Downlink MIMO Diversity with Maximal-Ratio Combining in Heterogeneous Cellular Networks
Ralph Tanbourgi and Friedrich K. Jondral
Abstract—We present a stochastic model for analyzing the see for instance [3], [4], and recently also for ad hoc networks performance of multiple-input multiple-output (MIMO) diversity [5]–[8]. IA-MRC was studied in [9] for single-tier cellular in a downlink heterogeneous cellular network. Multi-antenna re- networks with one transmit (Tx) antenna. ceivers are assumed to perform maximal-ratio combining (MRC). We consider interference-blind (IB) MRC and interference-aware Unfortunately, analyzing MIMO diversity in the context of (IA) MRC, where the latter takes the interference power at heterogeneous cellular networks (HetNets) is difficult due to each antenna into account. Using tools from stochastic geometry, the deployment of multiple tiers, limited site-planning, etc. we derive the coverage probability for both types of MRC as [10]. In particular, spatial interference correlation across Rx a function of various tier-specific system parameters, including antennas, which is known to have a detrimental effect on di- the number of base station transmit antennas in each tier. The model is then used to compare the performance of IB-MRC and versity combining schemes, is difficult to characterize. In order IA-MRC. One important insight is that IA-MRC becomes less to properly assess the performance of MRC in HetNets under favorable than IB-MRC in a transmit-diversity system due to a different MIMO settings, a realistic model and meaningful larger interference correlation across receive antennas. analysis covering the above aspects is hence necessary. This is Index Terms —MIMO, space-time coding, maximal-ratio com- the main motivation of this work. In this paper, we study the bining, coverage probability, Poisson point process. coverage performance of a downlink MIMO heterogeneous cellular system, where mobile receivers employ IB-MRC or I.INTRODUCTION IA-MRC. Our main contributions are summarized below. One way to face the steadily increasing rate demands in Analytical Model: We develop a tractable stochastic model downlink cellular systems is adding more antennas to both for downlink MIMO diversity with orthogonal space-time base stations (BSs) and user devices, and using multiple- block codes (OSTBCs) and IB-MRC/IA-MRC in Section III. input multiple-output (MIMO) techniques [1]. MIMO schemes The model captures relevant tier-specific parameters, such as can be roughly divided into open-loop and closed-loop based. BS density and Tx power, path loss exponent, and number of Many works on MIMO cellular networks have been focusing Tx antennas. We derive the coverage probability for both types on closed-loop schemes, such as spatial-multiplexing or multi- of MRC, while for IA-MRC we focus on two Rx-antenna case. user MIMO, showing that tremendous gains can be harvested Design Insights: In a typical three-tier scenario with multi- when channel state information is available at the transmitter ple Tx antennas at the BSs and IB-MRC, the gain of doubling (CSI-T). However, CSI may not always be available/reliable the number of Rx antennas is roughly 2.5 dB at practical in practice and one has to resort to open-loop MIMO schemes target coverage probabilities around 80%. For IA-MRC, the in this case. For instance, in 3GPP LTE, transmission mode corresponding gain is around 3.6 dB. Besides, adding more 2 is used for transmit-diversity with space-frequency block Tx antennas has only a minor impact on the performance of codes over two or four antennas [2]. Besides, mobile devices IA-MRC; the coverage probability gain when adding a second typically have space/complexity limitations per design, thereby Tx antenna is less than 5%. Relative to IB-MRC, IA-MRC often not allowing more than two antennas and requiring improves the coverage probability by roughly 1.5% for the only simple linear combining schemes. One such combining single Tx-antenna