}$. Combining (13) and (36) yields

$\displaystyle\mathcal{P}_{\text{WL}-\text{MSIC},n}$
$\displaystyle\simeq\Pr\left(\max_{n}\left\\{\mathsf{snr}\cdot\lambda_{1}\frac{\xi_{n}}{v_{n1}^{2}}+\xi_{n}\eta_{\text{WL},n}\right\\}\leq\gamma_{\text{WLT}}\right)$
(46)
$\displaystyle\leq\Pr\left(\lambda_{1}\leq\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\min_{n}\left[\frac{v_{n1}^{2}}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}v_{n1}^{2}\right]\right),$
(47)

where for (47) we upper-bounded the asymptotic outage probability assuming an
ordering rule which does not necessarily maximize the SINR. Defining

$\vartheta_{n}\triangleq\left[\frac{v_{n1}^{2}}{\xi_{n}}-\frac{\eta_{\text{WL},n}}{\gamma_{\text{WLT}}}v_{n1}^{2}\right],\quad\vartheta_{\min}\triangleq\min\limits_{n}\vartheta_{n},$
(48)

and using Theorem 2 and the fact that $\lambda_{1}$ and $\vartheta_{\min}$ are
statistically independent,

$\begin{split}\mathrm{Pr}\left(\lambda_{1}\leq\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\vartheta_{\min}\right)\simeq\mathbb{E}_{\vartheta_{\min}}\left\\{\beta_{\text{WL}}\left(\left[\frac{\gamma_{\text{WLT}}}{\mathsf{snr}}\vartheta_{\min}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\end{split}$
(49)

Accordingly, the coding gain of the WL-MMSE-SIC receiver is lower bounded as

$C_{\text{WL}-\text{MSIC}}\geq\frac{\beta_{\text{WL}}^{-\text{1/}d_{\text{WL}}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{\vartheta_{\min}}\left\\{\left(\left[\vartheta_{\min}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-\text{1/}d_{\text{WL}}}$
(50)

For a comparison, coding gain of the WL-MMSE receiver can be re-expressed as

$C_{\text{WL}-\text{MMSE}}=\frac{\beta_{\text{WL}}^{-\text{1/}d_{\text{WL}}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{\vartheta_{n}}\left\\{\left(\left[\vartheta_{n}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-\text{1/}d_{\text{WL}}}$
(51)

which follows from (46) without the $\max$-operator. Comparing (50) and (51),
we observe that, similar to the WL-ZF case, SIC with channel-dependent
ordering increases the coding gain of the WL-MMSE receiver. The difference to
the WL-ZF case is the contribution of the SINR difference term
$\eta_{\text{WL,}n}$ in $\vartheta_{n}$, which creates stronger fluctuation in
the SINR. Hence, we expect that coding gain improvement due to SIC is more
pronounced for WL-MMSE than for WL-ZF.

####  IV-B3 WL-SIC Receivers With PPC

In the PPC case, we obtain the coding gain of the WL-ZF-SIC receiver as

$C_{\text{WL}-\text{ZSIC}}^{\text{PPC}}=\frac{\beta_{\text{WL}}^{-1/d_{\text{WL}}}\xi_{\text{PPC}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{u_{\min}}\left\\{(u_{\min})^{d_{\text{WL}}}\right\\}\right]^{-1/d_{\text{WL}}},$
(52)

where $u_{\min}\triangleq\min_{n}\\{u_{n}\\}$ and
$u_{n}\triangleq|v_{n,1}|^{2}$. Since
$\xi_{\min}=\xi_{\max}=\xi_{\text{PPC}}$, the lower and upper bounds in (44)
coincide, and thus the coding gain of the WL-MMSE-SIC receiver can be
calculated as

$C_{\text{WL}-\text{MSIC}}^{\text{PPC}}=\frac{\beta_{\text{WL}}^{-\text{1/}d_{\text{WL}}}\xi_{\text{PPC}}}{\gamma_{\text{WLT}}+1}\left[\mathbb{E}_{u_{\min}}\left\\{\left(\left[u_{\min}-\frac{1}{\gamma_{\text{WLT}}+1}\right]^{+}\right)^{d_{\text{WL}}}\right\\}\right]^{-\text{1/}d_{\text{WL}}}.$
(53)

For such a system, the SIC operation can only exploit the SINR fluctuation
from the small-scale fading, and we expect that resulting improvement in
coding gain is less significant than that for the system without power
control.

###  IV-C Comparative Analysis

As the diversity gain does not change when applying SIC, we only compare the
coding gains of WL-SIC and CL-SIC receivers. Furthermore, since for
$N_{\text{WL}}<2N_{\text{CL}}-1$, the WL-SIC receivers are expected to have a
better outage performance than the CL-SIC receivers in the high SNR regime due
to $d_{\text{WL}}>d_{\text{CL}}$ (see Section III-B), we are interested in the
case of $N_{\text{WL}}=2N_{\text{CL}}-1$. Finally, for analytical tractability
of the comparison, we focus on the PPC case and comment on the general cases
at the end of this section.

First, we combine (19b) and (52) for the coding gain of WL-ZF with PPC to
obtain the identity

$\beta_{\text{WL}}=2^{-d_{\text{WL}}}\left(d_{\text{WL}}\Gamma(d_{\text{WL}})\right)^{-1}\left[\mathbb{E}_{u_{n}}\\{(u_{n})^{d_{\text{WL}}}\\}\right]^{-1},$
(54)

with which we can further rewrite (52) as

$C_{\text{WL}-\text{ZSIC}}^{\text{PPC}}=\frac{2\left(d_{\text{WL}}\Gamma\left(d_{\text{WL}}\right)\right)^{1/d_{\text{WL}}}\xi_{\text{PPC}}}{\gamma_{\text{WLT}}}\left[\mathbb{E}_{u_{n}}\\{(u_{n})^{d_{\text{WL}}}\\}\right]^{1/d_{\text{WL}}}\left[\mathbb{E}_{u_{\min}}\\{(u_{\min})^{d_{\text{WL}}}\\}\right]^{-1/d_{\text{WL}}}.$
(55)

Then, following the derivations for the WL-ZF-SIC case, the coding gain for
the CL-ZF-SIC receiver is obtained as

$C_{\text{CL}-\text{ZSIC}}^{\text{PPC}}=\frac{\left(d_{\text{CL}}\Gamma(d_{\text{CL}})\right)^{1/d_{\text{CL}}}\xi_{\text{PPC}}}{\gamma_{\text{CLT}}}\left[\mathbb{E}_{\mu_{n}}\\{(\mu_{n})^{d_{\text{CL}}}\\}\right]^{1/d_{\text{CL}}}\left[\mathbb{E}_{\mu_{\min}}\\{(\mu_{\min})^{d_{\text{CL}}}\\}\right]^{-1/d_{\text{CL}}},$
(56)

where $\mu_{n}=|\nu_{n,1}|^{2}$, $\mu_{\min}=\min_{n}\\{\mu_{n}\\}$,
$\nu_{n,1}$ is the first entry of vector
${\mathchoice{\mbox{\boldmath$\displaystyle\nu$}}{\mbox{\boldmath$\textstyle\nu$}}{\mbox{\boldmath$\scriptstyle\nu$}}{\mbox{\boldmath$\scriptscriptstyle\nu$}}}_{n}$,
and
${\mathchoice{\mbox{\boldmath$\displaystyle\nu$}}{\mbox{\boldmath$\textstyle\nu$}}{\mbox{\boldmath$\scriptstyle\nu$}}{\mbox{\boldmath$\scriptscriptstyle\nu$}}}_{n}$
is a unit-length eigenvector of the complex Wishart matrix
$\boldsymbol{\bar{H}}^{H}\boldsymbol{\bar{H}}$.

