140 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Mixed-ADC/DAC Multipair Massive MIMORelaying Systems: Performance Analysis

and Power OptimizationJiayi Zhang , Member, IEEE, Linglong Dai , Senior Member, IEEE, Ziyan He,

Bo Ai , Senior Member, IEEE, and Octavia A. Dobre , Senior Member, IEEE

Abstract— High power consumption and expensive hardwareare two bottlenecks for practical massive multiple-input multiple-output (mMIMO) systems. One promising solution is to employlow-resolution analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). In this paper, we consider a generalmultipair mMIMO relaying system with a mixed-ADC/DACarchitecture, in which some antennas are connected to low-resolution ADCs/DACs, while the rest of the antennas areconnected to high-resolution ADCs/DACs. Leveraging on theadditive quantization noise model, both exact and approximateclosed-form expressions for the achievable rate are derived. It isshown that the achievable rate can approach the unquantized oneby using only 2–3 bits of resolutions. Moreover, a power scalinglaw is presented to reveal that the transmit power can be scaleddown inversely proportional to the number of antennas at therelay. We further propose an efficient power allocation schemeby solving a complementary geometric programming problem.In addition, a tradeoff between the achievable rate and powerconsumption for different numbers of low-resolution ADCs/DACsis investigated by deriving the energy efficiency. Our results revealthat the large antenna array can be exploited to enable the mixed-

Manuscript received June 4, 2018; revised July 30, 2018 andAugust 27, 2018; accepted September 1, 2018. Date of publicationSeptember 13, 2018; date of current version January 15, 2019. This workwas supported in part by the National Natural Science Foundation of China(Nos. 61601020 and 61725101), the Beijing Natural Science Foundation (Nos.4182049 and L171005), the Open Research Fund of National Mobile Com-munications Research Laboratory, Southeast University (No. 2018D04), KeyLaboratory of Optical Communication and Networks (No. KLOCN2018002),National Key Research and Development Program (No. 2016YFE0200900),Major Projects of Beijing Municipal Science and Technology Commission(No. Z181100003218010), National Key Research and Development Program(No. 2016YFE0200900), and the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) through its Discovery Program. The associateeditor coordinating the review of this paper and approving it for publicationwas S. Ma. (Corresponding author: Jiayi Zhang.)

J. Zhang is with the School of Electronic and Information Engineering,Beijing Jiaotong University, Beijing 100044, China, and also with theNational Mobile Communications Research Laboratory, Southeast University,Nanjing 210096, China (e-mail: jiayizhang@bjtu.edu.cn).

L. Dai is with the Department of Electronic Engineering, TsinghuaUniversity, Beijing 100084, China (e-mail: daill@tsinghua.edu.cn).

Z. He was with Department of Electronic Engineering, Tsinghua University,Beijing 100084, China. He is now with the School of Electrical and ComputerEngineering, Georgia Institute of Technology, Atlanta, GA 30332 USA(e-mail: heziyan@gatech.edu).

B. Ai is with the State Key Laboratory of Rail Traffic Controland Safety, Beijing Jiaotong University, Beijing 100044, China (e-mail:boai@bjtu.edu.cn).

O. A. Dobre is with the Faculty of Engineering and Applied Science,Memorial University, St. John’s, NL A1B 3X5, Canada (e-mail:odobre@mun.ca).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2018.2869596

ADC/DAC architecture, which significantly reduces the powerconsumption and hardware cost for practical mMIMO systems.

Index Terms— Massive MIMO, multipair relay, mixed-ADC/DAC, achievable rate, energy efficiency.

I. INTRODUCTION

AS ONE of the disruptive technologies for the fifth-generation (5G) wireless communications, massive

multiple-input multiple-output (mMIMO) has attracted exten-sive research interests in recent years [1]–[4]. By exploitingquasi-orthogonal random channel vectors between differentusers, mMIMO can mitigate the inter-user interference toprovide high spectral efficiency and energy efficiency viasimple linear signal processing, e.g., maximum-ratio combin-ing (MRC) and zero-forcing (ZF) precoding. On the otherhand, relaying is an important way of extending coverage andproviding uniform service. The inter-user interference of themultiuser relaying system can be suppressed by equipping therelay with a large number of antennas [5]–[7].

The practical implementation of an mMIMO relayingsystem with hundreds or even thousands of antennas isa significant challenge [8]–[10]. For example, the perfectsynchronization is difficult in mMIMO relaying system. Onepossible solution is to employ rateless network coding [11].Typically, each antenna in mMIMO systems is connectedto an analog-to-digital converter (ADC) and a digital-to-analog converters (DAC) in the radio frequency (RF) chain,respectively. It is well known that the power consumptionand hardware cost of ADCs and DACs linearly increase withthe bandwidth and exponentially increase with the number ofquantization bits [12]. Thus, high-resolution ADCs and DACs(e.g., 8-12 bits for commercial use) will result in high powerconsumption and hardware cost in practical mMIMO relayingsystems.

To solve this challenging problem, a promising solutionis to replace power-hungry high-resolution ADCs and DACs(e.g., 8-12 bits) with low power low-resolution ADCs andDACs (e.g., 1-3 bits) [13], [14]. However, significant signalprocessing challenges and complex front-end designs (e.g.,channel estimation, phase/frequency synchronization, andmultiuser detection) inevitably occur due to the strong non-linear characteristic of coarse quantization [15]. As recentlyreported in [16], a high signal-to-noise ratio (SNR) channel

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ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 141

estimation error floor exists due to the one-bit quantization.Furthermore, Liang and Zhang [17] proposed a mixed-ADCarchitecture, where only a small fraction of ADCs are high-resolution, to facilitate the aforementioned signal processingand the establishment of front-end designs. For example,the CSI for each antenna can be obtained by using thehigh-resolution ADCs in a round-robin manner, which hasbeen clearly explained in [17]. Moreover, the mixed-ADCarchitecture is economically beneficial and easier to implementcompared with architectures with uniform converter resolution,as it adds some antennas with low-resolution ADCs to theexisting high-resolution conventional MIMO system [18].

A. Related Works

Most of recent works focused on the single-hop mMIMOsystem with a mixed-ADC architecture. For instance,the mutual information of mixed-ADC mMIMO systems hasbeen investigated in [17] and [19], which reveals that themixed-ADC architecture is able to approach the ideal channelcapacity of unquantized systems over both frequency-flat [17]and frequency-selective fading channels [19]. In addition,the achievable rate performance of multi-user mMIMOsystems with a mixed-ADC architecture is comparablefor Rayleigh [20] and Rician fading channels [21]. Thisarchitecture can achieve a better energy-rate trade-offcompared with the ideal infinite-resolution and low-resolutionADC architectures. By applying probabilistic Bayesianinference, a family of detectors for mixed-ADC mMIMOsystems was developed in [22]. It proves that the availablehigh-resolution ADCs are practically essential since they caneffectively eliminate the error floor of a relaxed Bayesiandetector. Given the energy constraint at the base station (BS),the sum achievable rate has been maximized in [23], whichshows that the optimal rate can be obtained by using only one-bit ADCs in most realistic scenarios. Moreover, Xu et al. [24]considered the downlink mMIMO with both mixed-resolutionDACs at the BS and mixed-resolution ADCs at the user side.These important contributions have shown that the powerconsumption and hardware cost of the single-hop mMIMOsystem with a mixed-ADC architecture can be considerablyreduced while keeping most of the gains in the achievable rate.

In contrast to single-hop systems, very little attention hasbeen paid to the two-hop mMIMO relaying system with bothmixed-resolution ADCs and mixed-resolution DACs. Veryrecently, Kong et al. [25], Dong et al. [26], Liu et al. [27],Kong et al. [28], and Jia et al. [29] investigated the per-formance of a multipair mMIMO relaying system with low-resolution ADCs and DACs at the relay. The achievable rateof such a system is limited by using very coarse quantization(e.g., one-bit). In this paper, we consider a more generalarchitecture, where ADCs and DACs with arbitrary resolutionprofile are employed at the relay to achieve a possibly higherrate.

