# On the Ergodic Mutual Information of Keyhole MIMO Channels With Finite-
Alphabet Inputs

Chongjun Ouyang, Ali Bereyhi, Saba Asaad, Ralf R. Müller, Julian Cheng, and
Hongwen Yang C. Ouyang and H. Yang are with the School of Information and
Communication Engineering, Beijing University of Posts and Telecommunications,
Beijing, 100876, China (e-mail: {DragonAim,yanghong}@bupt.edu.cn).A. Bereyhi,
S. Asaad, and R. R. Müller are with the Institute for Digital Communications,
Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058, Erlangen, Germany
(e-mail: {ali.bereyhi,saba.asaad,ralf.r.mueller}@fau.de).J. Cheng is with the
School of Engineering, The University of British Columbia, Kelowna, BC V1V
1V7, Canada (email: julian.cheng@ubc.ca).

###### Abstract

This letter studies the ergodic mutual information (EMI) of keyhole multiple-
input multiple-output channels having finite-alphabet input signals. The EMI
is first investigated for single-stream transmission considering both cases
with and without the channel state information at the transmitter. Then, the
derived results are extended to the scenario of multi-stream transmission.
Asymptotic analyses are performed in the regime of high signal-to-noise ratio
(SNR). The high-SNR EMI is shown to converge to a constant with its rate of
convergence determined by the diversity order. On this basis, the influence of
the keyhole effect on the EMI is discussed. The analytical results are
validated by numerical simulations.

###### Index Terms:

Ergodic mutual information, finite-alphabet inputs, keyhole channel, multiple-
input multiple-output.

##  I Introduction

Multiple-input multiple-output (MIMO) systems are known to boost the spectral
efficiency (SE) of wireless channels in comparison to conventional single-
antenna systems. Yet, practical MIMO systems may suffer from severe
degradation of the SE, due to channel degeneration. One of such phenomena is
termed the keyhole effect which may arise in a hallway or tunnel with the
electromagnetic waves propagating through the same hole as shown in Figure 1;
see [1, 6, 4, 2, 3, 5] and the references therein. This effect is observed in
various applications; for instance, in vehicle-to-vehicle communications under
dense urban environments [7]. The existence of this effect was initially
predicted in theory [1] and then validated by empirical measurements [3]. In
contrast to traditional MIMO channels, keyhole channels generally characterize
rank-deficient MIMO channels, which may have sufficient scattering around the
transceivers, but due to other propagation effects, such as diffraction, the
channel matrix might exhibit only low rank.

Theoretically, the keyhole effect can remove the spatial multiplexing gain of
MIMO channels [1]. It hence models the worst-case propagation environment for
MIMO systems from the SE perspective. In general, the system SE is
proportional to the achievable input-output mutual information (MI) of the
channel [8]. Consequently, analyzing the MI of keyhole MIMO channels can
benchmark the worst-case SE of multiple-antenna systems. Motivated by this,
several studies analyzed the MI of keyhole MIMO channels for Gaussian
distributed input signals [2, 3, 4, 5, 7, 6]. Particularly, the MI achieved by
Gaussian inputs was analyzed in single-user ergodic case [2, 3, 4, 5], multi-
user ergodic case [6], and the single-user outage case [7]. Yet, practical
transmit signals are often taken from finite constellation alphabets, e.g.,
quadrature amplitude modulation (QAM). These finite constellations yield
reduced MI, especially in the high signal-to-noise ratio (SNR) regime [8, 14,
9, 16]. Despite its importance, analysis of the ergodic MI (EMI) for keyhole
MIMO channels with finite input constellations has been left open.

This letter studies the EMI of keyhole MIMO channels with finite-alphabet
inputs under Nakagami-$m$ fading. The main contributions of this work are as
follows: 1) We derive novel expressions of the EMI under single-stream
transmission (SST) by considering perfect CSI at the receiver and both cases
with and without the CSI at the transmitter (CSIT); 2) We extend the scenario
of SST to the scenario of multi-stream transmission (MST) and study the EMI
under three typical precoding schemes; 3) We characterize the EMI in the high-
SNR region and determine the diversity order of the system, which enables us
to estimate the influence of the keyhole effect111We comment that also for
MIMO channels without keyholes, there has been very limited work on
characterizing the high-SNR asymptotic behaviours of the EMI achieved by
finite-alphabet inputs. Yet, this can be done by using the approach proposed
in this work, which will be considered in the future.. Compared with our
previous work [9] that focused more on approximating the EMI in single-antenna
systems and neglected the high-SNR analyses, this letter gains more insights
into the influence of finite-alphabet on the EMI in MIMO keyhole channels.

Figure 1: Illustration of a keyhole MIMO channel

##  II System Model

Consider the point-to-point keyhole MIMO channel illustrated in Figure 1,
where an $N_{\rm{t}}$-antenna transmitter (Tx) sends wireless signals to an
$N_{\rm{r}}$-antenna receiver (Rx). The received signal is given by

$\displaystyle{\mathbf{y}}=\sqrt{\bar{\gamma}}{\mathbf{H}}{\mathbf{s}}+{\mathbf{n}},$
(1)

where ${\mathbf{H}}\in{\mathbbmss{C}}^{N_{\rm{r}}\times N_{\rm{t}}}$
represents the channel matrix with $N_{\rm{t}}>1$ and $N_{\rm{r}}>1$,
${\mathbf{s}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times 1}$ denotes the transmit
signal satisfying
${\mathbbmss{E}}\left\\{{\mathbf{s}}^{\mathsf{H}}{\mathbf{s}}\right\\}=1$,
$\bar{\gamma}$ denotes the transmit SNR, and
${\mathbf{n}}\sim{\mathcal{CN}}\left({\mathbf{0}},{\mathbf{I}}_{N_{\rm{r}}}\right)$
is additive white Gaussian noise (AWGN).

Considering the spatial structure of keyhole MIMO channels, we have
${\mathbf{H}}={\mathbf{h}}_{\rm{r}}{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}$ for
${\mathbf{h}}_{\rm{r}}\in{\mathbbmss{C}}^{N_{\rm{r}}\times 1}$ and
${\mathbf{h}}_{\rm{t}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times 1}$, where

$\displaystyle{\mathbf{h}}_{\rm{r}}=\left[\sqrt{\alpha_{1}}{\rm{e}}^{{\rm{j}}\phi_{1}},\ldots,\sqrt{\alpha_{N_{\rm{r}}}}{\rm{e}}^{{\rm{j}}\phi_{N_{\rm{r}}}}\right]^{\mathsf{T}}\in{\mathbbmss{C}}^{N_{\rm{r}}\times
1},$ (2)
$\displaystyle{\mathbf{h}}_{\rm{t}}=\left[\sqrt{\beta_{1}}{\rm{e}}^{{\rm{j}}\psi_{1}},\ldots,\sqrt{\beta_{N_{\rm{t}}}}{\rm{e}}^{{\rm{j}}\psi_{N_{\rm{t}}}}\right]^{\mathsf{T}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times
1},$ (3)

denote the keyhole-to-Rx and keyhole-to-Tx channel vectors, respectively,
which are statistically independent of each other [1]222It is worth mentioning
that the keyhole channel is also influenced by the size of the keyhole.
Intuitively, the keyhole effect is more pronounced when the keyhole’s physical
size approximately equals or is even smaller than the wavelength [3].
Unfortunately, a quantitative characterization of the influence of the
keyhole’s size on the channel is still open.. We assume that all entries in
the vector ${\mathbf{h}}_{\rm{r}}$ are independent and identically distributed
(i.i.d.), i.e., the phases $\phi_{a}$ for $a\in\\{1,\ldots,N_{\rm{r}}\\}$ are
uniformly distributed on $\left[0,2\pi\right)$ and the magnitudes
$\sqrt{\alpha_{a}}$ follow the Nakagami-$m$ distribution with the probability
density function (PDF) of $\alpha_{a}$ given by
$f\left(x;m_{\rm{r}},m_{\rm{r}}\right)$. Here,