Relating (55) and (56) for the case $N_{\text{WL}}=2N_{\text{CL}}-1$ and thus
$d_{\text{WL}}=d_{\text{CL}}\triangleq d$, we have

$\frac{C_{\text{WL}-\text{ZSIC}}}{C_{\text{CL}-\text{ZSIC}}}=\frac{2\gamma_{\text{CLT}}}{\gamma_{\text{WLT}}}\frac{\left[\mathbb{E}_{u_{n}}\\{(u_{n})^{d}\\}/\mathbb{E}_{u_{\min}}\\{(u_{\min})^{d}\\}\right]^{1/d}}{\left[\mathbb{E}_{\mu_{n}}\\{(\mu_{n})^{d}\\}/\mathbb{E}_{\mu_{\min}}\\{(\mu_{\min})^{d}\\}\right]^{1/d}}.$
(57)

Furthermore, we show in Appendix C that

$\displaystyle\mathbb{E}_{\mu_{n}}\\{(\mu_{n})^{d}\\}/\mathbb{E}_{\mu_{\min}}\\{(\mu_{\min})^{d}\\}$
$\displaystyle=(N_{\text{CL}})^{d},$ (58a)
$\displaystyle\mathbb{E}_{u_{n}}\\{(u_{n})^{d}\\}/\mathbb{E}_{\mu_{\min}}\\{(u_{\min})^{d}\\}$
$\displaystyle\gtrsim\left(N_{\text{WL}}\right)^{d},$ (58b)

where $\gtrsim$ means “asymptotically” for $N_{\text{WL}}\to\infty$. Thus, we
obtain

$\frac{C_{\text{WL}-\text{ZSIC}}}{C_{\text{CL}-\text{ZSIC}}}\gtrsim\frac{2\gamma_{\text{CLT}}}{\gamma_{\text{WLT}}}\frac{N_{\text{WL}}}{N_{\text{CL}}}=\frac{2\gamma_{\text{CLT}}}{\gamma_{\text{WLT}}}\frac{2N_{\text{CL}}-1}{N_{\text{CL}}}\approx
2\mathcal{L}\left(R\right).$ (59)

As $\mathcal{L}(R)\rightarrow 1$ for $R\rightarrow 0$ (see (23)), we have
$C_{\text{WL}-\text{ZSIC}}/C_{\text{CL}-\text{ZSIC}}\gtrsim 2$. That is, in
low data-rate scenarios, the WL-ZF-SIC receiver has a greater coding gain than
the CL-ZF-SIC receiver, which is contrary to our earlier result for the WL-ZF
receiver. This can be explained by the fact that the WL receiver has more
candidate users to choose from for ordering and thus the coding gain
improvement by SIC is considerably higher for WL than for CL receivers. On the
other hand, for high data-rate scenarios, it is hard to completely compensate
the decrease in coding gain of the WL-ZF receiver, even with the use of SIC.
Consequently, $C_{\text{WL}-\text{ZSIC}}<C_{\text{CL}-\text{ZSIC}}$ for large
$R$.

In summary, as the WL receivers are capable of detecting more users, they
benefit more from SIC than the CL receivers, which is manifested in terms of
the improvement in coding gain, especially in low-rate scenarios. This
improvement is analogous to the multi-user diversity gain in opportunistic
communications [2]. For this reason, we expect that WL-SIC compares even more
favourably to CL-SIC for transmission without or with limited power control,
since the instantaneous received powers fluctuate more and thus the system
provides more diversity.

##  V Numerical Results

In this section, we present numerical performance results to illustrate (a)
the convergence of the asymptotic results in Theorems 1 and 2, (b) the
advantages for various types of WL receivers compared to their CL counterparts
in terms of outage performance, packet drop-out probability and eventually the
system throughput.

###  V-A CDFs of Smallest Eigenvalues

Theorems 1 and 2 provide the asymptotic result

$F_{\lambda_{1}}(\epsilon)=\Pr(\lambda_{1}<\epsilon)=\frac{\sqrt{|{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}|}}{K_{nm}\cdot(m-n+1)/2}\epsilon^{(m-n+1)/2}+o(\epsilon^{(m-n+1)/2})$
(60)

for the smallest eigenvalue of real-valued central Wishart matrices with
$n\leq m$, where $\textstyle J$ and $K_{nm}$ are given in Theorem 2. Fig. 1
shows the asymptotic approximation together with the empirical result from
simulating $\lambda_{1}$ (curves labeled with $k=1$) for several combinations
of $n$ and $m$. For clarity, the $x$-axis is set to be $-\log_{10}(\epsilon)$.
We observe that the asymptotic approximations converge well to the empirical
results as $\epsilon\rightarrow 0$. Fig. 1 also includes the empirical CDF for
the second ($k=2$) smallest eigenvalue in the case of $n=m=2$. We note that
this CDF curve runs parallel to the one for the smallest eigenvalue ($k=1$)
and $n=3$ and $m=6$, which has the same polynomial order
$\frac{1}{2}k\left(m-n+k\right)=2$, as predicted by Theorem 1.

Figure 1: CDF of the $k$th smallest eigenvalue of real-valued central Wishart
matrix ${\mathchoice{\mbox{\boldmath$\displaystyle
XX$}}{\mbox{\boldmath$\textstyle XX$}}{\mbox{\boldmath$\scriptstyle
XX$}}{\mbox{\boldmath$\scriptscriptstyle XX$}}}^{T}$ with
${\mathchoice{\mbox{\boldmath$\displaystyle X$}}{\mbox{\boldmath$\textstyle
X$}}{\mbox{\boldmath$\scriptstyle X$}}{\mbox{\boldmath$\scriptscriptstyle
X$}}}\in\mathbb{R}^{n\times m}$. Solid lines: Asymptotic approximations from
Theorems 1 and 2. Dashed lines: empirical results from simulation.