B. Contributions

In this paper, we focus on a general two-hop mixed-ADC/DAC mMIMO relaying system, where some antennas

are connected to low-resolution ADCs/DACs, while the restof the antennas are connected to high-resolution ADCs/DACs.This study aims to analyze the performance analysis of themultipair mMIMO relaying system with arbitrary quantizationnoise, which is in contrast to the previous study [25] thatemploys only one-bit ADCs and DACs. We demonstrate thatthe achievable rate of the considered system can approach thatof the ideal unquantized system. The main contributions of thispaper are summarized as follows:

• Leveraging on AQNM, we present a unified frame-work to derive the exact closed-form expressions for theachievable rate of mixed-ADC/DAC mMIMO systems.Compared with the Bussgang theorem used in [25],the AQNM can offer analytical tractability for multi-bitADCs and DACs. Furthermore, approximate achievablerate expressions are derived by using asymptotic argu-ments. These results can provide insights into the effectsof the number of relay antennas, user transmit power,quantization bits, and the fraction of high-resolutionquantizers on the achievable rate, respectively.

• The power-scaling law of the mixed-ADC/DAC architec-ture is investigated for power saving in the data transmis-sion phase of the relay. Our results reveal that when thenumber of relay antennas, M , gets asymptotically large,the transmit power of each antenna can be scaled downby 1/M without any rate loss for the considered system.Moreover, the achievable rate gap between the mixed-ADC/DAC relay system and the unquantized system is aconstant in the low power regime.

• In order to compensate for the rate degradation causedby the coarse quantization, a low-complexity powerallocation algorithm is proposed for the consideredsystem. The power allocation problem can be solvedby transforming it into a sequence of geometricprogramming (GP) problems.

• Finally, using a generic power consumption model,we study the effects of the fraction of high-resolutionADCs and quantization bits on energy efficiency.In order to maximize energy efficiency, the optimalnumber of quantization bits is derived through numericalcomputations. Furthermore, our analysis proves thatthe considered system can significantly reduce powerconsumption and hardware cost while maintainingconsiderable performance.

C. Outline

The remaining parts of the paper are structured asfollows. The mixed-ADC/DAC multipair mMIMO relayingsystem model is briefly introduced in Section II. Both exactand asymptotic achievable rate expressions are derivedin Section III. Moreover, the power-scaling laws for theconsidered system is presented in Section III. In Section V,a simple power allocation scheme is proposed to compensatefor the rate loss. Furthermore, numerical results are providedin Section V to illustrate the effect of various systemparameters on the achievable rate and energy efficiency.Finally, key findings are concluded in Section VI. Most ofthe mathematical proofs are given in Appendices A and B.

142 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

D. Notations

In this paper, x and X in bold typeface are used to representvectors and matrices, respectively, while scalars are presentedin normal typeface, such as x. We use XT and XH to representthe transpose and conjugate transpose of a matrix X, respec-tively. IN stands for an N × N identity matrix, and ‖X‖F

denotes the Frobenius norm of a matrix X. Furthermore,E{·} denotes the expectation operator, and x ∼ CN (m, σ2I)represents a circularly symmetric complex Gaussian stochasticvector with mean vector m and covariance matrix σ2I. Finally,diag (X) denotes a diagonal matrix by keeping only thediagonal elements of matrix X.

II. SYSTEM MODEL

Let us consider a multipair relaying system with K single-antenna user pairs, denoted as Sk and Dk, k = 1, . . . , K ,applying the relay to exchange information with each other.We assume that the direct links between Sk and Dk do notexist because of large obstacles or severe shadowing. Thelarge-scale relay is equipped with M pairs of antennas, namelyreceive antennas and transmit antennas, with mixed-resolutionADCs and DACs. In the mixed-ADC/DAC architecture, onlyM0 pairs of costly high-resolution ADCs and DACs areconnected to M0 relay antennas, while the remaining M1

(= M − M0) pairs of less expensive low-resolution ADCsand DACs are connected to M1 relay antennas. Furthermore,we use κ

Δ= M0/M (0 ≤ κ ≤ 1) to denote the fraction ofhigh-resolution ADCs and DACs in the mixed architecture.The low-resolution ADCs bring about severe quantizationerrors for data reception and the low-resolution DACs leadto obvious signal distortion for data transmission. Therefore,the correlation of the quantization noise is taken into accountfor the multipair mMIMO relaying system. Furthermore,we assume that the relay operates in half-duplex mode, so itcannot receive and transmit signals simultaneously. Hence,information transmission from Sk to Dk is completed in twotime slots. In the first time slot, the K users in the sourceset Sk transmit xS ∈ C

K×1 data to the relay independently,and in the next time slot the relay transmits the correlated-quantized signals xR ∈ C

M×1 to K users in the destinationset Dk. The received signal yR ∈ C

M×1 at the relay and thereceived signal yD ∈ C

K×1 can be respectively given by

yR = GSRPS1/2xS + nR, (1)

yD = γGTRDxR + nD, (2)

where γ is the normalization factor in order to make the totalpower at the relay constrained to pR, i.e., E

{‖γxR‖2

}= pR.

Moreover, PS is a diagonal matrix representing the transmitpower of the K source users and its kth element is given by[PS ]kk = pS,k. nR ∼ CN (0, IM ), nD ∼ CN (0, IK) denotethe additive white Gaussian noise (AWGN) matrix withindependently and identically distributed (i.i.d.) componentsfollowing the distribution CN (0, 1). We further followthe general assumption that the transmit signal vectorxS is Gaussian distributed. The matrices GSR =[gSR,1, . . . ,gSR,K

]and GT

RD =[gRD,1, . . . ,gRD,K

]T

refer to the Rayleigh fading channels from the K sourcesto the relay with gSR,k ∼ CN (0, βSR,kIM ) and thechannels from the relay to the K destinations withgRD,k ∼ CN (0, βRD,kIM ), respectively. The terms βSR,k

and βRD,k stand for the large-scale fading and are assumedto be known at the relay.

Furthermore, we define GSR0 as the M0×K channel matrixfrom the K sources to the M0 relay antennas connected withhigh-resolution ADCs, and GSR1 as the M1 × K channelmatrix from the K sources to the remained M1 relay antennasconnected with low-resolution ADCs. Therefore, we have

GSR =[GSR0

GSR1

]. (3)

Similarly, we can also define GTRD0 as the K × M0 channel

matrix from the M0 relay transmit antennas connected withhigh-resolution DACs to the K destinations, and GT

RD1 as theM1 ×K channel matrix from the M1 relay transmit antennasconnected with low-resolution DACs to the K destinations.Then, GRD can be expressed as

GRD =[GRD0

GRD1

]. (4)

With the help of (3) and (4), (1) and (2) can be rewritten as

yR =[yR0

yR1

]=[GSR0PS

1/2xS + nR0

GSR1PS1/2xS + nR1

], (5)

yD =[yD0

yD1

]= γGT

RD0xR0 + γGTRD1xR1 + nD, (6)

where yR0 denotes the first M0 rows of the overall receivedsignals vector yR, and yR1 denotes the rest M1 rows of yR.Without loss of generality, the notations yD0, yD1, nR0, nR1,xR0 and xR1 can also be explained in a similar way.