$\displaystyle
f\left(x;c,d\right)\triangleq\frac{1}{\Gamma\left(c\right)}x^{c-1}{\rm{e}}^{-dx}d^{c},x\geq
0,$ (4)

where
$\Gamma\left(x\right)\triangleq\int_{0}^{\infty}t^{x-1}{\rm{e}}^{-t}{\rm{d}}t$
is the gamma function [10], and $m_{\rm{r}}\geq\frac{1}{2}$ indicates the
fading severity. Likewise, we assume that the keyhole-to-Tx channel undergoes
i.i.d. Nakagami-$m$ fading; thus, the PDF of the magnitudes $\beta_{b}$ for
$b\in\\{1,\ldots,N_{\rm{t}}\\}$ is given by
$f\left(x;m_{\rm{t}},m_{\rm{t}}\right)$ for some fading severity
$m_{\rm{t}}\geq\frac{1}{2}$ and the phases $\psi_{b}$ are uniformly
distributed on $\left[0,2\pi\right)$. It is worth noting that the Nakagami-$m$
model is a generalization of the statistical model used in [3, 7, 6], which
has been illustrated to fit better with empirical data.

##  III Single-Stream Transmission

We start the analysis by considering the SST. The transmitted signal is given
by ${\mathbf{s}}={\mathbf{w}}x$, where
${\mathbf{w}}\in{\mathbbmss{C}}^{N_{\rm{r}}\times 1}$ denotes the precoding
vector satisfying $\left\|{\mathbf{w}}\right\|^{2}=1$ and
$x\in{\mathbbmss{C}}$ is the transmitted symbol. We assume that $x$ satisfies
the power constraint ${\mathbbmss{E}}\\{\left|x\right|^{2}\\}=1$ and is taken
from a finite constellation alphabet $\mathcal{X}$ consisting of $M$ points,
i.e., ${\mathcal{X}}=\left\\{\mathsf{x}_{g}\right\\}_{g=1}^{M}$. The $g$th
symbol in $\mathcal{X}$, i.e, $\mathsf{x}_{g}$, is transmitted with
probability $p_{g}$, $0<p_{g}<1$, and the vector of probabilities
${\mathbf{p}}_{\mathcal{X}}\triangleq[p_{1},\cdots,p_{M}]\in{\mathbbmss{C}}^{1\times
M}$ is called the input distribution with $\sum_{g=1}^{M}p_{g}=1$.

The derivation of EMI for the SST (SST-EMI) in a fading keyhole MIMO channel
is best understood by specifying the MI of a scalar Gaussian channel with
finite-alphabet inputs. To this end, consider the scalar AWGN channel

$Y=\sqrt{\gamma}X+Z,$ (5)

where $Z\sim{\mathcal{CN}}\left(0,1\right)$ is AWGN, $X$ is the channel input
taken from the alphabet $\mathcal{X}$ subject to the input distribution
${\mathbf{p}}_{\mathcal{X}}$, and $\gamma$ is the SNR. For this channel, the
MI is given by [8]

$\begin{split}I_{M}^{\mathcal{X}}\left(\gamma\right)&=H_{{\mathbf{p}}_{\mathcal{X}}}-\frac{1}{\pi}\sum\nolimits_{g=1}^{M}\int_{\mathbbmss{C}}p_{g}{\rm
e}^{-\left|u-\sqrt{\gamma}{\mathsf{x}}_{g}\right|^{2}}\\\
&\times\log_{2}{\left(\sum\nolimits_{{g^{\prime}}=1}^{M}\frac{p_{g^{\prime}}}{p_{g}}{\rm
e}^{\left|u-\sqrt{\gamma}{\mathsf{x}}_{g}\right|^{2}-\left|u-\sqrt{\gamma}{\mathsf{x}}_{g^{\prime}}\right|^{2}}\right)}{\rm
d}u,\end{split}$ (6)

where $H_{{\mathbf{p}}_{\mathcal{X}}}$ is the entropy of the input
distribution ${\mathbf{p}}_{\mathcal{X}}$ in bits. By a straightforward
extension of this result to a single-input vectorized channel, it is shown
that the SST-EMI achieved by maximum ratio combining is given by

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}={\mathbbmss{E}}\\{I_{M}^{\mathcal{X}}({\bar{\gamma}}\left\|{\mathbf{h}}_{\rm{r}}\right\|^{2}\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{w}}\right|^{2})\\}.$
(7)

It is worth noting that the EMI is a function of the precoding vector
${\mathbf{w}}$. In the sequel, we will analyze the SST-EMI based on the
availability of CSIT.

###  III-A SST Without CSIT

With no CSIT, the transmitter applies uniform beamforming, i.e.,
${\mathbf{w}}=\frac{1}{\sqrt{N_{\rm{t}}}}{\mathbf{1}}$, where
${\mathbf{1}}\triangleq\left[1,\cdots,1\right]^{\mathsf{T}}$. In this case, we
have
${\mathcal{I}}_{M}^{\mathcal{X}}={\mathbbmss{E}}\left\\{I_{M}^{\mathcal{X}}\left(S_{1}{\bar{\gamma}}/{N_{\rm{t}}}\right)\right\\}$,
where
$S_{1}=\left\|{\mathbf{h}}_{\rm{r}}\right\|^{2}\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{1}}\right|^{2}$.
To characterize the EMI, we follow three major steps which are illustrated in
the sequel.

####  III-A1 Channel Statistics

At the first step, we derive the PDF of $S_{1}$. The statistical independence
of ${\mathbf{h}}_{\rm{t}}$ and ${\mathbf{h}}_{\rm{r}}$ concludes that
$A=\left\|{\mathbf{h}}_{\rm{r}}\right\|^{2}$ and
$B=\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{1}}\right|^{2}$ are
mutually independent. It follows that the PDF of the product $S_{1}=AB$ can be
calculated as
$f_{S_{1}}\left(x\right)=\int_{0}^{\infty}f_{B}\left(\frac{x}{y}\right)f_{A}\left(y\right)\frac{1}{y}{\rm{d}}y$,
where $f_{A}(\cdot)$ and $f_{B}(\cdot)$ denote the PDFs of $A$ and $B$,
respectively. Yet, due to the intractability of
$\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{1}}\right|^{2}$, a closed-
form expression for its PDF is only available when $m_{\rm{t}}$ is an integer
[11]. Accordingly, we let $m_{\rm{t}}$ be an integer in order to facilitate
the subsequent analyses. The following two lemmas are then employed to
characterize $S_{1}$.

######  Lemma 1.

Define an operator ${\mathcal{F}}\left\langle{\cdot}\right\rangle$ as

${\mathcal{F}}\left\langle{Q}\right\rangle\triangleq\sum_{i_{1}=0}^{m_{\rm{t}}-1}\cdots\sum_{i_{N_{\rm{t}}}=0}^{m_{\rm{t}}-1}\sum_{h=0}^{S_{N_{\rm{t}}}}\frac{\left(-S_{N_{\rm{t}}}\right)_{h}S_{N_{\rm{t}}}!Y_{N_{\rm{t}}}{Q}}{X_{N_{\rm{t}}}\left(h!\right)^{2}U_{N_{\rm{t}}}^{S_{N_{\rm{t}}}}},$
(8)

where
$X_{N_{\rm{t}}}=\prod_{k=1}^{N_{\rm{t}}}\left(\frac{\left(i_{k}!\right)^{2}}{\left(1-m_{\rm{t}}\right)_{i_{k}}}\right)$,
$S_{N_{\rm{t}}}=\sum_{k=1}^{N_{\rm{t}}}i_{k}$,
$Y_{N_{\rm{t}}}=\prod_{k=1}^{N_{\rm{t}}}\left(\frac{1}{4m_{\rm{t}}}\right)^{i_{k}}$,
$U_{N_{\rm{t}}}=\sum_{k=1}^{N_{\rm{t}}}\frac{1}{4m_{\rm{t}}}$, and
$\left(z\right)_{n}\triangleq\frac{\Gamma\left(z+n\right)}{\Gamma\left(z\right)}$
is the Pochhammer symbol [10, Eq. (5.2.5)] with
$\left(-z\right)_{n}=\left(-1\right)^{n}\left(z-n+1\right)_{n}$. Then, the PDF
of $B=\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{1}}\right|^{2}$ can be
written as
$f_{B}\left(x\right)={\mathcal{F}}\left\langle{{\rm{e}}^{-\frac{x}{4U_{N_{\rm{t}}}}}x^{h}\left(4U_{N_{\rm{t}}}\right)^{-h-1}}\right\rangle$.

