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The Interplay between Massive MIMO andUnderlaid D2D NetworkingXingqin Lin, Robert W. Heath Jr., and Jeffrey G. Andrews

Abstract—In a device-to-device (D2D) underlaid cellular net-work, the uplink spectrum is reused by the D2D transmis-sions, causing mutual interference with the ongoing cellulartransmissions. Massive MIMO is appealing in such a contextas the base station’s (BS’s) large antenna array can nearlynull the D2D-to-BS interference. The multi-user transmission inmassive MIMO, however, may lead to increased cellular-to-D2Dinterference. This paper studies the interplay between massiveMIMO and underlaid D2D networking in a single cell setting.We investigate cellular and D2D spectral efficiency under bothperfect and imperfect channel state information (CSI) at thereceivers that employ partial zero-forcing. We find that cellulartransmit power can be scaled down as Θ(1/M) and Θ(1/

√M)

under perfect and imperfect CSI respectively, where M is thenumber of BS antennas, while achieving non-vanishing cellularspectral efficiency. Compared to the case without D2D, thereis a loss in cellular spectral efficiency due to D2D underlay.With perfect CSI, the loss can be completely overcome if thenumber of canceled D2D interfering signals is scaled with M atan arbitrarily slow rate. In the non-asymptotic regime, simpleanalytical lower bounds are derived for both the cellular andD2D spectral efficiency.

I. INTRODUCTION

A. Background

Device-to-device (D2D) communication enables nearby mo-bile devices to establish direct links in cellular networks [1]–[3], unlike traditional cellular communication where all trafficis routed via base stations (BSs). D2D has the potential toimprove spectrum utilization, shorten packet delay, and reduceenergy consumption, while enabling new peer-to-peer andlocation-based applications and services [3], [4] and beinga required feature in public safety networks [5]. IntroducingD2D poses many challenges and risks to the existing cellulararchitecture. In particular, in a D2D underlaid cellular networkwhere the spectrum is reused D2D transmission may causeinterference to cellular transmission and vice versa. Existingoperator services may be severely affected if the newly intro-duced D2D interference is not appropriately controlled.

The distinctive traits of massive MIMO make it appealingto enable D2D communication in the uplink resources ofcellular networks. In a massive MIMO system, each BS uses avery large antenna array to serve multiple users in each time-frequency resource block [6]. If the number of antennas at theBS is significantly larger than the number of served users, thechannel of each user to/from the BS is nearly orthogonal to

Xingqin Lin, Robert W. Heath Jr., and Jeffrey G. Andrews are withDepartment of Electrical & Computer Engineering, The University of Texas atAustin, USA. (Email: {xlin, rheath}@utexas.edu, jandrews@ece.utexas.edu).Date revised: September 10, 2014.

that of any other user. This allows for very simple transmitor receive processing techniques like matched filtering to benearly optimal with enough antennas even in the presence ofinterference [6]–[10]. This implies that, with a large antennaarray at the BS, D2D signals possibly result in close-to-zerointerference at the uplink massive MIMO BS, making D2Dvery simple and appealing in massive MIMO systems.

Though D2D-to-cellular interference may be effectivelyhandled by the large antenna array at the BS, cellular-to-D2D interference persists and may be worse in a massiveMIMO system. Specifically, massive MIMO is a multi-usertransmission strategy designed to support multiple users ineach time-frequency block; the number of simultaneouslyactive uplink users is scalable with the number of antennasat the BS. With this increased number of uplink transmitters,the D2D links reusing uplink radio resources will experienceincreased interference. To protect D2D links, the number ofsimultaneously active uplink users might have to be limited,eating into massive MIMO gain. It is not a priori clearto what extent the D2D signals would be affected by themultiuser transmission and the tradeoff between supportingD2D communication and scaling up the uplink capacity ina massive MIMO system. Further, if cochannel D2D signalsare present when estimating massive MIMO channels, theestimated channel state information (CSI) would become lessaccurate, which may hurt massive MIMO performance. It isnot a priori clear however to what extent the D2D signalswould affect the channel estimation and consequently theperformance of the massive MIMO system.

Existing research on D2D networking is mainly focused onsingle-antenna networks (see e.g. [11]–[16]) while researchon the use of antenna arrays has just begun [17]–[21]. Tomitigate or avoid mutual interference between cellular andD2D transmissions, [17], [18] considered precoding while[19], [20] studied various relaying strategies. In contrast, [21]proposed not to schedule uplink users that may generateexcessive interference to D2D users. How D2D MIMO andcellular MIMO interact, especially in the massive MIMOcontext, is still largely open.

B. Contributions and OutcomesThe main contributions and outcomes of this paper are

summarized as follows.1) A tractable hybrid network model: We introduce a

tractable hybrid network model consisting of both ad hocnodes and cellular infrastructure, which extends our previoussingle-antenna D2D model [14], [16] to multi-antenna trans-mission. We consider a single macro cell and focus on the

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uplink which is better than the downlink for D2D underlay[3]. The spatial positions of the underlaid D2D transmittersare modeled by a Poisson point process (PPP). Such a randomPPP model is well motivated by the random and unpredictablemobile user locations [22], [23]. All the transmissions (bothcellular and D2D) in this model are SIMO (i.e., single-inputmultiple-output) with the BS having a very large antenna array.For the receive processing, we extend the partial zero-forcing(PZF) receiver studied in ad hoc networks [24] to the hybridnetwork in question. Spectral efficiency is used as the solemetric throughout this paper.

2) Spectral efficiency with perfect CSI: In the asymptoticregime where the number of BS antennas M → ∞ and withperfect CSI, we find that the received signal-to-interference-plus-noise ratio (SINR) of any cellular user increases un-boundedly and the effects of noise, fast fading, and theinterfering signals from the other co-channel cellular users andthe infinite D2D transmitters vanish completely. Equivalently,it is possible to reduce cellular transmit power as Θ(1/M)but still achieve a non-vanishing cellular spectral efficiency.Compared to the case without D2D, there is a loss in cellularspectral efficiency if a constant number of D2D interferingsignals is canceled. The loss can be overcome if the numberof canceled D2D interfering signals is scaled appropriately(e.g. Θ(logM)). In the non-asymptotic regime, we derivesimple analytical lower bounds for both cellular and D2Dspectral efficiency; the derived bounds allow for very efficientnumerical evaluation.

3) Spectral efficiency with imperfect CSI: We study pilot-based CSI estimation in which known training sequences aretransmitted and the receivers use minimum mean squared error(MMSE) estimator for channel estimation. In the asymptoticregime with the estimated CSI, we find that the received SINRof any cellular user can be unbounded as in the case of perfectCSI. To achieve a non-vanishing cellular spectral efficiency,however, with imperfect CSI we can only reduce cellulartransmit power as Θ(1/

√M). Further, compared to the case

without D2D, there is a loss in cellular spectral efficiency evenif the number of canceled D2D interfering signals is scaled toinfinity. We omit non-asymptotic results in this paper due tospace constraint.

II. MATHEMATICAL MODELS

A. Network Model

Consider a single-cell D2D underlaid massive MIMO sys-tem shown in Fig. 1. In this system, K single-antenna cellularuser equipments (UEs) transmit to a single BS that has Mantennas. We denote by K the set of the K cellular UEs andassume that their positions are fixed during the transmission.We are interested in the performance regime where M is largeand thus the assumption M � K is made throughout thispaper.

The cellular system is underlaid with D2D UEs. The loca-tions of the D2D transmitters are distributed as a homogeneousPPP Φ with density λ. Each D2D receiver is located at afixed distance of d meters from its associated D2D transmitterwith uniformly distributed direction. Each D2D receiver is

Macro BS

M  (~100)  Rx  antennas  at  BS    

N  Rx  antennas  at  UE    1  Tx  antenna  at  UE  

Out-­‐of-­‐cell  D2D  pair  

DRx  i   DTx  i  CTx  k  

DTx  r  

g(d)ir

g(d)iig

(c)ik

h(c)k

h(d)i

Fig. 1. A single-cell D2D underlaid massive MIMO system consisting ofboth cellular and D2D links. D2D pairs located outside of the cell contributeto out-of-cell D2D interference.

equipped with N receive antennas. Therefore, though thecellular system only consists of single cell, D2D pairs aredistributed throughout the plane and those D2D pairs locatedoutside the cell contribute to out-of-cell D2D interference.