A. Quantization With Mixed-Resolution ADCs

For the mixed-ADC architecture, the quantized receivedsignal at the relay can be written as

yR =[yR0

yR1

]=[

yR0

Q (yR1)

], (7)

where Q (·) is the scalar quantization function, yR0 denotes thequantized received signals at the output of M0 high-resolutionADCs, and yR1 is the quantized received signals at theoutput of M1 low-resolution ADCs. According to the AQNM[30, eq. (1)], the quantization operation can be expressed as

yR1 = Q (yR1) = αyR1 + nqa , (8)

where nqa refers to the additive Gaussian quantization noisevector which is uncorrelated with yR1, and α denotes a lineargain given by [31, eq. (13)]

α = 1 − ρ = 1 − E

{‖yR1 − yR1‖2

}/E

{‖yR1‖2

}, (9)

with ρ as the distortion factor of the low-resolution ADCs.The exact values of ρ are given in Table I with respectto different resolution bits [32]. For large quantization bits(e.g., b > 5), the distortion factor ρ can be approximated

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 143

TABLE I

DISTORTION FACTORS FOR DIFFERENT QUANTIZATION BITS

as ρ ≈ π√

32 2−2b [32]. With the help of (5), (8) and (9),

the covariance matrix of nqa is expressed as

Rnqa= αρdiag

(GSR1PSGH

SR1 + IM1

). (10)

Moreover, (7) can be rewritten as

yR =[yR0

yR1

]=[

GSR0PS1/2xS+nR0

αGSR1PS1/2xS+αnR1+nqa

]. (11)

B. Maximum Ratio (MR) Processing at the Relay

We assume that the relay adopts a simple amplify-and-forward (AF)1 protocol to process the quantized receivedsignals, yielding

xR = WyR, (12)

where W = G∗RDGH

SR denotes the MR processing. The MRprocessing is used at the relay due to its low-complexity, beingsuitable for the low-cost multipair mMIMO relaying system.Furthermore, according to some research, the MR processingcan achieve similar performance as zero-forcing receiver/zero-forcing transmission (ZFR/ZFT) or minimum mean squareerror (MMSE). By applying (3) and (4), (12) is rewritten as

xR =[G∗

RD0GHSR0yR0+G∗

RD0GHSR1yR1

G∗RD1G

HSR0yR0+G∗

RD1GHSR1yR1

]. (13)

C. Quantization With Mixed-Resolution DACs

For simplicity, we assume that the DACs and ADCs havethe same resolution. The analysis method can be extendedto arbitrary resolution cases. With mixed-DAC architecture atthe transmitter, the mixed-ADC/DAC signals from the relay’stransmit antennas can be expressed as

xR =[xR0

xR1

]=[

xR0

Q (xR1)

]=[

xR0

αxR1 + nqD

], (14)

where α is the distortion factor of the low-resolution DACs,and nqD denotes the quantization noise of low-resolutionDACs, which is uncorrelated with xR1. Note that xR0 = xR0

because of using high-resolution DACs. Similar as for (10),we can derive the covariance matrix of nqD as

RnqD= αρdiag (RxR1), (15)

where RxR1 is the covariance matrix of xR1. Accordingto (13), RxR1 can be written as

RxR1 = G∗RD1G

HSR0RyR0yR0

GSR0GTRD1

+G∗RD1G

HSR0RyR0yR1

GSR1GTRD1

+G∗RD1G

HSR1RyR1yR0

GSR0GTRD1

+G∗RD1G

HSR1RyR1yR1

GSR1GTRD1, (16)

1The AF protocol is considered herein due to its lower implementationcomplexity compared with the decode-and-forward (DF) protocol [5].

where

RyR0yR0= GSR0PSGH

SR0 + IM0 , (17)RyR0yR1

= αGSR0PSGHSR1, (18)

RyR1yR0= αGSR1PSGH

SR0, (19)

RyR1yR1= α2

(GSR1PSGH

SR1 + IM1

)

+ αρdiag(GSR1PSGH

SR1 + IM1

). (20)

Consequently, the normalization factor γ can be expressed as

γ =√

pR/E

{‖xR‖2

}. (21)

Lemma 1: For mixed-ADC/DAC multipair mMIMO relay-ing systems, the expectation of the total transmit power at therelay can be expressed as

E

{‖xR‖2

}= μ (M0 + αM1), (22)

where μ is given by

μ = α (1 − α)M1

K∑k=1

pS,kβ2SR,kβRD,k. (23)

Proof: Please refer to Appendix A.Substituting (22) into (21), the normalization factor γ can

be obtained as

γ =√

pR/μ (M0 + αM1). (24)

With the normalization factor in hand, we can derive theachievable rate in the following section.

III. ACHIEVABLE RATE ANALYSIS

A. Exact Achievable Rate Analysis

It is assumed that the destination Dk applies only statisticalCSI to decode the signal, due to the reason that instantaneousCSI leads to excessive high computational complexity for largeantenna arrays in a practical mMIMO system. Combining (1),(2), (11), (13) and (14), the received signal at the destinationDk can be expressed as

yD,k =√

pS,kE {Tk,k}xS,k︸ ︷︷ ︸desired signal

+ nD,k︸︷︷︸effective noise

, (25)

where

Ti,j = γgTRD0,iG

∗RD0G

HSR0gSR0,j

+ γαgTRD0,iG

∗RD0G

HSR1gSR1,j

+ γαgTRD1,iG

∗RD1G

HSR0gSR0,j

+ γα2gTRD1,iG

∗RD1G

HSR1gSR1,j ,

and

nD,k

=√

pS,k (Tk,k − E {Tk,k})xS,k︸ ︷︷ ︸estimation error

+∑i�=k

Tk,ixS,i

︸ ︷︷ ︸inter-pair interference

+ γ(gT

RD0,kG∗RD0G

HSR0nR0+αgT

RD1,kG∗RD1G

HSR0nR0

)︸ ︷︷ ︸

noise of high-resolution quantization at the relay

144 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

+ γ(αgT

RD0,kG∗RD0G

HSR1nR1+α2gT

RD1,kG∗RD1G

HSR1nR1

)︸ ︷︷ ︸

noise of low-resolution quantization at the relay

+ γ(gT

RD0,kG∗RD0G

HSR1nqa + αgT

RD1,kG∗RD1G

HSR1nqa

)︸ ︷︷ ︸

quantization noise of ADCs

+ γgTRD1,knqD︸ ︷︷ ︸

quantization noise of DACs

+ nD,k︸︷︷︸noise

, (26)

where nD,k is the kth element of the vector nD. We canderive the signal-to-interference-plus-noise ratio (SINR)expression by using [33, eq. (18)]. Since the “desired signal”and the “effective noise” in (25) are uncorrelated, the exactachievable rate for the k-th destination is given in Theorem 1.

Theorem 1: For mixed-ADC/DAC multipair mMIMO relay-ing systems and using the capacity bound in [34], the exactclosed-form achievable rate of the k-th destination is given as

Rk =τc−2τp

2τclog2

(1+

Ak

Bk+Ck+Dk+Ek+Fk+Gk+1

),

(27)

where τc denotes the length (in symbols) of each coherenceinterval, τp represents the length of the mutually orthogonalpilot sequences, and

Ak = pS,kγ2(M0 + αM1)4β2

SR,kβ2RD,k, (28)

Bk = pS,kγ2(M0 + α2M1

)βSR,kβRD,k

[2(M0+αM1)

2

× βSR,kβRD,k+(M0 + α2M1

) K∑m=1

βSR,mβRD,m

],

(29)

Ck = γ2(M0 + α2M1

)∑i�=k

pS,i

×[(M0+αM1)

2(βSR,kβ2RD,kβSR,i+βRD,kβ2

SR,iβRD,i

)

+(M0 + α2M1

)βRD,kβSR,i

K∑m=1

βSR,mβRD,m

], (30)

Dk = γ2M0(M0 + αM1)2βSR,kβ2

RD,k

+ γ2M0

(M0 + α2M1

)βRD,k

K∑m=1

βSR,mβRD,m, (31)