###### Proof:

Please refer to [11] for more details. ∎

######  Lemma 2.

The PDF of $S_{1}$ is given by

$\begin{split}f_{S_{1}}\left(x\right)&={\mathcal{F}}\left\langle\frac{2}{\Gamma\left(N_{\rm{r}}m_{\rm{r}}\right)}\left({m_{\rm{r}}x}/{\left(4U_{N_{\rm{t}}}\right)}\right)^{\frac{N_{\rm{r}}m_{\rm{r}}+h+1}{2}}\right.\\\
&\times\left.x^{-1}K_{N_{\rm{r}}m_{\rm{r}}-h-1}\left(2\sqrt{{m_{\rm{r}}x}/{\left(4U_{N_{\rm{t}}}\right)}}\right)\right\rangle,\end{split}$
(9)

where $K_{\nu}\left(\cdot\right)$ is the $\nu$th order modified Bessel
function of the second kind [10, Eq. (10.31.1)].

###### Proof:

Since $\left\\{\sqrt{\alpha_{a}}\right\\}_{a=1}^{N_{\rm{r}}}$ are $N_{\rm{r}}$
i.i.d. Nakagami-$m$ variables, the PDF of
${A}=\sum_{a=1}^{N_{\rm{r}}}\alpha_{a}$ can be written as
$f_{A}\left(x\right)=f\left(x;N_{\rm{r}}m_{\rm{r}},m_{\rm{r}}\right)$. Aided
with the integral identity in [10, Eq. (10.32.10)], we finally conclude the
desired PDF in (9). ∎

####  III-A2 Explicit Analysis

In the second step, we invoke Lemma 2 to derive an approximation for the SST-
EMI.

######  Theorem 1.

For SST-EMI achieved without CSIT, the following approximation becomes exact
as the complexity-vs-accuracy tradeoff parameter $V$ approaches infinity:

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}\approx{\mathcal{F}}\left\langle\sum_{k=1}^{V}\sum_{l=1}^{V}\frac{w_{k}w_{l}I_{M}^{\mathcal{X}}\left(\frac{4U_{N_{\rm{t}}}\bar{\gamma}t_{k}t_{l}}{m_{\rm{r}}N_{\rm{t}}}\right)}{\Gamma\left(N_{\rm{r}}m_{\rm{r}}\right)t_{l}^{-j}t_{k}^{1-N_{\rm{r}}m_{\rm{r}}}}\right\rangle,$
(10)

where $\left\\{w_{i}\right\\}$ and $\left\\{t_{i}\right\\}$ denote the weight
and abscissa factors of Gauss–Laguerre integration.

###### Proof:

The EMI can be calculated as

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}=\int_{0}^{\infty}\left(\int_{0}^{\infty}f_{B}\left(\frac{x}{y}\right)\frac{f_{A}\left(y\right)}{y}{\rm{d}}y\right)I_{M}^{\mathcal{X}}\left(\frac{x\bar{\gamma}}{N_{\rm{t}}}\right){\rm
d}x.$ (11)

We use the Gauss–Laguerre quadrature method [10, Eq. (3.5.27)] to calculate
the two integrals in (11) successively. This leads to the approximate
expression shown in (10). ∎

Note that given a target approximation precision, quantifying the relationship
between the required value of $V$ and other system parameters, such as $M$ and
$\bar{\gamma}$, is challenging. By numerical simulation, we find out that
setting $V=200$ can generally achieve an approximation precision of
$10^{-14}$.

####  III-A3 Asymptotic Analysis

In the last step, we investigate the asymptotic behaviour of the EMI. It is
worth noting that the MIMO keyhole channel does not always harden under the
asymptotic condition when
$N_{\rm{t}}~{}{\text{or}}~{}N_{\rm{r}}\rightarrow\infty$ [6]. This makes it
challenging to gain further insights into the EMI by setting
$N_{\rm{t}}~{}{\text{or}}~{}N_{\rm{r}}\rightarrow\infty$. As a compromise,
more attention will be paid to the asymptotic limit in which the SNR
approaches infinity, i.e., $\bar{\gamma}\rightarrow\infty$. The result is
given in Theorem 2.

######  Theorem 2.

Let $N_{\rm{r}}m_{\rm{r}}\neq h+1$ for
$h\in\left\\{0,\cdots,N_{\rm{t}}(m_{\rm{t}}-1)\right\\}$. When
$\bar{\gamma}\rightarrow\infty$, the EMI achieved without CSIT can be
characterized as
${\mathcal{I}}_{M}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-\left({\mathcal{G}}_{\rm{a}}{\bar{\gamma}}\right)^{-{\mathcal{G}}_{\rm{d}}}$,
where ${\mathcal{G}}_{\rm{d}}=1$ and

$\displaystyle{\mathcal{G}}_{\rm{a}}^{-1}\\!=\\!\sum\limits_{i_{1}=0}^{m_{\rm{t}}-1}\\!\\!\cdots\\!\\!\sum\limits_{i_{N_{\rm{t}}}=0}^{m_{\rm{t}}-1}\\!\frac{U_{N_{\rm{t}}}^{-S_{N_{\rm{t}}}}S_{N_{\rm{t}}}!Y_{N_{\rm{t}}}\hat{\mathcal{M}}\left(2\right)m_{\rm{t}}m_{\rm{r}}\log_{2}{\rm{e}}}{\left(N_{\rm{r}}m_{\rm{r}}-1\right)\prod_{k=1}^{N_{\rm{t}}}\left(\frac{\left(i_{k}!\right)^{2}}{\left(1-m_{\rm{t}}\right)_{i_{k}}}\right)}.$
(12)

Here,
$\hat{\mathcal{M}}\left({x}\right)\triangleq{\mathcal{M}}\left[{\mathrm{mmse}}_{M}^{\mathcal{X}}\left(t\right);{x}\right]$,
where ${\mathrm{mmse}}_{M}^{\mathcal{X}}\left(t\right)$ denotes the minimum
mean square error (MMSE) in estimating $X$ in (5) from $Y$. Moreover,
${\mathcal{M}}\left[p\left(t\right);z\right]\triangleq\int_{0}^{\infty}t^{z-1}p\left(t\right){\rm
d}t$ denotes the Mellin transform of $p\left(t\right)$ [12].

###### Proof:

The proof is given in Appendix A. ∎

######  Remark 1.

The results in Theorem 2 suggest that the EMI achieved by finite-alphabet
input signals converges to ${H_{{\mathbf{p}}_{\mathcal{X}}}}$ as the SNR
increases and its rate of convergence (ROC) is determined by the diversity
order ${\mathcal{G}}_{\rm{d}}$ and the array gain ${\mathcal{G}}_{\rm{a}}$.