case, while this relative gain decreases to no scheme is maximal-ratio combining (MRC) [3], which offers more than 1% when Alamouti space-time coding is performed an acceptable trade-off between performance and complexity, over two Tx antennas. This loss in relative performance is due and is therefore ubiquitously found in multi-antenna receivers. to the fading-averaging effect of transmit diversity, which in- There exist two types of MRC, namely interference-blind creases with the number of Tx antennas. Although interference (IB) and interference-aware (IA) MRC. The former —and power estimation required by IA-MRC can usually be realized mostly considered— ignores the different interference powers with acceptable effort, this result suggests that IA-MRC is less at the receive (Rx) antennas at the combining stage, while the favorable than IB-MRC in a MIMO HetNet with Tx diversity. latter takes them into account though still treating interference as white noise. Measuring the per-antenna interference power II.SYSTEM MODEL can be done within the channel estimation phase. Both types of A. Network Geometry and User Association MRC are well-understood for networks with fixed geometry, We consider a K-tier HetNet in the downlink with BSs irregularly scattered in the plane, see Fig. 1. We model the The authors are with the Karlsruhe Institute of Technology, Communica- tions Engineering Lab, Germany. Email: [email protected], irregular BS locations in tier k by an independent stationary [email protected]. planar Poisson point Process (PPP) Φk with density λk. We B. OSTBC MIMO Signal Model
All BSs of the k-th tier use an (Mk,Tk,rk)-OSTBC, where Tk ≥ 1 is the codeword length and rk ∈ (0, 1] is the code rate; Tk can be seen as the number of slots (number of channel uses) for conveying Sk = Tkrk symbols using Mk Tx antennas. For analytical tractability, we shall consider only power-balanced (Mk,Tk,rk)-OSTBCs, i.e., having the property that exactly Sk symbols are transmitted, or equivalently that Sk ≤ Mk Tx antennas are active, in every slot of the codeword. This will allow us to assign a constant power load of Pk/Sk to every symbol-antenna pair. Practical examples of balanced OSTBCs are (1, 1, 1) (single-antenna), (2, 2, 1) (Alamouti), (4, 4, 1/2), and (4, 4, 3/4), see [15], [16] for instance. We use the notation Mk Fig. 1. Example: Downlink HetNet with K =2. Tier-1 (macro) BSs have 4 vi,τ ∈ {0, 1} to indicate which Tx antennas of BS i are antennas Tier-2 (small-cell) BSs have 2 antennas. Typical user has 2 antennas. active in slot τ, i.e., the m-th entry of vi,τ is equal to one if Tx antenna m is active and zero otherwise. , K denote by Φ ∪k=1 Φk the entire set of BSs. The spatial Assume for the moment that the typical user associates with Poisson model is widely-accepted for analyzing (multi-tier) the ℓ-th tier. It will then be served by an (Mℓ,Tℓ,rℓ)-OSTBC. cellular networks, see for instance [11], [12]. All BSs in tier The interference-plus-noise corrupted received signal at the k transmit an OSTBC using Mk Tx antennas. Similarly, we typical user in slot τ ∈ {1,...,Tℓ} can then be expressed by assume that mobile receivers (users) are equipped with N Rx K antennas. The users are independently distributed on the plane yτ = Ho co,τ + Hi ci,τ + nτ , (3) according to some stationary point process. By Slivnyak’s o k=1 xi∈Φk Theorem [13] and due to the stationarity of Φ, we can focus X X where R2 the analysis on a typical user located at the origin o ∈ . N×M • H ∈ C k is the channel fading matrix between the At the typical user, the long-term power received from a tier i −αk i-th BS of the k-th tier and the typical user. The elements k BS located at xi ∈ Φk is Pkkxik , where Pk is the BS of Hi, denoted by hi,nm, are CN(0, 1) distributed and Tx power in tier k, which is equally divided across all Mk −αk remain constant over one codeword period. We assume Tx antennas, and k · k is the distance-dependent path loss E ∗ that [hi,nmhj,uv] = 0 unless i = j, n = u, and