Ek = γ2α2M1(M0 + αM1)2βSR,kβ2

RD,k

+ γ2α2M1

(M0 + α2M1

)βRD,k

K∑m=1

βSR,mβRD,m,

(32)

Fk = αργ2M1βRD,k

{(M0 + α2M1

) K∑m=1

[βSR,mβRD,m

×(

K∑i=1

pS,iβSR,i + pS,mβSR,m+1

)]+(M0 + αM1)

2

× βSR,kβRD,k

(K∑

i=1

pS,iβSR,i + pS,kβSR,k + 1

)},

(33)

Gk = αργ2M1 (M0 + αM1)βRD,k

{βSR,kβRD,k

×(

K∑i=1

pS,iβSR,i+1+(M0+αM1) pS,kβSR,k

)

+K∑

m=1

βSR,m × βRD,m

×(

K∑i=1

pS,iβSR,i + 1 + (M0 + αM1) pS,mβSR,m

)}

+ α2ρ2γ2M21 βRD,k

×(

K∑i=1

pS,iβ2SR,iβRD,i+pS,kβ

2SR,kβRD,k

). (34)

Proof: Please refer to Appendix B.With the help of (24) and after some simplifications, we can

derive the compact expression for the sum achievable rate as

R =τc − 2τp

2τc

K∑k=1

log2 (1 + νk), (35)

where

νk = pS,k/ξk, (36)

ξk =K∑

i=1

pS,iaki + p−1R

(K∑

i=1

pS,ibki + ck

)+ dk, (37)

bki =1

(M0 + αM1)β2

SR,iβRD,i

β2SR,kβ2

RD,k

[1 +

αρM1

(M0 + αM1)2

]

+1

(M0 + αM1)2

βSR,i

K∑m=1

βSR,mβRD,m

β2SR,kβ2

RD,k

, (38)

ck =1

(M0 + αM1)2

K∑m=1

βSR,mβRD,m

β2SR,kβ2

RD,k

, (39)

dk =1

(M0 + αM1)1

βSR,k+

αρM1

(M0 + αM1)3

1βSR,k

+1

(M0 + αM1)2

K∑m=1

βSR,mβRD,m

β2SR,kβRD,k

, (40)

with aki given by (41) at the top of next page.

B. Asymptotic Analysis

Note that Fk and Gk are respectively derived by using exactvalues of the covariance matrices Rnqa

and RnqD, which

make results in (33) and (34) cumbersome. In order to providemore insights into the effect of various parameters on theachievable rate, we consider a large number of antennas anduse the law of large numbers. The covariance matrix Rnqa

in (10) is given by

Rnqa≈ αρ

(K∑

k=1

pS,kβSR,k + 1

)IM1 . (42)

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 145

aki =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

βSR,i

βSR,k

(M0+αM1)

⎛⎜⎜⎝1+

αρM1

(M0+αM1)2 +

βSR,iβRD,i

βSR,kβRD,k

[1+

αρM1

(M0+αM1)2

]+

1(M0+αM1)

K∑m=1

βSR,mβRD,m

βSR,kβRD,k

⎞⎟⎟⎠ , i �= k

1(M0+αM1)

⎛⎜⎜⎝2+

αρM1

(M0 + αM1)

[2+

2(M0+αM1)

+αρM1

(M0+αM1)2

]+

1(M0+αM1)

K∑m=1

βSR,mβRD,m

βSR,kβRD,k

⎞⎟⎟⎠ , i = k

(41)

Similarly, the covariance matrix RnqDis approximated as

RnqD≈ E

{RnqD

}= αρμIM1 , (43)

where μ is given by (23). Note that the expression of theapproximate RnqD

is obtained in (94), in Appendix A. Basedon the aforementioned discussion, we can derive a moreconcise close-form approximation for the achievable rate bysimplifying Fk and Gk as in (42) and (43), respectively. Sim-ilar to Theorem 1, we can derive the approximate achievablerate Rk by calculating the power expectations of the signaland interference as shown in the following theorem.

Theorem 2: For mixed-ADC/DAC multipair mMIMO relay-ing systems, the approximate achievable rate of the k-thdestination is

Rk =τc−2τp

2τclog2

(1+

Ak

Bk+Ck+Dk+Ek+Fk+Gk+1

),

(44)

where Gk = γ2αρμM1βRD,k,

Fk = γ2αρ

(K∑

k=1

pS,kβSR,k + 1

)M1βRD,k

((M0 + αM1)

2

× βSR,kβRD,k +(M0 + α2M1

) K∑m=1

βSR,mβRD,m

),

(45)

and Ak , Bk, Ck, Dk and Ek are given by (28), (29), (30),(31), and (32), respectively.

Proof: From (111) in Appendix B, we have

Fk =ρ

α

(K∑

k=1

pS,kβSR,k + 1

)Ek. (46)

Substituting (42) into (46), we obtain (45). Similarly,we derive Gk = γ2αρμE

{∣∣∣gTRD1,kg

∗RD1,k

∣∣∣}

. Using the

fact that E

{∣∣∣gTRD1,kg

∗RD1,k

∣∣∣}

= M1βSR,k, we can then

derive Gk.Following a similar reasoning as in the exact achievable rate

analysis, we substitute (23) and (24) into Theorem 2 to deducethe compact expression for the approximate achievable rate as

Rk = ((τc − 2τp)/2τc)K∑

k=1

log2 (1 + νk), (47)

where

νk = pS,k/ξk, (48)

ξk =K∑

i=1

pS,iaki +

(K∑

i=1

pS,ibki + ck

)/pR + dk,

dk =1

(M0+αM1)βSR,k+

1(M0 + αM1)

2

K∑m=1

βSR,mβRD,m

β2SR,kβRD,k

,

(49)

with aki given as

aki

=1

(M0+αM1)βSR,i

βSR,k

{1+

βSR,iβRD,i

βSR,kβRD,k

×[1+

α2ρ2M21

(M0+αM1)3

]+

1(M0+αM1)

K∑m=1

βSR,mβRD,m

βSR,kβRD,k

}.

Note that bki and ck have been defined in (38) and (39),respectively.

It is clear to see from (47) that the approximate achievablerate Rk increases with the total power of the relay pR, anddecreases with the transmit power of other sources. The mixed-ADC/DAC multipair mMIMO relaying system is interference-limited, which is consistent with [25]. Moreover, we find thatincluding more low-resolution ADCs and DACs decreases Rk.This is reasonable since the quantization noise increases.

C. Power Scaling Law

In this subsection, we investigate the potential for powersaving in the data transmission phase due to the deploymentof a very large antenna array at the relay. Here, let pS =ES/M (i.e., the power of all sources is the same, pS = pS,k,k = 1, . . . , K) and pR = ER/M , where the transmit powerof the source ES and of the relay ER are fixed. As M → ∞,the exact and approximate achievable rate for the consideredsystem is provided in the following Corollary.

Corollary 1: With pS = ES/M , pR = ER/M and ES, ER

fixed, the achievable rate limit of mixed-ADC/DAC multipairmMIMO relaying systems is given by (50) at the top of thenext page.

Proof: We start with the approximate achievable rateRk. Let pS = ES/M , pR = ER/M , and with the help

146 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Rk, Rk → τc − 2τp

2τclog2

[1 +

ESER(α + ρκ)2

ES (α + ρκ)K∑

i=1

β2SR,iβRD,i

β2SR,kβ2

RD,k+

K�

m=1βSR,mβRD,m

β2SR,kβ2

RD,k+ ER(α+ρκ)

βSR,k

]. (50)

of (48) and (49), we can obtain

νk =ES

ES

K∑i=1

aki + E−1R

(K∑

i=1

ESMbki + M2ck

)+ Mdk

.

