### Performance of ED-Based Non-Coherent Massive

### SIMO Systems in Correlated Rayleigh Fading

HUIQIANG XIE1, WEIYANG XU 1, (Member, IEEE),

WEI XIANG 2, (Senior Member, IEEE), BING LI1, AND RUI WANG3

1_{School of Microelectronics and Communication Engineering, Chongqing University, Chongqing 400044, China}
2_{College of Science, Technology and Engineering, James Cook University, Cairns, QLD 4870, Australia}
3_{Instrumentation Engineering Technology Research Center of Hebei Province, Chengde 067000, China}
Corresponding author: Weiyang Xu (weiyangxu@cqu.edu.cn)

This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant 2018CDXYTX0011, and in part by the Key Program of Natural Science Foundation of Chongqing under Grant CSTC2017JCYJBX0047.

**ABSTRACT** In recent studies, a simple non-coherent communications’ scheme based on energy detection
is proposed for massive single-input multiple-output systems, where transmit symbols are decoded by
averaging the received signal energy across all antennas. In this paper, we investigate the effect of correlated
Rayleigh fading on the performance of the aforementioned systems. Specifically, we derive the analytical
expressions of the symbol error rate, achievable rate, and outage probability. Furthermore, the asymptotic
behaviors of these expressions in regimes of large numbers of receive antennas, high channel correlation, and
high signal-to-noise ratio are investigated. The analytical results demonstrate the adverse impact of channel
correlation on the error probability, which can be attributed to the fact that channel correlation reduces the
degrees of freedom. Our analysis also shows that channel correlation poses little influence on the achievable
rate. Besides, an upper bound of the achievable rate is given when the number of receive antennas goes to
infinity. Finally, the numerical results are presented to verify our analysis.

**INDEX TERMS** Non-coherent massive single-input multiple-output (SIMO), correlated Rayleigh fading,
symbol-error rate (SER), achievable rate.

**I. INTRODUCTION**

In meeting the rapid growth of data traffic in the fifth genera-tion (5G) cellular networks, technological innovagenera-tion plays an essential role. Among those candidates that have been proposed, the massive multiple-input multiple-out (MIMO) technology, which deploys a large number of antennas at the base station (BS) to serve a relatively small number of users, can greatly enhance capacity due to its increased degrees of freedom [1], [2]. In addition, massive MIMO is energy effi-cient since the transmit power scales down with the number of antennas at BS. Moreover, the channel vectors associated with different users are asymptotically orthogonal, and thus both intra- and inter-cell interference can be eliminated with simple detection and precoding schemes [3]–[5].

To take full advantage of spatial multiplexing gains pro-vided by a large amount of antennas [6], the channel state information at the transmitter (CSIT) is essential in coherent massive MIMO systems. However, with large-scale antenna arrays, obtaining accurate CSIT can be cumbersome in real applications. In fact, the complexity of channel estimation is

so high that estimating the CSIT in a timely manner becomes intractable. Moreover, non-orthogonal pilots among adjacent cells would make the problem even worse, because channel estimates may be corrupted by the pilots transmitted from other cells [7]. This is dubbed pilot contamination, which has become a fundamental performance bottleneck in massive MIMO systems. [8] and references therein provide a good overview of schemes to tackle this problem, such as protocol-based, precoding and blind or semi-blind methods [9]–[12].

As an alternative, non-coherent communications systems
based on energy detection (ED) have attracted a great
atten-tion [13]–[16], since it requires no *prior* knowledge of
instantaneous channel state information (CSI) at either the
transmitter or receiver, In spite of a sub-optimal performance,
non-coherent receivers enjoy the benefits of low
complex-ity, low power consumption and simple structures compared
to their coherent counterparts [17]. In order to optimize
system performance, a minimum distance constellation and
asymptotically optimal constellation based on upper bound
SER is presented in [18] and [19] respectively. However,

14058

2169-35362019 IEEE. Translations and content mining are permitted for academic research only.

above two constellation design both base on approximate SER rather than exact SER. Besides, taking both the average and instantaneous channel energy into consideration, two non-coherent receiver architectures are designed for massive SIMO in [20]. Nevertheless, both two non-coherent receivers do not consider the effect of channel correlation. In addition, given that the number of receive antennas is asymptotically infinite, the ED-based non-coherent massive SIMO system can provide the same error performance as that of its coherent counterparts [21].

While in real-world application, communication system may contain variety of channel correlation, which could adversely affect the SER and system capacity. The impact of channel correlation on conventional MIMO has been investi-gated thoroughly. In [22] and [23], the effects of spatial cor-relation and mutual antenna coupling are separately studied. It is shown that energy efficiency does not increase unbound-edly in massive MIMO system when antennas are accommo-dated within a fixed physical space [22]. Besides, [24] uses a more general correlation model to derive insight on how spectral efficiency, capacity and channel gain are impacted by correlation. In addition, utilizing the eigenvalues of input covariance and channel matrix, a closed-form expressions of the capacity in correlative channel are proposed in [25]. Experimental studies on the transmit correlation have been conducted in [27] and [28]. Through analyzing spatial corre-lation, a new reconfigurable antenna array is demonstrated for improving link capacity in MIMO system [27]. Furthermore, in order to efficiently measure spatial correlation in indoor environments, a measurement method is proposed based on a broadband MIMO channel platform [28]. However, above experiments only investigated under conventional MIMO system rather than Massive MIMO systems.

The aforementioned studies validate that channel corre-lation can pose an adverse impact on the error probability in coherent massive MIMO systems. Furthermore, it also reduces the system capacity [25]. However, for non-coherent massive SIMO systems, whether and how antenna correlation influences the achievable rate or error performance is still not clear. Inspired by this, our study presents a thorough per-formance analysis of ED-based non-coherent massive SIMO systems over correlated Rayleigh fading channels. The main contributions of this paper can be summarized as follows.

• We derive analytical expressions of the SER, achievable rate and outage probability for ED-based non-coherent massive SIMO systems with channel correlation. • Closed-form expressions of the achievable rate are

pre-sented, which are built upon polynomial methods and Gaussian approximation. Conversely, the channel cor-relation poses far less impact on the achievable rate than the SER. As expected, analytical result proves the achievable rate cannot increase unboundedly because of the non-coherent scheme.

The remainder of this paper is organized as follows. The considered system model is illustrated in Section II. The preliminary work and the derivation of the SER are detailed

in Section III. Section IV presents analytical results of achiev-able rate and outage probability. Numerical simulation results are presented to show the effectiveness of our analysis in Section V. Finally, Section VI concludes this paper.

