The Interplay between Massive MIMO and Underlaid D2D Networking

# The Interplay between Massive MIMO and Underlaid D2D Networking

Xingqin Lin, Robert W. Heath Jr., and Jeffrey G. Andrews are with Department of Electrical Computer Engineering, The University of Texas at Austin, USA. (Email: {xlin, rheath}@utexas.edu, jandrews@ece.utexas.edu). Xingqin Lin, Robert W. Heath Jr., and Jeffrey G. Andrews
###### Abstract

In a device-to-device (D2D) underlaid cellular network, the uplink spectrum is reused by the D2D transmissions, causing mutual interference with the ongoing cellular transmissions. Massive MIMO is appealing in such a context as the base station’s (BS’s) large antenna array can nearly null the D2D-to-BS interference. The multi-user transmission in massive MIMO, however, may lead to increased cellular-to-D2D interference. This paper studies the interplay between massive MIMO and underlaid D2D networking in a multi-cell setting. We investigate cellular and D2D spectral efficiencies under both perfect and imperfect channel state information (CSI) at the receivers that employ partial zero-forcing. Compared to the case without D2D, there is a loss in cellular spectral efficiency due to D2D underlay. With perfect CSI, the loss can be completely overcome if the number of canceled D2D interfering signals is scaled with the number of BS antennas at an arbitrarily slow rate. With imperfect CSI, in addition to pilot contamination, a new asymptotic effect termed underlay contamination arises. In the non-asymptotic regime, simple analytical lower bounds are derived for both the cellular and D2D spectral efficiencies.

## I Introduction

### I-a Background

Device-to-device (D2D) communication enables nearby mobile devices to establish direct links in cellular networks [1, 2, 3], unlike traditional cellular communication where all traffic is routed via base stations (BSs). D2D has the potential to improve spectrum utilization, shorten packet delay, and reduce energy consumption, while enabling new peer-to-peer and location-based applications and services [4, 3] and being a required feature in public safety networks [5]. Introducing D2D poses many challenges and risks to the existing cellular architecture. In particular, in a D2D underlaid cellular network where the spectrum is reused D2D transmission may cause interference to cellular transmission and vice versa. Existing operator services may be severely affected if the newly introduced D2D interference is not appropriately controlled.

The distinctive traits of massive MIMO make it appealing to enable D2D communication in the uplink resources of cellular networks. In a massive MIMO system, each BS uses a very large antenna array to serve multiple users in each time-frequency resource block [6]. If the number of antennas at a BS is significantly larger than the number of served users, the channel of each user to/from the BS is nearly orthogonal to that of any other user. This allows for very simple transmit or receive processing techniques like matched filtering to be nearly optimal with enough antennas even in the presence of interference [6, 7, 8, 9, 10]. This implies that, with a large antenna array at a BS, D2D signals possibly result in close-to-zero interference at the uplink massive MIMO BS, making D2D very simple and appealing in massive MIMO systems.

Though D2D-to-cellular interference may be effectively handled by the large antenna array at a BS, cellular-to-D2D interference persists and may be worse in a massive MIMO system. Specifically, massive MIMO is a multi-user transmission strategy designed to support multiple users in each time-frequency block; the number of simultaneously active uplink users is scalable with the number of antennas at the BS. With this increased number of uplink transmitters, the D2D links reusing uplink radio resources will experience increased interference. To protect D2D links, the number of simultaneously active uplink users might have to be limited, eating into massive MIMO gain. It is not a priori clear to what extent the D2D signals would be affected by the multiuser transmission and the tradeoff between supporting D2D communication and scaling up the uplink capacity in a massive MIMO system. Further, if cochannel D2D signals are present when estimating massive MIMO channels, the estimated channel state information (CSI) would become less accurate, which may hurt massive MIMO performance. It is not a priori clear however to what extent the D2D signals would affect the channel estimation and consequently the performance of the massive MIMO system.

