• No results found

Performance of ED-based non-coherent massive SIMO systems in correlated Rayleigh fading

N/A
N/A
Protected

Academic year: 2019

Share "Performance of ED-based non-coherent massive SIMO systems in correlated Rayleigh fading"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

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

1School of Microelectronics and Communication Engineering, Chongqing University, Chongqing 400044, China 2College of Science, Technology and Engineering, James Cook University, Cairns, QLD 4870, Australia 3Instrumentation 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.

(2)

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:Cn×mindicates a matrix composed of complex numbers of sizen×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=gx+n (1) whereyCM×1represents the received signal at the multi-antenna receiver,nCM×1indicates a complex Gaussian noise vector with its n-th element ni ∼ CN(0, σn2), g

CM×1refers to the correlated Rayleigh channel realization, xis the transmit symbol drawn from a certain non-negative constellationP = √

p1, √

p2, . . . , √

pK ,K indicates the

constellation size andM is the number of receive antennas. In our study,σn2is 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 thatgis independent of transmit symbolx and noise vectorn, differentgis’ could

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

g=81/2

r h8

1/2

t (2)

wherehCM×1is a complex channel vector whose entries are independent identically distributed (i.i.d.) with its i-th componenthi ∼CN(0,1).8r1/2and81t/2are deterministic

receive and transmit correlation matrices. In the case of a single transmit antenna, the correlation matrix81t/2can be omitted. Therefore, (2) reduces to

g=81r/2h. (3) For the structure of8r, 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

8ij= (

ρ|ji|, i

6j

ρ|ji|∗,

(3)
[image:3.576.37.279.61.144.2]

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

where| · |is the absolute value operation and8ij indicates

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

coef-ficient with 0 6 a < 1. Note that the eigenvalues of8r

only depend on a, while θ decides the eigenvectors of8r.

Since only eigenvalues of8r will be used in the following

analysis, we assumeρ =athroughout 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= kyk 2 2

M (5)

which is able to be approximated to one of theK Gaussian variables depending on which symbol was transmitted. For example, with a non-negative PAM ofK=4, the probability density function (PDF) ofzover 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}Kk=0to decide which symbol was

trans-mitted based on the observation ofz, i.e.

bx=

pk ifdk−16z<dk. (6)

For the first point√p1,d0 is−∞, and for the last point √

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

exploited in modulation and the decoding regions ofbx=√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 metricz, which is essential to the SER analysis, the received signal is first rewritten as

y=81r/2hx+n=UH321Uhx+n=UH312vx+n (8) where the eigendecomposition is used to transform81r/2 =

U312UH,3 1

2 is an eigenvalue diagonal matrix with itsi-th

item beingλ 1 2

i, andUindicates a unitary matrix consisting of

corresponding eigenvectors. In addition,v=Uhfollows the same distribution ashand entries ofvare mutually indepen-dent [32]. After left-multiplyingywith a unitary matrix U, one can arrive at the following result

Uy=UUH312vx+Un=312vx+q. (9) Then, (5) can be rewritten as

z= y

HUHUy

M =

1

M kyk 2 2=

1

M

M X

i=1

|yi|2 (10)

where|yi|22 ∼ λ

ix2+σn2

2 χ

2(2). In this paper, the exact SER

expression is derived utilizing the characteristic function of the PDF ofz.

First, the characteristic function of|yi|2

M is given by

ϕβi(t)=

1

(1−βit) (11)

whereβi= λi x2+σ2

n

M . Accordingly, the characteristic function

ofzis

ϕz(t)=

1

(1−β1t)v1 1

(1−β2t)v2 · · · 1

1−βgtvg (12)

wherePg

i=1viis the rank of the correlation matrix8randgis the number of distinct nonzero eigenvalues. Actually, the rank of the correlation matrix equals toM if and only ifρ 6= 1. Resolving (12) into its partial fractions with repeated roots yields

ϕz(t)= g X

i=1

vg X

j=1

Ai,jϕβji(t) (13)

where the coefficientsAi,jcan be found as

Ai,j=

1

(vij)!βvi

j i

vijtvij

g Y

q=1,q6=i ϕvq

βq(t) 

t=

1 βi

(14)

The linearity of the inverse of the characteristic function transform allows therefore rewriting the PDF ofzas

f (z)=

g X

i=1

vg X

j=1 Ai,j

zj−1

0 (j) βij

exp

z

βi

(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

γj,βz

i

(4)
[image:4.576.47.272.63.244.2]

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

where

γj, z

βi

=

Z βz i

0

tj−1etdt.