m = v. with path loss exponent αk > 2. We assume independent and M • c C k are the space-time coded symbols of the identically distributed (i.i.d) frequency-flat Rayleigh fading. i,τ ∈ Users are assumed to associate with the BS providing the i-th BS sent over the Mk Tx antennas in slot τ. We assume E c cH for all and E c strongest average received power. For the typical user, the serv- i,τ j,τ = 0 i 6= j [ i,τ ] = 0 −αk element-wise. Furthermore, it is reasonable to assume ing BS is hence the one maximizing Pkkxik . Without loss E H Pk [ci,τ ci,τ ] = αk diag(vi,τ ), where diag(vi,τ ) is a of generality, we label the location of this BS by xo and denote Skkxik diagonal matrix with entries vi,τ . The latter follows from by y , kxok its distance to the typical user. For convenience, o , o , the balanced-power property of the considered OSTBCs. we define Φ Φ \{xo} (similarly, Φk Φk \{xo}), i.e., N • n C the set of interfering BSs. From the association rule discussed τ ∈ is a vector describing the Rx noise with independent CN 2 distributed entries. above it follows that, given y = y and that the serving BS is (0,σ ) o R2 Upon receiving all Tℓ code symbols of the desired codeword from tier ℓ, Φk is a homogeneous PPP on \b(0,dk), where 1/αk to be decoded, the typical user stacks the vectors y y ˆ 1/αˆk ˆ , , 1,..., Tℓ dk = Pk y with Pk Pk/Pℓ and αˆk αk/αℓ. It will be useful in the later analysis to know the probability to form the new vector Ho co,1 Hi ci,1 n1 that a user associates with a certain tier k as well as the K conditional probability density function (PDF) of the distance . . . ¯y = . + . + . , (4) to the serving BS, which are given in the next lemma. o k=1 xi∈Φk Ho co,T X X Hi ci,T nT Lemma 1 (Association Probability and Distance PDF [14]). ℓ ℓ ℓ ¯i A user associates with the ℓ-th tier with probability i ¯ NTℓ ∞ K where ii ∈ C is the interference| {z signal} from the i-th ˆ2/αk 2/αˆk Aℓ = 2πλℓ y exp −π λkPk y dy. (1) BS received over the entire codeword period. With CSI-R, 0 k=1 ! ¯y is linearly processed using MRC to form the final decision Z X variable. Two types of MRC are considered, which differ in The PDF of the distance y , kxok to the serving BS, given that it belongs to tier ℓ, is the amount of CSI needed. Moe specifically, IB-MRC requires knowledge of Ho, while IA-MRC additionally needs to know 2πλ y K ℓ ˆ2/αk 2/αˆk the interference-plus-noise power at each Rx antenna. The fy(y)= exp −π λkPk y , y ≥ 0. (2) Aℓ ! following lemma will be useful in the later analysis. kX=1 Lemma 2 (Gaussian Matrices). Let the matrix X(u) ∈ Cv×w A. MIMO Diversity with interference-blind MRC CN have u ≤ vw (0, 1)-distributed entries and vw − u zeros. An extremely useful property of OSTBCs is that the MIMO Then X 2 is Erlang distributed with CDF k (u)kF input-output relation, i.e., (4), can be reduced to parallel u−1 j SISO channels [19]. At the typical user, having knowledge 2 −θ θ P(kX(u)k ≤ θ) = 1 − e . (5) H F j! of o, this is achieved by performing the linear combination j=0 N Mℓ ∗ H T ∗ X n=1 m=1 ho,nmAnm¯y + ho,nmBnm¯y , where Anm and Let h =[h ,..., h ] be the n-th row of H . Then, Bnm are the dispersion matrices describing the OSTBC em- i,n i,n1 i,nMk i P P the interference power in slot τ (we shall drop this index in ployed in the serving tier, see [20], [21] for further details. the following) measured at the n-th Rx antenna, averaged over The resulting equivalent channel model allows treating the the interfering code symbols ci, is detection of each of the Sℓ information symbols encoded in SINR H the current codeword separately. The corresponding at K K the symbol decoder can then be expressed as I Ec h c h c n = i i,n i i,n i o o Pℓ 2 k=1 xi∈Φ k=1 xi∈Φ α kHok X Xk X Xk SINR Sℓy ℓ F ℓ(y)= K , (8) K 2 k=1 x ∈Φo Ii,eqv + σ (a) E H H i k = hi,n cici hi,n o P P k=1 xi∈Φ where Ii,eqv is the interference power from the i-th BS in the X Xk K equivalent channel model. Ii,eqv is statistically the same for (b) Pk h diag v hH all Sℓ symbols. Thus, focusing