Existing research on D2D networking is mainly focused on single-antenna networks (see, e.g., [11, 12, 13, 14, 15, 16]) while research on the use of antenna arrays has just begun [17, 18, 19, 20, 21]. To mitigate or avoid mutual interference between cellular and D2D transmissions, [17, 18] considered precoding while [19, 20] studied various relaying strategies. In contrast, [21] proposed not to schedule uplink users that may generate excessive interference to D2D users. How D2D MIMO and cellular MIMO interact, especially in the massive MIMO context, is still largely open.

### I-B Contributions and Outcomes

The main contributions and outcomes of this paper are summarized as follows.

#### I-B1 A tractable hybrid network model

We introduce a tractable hybrid network model consisting of both ad hoc nodes and cellular infrastructure, which extends our previous single-antenna D2D model [16, 14] to multi-antenna transmission. We consider a multi-cell setting and focus on the uplink which is better than the downlink for D2D underlay [3]. The spatial positions of the underlaid D2D transmitters are modeled by a Poisson point process (PPP). Such a random PPP model is well motivated by the random and unpredictable D2D user locations [22, 23]. All the transmissions (both cellular and D2D) in this model are SIMO (i.e., single-input multiple-output) with each 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 hybrid network in question. Spectral efficiency is used as the sole metric throughout this paper.

#### I-B2 Spectral efficiency with perfect CSI

In the asymptotic regime where the number of BS antennas and with perfect CSI, we find that the received signal-to-interference-plus-noise ratio (SINR) of any cellular user increases unboundedly and the effects of noise, fast fading, and the interfering signals from the other co-channel cellular users and the infinite D2D transmitters vanish completely. Equivalently, it is possible to reduce cellular transmit power as but still achieve a non-vanishing cellular spectral efficiency, as in the case without D2D underlay [7]. Compared to the case without D2D, with scaled cellular transmit power , there is a loss in cellular spectral efficiency if a constant number of D2D interfering signals is canceled. The loss can be overcome if the number of canceled D2D interfering signals is scaled appropriately (e.g., ). In the non-asymptotic regime, we derive simple analytical lower bounds for both cellular and D2D spectral efficiencies; the derived bounds allow for very efficient numerical evaluation.

#### I-B3 Spectral efficiency with imperfect CSI

We study pilot-based CSI estimation in which known training sequences are transmitted and the receivers use minimum mean squared error (MMSE) estimators for channel estimation. In the asymptotic regime with the estimated CSI, it is known that the received SINR of any cellular user is bounded due to pilot contamination [6]. With D2D underlay, the bounded SINR is further degraded due to a new asymptotic effect which we term underlay contamination. Due to the underlay contamination, we find that scaling down cellular transmit power results in a vanishing cellular spectral efficiency, no matter how slow the scaling rate is. This is dramatically different from the case without D2D underlay, for which [7] shows that cellular transmit power can be scaled down as . To recover the power scaling law , one possible approach is to deactivate the D2D links in the training phase of massive MIMO; however, compared to the case without D2D, there is a loss in cellular spectral efficiency due to D2D-to-cellular interference in the data transmission phase. Instead, if the cellular transmit power is not scaled down and D2D links are deactivated in the training phase, massive MIMO automatically eliminates the effect of D2D-to-cellular interference in the data transmission phase.

## Ii Mathematical Models

### Ii-a Network Model

Consider a multi-cell D2D underlaid massive MIMO system shown in Fig. 1. In this system, there are cells; in each cell , cellular user equipments (UEs) transmit to the BS . We denote by the set of the cellular UEs in the cell , and the coverage area of the cell satisfying that . We assume that the cellular UEs are uniformly distributed in each cell; this assumption is not essential in the analysis but will be used in the simulation. Specifically, as spatial division multiple access would be challenging for cellular UEs of high mobility, it makes more sense to consider static or low-mobility scenarios. Therefore, we condition on cellular UE positions when studying the achievable performance of a particular cellular link. But we still average over all possible realizations of cellular UE positions in the simulation to compute an average overall performance.

The cellular system is underlaid with D2D UEs. The locations of D2D transmitters are distributed as a homogeneous PPP with density , as they are random and unpredictable. We partition into disjoint PPPs , where Each D2D receiver is located at a random distance of meters from its associated D2D transmitter with uniformly distributed direction.