(51)

As M → ∞, the terms related to M in (51) are derived as

aki → 0, (52)

Mbki → β2SR,iβRD,i/(α + ρκ)β2

SR,kβ2RD,k, (53)

M2ck → 1

(α + ρκ)2

K∑m=1

βSR,mβRD,m

β2SR,kβ2

RD,k

, (54)

Mdk → 1(α + ρκ)

1βSR,k

. (55)

Substituting (52), (53), (54) and (55) into (51), we can derivethe limit of Rk after some simple mathematical manipulations.Following a similar way, the limit of exact achievable rate Rk

can be derived. With pS = ES/M and pR = ER/M , (36) canbe rewritten as

νk =ES

ES

K∑i=1

aki+E−1R

(K∑

i=1

ESMbki+M2ck

)+Mdk

. (56)

As M → ∞, the terms related to M in (56) are given by

aki → 0, (57)

Mdk → 1(α + ρκ)

1βSR,k

, (58)

and the limits of Mbki and M2ck are given by (53) and (54),respectively. Since the limit of aki and dk are separately thesame as aki and dk with M → ∞, Rk approaches the sameconstant limit as Rk. After some simplifications, the proof isconcluded by deriving (50).

It is clear from (50) that both exact and approximate resultstend to a same constant value with M → ∞. We canfind that the proportion of the high-resolution ADCs/DACs κand the distortion factor of the low-resolution ADCs/DACsρ have effects on the limit rate when scaling down thetransmit power proportion to 1/M . More specifically, the limitcan be improved by increasing κ. Adopting the fact thatρκ + α = (1 − κ)α + κ is a monotonic increasing functionof α, the limit of achievable rate monotonically increaseswith α, which means that we can boost the achievable rateby using higher quantization bits in the M1 low-resolutionADCs/DACs.

Proposition 1: With pS = ES/M , pR = ER/M andES → 0, ER fixed, we can derive the factor of the sumrate gap between the mixed-ADC/DAC relay system and theunquantized one as

Rk

Rpk

→(α + ρκ)2

(ERβSR,kβ2

RD,k +K∑

m=1βSR,mβRD,m

)

(α + ρκ)ERβSR,kβ2RD,k +

K∑m=1

βSR,mβRD,m

.

Proposition 2: With pS = ES/M , pR = ER/M andES fixed, ER → 0, we can derive the factor of the sumrate gap between the mixed-ADC/DAC relay system and theunquantized one as

Rk

Rpk

→(α+ρκ)2

(ES

K∑m=1

β2SR,mβRD,m+

K∑m=1

βSR,mβRD,m

)

(α+ρκ) ES

K∑m=1

β2SR,mβRD,m+

K∑m=1

βSR,mβRD,m

.

(59)

From Propositions 1 and 2, it is clear that the sum rate gapbetween the mixed-ADC/DAC relay system and the unquan-tized system is a constant factor in the low power regime. Thefactors in the case ES → 0 or ER → 0 are both relatedto α + ρκ. Using the fact that ρκ + α = (1 − κ)α + κis a monotonic increasing function of α, we can find thatthe factors also increase with α. For the special case ofα = 1, i.e., α + ρκ = 1, the achievable rate of the mixed-ADC/DAC system is the same as that of the ideal unquantizedsystem.

IV. POWER ALLOCATION

In this section, we try to maximize the sum achievable rateof the mixed-ADC/DAC multipair mMIMO relaying system

constrained to a given total sum power PT, i.e.,K∑

k=1

pS,k +

pR ≤ PT, and formulate it as a power allocation problem.Let us define pS = [pS,1, . . . , pS,K]T , the power allocation

problem can be expressed as

P1 : maximizepS,pR

τc − 2τp

2τc

K∑k=1

log2 (1 + νk) (60)

subject toK∑

k=1

pS,k + pR ≤ PT (61)

pS ≥ 0, pR ≥ 0. (62)

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 147

Since log (·) is a monotonic increasing function, theproblem P1 can be reformulated as

P2 : minimizepS,pR

K∏k=1

(1 + νk)−1 (63)

subject toK∑

k=1

pS,k + pR ≤ PT (64)

pS ≥ 0, pR ≥ 0. (65)

Problem P2 is a general nonconvex complementary geometricprogram (CGP), which can be approximated by solving asequence of GP problems. After that, we can use standardconvex optimization tools (e.g., CVX) to solve the GP prob-lems [35]. The detailed steps of the power allocation algo-rithm are provided in Algorithm 1. Following the successiveapproximation algorithm in [25], it is efficient to solve thepower allocation problem.

Algorithm 1 Successive Approximation Algorithm for P2

1) Initialization. Define a tolerance � and parameter θ.Set j = 1 and set the initial value νk according to theSINR expression in Theorem 1 with pS,k = PT

2K andpR = PT

2 . 2) iteration j. Compute δk = νk

1+νk. Then

solve the GP problem P3:

P3 : minimizepS,pR

K∏k=1

νk−δk (66)

subject to θ−1νk ≤ νk ≤ θνk, k = 1, · · · , K (67)

νkpS,k−1ξk ≤ 1, k = 1, · · · , K (68)

K∑k=1

pS,k + pR ≤ PT (69)

pS ≥ 0, pR ≥ 0. (70)

Denote the optimal solutions by νk(j), for k = 1, · · · , K .

3) Stopping criterion. If maxk

∣∣νk(j) − νk

∣∣ < �, stop;otherwise, go to step 4). 4) Update initial values. Setνk = νk

(j), and j = j + 1. Go to step 2)

V. NUMERICAL RESULTS

In this section, we conduct numerous simulations to verifythe accuracy of the analytical results. Apart from that, insightsare also provided. Then, the benefit of the proposed powerallocation algorithm is demonstrated. Moreover, we investigatethe energy efficiency to show the advantage of the mixed-ADC/DAC architecture.

In the Monte Carlo simulation, we assume that the users aredistributed in a hexagonal cell with a radius of 1000 meters,while the minimum distance between the users and relay isrmin = 100 meters. The length of the coherence intervaland pilot sequence are set as τc = 20K and τp = K ,respectively. Furthermore, the large-scale fading coefficientsare arbitrarily generated by βSR,k = zk(rSR,k/rmin)−α andβRD,k = zk(rRD,k/rmin)−α, where zk is a log-normal randomvariable with standard derivation 8 dB, rSR,k and rRD,k

Fig. 1. Sum achievable rate against the number of relay antennas M forpS = 10 dB, pR = 10 dB, K = 10 and κ = 1/2.

represent the distances from the sources to the relay anddestinations to the relay, respectively, and α = 3.8 denotesthe pathloss exponent.

A. Achievable Rate

In Fig. 1, the simulated achievable rate, the analytical exactresult (35) and the approximate result (47) are plotted againstthe number of relay antennas. It can be seen that the analyticalexact and approximate results, as well as simulation resultsare close to each other, which validates the correctness of ourderived expressions. For a small number of antennas at therelay, the sum achievable rates for the cases of b = 1, 2,∞matches well with each other. While as the quantization bits bincrease, the gap between the approximate and simulatedcurves becomes small. Finally, better rate performance isachieved with a larger number of quantization bits (b > 1).

In Fig. 2, we investigate the power scaling law of the mixed-ADC/DAC multipair mMIMO relaying system. The fractionof the number of high-resolution ADCs/DACs in the mixed-ADC/DAC architecture is κ = 0, 1/2 and 1, respectively. It canbe seen that the exact and approximate expressions tend to aconstant for M → ∞, and a higher value of κ increases theachievable rate, which agrees with Corollary 1.

B. Power Allocation

We show the impact of the efficient power allocation schemeon the sum achievable rate in Fig. 3. The uniform powerallocation scheme, i.e., pS = PT

2K and pR = PTK , is also

investigated as a benchmark for comparison. It is clear thatthe proposed optimal power allocation scheme significantlyboosts the sum rate compared with the one of the unquantizedsystem with uniform power allocation. This important findingdemonstrates the significance of adopting an efficient powerallocation scheme in the mixed-ADC/DAC multipair mMIMOrelaying system.