###  V-B Outage Performance

For the outage performance evaluation, we consider transmission without power
control as an illustrative example. We generate the power variation term for
the $i^{\mathrm{th}}$ user as

$\xi_{i}=\beta_{i}+\psi_{i}\ \text{dB},$ (61)

where $\beta_{i}=-120.9-\text{37.6}\log_{10}(r_{i})\ 0\leq r_{i}\leq
r_{\text{cell}}$ is the pathloss model used in the third Generation
Partnership Project (3GPP) standardization [36] for cellular MTC systems, with
$r_{i}$ being the distance in kilometers between user $i$ and the base
station, and $r_{\text{cell}}=0.91$ km the cell radius. The shadowing fading
factor $\psi_{i}$ in dB is a normally distributed random variable with zero
mean and standard deviation $8$ dB. We assume uniformly randomly distributed
user locations, so that the statistical variation of $\xi_{i}$ comes from both
the shadowing factor $\psi_{i}$ and the distance-dependent pathloss factor
$\beta_{i}$. The per-user rate $R$ is fixed for both WL and CL receivers,
which corresponds to the target data rate of the mMTC user application. Recall
that the WL receiver processes one-dimensional modulated signals while the CL
receiver works with complex-valued modulation. Thus, to ensure the required
rate, if a WL receiver, for instance, adopts a channel code rate of $r$ with
BPSK modulation over a TTI equal to $T$, the corresponding CL receiver chosen
for comparison will adopt quaternary PSK (QPSK) with a code rate $r/2$ over
the same TTI ($T$).

Fig. 2(a) and Fig. 2(b) show the analytical and simulated results444When
calculating the asymptotic outage probability of WL receivers, we encounter
the statistical expectations in the coding gains (20a), (20b), (39b) and (50)
for the WL-ZF, WL-MMSE, WL-ZF-SIC and WL-MMSE-SIC receivers, respectively.
These expectations can be quickly computed via the Monte Carlo integration,
which needs to generate i.i.d. samples of $\xi_{n}$, $\eta_{\text{WL},n}$ and
$v_{n,1}$ for $1\leq n\leq N$. The i.i.d. samples of $\xi_{n}$ and
$\eta_{\text{WL},n}$ can be easily generated using (61) and (14),
respectively. The i.i.d. samples of $\left\\{v_{n,1}\right\\}$ can be
conveniently drawn from the standard normal distribution (see Appendix C). for
the outage probability (10) versus transmit SNR (3) for the investigated WL
receivers without power control and with perfect power control (PPC),
respectively. The scenario of $N=4$ users with a data rate $R=2$ bits/sec/Hz
and a base station with $M=2$ receive antennas is considered. The analytical
results for WL-ZF, WL-MMSE and WL-ZF-SIC receivers converge to the simulated
curves in performance range of interest, i.e., at the outage probability of
about $10^{-1}$ to $10^{-2}$. Note that for the WL-MMSE-SIC receiver, the
asymptotic result is obtained from the lower bound of the coding gain (50),
and hence the simulated curve is below the asymptotic curve in the high SNR
regime. We further observe that, as it has been shown by the analysis in
Section IV, the SIC operation does not provide an extra diversity gain but
increases the coding gain. Moreover, as pointed out in Section IV-C, the
improvement in coding gain is more significant for the WL-MMSE than for the
WL-ZF detection.

(a) Without power control (b) With perfect power control

Figure 2: Outage probability vs. transmit SNR for different WL receivers.
$N=4$ users with data rate $R=2$ bits/sec/Hz and a base station with $M=2$
receive antennas.

(a) $N_{\text{WL}}=2,N_{\text{CL}}=2,R=2$ bits/sec/Hz (b)
$N_{\text{WL}}=3,N_{\text{CL}}=2,R=4$ bits/sec/Hz (c)
$N_{\text{WL}}=3,N_{\text{CL}}=2,R=0.3$ bits/sec/Hz (d)
$N_{\text{WL}}=4,N_{\text{CL}}=2,R=0.3$ bits/sec/Hz

Figure 3: Asymptotic outage probabilities vs. transmit SNR for different WL
and CL receivers. The number of receiver antennas at the base station is fixed
to $M=2$. The number of users and the target rates vary. The asymptotic
results for both WL-MMSE-SIC and CL-MMSE-SIC receivers are obtained from the
corresponding asymptotic lower bounds for the coding gains.

Finally, we compare the performance of WL and CL receivers based on asymptotic
outage probabilities. Fig. 3 shows the results for $M=2$ receive antennas and
different numbers of users $N_{\text{WL}}$ for the WL and $N_{\text{CL}}$ for
the CL case, and different target rates $R$. The asymptotic results for the
WL-MMSE-SIC and CL-MMSE-SIC receivers are obtained from the corresponding
lower bounds of the coding gain (see (50) for the WL case).

For the case of equal number of users, $N_{\text{WL}}=N_{\text{CL}}=2$ in Fig.
3(a), we observe that the diversity gains of all WL receivers are greater than
those of the CL receivers. Therefore, one can expect that the WL receivers
will provide improved transmission reliability (i.e., lower outage
probability) over their CL counterparts at high SNR.

Next, we consider $N_{\text{WL}}=2N_{\text{CL}}-1$ such that the diversity
gains of WL and CL receivers become identical and compare the asymptotic
outage probabilities for different target data rates. Figs. 3(b) and 3(c) show
the results for $N_{\text{WL}}=2N_{\text{CL}}-1=3$ and a relatively high and a
low target rate, respectively. For the high-rate case with $R=4$ bits/sec/Hz,
we observe that the WL receivers experience a decrease in coding gain, which
is less pronounced for the WL-SIC receivers. The situation changes when the
target data rate is decreased to $R=0.3$ bits/sec/Hz. For WL-ZF and WL-MMSE
receivers, the decrease in coding gain compared to CL-ZF and CL-MMSE becomes
negligible (see Section III-B). Consequently, the WL-SIC receivers obtain even
improved coding gains compared to those for the CL-SIC counterparts, which was
predicted by the discussion in Section IV-C.

In Fig. 3(d), the number of WL users is further increased to
$N_{\text{WL}}=2N_{\text{CL}}=4$, while the target data rate is kept to be
$R=0.3$ bits/sec/Hz. We observe that in this case the the WL receivers have a
lower diversity gain than the CL receivers, which is clearly undesirable at
high SNR. We remark that for all the comparisons above, the rates per user
were chosen identical for WL and CL cases and thus the overall system rate is
constant when $N_{\text{WL}}=N_{\text{CL}}$ and higher for the WL scenarios
when $N_{\text{WL}}>N_{\text{CL}}$. In summary, the numerical results at high
SNR match well with our asymptotic analysis for both high and low rate
scenarios.

The outage probability analysis can be used to determine the packet drop-out
probability and hence the system throughput [19], which is described in the
following subsection.