We focus on SIMO in this paper, i.e., a transmitter (either cellular or D2D) uses one antenna for transmission, while a BS and a D2D receiver respectively use and antennas for receiving. The analysis and results in this paper can be extended to MIMO transmission, i.e., spatial multiplexing, by treating a UE with multiple data streams as multiple co-located virtual UEs, each sending one data stream. We are interested in the performance regime where is large and the assumption is made throughout this paper, as in the seminal work on massive MIMO [6]. Note that the scenario where the ratio converges to some constant has also been widely assumed when studying the asymptotic behaviors of MIMO performance [25, 26]. Studying this scenario is an interesting topic, which we leave to future work.

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 constant powers and respectively.

### Ii-B Baseband Channel Models

Without loss of generality, we focus on the central cell, whose BS is indexed by and located at the origin. This helps simplify the notation. The dimensional baseband received signal at the central BS is

 y(c)0=B∑b=0∑k∈Kb√PcΞ(c)bk∥x(c)bk∥−αc2h(c)bku(c)bk+∑i∈Φ√PdΞ(d)i∥x(%d)i∥−αc2h(d)iu(d)i+v(c)0, (1)

where denotes the shadowing from cellular transmitter in the cell to the BS , denotes the position of cellular transmitter in the cell , denotes the pathloss exponent of UE-BS links, is the vector channel from cellular transmitter in the cell to the BS , denotes the zero-mean unit-variance transmit symbol of cellular transmitter in the cell , and are similarly defined for D2D transmitter , and is complex Gaussian noise at the BS with covariance , where denotes the dimensional identity matrix.

Similarly, the dimensional baseband received signal at D2D receiver is

 y(d)r=B∑b=0∑k∈Kb√PcΞ(c)rbk(d(c)rbk)−αd2g(c)rbku(c)bk+∑i∈Φ√PdΞ(d)ri(d(d)ri)−αd2g(% d)riu(d)i+v(d)r, (2)

where are the shadowing from cellular transmitter in the cell to D2D receiver and from D2D transmitter to D2D receiver respectively, and with denoting the position of D2D receiver , denotes the pathloss exponent of UE-UE links, are the vector channels from cellular transmitter in the cell to D2D receiver and from D2D transmitter to D2D receiver respectively, and is complex Gaussian noise with covariance .

Note that we have used different pathloss exponents and for UE-BS and UE-UE links (cf. (1) and (2)) due to their different propagation characteristics. Specifically, the antenna height of a macro BS is tens of meters, while the typical antenna height at a UE is under 2 m. As a result, both terminals of a UE-UE link are low and see similar near street scattering environment, which is different from the radio environment around a macro BS [3].

In this paper, we assume Gaussian signaling, i.e., are i.i.d. complex Gaussian , and i.i.d. shadowing with mean . We also assume that all the vector channels have i.i.d. elements, independent across transmitters. It follows that the favorable propagation condition [27] desired in massive MIMO systems holds in our model:

 1Mh(s)∗brh(s′)b′ℓa.s.−−→{1if s=s′, b=b′ and r=ℓ;0otherwise,

where , denotes the almost sure convergence as , and when the first subindex in should be understood as null. Recent measurement campaigns have given evidence to validate favorable propagation for massive MIMO in practice [28].

### Ii-C Receive Filters

Denote by the filter used by the central BS for receiving the signal of cellular transmitter in the central cell, i.e., the central BS detects the symbol based on . Similarly, D2D receiver detects the symbol based on , where denotes the filter used by D2D receiver . The performance of the D2D underlaid massive MIMO system depends on the receive filters. In general, either the receive filters can be designed to boost desired signal power or they can be used to cancel undesired interference. In this paper, we focus on a particular type of linear filters: the PZF receiver, which uses a subset of the degrees of freedom for boosting received signal power and the remainder for interference cancellation.

