Massive MIMO-Enabled Full-Duplex Cellular Networks

In this paper, we provide a theoretical framework for the study of massive multiple-input multiple-output (MIMO)-enabled full-duplex (FD) cellular networks in which the self-interference (SI) channels follow the Rician distribution and other channels are Rayleigh distributed. To facilitate bi-directional wireless functionality, we adopt (i) a downlink (DL) linear zero-forcing with self-interference-nulling (ZF-SIN) precoding scheme at the FD base stations (BSs), and (ii) an uplink (UL) self-interference-aware (SIA) fractional power control mechanism at the FD user equipments (UEs). Linear ZF receivers are further utilized for signal detection in the UL. The results indicate that the UL rate bottleneck in the baseline FD single-antenna system can be overcome via exploiting massive MIMO. On the other hand, the findings may be viewed as a reality-check, since we show that, under state-of-the-art system parameters, the spectral efficiency (SE) gain of FD massive MIMO over its half-duplex (HD) counterpart largely depends on the SI cancellation capability of the UEs. In addition, the anticipated two-fold increase in SE is shown to be only achievable with an infinitely large number of antennas.


I. INTRODUCTION
The fifth-generation mobile network (5G) is expected to roll out from 2018 onwards as a remedy for tackling the existing capacity crunch [1]. A key 5G technology is massive multiple-input multiple-output (MIMO), or large scale antenna system (LSAS), where the base stations (BSs) equipped with hundreds of antennas simultaneously communicate with multiple mobile terminals (MTs) [2]. Massive MIMO, via spatial-multiplexing and directing power intently, can greatly outperform the stateof-the-art cellular standards jointly in terms of spectral efficiency (SE) and energy efficiency (EE) [3], [4]. Moreover, under increasingly scarce spectrum, the transcieving of information over the same radio-frequency (RF) resources, i.e. full-duplex (FD) mode [5], has become a topic of interest for 5G and beyond [6], [7]. In theory, FD can double the sum-rate of half-duplex (HD) systems, where orthogonal RF partitioning is typically employed to avoid the over-powering self-interference (SI). In practice, however, the FD over HD SE gain predominantly depends on the SI cancellation capability.
There has been major breakthroughs in SI cancellation using any combination of (i) spatial/angular isolation and (ii) subtraction in digital/analog domains [8]. FD, beyond point-to-point, remains in its infancy. In particular, the introduction of cross-mode interference (CI) between the downlink (DL) and the uplink (UL), as well as SI, significantly increases the complexity for FD cellular setups. Many relevant works have emerged recently, including FD for small-cell (SC) [9], [10], relay (RL) [11]- [13], cloud radio access network (CRAN) [14], and heterogeneous cellular network (HCN) [15]- [17]. A general consensus from early results is that the FD over HD SE gain mostly arises in the DL and that the UL is the main performance bottleneck. The authors in [18] have shown that bi-directional cellular systems with baseline single-input single-output (SISO), achieve double the DL rate at the cost of more than a thousand-fold reduction in the UL rate. A potential strategy for tackling this limitation is to exploit the large degrees of freedom (DoF) in massive MIMO for better resilience against SI and CI [19], [20].
In [21], the authors considered a FD BS with large scale antenna array serving multiple HD single-antenna MTs, and proposed a linear extended zero-forcing (ZF) precoder to suppress SI at the receiving antennas, subject to perfect channel state information (CSI). The sum-rate with FD mode was almost doubled versus that in the HD case, and the optimal ratio of k-th MT, respectively. Their respective Euclidean distance is therefore d l,k = l − k . The BSs are assumed to be equipped with N t transmit and N r receive antennas (N t + N r RF chains), respectively. The MTs are in turn assumed to be equipped with single transmit/receive antennas (two RF chains). In the DL, the BSs simultaneously serve U MTs per resource block using linear transmit precoding. In the UL, the scheduled MTs simultaneously transmit to their serving massive MIMO BS per resource block [31]. Linear receive filters are then used for UL signal detection. We assume the condition U ≤ min(N t , N r ) holds, thus scheduling is not necessary here.
Let g l,k ∈ C 1×Nt , G l,j ∈ C Nr×Nt , and G l,l ∈ C Nr×Nt denote the channel from the l-BS to the k-th MT, the channel from the l-th BS to the j-th BS, and the residual SI channel at the l-th BS, respectively. Moreover, h k,l ∈ C Nr×1 , h k,i , and h k,k are respectively the channel from the k-th MT to the l-th BS, the channel from the k-th MT to the i-th MT, and the residual SI channel at the k-th MT. The residual SI (hereafter, refered to as SI) channels are subject to Rician fading with independent and identically-distributed (i.i.d.) elements drawn from CN (µ, υ 2 ). All other channels are modeled using Rayleigh fading with i.i.d. elements drawn from CN (0, 1). We use the unbounded distance-dependent path-loss model with exponent β (> 2). CSI in time-division duplex (TDD)-based massive MIMO systems can be acquired based on channel reciprocity through UL training. In this work, we assume sophisticated channel estimation algorithms with sufficient training information are used to obtain perfect CSI [32].
By invoking the Campbell-Mecke theorem [33], the DL analysis is carried out for an arbitrary MT o assumed to be located at the center. We consider a cellular association strategy where the reference MT is exclusively served by a BS b which provides the greatest received signal power. For homogeneous deployments, this is equivalent to the cell selection approach based on the closest distance b = arg max d −β l,o , ∀l ∈ Φ (d) [34]. The UL analysis, on the other hand, is carried out for the reference MT at its serving BS. The reference transceiver distance p.d.f. is given by It should be noted that the alternative decoupling approach for cellular association [35] results in the loss of channel reciprocity in massive MIMO systems.