###  V-C Packet Drop-out Probability and System Throughput

####  V-C1 Simulation set-up

The simulation set-up for characterizing the performance of the proposed WL
receivers corresponds to an NB-IoT scenario operating over a system bandwidth
of 180 kHz (48 tones with a subcarrier spacing of 3.75 kHz). We adopt a grant-
free access mechanism, where users access the network in a contention based
manner. We consider open loop power control where each user transmits at the
maximum allowable transmit power (23 dBm) over a single subcarrier. The path-
loss/shadowing follows the 3GPP path-loss model in [Annex D, [36]]. The TTI
for a WL user is taken to be 32 ms for a transport block (packet) size is 32
bits. The data arrival process is Poisson with a rate $\lambda_{a}=4.16\times
10^{-6}$ packets/TTI/user. The target data rate for each user is $R=0.3$
bits/s/Hz. The quality of a link is considered to be good if the packet drop
probability is $\leq$1%. As mentioned in Section V-B, the WL and CL users
adopt BPSK and QPSK, respectively, and they transmit for the same TTI. Hence,
the code rate of the WL users is set to $r$, while that of the CL users is set
to $r/2$ (in order to maintain the same target rate $R$).

(a) Packet drop probability (b) Throughput

Figure 4: Performance comparison of CL and WL receivers with same TTI.

####  V-C2 Results

Fig. 4(a) and Fig. 4(b) demonstrate the results for packet drop probability
and system throughput, respectively, computed as per our previous work in
[19]. Specifically, we have considered the best performing receivers for CL
and WL, CL-MMSE-SIC and WL-MMSE-SIC, respectively, and the simplest WL
receiver (WL-ZF). It is evident from Fig. 4(a) that the number of users
supported by CL is around 50k users per cell, while that using WL is about
500k for $M=1$. When $M=2$, CL and WL support around 650k users and 3 million
users per cell, respectively. The improvement obtained in WL processing is
because of its ability to resolve collisions more effectively than CL.

(a) Packet drop probability (b) Throughput

Figure 5: Performance comparison with CL and WL receivers with CL users
occupying half TTI.

####  V-C3 Discussion

Hitherto we have considered that users transmit over the same duration. If
this restriction is relaxed, the CL users could operate in a mode where the
code rate is equal to that of the WL users, i.e., $r$, but the transmission
rate is $2R$ over half the duration (TTI/2) as that of WL users (in order to
maintain the same target rate $R$). Such a mode would be advantageous for CL
users, since it helps to reduce collision rates owing to the shorter
transmission interval. In this case, the outage probability analysis can be
done by replacing $R$ with $2R$ in the equations corresponding to CL
receivers. The packet-drop and throughput in this case can be analyzed by
using the packet transmission rate as $\lambda_{a}/2$ per TTI. The results
corresponding to this mode of operation are indicated by the label CL (half
TTI) in Fig. 5. As expected, CL (half TTI) results in lower packet drop
probability when compared to CL using the same TTI. Consequently, for a packet
drop probability of $1\%$, the CL (half TTI) supports around 80k users per
cell and 1.2 million users per cell for $M=1$ and $M=2$, respectively.

We note that one could choose a larger constellation and reduce the
transmission interval for WL users too, e.g. the combination of 4-PAM and
TTI/2 for WL users. However, even without this alteration, WL processing
supports higher number of users (500k and 3 million users per cell,
respectively) than CL (half TTI), since it is better at resolving collisions.
Furthermore, we would like to point that an optimization of the transmission
duration would not be suitable for grant-free transmission.

Thus, our analysis and the results above indicate that the use of real-valued
transmission with WL detection for uplink transmission would support a larger
number of users, i.e., $N>M$ per time-frequency resource.

##  VI Conclusions

In this paper, we have analyzed the outage performance of various WL receivers
in an uplink multi-user MIMO system with real-valued modulations, in terms of
diversity and coding gains. We prove that WL receivers with $N_{\textrm{WL}}$
real-valued user signals have a higher diversity gain than their CL
counterparts with $N_{\textrm{CL}}$ complex-valued user signals when
$N_{\textrm{WL}}<2N_{\textrm{CL}}-1$ and provide the same diversity gain when
$N_{\textrm{WL}}=2N_{\textrm{CL}}-1$. We show that WL receivers with $M$
antennas can support a maximum number of $N_{\textrm{WL}}=2M$ users with a
positive diversity gain. Amongst the WL receivers serving $N$ users, we show
that the WL-ZF and WL-MMSE receivers have the same diversity gain
($d_{\mathrm{WL}}=M-(N-1)/2$), but different coding gains. Moreover, the SIC
operation with channel-dependent ordering brings no additional diversity gain
to the WL-ZF and WL-MMSE receivers but increases their coding gains. The
increase in coding gain due to SIC grows as the number of users increases.
Furthermore, through analytical comparisons and numerical results, we show
that the WL receivers are superior to the CL receivers. For grant-free mMTC
(with a fixed TTI), WL processing results in improved transmission reliability
(higher diversity gain) and user-multiplexing capability (supporting more
users). Finally, we note that the WL receiver performance has been analyzed
for Rayleigh fading channels. The analysis for more general cases, such as
Ricean fading channels, is expected to be more intricate and will thus be an
interesting direction of work in the near future. We conclude that WL
processing would be beneficial for numerous mMTC applications, such as, data
transfer from smart metering systems, industrial asset tracking, transport
fleet management, smart buildings, etc. that demand low per-user data rates
and are also delay tolerant while supporting high user densities.

##  Appendix A Proof of Theorem 1

We first present three Lemmas that we will use for proving Theorem 1.

######  Lemma 1.

[26] For an arbitrary $n\times n$ matrix
${\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}({\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}})$ whose $(i,j)$th entry can be
expressed as
$\left[{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}({\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}})\right]_{i,j}=\Phi_{i}(x_{j})$
for $1\leq i,j\leq n$, the following identity holds:

$\int_{\mathcal{D}_{\text{ord}}}{\left|{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}({\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}})\right|\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}}=\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}],$ (62)

where $\mathcal{D}_{\text{ord}}=\left\\{a\leq x_{1}\leq\ldots\leq x_{n}\leq
b\right\\}$ stands for the integral domain with ordered entries of
${\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle
x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}=[x_{1}\ x_{2}\ \ldots\ x_{n}]^{T}$,
$\left|{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}({\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}})\right|$ denotes the determinant
of
${\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}({\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}})$ and
$\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}]$ is the Pfaffian of the skew-
symmetric matrix $\textstyle A$. The Pfaffian of skew-symmetric matrix
$\textstyle A$ is a polynomial in the entries of $\textstyle A$ whose square
is the determinant $|{\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}|$, i.e.,
$\left[\text{Pf}({\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle
A$}}})\right]^{2}=|{\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}|$.