*Notation:*C*n*×*m*indicates a matrix composed of complex
numbers of size*n*×*m. Bold-font variables represent matrices*
and vectors. P indicates a set of constellation points. For
a random vector **x**, **x** ∼ CN(µ, σ2) means it follows a
complex Gaussian distribution with meanµand covariance

σ2_{.}_{(}_{·}_{)}*H*_{,}

E[·] andk·k2denote the Hermitian, expectation and

L2norm operators, respectively. <(·) refers to the real part of a complex number. sup(·) is the least upper bound. Finally, erf(·) and erfc(·) are taken to mean the Gaussian error function and complementary Gaussian error function, respectively.

**II. SYSTEM MODEL**

We consider a massive SIMO configuration with one transmit antenna and a large number of receive antennas. The cor-related Rayleigh flat-fading channels of different transmit-receive pairs are assumed. As such, the transmit-received signal vector is represented by

**y**=**g***x*+**n** (1)
where**y**∈_{C}*M*×1represents the received signal at the
multi-antenna receiver,**n** ∈ _{C}*M*×1indicates a complex Gaussian
noise vector with its *n-th element* *ni* ∼ CN(0, σ*n*2), **g** ∈

C*M*×1refers to the correlated Rayleigh channel realization,
*x*is the transmit symbol drawn from a certain non-negative
constellationP = √

*p*1,
√

*p*2, . . . ,
√

*pK* ,*K* indicates the

constellation size and*M* is the number of receive antennas.
In our study,σ* _{n}*2is assumed to be known as a prior, which can
be achieved by sending a sequence of training symbols before
data transmission [20].

A. CORRELATED RAYLEIGH CHANNEL MODEL

Although it is reasonable to assume that**g**is independent of
transmit symbol*x* and noise vector**n**, different*gi*s’ could

be mutually correlated because of closely-spaced diversity branches. The correlated channel can be constructed by [29]

**g**=81/2

*r* **h**8

1/2

*t* (2)

where**h**∈_{C}*M*×1is a complex channel vector whose entries
are independent identically distributed (i.i.d.) with its *i-th*
component*hi* ∼CN(0,1).8*r*1/2and81*t*/2are deterministic

receive and transmit correlation matrices. In the case of a
single transmit antenna, the correlation matrix81* _{t}*/2can be
omitted. Therefore, (2) reduces to

**g**=81* _{r}*/2

**h**. (3) For the structure of8

*r*, the exponential correlation model

is often utilized to quantify the level of spatial
correla-tion [29]. Accordingly, the receive correlacorrela-tion matrix can be
constructed by correlation coefficientρ∈_{C}, namely

8*ij*=
(

ρ|*j*−*i*|_{,} _{i}

6*j*

ρ|*j*−*i*|∗_{,}

**FIGURE 1.** Decoding regions for non-coherent massive SIMO systems
with a non-negative PAM ofK=4.

where| · |is the absolute value operation and8*ij* indicates

the (i,*j)-th entry of* 8*r*,ρ = *aej*θ is the correlation

coef-ficient with 0 _{6} *a* < 1. Note that the eigenvalues of8*r*

only depend on *a, while* θ decides the eigenvectors of8*r*.

Since only eigenvalues of8*r* will be used in the following

analysis, we assumeρ =*a*throughout this paper. In addition,
the correlation coefficient is supposed to be constant, due to
the fact that it is far less frequently varying than the channel
matrix [30].

B. ENERGY DETECTION

Based on the ED principle, after the received signal having been filtered, squared and integrated, the decision metric can be written as

*z*= k**y**k
2
2

*M* (5)

which is able to be approximated to one of the*K* Gaussian
variables depending on which symbol was transmitted. For
example, with a non-negative PAM of*K*=4, the probability
density function (PDF) of*z*over an additive white Gaussian
noise (AWGN) channel is shown in Fig. 1, where *M* =
100 and SNR = 4 dB. Four distinct Gaussian-like curves
can be observed, corresponding to four constellation points.
Accordingly, provided with the knowledge of channel and
noise statistics, the positive line is partitioned into multiple
decoding regions{*dk*}*K _{k}*

_{=0}to decide which symbol was

trans-mitted based on the observation of*z, i.e.*

b*x*=

√

*pk* if*dk*−16*z*<*dk*. (6)

For the first point√*p*1,*d*0 is−∞, and for the last point
√

*pK*,*dK* is+∞. In this system, the non-negative PAM is

exploited in modulation and the decoding regions of_{b}*x*=√*pk*

can be given by [31]

*dk* =

*pk*+*pk*+1

2 +σ

2

*n*. (7)

With (7), the process of obtaining the decision threshold can be largely simplified.

**III. SYMBOL-ERROR RATE ANALYSIS**

In this section, two closed-form expressions of SER are derived. Specifically, the first result is accurate but difficult to analyze, thus the Gaussian approximation is employed to obtain the second expression. The asymptotic behaviors in regimes of large numbers of receive antennas and high SNRs are also discussed.

A. EXACT SYMBOL-ERROR RATE

To obtain the PDF of decision metric*z, which is essential to*
the SER analysis, the received signal is first rewritten as

**y**=81* _{r}*/2

**h**

*x*+

**n**=

**U**

*H*321

**Uh**

*+*

_{x}**n**=

**U**

*H*312

**v**

*+*

_{x}**n**(8) where the eigendecomposition is used to transform81

*r*/2 =

**U**312_{U}*H*_{,}3
1

2 _{is an eigenvalue diagonal matrix with its}_{i-th}

item beingλ 1 2

*i*, and**U**indicates a unitary matrix consisting of

corresponding eigenvectors. In addition,**v**=**Uh**follows the
same distribution as**h**and entries of**v**are mutually
indepen-dent [32]. After left-multiplying**y**with a unitary matrix **U**,
one can arrive at the following result