on an arbitrary symbol, i.e., = α i,n ( i) i,n Skkxik k k=1 x ∈Φo considering a single arbitrary column of Anm, Bnm, say anm, X iXk bnm, the interference power Ii,eqv is K P = k kh (S )k2 , (6) α i,n k F N Mℓ ∗ Skkxik k h k=1 x ∈Φo o,nm H ho,nm T ∗ i k I = Varc a ¯i + b ¯i . (9) X X i,eqv i kH k nm i kH k nm i where (a) follows from the independence between the c across "n=1 m=1 o F o F # i X X BSs and (b) follows from the correlation properties of the ci. Note that the Rx noise statistics remain unaffected by the Remark 1 (Feasibility of Interference Power Estimation). linear combination [19], [21]. However, the distribution of When the set of active antennas of interfering BSs changes Ii,eqv is more complicated, particularly due to its dependence in every slot τ, In varies unpredictably from slot to slot. on Ho. This was already observed in [5] for a similar MIMO This is the case when Sk < Mk. Unfortunately, such rapid network model, where the authors also showed that ignoring variations over τ are imperceptible to CSI estimation since the this dependence and assuming Ii,eqv to be Gamma distributed latter is usually designed to track channel-fading variations, yields a valid approximation. We shall thus follow the same Pk 2 approach and assume I ≃ α kH (S )k with I which happen on a larger time scale. However, when full-rate i,eqv Skkxik k i k F i,eqv OSTBCs are used (rk = 1 for all k), In is identical across being independent from Ho. The following two facts support τ. In that case, the receiver can obtain knowledge of In with this approximation: acceptable complexity, e.g., by estimating it once within one • It can be shown that the approximation is moment match- frame using techniques from [17], [18]. E ing irrespective of the realization of Ho, i.e., Hi [Ii,eqv]= Pk 2 Pk α EH [kHi(Sk)k ]= α in (9). III.COVERAGE PROBABILITY ANALYSIS Skkxik k i F kxik k • Whenever Mk = 1, it follows from [15] that the above We now study the downlink performance at the typical user approximation becomes exact. In this case I is also for both IB-MRC and IA-MRC. As explained in Remark 1, i,eqv truly independent from Ho. IA-MRC is practical only for full-rate OSTBCs (rk = 1 for all k). A common way for studying the performance of diversity- Lemma 3 (Interference Laplace Transform). Consider the K Pk 2 combining techniques is to analyze the post-combiner signal- interference field I = o αk kHi(Sk)k . Its k=1 xi∈Φk Skkxik F to-interference-plus-noise ratio (SINR). The specific form of Laplace transform is given by SINR P P the depends on the considered scheme and will be K 2 2 2 sP 2 1 k −π λkdk F − α ,Sk,1− α ;− αk −1 derived in Sections III-A and III-B. =1 k k S d LI(s)= e kP k k . (10) Definition 1 (Coverage Probability Pc). The coverage proba- bility at the typical user is defined as Proof: We write
Pc , P (SINR ≥ T ) (7) K Pk 2 E exp −s kHi(Sk)k αk F for a coding and modulation specific threshold T > 0. o Skkxik k=1 xi∈Φk P SINR X X The c can be interpreted as the distribution at the K (a) sPk typical user, or alternatively as the average fraction of users = E L H 2 k i(Sk)kF α SINR Skkxik k in the network covered by an no less than T [14]. k=1 x ∈Φo Y iYk NM −1 K ℓ (−1)mλ ∞ dm sS T K 2 2 sT PIB = 2π ℓ y exp − ℓ − π λ Pˆ2/αk y2/αˆk F − ,S , 1 − ; − dy (12) c m! dsm SNR (y) k k 2 1 α k α ˆ m=0 0 " ℓ k k Sk !# Xℓ=1 X Z kX=1 s=1
M −1 K ℓ m+Mℓ ∞ ∞ m Mℓ PIA (−1) λℓ −1 d d Mℓ + c = 2π yz m M exp − s (T − z) + tz m! Γ(Mℓ) ds dt ℓ SNRℓ(y) ℓ=1 m=0 Z0 Z0 " X X K + 2/αk 2/αˆk s (T − z) tz × exp −π λkPˆ y 1+Ψ , ,Mk,αk dy dz, 1 ≤ Mk ≤ 2,N = 2 (20) k Mˆ Mˆ k=1 k k !#s=1 X t=1