148 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Fig. 2. Sum achievable rate against the number of relay antennas forpS = 10 dB, pR = 10 dB, K = 10 and b = 1.

Fig. 3. Sum achievable rate against the number of relay antennas M forK = 10, κ = 1/2 and PT = 10 dB.

C. Energy Efficiency

Up to now, we have investigated the achievable rateof mixed-ADC/DAC multipair mMIMO relaying systems.As expected, under the same power allocation, the sum rateof unquantized system outperforms the one with mixed-resolution ADCs/DACs, at the cost of expensive hardwareand power consumption. There should be a fundamentaltrade-off between the achievable rate and energy efficiency,and therefore, we also study the energy efficiency ofmixed-ADC/DAC multipair mMIMO relaying systems.

According to [21], the energy efficiency can be defined as

ηEE =R × B

Ptotalbit/Joule, (71)

where R denotes the sum achievable rate, B refers to thetransmission bandwidth assumed to be 20 MHz, and Ptotal isthe total power consumption. Combining [21, eq. (43)] and

[36, eq. (9)], Ptotal can be expressed as

Ptotal = M (Pmix + Pfilt) + 2Psyn + M1

(cPAGC + PL

DAC

)

+ M (PLNA + Pmix + PIFA + Pfilr)+ M0

(PAGC + PH

ADC

)+ M1

(cPAGC + PL

ADC

)

+ M0

(PAGC + PH

DAC

), (72)

where Pmix, Pfilt, Psyn, PLNA, PIFA, Pfilr, PAGC, PHADC,

PLADC, PH

DAC and PLDAC are the power consumption val-

ues for the mixer, the active filters at the transmitterside, the frequency synthesizer, low-noise amplifiers (LNA),the intermediate frequency amplifier (IFA), the active filtersat the receiver side, the automatic gain control (AGC), high-resolution ADCs, low-resolution ADCs, high-resolution DACsand low-resolution DACs, respectively. In addition, c denotesthe flag related to quantization bits of low-resolution ADCs,which is given by

c =

{0, b = 1,

1, b > 1.(73)

According to [37], the power consumed in DACs and ADCscan be respectively expressed in terms of the number ofquantization bits as

PDAC =12VddI0

(2b − 1

)+ bCp (2B + fcor)V 2

dd, (74)

PADC =3V 2

ddLmin (2B + fcor)10−0.1525b+4.838

, (75)

where b denotes the quantization bits, B is the bandwidth ofthe original signal assumed to be 20 MHz, Vdd is the powersupply of converter, I0 is the unit current source correspondingto the least significant bit (LSB), Cp is the parasitic capaci-tance of each switch in the converter, Lmin is the minimumchannel length for the given CMOS technology, fcor is thecorner frequency of the 1/f noise and all those parametersare specifically defined in [37]. (74) holds for binary-weightedcurrent-steering DACs [37] and (75) is established for thecomplete class of CMOS Nyquist-rate high speed ADCs [38].In numerical examples, we consider the following classicalvalues: Pmix = 30.3 mW, Pfilt = Pfilr = 2.5 mW, Psyn =50.0 mW, PLNA = 20 mW, PIFA = 3 mW and PAGC = 2 mWas in [21] and [36], and the power consumption values of othervarious circuit blocks have been discussed in [37].

The energy efficiency of mixed-ADC/DAC multipairmMIMO relaying systems against the quantization bits isillustrated in Fig. 4. It is clear from the figure that therelay adopting pure low-resolution ADCs/DACs attains thebest energy efficiency. That means the energy efficiencyincreases with the proportion of the number of low-resolutionADCs/DACs (1 − κ) in the mixed-ADC/DAC architecture.Although the pure low-resolution ADC/DAC architecture canachieve better energy efficiency than the mixed-ADC/DACarchitecture, as shown in 1, the spectral efficiency of the low-resolution ADC/DAC architecture is much lower than that ofthe mixed-ADC/DAC architecture. Moreover, the channel esti-mation in the mixed-ADC/DAC architecture is more tractablethan that in the low-resolution ADC/DAC architecture due tothe use of partial high-resolution ADCs. Moreover, the pure

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 149

Fig. 4. Energy efficiency of multipair mMIMO relaying systems with mixed-resolution ADCs/DACs against the number of low-resolution quantizationbits b for pS = 10 dB, pR = 10 dB and K = 10.

low-resolution ADCs/DACs has low spectrum efficiency.Fig. 4 indicates that we can achieve a better spectralefficiency by reducing the burden of power consumptionconsiderably by using the mixed-ADC/DAC architecture.

VI. CONCLUSIONS

In this paper, we investigated the achievable rate of amultipair mMIMO relaying system with mixed-resolutionADCs and DACs at the relay. Both exact and approximateclosed-form expressions for the achievable rate were derived.Then, we proved that the transmit power of each users can bescaled down as 1/M for the considered system. Despite therate loss due to the use of low-resolution ADCs and DACs,employing massive antenna arrays still enables high achievablerate and large power saving. Furthermore, we proposed anefficient power allocation scheme, which can compensatefor the rate degradation caused by low-resolution ADCs andDACs. Finally, the energy efficiency was investigated, andshowed that the mixed-ADC/DAC architecture can attain aconsiderable rate and energy efficiency simultaneously, whichis promising for practical mMIMO relaying systems.

APPENDIX APROOF OF LEMMA 1

For E

{‖xR‖2

}= E

{‖xR0‖2

}+ E

{‖xR1‖2

}, we first

A. Calculate E

{‖xR0‖2

}

RxR0xR0 = G∗RD0G

HSR0RyR0yR0

GSR0GTRD0

+G∗RD0G

HSR0RyR0yR1

GSR1GTRD0

+G∗RD0G

HSR1RyR1yR0

GSR0GTRD0

+G∗RD0G

HSR1RyR1yR1

GSR1GTRD0

= Q1 + Q2 + Q3 + Q4, (76)

where RyR0yR0, RyR0yR1

, RyR1yR0and RyR1yR1

are sepa-rately given by (17) to (20), and Q1, Q2, Q3 and Q4 aredefined as the four parts of RxR0xR0 , respectively.

E {Q1} = E

{G∗

RD0GHSR0GSR0PSGH

SR0GSR0GTRD0

}

+ E

{G∗

RD0GHSR0GSR0GT

RD0

}. (77)

Using E

{∥∥gSR0,m

∥∥4} = M0 (M0 + 1)β2SR,m, we have

E

{G∗

RD0GHSR0GSR0GT

RD0

}

= M0

K∑m=1

βSR,mβRD,mIM0, (78)

E

{G∗

RD0GHSR0GSR0PSGH

SR0GSR0GTRD0

}

= M0

K∑m=1

βSR,mβRD,m

(K∑

i=1

pS,iβSR,i+M0pS,mβSR,m

)

× IM0. (79)

Then substituting (79) and (78) into (77), we directly obtain

E {Q1} = M0

K∑m=1

βSR,mβRD,m

×(

K∑i=1

pS,iβSR,i + M0pS,mβSR,m + 1

)IM0.