**Uy**=**UU***H*312_{v}* _{x}*+

**Un**=312

_{v}*+*

_{x}**q**. (9) Then, (5) can be rewritten as

*z*= **y**

*H*_{U}*H*_{Uy}

*M* =

1

*M* k**y**k
2
2=

1

*M*

*M*
X

*i*=1

|*yi*|2 (10)

where|*yi*|22 ∼ λ

*ix*2+σ*n*2

2 χ

2_{(}_{2}_{)}_{. In this paper, the exact SER}

expression is derived utilizing the characteristic function of
the PDF of*z.*

First, the characteristic function of|*yi*|2

*M* is given by

ϕβ*i*(t)=

1

(1−β* _{i}t*) (11)

whereβ*i*= λ*i*
*x*2_{+}_{σ}2

*n*

*M* . Accordingly, the characteristic function

of*z*is

ϕ*z*(*t*)=

1

(1−β_{1}*t*)*v*1
1

(1−β_{2}*t*)*v*2 · · ·
1

1−β* _{g}_{t}vg* (12)

whereP*g*

*i*=1*vi*is the rank of the correlation matrix8*r*and*g*is
the number of distinct nonzero eigenvalues. Actually, the rank
of the correlation matrix equals to*M* if and only ifρ 6= 1.
Resolving (12) into its partial fractions with repeated roots
yields

ϕ*z*(*t*)=
*g*
X

*i*=1

*vg*
X

*j*=1

*Ai*,*j*ϕ_{β}*j _{i}*(

*t*) (13)

where the coefficients*Ai*,*j*can be found as

*Ai*,*j*=

1

(*vi*−*j*)!β*vi*

−*j*
*i*

∂*vi*−*j*
∂*tvi*−*j*

*g*
Y

*q*=1,*q*6=*i*
ϕ*vq*

β*q*(*t*)

*t*=

1
β*i*

(14)

The linearity of the inverse of the characteristic function
transform allows therefore rewriting the PDF of*z*as

*f* (*z*)=

*g*
X

*i*=1

*vg*
X

*j*=1
*Ai*,*j*

*zj*−1

0 (*j*) β*ij*

exp

−*z*

β*i*

(15)

where0 (·) is the Gamma function. Then, the cumulative distribution function (CDF) of (15) is presented by

*F*(*z*)=

*g*
X

*i*=1

*vg*
X

*j*=1
*Ai*,*j*

γ*j*,_{β}*z*

*i*

**FIGURE 2.** PDFs ofzwith SNR=6 dB andK=4, which are shown using
the linear scale (left) and logarithmic scale (right).

where

γ*j*, *z*

β*i*

=

Z _{β}*z*
*i*

0

*tj*−1*e*−*tdt*.

Eventually, the SER expression is obtained by integrating the former equation, namely

*Pe*=
*K*
X

*k*=1

(*F*(*dk*)−*F*(*dk*−1)) (17)

B. PDF BASED ON GAUSSIAN APPROXIMATION

Since the SER in (17) is difficult to analyze, an approximate
result is given here to shed light on the relationship between
SER and key parameters. To facilitate the derivation of PDF
of*z, we resort to the following lemma.*

*Lemma 1:* If the number of receive antennas *M* grows
large, then the following approximations are attainable thanks
to Lyapunov Central Limit Theorem (CLT).

*z*∼N(µ*z*, σ*z*2) (18)

where

µ*z* =*x*2+σ*n*2

σ2

*z* =

1
*M*

*x*2+σ2

*n*
2

+*f*(ρ)x
4

*M*2 (19)

with

*f*(ρ)=2M(ρ

2_{−}_{ρ}4_{)}_{−}_{2}_{ρ}2

(1−ρ2_{)}2 , 06ρ <1 (20)

*Proof:* The proof can be found in Appendix A.
It can be shown that the decision metric*z*approximates to
a real Gaussian distribution. Besides, it is clear from (19) that
the channel correlation does not affectµ*z*. However, the

vari-ance σ* _{z}*2 is an increasing function ofρ and can be greatly
influenced. Intuitively, when the correlation coefficient varies
from 0 to 1, the overlap between neighboring PDFs of

*z*would increase. As a result, the final decision becomes more prone to errors.

In addition, Fig.2 compares the analytical result in (18) with the distribution obtained by numerical simulations at SNR=6 dB, where 4-PAM constellation is employed. The left figure verifies that our derivation is quite accurate due to the well match between these two results. However, although this difference is small in absolute value, it would be ampli-fied if the logarithmic scale is utilized, as shown in the right figure. This observation could results in a mismatch between analytical and numerical SER, which will be encountered in SectionV.

C. SER DERIVATION AND ANALYSIS

In this part, the approximate SER is derived based on the PDF
shown in (18). There are *K* PDFs of*z*depending on which
symbol inP was transmitted. The expectation and variance
of*z*when√*pk*is selected at the transmitter can be written as

µ*z*(p*k*)=*pk*+σ*n*2

σ2

*z*(p*k*)=

1
*M*

*pk*+σ*n*2
2

+*f*(ρ)p
2

*k*

*M*2 . (21)

*Proposition 1:* In the presence of correlated Rayleigh
fad-ing, the SER of the ED-based non-coherent massive SIMO
system employing Gaussian approximation is given by

*P*G* _{e}* =1− 1

*K*

*K*
X

*k*=1
*P(pk*)

=1− 1 2K

*K*
X

*k*=1

erf

1

*k*,*L*

√
2σ*z*(p*k*)

+erf

1

*k*,*R*

√
2σ*z*(p*k*)

(22)

where1*k*,*L* =µ*z*(p*k*)−*dk*−1and1*k*,*R*=*dk*−µ*z*(p*k*).

*Proof:* The proof can be found in Appendix A.
It is worth noting that (22) is a generalized result suitable
for a variety of non-negative constellations. Given variance

σ*z*(p*k*) and decoding regions, one can obtain the

correspond-ing error probability. Moreover, the result in (22) reveals how
the channel correlation affects the error performance. When
*M*, SNR and constellation size are fixed,σ* _{z}*2(p

*k*) increases

along withρ. Since SER is an increasing function ofσ* _{z}*2(p

*k*),

the error probability will increase if channels of different transmit-receive pairs are more correlated. Hence, the follow-ing analysis, which concerns the behaviors of error rate in the events of larger numbers of receive antennas and high SNRs, is expected to find out how to combat the adverse impact brought by the correlated Rayleigh fading.

1) THE INFLUENCE OF THE NUMBER OF RECEIVE ANTENNAS

In order to demonstrate the impact of *M* on the SER,
the closed-form expression in (22) is first transformed into

*P*G* _{e}* = 1
2K

*K*
X

*k*=1

erfc

1

*k*,*L*

√
2σ*z*(p*k*)

+erfc

1

*k*,*R*

√
2σ*z*(p*k*)

.

Because *M* influences erfc_{√}1*k*,*R*

2σ*z*(*pk*)

and erfc_{√}1*k*,*L*

2σ*z*(*pk*)

in the same way, only erfc_{√}1*k*,*R*

2σ*z*(*pk*)

is considered here. To make the SER analysis feasible, we employ the result in [33] to approximate function erfc(x).