K ∞ −Sk (b) sPk The coverage probability in (14) does not depend on the = exp −π λk 2r 1 − 1+ S rαk dr , k BS densities λk and powers Pk, nor on the total number of ( dk ) k=1 Z tiers , which is consistent with the literature, see for instance X (11) K [12], [14]. Note that the first term m = 0 in (14) corresponds o where (a) follows from the independence of the Φk across k to the coverage probability for the SISO case [25]. and of the independence of the kH (S )k2 across i, and (b) i k F B. MIMO Diversity with interference-aware MRC follows from the probability generating functional (PGFL) for PPPs, see [13]. Solving the integral yields the result We now assume Mk ≤ 2 for all K tiers. This ensures We now have the tools required to characterize the cov- that the receiver can estimate the interference power with erage probability for IB-MRC. This task is addressed in the acceptable complexity once within the current block/frame, following theorem. see Remark 1. In this case, we have Sk ≡ Mk. When Mk ≤ 2, tier k uses either Alamouti space-time coding Theorem 1 (P for Interference-Blind MRC). The coverage c (Mk = 2) or no space-time coding (Mk = 1). In both cases, PIB probability c for IB-MRC in the described setting is given the interference power remains constant in each codeword slot. SNR −αℓ 2 by (12) at the top of the page, where ℓ(y) , Pℓ y /σ Its value at the n-th antenna is given by (6). We assume that the ˆ and Sk , Sk/Sℓ. receiver perfectly knows the current per-antenna interference- 2 Proof: See Appendix A. plus-noise power In + σ at each antenna, in addition to The differentiation dm/dsm in (12) can be calculated us- knowing Ho. Interference is still treated as white noise. ing Faa` di Bruno’s formula for higher-order derivatives of In IA-MRC, the phase-corrected and channel-weighted re- composite functions [22]. While the outer function is simple ceived signals are additionally scaled by the interference-plus- due to the exp-term, the inner function, more specifically noise power experienced at each antenna, thereby following the original MRC approach from [3]. The linear combination the 2F1 −2/αk,Sk, 1 − 2/αk; −sT/Sˆk expression, is more N Mℓ ∗ involved. With [22], its derivative is obtained as ho,nm ho,nm AH ¯y + BT ¯y∗, (15) m 2 nm 2 nm d 2 2 sT In + σ In + σ F − ,S , 1 − ; − n=1 m=1 m 2 1 k ˆ X X ds αk αk Sk s=1 yields the equivalent channel model for IA-MRC. Note that m T −2/α Γ(S + m) now, the interference-plus-noise power is factored in. Similar = − k k ˆ (m − 2/α ) Γ(S ) to Section III-A, we focus again on an arbitrary symbol and Sk k k 2 2 T therefore consider an arbitrary column amn, bnm of Anm, × F − + m, S + m, 1 − + m; − . (13) SINR 2 1 α k α ˆ Bnm. The for IA-MRC can then be expressed as k k Sk h 2 2 Pℓ N k o,nkF In dense deployments the performance is typically limited α 2 SINR Mℓ y ℓ n=1 In+σ by interference rather than noise [23]. In this case, one can set ℓ(y)= h 2 2 , (16) K N k o,nkF σ 2 SNR o IPi,eqv + 2 2 σ = 0 ⇔ 1/ ℓ(y) = 0. In addition, the path loss exponent k=1 xi∈Φk n=1 (In+σ ) does not vary significantly across tiers in practice with typical where now Ii,eqvPis P P values around αk ≈ 3.7 [24]. When also the number of Tx N Mℓ ∗ antennas is equal across tiers, the following corollary applies. ho,nm H ho,nm T ∗ I Varc a ¯i b ¯i (17) i,eqv = i 2 nm i + 2 nm i . 2 I + σ I + σ Corollary 1 (Special Case). In the absence of Rx noise (σ = "n=1 m=1 n n # X X 0) and with equal path loss exponents (αk = α) and the same Using the orthogonality property of Anm, Bnm, see for PIB number of Tx antennas (Mk = M, Sk = S), c simplifies to instance [15], we can rewrite (17) as NM−1 m m N h 2 P (−1) d 1 Pk 2 k i,nkF Zi,n c = . (14) Ii,eqv = kho,nkF + , (18) m! dsm F − 2 ,S, 1 − 2 ; −sT M kx kαk (I + σ2)2 (I + σ2)2 m=0 " 2 1 α α # k i n=1 n n X s=1 X 1 1 1 Exact Sim. Zoom 0.9 0.9 Approx. Sim. 0.9 Theorem 1 0.8 0.8 0.8
0.7 0.7 0.7
0.6 0.6 0.6
0.5 0.5 5 0 5 T [dB] 0.4 0.4
0.3 0.3 Coverage Probability Coverage Coverage Probability Coverage 0.2 0.2 Exact Sim. 0.1 0.1 Corollary 2, M=1 Corollary 2, M=2 0 0 20 10 0 10 20 30 10 5 0 5 10 15 20 25 30 T [dB] T [dB] (a) Coverage probability IB-MRC (b) Coverage probability IA-MRC PIB =3 76 =3 67 =3 5 2 = −104 Fig. 2. (a) Coverage probability c for path loss exponents α1 . , α2 . , α3 . . Receiver noise is σ dBm. (b) Coverage probability PIA =3 7 2 =0 =1 2 c for α . , σ , and M , .