The central BS uses and degrees of freedom to cancel the interference from the nearest cellular interferers and the nearest D2D interferers. A feasible choice of needs to be in the following set:

 Zc={(mc,md)∈N×N:mc≤(B+1)K−1,mc+md≤M−1}. (3)

The PZF filter is the projection of the channel vector onto the subspace orthogonal to the one spanned by the channel vectors of canceled interferers. For ease of reference, we denote by the set of uncanceled cellular interferers in the cell and the set of uncanceled D2D interferers when detecting the symbol of cellular transmitter in the central cell.

Similarly, each D2D receiver uses and degrees of freedom to cancel the interference from the nearest cellular interferers and the nearest D2D interferers, and needs to be in the following set:

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

The PZF filter of D2D receiver is the projection of the channel vector onto the subspace orthogonal to the one spanned by the channel vectors of canceled interferers. For ease of reference, we denote by the set of uncanceled cellular interferers in the cell and the set of uncanceled D2D interferers at D2D receiver .

## Iii Spectral Efficiency with Perfect Channel State Information

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

### Iii-a Asymptotic Cellular Spectral Efficiency

For cellular UE in the central cell, the post-processing SINR with the PZF filter is

 SINR(c)k=S(c)kI(c→c)k+I(d→c)k+∥w(c)k∥2N0, (5)

where denotes the desired signal power of cellular UE , and respectively denote the cochannel cellular and D2D interference powers experienced by cellular UE and are given by

 I(c→c)k =B∑b=0∑ℓ∈K(c)bkPcΞ(c)bℓ∥x(c)bℓ∥−αc|w(c)∗kh(c)bℓ|2 (6) I(d→c)k =∑i∈Φ(c)kPdΞ(d)i∥x(d)i∥−αc|w(c)∗kh(d)i|2. (7)

The spectral efficiency of cellular UE in the central cell is defined as

 R(c)k=E[log(1+SINR(c)k)], (8)

where the expectation is taken with respect to the fast fading, shadowing and random locations of UEs.

###### Proposition 1.

With perfect CSI and conditioned on , as , the desired signal power when normalized by and conditioned on converges to

 limM→∞1M2S(c)ka.s.−−→PcΞ(c)0k∥x(c)0k∥−αc, (9)

and the cellular interference power , the D2D interference power , and the noise power when normalized by converge as follows.

 limM→∞1M2I(c→c)ka.s.−−→0,limM→∞1M2I(d→c)kp.→0,limM→∞1M2∥w(c)k∥2N0a.s.−−→0, (10)

where denotes the convergence in probability.

###### Proof.

See Appendix -A. ∎

Prop. 1 shows that with perfect CSI, as , the post-processing increases unboundedly in probability (as almost sure convergence implies convergence in probability). More specifically, a deterministic received power of the desired signal from cellular UE (conditioned on its pathloss and shadowing) can be achieved and the effects of noise, fast fading, and the interfering signals from the other cellular UEs and the infinite D2D transmitters vanish completely. Therefore, Prop. 1 validates the intuition that with perfect CSI D2D-to-cellular interference can be made arbitrarily small with a large enough antenna array at the BS. Note that the D2D-to-cellular interference can be completely nulled out, even though (i) the number of the PPP distributed D2D interferers is infinite and (ii) the mean of the aggregate D2D interference is infinite. Further, the proof of Prop. 1 shows that a simple MRC filter with suffices.

Though Prop. 1 shows that arbitrarily large received SINR can be achieved with massive MIMO, this in practice will ultimately fail since the received power cannot be larger than the transmit power. Further, it may not be possible to fully exploit a very high SINR due to practical constraints such as the highest order of modulation and coding schemes. Nevertheless, the large array gains may be translated into power savings for cellular UEs: with a given SNR target we can lower the transmit powers of cellular UEs and thus improve their energy efficiency, as shown in the following proposition.

###### Proposition 2.

With perfect CSI, fixed PZF parameters , scaled cellular transmit power , and conditioned on and , as , the spectral efficiency of cellular UE in the central cell converges to

 R(c)k →EΦ,η⎡⎢ ⎢ ⎢⎣log⎛⎜ ⎜ ⎜⎝1+SNR(c)0k∑i∈Φ(c)kPdN0∥x(d)i∥−αcηi+1⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥⎦, (11)

where , are i.i.d. random variables distributed as . Further, if ,

 limM→∞R(c)k ≥log⎛⎝1+SNR(c)0kρ(md,αc)+1⎞⎠, (12)

where

 ρ(m,α)=2(πλ)α2Pd% ¯ΞΓ(m+1−α2)(α−2)N0Γ(m), (13)

where the Gamma function .