III. SIGNALS DEFINITIONS
Let G l = [g T l,k ] T 1≤k≤U ∈ C U ×Nt denote the combined DL channels from the l-th BS to its U MTs. We use s l = [s l,k ] T 1≤k≤U ∈ C U ×1 , E |s l,k | 2 = 1, to denote the DL complex symbol vector from the l-th BS to its U MTs. Here, we consider the case where each BS equally allocates its total transmit power p (d) among its U MTs. The normalized precoding matrix at the l-th The DL received signal can be represented as ) is the set of scheduled MTs in the cell of BS l, p (u) k,l is the k-th scheduled MT transmit power for sending s k,l to BS l, and η o is the complex additive white Gaussian noise (AWGN) with mean zero and variance σ 2 d , respectively. Next, let H l = [h k,l ] 1≤k≤U ∈ C Nr×U represent the compound UL channel matrix at the l-BS from its U scheduled MTs. The linear receiver filter at the l-th BS is denoted using The corresponding post-processing UL signal can be written as where η b ∈ C Nr×1 is the AWGN vector with mean zero and covariance matrix σ 2 u I Nr . It is important to note that the set of scheduled MTs is strictly not an indpendent process due to the spatial dependencies arising from (i) the cellular association strategy, and (ii) the constraint of each BS serving multiple MTs per resource block [36], [37]. For the sake of mathematical tractability, in the same spirit as in [38], we invoke the following assumption. Assumption 1. The set of scheduled MTs, conditioned on the spatial constraints imposed by the cellular association strategy and the number of MTs being served by each BS per resource block, is modeled as an independent stationary PPP with density λ (u) . Proposition 1. In the DL, we adopt a linear ZF-SIN precoder where the transmit antenna array (conditioned on N t ≥ N r + U) is utilized to jointly suppress SI and multi-user interference at the receiving antennas. This is achieved at the BS l by setting the columns of V l equal to the normalized columns ofĜ Note that the proposed interference nulling-based precoder differs from the extended ZF scheme in [21] where 'all-zero' streams are sent for suppressing SI, i.e., in [21] the (normalized) transmit signal vector V l In the UL, a linear ZF decoder, eliminating multi-user interference, is employed with the normalized rows of T 1≤k≤U ∈ C U ×Nr set as the row vectors of W l , at the BS l. The received signal-to-interference-plus-noise ratio (SINR) in the DL is given by The nullspace spanned by the SI and multiuser interference is D d N t − N r − U + 1 dimensional. For the sake of analytical tractability, we assume that the outer-cell precoding matrices have independent column vectors [38], [39]. As a result, the channel power gain from each interfering BS in the DL is interpreted as the aggregation of multiple separate beams from the projection of the cross-link channel vector g l,o onto the one-dimensional precoding vectors v l,k . The scheduled MTs, on the other hand, transmit using single-antennas (in all directions). The Rayleigh fading model applies to cases without line-of-sight (LOS), e.g., with afar transceiver distances. In FD setups, however, the nodes' transmit and receive antennas are co-located. Hence, the Rician fading model, which takes into account the different LOS and scattered paths, can be invoked to capture performance under generalized SI cancellation capability [8].
Assumption 3. The SI channel power gain at the reference MT o is a non-central Chi-squared random variable with Rician factor K and fading attenuation Ω such that µ KΩ K+1 and ν Ω K+1 . The corresponding p.d.f. and m.g.f. are respectively given by and Remark 2. The SI channel power gain at the reference MT o can be approximated using Gamma moment matching as Next, we express the received SINR from the reference MT at its serving BS as We recall the assumption that V l has independent column vectors. Assumption 4. The channel power gains at the reference BS b, from the intended MT o, interfering MT k, and interfering BS l are respectively modeled using It is important to note that it is feasible to apply other linear precoding schemes such as CB in FD massive MIMO systems [26]. In such cases, at the reference BS b, the SI channel power gain needs to be characterized. In [40], the distribution of the SI with linear processing over FD multi-user MIMO Rician fading channels was derived using Gamma moment matching. In particular, for FD multi-user massive MIMO systems, the following holds.
Remark 3. With conventional linear precoders in the DL (such as CB and ZF), the SI channel power gain at the reference massive MIMO BS b can be approximated using Gamma moment matching as