In this system, all the transmitters use the same time-frequency resource block, leading to cochannel interference.We assume that cellular and D2D UEs transmit at constantpowers Pc and Pd respectively.

B. Baseband Channel Models

The M ×1 dimensional baseband received signal at the BSis

y(c) =∑k∈K

√Pc‖x(c)

k ‖−αc

2 h(c)k u

(c)k

+∑i∈Φ

√Pd‖x(d)

i ‖−αc

2 h(d)i u

(d)i + v(c), (1)

where x(c)k denotes the position of cellular transmitter k, αc >

2 denotes the pathloss exponent of UE-BS links, h(c)k ∈ CM×1

is the vector channel from cellular transmitter k to the BS,u

(c)k denotes the zero-mean unit-variance transmit symbol of

cellular transmitter k, x(d)i ,h

(d)i ∈ CM×1 and u(d)

i are similarlydefined for D2D transmitter i, and v(c) ∈ CM×1 is complexGaussian noise with covariance N0IM with IM denoting theM dimensional identity matrix. For ease of reference, wedefine UE-BS channel matrices H(c) = [h

(c)1 , ...,h

(c)K ] and

H(d) = [h(d)1 ,h

(d)2 , ...].

Similarly, the N × 1 dimensional baseband received signalat the D2D receiver r is

y(d)r =

∑k∈K

√Pc(d

(c)rk )−

αd2 g

(c)rk u

(c)k

+∑i∈Φ

√Pd(d

(d)ri )−

αd2 g

(d)ri u

(d)i + v(d)

r , (2)

where d(s)rk = ‖x(s)

k − z(d)r ‖ with s ∈ {c, d} and z

(d)r

denoting the position of D2D receiver r, αd > 2 denotesthe pathloss exponent of UE-UE links, g

(c)rk ,g

(d)ri ∈ CN×1

are the vector channels from the cellular transmitter k tothe D2D receiver r and from the D2D transmitter i to theD2D receiver r respectively, and v

(d)r ∈ CN×1 is complex

Gaussian noise with covariance N0IN . For ease of reference,we define UE-UE channel matrices G(c)

r = [g(c)r,1, ...,g

(c)r,K ] and

G(d)r = [g

(d)r,1,g

(d)r,2, ...].

Note that we have used different pathloss exponents αc andαd for UE-BS and UE-UE links (cf. (1) and (2)) due to their

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different propagation characteristics. Specifically, the antennaheight of a macro BS is tens of meters, while the typicalantenna height at a UE is under 2 m. As a result, both terminalsof a UE-UE link are low and see similar near street scatteringenvironment, which is different from the radio environmentaround a macro BS [3].

In this paper, we assume Gaussian signaling, i.e., {u(s)k }, s ∈

{c, d}, are i.i.d. zero-mean unit-variance complex GaussianCN (0, 1), and that all the vector channels, h(s)

k and g(s)rk , s ∈

{c, d}, have i.i.d. CN (0, 1) elements, independent across trans-mitters. It follows that the favorable propagation condition[25] desired in massive MIMO systems holds in our model:

1

Mh(s)∗r h

(s′)`

a.s.−−→{

1 if s = s′ and r = `;0 otherwise,

where a.s.−−→ denotes the almost sure convergence, as M →∞. Recent measurement campaigns have given evidence tovalidate favorable propagation for massive MIMO in practice[26].

C. Receive Filters

Denote by w(c)k the filter used by the BS for receiving the

signal of the k-th cellular transmitter, i.e., the BS detects thesymbol u(c)

k based on w(c)∗k y(c). Similarly, the D2D receiver

r detects the symbol u(d)r based on w

(d)∗r y

(d)r , where w

(d)r

denotes the filter used by the D2D receiver r. The performanceof the D2D underlaid massive MIMO system depends on thereceive filters. In general, either the receive filters can bedesigned to boost desired signal power or they can be usedto cancel undesired interference. In this paper, we focus on aparticular type of linear filters: the PZF receiver, which uses asubset of the degrees of freedom for boosting received signalpower and the remainder for interference cancellation.

The BS uses mc and md degrees of freedom to cancel theinterference from the nearest mc cellular interferers and mdD2D interferers. A feasible choice of (mc,md) needs to be inthe following set:

Zc = {(mc,md) ∈ N× N :

mc ≤ K − 1,mc +md ≤M − 1}. (3)

The PZF filter w(c)k is the projection of the channel vector

h(c)k onto the subspace orthogonal to the one spanned by the

column vectors of the matrix[H(c)\h(c)

k (1:mc), H(d)(1:md)

],

where A\ak denotes the matrix formed by removing the k-th column vector ak from A, and A(1:m) denotes the matrixformed by the first m column vectors of the matrix A. For easeof reference, we denote by K(c)

k the set of uncanceled cellularinterferers and Φ

(c)k the set of uncanceled D2D interferers

when detecting the symbol u(c)k of the k-th cellular transmitter.

Similarly, each D2D receiver uses nc and nd degrees offreedom to cancel the interference from the nearest nc cellularinterferers and nd D2D interferers, and (nc, nd) needs to bein the following set:

Zd = {(nc, nd) ∈ N× N : nc ≤ K,nc + nd ≤ N − 1}. (4)

The PZF filter w(d)r at the D2D receiver r is the pro-

jection of the channel vector g(d)rr onto the subspace or-

thogonal to the one spanned by the column vectors of[G

(c)r (1:nc), G

(d)r \g(d)

rr (1:nd)]. For ease of reference, we de-

note by K(d)r the set of uncanceled cellular interferers and Φ

(d)r

the set of uncanceled D2D interferers at the D2D receiver r.Remark on PZF receiver. Although suboptimal, PZF

receivers have several advantages that motivate us to focuson them in this paper. On the one hand, PZF receivers arerelatively general: they reduce to maximum ratio combining(MRC) receivers when mc + md = 0 and nc + nd = 0 andto conventional fully ZF receivers when mc + md = M − 1and nc + nd = N − 1. It has been shown that PZF receiverscan achieve the same scaling law in terms of transmissioncapacity as MMSE receivers [24], which is not true for eitherMRC or fully ZF receivers. On the other hand, PZF receiversare analytically more tractable than other more sophisticatedreceivers like MMSE receivers from a system point of view.This analytical tractability allows us to develop an explicitcharacterization on the performance of the massive MIMOsystem with D2D underlay. Nevertheless, we would like topoint out that, as noted in [24], MMSE fitlers should be usedin practice because they have less stringent CSI requirementwhile being the best linear filters.

III. SPECTRAL EFFICIENCY WITH PERFECT CHANNELSTATE INFORMATION

In this section, we derive the spectral efficiency of cellularand D2D links under the assumption of perfect CSI; the caseof imperfect CSI will be treated in the next section.

A. Asymptotic Cellular Spectral Efficiency

For the k-th cellular UE, the post-processing SINR with thePZF filter w(c)

k is

SINR(c)k =

S(c)k

I(c→c)k + I

(d→c)k + ‖w(c)

k ‖2N0

, (5)

where S(c)k = Pc‖x(c)

k ‖−αc‖w(c)∗k h

(c)k ‖2 denotes the desired

signal power of cellular UE k, I(c→c)k and I(d→c)

k respectivelydenote the cochannel cellular and D2D interference powersexperienced by cellular UE k and are given by

I(c→c)k =

∑`∈K(c)

k

Pc‖x(c)` ‖−αc |w(c)∗

k h(c)` |

2

I(d→c)k =

∑i∈Φ

(c)k

Pd‖x(d)i ‖−αc |w(c)∗

k h(d)i |

2. (6)

The spectral efficiency of the k-th cellular UE is defined as

R(c)k = E

[log(1 + SINR(c)

k )], (7)

where the expectation is taken with respect to the fast fadingand the random locations of D2D transmitters.