(80)

Similar to the computation of Q1, the expectation of Q2, Q3

and Q4 can be derived respectively as

E {Q2} = αM0 M1

K∑m=1

pS,mβ2SR,mβRD,mIM0, (81)

E {Q3} = αM0 M1

K∑m=1

pS,mβ2SR,mβRD,mIM0. (82)

E {Q4} = α2M1

K∑m=1

βSR,mβRD,m

×(

K∑i=1

pS,iβSR,i + M1pS,mβSR,m + 1

)IM0

+ αρE

{G∗

RD0GHSR1Rnqanqa

GSR1GTRD0

}. (83)

where

E

{G∗

RD0GHSR1Rnqanqa

GSR1GTRD0

}

= E

{G∗

RD0E

[GH

SR1diag(GSR1PSGH

SR1

)GSR1

]GT

RD0

}

+ M1

K∑m=1

βSR,mβRD,mIM0. (84)

150 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

The expectation of the diagonal term can be decomposed as

E

{gH

SR1,idiag(GSR1PSGH

SR1

)gSR1,j

}

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, i �= j,M1∑

m=1

pS,nE

{|gSR1,mn|4

}

+M1∑

m=1

K∑i�=n

pS,iE

{|gSR1,mn|2

}E

{|gSR1,mi|2

},

i = j = n.

(85)

Applying the fact that E

{|gSR1,mn|4

}= 2β2

SR,n, (85) can beexpressed as

E

{gH

SR1,idiag(GSR1PSGH

SR1

)gSR1,j

}

=

⎧⎨⎩

0, i �= j,

M1βSR,n

(K∑

i=1

pS,iβSR,i + pS,nβSR,n

), i = j = n.

(86)

Substituting (86) into (84), we can obtain

E

{G∗

RD0GHSR1Rnqanqa

GSR1GTRD0

}

= M1

K∑m=1

βSR,mβRD,m

(K∑

i=1

pS,iβSR,i+pS,mβSR,m+1

)

× IM0. (87)

With the help of (83) and (87), we derive E {Q4} as

E {Q4}

= αM1

K∑m=1

βSR,mβRD,m

×(

K∑i=1

pS,iβSR,i+αM1pS,mβSR,m+ρpS,mβSR,m+1

)

× IM0. (88)

Combining (80), (81), (82) and (88), we can derive

E {RxR0xR0} = E {RxR0xR0} = μIM0. (89)

E

{‖xR0‖2

}= E

{‖xR0‖2

}= μM0. (90)

B. Calculate E

{‖xR1‖2

}

E

{‖xR1‖2

}= α2

E

{‖xR1‖2

}+ E

{nH

qDnqD

}. (91)

Similar to the calculation of (89) and (90), E {RxR0xR0} and

E

{‖xR1‖2

}can be respectively expressed as

E {RxR1xR1} = μIM1, (92)

E

{‖xR1‖2

}= μM1. (93)

As for E{nH

qDnqD

}, considering (15) and (92), we can derive

RnqDas

E

{RnqD

}= αρE {RxR1xR1} = αρμIM1. (94)

Hence,

E{nH

qDnqD

}= αρμM1. (95)

Substituting (93) and (95) into (91), we can directly obtain

E

{‖xR1‖2

}= μαM1. (96)

Therefore, E

{‖xR‖2

}= E

{‖xR0‖2

}+ E

{‖xR1‖2

}=

μ (M0 + αM1). The proof is concluded.

APPENDIX BPROOF OF THEOREM 1

The argument of the log function in the right-hand sideof (27) consists of six terms: 1) desired signal power Ak;2) estimation error Bk; 3) inter-pair interference Ck; 4) noiseat the relay Dk and Ek; 5) quantization noise of ADCs Fk;6) quantization noise of DACs Gk.

A. Compute Ak

Since

E {Tk,k} = γ (M0 + αM1)2βSR,kβRD,k, (97)

we have

Ak = pS,kγ2(M0 + αM1)4β2

SR,kβ2RD,k. (98)

B. Compute Bk

Bk = pS,kVar (Tk,k) = pS,kE

{|Tk,k|2

}− Ak, (99)

We define t1 to t10 as the decomposed terms of E

{Tk,iT

Hk,i

}

in order. Note that the undefined terms in tk meanthat they are included in the expressions if and onlyif i = k.

t1 = E

{gT

RD0,kG∗RD0G

HSR0gSR0,ig

HSR0,iGSR0GT

RD0g∗RD0,k

}

= M20 βSR,iβRD,k(M0βSR,kβRD,k+

K∑m=1

βSR,mβRD,m

+ M0βSR,iβRD,i) + M40 β2

SR,kβ2RD,k, (100)

t2 = E

{α2gT

RD0,kG∗RD0G

HSR1gSR1,ig

HSR1,iGSR1

×GTRD0g

∗RD0,k

}

= α2M0M1βSR,iβRD,k(M0βSR,kβRD,k

+K∑

m=1

βSR,mβRD,m + M1βSR,iβRD,i)

+ α2M20 M2

1 β2SR,kβ2

RD,k, (101)

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 151

t3 = E

{α2gT

RD1,kG∗RD1G

HSR0gSR0,ig

HSR0,i

×GSR0GTRD1g

∗RD1,k

}

= α2M0M1βSR,iβRD,k(M1βSR,kβRD,k

+K∑

m=1

βSR,mβRD,m + M0βSR,iβRD,i)

+ α2M20M2

1 β2SR,kβ2

RD,k, (102)

t4 = E

{α4gT

RD1,kG∗RD1G

HSR1gSR1,ig

HSR1,i

×GSR1GTRD1g

∗RD1,k

}

= α4M21 βSR,iβRD,k

(M1βSR,kβRD,k +

K∑m=1

βSR,mβRD,m

+ M1βSR,iβRD,i

)+ α4M4

1 β2SR,kβ2

RD,k, (103)

t5 = E

{αgT

RD0,kG∗RD0G

HSR0gSR0,ig

HSR1,iGSR1

×GTRD0g

∗RD0,k

}

= αM20 M1β

2SR,iβRD,k (βRD,i + M0βRD,k) , (104)

t6 = E

{αgT

RD0,kG∗RD0G

HSR0gSR0,ig

HSR0,iGSR0

×GTRD1g

∗RD1,k

}

= αM20 M1βSR,kβ2

RD,k(βSR,i + M0βSR,k), (105)

t7 = E

{α2gT

RD0,kG∗RD0G

HSR0gSR0,ig

HSR1,iGSR1

×GTRD1g

∗RD1,k

}

= α2M20 M2

1 β2SR,kβ2

RD,k, (106)

t8 = E

{α2gT

RD0,kG∗RD0G

HSR1gSR1,ig

HSR0,iGSR0

×GTRD1g

∗RD1,k

}

= α2M20 M2

1 β2SR,kβ2

RD,k, (107)

t9 = E

{α3gT

RD0,kG∗RD0G

HSR1gSR1,ig

HSR1,iGSR1

×GTRD1g

∗RD1,k

}

= α3M0M21 βSR,kβ2

RD,k(βSR,i + M1βSR,k), (108)

t10 = α3M0M21 β2

SR,iβRD,k(βRD,i + M1βRD,k). (109)

With the help of (100)–(109), we have

E{Tk,iT

Hk,i

}= γ2

(t1 + t2 + t3 + t4 + 2t5 + 2t6

+ 2t7 + 2t8 + 2t9 + 2t10

). (110)

After some simplifications with i = k, we can obtain (29).

C. Compute Ck

Similar to the calculation of Bk, We can derive theexpression (31) of Ck .

D. Compute Dk and Ek

Similar to the calculation of Bk, we can obtain(32) and (33).

E. Compute Fk

Fk = γ2E

{∣∣∣gTRD0,kG

∗RD0E

{GH

SR1RnqanqaGSR1

}

×GTRD0g

∗RD0,k + αgT

RD0,kG∗RD0

×E

{GH

SR1RnqanqaGSR1

}GT

RD1g∗RD1,k

+ αgTRD1,kG

∗RD1E

{GH

SR1RnqanqaGSR1

}

×GTRD0g

∗RD0,k + α2gT

RD1,kG∗RD1

×E

{GH

SR1RnqanqaGSR1

}GT

RD1g∗RD1,k

∣∣∣}

. (111)

Using results in (86), we have

E

{GH

SR1RnqanqaGSR1

}= diag (a1, . . . , aK), (112)

where an = M1βSR,n

(K∑

i=1

pS,iβSR,i + pS,nβSR,n + 1)

.