###  III-B SST With CSIT

With CSIT, we can apply maximal ratio transmission (MRT) at the transmitter,
i.e.,
${\mathbf{w}}=\frac{1}{\left\|{\mathbf{h}}_{\rm{t}}\right\|}{\mathbf{h}}_{\rm{t}}$.
Hence, the EMI is given by
${\mathcal{I}}_{M}^{\mathcal{X}}={\mathbbmss{E}}\left\\{I_{M}^{\mathcal{X}}\left({\bar{\gamma}}S_{2}\right)\right\\}$,
where
$S_{2}=\left\|{\mathbf{h}}_{\rm{r}}\right\|^{2}\left\|{\mathbf{h}}_{\rm{t}}\right\|^{2}$.
Similar to the previous case, we characterize the SST-EMI in three steps.

####  III-B1 Channel Statistics

Using similar steps as those outlined in the proof of Lemma 2, we arrive at
the following lemma.

######  Lemma 3.

The PDF of $S_{2}$ is given by

$\displaystyle f_{S_{2}}\left(x\right)\\!=\\!\frac{2\\!\left(m_{\rm t}m_{\rm
r}\right)^{\frac{N_{\rm t}m_{\rm t}+N_{\rm r}m_{\rm r}}{2}}\\!K_{N_{\rm
r}m_{\rm r}-N_{\rm t}m_{\rm t}}\left(2\sqrt{m_{\rm t}m_{\rm
r}x}\right)}{\Gamma\left(N_{\rm t}m_{\rm t}\right)\Gamma\left(N_{\rm r}m_{\rm
r}\right)x^{1-\frac{N_{\rm t}m_{\rm t}+N_{\rm r}m_{\rm r}}{2}}}.$

###### Proof:

The proof is similar to the one given for Lemma 2. We hence omit it. ∎

####  III-B2 Explicit Analysis

The EMI with CSIT is given by
${\mathcal{I}}_{M}^{\mathcal{X}}=\int_{0}^{\infty}f_{S_{2}}\left(x\right)I_{M}^{\mathcal{X}}\left({\bar{\gamma}}x\right){\rm{d}}x$.
By following the same steps as those taken in the proof of Theorem 1, we
conclude the following approximation for ${\mathcal{I}}_{M}^{\mathcal{X}}$:

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}\approx\sum_{k=1}^{V}\sum_{l=1}^{V}\frac{w_{k}w_{l}t_{j}^{N_{\rm{r}}m_{\rm{r}}-1}t_{l}^{N_{\rm{t}}m_{\rm{t}}-1}}{\Gamma\left(N_{\rm{r}}m_{\rm{r}}\right)\Gamma\left(N_{\rm{t}}m_{\rm{t}}\right)}I_{M}^{\mathcal{X}}\left(\frac{\bar{\gamma}t_{k}t_{l}}{m_{\rm{r}}m_{\rm{t}}}\right).$
(13)

Similar to (10), the approximation in (13) becomes exact as the complexity-vs-
accuracy tradeoff parameter $V$ tends infinity.

####  III-B3 Asymptotic Analysis

The limiting EMI with CSIT for asymptotically high SNRs is characterized as
follows.

######  Theorem 3.

Let $N_{\rm r}m_{\rm r}\neq N_{\rm t}m_{\rm t}$. When
$\bar{\gamma}\rightarrow\infty$, the asymptotic EMI with CSIT satisfies
${\mathcal{I}}_{M}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-\left({\mathcal{G}}_{\rm{a}}{\bar{\gamma}}\right)^{-{\mathcal{G}}_{\rm{d}}}$,
where
${\mathcal{G}}_{\rm{d}}=\min\left\\{N_{\rm{t}}m_{\rm{t}},N_{\rm{r}}m_{\rm{r}}\right\\}$
and

$\displaystyle{\mathcal{G}}_{\rm{a}}=\frac{1}{m_{\rm{r}}m_{\rm{t}}}\left(\frac{\Gamma\left(N_{\rm{t}}m_{\rm{t}}\right)\Gamma\left(N_{\rm{r}}m_{\rm{r}}\right){\mathcal{G}}_{\rm{d}}\ln{2}}{\Gamma\left(\left|N_{\rm{t}}m_{\rm{t}}-N_{\rm{r}}m_{\rm{r}}\right|\right)\hat{\mathcal{M}}\left({\mathcal{G}}_{\rm{d}}+1\right)}\right)^{1/{{\mathcal{G}}_{\rm{d}}}}.$

###### Proof:

The proof is given by directly applying the method detailed in Appendix A. We
hence skip the details. ∎

######  Remark 2.

The above result suggest that the diversity order in this case is a function
of $\left\\{N_{\rm{t}},N_{\rm{r}},m_{\rm{t}},m_{\rm{r}}\right\\}$. By
increasing the number of antennas, this expression can become larger than the
one derived for the case without CSIT.

###  III-C Discussions on Keyhole Rank-Deficiency

Consider a special case, in which the amplitudes of the channel coefficients
follow the Rayleigh distribution, namely $m_{\rm{t}}=m_{\rm{r}}=1$. The MIMO
channel matrix in this case has full rank, if there exist no keyholes [1].
Using the method presented in Appendix A, we can characterize the high-SNR
SST-EMI in the keyhole and full-rank MIMO channels, respectively.
Particularly, in the keyhole MIMO channel, the high-SNR SST-EMI achieved with
and without CSIT can be written as
${\mathcal{I}}_{M,{\rm{c}},{\rm{r}}}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-\min\left\\{N_{\rm{t}},N_{\rm{r}}\right\\}}\right)$333The
notation $f(x)={\mathcal{O}}\left(g(x)\right)$ means that
$\limsup_{x\rightarrow\infty}\frac{\left|f(x)\right|}{g(x)}<\infty$. and
${\mathcal{I}}_{M,{\rm{n}},{\rm{r}}}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$,
respectively. Moreover, in full-rank MIMO channels, the high-SNR SST-EMI
achieved with and without CSIT can be expressed as
${\mathcal{I}}_{M,{\rm{c}},{\rm{nk}}}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-N_{\rm{r}}N_{\rm{t}}}\right)$
and
${\mathcal{I}}_{M,{\rm{n}},{\rm{nk}}}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-N_{\rm{r}}}\right)$,
respectively.

######  Remark 3.

Comparing ${\mathcal{I}}_{M,{\rm{c}},{\rm{r}}}^{\mathcal{X}}$ (or
${\mathcal{I}}_{M,{\rm{n}},{\rm{r}}}^{\mathcal{X}}$) with
${\mathcal{I}}_{M,{\rm{c}},{\rm{nk}}}^{\mathcal{X}}$ (or
${\mathcal{I}}_{M,{\rm{n}},{\rm{nk}}}^{\mathcal{X}}$), we conclude that the
keyhole effect can reduce the diversity order of the SST-EMI.

We can extend the above results to a more generic case, where the rank of the
channel matrix is smaller than $\min\left\\{N_{\rm{r}},N_{\rm{t}}\right\\}$,
i.e., the channel matrix is rank-deficient. One example of such rank-deficient
channels is a multi-keyhole MIMO channel whose number of keyholes is smaller
than $\min\left\\{N_{\rm{r}},N_{\rm{t}}\right\\}$. Particularly, for fixed
$N_{\rm{r}}$ and $N_{\rm{t}}$, the SST-EMI achieved by a finite-alphabet input
in a rank-deficient channel yields a lower diversity order than the one
achieved in a full-rank channel. This is similar to MIMO channels with
Gaussian inputs under SST. Due to the page limitations, further discussions
are skipped here and left as a potential direction for future work.