Proposition 1. With perfect CSI, as M → ∞, the desired

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signal power S(c)k when normalized by M2 converges to

limM→∞

1

M2S

(c)k

a.s.−−→ Pc‖x(c)k ‖−αc , (8)

and the cellular interference power I(c→c)k , the D2D interfer-

ence power I(d→c)k , and the noise power ‖w(c)

k ‖2N0 whennormalized by M2 converge as follows.

limM→∞

1

M2I

(c→c)k

a.s.−−→ 0

limM→∞

1

M2I

(d→c)k

p.−→ 0

limM→∞

1

M2‖w(c)

k ‖2N0

a.s.−−→ 0, (9)

wherep.−→ denotes the convergence in probability.

Proof: See Appendix A.

Prop. 1 shows that with perfect CSI, as M →∞, the post-processing SINR(c)

k increases unboundedly with probability 1(as almost sure convergence implies convergence in probabil-ity). More specifically, a deterministic received power of thedesired signal from cellular UE k can be achieved and theeffects of noise, fast fading, and the interfering signals fromthe other K−1 cellular UEs and the infinite D2D transmittersvanish completely. Therefore, Prop. 1 validates the intuitionthat with perfect CSI, D2D-to-cellular interference can bemade arbitrarily small with a large enough antenna array atthe BS. Perhaps the most surprising result from Prop. 1 isthat the D2D-to-cellular interference can be completely nulledout (with probability 1) even though the number of the PPPdistributed D2D interferers is infinite and the mean of theaggregate D2D interference is infinite. Further, the proof ofProp. 1 shows that a simple MRC filter with mc = md = 0suffices.

Though Prop. 1 shows that arbitrarily large received SINRand thus arbitrarily large rate (at least in theory) can beachieved with massive MIMO, it is not possible to fully exploita very high SINR due to practical constraints such as thehighest order of modulation and coding schemes. Nevertheless,the large array gains may be translated into power savingsfor cellular UEs: with a given SNR target we can lower thetransmit powers of cellular UEs and thus improve their energyefficiency, as shown in the following proposition.

Proposition 2. With perfect CSI, fixed PZF parameters(mc,md), and scaled cellular transmit power Pc/M , as M →∞, the spectral efficiency R(c)

k of cellular UE k converges to

R(c)k → EΦ,η

log

1 +SNR(c)

k∑i∈Φ

(c)k

PdN0‖x(d)

i ‖−αcηi + 1

,(10)

where SNR(c)k = Pc‖x(c)

k ‖−αc/N0, {ηi} are i.i.d. randomvariables distributed as ηi ∼ Exp(1). Further, if md + 1 > αc

2 ,

limM→∞

R(c)k ≥ log

(1 +

SNR(c)k

ρ(md, αc) + 1

), (11)

where

ρ(m,α) =2(πλ)

α2 PdΓ(m+ 1− α

2 )

(α− 2)N0Γ(m), (12)

where the Gamma function Γ(x) =∫∞

0tx−1e−tdt.

Proof: See Appendix B.Note that in Prop. 2, if the underlaid D2D transmitters

did not exist, the spectral efficiency R(c)k of cellular UE k

would converge to log(

1 + SNR(c)k

), the maximum achiev-

able spectral efficiency of a point-to-point SISO (single-inputsingle-output) Gaussian channel. It is as if massive MIMOcould simultaneously support K interference-free SISO linkswhile reducing the power of each cellular UE by 10 log10MdB. This result is consistent with Prop. 1 in [7] without D2Dunderlay.

With D2D underlay, the asymptotic result (10) shows thatthere is a loss in cellular spectral efficiency due to theuncanceled interfering signals from the D2D transmitters inΦ

(c)k , i.e., D2D transmitters in Φ except the nearest md ones

whose signals are canceled by the PZF filter. Though it ispossible to derive an exact analytical expression (involvingintegrals) for (10), we give a more intuitive lower bound(11), which succinctly characterizes the loss due to the D2Dunderlay through a single term ρ(md, αc). Several remarks arein order.

Remark 1. The term ρ(md, αc) corresponding to the un-canceled D2D interference increases with Pd and λ and de-creases with md, agreeing with intuition: larger transmit poweror larger density of D2D interferers or smaller number ofcanceled D2D interferers leads to higher D2D-to-cellular inter-ference, thus lowering the cellular spectral efficiency. Further,ρ(md, αc) ∼ λ

αc2 because a linear increase in λ implies that

the distances of the PPP distributed D2D transmitters to theBS decrease as λ

12 and thus the D2D-to-cellular interference

power increases as λαc2 .

Remark 2. Note that the lower bound (11) is meaningfulonly if md + 1 > αc

2 . As md → αc2 − 1, Γ(md + 1− αc

2 )→∞and thus ρ(md, αc) → ∞. In fact, from the proof of Prop.2, we can see that if this condition is violated, the expectedinterference from the (md + 1)-th nearest D2D transmitter tothe BS would be infinite.

It can be shown that as md becomes large, Γ(md) ∼ (md−1)

αc2 Γ(md − αc

2 ) and thus

Γ(md + 1− αc2 )

Γ(md)∼md + 1− αc

2

(md − 1)αc2

. (13)

Therefore, ρ(md, αc) decreases at a polynomial rate αc2 − 1 as

md increases. To completely null out the D2D-to-cellular inter-ference, we need only appropriately increase md as M grows.The growth rate of md can be arbitrarily slow. In particular, ifM increases linearly, asymptotically md = Θ(logM) degreesof freedom are sufficient to null out the D2D-to-cellularinterference caused by the infinite D2D transmitters. Thisleaves the BS Θ(M − logM) = Θ(M) degrees of freedom toboost the signal power. Therefore, the transmit power of eachcellular UE can still be scaled down as Θ(1/M). In otherwords, there is no loss of spectral efficiency and power saving

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due to the D2D underlay as long as md scales as Θ(logM).We summarize this result in the following proposition, whichstates that massive MIMO can asymptotically null out all theinterfering signals from the infinite D2D transmitters but stillmaintain a linear scaling in the desired signal power, i.e., wecan reduce cellular transmit power as Θ(1/M) but still achievethe spectral efficiency of an interference-free cellular link.

Proposition 3. With perfect CSI, arbitrary but fixed mc, andscaled cellular transmit power Pc/M , ifmd scales as o(M) (e.g.md = Θ(logM)), the spectral efficiency R(c)

k of cellular UE kconverges as follows.

R(c)k → log

(1 + SNR(c)

k

), as M →∞. (14)

B. Non-asymptotic Cellular Spectral Efficiency

Next we analyze the cellular spectral efficiency in the non-asymptotic regime to generate more insights into the impactof the various system parameters. To this end, using Jensen’sinequality we derive a lower bound for R(c)

k in the followingproposition.

Proposition 4. With perfect CSI, M ≥ mc +md +1 and md >αc2 − 1, the spectral efficiency R(c)

k of cellular UE k is lowerbounded by

R(c,lb)k = log

1 +(M −mc −md − 1)SNR(c)

k∑`∈K(c)

k

SNR(c)` + ρ(md, αc) + 1

,

(15)

where ρ(m,α) is defined in (12).

Proof: See Appendix C.Note that the first term in the denominator of (15) corre-

sponds to the uncanceled cellular interference; it decreases asmc increases. Similarly, the second term in the denominatorof (15) corresponds to the uncanceled D2D interference; itdecreases as md increases. In contrast, the numerator of (15)corresponds to the desired signal power; it decreases as mcand/or md increase. The lower bound (15) demonstrates thevarious tradeoffs when choosing the PZF parameters mc andmd. Note that such tradeoffs disappear in the asymptoticregime (cf. Prop. 2 and 3). If the PZF parameter mc = K−1,then all the cochannel cellular interference will be nulled out,leading to the following specialized lower bound.