*Lemma 2:* When*x* → ∞, the complementary error
func-tion erfc(x) approaches its limit as follows

erfc(x)≈ *e*
−*x*2

√

π*x*, if*x*→ ∞. (24)

Using Lemma2, erfc_{√}1*k*,*R*

2σ*z*(*pk*)

can be approximated to

erfc

1

*k,R*

√
2σ*z*(p*k*)

≈ √

2σ*z*(p*k*)e

− 1 2

*k*,*R*

2σ2

*z*(*pk*)

√

π1*k*,*R*

. (25)

Afterwards, the logarithmic operation is applied on both sides, i.e.

log erfc

1

*k*,*R*

√
2σ*z*(p*k*)

= − 1 2

*k*,*R*

2σ2

*z*(p*k*)

log*e*−1
2log

12

*k*,*R*
σ2

*z*(p*k*)

+ 1 2log

2

π (26)

where

12

*k,R*
σ2

*z*(p*k*)

= 1

2

*k,R*

(p*k*+σ*n*2)

2 +2ρ

2* _{p}*2

*k*

1−ρ2

*M*+*A* (27)

with

*A*= 2ρ

2* _{p}*2

*k*1

2

*k,R*

(1−ρ2_{)(p}

*k*+σ*n*2)

2_{+}
2ρ2* _{p}*2

*k*
2.

The derivation of (27) is shown in Appendix B.

The first component in (26) plays a far more important role
than the second due to the logarithmic operation. Here we
pay attention to the coefficient of *M* in (27). Although the
logarithm of SER decreases almost linearly with*M*, the rate
of descent is subject to ρ. As ρ increases, the coefficient
of *M* gradually reduces to zero. Hence, when ρ → 1, (27)
approximates to

lim

ρ→1

12

*k,R*
σ2

*z*(p*k*)

= (d*k*−µ*z*(p*k*))
2

2p2* _{k}* (28)

which is a constant independent of*M*. It is readily observed
that no matter how large the number of antennas is, it will not
be helpful to reduce SER. Moreover, the following findings
can be obtained from the above analysis.

• Deploying more antennas at the receive end is highly effective to avoid decision error;

• The channel correlation would reduce the coefficient
of *M* in (27), which harms the performance
improve-ment provided by extra antennas. In this situation,
the receive diversity of massive antenna array cannot be
fully exploited;

• For a certainρ, there exists an optimal number of receive antennas which strikes the balance between the perfor-mance gain brought by extra antennas and perforperfor-mance loss incurred by an increased exponential correlation. • In the limit ofρ → 1, which means all the channels

are identical, increasing *M* is of no use in enhancing
the system performance. In this scenario, the receive
diversity of massive antenna array is completely lost.

2) THE INFLUENCE OF SNRS

In the presence of low SNRs and a fixedρ, the variance in (21)
converges to zero if*M* is large enough, which is beneficial
to the final decision. As a result, although low SNRs and a
non-zeroρcan incur more symbol errors, decent performance
can be achieved in non-coherent massive SIMO systems if*M*
increases unboundedly.

In the presence of high SNRs and a fixedρ, we haveσ* _{n}*2≈
0. Thus, the variance in (21) is simplified as

lim SNR→∞σ

2

*z*(p*k*)≈

*p*2_{k}

*M* +
*f*(ρ)p2_{k}

*M*2 . (29)

According to 0_{6}*f*(ρ)<*M*2−*M*, the range of result in (29)
is computed as follows

*p*2_{k}

*M* 6SNR→∞lim σ
2

*z*(p*k*)<*p*2*k*. (30)

Interestingly, even thoughσ* _{n}*2≈0 at high SNRs, the variance
is nonzero and grows with ρ. This observation illustrates
that the channel correlation incurs additional interference to
the non-coherent massive SIMO system, making it become
interference-limited instead of noise-limited.

In general, increasing*M* and transmit power would help
to reduce the SER of non-coherent SIMO systems, but the
performance gain can be greatly reduced by channel
corre-lation. If further reduction of SER is expected, constellation
design and optimization is regarded as an effective way [18].
Through adaptively optimizing*pi* according to the channel

and noise statistics, the error rate can be minimized.

**IV. ACHIEVABLE RATE ANALYSIS**

In this subsection, a closed-form expression of the achievable rate is derived for ED-based non-coherent massive SIMO sys-tems in correlated Rayleigh fading. First of all, the decision metric is rewritten as

*z*= 1
*M*[**h**

*H*_{8}

*r***h**]x2+

1
*M***n**

*H*_{n}_{+} 2

*M*<

**n***H*81* _{r}*/2

**h**

*x*

= 1
*M*

*M*
X

*i*=1

λ*i*|*vi*|2*x*2+

1

*M*

*M*
X

*i*=1
|*ni*|2+

2

*M*

*M*
X

*i*=1

λ12

*i*<(*qivi*)*x*.

(31)

Before further analysis, we introduce the following corollary.
*Corollary 1:* |*vi*|2and|*ni*|2are uncorrelated to*qivi*.

*Proof:* We only need to prove |*vi*|2 is uncorrelated to

be carried out in the same way. The covariance between|* _{v}_{i}*|2
and

*qivi*is defined as

Cov[|*vi*|2,*qivi*]=E[|*vi*|2*qivi*]−E[|*vi*|2]E[q*ivi*]

1

=_{E}_{[}|*vi*|2*viqi*]

2

=_{E}[|*vi*|2*vi*]E[q*i*]

3

=_{0} _{(32)}

where step 1 is based on that the expectation of*qivi*is zero,

step 2 utilizes the independence between*qi*and*vi*, while step

3 is derived from the fact that*qi*is of zero mean. According

to the above results, Corollary1is achieved.

It is worth noting that decision metric*z*is different from
that in coherent receivers. Since the last two elements in (31)
can be considered as the effective noise, the de facto SNR of
*z*is defined as

γ =

1

*M*

*M*
X

*i*=1

λ*i*|*vi*|2
!2

*x*4

1
*M*

*M*
X

*i*=1
|* _{n}_{i}*|2

!2

+ 2

*M*

*M*
X

*i*=1

λ12

*i*<(q*ivi*)
!2

*x*2

. (33)

A. ACHIEVABLE RATE ANALYSIS BASED ON GAUSSIAN DISTRIBUTION APPROXIMATION

In order to obtain a closed-form expression of the achievable rate, we resort to Gaussian distribution approximation. First of all, the representation of SNR is required to be transformed to facilitate analysis.