IV. SELF-INTERFERENCE-AWARE POWER CONTROL
In long-term-evolution (LTE), UL fractional power control is defined to account for the path-loss effect [41]. Recently, interference-aware fractional power control has been proposed to ensure that the power adjustment intended for path-loss compensation does not cause undesired interference to neighboring nodes [42]. In this work, we propose an LTE-compliant SIA fractional power control mechanism where the MTs adjust their transmit power based on the distance-dependent path-loss, SI, and maximum available transmit power. Here, specifically, an arbitrary scheduled MT k transmits to its serving BS l using where p 0 , ψ (∈ (0, 1]), I SI , and p (u) are respectively the normalized power density, compensation factor, tolerable SI level, and maximum transmit power at the MT [38]. The value of I SI can be set as the difference in the noise floor power from the gain of the MT SI cancellation capability.
Lemma 1. The c.d.f. and p.d.f. of the transmit power of a typical MT under the SIA fractional power control mechanism are respectively given by and where Ξ I (p) = πλ (d) p p0 2 ψβ and Ξ II (p) = ISI pΩ . Proof: See Appendix B.
Remark 4. The proposed SIA fractional power control mechanism is a generalization of the existing approaches for UL power control including total (without I SI ) and truncated (without I SI and p (u) ) fractional power control schemes.
The computation of SE can be greatly simplified with a non-direct methodology requiring only the moments of the random variables involved [43]. Next, we develop results for the moments of the SIA power control in certain special cases. It should be noted that a Meijer-G function can be readily calculated using common software for numerical computation. A Meijer-G function can also be expressed in terms of a hypergeometric function based on the results from [44]. Corollary 1. The p.d.f. of the transmit power of a typical MT under the SIA fractional power control mechanism can be simplified in certain special cases.
For K = 0 (Rayleigh SI channel), For p 0 → +∞ (no path-loss compensation), For I SI → +∞ (no constraint on the SI), Lemma 2. The -th positive moment of the transmit power of a typical MT under the SIA fractional power control mechanism admits a closed-form expression in certain special cases. For p (u) → +∞ (no constraint on the maximum transmit power), K = 0 (Rayleigh SI channel), ψ = 1 (compensation factor), and β = 4 (path-loss exponent), For p 0 → +∞ (no path-loss compensation) and p (u) → +∞ (no constraint on the maximum transmit power), Further, for K = 0 (Rayleigh SI channel), For p (u) → +∞) (no constraint on the maximum transmit power) and I SI → +∞ (no constraint on the SI), Further, for ψ = 1 (compensation factor), and β = 4 (path-loss exponent), Proof: See Appendix C.

V. SPECTRAL EFFICIENCY ANALYSIS
In order to facilitate performance analysis and optimization, we provide a framework for the computation of the DL and UL SEs in the FD massive MIMO cellular network. We utilize a m.g.f.-based methodology, which avoids the need for the direct computation of the SINR p.d.f. and only requires the m.g.f.s of the different signals involved [45], [46].