##  IV Extension to Multi-Stream Transmission

For MST, the received signal vector is given by

$\displaystyle{\mathbf{y}}=\sqrt{\bar{\gamma}}{\mathbf{H}}{\mathbf{P}}{\mathbf{x}}+{\mathbf{n}},$
(14)

where ${\mathbf{P}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times N}$ denotes the
precoding matrix satisfying
${\mathsf{tr}}\left\\{{\mathbf{P}}{\mathbf{P}}^{\mathsf{H}}\right\\}=1$ with
$N$ being the number of data streams, and
${\mathbf{x}}\in{\mathbbmss{C}}^{N\times 1}$ is the data vector with i.i.d.
elements drawn from the $M$-ary constellation $\mathcal{X}$. Hence, the input
signal $\mathbf{x}$ is taken from a multi-dimensional constellation
${\mathcal{Y}}$ consisting of $M^{N}$ points, i.e.,
$\mathbf{x}\in{\mathcal{Y}}=\left\\{{\bm{\mathsf{x}}}_{g}\in{\mathbbmss{C}}^{N\times
1}\right\\}_{g=1}^{M^{N}}$, with
$\mathbbmss{E}\left\\{{\mathbf{x}}{\mathbf{x}}^{\mathsf{H}}\right\\}={\mathbf{I}}_{N}$.
Assume ${\bm{\mathsf{x}}}_{g}$ is sent with probability $q_{g}$, $0<q_{g}<1$,
and the input distribution is given by
${\mathbf{q}}_{{\mathcal{Y}}}\triangleq[q_{1},\cdots,q_{M^{N}}]\in{\mathbbmss{C}}^{1\times
M^{N}}$ with $\sum_{g=1}^{M^{N}}q_{g}=1$. The MI in this case can be written
as
$\texttt{I}\left({\bar{\gamma}};{\mathbf{H}}{\mathbf{P}}\right)=H_{{\mathbf{q}}_{{\mathcal{Y}}}}-N_{\rm{r}}\log_{2}{\rm{e}}-\sum\nolimits_{g=1}^{M^{N}}p_{g}f_{g}\left({\bar{\gamma}};{\mathbf{H}}{\mathbf{P}}\right)$,
where
$f_{g}\left({\bar{\gamma}};{\mathbf{H}}{\mathbf{P}}\right)\\!\triangleq\\!{\mathbbmss{E}}_{\mathbf{n}}\left\\{\log_{2}\left({\sum_{g^{\prime}=1}^{M^{N}}\frac{p_{g^{\prime}}}{p_{g}}{\rm{e}}^{-\left\|{\mathbf{n}}+\sqrt{\bar{\gamma}}{\mathbf{H}}{\mathbf{P}}{\mathbf{b}}_{g,g^{\prime}}\right\|^{2}}}\right)\right\\}$
with
${\mathbf{b}}_{g,g^{\prime}}={\bm{\mathsf{x}}}_{g}-{\bm{\mathsf{x}}}_{g^{\prime}}=\left[b_{g,g^{\prime},1},\cdots,b_{g,g^{\prime},N}\right]^{\mathsf{H}}\in{\mathbbmss{C}}^{N\times
1}$ [8]. Note that although the authors in [5] derived a closed-form
expression for the MST-EMI achieved by Gaussian inputs, it is challenging to
extend the results in [5] to systems with finite-alphabet inputs. We hence
consider high-SNR limit while analyzing the MST-EMI achieved by finite-
alphabet inputs.

###  IV-A MST Without CSIT

For the case without CSIT, the preocding matrix can be set to
${\mathbf{P}}=1/\sqrt{N_{\rm{t}}}{\mathbf{I}}_{N_{\rm{t}}}\triangleq{\mathbf{P}}_{\rm{no}}$
and the number of data streams is given by $N=N_{\rm{t}}$. The corresponding
high-SNR MST-EMI is characterized in the following theorem.

######  Theorem 4.

Let $\bar{\gamma}\rightarrow\infty$. Then, the MST-EMI without CSIT can be
characterized as ${\mathscr{I}}_{M}^{\mathcal{Y}}\simeq
H_{{\mathbf{q}}_{{\mathcal{Y}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$.

###### Proof:

The proof is given in Appendix B. ∎

######  Remark 4.

The results in Theorem 4 suggest that the diversity order of the MST-EMI
without CSIT is given by ${\mathcal{G}}_{\rm{d}}=1$, which is the same as that
of the SST-EMI without CSIT.

###  IV-B MST With CSIT

In this case, we consider two main precoding techniques.

####  IV-B1 MRT Precoding

By MRT precoding, we have

$\displaystyle{\mathbf{P}}=\left[{\mathsf{tr}}\left({\mathbf{H}}^{\mathsf{H}}{\mathbf{H}}\right)\right]^{-1/2}{\mathbf{H}}^{\mathsf{H}}\triangleq{\mathbf{P}}_{\rm{mrt}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times
N_{\rm{r}}}$ (15)

and $N=N_{\rm{r}}$, which yields
${\mathbf{y}}=\sqrt{\bar{\gamma}}{\mathbf{G}}{\mathbf{x}}+{\mathbf{n}}$ with
${\mathbf{G}}=\left\|{\mathbf{h}}_{\rm{t}}\right\|\frac{{\mathbf{h}}_{\rm{r}}}{\left\|{\mathbf{h}}_{\rm{r}}\right\|}{\mathbf{h}}_{\rm{r}}^{\mathsf{H}}\in{\mathbbmss{C}}^{N_{\rm{r}}\times
N_{\rm{r}}}$. The high-SNR EMI in this case is characterized as follows.

######  Theorem 5.

Let $\bar{\gamma}\rightarrow\infty$. Then, the MST-EMI achieved by the MRT
precoding satisfies ${\mathscr{I}}_{M}^{\mathcal{Y}}\simeq
H_{{\mathbf{q}}_{{\mathcal{Y}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$.

###### Proof:

Similar to the proof of Theorem 4. ∎

It is worth noting that the diversity order achieved by the MRT precoding is
${\mathcal{G}}_{\rm{d}}=1$, which is the same as that achieved without CSIT.
To address this issue, we proceed to the max-$d_{\min}$ precoding scheme which
enhances the diversity order.

####  IV-B2 Max-$d_{\min}$ Precoding

Optimizing $\texttt{I}\left({\bar{\gamma}};{\mathbf{H}}{\mathbf{P}}\right)$ at
high-SNRs is equivalent to maximizing the minimum distance [14]

$\displaystyle d_{\min}\triangleq\min\nolimits_{g\neq
g^{\prime}}\left\|{\mathbf{H}}{\mathbf{P}}{\mathbf{b}}_{g,g^{\prime}}\right\|=\min\nolimits_{g\neq
g^{\prime}}\left|{\mathbf{h}}_{\rm{t}}^{\mathsf{H}}{\mathbf{P}}{\mathbf{b}}_{g,g^{\prime}}\right|.$
(16)

The resulting max-$d_{\min}$ precoder is given by

$\displaystyle\mathbf{P}_{\star}=\operatorname*{argmax}\nolimits_{{\mathbf{P}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times
N},{\mathsf{tr}}\\{{\mathbf{P}}{\mathbf{P}}^{\mathsf{H}}\\}=1}d_{\min}.$ (17)

Yet, finding a closed-form solution to $\mathbf{P}_{\star}$ is a challenging
task, which makes the subsequent analyses intractable. As a compromise, we
propose a heuristic precoding design by exploiting the structure of
$d_{\min}$. Specifically, by observing (16), we design the heuristic
max-$d_{\min}$ precoder as a rank-one matrix that satisfies
${\mathbf{P}}_{\rm{mm}}={\left\|{\mathbf{h}}_{\rm{t}}\right\|}^{-1}{\mathbf{h}}_{\rm{t}}{\mathbf{d}}_{\star}^{\mathsf{H}}\in{\mathbbmss{C}}^{N_{\rm{t}}\times
N}$ with

$\displaystyle{\mathbf{d}}_{\star}={\operatorname*{argmax}}_{{\mathbf{x}}\in{{\mathbbmss{C}}^{N\times
1}},\left\|{\mathbf{x}}\right\|=1}\min\nolimits_{g\neq
g^{\prime}}\left|{\mathbf{x}}^{\mathsf{H}}{\mathbf{b}}_{g,g^{\prime}}\right|.$
(18)

Note that ${\mathbf{d}}_{\star}$ can be obtained via an off-line exhaustive
search, since it is fixed in $\mathbf{H}$. The corresponding high-SNR MST-EMI
is characterized in the following theorem.