###### Proof.

See Appendix -B. ∎

Note that in Prop. 2, if the underlaid D2D transmitters did not exist, the spectral efficiency of cellular UE (conditioned on its pathloss and shadowing) in the central cell would converge to , the maximum achievable spectral efficiency of a point-to-point SISO (single-input single-output) Gaussian channel. It is as if massive MIMO could simultaneously support interference-free SISO links while reducing the power of each cellular UE by dB. This result is consistent with Prop. 1 in [7] without D2D underlay.

With D2D underlay, the asymptotic result (11) shows that there is a loss in cellular spectral efficiency due to the uncanceled interfering signals from the D2D transmitters in , i.e., D2D transmitters in except the nearest ones whose signals are canceled by the PZF filter. Though it is possible to derive an exact analytical expression (involving integrals) for (11), we give a more intuitive lower bound (12), which succinctly characterizes the loss due to the D2D underlay through a single term . Several remarks are in order.

Remark 1. The term corresponding to the uncanceled D2D interference increases with and and decreases with , agreeing with intuition: larger transmit power or larger density of D2D interferers or smaller number of canceled D2D interferers leads to higher D2D-to-cellular interference, thus lowering the cellular spectral efficiency. Further, , because a linear increase in implies that the distances of the PPP distributed D2D transmitters to the BS decrease as and thus the D2D-to-cellular interference power increases as .

Remark 2. Note that the lower bound (12) is meaningful only if . As , and thus . In fact, from the proof of Prop. 2, we can see that denotes the mean residual D2D-to-cellular interference after canceling the nearest D2D intererers. If , the mean residual D2D-to-cellular interference is infinite, and the lower bound (12) becomes , which is trivially true.

Next we show that the loss of cellular spectral efficiency due to D2D underlay can be recovered if we scale the number of canceled D2D interferers to infinity as . Further, the growth rate of can be arbitrarily slow.

###### Proposition 3.

With perfect CSI, arbitrary but fixed , scaled cellular transmit power , and conditioned on and , if and , i.e., increases to infinity at a rate slower than , the spectral efficiency of cellular UE in the central cell converges as follows.

 R(c)k →log(1+SNR(c)0k), as M→∞. (14)
###### Proof.

According to Stirling’s formula, , when is large. It follows that

 Γ(md+1−αc2)Γ(md) ∼√2π(md−αc2)(md−αc2e)md−αc2√2π(md−1)(m%d−1e)md−1 =⎛⎜⎝emd−αc2⎞⎟⎠αc2−1⎛⎜⎝md−αc2md−1⎞⎟⎠md−12∼⎛⎜⎝1md−αc2⎞⎟⎠αc2−1. (15)

Therefore, as , and thus

 log(1+SNR(c)0k)≥limM→∞R(c)k ≥log⎛⎝1+SNR(c)0kρ(md,αc)+1⎞⎠→log(1+SNR(c)0k). (16)

This completes the proof. ∎

Before ending this subsection, we would like to point out that the power scaling law does not imply that cellular UEs can transmit at a vanishing power level as increases to infinity. A more appropriate understanding is that large power savings are possible when is large, and there exists a limiting cellular spectral efficiency. For example, numerical results (e.g., Fig. 3) show that when is on the order of several hundred, the limiting cellular spectral efficiency is reached and the power savings are about 20 - 30 dB. Increasing further theoretically leads to more power savings, but the conclusion becomes fragile since non-ideal effects like spurious emission may not be negligible when becomes too small. The power scaling law for imperfect CSI case studied in Section IV should also be understood in a similar fashion.

### Iii-B Non-asymptotic Cellular Spectral Efficiency

Next we analyze the cellular spectral efficiency in the non-asymptotic regime to generate more insights into the impact of the various system parameters. To this end, using Jensen’s inequality we derive a lower bound for in the following proposition.