where Zi,n describes the part resulting from the non-zero expression in (20) can be evaluated with acceptable complexity E ¯¯H off-diagonal elements of the covariance matrix ci [iiii ]. It using semi-analytical tools, see for instance [8]. Moreover, E can be shown that Hi [Zi,n] = 0 irrespective of Ho. To Ψ(·, ·, ·, ·) can be given in terms of the Gaussian hypergeo- obtain a more tractable SINR expression, we hence ignore metric function, which reduces to elementary functions for the Zi,n term. After some simple algebraic manipulations, the suitable αk. Besides, (20) covers the general case and the simplified SINR from (16) becomes expression can be further simplified for certain scenarios as discussed next. The counterpart to Corollary 1 is given next. N 2 Pℓ kho,nkF SINRℓ(y)= . (19) 2 M yαℓ I + σ2 Corollary 2 (Special Case). In the absence of Rx noise (σ = ℓ n=1 n X 0), and with equal path loss exponents (αk = α) and the same Remark 2 (SINR-Approximation). It follows by Jensen’s In- PIA number of Tx antennas (Mk = M ≤ 2), c reduces to equality [26] that the approximate SINR in (19) underestimates the true SINR in (16). The resulting error, however, is barely M−1 (−1)m+M ∞ PIA = z−1 noticeable as confirmed by simulations, see Sec. IV. c m! Γ(M) m=0 Z0 X m M Although the hi,n in (19) are mutually independent, the d d 1 (21) × m M + dz. interference terms In are correlated across the N Rx antennas ds dt 1+Ψ(s (T − z) ,tz,M,α) s=1 due to the common locations of interfering BSs. More specif- t=1 ically, the expression in (19) is a sum of correlated random The expression in (21) is less complicated than (20). When variables exhibiting a complicated correlation structure. This the SINR threshold T is not large, the Ψ(s (T − z)+,tz,M,α) renders the characterization of the coverage probability of IA- term can be further simplified as shown next. MRC for general number of Rx antennas N challenging. In Corollary 3 (Small-T Approximation). When T is small, the practical systems, however, the number of antennas mounted following approximation becomes tight on mobile devices is limited due to space/complexity limi- tations, thereby often not exceeding N = 2 antennas. This s (T − z)+ tz 1+Ψ , ,M ,α practically relevant case is addressed in the following theorem. ˆ ˆ k k Mk Mk Theorem 2 (P for Interference-Aware MRC). The coverage 2 2 s (T − z)+ + tz c ≃ F − ,M , 1 − ; − . (22) probability PIA for IA-MRC in the described setting with M ≤ 2 1 α k α ˆ c k k k Mk 2 is given by (20) at the top of the last page, where Ψ(·, ·, ·, ·) The right-hand side of (22) may be easier to evaluate than is given by (32) and Mˆ , M /M . k k ℓ the original expression since the Gaussian hypergeometric Proof: See Appendix B. function is available in most numerical software programs. PIB PIA Compared to c in (12), c is analytically more involved Moreover, its higher-order derivatives with respect to both s due to the mathematical form of (19), which translates into the and t appearing in (20) can be evaluated fairly easily following convolution-type integration over z in (20). Nevertheless, the the same idea as in (13). Remark 3 (Single-Tier, Single-Tx-Antenna). Setting K = 1 2 and M = 1, we recover the coverage probability result from M=1 1.8 M=2 [9] for single-tier single-Tx-antenna cellular networks. 1.6
IV. DISCUSSIONAND NUMERICAL EXAMPLES 1.4 We assume a typical three-tier HetNet setting (K = 3), 1.2 2 2 where BSs have densities λ1 = 4 BS/km , λ2 = 16 BS/km , 1 2 λ3 = 40 BS/km , and transmit powers P1 = 46 dBm, P2 = 0.8 30 dBm, P3 = 24 dBm. The dispersion matrices Anm, Bnm are obtained using [15, Sec. 2.2.3]. 0.6