For $n$ even, matrix $\textstyle A$ is $n\times n$ with its $(i,j)$th entry

$a_{i,j}=\int_{a}^{b}{\int_{a}^{b}{\mathcal{R}(y-x)\Phi_{i}(x)\Phi_{j}(y)\mathrm{d}x\mathrm{d}y}},\quad
1\leq i<j\leq n.$ (63)

where $\mathcal{R}(y-x)$ is equal to $1$ for $y\geq x$ and to $-1$ otherwise.
For $n$ odd, the skew-symmetric matrix $\textstyle A$ is $(n+1)\times(n+1)$
with $a_{i,j}$ given by (63) for $1\leq i<j\leq n$. The additional entries are
$a_{i,n+1}=-a_{n+1,i}=\int_{a}^{b}{\Phi_{i}(x)\mathrm{d}x}$ for $1\leq i\leq
n$ and $a_{n+1,n+1}=0$.

######  Lemma 2.

[37] Let ${\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}\in\mathbb{R}^{2m\times 2m}$ be a
skew-symmetric matrix with its $(i,j)$th entry $a_{i,j}$, $1\leq i,j\leq 2m$.
The Pfaffian of $\textstyle A$ can be expressed as

$\displaystyle\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}]$
$\displaystyle=\sum_{\sigma\in\mathcal{S}}{\text{sgn}(\sigma)\prod_{i=1}^{m}{a_{\sigma_{2i-1},\sigma_{2i}}}}$
(64a)
$\displaystyle=\sum_{k=1}^{m}{\sum_{\sigma\in\mathcal{S}_{k}}{\text{sgn}(\sigma)a_{\sigma_{2k-1},2m}\prod_{i=1,i\neq
k}^{m}{a_{\sigma_{2i-1},\sigma_{2i}}}}}$ (64b)

where $\sigma=\sigma_{1},\sigma_{2},\cdots,\sigma_{2m}$ is a permutation of
the integers $1,2,...,2m$, ${\cal S}$ is the set of all permutations $\sigma$
satisfying $\sigma_{1}<\sigma_{3}<\cdots<\sigma_{2m-1}$ and
$\sigma_{2i-1}<\sigma_{2i}$ for $1\leq i\leq m$, and ${\cal S}_{k}$ is the
$k$th subset of ${\cal S}$ including all permutations satisfying
$\sigma_{2k}=2m$. In (64a) and (64b), the sums are over all permutations in
the set ${\cal S}$ and ${\cal S}_{k}$, respectively. $\text{sgn}(\sigma)$
denotes the sign of permutation $\sigma$, which equals to either $+1$ or $-1$
and is determined by the number of transpositions to obtain the permutation.

######  Lemma 3.

[25] Let $0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}<\infty$ be
the ordered eigenvalues of the real-valued central Wishart matrix
${\mathchoice{\mbox{\boldmath$\displaystyle XX$}}{\mbox{\boldmath$\textstyle
XX$}}{\mbox{\boldmath$\scriptstyle XX$}}{\mbox{\boldmath$\scriptscriptstyle
XX$}}}^{T}\in\mathbb{R}^{n\times n}$, where each entry of $\textstyle X$ is
i.i.d. drawn from the standard normal distribution. The joint PDF of
${\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}=[\lambda_{1},\lambda_{2},\cdots,\lambda_{n}]^{T}$
is expressed as

$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})=K_{nm}^{-1}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right|\left[\prod_{i=1}^{n}{{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}(m-n-1)}}\right],$
(65)

where
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}=\left[\phi_{1}\
\phi_{2}\ldots\phi_{n}\right]^{T}$ with
$0\leq\phi_{1}\leq\phi_{2}\leq\ldots\leq\phi_{n}<\infty$,
${\mathchoice{\mbox{\boldmath$\displaystyle U$}}{\mbox{\boldmath$\textstyle
U$}}{\mbox{\boldmath$\scriptstyle U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\in\mathbb{R}^{n\times
n}$ is a Vandermonde matrix with the $(i,j)$th entry
$\left[{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right]_{i,j}=\phi_{j}^{i-1}$
and $\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right|$
is its determinant, given by

$\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right|=\prod_{i=1}^{n-1}{\prod_{j=i+1}^{n}{(\phi_{j}-\phi_{i})}}.$
(66)

From Lemma 3, we can express the marginal PDF of the $k$th ordered eigenvalue
as

$f_{\lambda_{k}}(\phi_{k})=\int\limits_{\mathcal{D}_{k}^{+}}{\int\limits_{\mathcal{D}_{k}^{-}}{f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}}}=\int\limits_{\mathcal{D}_{k}^{+}}{\int\limits_{\mathcal{D}_{k}^{-}}{K_{nm}^{-1}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right|\left[\prod_{i=1}^{n}{\text{e}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}}\right]\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}}}$
(67)

where
$\mathcal{D}_{k}^{-}=\left\\{0\leq\phi_{1}\leq\ldots\leq\phi_{k-1}\leq\phi_{k}\right\\}$
and
$\mathcal{D}_{k}^{+}=\left\\{\phi_{k}\leq\phi_{k+1}\leq\ldots\leq\phi_{n}<\infty\right\\}$
are the integral domains with ordered eigenvalues
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}=[\phi_{1}\
\phi_{2}\ldots\phi_{k-1}]^{T}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}=[\phi_{k+1}\
\phi_{k+1}\ldots\phi_{n}]^{T}$, respectively. Now we can proceed with the
proof of Theorem 1 as follows.

  1. Step 1: Express the joint PDF $f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})$ as a product of three parts such that terms only depending on ${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}$ and ${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}$ are separated.

  2. Step 2: Calculate the integrals in ${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}$ and ${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}$ using Lemmas 1 and 2.

  3. Step 3: Simplify the result using the polynomial approximation around zero.

##### Step 1

We can write (66) as

$\begin{split}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\right|=\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-})\right|\prod_{i=1}^{k-1}{\prod_{j=k}^{n}{\left(\phi_{j}-\phi_{i}\right)}}\times\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right|\prod_{j=k+1}^{n}{(\phi_{j}-\phi_{k})}\end{split}$
(68)

and accordingly rewrite the joint PDF in (65) as

$\begin{split}f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})=&\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-})\right|\prod_{i=1}^{k-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}\prod_{j=k}^{n}{(\phi_{j}-\phi_{i})}\\\
&\times
K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}\left(m-n-1\right)}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right|\prod_{i=k+1}^{n}{{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}(\phi_{i}-\phi_{k})},\end{split}$
(69)

for which the product terms in the second line only depend on $\phi_{k}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}$.