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 ofz, 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(µz, σz2) (18)

where

µz =x2+σn2

σ2

z =

1 M

x2+σ2

n 2

+f(ρ)x 4

M2 (19)

with

f(ρ)=2M(ρ

2ρ4)2ρ2

(1−ρ2)2 , 06ρ <1 (20)

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

vari-ance σz2 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 ofzwould 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 ofzdepending on which symbol inP was transmitted. The expectation and variance ofzwhen√pkis selected at the transmitter can be written as

µz(pk)=pkn2

σ2

z(pk)=

1 M

pkn2 2

+f(ρ)p 2

k

M2 . (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

PGe =1− 1 K

K X

k=1 P(pk)

=1− 1 2K

K X

k=1

erf

1

k,L

√ 2σz(pk)

+erf

1

k,R

√ 2σz(pk)

(22)

where1k,Lz(pk)−dk−1and1k,R=dk−µz(pk).

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

σz(pk) 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,σz2(pk) increases

along withρ. Since SER is an increasing function ofσz2(pk),

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

PGe = 1 2K

K X

k=1

erfc

1

k,L

√ 2σz(pk)

+erfc

1

k,R

√ 2σz(pk)

.

(5)

Because M influences erfc1k,R

z(pk)

and erfc1k,L

z(pk)

in the same way, only erfc1k,R

z(pk)

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

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

erfc(x)≈ ex2

πx, ifx→ ∞. (24)

Using Lemma2, erfc1k,R

z(pk)

can be approximated to

erfc

1

k,R

√ 2σz(pk)

≈ √

z(pk)e

− 1 2

k,R

2σ2

z(pk)

π1k,R

. (25)

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

log erfc

1

k,R

√ 2σz(pk)

= − 1 2

k,R

2σ2

z(pk)

loge−1 2log

12

k,R σ2

z(pk)

+ 1 2log

2

π (26)

where

12

k,R σ2

z(pk)

= 1

2

k,R

(pkn2)

2 +2ρ

2p2

k

1−ρ2

M+A (27)

with

A= 2ρ

2p2

k1

2

k,R

(1−ρ2)(p

kn2)

2+ 2ρ2p2

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 withM, 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(pk)

= (dk−µz(pk)) 2

2p2k (28)

which is a constant independent ofM. 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 ifM 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 ifM increases unboundedly.

In the presence of high SNRs and a fixedρ, we haveσn2≈ 0. Thus, the variance in (21) is simplified as

lim SNR→∞σ

2

z(pk)≈

p2k

M + f(ρ)p2k

M2 . (29)

According to 06f(ρ)<M2−M, the range of result in (29) is computed as follows

p2k

M 6SNR→∞lim σ 2

z(pk)<p2k. (30)

Interestingly, even thoughσn2≈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, increasingM 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 optimizingpi 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

H8

rh]x2+

1 Mn

Hn+ 2

M<

nH81r/2hx

= 1 M

M X

i=1

λi|vi|2x2+

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 toqivi.

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

(6)

be carried out in the same way. The covariance between|vi|2 andqiviis defined as

Cov[|vi|2,qivi]=E[|vi|2qivi]−E[|vi|2]E[qivi]

1

=E[|vi|2viqi]

2

=E[|vi|2vi]E[qi]

3

=0 (32)

where step 1 is based on that the expectation ofqiviis zero,

step 2 utilizes the independence betweenqiandvi, while step

3 is derived from the fact thatqiis of zero mean. According

to the above results, Corollary1is achieved.