A. Downlink
We proceed by providing an explicit expression for the calculation of the SE in the DL. Theorem 1. The DL SE in the FD massive MIMO cellular network is given by Proof: The result follows directly from [45, Lemma 1].
Next, we provide explicit expressions for the conditional m.g.f.s of the different DL signals.
Lemma 3. The conditional m.g.f.s of the different DL signals in the FD massive MIMO cellular network are given by and Proof: See Appendix D.
Remark 5. The CI at the reference MT in (21), is from the aggregation of interference from all other scheduled MTs; hence, it is approximated using an independent PPP with spatial density Uλ (d) based on Assumption 1.

B. Uplink
Here, we provide a systematic approach for the calculation of the SE in the UL.
Theorem 2. The UL SE in the FD massive MIMO cellular network is given by Proof: The result follows directly from [45, Lemma 1].
The conditional m.g.f.s of the signals required for the calculation of the UL SE can be derived in closed-form as in the following lemma. and Proof: See Appendix D.
Remark 6. The UL inter-cell interference in (25) is derived using a spatially-thinned PPP with spatial density Uλ (d) and circular exclusion region with radius r based on the spatial constraints from Assumption 1 [38], [47].

C. Special Cases
In certain special cases, the conditional m.g.f.s of the signals can be simplified further as shown in the following results.
Corollary 2. The conditional m.g.f. of the inter-cell interference in the DL for U = 1 (single-user), and β = 4 (path-loss exponent) is given by Corollary 3. The conditional m.g.f. of the CI in the DL with β = 4 (path-loss exponent) is given by The above can be further simplified under special cases of UL power control. For p (u) → +∞ (no constraint on the maximum transmit power), K = 0 (Rayleigh SI channel), ψ = 1 (compensation factor), and β = 4 (path-loss exponent), Further, for p 0 → +∞ (no path-loss compensation), On the other hand, for I SI → +∞ (no constraint on the SI), Corollary 4. The conditional m.g.f. of the inter-cell interference in the UL for β = 4 (path-loss exponent) is given by Corollary 5. The conditional m.g.f. of the CI in the UL for U = 1 (single-user) and β = 4 (path-loss exponent) is given by The SE expressions in (18) and (23) require three-fold integral computations -as opposed to the manifold integrals involved in the direct p.d.f.-based approach. It is possible to further reduce the computational complexity in certain special cases with UL power control. For baseline SISO, we utilize linear ZF precoding, and take into account the SI at the BS side in FD mode. Lemma 5. For N t , N r , U = 1 (baseline SISO), K = 0 (Rayleigh SI channel), σ 2 d , σ 2 u = 0 (interference-limited region), and β = 4 (path-loss exponent), the SEs (ω = p (d) p for DL and ω = p p (d) for UL) can be reduced to double-integral expressions k,l (p) ds dp. (1 + s 2 ) π 2 √ ω + s + arccot(s) k,l (p) ds dp.

VI. FULL-DUPLEX VERSUS HALF-DUPLEX
The SE expressions developed facilitate performance analysis and optimization for generalized FD cellular deployments. At the same time, the proposed framework can serve as a benchmark tool for comparing the performance of FD over HD systems. Although an explicit expression for the corresponding gain cannot be obtained due to the highly complex SE expressions involving multiple improper integrals, we do provide results in certain special cases.
In what follows, S d,h and S u,h respectively denote the SEs of a typical MT in the DL and the UL of a HD cellular system.
For comparison of FD and HD systems, we consider the SE over two time slots, i.e., S f = 2(S d,f + S u,f ) for FD, and S h = S d,h + S u,h for HD, respectively. We first study the baseline SISO case and then derive results for multi-user massive MIMO setups.

A. Baseline SISO
Here, we compare the FD versus HD performance for the baseline SISO case.
Further, bounded closed-form expressions of the FD and HD SEs are respectively given bỹ . (44) Proof: See Appendix F. Lemma 7. Consider the bounded FD and HD SEs for N t , N r , U = 1 (baseline SISO), p (u) (MT transmit power), SI = 0 (perfect SI subtraction), σ 2 d , σ 2 u = 0 (interference-limited region), and β = 4 (path-loss exponent) from Lemma 6. We can formulate the following optimization problem for the highest FD over HD SE gain maximize x= p (d) The solution to the above problem is given by Proof: See Appendix G.
Remark 8. From Lemma 7, the highest exact and bounded SE percentage gains of FD over HD for baseline SISO are respectively ∼9% and ∼13%. We can infer that without advanced techniques for tackling the CI, even with perfect SI subtraction, the FD baseline SISO system achieves only modest improvements over its HD counterpart.