######  Theorem 6.

Let $\bar{\gamma}\rightarrow\infty$. Then, the MST-EMI achieved by the
heuristic max-$d_{\min}$ precoder can be characterized as
${\mathscr{I}}_{M}^{\mathcal{X}}\simeq
H_{{\mathbf{q}}_{{\mathcal{Y}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-{\mathcal{G}}_{\rm{d}}}\right)$
with
${\mathcal{G}}_{\rm{d}}=\min\left\\{N_{\rm{t}}m_{\rm{t}},N_{\rm{r}}m_{\rm{r}}\right\\}$.

###### Proof:

Similar to the proof of Theorem 4. ∎

######  Remark 5.

In contrast to ${\mathbf{P}}_{{\rm{no}}}$ and ${\mathbf{P}}_{{\rm{mrt}}}$, the
diversity order achieved by ${\mathbf{P}}_{{\rm{mm}}}$ can be improved by
increasing $N_{\rm{r}}$ and $N_{\rm{t}}$, which highlights the superiority of
the max-$d_{\min}$ precoder.

###  IV-C Discussions on Keyhole Effect

Consider the Rayleigh fading model. Since it is challenging to obtain a
closed-form $\mathbf{P}$ that can maximize the MI with CSIT being available,
we only consider the case without CSIT. Using the approach in deriving Theorem
4, we find that the high-SNR MST-EMI without CSIT in full-rank and keyhole
MIMO channels can be written as
${\mathscr{I}}_{M,{\rm{nk}}}^{\mathcal{X}}\simeq
H_{{\mathbf{q}}_{{\mathcal{Y}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-N_{\rm{r}}}\right)$
and ${\mathscr{I}}_{M,{\rm{k}}}^{\mathcal{X}}\simeq
H_{{\mathbf{q}}_{{\mathcal{Y}}}}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$,
respectively.

######  Remark 6.

Comparing ${\mathscr{I}}_{M,{\rm{nk}}}^{\mathcal{X}}$ with
${\mathscr{I}}_{M,{\rm{k}}}^{\mathcal{X}}$, we find the keyhole effect can
reduce the diversity order of the MST-EMI. Similar conclusions can be given
for other rank-deficient MIMO channels with finite-alphabet and Gaussian
inputs under MST. We skip further details for sake of brevity.

##  V Numerical Results

We now validate our analyses through numerical simulations. Here, we set
$N_{\rm{t}}=N_{\rm{r}}=2$, $m_{\rm{t}}=2$, $m_{\rm{r}}=3$, $p_{g}=\frac{1}{M}$
for $g\in\\{1,\cdots,M\\}$, and $q_{g}=\frac{1}{M^{N}}$ for
$g\in\\{1,\cdots,M^{N}\\}$. As a result, we have
$H_{{\mathbf{p}}_{\mathcal{X}}}=\log_{2}{M}$ and
$H_{{\mathbf{q}}_{\mathcal{Y}}}=N\log_{2}{M}$. The simulation results are
gathered via $10^{6}$ channel realizations.

#### SST-EMI

Figure 2(a) shows the SST-EMI achieved by $M$-QAM signals for
$M\in\\{4,16,64,256\\}$ against the SNR, where the analytical EMI (denoted by
solid lines) is calculated by (10) or (13) by setting $V=200$. As Figure 2(a)
shows, the analytical results closely track the simulations (denoted by
symbols). This verifies the accuracy of (10) and (13). For comparison, we also
plot the EMI achieved by Gaussian signaling in Figure 2(a). As shown, the EMI
of Gaussian inputs grows unboundedly as $\bar{\gamma}$ increases, whereas the
EMI of finite-alphabet inputs converges to the entropy of the input, in the
large limit of $\bar{\gamma}$. Moreover, we observe that the EMI with CSIT is
higher than that without CSIT (denoted by NCSIT). By Remark 1, the rate of the
EMI (${{{\mathcal{I}}}}_{M}^{\mathcal{X}}$) converging to
$H_{{\mathbf{p}}_{\mathcal{X}}}$ equals the rate of
${\mathcal{I}}_{M}^{\rm{con}}=H_{{\mathbf{p}}_{\mathcal{X}}}-{{{\mathcal{I}}}}_{M}^{\mathcal{X}}$
converging to zero. To show this ROC, we plot ${\mathcal{I}}_{M}^{\rm{con}}$
versus $\bar{\gamma}$ in Figure 2(b). As shown, the derived asymptotic results
almost perfectly match the numerical results in the high-SNR regime. This
means that the diversity order derived in previous part is tight. It is
further seen that the EMI with CSIT yields a faster ROC (or a higher diversity
order) than that without CSIT. This agrees with the conclusion in Remark 2.

(a) Explicit results.

(b) Asymptotic results.

Figure 2: EMI of single-stream transmission.

(a) Explicit results.

(b) Asymptotic results.

Figure 3: EMI of multi-stream transmission with $N=2$.

#### MST-EMI

Turn now to the MST-EMI. Figure 3(a) compares the MST-EMI and SST-EMI achieved
by 4-QAM and Gaussian signals. In both cases of with and without CSIT, the
MST-EMI is higher than the SST-EMI and the Gaussian input achieves a higher
EMI than finite-alphabet inputs. We further observe that the max-$d_{\min}$
precoding yields virtually the same EMI as the MRT precoder in the low-SNR
regime but outperforms the latter one in the high-SNR regime. To show the ROC
of the EMI, we plot
${\mathcal{I}}_{M}^{\rm{con}}=H_{{\mathbf{p}}_{\mathcal{X}}}-{{{\mathcal{I}}}}_{M}^{\mathcal{X}}$
(for SST) and
${\mathcal{I}}_{M}^{\rm{con}}=H_{{\mathbf{q}}_{\mathcal{Y}}}-{\mathscr{I}}_{M}^{\mathcal{X}}$
(for MST) versus $\bar{\gamma}$ in Figure 3(b). The curves for
${\bar{\gamma}}^{{-{\mathcal{G}}_{\rm{d}}}}$ are further provided to
demonstrate the achievable diversity order. In the high-SNR regime, the curves
for ${\mathcal{I}}_{M}^{\rm{con}}$ are parallel to
${\bar{\gamma}}^{{-{\mathcal{G}}_{\rm{d}}}}$. This indicates that the derived
achievable diversity order is tight. Moreover, as Figure 3(b) shows, the
max-$d_{\min}$ precoder yields a faster ROC (or a higher diversity order) than
the MRT precoder, which is consistent with the conclusion in Remark 5.

##  VI Conclusion

For keyhole MIMO channels with finite-alphabet inputs, irrespective of the
number of streams, theoretical analyses indicate that the ROC of the EMI is
determined by the array gain and the diversity order. It is further found that
the keyhole effect can reduce the diversity order of the EMI achieved by
finite-alphabet inputs.