##### Step 2

We first consider the integration of
$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})$
with respect to
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}$.
In order to apply Lemma 1, we define

$\displaystyle\begin{split}g(\phi_{i},\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})&\triangleq{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}\prod_{j=k}^{n}{(\phi_{j}-\phi_{i})}\end{split}$
(70)
$\displaystyle\begin{split}h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})&\triangleq
K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}\left(m-n-1\right)}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right|\left[\prod_{i=k+1}^{n}{{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}(\phi_{i}-\phi_{k})}\right].\end{split}$
(71)

and then express the joint PDF as

$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})=h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-})\right|\prod_{i=1}^{k-1}g(\phi_{i},\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}).$
(72)

By identifying $\Phi_{i}(x_{j})$ in Lemma 1 with
$x_{j}^{i-1}g(x_{j},\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})$,
we can calculate the joint PDF of $\lambda_{k}$ and
${\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}_{k}^{+}=[\lambda_{k+1},...,\lambda_{n}]^{T}$
as

$\begin{split}f_{\lambda_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}_{k}^{+}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})&=\int_{\mathcal{D}_{k}^{-}}{f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}}=h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\int_{\mathcal{D}_{k}^{-}}{\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-})\right|\prod_{i=1}^{k-1}{g(\phi_{i},\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{-}}}\\\
&=h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})],\end{split}$
(73)

where the entries of the skew-symmetric matrix
${\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})$
are

$\begin{split}\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right]_{i,j}&=\int_{0}^{\phi_{k}}{\int_{0}^{\phi_{k}}{\mathcal{R}(y-x)x^{i-1}g(x,\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})y^{j-1}g(y,\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\mathrm{d}x\mathrm{d}y}}\\\
&=\int_{0}^{\phi_{k}}{\int_{0}^{\phi_{k}}{e_{i,j}(x,y)\prod_{s=k}^{n}{(\phi_{s}-x)(\phi_{s}-y)}\mathrm{d}x\mathrm{d}y}}\end{split}$
(74)

for $1\leq i<j\leq k-1$ and

$e_{i,j}(x,y)=\mathcal{R}(y-x){\mathrm{e}}^{-\frac{1}{2}x}{\mathrm{e}}^{-\frac{1}{2}y}x^{\frac{1}{2}(m-n-1)+(i-1)}y^{\frac{1}{2}(m-n-1)+(j-1)}\;.$
(75)

For odd $k$, ${\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})$
has size $(k-1)\times(k-1)$, and for even $k$,
${\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})$
has size $k\times k$, and the additional entries are given by

$\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right]_{i,k}=-\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right]_{k,i}=\int_{0}^{\phi_{k}}{x^{i-1}g(x,\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}){\mathrm{d}}x}$
(76)

for $i<k$ and $\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right]_{k,k}=0$.

Inserting (73) into (67) yields the marginal PDF of $\lambda_{k}$ as

$f_{\lambda_{k}}(\phi_{k})=\int_{\mathcal{D}_{k}^{+}}{f_{\lambda_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}_{k}^{+}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}}=\int_{\mathcal{D}_{k}^{+}}{h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})]{\mathrm{d}}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}},$
(77)

and we turn to the integration with respect to
${\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}$.
For this, it is necessary to expand
$\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}_{k}^{-}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})]$
via Lemma 2.

For odd $k$, we apply (64a) to (77) and obtain

$f_{\lambda_{k}}(\phi_{k})=\sum_{\sigma\in\mathcal{S}}{\text{sgn}(\sigma)\int_{\mathcal{D}_{k}^{+}}{h(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\prod_{i=1}^{\left(k-1\right)/2}{\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})\right]_{\sigma_{2i-1},\sigma_{2i}}}\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+}}}.$
(78)

Next, to apply Lemma 1, we consider the expression for
${\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{k},{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{k}^{+})$
in (74) and define ${\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}\triangleq[x_{1},\cdots,x_{\left(k-1\right)/2}]^{T}$,
${\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle
y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle
y$}}}\triangleq[y_{1},\cdots,y_{\left(k-1\right)/2}]^{T}$ and
$\mathcal{D}_{{\mathchoice{\mbox{\boldmath$\displaystyle
xy$}}{\mbox{\boldmath$\textstyle xy$}}{\mbox{\boldmath$\scriptstyle
xy$}}{\mbox{\boldmath$\scriptscriptstyle
xy$}}}}\triangleq\left\\{{\mathchoice{\mbox{\boldmath$\displaystyle
0$}}{\mbox{\boldmath$\textstyle 0$}}{\mbox{\boldmath$\scriptstyle
0$}}{\mbox{\boldmath$\scriptscriptstyle
0$}}}\leq{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}\leq\phi_{k}{\mathchoice{\mbox{\boldmath$\displaystyle
1$}}{\mbox{\boldmath$\textstyle 1$}}{\mbox{\boldmath$\scriptstyle
1$}}{\mbox{\boldmath$\scriptscriptstyle
1$}}},{\mathchoice{\mbox{\boldmath$\displaystyle
0$}}{\mbox{\boldmath$\textstyle 0$}}{\mbox{\boldmath$\scriptstyle
0$}}{\mbox{\boldmath$\scriptscriptstyle
0$}}}\leq{\mathchoice{\mbox{\boldmath$\displaystyle
y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle
y$}}{\mbox{\boldmath$\scriptscriptstyle
y$}}}\leq\phi_{k}{\mathchoice{\mbox{\boldmath$\displaystyle
1$}}{\mbox{\boldmath$\textstyle 1$}}{\mbox{\boldmath$\scriptstyle
1$}}{\mbox{\boldmath$\scriptscriptstyle 1$}}}\right\\}$, where $\textstyle 0$
and $\textstyle 1$ are the all-zero and all-one vector of length $(k-1)/2$,
respectively, and we obtain

$\displaystyle\prod_{i=1}^{\left(k-1\right)/2}{\left[\boldsymbol{J}(\phi_{k},\boldsymbol{\phi}_{k}^{+})\right]_{\sigma_{2i-1},\sigma_{2i}}}=\int_{\mathcal{D}_{\boldsymbol{xy}}}\prod_{i=1}^{(k-1)/2}{e_{\sigma_{2i-1},\sigma_{2i}}(x_{i},y_{i})}\prod_{s=k}^{n}(\phi_{s}-x_{i})(\phi_{s}-y_{i})\mathrm{d}\boldsymbol{x}\mathrm{d}\boldsymbol{y}$
$\displaystyle\qquad=\int\limits_{\mathcal{D}_{\boldsymbol{xy}}}{\prod_{i=1}^{(k-1)/2}{e_{\sigma_{2i-1},\sigma_{2i}}(x_{i},y_{i})(\phi_{k}-x_{i})(\phi_{k}-y_{i})}}\prod_{i=1}^{(k-1)/2}{\prod_{s=k+1}^{n}{(\phi_{s}-x_{i})(\phi_{s}-y_{i})}}\mathrm{d}\boldsymbol{x}\mathrm{d}\boldsymbol{y}.$
(79)