It is worth noting that decision metriczis 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 zis defined as

γ =

1

M

M X

i=1

λi|vi|2 !2

x4

1 M

M X

i=1 |ni|2

!2

+ 2

M

M X

i=1

λ12

i<(qivi) !2

x2

. (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∼ X1 X2

(34)

where X1 and X2 are mutually independent real Gaussian random variables, namely

X1∼N

µX1, σ

2

X1

X2∼N

µX2, σ

2

X2

(35)

with

µX1 =p 2

k

1+M +f(ρ) M2

,

µX2 =σ 4

n

1+ 1 M

+2pkσ 2

n

M ,

σ2

X1 =p

4

k

2+M +f(ρ)

M2 +

4Mf(ρ)+2f2(ρ)

M4

,

σ2

X2 =2σ

8

n

1+2M

M2

+8p 2

kσn4

M2 .

From Proposition2, the achievable rate when√pkis

trans-mitted can be computed through averaging overX1 andX2, i.e.

RG=EX1,X2

log2

1+X1 X2

. (36)

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

RG

= log√2e

π n X

k=0

Wkln(1+vk)K(vk)

−log2e m√π

m−1

X

k=1 ln

1+ k m

K

k m

− 1

2m√πK(1)

+log√2e

π n X

k=0 Ak

2√sk

ln µZµX2+ √

XZ

sk µZµX2+

√ 2µZσX2

sk

!

+log2µZ 2 erfc

−√µZZ

−log2µX2

2 erfc −

µX2 √

X2

!

(37)

where

µZX1+µX2

σ2

ZX21 +σ 2

X2

K(x)= õZZ

e

µZ

Zx 2

−√µX2 2σX2

e

µ

X2

X2x 2

andWk andvk are derived from Gauss-Legendre quadrature

formula,AKandskare derived from Gauss-Laguerre

quadra-ture formula, 1/mis 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.

¯ RG= 1

K

K X

k=1

RG(pk) (38)

whereRG(pk) indicates the rate achieved when

pkis sent.

B. LIMITS ANALYSIS

1) THE INFLUENCE OF CHANNEL CORRELATION

By inspecting (35), it is found that the correlation coefficient only exists inX1. While for (37),ρ only presents itself in

µZ andσZ2. In order to concentrate on the impact of channel

correlation, we assume high SNR conditions in the following analysis, i.e.,σn2 ≈ 0. Then, one can arrive at the following approximations

µZ ≈µX1

σ2

Z ≈σX21. (39)

(7)

In the first place, the component in µX1 that relates to correlated Rayleigh fading can be expanded as

f(ρ)

M2 = 2ρ2

M(1−ρ2)

2ρ2

M2(1ρ2)2. (40) Since f(ρ) grows withρ, (40) is an increasing function of

ρ for a fixedM. 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 inµX1, it is small enough and can be safely ignored. Not to mention when 0 6ρ < 1, (40) will become much more smaller. Hence, the influence of channel correlation can be safely neglected inµX1.

ForµX1/σX1, it can be expressed as

µX1

σX1 =

1+M+f(ρ) M2

r

2+M+f(ρ)

M2 +

4Mf(ρ)+2f2(ρ) M4

. (41)

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

1<µX1

σX1

≈ √ M M+f(ρ) 6

M, 06ρ <1. (42)

Putting the result of (42) into (37) and (38), R¯G can be calculated whenM =400 and SNR=10 dB, i.e.

4.2812<R¯G64.3013, 06ρ <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 ofX1and X2 approach zero ifM → ∞. Meanwhile, the mean of X1 andX2will converge to constants, i.e.

µX1 ≈p 2

k, µX2 ≈σ

4

n σ2

X1 ≈0, σ

2

X2 ≈0. (44)

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

lim

M→∞RG(pk)

=log2 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 ofM. 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 6γth) (46)

whereγth 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

√ 2σXth

q

γthµX2−µX1

2

+2σX2 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 andM 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]
(8)
[image:8.576.312.522.65.223.2]

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 thatPeconverges 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 andK = 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 asM 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.

[image:8.576.53.262.66.222.2]

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

[image:8.576.312.521.259.418.2]

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]
(9)
[image:9.576.54.260.64.222.2] [image:9.576.53.261.258.416.2]

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 withK = 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 atM =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{X1,X2, . . . ,Xn}is a sequence of independent

ran-dom variables, each with finite meanµiand varianceσi2. If δ >0, the Lyapunov’s condition is given by

lim

n→∞

1

s2+n δ n X

i=1 E

h

|Xi−µi|2+δ i

=0. (48)

wheres2n = Pn

i=1σi2. If Lyapunov’s condition is satisfied,

then a sum of Xi−µi

sn converges in distribution to a standard

normal random variable, asngoes to infinity:

n X

i=1 Xi

d

− →N

n X

i=1

µi,s2n !