B. Massive MIMO
Next, we analyze the FD versus HD SE for massive MIMO-enabled cellular networks.

ds. (48)
For U = 1 (single-user), the above expressions can be respectively reduced to and Proof: The proof follows from a similar approach to that in Appendix F. The SE expressions for the case of multi-user massive MIMO in Lemma 8 involve single-fold integrals. Next, we employ non-linear curve-fitting in order to provide a closed-form approximation for the FD versus HD massive MIMO SE gain.
Corollary 6. The FD over HD massive MIMO SE gain with ZF beamforming, N (number of transmit and receive antennas), Up (BS transmit power), p (MT transmit power), SI = 0 (perfect SI subtraction), σ 2 d , σ 2 u = 0 (interference-limited region), and β = 4 (path-loss exponent) can be approximated using non-linear curve fitting as 1 Remark 10. We infer from the results of Lemma 8 and Corollary 6 that the FD over HD multi-user massive MIMO SE gain logarithmically increases (decreases) in the number of antennas (users). Furthermore, the anticipated two-fold increase in SE from massive MIMO-enabled FD versus HD cellular networks is achieved as the number of antennas tends to be infinitely large (N → +∞).
Note that EE can be studied (i) using the proposed framework for the calculation of SE and, (ii) adopting an appropriate power consumption model [48]. A rigorous assessment of EE in FD massive MIMO cellular networks is left for future work.

VII. NUMERICAL RESULTS
In this section, we provide numerical examples to assess the performance of the FD massive MIMO cellular network under different system settings. MC simulations are accordingly provided for the purpose of examining the validity of the proposed analytical framework. The BS deployment density is considered to be λ (d) = 4 π per unit area (km×km). The total system bandwidth is B = 20 MHz. The noise power is calculated using σ 2 = −170 + 10 log 10 (B) + N f (dBm), where N f is the noise figure [49]. The maximum available transmit powers at the BSs and MTs are set to 43 dBm and 23 dBm, respectively. Note that the results from the MC simulations are obtained from 10 4 trials in a circular region of radius 10 2 km, considering (i) PPP under spatial constraints for the scheduled MTs (Assumption 1), (ii) non-central Chi-squared distribution for the SI channel power gain (Assumption 3), and (iii) Gamma distribution for the other channels power gains (Assumptions 2 and 4).
1) Different FD Cellular Setups: We compare the DL and UL performance of different FD cellular networks, namely, massive MIMO (with ZF-SIN precoder and ZF decoder) and baseline SISO (with ZF precoder/decoder) in Fig. 1. The results confirm prior findings that the UL rate is the main performance bottleneck in FD cellular systems with baseline SISO. Here, the UL performance is severely limited due to the imperfect SI subtraction and large disparity in the BS and MT transmit power levels. Furthermore, it can be observed that significant performance gains can be achieved by exploiting the large scale antenna array with linear ZF-SIN precoding and linear ZF receive combining. For example, with β = 4, the UL SE in the proposed FD massive MIMO cellular setup is more than 96 times greater than that in the baseline SISO case. By increasing the antenna array size, further improvements can be realized from (i) the added DoF, and (ii) the potential to linearly reduce the transmit powers of the BSs and MTs without degrading the received signal-to-noise ratio (SNR). It should be noted that the MC results confirm the validity of the proposed analytical framework. 2) FD versus HD Massive MIMO: Next, we investigate the performance of the FD massive MIMO cellular network with respect to its HD counterpart over a wide range of MT SI channel attenuations in Fig. 2. It can be seen that any potential improvements from the FD operation occurs for SI channel attenuation well below −80 dB. This trend indicates that without advanced SI mitigation solutions being available at the MTs, the conventional HD massive MIMO cellular network is arguably the more sensible deployment choice. With nearly perfect SI cancellation at the reference MT, on the other hand, the maximum system SE gain in the FD massive MIMO cellular network over its analogous HD variant here is 47% with U = 1 (resulting from 73% and 1% increase in the DL and UL SEs, respectively). It should be noted that the small improvement in UL SE is due to the CI from the BSs transmitting in the DL in FD mode.
3) Transmit/Receive Antenna Array Size: The impact of different number of transmit and receive antennas on the SE gain of the massive MIMO-enabled FD system over its HD counterpart is depicted in Fig. 3. It can be observed that the corresponding performance gain increases by adding more transmit antennas due to the reduced impact of the DL array gain from the ZF-SIN precoding scheme and improved resilience against interference. For smaller transmit antenna arrays, however, increasing the number of receive antennas may degrade the relative improvement from the FD operation over HD mode. It is also important to note that the relative FD over HD massive MIMO gain improves in the path-loss severity. The optimal ratio of transmit over receive antennas, on the other hand, depends on the particular system settings. A rigorous study of this aspect is postponed to future work. 4) Transmit Power Budget: It has been shown in [2] that under perfect CSI, the transmit energy can be linearly conserved in the number of antennas. This is a contributing factor in tackling the UL rate bottleneck through massive MIMO, as was demonstrated in Fig. 2. We depict the impact of different BS transmit power and MT maximum transmit power in Fig. 4. Intuitively, increasing p (d) , or p (u) , respectively improves the corresponding SE in the DL, or the UL; the relative gain however decreases for larger power budgets. Further, the DL and UL SEs are conflicting functions in the BS/MT transit powers. In general, we observe that a large difference in the DL/UL power levels is deteriorating to the overall performance in FD mode. Due to the large disparity in the DL/MT SEs, however, the direct optimization of the FD sum-rate places the focus on the DL, see Fig. 4. It can therefore be inferred that the optimization problem should be tackled by applying a weight to the UL SE. This is however beyond the scope of this work.

5) Uplink Power Control:
The results presented so far were based on the conventional UL fractional power control mechanism defined in the existing LTE standards for HD cases. Next, we study the performance of the massive MIMOenabled FD cellular network with different fixed (at maximum power), conventional, and proposed SIA fractional power control protocols under different SI channel attenuations in Fig. 5. It can be observed that the lack of self-interference-awareness in the case of fixed as well as fractional power allocation strategies means that the DL performance significantly suffers in the case of large SI channel attenuations. The proposed scheme can therefore serve as an important safe-guard mechanism for ensuring that a certain maximum SI level is not exceeded.

VIII. SUMMARY
We provided a theoretical framework using tools from stochastic geometry and point processes for the study of massive MIMO-enabled FD multi-cell multi-user cellular networks. The DL and UL SEs with linear ZF-SIN precoding, SIA fractional power control, and linear ZF receiver were characterized over Rician SI and Rayleigh intended and other-interference fading channels. The results highlighted the promising potential of massive MIMO towards unlocking the UL rate bottleneck in FD communication systems. On the other hand, the results demonstrated that the anticipated two-fold increase in SE of the massive MIMO-enabled FD cellular network overs its HD variant is only achievable in the asymptotic antenna region. . (C.6) By utilizing the integral identity (C.2), we can obtain (16).
APPENDIX D Let Φ, λ, and E denote the PPP, density, and exclusion region radius, respectively. Hence, I = x∈Φ Q x x −β where x and Q x are respectably the location and channel power gain of an arbitrary interferer with respect to a typical receiver at the origin. We proceed as follows where (i) is written considering N conditional interferers are i.i.d. in a circular region of radius D (> E ) around the center with the limit as D → +∞; (ii) is from characterizing N as a Poisson random variable with expected value πλ D 2 − E 2 and hence utilizing the Poisson identity E ζ N = exp (E {N } (ζ − 1)); (iii) is written using the p.d.f. of the distance R (where E < R < D) (iv) is obtained by invoking the integral identity (where α > 0 and β > 2) and taking the limit as D → +∞.
To proceed, the distribution of the channel power gain should be specified. Here, we consider the general case where Q x ∼ Γ(U, V ), which can be used to capture a wide range of MIMO setups [46]. Hence, through utilizing the following integral identities (where α > 0 and β > 2) (1) and basic algebraic manipulation, we derive (42) and (43).

APPENDIX G
We proceed by defining the super-level set of S f S h in x as For the case in which K < 0, the total SE gain of FD over HD is always positive, hence, there are no points on the counter, S f S h * = K . On the other hand, for K > 0, we can write 3) The joint HD DL/UL SE in the above is affine in x. It can be shown that the second derivative of the FD mode SE function, d 2 dx 2 (2 (S d,f + S d,f )), is always positive for π π−8 2 < x < 1, always negative for 1 < x < π−8 π 2 , and zero for x = 1.
Consequently, L is convex in x and the objective function in (45) is strictly quasi-concave in x, with a maximum point at x * = 1.