*Proposition 2:* With the Gaussian approximation, the SNR
of received signal at multi-antennas receiver can be
repre-sented as

γG∼
*X*1
*X*2

(34)

where *X*1 and *X*2 are mutually independent real Gaussian
random variables, namely

*X*1∼N

µ*X*1, σ

2

*X*1

*X*2∼N

µ*X*2, σ

2

*X*2

(35)

with

µ*X*1 =*p*
2

*k*

1+*M* +*f*(ρ)
*M*2

,

µ*X*2 =σ
4

*n*

1+ 1
*M*

+2p*k*σ
2

*n*

*M* ,

σ2

*X*1 =*p*

4

*k*

_{2}_{+}_{M}_{+}_{f}_{(}ρ_{)}

*M*2 +

4Mf(ρ)+2f2(ρ)

*M*4

,

σ2

*X*2 =2σ

8

*n*

_{1}_{+}_{2M}

*M*2

+8p 2

*k*σ*n*4

*M*2 .

From Proposition2, the achievable rate when√*pk*is

trans-mitted can be computed through averaging over*X*1 and*X*2,
i.e.

*R*G=E*X*1,*X*2

log_{2}

1+*X*1
*X*2

. (36)

*Proposition 3:* The achievable rate applying the Gaussian
distribution approximation in the presence of channel
corre-lation is given by

*R*_{G}

= log√2*e*

π
*n*
X

*k*=0

*Wk*ln(1+*vk*)K(v*k*)

−log2*e*
*m*√π

*m*−1

X

*k*=1
ln

1+ *k*
*m*

*K*

*k*
*m*

− 1

2m√π*K*(1)

+log√2*e*

π
*n*
X

*k*=0
*Ak*

2√*sk*

ln µ*Z*µ*X*2+
√

2µ*X*2σ*Z*
√

*sk*
µ*Z*µ*X*2+

√
2µ*Z*σ*X*2

√
*sk*

!

+log2µ*Z*
2 erfc

−√µ*Z*
2σ*Z*

−log2µ*X*2

2 erfc −

µ*X*2
√

2σ*X*2

!

(37)

where

µ*Z* =µ*X*1+µ*X*2

σ2

*Z* =σ*X*21 +σ
2

*X*2

*K*(x)= √µ*Z*
2σ*Z*

*e*
−

µ*Z*

√

2σ*Zx*
2

−√µ*X*2
2σ*X*2

*e*
−

_{µ}

*X*_{2}

√

2σ*X*_{2}*x*
2

and*Wk* and*vk* are derived from Gauss-Legendre quadrature

formula,*AK*and*sk*are derived from Gauss-Laguerre

quadra-ture formula, 1/*m*is the step in compound trapezoid formula.
*Proof:* The proof can be found in Appendix C.

Besides, the average achievable rate is obtained by
averag-ing over all√*pk*, i.e.

¯
*R*_{G}= 1

*K*

*K*
X

*k*=1

*R*_{G}(p*k*) (38)

where*R*_{G}(p*k*) indicates the rate achieved when

√

*pk*is sent.

B. LIMITS ANALYSIS

1) THE INFLUENCE OF CHANNEL CORRELATION

By inspecting (35), it is found that the correlation coefficient
only exists in*X*1. While for (37),ρ only presents itself in

µ*Z* andσ* _{Z}*2. In order to concentrate on the impact of channel

correlation, we assume high SNR conditions in the following
analysis, i.e.,σ* _{n}*2 ≈ 0. Then, one can arrive at the following
approximations

µ*Z* ≈µ*X*1

σ2

*Z* ≈σ*X*21. (39)

In the first place, the component in µ*X*1 that relates to
correlated Rayleigh fading can be expanded as

*f*(ρ)

*M*2 =
2ρ2

*M*(1−ρ2_{)}−

2ρ2

*M*2_{(1}_{−}_{ρ}2_{)}2. (40)
Since *f*(ρ) grows withρ, (40) is an increasing function of

ρ for a fixed*M*. Here, we analyze the asymptotic behavior
of (40) when ρ → 1. For example, when ρ = 0.95 and
*M* = 400, (40) equals to 0.045. Compared with constant
1 inµ*X*1, it is small enough and can be safely ignored. Not
to mention when 0 _{6}ρ < 1, (40) will become much more
smaller. Hence, the influence of channel correlation can be
safely neglected inµ*X*1.

Forµ*X*1/σ*X*1, it can be expressed as

µ*X*1

σ*X*1
=

1+*M*+*f*(ρ)
*M*2

r

2+*M*+*f*(ρ)

*M*2 +

4Mf(ρ)+2f2(ρ)
*M*4

. (41)

According to (40), the boundaries of (41) can be written as

1<µ*X*1

σ*X*1

≈ √ *M*
*M*+*f*(ρ) 6

√

*M*, 0_{6}ρ <1. (42)

Putting the result of (42) into (37) and (38), *R*¯_{G} can be
calculated when*M* =400 and SNR=10 dB, i.e.

4.2812<*R*¯_{G}_{6}4.3013, 0_{6}ρ <0.9;

4.3013<* _{R}*¯

G64.8212, 0.96ρ <1. (43)

It is found that although the exponential correlation can reduce the achievable rate, this degradation is negligible for most range ofρ, especially from 0 to 0.9. As a result, it comes to the conclusion that correlated channel has little effect on the achievable rate, which is quite different from the scenario of error performance analysis.

2) THE INFLUENCE OF NUMBERS OF RECEIVE ANTENNAS

For a fixed exponential correlationρ, the variance of*X*1and
*X*2 approach zero if*M* → ∞. Meanwhile, the mean of *X*1
and*X*2will converge to constants, i.e.

µ*X*1 ≈*p*
2

*k*, µ*X*2 ≈σ

4

*n*
σ2

*X*1 ≈0, σ

2

*X*2 ≈0. (44)

Substituting (44) into (37), the following concise result can be obtained

lim

*M*→∞*R*G(p*k*)

=log_{2} 1+ *p*
2

*k*
σ4

*n*
!

. (45)

Some interesting observations can be recovered from (45).
In the presence of non-zero channel correlation, the
achiev-able rate cannot grow unboundedly with *M* and converges
to a constant independent of*M*. This finding coincides with
that capacity of non-coherent systems is only decided by the
SNR when the number of antennas is sufficiently large [34].
If one wish to further enhance the system capacity, increasing
transmit power to improve SNR and optimizing constellation
will be the options.