(49)

In order to attest whether z = 1

M

PM

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 limM→∞

1

s3

n M P

i=1

|Xi−µi|3

is

lim

M→∞

4

M

P

i=1

λix2+σn2 3

2

M

P

i=1

λix2+σn2 2

3

= lim

M→∞

4M x2+σ2

n

3+

f(ρ)(x2+2σn2)x4+v(ρ)x62

M x2+σ4

n

+f (ρ)x43

= lim

M→∞

4

x2+σ2

n 3

M +

f(ρ)x6+2x4σ2

n

+v(ρ)x6

M

M

2

x2+σ4

n

+f(ρ)x4

M 3

[image:9.576.299.540.493.731.2]
(10)

wherePM

i=1λ2i =M+f(ρ) and

PM

i=1λ3i =M+f(ρ)+v(ρ).

According to (50),zwill follow a real Gaussian distribution with meanµzandσz2as follows

µz = M X

i=1

λiσh2+Mσn2

σ2 z = M X i=1 λ2

iσh4+2 M X

i=1

λiσhn2+Mσn4 (51)

wherePM

i=1λi=M and

M X

i=1

λ2

i =M+2

M−1

X

i=1

(M−i)ρ2i

=M+f(ρ) (52)

with

f(ρ)=2·ρ

2M+2+M(ρ2ρ4)ρ2

(1−ρ2)2 , 06ρ <1. (53)

Since 06ρ <1,ρ2M+2can be removed from the numerator. Therefore, (53) is further simplified to

f(ρ)=2M(ρ

2ρ4)2ρ2

(1−ρ2)2 , 06ρ <1 (54)

with 06f(ρ)<M2−M.PM

i=1λiequals to the trace of8r

andPM

i=1λ2i is the trace of82r. Thus, Lemma1is proved and

the results ofµzandσz2is able to be obtained.

B. PROOF OF PROPOSITION 1

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

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

pk, the probability

of correct decision is

P(pk)=Pr(dk−16Z <dk) (55)

whereZrepresents the distribution ofz. 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

√ 2σz(pk)

+erf

1

k,R

√ 2σz(pk)

(56)

where1k,Lz(pk)−dk−1and1k,R=dk−µz(pk),µz(pk)

and σz(pk) 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(pk)

= M

212

k,R

(pkn2)

2 +2ρ2p

2

k

1−ρ2

M− 2ρ2p 2

k

(1−ρ2)2

= M1

2

k,R

(pkn2)

2+2ρ2p2

k

1−ρ2

+g(M) (58)

where

g(M)

= 2ρ

2p2

k1

2

k,R

(1−ρ2)(p

kn2)

2 +2ρ2p2

k 2

− 2ρ2p2

k

(pkn2)

2+ 2 1−ρ2

M

.

(59)

The following approximation can be obtained ifM → ∞

lim

M→∞

2ρ2p2k pkn2 2

+4ρ 2p2

k

1−ρ2

M ≈0. (60)

This approximation is reasonable since the numerator above is far less thanM if ρ 6 0.9. Therefore, (59) becomes a constant independent ofM, and then (27) is proved.

APPENDIX C

PROOF OF PROPOSITION 3

From (36), we can get the following transformation

RG=EX 1,X2

log2(X1+X2)−log2(X2)

=log2e EZ[ln(Z)]−EX2[ln(X2)]

(61)

whereZ =X1+X2. The correlation betweenX1andX2has already been proved in Corollary1.