###### Proposition 4.

With perfect CSI, and , and conditioned on and , the spectral efficiency of cellular UE in the central cell is lower bounded as

 R(c)k≥R(c,lb)k=log⎛⎜ ⎜⎝1+(M−mc−md−1)SNR(c)0k∑Bb=0∑ℓ∈K(c)bkSNR(c)bℓ+ρ(md,αc)+1⎞⎟ ⎟⎠, (17)

where is defined in (13).

###### Proof.

See Appendix -C. ∎

Note that the first term in the denominator of (17) corresponds to the uncanceled cellular interference; it decreases as increases. Similarly, the second term in the denominator of (17) corresponds to the uncanceled D2D interference; it decreases as increases. In contrast, the numerator of (17) corresponds to the desired signal power; it decreases as and/or increase. The lower bound (17) demonstrates the various tradeoffs when choosing the PZF parameters and . Note that such tradeoffs disappear in the asymptotic regime (cf. Prop. 2 and 3).

We point out that the received signal power gain is only proportional to in the lower bound (17). One might think the power gain should be proportional to , the number of degrees of freedom left for power boosting after using degrees of freedom for interference cancellation. The fallacy of the above argument is that it ignores the effect of fading, which makes a power gain proportional to unachievable.

We may optimize the PZF filter by choosing () such that they maximize the sum spectral efficiency in the central cell, i.e.,

 (m⋆c,m⋆d)=argmax(mc,md)∈ZcK∑k=1R(c)k. (18)

This is a combinatorial optimization, and finding the global optimum involves exhaustive search over the feasible space . In practice, each BS only cancels intra-cell cellular interference, leading to . Further, existing studies (see, e.g., [24]) show that canceling a few Poisson distributed interferers provides close-to-optimal performance. This implies that it suffices to consider a few small values for . These two facts greatly reduce the search space for (). A demonstrative numerical result is given in Fig. 6 in Section V.

### Iii-C D2D Spectral Efficiency

For D2D receiver , the post-processing SINR with the PZF filter is

 SINR(d)r=S(d)rI(c→d)r+I(d→d)r+∥w(d)r∥2N0, (19)

where denotes the desired signal power of D2D Tx-Rx pair , and respectively denote the cochannel cellular and D2D interference powers experienced by D2D receiver and are given by

 I(c→d)r =B∑b=0∑k∈K(d)brPcΞ(c)rbk(d(c)rbk)−α% d|w(d)∗rg(c)rbk|2 I(d→d)r =∑i∈Φ(d)rPdΞ(d)ri(d(d)ri)−αd|w(d)∗rg(d)ri|2. (20)

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

 R(d)r=E[log(1+SINR(d)r)], (21)

where the expectation is taken with respect to the fast fading, shadowing and random locations of UEs.

As the number of antennas at UEs is often limited due to hardware constraints, it is not very meaningful to study the asymptotic performance with . Instead, as in the case of cellular spectral efficiency, we provide a lower bound for in the non-asymptotic regime, which characterizes the impact of the various system parameters on the D2D spectral efficiency.

###### Proposition 5.

With perfect CSI, and , and conditioned on , , and , the spectral efficiency of D2D Tx-Rx pair is lower bounded as

 R(d)r≥R(d,lb)r=log⎛⎜ ⎜ ⎜⎝1+(N−nc−nd−1)SNR(d)r∑Bb=0∑k∈K(d)brPcN0Ξ(c)rbk(d(c)rbk)−αd+ρ(nd,αd)+1⎞⎟ ⎟ ⎟⎠, (22)

where , and is defined in (13).

###### Proof.

The proof is similar to that of Prop. 4 and is omitted for brevity. ∎

Many of the remarks on Prop. 4 apply to Prop. 5 as well and are not repeated here. One additional remark is that the cellular-to-D2D interference is not homogeneous: the D2D receivers located in the boundary of the cellular network experience less cellular interference than the D2D receivers located in the central cell. But if we focus on the D2D performance in the central cell and choose the number of cellular cells large enough, this heterogeneity can be made negligible.