**FIGURE 3.** SER versus SNR for different numbers of receive antennas
without channel correlation.

C. OUTAGE PROBABILITY ANALYSIS

For nonergodic channels, the outage probability is often uti-lized to evaluate the system performance. Outage probability is defined as the probability that information rate is less than the required threshold information rate, i.e.

*Pout*,Pr(γ _{6}γ*th*) (46)

whereγ*th* is defined as the threshold. With this definition,

we can present the following exact result based on Gaussian distribution approximation.

*Proposition 4:* The outage probability of non-coherent
massive SIMO systems with correlated Rayleigh fading is
given by

*Pout*= 1
2 +

1 2erf

√
2σ*X*2γ*th*

q

γ*th*µ*X*2−µ*X*1

2

+2σ* _{X}*2
1

. (47)

*Proof:* The proof can be found in Appendix D.

**V. NUMERICAL RESULTS**

Numerical simulations are performed to verify our analysis.
One transmit antenna and*M* receive antennas are deployed
in the considered model. Besides, we assume that the
non-negative PAM is utilized and channels between antenna pairs
are correlated Rayleigh-fading.

Fig.3 shows SER versus SNR for different numbers of receive antennas without channel correlation, of which exact SER and approximate SER are computed by (17) and (22) respectively, upper bound SER is derived from [21]. From the figure, obviously, the exact SER and approximate SER pro-posed in our paper are closer the simulation than upper bound SER, which means that more efficient constellation or cod-ing method can be designed by the exact SER. In addition, the upper bound SER analysis cannot be used in correlated Rayleigh fading. Therefore, our performance analysis can be applied in wider and more realistic environment.

[image:7.576.312.521.66.223.2] [image:7.576.38.280.227.392.2]**FIGURE 4.** SER versus SNR for different numbers of receive antennas,
whereK=4 andρ=0.5.

thus massive antenna array is effective to avoid decoding
error in both coherent and non-coherent massive systems.
In addition, the analytical expression in (17) matches the
simulation results quite well. However, a discrepancy appears
between simulation results and (22). This performance gap is
attributed to the CLT approximation in Lemma1, where the
tail distribution of a Gaussian variable shows a slight
differ-ence with the actual distribution. Although it is quite small in
absolute value, the logarithmic representation in Fig. 4will
largely amplify this difference. Nevertheless, the tendency
of numerical and analytical curves is quite similar. It is also
found that*Pe*converges to a non-zero constant as SNR keeps

on growing. Therefore, high SNRs may not be necessary depending on the levels ofρ.

The influence of channel correlation on the SER can be
further verified in Fig.5, where SNR = 6 dB and*K* = 4.
This figure clearly demonstrates the adverse effect of channel
correlation on the error performance. Meanwhile, the
per-formance gain provided by massive antenna array would be
counteracted by Rayleigh correlated fading. For example,
the SER at *M* = 400 and ρ = 0.7 is identical to that
when *M* = 200 andρ = 0.3. As expected, although the
analytical expression in (17) is fairly accurate, a remarkable
gap between numerical results and (22) exists, and it enlarges
as*M* varies from 50 to 400.

Fig.6reports the SER versus the number of antennas with
*K* =4 and SNR=9 dB, whereρvaries. It is demonstrated
from this figure that the logarithmic SER decreases almost
linearly with *M*. Besides, as the correlation coefficient ρ
increases, the slope of SER curves gradually reduces, which
proves the results in (27). From the point of view of system
design, although increasing more antennas can lower the error
rate, the resulted more limited separation among antennas
can incur higher correlation. Therefore, the gain provided by
extra antennas would be largely offset by channel correlation,
thus a balance exists between performance improvement and
degradation.

Fig. 7 plots the achievable rate versus SNR with
*K* =4 andρ =0.5. Clearly, the analytical results match the

**FIGURE 5.** SER versus channel correlation coefficient for various numbers
of antennas withK=4 and SNR=6 dB.

**FIGURE 6.** SER versus numbers of receive antennas at various channel
correlation coefficients withK=4 and SNR=9 dB.

**FIGURE 7.** Achievable rate versus SNR at various channel correlation
coefficients withK=4 andρ=0.5 (Gaussian distribution

approximation).

[image:8.576.314.521.445.611.2]**FIGURE 8.** Achievable rate versus numbers of antennas at various SNR
withK=4 andρ=0.5 (Gaussian distribution approximation).

**FIGURE 9.** Achievable rate versus channel correlation coefficients with

K=4 and SNR=10 dB (Gaussian distribution approximation).

Fig.8shows the achievable rate versus numbers of receive
antennas at various SNRs with*K* = 4 and ρ = 0.5. The
upper boundary is computed by (45). As *M* grows large,
the achievable rate gradually converges to a constant. This
is in accordance with the results in Fig. 7. For instance,
the achievable rate is almost fixed as *M* varies from 50 to
850 at SNR = −3 dB. This is able to be understood that
additive noise plays a more vital role in achievable rate than
*M* at low SNRs. On the other hand, deploying more antennas
is beneficial to increase the achievable rate at high SNRs,
as can be observed from the scenario of SNR=15 dB.

Fig. 9 shows how the achievable rate changes with
cor-relation coefficient for different numbers of antennas with
*K* = 4 and SNR = 10 dB. The remarkable gap between
analytical and numerical results at*M* =100 arises because
the number of antennas is insufficient and the resulting
Gaus-sian approximation by using CLT is not accurate enough.
Most importantly, for a large range of correlation coefficients,
the achievable rate remains almost unchanged, especially
when *M* _{>} 200. Compared with the results in Fig.5, it is
concluded that Rayleigh correlated fading poses a far more
severe impact on the SER than on the achievable rate, which
needs to be emphasized during system design.

**VI. CONCLUSION**

Non-coherent receivers are attractive in massive SIMO sys-tems, due primarily to their low complexity and cost. This paper has analyzed in detail the system performance of non-coherent massive SIMO systems using the ED-based receiver in Rayleigh correlated channels.

We have presented two types of closed-form expressions of the average SER. Also, the achievable rate and outage probability are derived. To provide a deeper insight, we fur-ther investigate the system performance in different aspects. Concretely, the channel correlation degrades the SER perfor-mance because the diversity order of non-coherent massive SIMO systems is reduced over correlated channels. Conse-quently, the SER presents an error floor at high SNRs. On the other hand, channel correlation poses far less adverse impact on the achievable rate than the error probability. Although increasing the transmit power or deploying more antennas could be helpful to lower the SER and improve the achievable rate, constellation design and optimization maybe the option with the minimum cost to reduce the impact of Rayleigh correlated correlation.