Since bothZ andX2 can be approximated as Gaussian, we only considerZin the following analysis. First,Ez[ln(Z)]

is expanded as follows

Ez[ln(Z)] =

1 √

2πσz

Z +∞

0

lnze −(z−µz)2

2σ2

z dz

1 = √1

π

Z +∞

−√µz

z

lnµzeu

2 du

+√1

π

Z +∞

−√µz

z

ln 1+ √

z µz

u

!

eu2du

2 = õz

2πσz

Z +∞

−1

ln(1+v)e

µ

z

zv 2

dv

+lnµz 2 erfc

−√µzz

= √µz 2πσz

Z 0

−1

ln(1+v)e

µ

z

zv 2

dv

| {z }

A

+√µz 2πσz

Z +∞

0

ln(1+v)e

µ

z

zv 2

dv

| {z }

B

+lnµz 2 erfc

−√µzz

(62)

(11)

First, the integral inAcan be expanded as

Z 0

−1

ln(1+v)e

µ

z

zv 2

dv

=

Z 1

−1

ln(1+v)e

µ

z

zv 2

dv

| {z }

A1

Z 1

0

ln(1+v)e

µ

z

zv 2

dv

| {z }

A2

(63)

Applying the Gauss-Legendre quadrature formula, the inte-gral inA1is obtained by

Z 1

−1

ln(1+v)e

µ

z

zv 2

dv=

n X

k=0

Wkln(1+vk)e

−√µz

zvk 2

(64)

wherevk is thekthroot of Legendre polynomialsPn(v), and

the expressions ofPn(v) andWk are

Pn(v)=

1 2nn!·

dn dvn

v2−1n

Wk =

2

1−v2k[Pn+1(vk)]2 .

For the integral inA2, we resort to the compound trapezoid formula, i.e.

Z 1

0

ln(1+v)e

µ

z

zv 2 dv = 1 m m−1 X k=1 ln

1+k n ekµ z

2mσz 2

−ln 2 2me

−√µz

z 2

(65)

wheremis the step size.

Then the integral inBis computed as follows

Z +∞

0

ln(1+v)e

µ

z

zv 2

dv

1 =

√ 2σz µz

Z +∞

0 es

ln1+ √

z µz

s

2√s ds

2 =

√ 2σz µz

n X

k=0 Ak

ln1+ √

z µz

sk

2√sk

(66)

where step 1 utilizes the variable substitution, step 2 exploits the Gauss-Laguerre quadrature.skis thek-th root of Legendre

polynomialsLn(s), the expressions ofLn(s) andAkare

Ln(s)=ex

dn d xn(s

nex)

Ak =

sk

(n+1)2[Ln+1(sk)]2 .

According to (64), (65) and (66), the expectation of lnZ is obtained. On the other hand, EX2[lnX2] 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

Pout,Pr

X1 X2 6γth

(67)

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

Pout,

Z +∞

−∞

Pr(X16γthX2|X2)pX2(x2)d x2. (68)

Besides, the CDF ofX1equals to

FX1(x)= 1 2

"

1+erf x√−µX1 2σX1

!#

. (69)

Combining (68) with (69),Poutcan be rewritten as

Pout− 1 2

= 1

2 √

2πσX2 +∞

Z

−∞

erf γthx2−µX1 2σX1

!

e

x2−µX

2

2

2σ2

X2 d x

2

1 = 1

2√π +∞ Z −∞ erf   √

X2uX2

γth−µX1 √

X1

e

u2du

2 = 1 2erf   √ 2σXth

q

γthµX2 −µX1

2

+2σX2 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.

REFERENCES

[1] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, ‘‘An overview of massive MIMO: Benefits and challenges,’’IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 742–758, Oct. 2014.

[2] T. L. Marzetta, ‘‘Noncooperative cellular wireless with unlimited numbers of base station antennas,’’IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.

[3] F. Ruseket al., ‘‘Scaling up MIMO: Opportunities and challenges with very large arrays,’’IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2012.

[4] T. S. Rappaport et al., ‘‘Millimeter wave mobile communications for 5G cellular: It will work!’’IEEE Access, vol. 1, pp. 335–349, May 2013. [5] J. Hoydis, S. ten Brink, and M. Debbah, ‘‘Massive MIMO in the UL/DL of

cellular networks: How many antennas do we need?’’IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013.

[6] H. Q. Ngo and E. G. Larsson, ‘‘EVD-based channel estimation in multi-cell multiuser MIMO systems with very large antenna arrays,’’ inProc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Kyoto, Japan, Mar. 2012, pp. 3249–3252.