##  Appendix A Proof of Theorem 2

To facilitate the derivation, we rewrite the EMI as

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}\\!=\\!\left.I_{M}^{\mathcal{X}}\left(t\right)F_{S_{1}}\left(\frac{N_{\rm{t}}t}{\bar{\gamma}}\right)\right|_{0}^{\infty}\\!-\\!\int_{0}^{\infty}\\!\\!F_{S_{1}}\left(\frac{N_{\rm{t}}t}{\bar{\gamma}}\right){\rm
d}I_{M}^{\mathcal{X}}\left(t\right)$ (19)

with $F_{S_{1}}\left(\cdot\right)$ denoting the cumulative distribution
function of $S_{1}$. According to [8], we rewrite
${\mathcal{I}}_{M}^{\mathcal{X}}$ as

$\displaystyle{\mathcal{I}}_{M}^{\mathcal{X}}\\!=\\!H_{{\mathbf{p}}_{\mathcal{X}}}\\!-\\!\int_{0}^{\infty}\int_{0}^{{N_{\rm{t}}t}/{\bar{\gamma}}}f_{S_{1}}\left(x\right){\rm{d}}x\frac{{\rm{mmse}}_{M}^{\mathcal{X}}\left(t\right)}{\ln{2}}{\rm
d}t,$ (20)

where ${\rm{mmse}}_{M}^{\mathcal{X}}\left(\gamma\right)=\frac{{\rm
d}I_{M}^{\mathcal{X}}\left(\gamma\right)}{{\rm d}\gamma}\ln{2}$ is the MMSE in
estimating $X$ in (5) by observing $Y$ [8]. When
$\bar{\gamma}\rightarrow\infty$, we have $\frac{1}{\bar{\gamma}}\rightarrow
0$, which together with the facts of
$K_{\nu}\left(z\right)\\!=\\!K_{-\nu}\left(z\right)$ [10, Eq. (10.27.3)] and
$\lim_{z\rightarrow
0}K_{\nu}\left(z\right)\\!=\\!\frac{1}{2}\Gamma\left(\nu\right)\left(\frac{1}{2}z\right)^{-\nu}$
($\nu>0$) [10, Eq. (10.30.2)], yields
$\lim_{\bar{\gamma}\rightarrow\infty}{\mathcal{I}}_{M}^{\mathcal{X}}=\dot{\mathcal{I}}_{M}^{\mathcal{X}}$,
where

$\displaystyle\dot{\mathcal{I}}_{M}^{\mathcal{X}}\triangleq
H_{{\mathbf{p}}_{\mathcal{X}}}\\!-\\!{\mathcal{F}}\\!\left\langle\\!{\frac{\Gamma\left(\left|N_{\rm{r}}m_{\rm{r}}-h-1\right|\right)\hat{\mathcal{M}}\left(\bar{h}+1\right)}{\Gamma\left(N_{\rm{r}}m_{\rm{r}}\right)\bar{h}\left({4U_{N_{\rm{t}}}\bar{\gamma}}/({N_{\rm{t}}m_{\rm{r}})}\right)^{\bar{h}}\ln{2}}\\!}\right\rangle$
(21)

and $\bar{h}\triangleq\min\left\\{N_{\rm{r}}m_{\rm{r}},h+1\right\\}$. Then, we
introduce the following two lemmas for further discussion.

######  Lemma 4.

Given the constellation
${\mathcal{X}}=\left\\{\mathsf{x}_{g}\right\\}_{g=1}^{M}$, the MMSE function
satisfies
$\lim_{\gamma\rightarrow\infty}{\rm{mmse}}_{M}^{\mathcal{X}}\left(\gamma\right)={\mathcal{O}}(\gamma^{-\frac{1}{2}}{\rm
e}^{-\frac{\gamma}{8}d_{{\mathcal{X}},{\min}}^{2}})$, where
$d_{\mathcal{X},\min}\triangleq\min_{g\neq
g^{\prime}}\left|{\mathsf{x}_{g}}-{\mathsf{x}_{g^{\prime}}}\right|$ [16].

######  Lemma 5.

If $p\left(t\right)$ is ${\mathcal{O}}\left(t^{a}\right)$ as $t\rightarrow
0^{+}$ and ${\mathcal{O}}\left(t^{b}\right)$ as $t\rightarrow+\infty$, then
$\left|{\mathcal{M}}\left[p\left(t\right);z\right]\right|<\infty$ when
$-a<z<-b$ [12].

Particularly, $\lim_{t\rightarrow
0^{+}}{\rm{mmse}}_{M}^{\mathcal{X}}\left(t\right)=1$ [8], which together with
Lemma 4, suggests that ${\rm{mmse}}_{M}^{\mathcal{X}}\left(t\right)$ is
${\mathcal{O}}\left(1\right)$ as $t\rightarrow 0^{+}$ and
${\mathcal{O}}\left(t^{-\infty}\right)$ as $t\rightarrow\infty$. Using this
fact and Lemma 5, we find that $|\hat{\mathcal{M}}\left(x\right)|<\infty$
holds for $0<x<\infty$, which in combination with the fact that
${\rm{mmse}}_{M}^{\mathcal{X}}\left(x\right)>0$ ($x>0$) [8], suggests that
$\hat{\mathcal{M}}\left(x\right)\in\left(0,\infty\right)$ holds for
$0<x<\infty$. It follows from
$\bar{h}=\min\left\\{N_{\rm{r}}m_{\rm{r}},h+1\right\\}>0$ that
$\hat{\mathcal{M}}\left(\bar{h}+1\right)\in\left(0,\infty\right)$. As
previously assumed, $N_{\rm{r}}m_{\rm{r}}\neq h+1$,
$m_{\rm{r}}\geq\frac{1}{2}$, and $N_{\rm{r}}>1$, which yields
$N_{\rm{r}}m_{\rm{r}}>1$. We then neglect the higher order terms in (21) to
derive the asymptotic EMI as
${\mathcal{I}}_{M}^{\mathcal{X}}\simeq{H_{{\mathbf{p}}_{\mathcal{X}}}}-{\mathcal{G}}_{\rm{a}}^{-1}{\bar{\gamma}}^{-1}$,
where ${\mathcal{G}}_{\rm{a}}^{-1}$ is shown in (12).

##  Appendix B Proof of Theorem 4

###### Proof:

The MI satisfies [14]

$\displaystyle\texttt{I}\left({\bar{\gamma}};{\mathbf{H}}{\mathbf{P}}\right)=L\log_{2}{M}-\frac{1}{\ln{2}}\int_{{\bar{\gamma}}}^{\infty}{\text{mmse}}_{M}^{\mathcal{X}}\left(x;{\mathbf{H}}{\mathbf{P}}\right){\rm{d}}x,$
(22)

where
${\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right)$
denotes the MMSE in estimating $\mathbf{x}$ in (14) by observing $\mathbf{y}$.
Moreover, for any MIMO channels, the MMSE is bounded by [14]

$\displaystyle\underline{\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right)\\!\leq\\!{\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right)\\!\leq\\!\overline{\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right).$
(23)

Defining $f_{l}\left(x\right)\triangleq
1-\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\tanh\left(\sqrt{x}a\right){\emph{e}}^{-{\left(a-\frac{\sqrt{x}}{2}\right)^{2}}}{\rm{d}}a$,
$f_{u}\left(x\right)\triangleq Q\left(\sqrt{\frac{{x}}{2}}\right)$ with
$Q\left(x\right)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}{\emph{e}}^{-u^{2}/2}{\rm{d}}u$
being the Q-function, and
$d_{i,k}\triangleq\left\|{\mathbf{H}}{\mathbf{P}}{\mathbf{b}}_{i,k}\right\|^{2}$,
we have [14, Appendix III]

$\displaystyle\underline{\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right)=\sum\nolimits_{i,k=1,k\neq
i}^{M^{L}}\frac{d_{i,k}}{4M^{L}}\frac{f_{l}\left(\bar{\gamma}d_{i,k}\right)}{M^{L}-1},$
(24)
$\displaystyle\overline{\text{mmse}}_{M}^{\mathcal{X}}\left(\bar{\gamma};{\mathbf{H}}{\mathbf{P}}\right)=\sum\nolimits_{i,k=1,k\neq
i}^{M^{L}}\frac{d_{i,k}}{M^{L}}f_{u}\left(\bar{\gamma}d_{i,k}\right).$ (25)

Therefore, the EMI is upper bounded by

$\displaystyle{\mathscr{I}}_{M}^{\mathcal{X}}\leq
L\log_{2}{M}-\frac{1}{\ln{2}}\int_{{\bar{\gamma}}}^{\infty}\underline{\text{mmse}}_{M}^{\mathcal{X}}\left(x;{\mathbf{H}}{\mathbf{P}}\right){\rm{d}}x\triangleq\overline{\mathscr{I}}_{M}^{\mathcal{X}}.$
(26)

After some manipulations, we can get

$\displaystyle\overline{\mathscr{I}}_{M}^{\mathcal{X}}=L\log_{2}{M}-\sum\nolimits_{i,k=1,k\neq
i}^{M^{L}}\\!\frac{\underline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}\log_{2}{\emph{e}}}{4\left(M^{L}-1\right)M^{L}},$
(27)

where
$\underline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}\triangleq\int_{0}^{\infty}\int_{{\bar{\gamma}}}^{\infty}{y}f_{l}\left(\bar{\gamma}y\right)f_{i,k}\left(y\right){\rm{d}}x{\rm{d}}y$
with $f_{i,k}\left(y\right)$ denoting the PDF of $d_{i,k}$. It follows that

$\displaystyle\underline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}=\int_{0}^{\infty}\frac{1}{\bar{\gamma}}f_{i,k}\left(\frac{y}{\bar{\gamma}}\right)\int_{y}^{\infty}f_{l}\left(x\right){\rm{d}}x{\rm{d}}y.$
(28)

When ${\mathbf{P}}=1/\sqrt{N_{\text{t}}}{\mathbf{I}}_{N_{\text{t}}}$, we have
$d_{i,k}=\left\|{\mathbf{h}}_{\text{r}}\right\|^{2}\left|1/\sqrt{N_{\text{t}}}{\mathbf{h}}_{\text{t}}^{\mathsf{H}}{\mathbf{b}}_{i,k}\right|^{2}$,
whose PDF presents the same form as (9) by setting
$U_{N_{\text{t}}}=\sum_{a=1}^{N_{\text{t}}}\frac{\left|b_{i,k,a}\right|}{4m_{\text{t}}{N_{\text{t}}}}$
and
$Y_{N_{\text{t}}}=\prod_{a=1}^{N_{\text{t}}}\left(\frac{\left|b_{i,k,a}\right|}{4m_{\text{t}}{N_{\text{t}}}}\right)^{i_{a}}$.
Following similar steps as those outlined in Appendix A, we find that when
$\bar{\gamma}\rightarrow\infty$, it has

$\displaystyle\underline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}\\!\simeq\\!\sum_{i_{1}=0}^{m_{\text{t}}-1}\\!\\!\cdots\\!\\!\sum_{i_{N_{\text{t}}}=0}^{m_{\text{t}}-1}\\!\frac{S_{N_{\text{t}}}!Y_{N_{\text{t}}}m_{\text{r}}{\mathcal{M}}_{l}\left(1\right){\bar{\gamma}}^{-1}}{4\left(N_{\text{r}}m_{\text{r}}-1\right)X_{N_{\text{t}}}U_{N_{\text{t}}}^{S_{N_{\text{t}}}+1}}$
(29)

with
${\mathcal{M}}_{l}\left(t\right)\\!\triangleq\\!{\mathcal{M}}\\!\left[\int_{y}^{\infty}\\!f_{l}\left(x\right){\rm{d}}x;{t}\right]$.
Define
$\underline{f}\left(y\right)=\int_{y}^{\infty}f_{l}\left(x\right){\rm{d}}x$,
and we have $\lim_{x\rightarrow 0^{+}}f_{l}\left(x\right)=0$ and
$\lim_{x\rightarrow\infty}f_{l}\left(x\right)={\mathcal{O}}\left({\emph{e}}^{-\frac{x}{4}}x^{-\frac{1}{2}}\right)$
[15, Theorem 3, Appendix B], indicating that $f_{l}\left(x\right)$ is
${\mathcal{O}}\left(x^{a}\right)$ ($a\geq 0$) as $t\rightarrow 0^{+}$ and
${\mathcal{O}}\left(x^{-\infty}\right)$ as $x\rightarrow\infty$. It follows
form this fact and Lemma 5 that $\lim_{y\rightarrow
0^{+}}\underline{f}\left(y\right)=\int_{0}^{\infty}f_{l}\left(x\right){\rm{d}}x={\mathcal{O}}\left(y^{0}\right)\in\left(0,\infty\right)$.
Moreover, based on L’Hôspital’s rule and [15, Appendix B], we can get
$\lim_{y\rightarrow\infty}\underline{f}\left(y\right)={\mathcal{O}}\left({\emph{e}}^{-\frac{x}{4}}x^{-\frac{1}{2}}\right)$.
By continuously using Lemma 5, we find that
${\mathcal{M}}_{l}\left(1\right)\in\left(0,\infty\right)$ and thus
$\underline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}={\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$.
It follows that

$\displaystyle\lim_{\bar{\gamma}\rightarrow\infty}{\mathscr{I}}_{M}^{\mathcal{X}}\leq\lim_{\bar{\gamma}\rightarrow\infty}\overline{\mathscr{I}}_{M}^{\mathcal{X}}=N_{\text{t}}\log_{2}{M}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right).$
(30)

Turn now to the EMI’s lower bound given by

$\displaystyle{\mathscr{I}}_{M}^{\mathcal{X}}\geq
L\log_{2}{M}-\sum\nolimits_{i,k=1,k\neq
i}^{M^{L}}\\!\frac{\overline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}}{M^{L}\ln{2}}\triangleq\underline{\mathscr{I}}_{M}^{\mathcal{X}}$
(31)

with
$\overline{\mathscr{I}}_{M,i,k}^{\mathcal{X}}\triangleq\int_{0}^{\infty}\int_{{\bar{\gamma}}}^{\infty}{y}f_{u}\left(\bar{\gamma}y\right)f_{i,k}\left(y\right){\rm{d}}x{\rm{d}}y$.
We find that when $\bar{\gamma}\rightarrow\infty$,

$\displaystyle\overline{\mathscr{I}}_{M,i,j}^{\mathcal{X}}\\!\simeq\\!\sum_{i_{1}=0}^{m_{\text{t}}-1}\\!\\!\cdots\\!\\!\sum_{i_{N_{\text{t}}}=0}^{m_{\text{t}}-1}\\!\frac{S_{N_{\text{t}}}!Y_{N_{\text{t}}}m_{\text{r}}{\mathcal{M}}_{u}\left(1\right)U_{N_{\text{t}}}^{-S_{N_{\text{t}}}-1}{\bar{\gamma}}^{-1}}{4\left(N_{\text{r}}m_{\text{r}}-1\right)\prod_{k=1}^{N_{\text{t}}}\left(\frac{\left(i_{k}!\right)^{2}}{\left(1-m_{\text{t}}\right)_{i_{k}}}\right)},$
(32)

where
${\mathcal{M}}_{u}\left(t\right)\triangleq{\mathcal{M}}\left[\int_{y}^{\infty}f_{u}\left(x\right){\rm{d}}x;{t}\right]$.
Then, following similar steps in proving
${\mathcal{M}}_{l}\left(1\right)\in\left(0,\infty\right)$, we can prove
${\mathcal{M}}_{u}\left(1\right)\in\left(0,\infty\right)$ and thus

$\displaystyle\lim_{\bar{\gamma}\rightarrow\infty}{\mathscr{I}}_{M}^{\mathcal{X}}\geq\lim_{\bar{\gamma}\rightarrow\infty}\underline{\mathscr{I}}_{M}^{\mathcal{X}}=N_{\text{t}}\log_{2}{M}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right),$
(33)

which together with (30), yields
$\lim_{\bar{\gamma}\rightarrow\infty}{\mathscr{I}}_{M}^{\mathcal{X}}=N_{\text{t}}\log_{2}{M}-{\mathcal{O}}\left({\bar{\gamma}}^{-1}\right)$.
∎

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