R(c,lb)k = log

(1 +

(M −K −md)SNR(c)k

ρ(md, αc) + 1

). (16)

We point out that the received signal power gain is onlyproportional to M −mc −md − 1 in the lower bound (15).One might think the power gain should be proportional toM − mc − md, the number of degrees of freedom left forpower boosting after using mc + md degrees of freedom forinterference cancellation. The fallacy of the above argumentis that it ignores the effect of fading, which makes a powergain proportional to M −mc −md unachievable.

C. D2D Spectral Efficiency

For D2D receiver r, the post-processing SINR with PZFfilter is

SINR(d)r =

S(d)r

I(c→d)r + I

(d→d)r + ‖w(d)

r ‖2N0

, (17)

where S(d)r = Pdd

−αd‖w(d)∗r g

(d)rr ‖2 denotes the desired signal

power of D2D Tx-Rx pair r, I(c→d)r and I

(d→d)r respectively

denote the cochannel cellular and D2D interference powersexperienced by D2D receiver r and are given by

I(c→d)r =

∑k∈K(d)

r

Pc(d(c)rk )−αd |w(d)∗

r g(c)rk |

2

I(d→d)r =

∑i∈Φ

(d)r

Pd(d(d)ri )−αd |w(d)∗

r g(d)ri |

2. (18)

The spectral efficiency of the D2D Tx-Rx pair r is defined as

R(d)r = E

[log(1 + SINR(d)

r )], (19)

where the expectation is taken with respect to the fast fadingand the random locations of D2D transmitters.

As the number N of antennas at the UE is often limiteddue to hardware constraints, it is not very meaningful to studythe asymptotic performance with N → ∞. Instead, as in thecase of cellular spectral efficiency, we provide a lower boundfor R(d)

r in the non-asymptotic regime, which characterizes theimpact of the various system parameters on the D2D spectralefficiency.

Proposition 5. With perfect CSI, N ≥ nc + nd + 1 and nd >αd2 −1, the spectral efficiencyR(d)

r of D2D Tx-Rx pair r is lowerbounded by

R(d,lb)r = log

1 +(N − nc − nd − 1)SNR(d)∑

k∈K(d)r

PcN0

(d(c)rk )−αd + ρ(nd, αd) + 1

,

(20)

where SNR(d) = Pdd−αd/N0 and ρ(m,α) is defined in (12).

Proof: The proof is similar to that of Prop. 4 and isomitted for brevity.

Many of the remarks on Prop. 4 apply to Prop. 5 as welland are not repeated here. One additional remark is that thecellular-to-D2D interference is not homogeneous: differentD2D receivers experience different levels of cellular interfer-ence depending on their locations relative to the locations ofthe cellular transmitters (which are fixed in our model).

IV. SPECTRAL EFFICIENCY WITH IMPERFECT CHANNELSTATE INFORMATION

A. Estimating UE-BS Channels

We consider pilot-based CSI estimation in which knowntraining sequences are transmitted and used for estimationpurpose. During the training phase, the BS requires the Kcellular UEs and the md nearest D2D transmitters (w.r.t. theBS) to simultaneously transmit orthogonal training sequences.

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The BS does not need to coordinate the other D2D trans-mitters. In particular, the uncoordinated transmitters can sendindependent symbols during the training phase.

Denoting by Tc ≥ K+md the length of a training sequence,we can represent the training sequences as a Tc × (K + md)

dimensional matrix√TcQ

(c) =√Tc(q

(c)1 , ...,q

(c)K+md

) satisfy-ing Q(c)∗Q(c) = IK+md . In the training phase, the M × Tcdimensional baseband received signal Y(c) at the BS is

Y(c) =∑k∈K

√TcPc‖x(c)

k ‖−αc

2 h(c)k q

(c)∗k

+

md∑i=1

√TcPc‖x(d)

i ‖−αc

2 h(d)i q

(c)∗K+i

+∑r∈Φ

(c)k

√Pd‖x(d)

r ‖−αc2 h(d)

r u(d)∗r + V(c), (21)

where the Tc × 1 dimensional vector u(d)r contains the data

symbols sent by D2D interferers, and the M×Tc dimensionalnoise matrix V(c) consists of i.i.d. CN (0, N0) elements. Notethat the D2D transmitters i = 1, ...,md also use power Pcduring the training phase since they now transmit to the BS.

We assume that the BS uses linear MMSE estimator for thechannel estimation. To this end, we first project the receivedsignal Y(c) in the direction of q(c)

kand normalize it to obtain

y(s)k =

1√TcPc‖x(s)

k ‖−αc2

Y(c)q(c)k

= h(s)k + v

(s)k , (22)

where (s, k) ∈ {(c, k), (d,K+k)}, v(s)k denotes the equivalent

channel estimation “noise” and is given by

v(s)k =

1√TcPc‖x(s)

k ‖−αc2

× ∑r∈Φ

(c)k

√Pd‖x(d)

r ‖−αc2 h(d)

r u(d)∗r q

(c)k

+ V(c)q(c)k

. (23)

Lemma 1. The linear MMSE estimate of h(s)k , s ∈ {c, d}, is

given by h(s)k = ξ

(s)k y

(s)k , where

ξ(s)k =

TcPc‖x(s)k ‖−αc

TcPc‖x(s)k ‖−αc +

∑r∈Φ

(c)k

Pd‖x(d)r ‖−αc +N0

. (24)

Further, E[h(s)k ] = 0 and E[h

(s)k h

(s)∗k ] = ξ

(s)k IM . As for

the estimation error ε(s)k = h

(s)k − h

(s)k , E[ε

(s)k ] = 0 and

E[ε(s)k ε

(s)∗k ] = (1− ξ(s)

k )IM .

Proof: We omit the proof due to page limit.Lemma 1 shows that the longer the length Tc of a training

sequence, the smaller the covariance of the estimation errorε

(s)k and thus the more accurate the channel estimation h

(s)k ,

agreeing with intuition. In particular, E[ε(s)k ε

(s)∗k ] → 0 as

Tc → ∞. But longer training sequence consumes moretransmission resources in terms of power and bandwidth.

From the proof of Lemma 1, we know that the equiva-lent channel estimation “noise” v

(s)k is zero mean and its

covariance E[v(s)k v

(s)∗k ] = (1/ξ

(s)k − 1)IM , implying that the

elements of v(s)k are uncorrelated zero-mean random variables.

Since the definition of v(s)k (cf. (23)) involves the products

of complex Gaussian random vectors h(d)r u

(d)∗r , the elements

of v(s)k are neither Gaussian distributed nor independent. This

makes the exact analysis very complicated. Nevertheless, sincethere are infinite D2D interferers in Φ

(c)k and the contribution

of any one interferer is small (when md � 1), it is reasonableto apply the Central Limit Theorem to approximate v

(s)k as a

Gaussian random vector as follows.

Assumption 1. The elements of v(s)k defined in (23) are treated

as i.i.d. CN (0, 1/ξ(s)k − 1).

Under Assumption 1, the elements of the linear MMSEestimate h

(s)k are i.i.d. CN (0, ξ

(s)k ) and the elements of the

estimation error ε(s)k are i.i.d. CN (0, 1−ξ(s)

k ). Further, h(s)k and

ε(s)k are independent, and the linear MMSE estimator becomes

the MMSE estimator.With the estimated channel vectors, the M × 1 dimensional

baseband received signal at the BS (cf. (1)) in the normal datatransmission phase can be equivalently written as

y(c) =∑`∈K

√Pc‖x(c)

` ‖−αc

2 h(c)` u

(c)` +

∑i∈Φ\Φ(c)

k

√Pd‖x(d)

i ‖−αc

2 h(d)i u

(d)i

+∑`∈K

√Pc‖x(c)

` ‖−αc

2 ε(c)` u

(c)` +

∑i∈Φ\Φ(c)

k

√Pd‖x(d)

i ‖−αc

2 ε(d)i u

(d)i

+∑i∈Φ

(c)k

√Pd‖x(d)

i ‖−αc

2 h(d)i u

(d)i + v(c). (25)

When decoding the symbol of cellular UE k, the BS appliesthe PZF filter w(c)

k designed based on the estimated channelsh

(s)k . The last four terms in (25) represent unknown random

variables and are treated as independent interference and noise.Then we can obtain the post-processing SINR of cellular UEk as in Lemma 2.

Lemma 2. With imperfect CSI at the BS and the correspondingPZF filter w(c)

k , the post-processing SINR of cellular UE k is

ˆSINR(c)k =

S(c)k

I(c→c)k + I

(d→c)k + ‖w(c)

k ‖2N0

, (26)

where S(c)k = Pc‖x(c)

k ‖−αc‖w(c)∗k h

(c)k ‖2 denotes the desired

signal power, I(c→c)k and I(d→c)

k respectively denote the cellularand D2D interference powers and are given by

I(c→c)k =

∑`∈K(c)

k

Pc‖x(c)` ‖−αc‖w(c)∗

k h(c)` ‖

2

+ ‖w(c)k ‖

2∑`∈K

Pc‖x(c)` ‖−αc(1− ξ(c)

` ) (27)

I(d→c)k =‖w(c)

k ‖2∑i∈Φ

(c)k

Pd‖x(d)i ‖−αc

+ ‖w(c)k ‖

2∑

i∈Φ\Φ(c)k

Pd‖x(d)i ‖−αc(1− ξ(d)

i ). (28)

Comparing I(c→c)k to its counterpart I(c→c)

k defined in (6),

7

we can see that I(c→c)k has one additional term representing

the residual cellular interference due to the imperfect CSIbesides the term representing the uncanceled cellular inter-ference. Similar observation holds for I(d→c)

k . With ˆSINR(c)k ,

the spectral efficiency of cellular UE k is given by

R(c)k = E

[log(1 + ˆSINR

(c)k )]. (29)

B. Asymptotic Cellular Spectral Efficiency

In this subsection, we examine the asymptotic performanceof the cellular links as M → ∞. We first have the followingproposition.

Proposition 6. With imperfect CSI at the BS, as M → ∞,the desired signal power S(c)

k when normalized by M2 andconditioned on Φ converges to

limM→∞

1

M2S

(c)k

a.s.−−→ Pc‖x(c)k ‖−αc(ξ

(c)k )2, (30)

and the cellular interference power I(c→c)k , the D2D interfer-

ence power I(d→c)k , and the noise power ‖w(c)

k ‖2N0 whennormalized by M2 converge as follows.

limM→∞

1

M2I

(c→c)k

a.s.−−→ 0

limM→∞

1

M2I

(d→c)k

p.−→ 0

limM→∞

1

M2‖w(c)

k ‖2N0

a.s.−−→ 0. (31)

Further, the above convergence results can be achieved withmc = md = 0, i.e., the MRC receiver.

Proof: The derivation is similar to the proof of Prop. 1.We omit the details for brevity.

Note that the interference generated by the D2D transmittersis finite almost surely (although its mean is infinite). Itfollows that ξ(c)

k (c.f. (24)) is positive almost surely and sois limM→∞

1M2 S

(c)k in Prop. 6. Therefore, with imperfect

CSI and a simple MRC filter, as M → ∞, the post-processing SINR increases unboundedly. Specifically, as inthe case of perfect CSI, (conditioning on Φ) a deterministicreceived power of the desired signal of cellular UE k canbe achieved and the effects of noise, fast fading, and theinterfering signals from the other K − 1 cellular UEs andthe infinite D2D transmitters vanish completely. Note thatlimM→∞ S

(c)k /S

(c)k = (ξ

(c)k )2 < 1, i.e., there is a loss in signal

power with imperfect CSI compared to the case with perfectCSI. This loss comes from the fact that with imperfect CSIit is impossible to design a perfect PZF filter to coherentlycombine the received signal branches at the BS.

In spite of the imperfect CSI, with a given SINR target, thelarge received power gain promised by the massive MIMOhelps lower the transmit powers of cellular UEs and thusimprove their energy efficiency, as shown in the followingproposition.

Proposition 7. With imperfect CSI at the BS, fixed PZF pa-rameters (mc,md), and scaled cellular transmit power Pc/

√M ,

as M → ∞, the spectral efficiency R(c)k of cellular UE k

converges as follows.

R(c)k → EΦ

log

1 +Tc(SNR(c)

k )2(IΦ

(c)k

+ 1)

(IΦ + 1)

, (32)

where IΦ =∑i∈Φ

PdN0‖x(d)

i ‖−αc .

Proof: The derivation is similar to the proof of Prop. 2.We omit the details for brevity.

Note that in Prop. 7, if the underlaid D2D transmittersdid not exist, the spectral efficiency R

(c)k of cellular UE k

would converge to log(1 + Tc(SNR(c)k )2), which is consistent

with Prop. 5 in [7] without D2D underlay. It seems that thespectral efficiency R(c)

k with imperfect CSI is higher than itscounterpart R(c)

k = log(1 + SNR(c)k ) in the case of perfect

CSI. This artificial “gain” however comes at the cost of 1) theextra transmission resources consumed in the channel trainingand 2) the worse power scaling Θ(1/

√M) (vs. Θ(1/M)).

In fact, Θ(1/√M) is the best power scaling law that can

be achieved: the cellular spectral efficiency R(c)k tends to 0

if the cellular transmit power scales as Θ(1/M0.5+ε) for anyε > 0. The intuition behind the scaling law is that both cellulardata and training transmit powers decrease as Θ(1/

√M) and

their product determining the received signal power scales asΘ(1/M), which is exactly compensated by the array gainΘ(M) at the BS. As a result, a constant post-processingreceived signal power is achievable.

With D2D underlay, the asymptotic result (32) shows thatthere is a loss in cellular spectral efficiency due to the D2Dunderlay. The loss may be decomposed into two parts. Thefirst part, reflected in the first term in the denominator of(32), indicates the effect of the undesired D2D interferenceduring the channel training phase. Specifically, the uncon-trolled signals of the D2D interferers in the set Φ

(c)k during

the training phase affects the accuracy of the CSI h(c)k of the

desired channel, leading to the imperfect coherent combinationof the signal branches and thus a loss of signal power10 log10(

∑i∈Φ

(c)k

PdN0‖x(d)

i ‖−αc + 1) dB.

The second part, reflected in the second term in the denom-inator of (32), indicates the effect of the undesired D2D inter-ference during the data transmission phase. Note that the unde-sired D2D interference power equals

∑i∈Φ

PdN0‖x(d)

i ‖−αc + 1,including the interfering signals from all the D2D transmit-ters, although we design the received filter w

(c)k to cancel

the interfering signals from the D2D transmitters in the setΦ\Φ(c)

k . This is because the cellular transmit power scalesas Θ(1/

√M); this scaled power is also used by the D2D

transmitters in the set Φ\Φ(c)k during the training phase. As

a result, as M → ∞, the estimated D2D UE-BS channelsbecome totally inaccurate and thus asymptotically no D2Dinterfering signals would be nulled out. So in this case, it isnot useful to estimate the D2D UE-BS channels. A simpleapproach to improve the cellular spectral efficiency is toprohibit the D2D transmissions when the BS is estimatingcellular UE-BS channels. This would reduce the requirementon the length of cellular training sequence from Tc ≥M+md

8

to Tc ≥M and also result in better cellular spectral efficiency

R(c)k → EΦ

[log

(1 +

Tc(SNR(c)k )2∑

i∈ΦPdN0‖x(d)

i ‖−αc + 1

)]. (33)

Certainly, the above benefits come at the cost of (slightly)reduced time resources for D2D communication.