Substituting (112) into (111), we can obtain (33).

F. Compute Gk

With the help of (16), we can obtain (34) by applying similarapproaches in the derivations of E {RxR0xR0}.

Combining all derived terms completes the proof.

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Jiayi Zhang (S’08–M’14) received the B.Sc. andPh.D. degrees in communication engineering fromBeijing Jiaotong University, China, in 2007 and2014, respectively. From 2012 to 2013, he was aVisiting Ph.D. Student with the Wireless Group,The University of Southampton, U.K. From 2014 to2015, he was a Humboldt Research Fellow with theInstitute for Digital Communications, University ofErlangen-Nuremberg, Germany. From 2014 to 2016,he was a Post-Doctoral Research Associate withthe Department of Electronic Engineering, Tsinghua

University, China. Since 2016, he has been a Professor with the Schoolof Electronic and Information Engineering, Beijing Jiaotong University.His current research interests include massive multiple input multiple output,unmanned aerial vehicle systems, and performance analysis of generalizedfading channels.

He was a recipient of the WCSP and IEEE APCC Best Paper Award in 2017.He serves as an Associate Editor for the IEEE COMMUNICATIONS LETTERS

and the IEEE ACCESS.He was recognized as an Exemplary Reviewer forthe IEEE COMMUNICATIONS LETTERS in 2015 and 2016, and the IEEETRANSACTIONS ON COMMUNICATIONS in 2017.

Linglong Dai (M’11–SM’14) received the B.S.degree from Zhejiang University in 2003, the M.S.degree (Hons.) from the China Academy ofTelecommunications Technology in 2006, and thePh.D. degree (Hons.) from Tsinghua University,Beijing, China, in 2011. From 2011 to 2013, he wasa Post-Doctoral Research Fellow with the Depart-ment of Electronic Engineering, Tsinghua Univer-sity, where he was an Assistant Professor from2013 to 2016 and has been an Associate Professorsince 2016. He has co-authored the book mmWave

Massive MIMO: A Paradigm for 5G (Academic Press, Elsevier, 2016).He has authored over 60 IEEE journal papers and over 40 IEEE conferencepapers. He also holds 15 granted patents. His current research interestsinclude 5G/6G key technologies and machine learning for wireless com-munications. He was a recipient of the Tsinghua University OutstandingPh.D. Graduate Award in 2011, the Beijing Excellent Doctoral DissertationAward in 2012, the China National Excellent Doctoral Dissertation Nomi-nation Award in 2013, the URSI Young Scientist Award in 2014, the IEEETransactions on Broadcasting Best Paper Award in 2015, the Second Prize ofScience and Technology Award of China Institute of Communications in 2016,the Electronics Letters Best Paper Award in 2016, the IEEE CommunicationsLetters Exemplary Editor Award in 2017, the National Natural ScienceFoundation of China for Outstanding Young Scholars in 2017, and the IEEEComSoc Asia-Pacific Outstanding Young Researcher Award in 2017. He hasreceived Conference Best Paper Award at the IEEE ICC 2013, the IEEEICC 2014, the IEEE ICC 2017, the IEEE VTC 2017-Fall, and the IEEEICC 2018. He currently serves as an Editor for the IEEE TRANSACTIONS

ON COMMUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECH-NOLOGY, and the IEEE COMMUNICATIONS LETTERS. Particularly, he isdedicated to reproducible research and has made a large amount of simulationcodes publicly available (http://oa.ee.tsinghua.edu.cn/dailinglong).

Ziyan He received the B.S. degree in electronicengineering from Tsinghua University in 2018.He is currently pursuing the Ph.D. degree with theInformation Transmission and Processing Labora-tory, School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta, GA, USA.His research interests include massive multiple-inputmultiple-output systems and machine learning forsignal processing in communications.

ZHANG et al.: MIXED-ADC/DAC MULTIPAIR MASSIVE MIMO RELAYING SYSTEMS 153

Bo Ai (M’00–SM’10) received the master’s andPh.D. degrees from Xidian University, China.He received the honor of Excellent PostdoctoralResearch Fellow from Tsinghua University in 2007.He was a Visiting Professor with the Electrical Engi-neering Department, Stanford University, Stanford,CA, USA, in 2015. He is currently a Full Professorwith Beijing Jiaotong University, where he is alsothe Deputy Director of the State Key Laboratoryof Rail Traffic Control and Safety and the DeputyDirector of the International Joint Research Center.

He is one of the directors for Beijing “Urban Rail Operation Control System”International Science and Technology Cooperation Base, and the backbonemember of the Innovative Engineering based jointly granted by the ChineseMinistry of Education and the State Administration of Foreign ExpertsAffairs.

He is the Research Team Leader of 26 national projects. He holds 26 inven-tion patents. His interests include the research and applications of channelmeasurement and channel modeling and dedicated mobile communications forrail traffic systems. He has authored or co-authored eight books and authoredover 300 academic research papers in his research area. Five papers have beenthe ESI highly cited paper. He has won some important scientific researchprizes. He has been notified by the Council of Canadian Academies that basedon the Scopus database, he has been listed as one of the Top 1% authors inhis field all over the world. He has also been feature interviewed by theIET Electronics Letters.

Dr. Ai is a fellow of The Institution of Engineering and Technologyand an IEEE VTS Distinguished Lecturer. He received the DistinguishedYouth Foundation and Excellent Youth Foundation from the National NaturalScience Foundation of China, the Qiushi Outstanding Youth Award by theHong Kong Qiushi Foundation, the New Century Talents by the ChineseMinistry of Education, the Zhan Tianyou Railway Science and TechnologyAward by the Chinese Ministry of Railways, and the Science and TechnologyNew Star by the Beijing Municipal Science and Technology Commission.He is an IEEE VTS Beijing Chapter Vice Chair and an IEEE BTS Xi’anChapter Chair. He was a Co-Chair or a Session/Track Chair of manyinternational conferences. He is an Associate Editor of the IEEE ANTENNAS

AND WIRELESS PROPAGATION LETTERS, the IEEE TRANSACTIONS ON

CONSUMER ELECTRONICS, and an Editorial Committee Member of theWireless Personal Communications Journal. He is the Lead Guest Editor ofspecial issues on the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,the IEEE ANTENNAS AND PROPAGATIONS LETTERS, and the InternationalJournal on Antennas and Propagations.

Octavia A. Dobre (M’05–SM’07) received theDipl.Ing. and Ph.D. degrees from the PolitehnicaUniversity of Bucharest, Romania, in 1991 and2000, respectively. Between 2002 and 2005, she waswith the Politehnica University of Bucharest andthe New Jersey Institute of Technology, Newark,NJ, USA. In 2005, she joined Memorial Univer-sity, Canada. She was a Visiting Professor with theUniversité de Bretagne Occidentale, France, and theMassachusetts Institute of Technology, Cambridge,MA, USA, in 2013. She is currently a Professor and

a Research Chair with Memorial University. Her research interests includefifth-generation enabling technologies, blind signal identification and parame-ter estimation techniques, as well as optical and underwater communications.

Dr. Dobre is a fellow of The Engineering Institute of Canada. She was aRoyal Society Scholar in 2000 and a Fulbright Scholar in 2001. She servesas the Editor-in-Chief for the IEEE COMMUNICATIONS LETTERS, as well asan Editor for the IEEE COMMUNICATIONS SURVEYS AND TUTORIALS andthe IEEE SYSTEMS. She was an editor, a senior editor, and a guest editor ofother prestigious journals. She served as a general chair, a tutorial co-chair,and a technical co-chair for numerous conferences.