**APPENDIX A**

**PROOF OF LEMMA1and Proposition 1**

A. PROOF OF LEMMA1

Suppose{*X*1,*X*2, . . . ,*Xn*}is a sequence of independent

ran-dom variables, each with finite meanµ*i*and varianceσ*i*2. If
δ >0, the Lyapunov’s condition is given by

lim

*n*→∞

1

*s*2+*n* δ
*n*
X

*i*=1
E

h

|*Xi*−µ*i*|2+δ
i

=0. (48)

where*s*2* _{n}* = P

*n*

*i*=1σ*i*2. If Lyapunov’s condition is satisfied,

then a sum of *Xi*−µ*i*

*sn* converges in distribution to a standard

normal random variable, as*n*goes to infinity:

*n*
X

*i*=1
*Xi*

*d*

− →N

*n*
X

*i*=1

µ*i*,*s*2*n*
!

(49)

In order to attest whether *z* = 1

*M*

P*M*

*i*=1|*yi*|2 fits with

Lyapunov’s condition, it is convenient to check Lyapunov’s
condition whenδ = 1. And *Xi* indicate the distribution of

|*yi*|2

*M* , thus the value of lim_{M}_{→∞}

1

*s*3

*n*
*M*
P

*i*=1

|*Xi*−µ*i*|3

is

lim

*M*→∞

4

_{M}

P

*i*=1

λ*ix*2+σ*n*2
3

2

_{M}

P

*i*=1

λ*ix*2+σ*n*2
2

3

= lim

*M*→∞

4*M x*2+σ2

*n*

3_{+}

*f*(ρ)(x2+2σ* _{n}*2)x4+

*v(*ρ)x62

*M x*2_{+}_{σ}4

*n*

+*f* (ρ)*x*43

= lim

*M*→∞

4

*x*2_{+}_{σ}2

*n*
3

√

*M* +

*f*(ρ)*x*6_{+2}* _{x}*4

_{σ}2

*n*

+*v*(ρ)*x*6

*M*

√

*M*

2

*x*2_{+}_{σ}4

*n*

+*f*(ρ)*x*4

*M*
3

whereP*M*

*i*=1λ2*i* =*M*+*f*(ρ) and

P*M*

*i*=1λ3*i* =*M*+*f*(ρ)+*v(*ρ).

According to (50),*z*will follow a real Gaussian distribution
with meanµ*z*andσ*z*2as follows

µ*z* =
*M*
X

*i*=1

λ*i*σ*h*2+*M*σ*n*2

σ2
*z* =
*M*
X
*i*=1
λ2

*i*σ*h*4+2
*M*
X

*i*=1

λ*i*σ*h*2σ*n*2+*M*σ*n*4 (51)

whereP*M*

*i*=1λ*i*=*M* and

*M*
X

*i*=1

λ2

*i* =*M*+2

*M*−1

X

*i*=1

(M−*i)*ρ2*i*

=*M*+*f*(ρ) (52)

with

*f*(ρ)=2·ρ

2*M*+2_{+}_{M}_{(}_{ρ}2_{−}_{ρ}4_{)}_{−}_{ρ}2

(1−ρ2_{)}2 , 06ρ <1. (53)

Since 06ρ <1,ρ2*M*+2can be removed from the numerator.
Therefore, (53) is further simplified to

*f*(ρ)=2M(ρ

2_{−}_{ρ}4_{)}_{−}_{2}_{ρ}2

(1−ρ2_{)}2 , 06ρ <1 (54)

with 0_{6}*f*(ρ)<*M*2−*M*.P*M*

*i*=1λ*i*equals to the trace of8*r*

andP*M*

*i*=1λ2*i* is the trace of82*r*. Thus, Lemma1is proved and

the results ofµ*z*andσ*z*2is able to be obtained.

B. PROOF OF PROPOSITION 1

From (6) and Fig.1, the decoding region of√*pk*is denoted by

{*dk*−1,*dk*}. For a single transmit symbol

√

*pk*, the probability

of correct decision is

*P(pk*)=Pr(d*k*−16*Z* <*dk*) (55)

where*Z*represents the distribution of*z. Through invoking the*
Gaussian approximations, the probability of correct decision
when√*pk* is transmitted can be written as

*P(pk*)=

1

2

erf

1

*k*,*L*

√
2σ*z*(p*k*)

+erf

1

*k*,*R*

√
2σ*z*(p*k*)

(56)

where1*k*,*L* =µ*z*(p*k*)−*dk*−1and1*k*,*R*=*dk*−µ*z*(p*k*),µ*z*(p*k*)

and σ*z*(p*k*) denotes the mean and variance shown in (21).

Moreover, the average probability of error is given by

*Pe*=1−

1
*K*

*K*
X

*k*=1

*P(pk*). (57)

Therefore, the proof of Proposition1is concluded.

**APPENDIX B**
**PROOF OF (26)**

Substituting (54) and (21) into (26), one can get

12

*k*,*R*
σ2

*z*(p*k*)

= *M*

2_{1}2

*k*,*R*

(p*k*+σ*n*2)

2
+2ρ2*p*

2

*k*

1−ρ2

*M*− 2ρ2*p*
2

*k*

(1−ρ2_{)}2

= *M*1

2

*k*,*R*

(p*k*+σ*n*2)

2_{+}2ρ2* _{p}*2

*k*

1−ρ2

+*g(M*) (58)

where

*g(M*)

= 2ρ

2* _{p}*2

*k*1

2

*k,R*

(1−ρ2_{)(p}

*k*+σ*n*2)

2
+2ρ2* _{p}*2

*k*
2

−
2ρ2* _{p}*2

*k*

(*pk*+σ*n*2)

2_{+} 2
1−_{ρ}2

*M*

.

(59)

The following approximation can be obtained if*M* → ∞

lim

*M*→∞

2ρ2*p*2_{k}*pk*+σ*n*2
2

+4ρ
2* _{p}*2

*k*

1−ρ2

*M* ≈0. (60)

This approximation is reasonable since the numerator above
is far less than*M* if ρ _{6} 0.9. Therefore, (59) becomes a
constant independent of*M*, and then (27) is proved.

**APPENDIX C**

**PROOF OF PROPOSITION 3**

From (36), we can get the following transformation

*R*_{G}=_{E}* _{X}*
1,

*X*2

log_{2}(*X*1+*X*2)−log2(*X*2)

=log_{2}*e* E*Z*[ln(Z)]−E*X*2[ln(X2)]

(61)

where*Z* =*X*1+*X*2. The correlation between*X*1and*X*2has
already been proved in Corollary1.

Since both*Z* and*X*2 can be approximated as Gaussian,
we only consider*Z*in the following analysis. First,E*z*[ln(Z)]

is expanded as follows

E*z*[ln(Z)] =

1 √

2πσ*z*

Z +∞

0

ln*ze*
−(*z*−µ*z*)2

2σ2

*z* _{dz}

1 = √1

π

Z +∞

−√µ*z*

2σ*z*

lnµ*ze*−*u*

2
*du*

+√1

π

Z +∞

−√µ*z*

2σ*z*

ln 1+ √

2σ*z*
µ*z*

*u*

!

*e*−*u*2*du*

2
= √µ*z*

2πσ*z*

Z +∞

−1

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

+lnµ*z*
2 erfc

−√µ*z*
2σ*z*

= √µ*z*
2πσ*z*

Z 0

−1

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

| {z }

*A*

+√µ*z*
2πσ*z*

Z +∞

0

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

| {z }

*B*

+lnµ*z*
2 erfc

−√µ*z*
2σ*z*

(62)

First, the integral in*A*can be expanded as

Z 0

−1

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

=

Z 1

−1

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

| {z }

*A*1

−

Z 1

0

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

| {z }

*A*2

(63)

Applying the Gauss-Legendre quadrature formula, the
inte-gral in*A*1is obtained by

Z 1

−1

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*=

*n*
X

*k*=0

*Wk*ln(1+*vk*)e

−√µ*z*

2σ*zvk*
2

(64)

where*vk* is the*kth*root of Legendre polynomials*Pn*(v), and

the expressions of*Pn*(v) and*Wk* are

*Pn*(v)=

1
2*n _{n}*

_{!}·

d*n*
dv*n*

*v*2−1*n*

*Wk* =

2

1−*v*2* _{k}*[P

*n*+1(v

*k*)]2 .

For the integral in*A*2, we resort to the compound trapezoid
formula, i.e.

Z 1

0

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2
*dv*
= 1
*m*
*m*−1
X
*k*=1
ln

1+*k*
*n*
*e*−
_{k}_{µ}
*z*
√

2*m*σ*z*
2

−ln 2
2m*e*

−√µ*z*

2σ*z*
2

(65)

where*m*is the step size.

Then the integral in*B*is computed as follows

Z +∞

0

ln(1+*v)e*−

_{µ}

*z*

√

2σ*zv*
2

*dv*

1 =

√
2σ*z*
µ*z*

Z +∞

0
*e*−*s*

ln1+ √

2σ*z*
µ*z*

√
*s*

2√*s* *ds*

2 =

√
2σ*z*
µ*z*

*n*
X

*k*=0
*Ak*

ln1+ √

2σ*z*
µ*z*

√
*sk*

2√*sk*

(66)

where step 1 utilizes the variable substitution, step 2 exploits
the Gauss-Laguerre quadrature.*sk*is the*k-th root of Legendre*

polynomials*Ln*(s), the expressions of*Ln*(s) and*Ak*are

*Ln*(s)=*ex*

*dn*
*d xn*(s

*n _{e}*−

*x*

_{)}

*Ak* =

*sk*

(n+1)2[L*n*+1(s*k*)]2
.

According to (64), (65) and (66), the expectation of ln*Z* is
obtained. On the other hand, E*X*2[ln*X*2] can be calculated
similarly, we omit the derivation for the purpose of
con-ciseness. Summing up all the results above, Proposition3is
proved.

**APPENDIX D**

**PROOF OF PROPOSITION 4**

From Proposition2and (46), we have

*P*out,Pr

*X*1
*X*2 6γ*th*

(67)

which can be rewritten as an integral form, i.e.

*P*out,

Z +∞

−∞

Pr(*X*16γ*thX*2|*X*2)*pX*2(x2)d x2. (68)

Besides, the CDF of*X*1equals to

*FX*1(x)=
1
2

"

1+erf *x*√−µ*X*1
2σ*X*1

!#

. (69)

Combining (68) with (69),*P*_{out}can be rewritten as

*P*out−
1
2

= 1

2 √

2πσ*X*2
+∞

Z

−∞

erf γ*th*_{√}*x*2−µ*X*1
2σ*X*1

!

*e*
−

*x*_{2}−µ_{X}

2

2

2σ2

*X*_{2} _{d x}

2

1 = 1

2√π +∞ Z −∞ erf √

2σ*X*2*u*+µ*X*2

γ*th*−µ*X*1
√

2σ*X*1

*e*

−*u*2_{du}

2
= 1
2erf
√
2σ*X*2γ*th*

q

γ*th*µ*X*2 −µ*X*1

2

+2σ* _{X}*2
1

(70)

where step 1 denotes variable substitution and step 2 utilizes the result in [35, eq. (8.259.1)]. Hence, the proof of Proposi-tion4is concluded.

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HUIQIANG XIE received the B.S. degree in electronic science and technology from North-western Polytechnical University, China, in 2016. He is currently pursuing the master’s degree with Chongqing University, China. His research interests include massive input multiple-output and machine learning.

WEIYANG XU(M’16) received the B.S.E. and M.S.E. degrees from Xi’an Jiaotong University, Xi’an, China, in 2004 and 2007, respectively, and the Ph.D. degree from Fudan University, Shanghai, China, in 2010. In 2014, he was a Visiting Scholar with the University of Southern Queensland, Australia. He is currently an Associate Professor with the School of Microelectronics and Com-munication Engineering, Chongqing University, China. His research interests include massive multiple-input multiple-output and cognitive radio techniques.

WEI XIANG (S’00–M’04–SM’10) received the Ph.D. degree in telecommunications engineering from the University of South Australia, Adelaide, Australia, in 2004. He is currently the Head of Internet of Things engineering with the College of Science and Engineering, James Cook University, Cairns, Australia. His research interests include communication and information theory, particu-larly coding and signal processing for multimedia communication systems. He is a Fellow of the IET and the Engineers Australia. He was a co-recipient of three best paper awards from WCSP 2015, the IEEE WCNC 2011, and ICWMC 2009.

BING LIreceived the B.S. degree in electronic and information engineering from Zhengzhou University, China, in 2018. He is currently pursu-ing the master’s degree with Chongqpursu-ing Univer-sity, China. His research interests include massive multiple-input multiple-output, machine learning, and physical layer security.