## Iv Spectral Efficiency with Imperfect Channel State Information

### Iv-a Estimating UE-BS Channels

We consider pilot-based CSI estimation in which known training sequences are transmitted and used for estimation purpose. To alleviate the training overhead and coordination complexity, we assume that each BS does not estimate the channels from other-cell transmitters (either cellular or D2D). Note that as the number of D2D transmitters in the cell is Poisson distributed, there may be less than D2D transmitters in the cell . Therefore, during the training phase, each BS requires the cellular UEs and the nearest D2D transmitters (w.r.t. the BS ) in its cell to simultaneously transmit orthogonal training sequences. The BSs do not coordinate the other D2D transmitters, which can send independent symbols during the training phase.

Unlike the perfect CSI case, other-cell transmissions (both cellular and D2D) now have a more delicate impact on the performance of the central cell. To accommodate this, in this subsection we extend the previous notation as follows. We add an additional subscript to and , and obtain and , indicating that they are associated with D2D transmitter in the cell . Similarly, we use to denote the set of uncanceled D2D interferers in the cell . Note that the coverage of the “cell” is simply the complement (w.r.t. ) of the coverage areas of the cells , and the “cell” does not contain a BS.

Denoting by the length of a training sequence, we can represent the training sequences as a dimensional matrix satisfying . These pilots are reused over different cells. In the training phase, the dimensional baseband received signal at the central BS is

 Y(c)0= B∑b=0∑k∈Kb√TcPcΞ(c)bk∥x(c)bk∥−αc2h(c)bkq(c)∗k+B∑b=0md,b∑i=1√TcPc% Ξ(d)bi∥x(d)bi∥−αc2h(d)biq(c)∗K+i +B+1∑b=0∑r∈Φ(c)bk√PdΞ(d)br∥x(d)br∥−αc2h(d)bru(d)∗br+V(c)0, (23)

where the dimensional vector contains the data symbols sent by D2D interferer in the cell , and the dimensional noise matrix consists of i.i.d. elements. Note that the coordinated D2D transmitters also use power during the training phase since they now transmit to their associated BSs.

We assume that the central BS uses linear MMSE estimator for the channel estimation. To this end, we first project the received signal in the direction of and normalize it to obtain

 ~y(s)k =1√TcPcΞ(s)0k∥x(s)0k∥−αc2Y(c)0q(c)~k =h(s)0k+B∑b=1√β(s)bkh(s)bk+~v(s)k,(s,~k)∈{(c,k),(d,K+k)}, (24)

where

 β(s)bk≜⎧⎪ ⎪⎨⎪ ⎪⎩0if s= d and k>md,b ;Ξ(s)bk∥x(s)bk∥−αcΞ(s)0k∥x(s)0k∥−αcotherwise,

and denotes the equivalent channel estimation “noise” and is given by

 ~v(s)k=1√TcPcΞ(s)0k∥x(s)0k∥−αc2⎛⎜ ⎜⎝B+1∑b=0∑r∈Φ(c)bk√PdΞ(d)br∥x(d)br∥−αc2h(d)br¯u(d)br+¯v(c)k⎞⎟ ⎟⎠. (25)

where and .

###### Lemma 1.

The linear MMSE estimate of , is given by , where

 ξ(s)k =⎛⎜ ⎜⎝1+B∑b=1β(s)bk+∑B+1b=0∑r∈Φ(c)bkPdΞ(d)br∥x(d)br∥−αc+N0TcPcΞ(s)0k∥x(s)0k∥−αc⎞⎟ ⎟⎠−1. (26)

Further, and . As for the estimation error , and .

###### Proof.

See Appendix -D. ∎

Lemma 1 shows that the longer the length of a training sequence, the smaller the covariance of the estimation error and thus the more accurate the channel estimation , agreeing with intuition. In particular, , as . This shows that even with infinitely long training sequences the channel estimation cannot be perfect due to pilot contamination.

### Iv-B Asymptotic Cellular Spectral Efficiency

In this subsection, we examine the asymptotic performance of the cellular links as . For simplicity, we focus on . Then