We first focus on IB-MRC and consider a typical scenario Probability Gain [%] Coverage 0.4 with M1 = 4, M2 = 2 (Alamouti), and M3 = 1 (no space- time coding). The (4, 4, 3/4)-OSTBC from [16, 7.4.10] is 0.2 chosen for tier one. Fig. 2a shows the coverage probability 0 PIB for IB-MRC and different number of Rx antennas . It 20 10 0 10 20 30 c N T [dB] can be seen that the theoretical expressions perfectly match the simulation results. Furthermore, the simulation results for the Fig. 3. Relative coverage probability gain of IA-MRC over IB-MRC for 2 interference Gamma approximation explained in Section III-A α =3.7, σ =0, and M =1, 2. (Approx. Sim.) and for the exact case (Exact Sim.) are hardly distinguishable, which is consistent with [5]. As expected, a considerable impact on the relative performance of IB- increasing improves PIB since the typical user enjoys a N c MRC and IA-MRC. Our results revealed that IA-MRC is larger array gain. For operation points of practical relevance, only slightly better than IB-MRC when multiple Tx antennas i.e. around 80% of covered users, the horizontal gap between are used. Future work may include extending the model to the curves in Fig. 2a is roughly 2.5 dB. For IA-MRC, this gain incorporate also other linear combing schemes in combination is about 3.6 dB (verified through simulations). with transmit diversity and spatial multiplexing. PIA Figure 2b shows the coverage probability c for IA-MRC. Here, we consider the interference-limited case (σ2 = 0) APPENDIX with equal path loss exponents (αk = 3.7) and the same A. Proof of Theorem 1 number of Tx antennas (M = M). Again, simulation results k Applying the law of total probability and making use of and theoretical expressions (Corollary 2) are fairly close over Lemma 1, we can express (7) by the whole T -range. It can be further observed that, for IA- MRC, adding a second Tx antenna and performing Alamouti K ∞ P P SINR space-time coding slightly improves performance only at low c = Aℓ fy,ℓ(y) ( ℓ(y) ≥ T ) dy, (23) 0 T . This gain, about 4% in the practically relevant regime, Xℓ=1 Z is comparable to the gain due to frequency-diversity based where P (SINRℓ(y) ≥ T ) can be seen as the conditional Pc, resource allocation in HetNets with single Tx antenna [27]. given ℓ,y. With Lemma 2, the conditional Pc is rewritten as Next, we compare the performance between IB-MRC and P SINR IA-MRC for the same scenario as for Fig. 2b. In Fig. 3, the ( ℓ(y) ≥ T ) relative coverage probability gain of IA-MRC over IB-MRC S T K P H 2 ℓ I 2 is shown for M = 1 and M = 2 Tx antennas. In line with = k okF ≥ −α i,eqv + σ Pℓy ℓ k=1 x ∈Φo the observations from [9] for single-tier single-Tx-antenna X iXk cellular networks, the gain is insignificant (< 2% in this NMℓ−1 m (a) (−1) m m −Y example). In fact, IA-MRC becomes even less favorable over = EY (−1) Y e m! IB-MRC when adding more Tx antennas. This due to the fact m=0 X NMℓ−1 m m that adding more antennas effectively averages out the fading (b) (−1) d = LY(s) , (24) on the interfering channels, which implies that interference m! dsm s=1 m=0 power becomes even more correlated across Rx antennas. X h i Thus, with increasing similarity of the interference level across SℓT K where in (a) we define Y , −α ( o Ii,eqv + Pℓy ℓ k=1 xi∈Φk Rx antennas, the performance of IA-MRC eventually becomes σ2) and (b) follows from the differentiation rule for Laplace equal to that of IB-MRC. P P transforms. With Lemma 3, LY(s) can be obtained as
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