One might also wonder what would happen if we keep thetraining power constant but reduce the data transmission poweras Θ(1/

√M). In this case, the overall power consumption

scales as Θ(1) as the time spent in the training phase occupiesa non-vanishing fraction of the overall time resource. But thebenefit is that the D2D interference is reduced, as formalizedin the following corollary.Corollary 1. With imperfect CSI at the BS, fixed PZF param-eters (mc,md), and scaled cellular transmit power Pc/

√M in

the data transmission phase but constant transmit power Pc inthe training phase, as M → ∞, the spectral efficiency R(c)

k ofcellular UE k converges as follows.

R(c)k → EΦ

[log

(1 +

SNR(c)k ξ

(c)k

IΦ

(c)k

+ IΦ\Φ(c)

k ,ξ+ 1

)], (34)

where IΦ,ξ =∑i∈Φ

PdN0‖x(d)

i ‖−αc(1− ξ(d)i ).

Corollary 1 is intuitive. The loss in cellular spectral effi-ciency due to the D2D underlay may be decomposed intotwo parts. The first part is the 10 log10 ξ

(c)k dB loss of the

signal power, caused by the uncoordinated D2D interferenceduring the channel training phase. The second part (i.e., thedenominator of (34)) is the undesired D2D interference duringthe data transmission phase. Note that as we keep the trainingpower constant, the estimated D2D UE-BS channels are usefulfor canceling D2D interference: the interference from the D2Dtransmitter i ∈ Φ\Φ(c)

k is reduced by 100ξ(d)i %. Further, the

residual D2D interference decreases with the length Tc ofcellular training sequence.

V. SIMULATION AND NUMERICAL RESULTS

In this section, we provide simulation and numerical re-sults to demonstrate the analytical results and obtain insightsinto how the various system parameters affect the cellularand D2D spectral efficiency. The specific parameters usedare summarized in Table I unless otherwise specified. Thepathloss parameters given in Table I corresponds to a carrierfrequency of 2 GHz. Specifically, we use the 3GPP macrocellpropagation model (urban area) for UE-BS channels [27] andthe revised Winner + B1 model (non-light-of-sight with −5 dBoffset) for UE-UE channels [1]. Note that different pathlossreference values Cc,0 and Cd,0 are used in the UE-BS andUE-UE channels. Therefore, when evaluating the analyticalexpressions using the parameters in Table I, Pc = 23 − Cc,0(dBm) and Pd = 13 − Cc,0 (dBm) for the UE-BS channelswhile Pc = 23 − Cd,0 (dBm) and Pd = 13 − Cd,0 (dBm) forthe UE-UE channels. In the simulation, the BS is located atthe origin. As a benchmark, we fix the 4 cellular UE positionsat (Rc/2, 0

o), (Rc, 0o), (Rc, 120o), and (Rc, 240o) unless

otherwise stated; here Rc is the coverage radius of the BS.The cellular spectral efficiency plotted is the average cellular

BS coverage radius Rc 500 mD2D link length d 20 m# cellular UEs K 4Density of D2D UEs λ 4

πR2c

m−2

# BS antennas M 100# UE Rx antennas N 6Training seq. Tc 8UE-BS PL exponent αc 3.76UE-UE PL exponent αd 4.37UE-BS PL reference Cc,0 15.3 dBUE-UE PL reference Cd,0 38.5 dBCellular Tx power Pc 23 dBmD2D Tx power Pd 13 dBmChannel bandwidth 10 MHzNoise PSD −174 dBmBS noise figure 6 dBUE noise figure 9 dB

TABLE ISIMULATION/NUMERICAL PARAMETERS

spectral efficiency of the 4 cellular UEs, while the D2D spec-tral efficiency plotted is from the D2D pair with the transmitterand the receiver respectively located at (Rc/2− d, 180o) and(Rc/2, 180o).

We first compare the simulated cellular spectral efficiency tothe corresponding analytical lower bound (15) under variousPZF parameters (mc,md) in Fig. 2. As shown in Fig. 2, theanalytical lower bound (15) closely matches the simulationresults. The larger md, the closer match between the simulationand the analytical lower bound (15). This is because largermd implies less D2D-to-cellular interferers and thus smallerinterference variance. As a result, the lower bound based onJensen’s inequality becomes more accurate with larger md.Note that mc = 0 and mc = 3 correspond to MRC andZF (w.r.t. cellular UEs), respectively. Comparing the spectralefficiency with (mc,md) = (0, 2) to that of (mc,md) = (3, 2),we can see that ZF has much better performance and thespectral efficiency gain is about 3 bps/Hz. This observationimplies that although asymptotically ZF and MRC have sim-ilar performance, it is still quite beneficial to appropriatelysuppress the co-channel cellular interference in practical non-asymptotic regime.

Since the lower bound (15) is accurate, next we use itto demonstrate the cellular spectral efficiency with scaledcellular transmit power (i.e., Pc → Pc/M ) in Fig. 3. Weconsider two PZF choices: PZF with constant md and PZFwith scaled md = Θ(

√M). As a benchmark, we also include

the curves corresponding the scenarios without D2D underlay.Several observations about Fig. 3 are in order. First, unlikethe case with unscaled cellular transmit power, Fig. 3 showsthat ZF and MRC have similar performance even with amoderate number of BS antennas. Second, adopting a constantmd results in a fixed loss in the cellular spectral efficiencydue to the underlaid D2D interference; this loss cannot beovercome by increasing the number of BS antennas when

9

50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

M: # of BS Antennas

Sp

ectr

al E

ffic

ien

cy (

bit/s

/Hz)

Lower bound: (mc,m

d)=(0,2)

Simulation: (mc,m

d)=(0,2)

Lower bound: (mc,m

d)=(1,2)

Simulation: (mc,m

d)=(1,2)

Lower bound: (mc,m

d)=(3,2)

Simulation: (mc,m

d)=(3,2)

Lower bound: (mc,m

d)=(3,3)

Simulation: (mc,m

d)=(3,3)

Fig. 2. Simulated cellular spectral efficiency vs. analytical lower bound (15)with perfect CSI.

50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

M: # of BS Antennas

Spectr

al E

ffic

iency (

bit/s

/Hz)

mc = 0, No D2D

(mc,m

d) = (0,2)

(mc,m

d) = (0, M

1/2)

mc=3, No D2D

(mc,m

d) = (3,2)

(mc,m

d) = (3, M

1/2)

Fig. 3. Cellular spectral efficiency with scaled cellular transmit power withperfect CSI.

the cellular transmit power is also scaled down as Θ(1/M).This observation confirms the analytical results in Prop. 2.Third, the loss in the cellular spectral efficiency due to D2Dunderlay can be overcome by scaling md at a sublinear rateΘ(√M), validating the theoretical finding in Prop. 3. Further,

Fig. 3 shows that the convergence rate is relatively slow as Mincreases.

Fig. 4 compares the simulated D2D spectral efficiency tothe corresponding analytical lower bound (20) under differentD2D distances and (nc, nd) = (0, 2). As shown in Fig. 4, theanalytical lower bound (20) closely matches the simulationresults when N ≥ 6 while being a bit loose when N < 6. Theaccuracy of the lower bound obtained from Jensen’s inequalityimplies that after canceling 2 nearest D2D interferers, thevariance of the remaining interference is relatively small. Fig.4 also shows that D2D spectral efficiency is quite sensitive toits communication range: there is a loss of about 3 bps/Hz inspectral efficiency if D2D range is increased from 20 m to 35m.

Next we use the analytical lower bound (20) to evaluate theeffect of multi-user cellular transmission on D2D spectral effi-ciency (averaged over all D2D pairs in the cell). Fig. 5 showsthe D2D spectral efficiency vs. the number K of co-channelcellular UEs. The cellular UEs are uniformly distributed in the

4 6 8 10 12 140

1

2

3

4

5

6

7

8

N: # of UE Rx Antennas

Sp

ectr

al E

ffic

ien

cy (

bit/s

/Hz)

Lower bound: d=20 mSimulation: d=20 mLower bound: d=25 mSimulation: d=25 mLower bound: d=30 mSimulation: d=30 mLower bound: d=35 mSimulation: d=35 m

Fig. 4. Simulated D2D spectral efficiency vs. analytical lower bound (20)with perfect CSI and (nc, nd) = (0, 2).

10 20 30 40 50 60 70 800

1

2

3

4

5

6

K: # of Cellular UEs/Cell

Sp

ectr

al E

ffic

ien

cy (

bit/s

/Hz)

d=20 m, N=4d=35 m, N=4d=50 m, N=4d=20 m, N=6d=35 m, N=6d=50 m, N=6

Fig. 5. Effect of multi-user cellular transmission on D2D spectral efficiencywith perfect CSI and (nc, nd) = (0, 2)

coverage region of the BS. Not surprisingly, as K increases,D2D spectral efficiency decreases due to the increased cellular-to-D2D interference. The interesting observation from Fig. 5 isthat D2D spectral efficiency is not severely affected by scalingup the number of cellular UEs. For example, when K increasesfrom 10 to 20, the loss in D2D spectral efficiency is less than0.5 bps/Hz. This implies that we may be able to scale up theuplink capacity in a massive MIMO system without noticeablydegraded D2D performance.

Fig. 6 illustrates the cellular spectral efficiency with scaledcellular transmit power (i.e., Pc → Pc/

√M ) and imperfect

CSI. As in Fig. 3, Fig. 6 shows that adopting a constant mdresults in a fixed loss in the cellular spectral efficiency due tothe underlaid D2D interference. Unlike the case of perfect CSI,the loss cannot be overcome even by scaling md at a sublinearrate Θ(

√M) (which comes at the cost of a very large overhead

in the training). Note that compared to Fig. 3, the convergencerate in Fig. 6 is much slower as the cellular transmit power isscaled down at a rate Θ(1/

√M) (vs. Θ(1/M) in Fig. 3).

VI. CONCLUSIONS

In this paper, we have studied the spectral efficiency ofa D2D underlaid massive MIMO system under perfect and

10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

1

2

3

4

5

6

M: # of BS Antennas

Sp

ectr

al E

ffic

ien

cy (

bit/s

/Hz)

mc=0,No D2D

(mc,m

d)=(0,2)

(mc,m

d)=(0,M

1/2)

mc=3,No D2D

(mc,m

d)=(3,2)

(mc,m

d)=(3,M

1/2)

Fig. 6. Cellular spectral efficiency with scaled cellular transmit power andimperfect CSI.

imperfect CSI. We have found that massive MIMO can ef-ficiently handle the D2D-to-cellular interference. Meanwhile,from an average perspective, D2D links are relatively robustto the cellular-to-D2D interference even if there are quitemany cochannel cellular users. D2D interference does makethe estimated CSI in massive MIMO less accurate and thus inturn hurts the cellular spectral efficiency. One simple approachto alleviating this effect is to deactivate D2D links in thecellular training phase. Overall, our study suggests that D2Dmay be much simpler in massive MIMO cellular systems thanin current cellular systems.

Spectral efficiency is used as the sole metric throughoutthis paper. Future work may carry out throughput analysisby taking into account the overhead cost in channel training,scheduling and possibly retransmissions. Also, it is of interestto consider other more sophisticated receivers like MMSEreceivers and linear receivers with successive interferencecancellation and compare their system-level performance withthat of PZF receivers studied in this paper.

APPENDIX

A. Proof of Proposition 1

We show that a PZF receiver with mc = md = 0, i.e.,MRC receiver, at the BS suffices. With mc = md = 0, thePZF receiver w

(c)k = h

(c)k . By the law of large numbers,

1M ‖h

(c)k ‖2

a.s.−−→ 1, 1M h

(c)∗k h

(c)`

a.s.−−→ 0, ` 6= k. It follows that

1

M2Pc‖x(c)

k ‖−αc‖h(c)

k ‖4 a.s.−−→ Pc‖x(c)

k ‖−αc ,

1

M2N0‖h(c)

k ‖2 a.s.−−→ 0. (35)

Further, interchanging the order of the limit and the finite sum,

limM→∞

1

M2

∑` 6=k

Pc‖x(c)` ‖−αc |h(c)∗

k h(c)` |

2

=∑` 6=k

Pc‖x(c)` ‖−αc

(limM→∞

1

M2|h(c)∗k h

(c)` |

2

)a.s.−−→ 0. (36)

As for limM→∞1M2 I

(d→c)k , we cannot directly interchange

the order of the limit and the infinite sum to conclude that

it converges to 0 almost surely. Instead, we can prove itsconvergence in probability, i.e., for any ε > 0,

limM→∞

P

(1

M2

∑i∈Φ

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 < ε

)= 1. (37)

To this end, we partition the D2D transmitters into two groups:one group is composed of those transmitters located withindistance ro from the BS and the other group is composed ofthose transmitters located with distance grater than ro fromthe BS. Then using the inequalities

P(X + Y ≥ ε) ≤ P(X ≥ ε

2or Y ≥ ε

2

)≤ P

(X ≥ ε

2

)+ P

(Y ≥ ε

2

), (38)

where X and Y are two arbitrary random variables, we have

P

(1

M2

∑i∈Φ

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

)≤

P

1

M2

∑i∈Φ∩Bc(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

+

P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

. (39)

Next we show in two steps that the two terms on the righthand side of (39) can be made arbitrarily small by choosingM large enough.

Step 1. For the first term on the right hand side of (39), wehave

P

1

M2

∑i∈Φ∩Bc(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

≤

E[

1M2

∑i∈Φ∩Bc(o,ro) Pd‖x(d)

i ‖−αc |h(c)∗k h

(d)i |2

]ε/2

(40)

=2

εME[

∑i∈Φ∩Bc(o,ro)

Pd‖x(d)i ‖−αc ] (41)

=2λPd

εM

∫Bc(o,ro)

‖x‖−αcdx (42)

=4πλPd

εM

∫ ∞ro

r1−αcdr

=4πλPd

εM

1

(αc − 2)rαc−2o

, (43)

where (40) is due to the Markov inequality, (41) is due toE[|h(c)∗

k h(d)i |2] = M , and (42) is due to Campbell’s formula

[28], and we use our assumption that αc > 2 in (43). It followsthat that there exists M1 large enough such that for all M ≥M1,

P

1

M2

∑i∈Φ∩Bc(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

<δ

2, (44)

where δ is an arbitrary small positive constant.

11

Step 2. For the second term on the right hand side of (39),

P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

= P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

∣∣EP (E)

+ P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

∣∣EcP (Ec) .

where E = {|Φ ∩ B(o, ro)| ≤ C} and Ec is the complementof E.

Step 2(a). Note that the number of D2D transmitters inB(o, ro), denoted as |Φ∩B(o, ro)|, is Poisson distributed withmean λπr2

o . We can choose C large enough such that

P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

∣∣EcP (Ec)

≤ P (Ec) = 1−C∑n=0

(λπr2o)n

n!e−λπr

2o <

δ

4. (45)

Step 2(b). Since 1M h

(c)∗k h

(d)i

a.s.−−→ 0, conditioning on |Φ ∩B(o, ro)| ≤ C, we have

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 a.s.−−→ 0.

It follows that there exists M2 large enough such that for allM ≥M2,

P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

∣∣EP (E)

≤ P

1

M2

∑i∈Φ∩B(o,ro)

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

2

∣∣E

<δ

4. (46)

Combining (44), (45) and (46) obtained in Steps 1, 2(a) and2(b) respectively, we have for all M ≥ max{M1,M2},

P

(1

M2

∑i∈Φ

Pd‖x(d)i ‖−αc |h(c)∗

k h(d)i |

2 ≥ ε

)≤ δ

2+δ

4+δ

4= δ.

As δ is an arbitrary positive constant, we conclude that (37)holds. This completes the proof.

B. Proof of Proposition 2

When the transmit powers of cellular UEs scale as Pc/M ,as in the proof of Prop. 1, we can show that as M → ∞,the desired signal power S(c)

k , the cellular interference powerI

(c→c)k , and the noise power ‖w(c)

k ‖2N0 normalized by M

converge as follows.

limM→∞

1

MS

(c)k

a.s.−−→ Pc‖x(c)k ‖−αc

limM→∞

1

MI

(c→c)k

a.s.−−→ 0

limM→∞

1

M‖w(c)

k ‖2N0

a.s.−−→ N0. (47)

Next we show that 1M I

(d→c)k

d.−→∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi,

where d.−→ denotes the convergence in distribution. To this end,we show that the Laplace transform of the former convergesto that of the latter as follows. As the BS uses md degreesof freedom to cancel the interference from the md nearestD2D transmitters when detecting the signal of cellular UE k,Φ

(c)k consists of the points from the original PPP Φ except the

nearest md points to the origin. Let us order the points in Φbased on their distances to the BS in an ascending manner, i.e.,‖x(d)

1 ‖ ≤ ‖x(d)2 ‖ ≤ .... Conditioning on the location x

(d)md =

(r, θ) of the md-th nearest point in Φ, Φ(c)k = {x(d)

i }i>md aredistributed as a PPP of density λ outside the ball B(o, r) bythe independence property of PPP [28]. Then

limM→∞

E[exp

(−s 1

MI

(d→c)k

)|x(d)md

]

= limM→∞

E

exp

−s 1

M

∑i∈Φ

(c)k

Pd‖x(d)i ‖−αc |w(c)∗

k h(d)i |

2

∣∣∣∣x(d)md

= limM→∞

exp

(− λ

∫R2\B(o,r)

(1−

E[exp

(−s 1

MPd‖x‖−αc |w(c)∗h(d)|2

)])dx

)= exp

(− λ

∫R2\B(o,r)

(1−

E[

limM→∞

exp

(−s 1

MPd‖x‖−αc |w(c)∗h(d)|2

)])dx

)= exp

(− λ

∫R2\B(o,r)

(1−

E[exp

(−sPd‖x‖−αc

∣∣ limM→∞

1√M

w(c)∗h(d)∣∣2)])dx

)= exp

(−λ∫R2\B(o,r)

(1− E

[exp(−sPd‖x‖−αcη)

])dx

)

= E

exp

−s ∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi

∣∣∣∣x(d)md

, (48)

where the second equality is due to the Laplace transform ofthe PPP Φ

(c)k [28], the interchange of limit and expectation

in the third equality is justified by Lebesgue’s DominatedConvergence Theorem, the fifth equality is due to Cen-tral Limit Theorem: 1√

Mw(c)∗h(d) d.−→ CN (0, 1), and thus

1M |w

(c)∗h(d)|2 d.−→ Exp(1)d.= η. Here X

d.= Y denotes that

the two random variables X and Y are equal in distribution.Deconditioning on x(d)

md = (r, θ), we can see the Laplace trans-

12

form of 1M I

(d→c)k converges to that of

∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi.

Therefore, the spectral efficiency of cellular UE k convergesas in (10).

The lower bound (11) is due to Jensen’s inequality:

E

log

1 +Pc‖x(c)

k ‖−αc∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi +N0

≥ log

1 +Pc‖x(c)

k ‖−αc

E[∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi] +N0

. (49)

With a slight abuse of notation, we also denote by I(d→c)k the

asymptotic interference power∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcηi. Then

E[I(d→c)k ] = EΦ[

∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcE[ηi]]

= EΦ[∑i∈Φ

(c)k

Pd‖x(d)i ‖−αc ] = EΦ[

∞∑i=md+1

Pd‖x(d)i ‖−αc ]. (50)

Conditioning on the location x(d)md = (r, θ) of the md-th nearest

point in Φ,

E[I(d→c)k |x(d)

md] = EΦ

[ ∞∑i=md+1

Pd‖x(d)i ‖−αc∣∣x(d)md

]

= Pd2πλ

∫ ∞r

t1−αcdt =Pd2πλ

αc − 2r2−αc , (51)

where the second equality is due to Campbell formula [28].To decondition on x

(d)md = (r, θ), we need the PDF of ‖x(d)

md‖derived in [29]:

f‖x(d)md‖

(r) =2(λπr2)md

r(md − 1)!e−λπr

2

, r ≥ 0. (52)

Using the fact that x(d)md is uniform in direction and

f‖x(d)md‖

(r), we can now decondition on x(d)md in (51) and obtain

E[I(d→c)k ] =

Pd2πλ

αc − 2

∫ ∞0

r2−αcf‖x(d)md‖

(r)dr

=Pd2πλ

αc − 2· 1

(md − 1)!(λπ)

αc2 −1

∫ ∞0

tmd−αc2 e−tdt, (53)

where we have changed variable t = λπr2 in (53). By thedefinition of the Gamma function,

E[I(d→c)k ] =

2Pd

αc − 2(πλ)

αc2

Γ(md + 1− αc2 )

Γ(md), (54)

Plugging (54) into (49) yields the desired lower bound (11).

C. Proof of Proposition 4

Using the convexity of the function log(1+ 1x ) and applying

Jensen’s inequality [7],

R(c)k ≥ R

(c,lb)k = log

1 +

(E

[1

SINR(c)k

])−1 , (55)

where

E

[1

SINR(c)k

]= E

[1

S(c)k

](E[I

(c→c)k ] + E[I

(d→c)k ] +N0. (56)

In the following three steps, we calculate E[

1

S(c)k

], E[I

(c→c)k ],

and E[I(d→c)k ], respectively. Without loss of generality, we

assume that w(c)k is normalized, i.e., ‖w(c)

k ‖ = 1.

Step 1: calculating E[

1

S(c)k

]. By definition ‖w(c)∗

k h(c)k ‖2 is

the squared norm of the projection of the vector h(c)k onto

the subspace orthogonal to the one spanned by the columnvectors of the matrix

[H(c)\h(c)

k (1:mc), H(d)(1:md)

]. The

space is of M − mc − md dimensions and is independentof h

(c)k . It follows that ‖w(c)∗

k h(c)k ‖2 ∼ χ2

2(M−mc−md), i.e.,

‖w(c)∗k h

(c)k ‖2 ∼ Γ(M−mc−md, 1). Therefore, 1

S(c)k

is inverse-Gamma distributed and its mean equals

E

[1

S(c)k

]=

1

Pc‖x(c)k ‖−αc(M −mc −md − 1)

. (57)

Step 2: calculating E[I(c→c)k ]. Since ‖w(c)

k ‖ = 1 and w(c)k

is independent of h(c)` ,∀` ∈ K(c)

k , w(c)∗k h

(c)` is a linear

combination of complex Gaussian random variables and thusis complex Gaussian distributed as CN (0, 1). It follows that|w(c)∗

k h(c)` |2 ∼ Exp(1) and

E[I(c→c)k ] = E[

∑`∈K(c)

k

Pc‖x(c)` ‖−αc |w(c)∗

k h(c)` |

2]

=∑`∈K(c)

k

Pc‖x(c)` ‖−αc . (58)

Step 3: calculating E[I(d→c)k ]. With a similar argument as

in Step 2, we have |w(c)∗k h

(d)i |2 ∼ Exp(1) and

E[I(d→c)k ] = EΦ[

∑i∈Φ

(c)k

Pd‖x(d)i ‖−αcEh[|w(c)∗

k h(d)i |

2]]

= EΦ[∑i∈Φ

(c)k

Pd‖x(d)i ‖−αc ]. (59)

The remaining steps for calculating E[I(d→c)k ] follow the same

steps in the proof of Prop. 2, i.e., the steps after (50), andE[I

(d→c)k ] is given in (54).

Finally, plugging (57), (58) and (54) into (55) completes theproof.

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