[7] A. Ashikhmin and T. Marzetta, ‘‘Pilot contamination precoding in multi-cell large scale antenna systems,’’ in Proc. IEEE Int. Symp. Inf. The-ory (ISIT), Cambridge, MA, USA, Jul. 2012, pp. 1137–1141.

[8] O. Elijah, C. Y. Leow, T. A. Rahman, S. Nunoo, and S. Z. Iliya, ‘‘A comprehensive survey of pilot contamination in massive MIMOâĂŤ5G system,’’IEEE Commun. Surveys Tuts., vol. 18, no. 2, pp. 905–923, 2nd Quart., 2016.

[9] F. Fernandes, A. Ashikhmin, and T. L. Marzetta, ‘‘Inter-cell interference in noncooperative TDD large scale antenna systems,’’IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 192–201, Feb. 2013.

(12)

[11] R. R. Müller, L. Cottatellucci, and M. Vehkaperä, ‘‘Blind pilot decontam-ination,’’IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 773–786, Oct. 2014.

[12] W. Xu, W. Xiang, Y. Jia, Y. Li, and Y. Yang, ‘‘Downlink performance of massive-MIMO systems using EVD-based channel estimation,’’IEEE Trans. Veh. Technol., vol. 66, no. 4, pp. 3045–3058, Apr. 2017.

[13] K. Witrisalet al., ‘‘Noncoherent ultra-wideband systems,’’IEEE Signal Process. Mag., vol. 26, no. 4, pp. 48–66, Jul. 2009.

[14] H. Urkowitz, ‘‘Energy detection of unknown deterministic signals,’’Proc. IEEE, vol. 55, no. 4, pp. 523–531, Apr. 1967.

[15] R. Moorfeld and A. Finger, ‘‘Multilevel PAM with optimal amplitudes for non-coherent energy detection,’’ inProc. Int. Conf. Wireless Commun. Signal Process. (WCSP), Nanjing, China, Nov. 2009, pp. 1–5.

[16] A. Manolakos, M. Chowdhury, and A. J. Goldsmith, ‘‘CSI is not needed for optimal scaling in multiuser massive SIMO systems,’’ in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Honolulu, HI, USA, Jun./Jul. 2014, pp. 3117–3121.

[17] F. Wang, Z. Tian, and B. M. Sadler, ‘‘Weighted energy detection for non-coherent ultra-wideband receiver design,’’IEEE Trans. Wireless Commun., vol. 10, no. 2, pp. 710–720, Feb. 2011.

[18] M. Chowdhury, A. Manolakos, and A. J. Goldsmith, ‘‘Design and perfor-mance of noncoherent massive SIMO systems,’’ inProc. 48th Annu. Conf. Inf. Sci. Syst. (CISS), Princeton, NJ, USA, Mar. 2014, pp. 1–6.

[19] A. Manolakos, M. Chowdhury, and A. J. Goldsmith, ‘‘Constellation design in noncoherent massive SIMO systems,’’ inProc. IEEE Global Commun. Conf. (GLOBECOM), Austin, TX, USA, Dec. 2014, pp. 3690–3695. [20] L. Jing, E. De Carvalho, P. Popovski, and À. O. MartÃnez, ‘‘Design

and performance analysis of noncoherent detection systems with mas-sive receiver arrays,’’ IEEE Trans. Signal Process., vol. 64, no. 19, pp. 5000–5010, Oct. 2016.

[21] A. Manolakos, M. Chowdhury, and A. Goldsmith, ‘‘Energy-based mod-ulation for noncoherent massive SIMO systems,’’IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7831–7846, Nov. 2015.

[22] S. Biswas, C. Masouros, and T. Ratnarajah, ‘‘Performance analysis of large multiuser MIMO systems with space-constrained 2-D antenna arrays,’’

IEEE Trans. Wireless Commun., vol. 15, no. 5, pp. 3492–3505, May 2016. [23] C. Masouros, M. Sellathurai, and T. Ratnarajah, ‘‘Large-scale MIMO transmitters in fixed physical spaces: The effect of transmit correla-tion and mutual coupling,’’ IEEE Trans. Commun., vol. 61, no. 7, pp. 2794–2804, Jul. 2013.

[24] A. M. Tulino, A. Lozano, and S. Verdu, ‘‘Impact of antenna correlation on the capacity of multiantenna channels,’’IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2491–2509, Jul. 2005.