Substituting (79) into (78) yields

$f_{\lambda_{k}}(\phi_{k})=\sum_{\sigma\in\mathcal{S}}{\text{sgn}(\sigma)\int_{\mathcal{D}_{{\mathchoice{\mbox{\boldmath$\displaystyle
xy$}}{\mbox{\boldmath$\textstyle xy$}}{\mbox{\boldmath$\scriptstyle
xy$}}{\mbox{\boldmath$\scriptscriptstyle
xy$}}}}}{\prod_{i=1}^{(k-1)/2}{e_{\sigma_{2i-1},\sigma_{2i}}(x_{i},y_{i})}(\phi_{k}-x_{i})(\phi_{k}-y_{i})p(\phi_{k},x_{i},y_{i})\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}}\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle
y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle
y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}}}},$ (80)

where

$\begin{split}p\left(\phi_{k},x_{i},y_{i}\right)&=\int_{\mathcal{D}_{k}^{+}}{h\left(\phi_{k},\boldsymbol{\phi}_{k}^{+}\right)\prod_{i=1}^{\left(k-1\right)/2}{\prod_{s=k+1}^{n}{\left(\phi_{s}-x_{i}\right)\left(\phi_{s}-y_{i}\right)d\boldsymbol{\phi}_{k}^{+}}}}\\\
&=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}(m-n-1)}\int_{\mathcal{D}_{k}^{+}}{\left|\boldsymbol{U}(\boldsymbol{\phi}_{k}^{+})\right|\prod_{s=k+1}^{n}{q(\phi_{k},\phi_{s},x_{i},y_{i})\mathrm{d}\boldsymbol{\phi}_{k}^{+}}}\end{split}$
(81)

with

$q(\phi_{k},\phi_{s},x_{i},y_{i})={\mathrm{e}}^{-\frac{1}{2}\phi_{s}}\phi_{s}^{\frac{1}{2}(m-n-1)}(\phi_{s}-\phi_{k})\prod_{i=1}^{(k-1)/2}{(\phi_{s}-x_{i})(\phi_{s}-y_{i})}.$
(82)

We can now apply Lemma 1 to (81) and then obtain

$p(\phi_{k},x_{i},y_{i})=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}(m-n-1)}\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle\tilde{J}$}}{\mbox{\boldmath$\textstyle\tilde{J}$}}{\mbox{\boldmath$\scriptstyle\tilde{J}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{J}$}}}(\phi_{k},x_{i},y_{i})],$
(83)

where the concrete expression of the elements of the skew-symmetric matrix
${\mathchoice{\mbox{\boldmath$\displaystyle\tilde{J}$}}{\mbox{\boldmath$\textstyle\tilde{J}$}}{\mbox{\boldmath$\scriptstyle\tilde{J}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{J}$}}}(\phi_{k},x_{i},y_{i})$
is not required for subsequent derivations and thus omitted. Substituting (83)
into (80) and reverting back to single-variate integrals, we finally get

$f_{\lambda_{k}}(\phi_{k})=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}(m-n-1)}\sum_{\sigma\in\mathcal{S}}{\text{sgn}(\sigma)\prod_{i=1}^{(k-1)/2}{z_{\sigma_{2i-1},\sigma_{2i}}(\phi_{k})}},$
(84)

where

$z_{\sigma_{2i-1},\sigma_{2i}}(\phi_{k})=\int_{0}^{\phi_{k}}{\int_{0}^{\phi_{k}}{e_{\sigma_{2i-1},\sigma_{2i}}(x,y)}}(\phi_{k}-x)(\phi_{k}-y)\text{Pf}[{\mathchoice{\mbox{\boldmath$\displaystyle\tilde{J}$}}{\mbox{\boldmath$\textstyle\tilde{J}$}}{\mbox{\boldmath$\scriptstyle\tilde{J}$}}{\mbox{\boldmath$\scriptscriptstyle\tilde{J}$}}}(\phi_{k},x,y)]\mathrm{d}x\mathrm{d}y.$
(85)

For even $k$, we apply (64b) to (77) and following the analogous steps as
above, obtain

$f_{\lambda_{k}}(\phi_{k})=K_{nm}^{-1}\text{e}^{-\frac{1}{2}\phi_{k}}\phi_{k}^{\frac{1}{2}(m-n-1)}\sum_{\ell=1}^{k/2}{\sum_{\sigma\in\mathcal{S}_{\ell}}{\text{sgn}(\sigma)\tilde{z}_{\sigma_{2\ell-1},k}(\phi_{k})\prod_{i=\text{1,}i\neq\ell}^{k/2}{z_{\sigma_{2i-1},\sigma_{2i}}(\phi_{k})}}},$
(86)

where

$\tilde{z}_{\sigma_{2\ell-1},k}(\phi_{k})=\int_{0}^{\phi_{k}}{{\mathrm{e}}^{-\frac{1}{2}t}t^{\frac{1}{2}(m-n-1)+\sigma_{2\ell-1}-1}(\phi_{k}-t)\mathrm{d}t}.$
(87)

##### Step 3

We consider the polynomial approximation of $f_{\lambda_{k}}(\phi_{k})$ in
(84) and (86) around $\phi_{k}=0$. We write the first-order Taylor-series
expansion of $\text{Pf}[\boldsymbol{\tilde{J}}(\phi_{k},x,y)]$ as

$\text{Pf}\left[\boldsymbol{\tilde{J}}\left(\phi_{k},x,y\right)\right]=\text{Pf}\left[\boldsymbol{\tilde{J}}\left(0,0,0\right)\right]+D\left(\phi_{k},x,y\right)$
(88)

where $D\left(\phi_{k},x,y\right)\triangleq
D_{\phi}\phi_{k}+D_{x}x+D_{y}y+o\left(\phi_{k}\right)+o\left(x\right)+o\left(y\right)$,
with $D_{\phi}$, $D_{x}$ and $D_{y}$ being the partial derivatives of
$\text{Pf}[\boldsymbol{\tilde{J}}(\phi_{k},x,y)]$ with respect to $\phi_{k}$,
$x$ and $y$, all evaluated at $0$. As $\phi_{k}\rightarrow 0$, it is not hard
to show that

$\int_{0}^{\phi_{k}}{\int_{0}^{\phi_{k}}{e_{i,j}(x,y)(\phi_{k}-x)(\phi_{k}-y){\mathrm{d}}x{\mathrm{d}}y}}=\alpha_{i,j}\phi_{k}^{(m-n+1)+i+j}+o\left(\phi_{k}^{(m-n+1)+i+j}\right),$
(89)

where $\alpha_{i,j}$ is a coefficient determined by $e_{i,j}(x,y)$.
Consequently, we have

$z_{\sigma_{2i-1},\sigma_{2i}}(\phi_{k})=\text{Pf}[\boldsymbol{\tilde{J}}(0,0,0)]\alpha_{\sigma_{2i-1},\sigma_{2i}}\phi_{k}^{(m-n+1)+\sigma_{2i-1}+\sigma_{2i}}+o\left(\phi_{k}^{(m-n+1)+\sigma_{2i-1}+\sigma_{2i}}\right).$
(90)

Furthermore, the first-order expansion of
$\tilde{z}_{\sigma_{2i-1},\sigma_{k}}\left(\phi_{k}\right)$ around
$\phi_{k}=0$ is given by

$\tilde{z}_{\sigma_{2\ell-1},k}\left(\phi_{k}\right)=\tilde{\alpha}_{\sigma_{2\ell-1},k}\phi_{k}^{\frac{1}{2}\left(m-n-1\right)+\sigma_{2\ell-1}+1}+o\left(\phi_{k}^{\frac{1}{2}\left(m-n-1\right)+\sigma_{2\ell-1}+1}\right)$
(91)

where $\tilde{\alpha}_{\sigma_{2\ell-1},k}$ is determined by the integrand in
(87).

Then, inserting (90) into (84) yields the polynomial approximation of the
marginal PDF of $\lambda_{k}$ for odd $k$, and using (90) and (91) in (86)
yields the polynomial approximation of the marginal PDF of $\lambda_{k}$ for
even $k$.

Next, noting that ${\mathrm{e}}^{-\frac{1}{2}\phi_{k}}=1+o\left(1\right)$ and

$\displaystyle\prod_{i=1}^{(k-1)/2}{\phi_{k}^{(m-n+1)+\sigma_{2i-1}+\sigma_{2i}}}=\phi_{k}^{\frac{1}{2}(k-1)(m-n+1)+\frac{1}{2}k(k-1)}$
(92a)
$\displaystyle\phi_{k}^{\frac{1}{2}(m-n-1)+\sigma_{2\ell-1}+1}\prod_{i=\text{1,}i\neq\ell}^{k/2}{\phi_{k}^{(m-n+1)+\sigma_{2i-1}+\sigma_{2i}}}=\phi_{k}^{\frac{1}{2}(k-1)(m-n+1)+\frac{1}{2}k(k-1)}$
(92b)

we obtain

$f_{\lambda_{k}}(\phi_{k})=K_{nm}^{-1}\text{Pf}[\boldsymbol{J}_{\alpha}]\phi_{k}^{\tilde{d}_{k}}+o(\phi_{k}^{\tilde{d}_{k}}),$
(93)

where, for odd $k$, ${\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}_{\alpha}$ is a
$(k-1)\times(k-1)$ skew-symmetric matrix with its $(i,j)$th entry given by
$\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}_{\alpha}\right]_{i,j}=\text{Pf}[\boldsymbol{\tilde{J}}(0,0,0)]\alpha_{i,j}$,
and for even $k$, ${\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}_{\alpha}$ is $k\times k$, with
the additional entries
$[\boldsymbol{J}_{\alpha}]_{i,k}=-[\boldsymbol{J}_{\alpha}]_{k,i}=\tilde{\alpha}_{i,k}$
for $1\leq i\leq k-1$ and $[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}_{\alpha}]_{k,k}=0$, and
$\tilde{d}_{k}=\frac{1}{2}k\left(m-n+k-2\right)+\left(k-1\right)$.

Hence, the marginal CDF of $\lambda_{k}$ is polynomially expanded as

$F_{\lambda_{k}}(\phi_{k})=\beta_{k}\phi_{k}^{d_{k}}+o(\phi_{k}^{{d}_{k}}),$
(94)

where

$\displaystyle\beta_{k}$
$\displaystyle=d_{k}^{-1}K_{nm}^{-1}\text{Pf}[\boldsymbol{J}_{\alpha}]$ (95a)
$\displaystyle d_{k}$ $\displaystyle=\frac{1}{2}k(m-n+k).$ (95b)

##  Appendix B Proof of Theorem 2

Similar to the proof of Theorem 1, this proof makes use of Lemma 1 and
performs the three steps of expressing the joint PDF
$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})$
as a product of terms (Step 1), marginalizing the PDF (Step 2), and developing
the polynomial expansion (Step 3). A final 4th step is added to obtain the
coefficients in (29).

##### Step 1

Starting from (69) for $k=1$, and defining

$g(\phi_{1},\phi_{i})\triangleq{\mathrm{e}}^{-\frac{1}{2}\phi_{i}}\phi_{i}^{\frac{1}{2}\left(m-n-1\right)}(\phi_{i}-\phi_{1}).$
(96)

the joint PDF
$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})$
is

$f_{{\mathchoice{\mbox{\boldmath$\displaystyle\lambda$}}{\mbox{\boldmath$\textstyle\lambda$}}{\mbox{\boldmath$\scriptstyle\lambda$}}{\mbox{\boldmath$\scriptscriptstyle\lambda$}}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}})=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{1}}\phi_{1}^{\frac{1}{2}(m-n-1)}\left|{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{1}^{+})\right|\prod_{i=2}^{n}{g(\phi_{1},\phi_{i})}.$
(97)

##### Step 2

The marginal PDF of $\lambda_{1}$ is calculated as

$f_{\lambda_{1}}(\phi_{1})=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{1}}\phi_{1}^{\frac{1}{2}(m-n-1)}\int_{\mathcal{D}_{1}^{+}}{{\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle
U$}}}({\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{1}^{+})\prod_{i=2}^{n}{g(\phi_{1},\phi_{i})}\mathrm{d}{\mathchoice{\mbox{\boldmath$\displaystyle\phi$}}{\mbox{\boldmath$\textstyle\phi$}}{\mbox{\boldmath$\scriptstyle\phi$}}{\mbox{\boldmath$\scriptscriptstyle\phi$}}}_{1}^{+}}.$
(98)

By identifying $\Phi_{i}(x_{j})$ in Lemma 1 with
$x_{j}^{i-1}g(\phi_{1},x_{j})$ in (98), we can solve the marginal PDF as

$f_{\lambda_{1}}(\phi_{1})=K_{nm}^{-1}{\mathrm{e}}^{-\frac{1}{2}\phi_{1}}\phi_{1}^{\frac{1}{2}(m-n-1)}\text{Pf}\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}(\phi_{1})\right],$ (99)

where ${\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}(\phi_{1})$ is a skew-symmetric
matrix of size $(n-1)\times(n-1)$ for odd $n$ and of size $n\times n$ for even
$n$. For odd $n$, the elements of ${\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}(\phi_{1})$ are

$\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{1})\right]_{i,j}=\int_{\phi_{1}}^{\infty}{\int_{\phi_{1}}^{\infty}{\mathcal{R}(y-x)x^{i-1}g(\phi_{1},x)y^{j-1}g(\phi_{1},y)\mathrm{d}x\mathrm{d}y}},$
(100)

for $1\leq i<j\leq n-1$. For $n$ even, the additional entries of
${\mathchoice{\mbox{\boldmath$\displaystyle J$}}{\mbox{\boldmath$\textstyle
J$}}{\mbox{\boldmath$\scriptstyle J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{1})$ are

$\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{1})\right]_{i,n}=-\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}}(\phi_{1})\right]_{n,i}=\int_{\phi_{1}}^{\infty}{{\mathrm{e}}^{-\frac{1}{2}x}x^{\frac{1}{2}\left(m-n-1\right)+\left(i-1\right)}\left(x-\phi_{1}\right)\mathrm{d}x}$
(101)

for $1\leq i\leq n-1$ and $\left[{\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}}(\phi_{1})\right]_{n,n}=0$.

##### Step 3