[25] G. Alfano, A. M. Tulino, A. Lozano, and S. Verdu, ‘‘Capacity of MIMO channels with one-sided correlation,’’ in Proc. 8th IEEE Int. Symp. Spread Spectr. Techn. Appl., Sydney, NSW, Australia, Aug./Sep. 2004, pp. 515–519.

[26] H. Liu, Y. Song, and R. C. Qiu, ‘‘The impact of fading correlation on the error performance of MIMO systems over Rayleigh fading channels,’’

IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2014–2019, Sep. 2005. [27] D. Piazza, N. J. Kirsch, A. Forenza, R. W. Heath, Jr., and K. R. Dandekar,

‘‘Design and evaluation of a reconfigurable antenna array for MIMO systems,’’IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 869–881, Mar. 2008.

[28] P. L. Kafle, A. Intarapanich, A. B. Sesay, J. Mcrory, and R. J. Davies, ‘‘Spatial correlation and capacity measurements for wideband MIMO channels in indoor office environment,’’IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1560–1571, May 2008.

[29] S. Chatzinotas, M. A. Imran, and R. Hoshyar, ‘‘On the multicell processing capacity of the cellular MIMO uplink channel in correlated Rayleigh fading environment,’’IEEE Trans. Wireless Commun., vol. 8, no. 7, pp. 3704–3715, Jul. 2009.

[30] D. Mi, M. Dianati, S. Muhaidat, and Y. Chen, ‘‘A novel antenna selection scheme for spatially correlated massive MIMO uplinks with imperfect channel estimation,’’ inProc. IEEE 81st Veh. Technol. Conf. (VTC Spring), May 2015, pp. 1–6.

[31] L. Jing and E. D. Carvalho, ‘‘Energy detection in ISI channels using large-scale receiver arrays,’’ in Proc. IEEE Int. Conf. Acoust.,

Speech Signal Process. (ICASSP), Shanghai, China, Mar. 2016,

pp. 3431–3435.

[32] D. Park and S. Y. Park, ‘‘Performance analysis of multiuser diversity under transmit antenna correlation,’’IEEE Trans. Commun., vol. 56, no. 4, pp. 666–674, Apr. 2008.

[33] K. B. Oldham, J. C. Myland, and J. Spanier,The Error Function erf(x) and its Complement erfc(x). New York, NY, USA: Springer, 2008.

[34] Y. Liang and V. V. Veeravalli, ‘‘Capacity of noncoherent time-selective Rayleigh-fading channels,’’IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3095–3110, Dec. 2004.

[35] I. S. Gradshteuin, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger,Table of Integrals, Series, and Products, 7th ed. San Diego, CA. USA: Academic, 2007.

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.

Figure

FIGURE 1. Decoding regions for non-coherent massive SIMO systemswith a non-negative PAM of K = 4.
FIGURE 2. PDFs of z with SNR = 6 dB and K = 4, which are shown usingthe linear scale (left) and logarithmic scale (right).
Fig. 3 shows SER versus SNR for different numbers of
FIGURE 5. SER versus channel correlation coefficient for various numbersof antennas with K = 4 and SNR = 6 dB.
+2

References

Related documents

Auld said written staff reports were also provided in the meeting materials but only two staff reports would be presented... Public Services and Branch

In this paper, we have derived the exact and approximate closed-form expressions of the user outage probability and the system outage probability of a two-way DF relay system where

Guiding Tourists through Haptic Interaction: Vibration Feedback in the Lund Time Machine.. Springare (Eds.), Lecture Notes in

Ş anl ı urfa Chamber of Commerce and Industry Çaycuma Chamber of Commerce and Industry Fethiye Chamber of Commerce and Industry Kayseri Chamber of Commerce and Industry

[r]

Motivated by the ob- servation that individual �ows with �xed priority may not be the ideal basis for bandwidth allocation, we present the design and implementation of

The article uses fluid flow approximation to investigate the influence of the two above-mentioned TCP congestion control mechanisms on CWND evolution, packet loss probability, queue

Transactional focus React to requisitions Not involved in key source selections Emphasis: purchase price Relationships: transactional and adversarial Bottom line impact: