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(SINR). and SINR, function distribu of upper ratio distribution c.d.f.) are derive power cumulative we noise the random, interfering plus at where interference uniformly to scenarios distr probability signal For the the derive dire and similar of a the emission, in side-lobe signals pointing via main-beam or whose the via transmissions cause accumulate either power leaked loss concurrent interference power neighboring 60 the signal and of from the misalignment performance quantify beam system We by impac access. the the wireless on analyze GHz misalignment we practic paper beam in this ca antenna rare may In transmissions is interference. betwe se concurrent significant beams from alignment the signals antenna overheard overcome perfect and receiving However, to and antennas attenuation. transmitting directional signal high-gain vere on relies nlsson Analysis igXa swt h omncto hoyDprmn,Roya Department, Theory Communication the with is Xiao Ming ifn ui ihNkaBl as ome,N 73,United 07733, NJ Holmdel, Labs, Bell Nokia with is Du Jinfeng un agi ihteCmuiainTer eatet Roy Department, Theory Communication the with is Yang Guang h rlfrto fdvreapiain n ead of demands and applications diverse of proliferation The ne Terms Index Abstract un Yang, Guang Hg pe ieesacs n6 H spectrum GHz 60 on access wireless speed —High — 60 emit-eedn Misalignment Beamwidth-Dependent H,Mi-oeBawdh emMisalign- Beam Beamwidth, Main-lobe GHz, tdn ebr IEEE Member, Student .I I. NTRODUCTION 60 60 H ad doae by advocated band, GHz H ieesCmuiain with Communications Wireless GHz ifn Du, Jinfeng , 60 H band, GHz lInstitute al Institute l ibution tthe st States ction igh- ions of t igh use ted em lar en nt re ebr IEEE Member, (3 d d n d r e - t atr salse nte3P tnad[9,weetenon the where [19], standard 3GPP the in established pattern oe n[7.Ohrrltdefrscnb eni [22]–[24] in seen be can line efforts piecewise related a Other using [17]. studied in been model has Bes effect discussed. side-lobe are fairness the im and gain, efficiency radiation multiplexing of persona spatial pact the wireless and mmWave (WPAN), considering for networks For antennas area exploited publications. are directional recent effect [21], side-lobe some [20], pattern in trans- in radiation studied directional instance, of of been performance impact has the The missions on [19]. alignment beam model [ and [17], 3GPP model the piece-wised the plausib and e.g., some established, and are antennas models directional practical mimicking their quantify degradation. and error performance alignment on beam impacts and to pattern crucial is beam trackin it the Therefore, invokes delay. reaction which system and [16], error mobili [15], errors; terminals array estimation communication impairments, (DoA) erro direction-of-arrival phase [13], the based analog locking-range [7], the imperfection oscillator techniques perturbations, as categories: beamforming such two and [14], into antenna divided existing of coarsely misalignme be beam of can and origin pra transmitting to The due constraints. directional rare implementation achieved is of aligned, be perfectly beams are antennas can main-lobe receiving more. the which any if gain, gain ignored only side- beam-forming main-lobe be of cannot maximum of effect misalignment reduction The the beam gradual non- by increases, the is caused nodes and radiation of radiation side-lobe density the lobe the and their As constant on not zero. depend is main- the gain largely complex: more lobe antennas more usually are of practic and implementation patterns in However, , analysis. the radiation performance [10]–[ used level idealized widely system is main- This model”, for narrow “flat-top where. the the as else referred within often zero gain of antenna and pattern large lobe radiation constant fas the idealized a an study, e.g., in previous modeled usually of is antennas most directional u In per [9]. capacity con strong network area in may experiencing improvement significant which higher of to transmissions, tribute a probability concurrent to from lower corresponds interference and beamwidth patterns. gain beam narrower antenna simplified a using general, [6]–[9] networks In in wireless studied of been performance have the on misalignment beam probabili transmissions. the current lowers among it interference area; strong unit of a within capacity network nti ae,w dp ls-oraiyatnaradiatio antenna close-to-reality a adopt we paper, this In nrcn er,nmru fot aebe eoe in devoted been have efforts numerous years, recent In of impact the and antennas directional of benefits The n igXiao, Ming and eirMme,IEEE Member, Senior ,and r, study ctical yof ty hion, ides, 18] 12] the nit ar nt ty le e, n g 1 - - - l . 2

t constant main-lobe gain and the nonzero side-lobe radiation ε 2 εr Rx2

gain are correlated via a total radiated power constraint. 2 t We measure the beam misalignment and the half-power (3- Tx φ 21 r 2 φ 12 dB) beamwidth by the ratio between their absolute value φr and the main-lobe beamwidth, and investigate the effects of φt 21 radiation pattern and misalignment on performance degrada- 12 tion of 60 GHz wireless systems. We derive the probability t r ε 1 ε 1 distribution of the signal to interference plus noise power ratio Tx Rx (SINR), where the received signal power degrades owing to the 1 1 imperfection of beam alignment, and the interference power is Fig. 1. Illustration of the two dimensional model with beam misalignment, accumulated through signals leaked from either the side-lobe concurrent transmission interference, and side-lobe signal leakage. radiation or the main-lobe beam of surrounding concurrent links. We also establish upper and lower bounds for the CDF of SINR to facilitate the computation in characterizing the where V (θm,ω) is given by network performance. We evaluate via simulations the average 2 − 2 θm 3 θm θ 10  ω2  throughput and outage probability of an indoor 60 GHz V (θm,ω)= 10 dθ. wireless communication system and quantify the impact of Z0 beam misalignment and beam pattern, and demonstrate the To highlight the main-lobe radiation pattern, we introduce trade-off in beam pattern design to balance the robustness the parameter half-power to main-lobe beamwidth ratio against interference and beam misalignment. ω The rest of the paper is organized as follows. We present the η , (0, 1) (4) θm ∈ system model in Section II and derive the probability distribu- tion of SINR in the presence of random beam misalignment to quantify the attenuation speed of the main beam gain. η 1 indicates an idealized constant-gain beam and η 0 mimics→ in Section III. In Section IV we derive the bounds for the → probability distribution of SINR performance. Performance a fast-attenuating pencil beam. evaluations are performed in Section V and conclusions are Throughout the paper we assume that the random misalign- in Section VI. ment, denoted by ε, is bounded within the range of the main- θm lobe beamwidth θm, namely, 0 ε 2 . This assumption is intuitively based on the fact≤ that | |≤ deviation II. SYSTEM MODEL exceeding the main-lobe beamwidth should be treated as align- A. Antenna Model with Beam Misalignment ment failure rather than merely an misalignment. Furthermore, The 3GPP two-dimension pattern [19] we assume that the misalignment ε follows a truncated normal 2 is adopted in our study, where the G(θ), with distribution with zero mean and variance σε , that is, respect to the relative angle θ to its boresight, is given by x2 exp 2σ2 θ 3 2θ 2 θm − ε m 10 ( ω ) fε(x)= , x , (5) Gm 10− , θ ,  θ · | |≤ 2 σ √2π erf m | |≤ 2 G (θ)= (1) ε 2√2σ  θ · ε m   Gs, θ π, θm 2 ≤ | |≤ where erf ( ) denotes the error function, and σε 0, , ∗ ∈ 6 i.e., 0 3σ θm , mimicking the 3σ-rule. The misalignment where ω denote the half-power (3 dB) beamwidth, and θm is ε 2   deviation≤ to main-lobe≤ beamwidth ratio is therefore defined as the main-lobe beamwidth. Gm and Gs represent the maximum main-lobe gain and averaged side-lobe gain, respectively. σ 1 ρ , ε 0, . (6) The total radiated power constraint [9], [20] requires that θ ∈ 6 π m   π G(θ)dθ =2π, that is, − θm R 2 π B. Network Setting 3 ( 2θ )2 Gm10− 10 ω dθ + Gsdθ = π, (2) θ We consider a network that consists of N active communi- 0 m Z Z 2 cation pairs deployed randomly within an area of interest on and the continuity of the radiation pattern (1) at the critical a two dimensional plane, where for each communication pair value θm requires θ = 2 i 1, 2,...,N , the main beam of the TX and ∈{ } i 3 θm 2 the main beam of its intended receiver RXi are approximately G = G 10 10 ( ω ) . (3) m s · aligned after appropriate channel/DoA estimation, position Combining (2) and (3), we can determine Gm and Gs analyt- tracking, and beam steering. To highlight the impact of beam ically, in terms of θm and ω, as misalignment and to simplify presentation, we assume that all the transmitters and receivers have the same antenna radiation 2π G = pattern as described in (1), and extension to heterogeneous s V (θ ,ω)+2π θ  m − m antenna patterns is straightforward. In Fig. 1 we illustrate 3 ( θm )2 ,  2π 10 10 ω a snapshot of the beam misalignment and concurrent trans- G = · m V (θ ,ω)+2π θ mission interference between two neighboring communication m − m   3

t r (n) (n) r pairs. We denote by ε and ε the beam alignment errors (i.e., f (n) r q ,e denotes the joint p.d.f. of Q ,ε , which i i Q ,ε1 1 the angle between the transmission path1 and misaligned bore- can be reduced to (due to the independence of Q(n) and εr)   1 sight) of the ith link at the transmitter and the receiver sides, (n) (n) f (n),εr q ,e = f (n) q f r (e) . respectively. The incident angle of interference (with respect Q 1 Q ε1 to the boresight of the receiver) from TXj to RXi, i = j, is     r 6 (n) r denoted by ϕji, and the departure angle of interference (w.r.t. Proposition 1. Let Q and ε1 be the set of random location t the boresight of the transmitter) is represented by ϕji. information vectors for n links and the beam misalignment The desired signal strength can therefore be represented as at RX1, respectively, the conditional p.d.f. of SINR γ1 by (7) t r (n) (n) r a function of the beam alignment errors εi and εi , and the given Q = q and ε1 = e is obtained as interference power can be written as a function of the incident (n) r t fγ Q(n),εr x q ,e = (9) angles ϕji and ϕji. The SINR at receiver RXi is written as 1| 1 | t r ∞   (n) P P G (ε ) G (ε ) L (d ) r (n) r r,i t i i i yfPr,1 Q1,ε (xy q1,e) fI Q ,ε y N0 q ,e dy, γ , = , (7) | 1 1 1 i t r N0 | | − | N0 + Ii N0 + Pt G (ϕki) G (ϕki) L (dki) Z   k=i (n) r n r 6 where fPr,1 Q1,ε ( q1,e) and fI Q( ),ε q ,e denote P | 1 ∗| 1| 1 ∗| where Pt is the transmit power, N0 is the noise power, and the conditional p.d.f. of P and I , respectively. r,1 1  G(θ) represents the antenna gain with respect to angle θ. Pr,i represents the power of the received signal, Ii is the aggregate Proof: Given two independent positive random variables interference power at RXi, di is the transmission distance from Y and W with p.d.f. fY (y) and fW (w), respectively, by TXi to RXi, and dki is the distance from TXk to RXi, k = i. applying the p.d.f. computation for the product of two random L(d) denotes the path loss at distance d, which is given by6 variables (see Appendix), it is straightforward to derive the p.d.f. of 2 λ α Y 1 L (d)= d− , X , = Y (c + W )− , 4π c + W ·   where λ is the carrier wavelength, and α is the path loss where c is a positive constant. Note that Pr,1 and I1 are (n) (n) r attenuation exponent. We assume that d d0 = 0.5 meter conditionally independent given Q = q and ε1 = e, to ensure the far field for propagation.≥ we have

(n) ∞ (n) (n) r (n) r fγ Q ,ε x q ,e = yfPr, Q ,ε xy q ,e 1| 1 1| 1 III. BEAM MISALIGNMENT AND INTERFERENCE | N0 |   Z   (n) When the mobility of user terminals is small, the SINR fI Q(n),εr y N0 q ,e dy. · 1| 1 − | observed during a small period of time relies on the positions   (n) (n) of all the active nodes. We describe the positions of an Since Pr,1 depends on Q = q only through Q1 = q1, active communication pair in the two dimensional plane by the p.d.f. of SINR γ1 can be obtained as (9). a complex vector Q = [Qt,Qr]T C2, where Qt and Qr i i i ∈ i i represent the location information of TXi and RXi, respec- tively. Likewise, all the neighboring concurrent transmissions can be captured by vectors Qj , j = i, based on which the 6 A. Distribution of Signal Power with Beam Misalignment aggregate interference Ii can be computed. For the sake of simplicity, we take the node pair (TX1, RX1) as the typical object for investigation. Let Pt denote the transmit signal power and assume that the t It is worth pointing out that, the received signal power transmit beam gain g t = G (ε ) is a random variable with Pr,1 ε1 1 r t depends on both ε1 and ε1, and the interference power I1 associated p.d.f. fg (x), x [Gs, Gm]. The received signal r t ∈, t r T C2 r depends on ε1 and εj, for j = 2, ..., n. Therefore, Pr,1 is power Pr,1 given Q1 = q1 [q1, q1] and ε1 = e can r ∈ correlated with I1 through ε1. Given the set of n random lo- be reformulated as (n) cation information vectors, namely, Q , (Q1, Q2,..., Qn) r P r =P L (d ) G (e) g t , r,1 Q =q ,ε =e t 11 ε1 (10) and the beam misalignment ε1 at RX1, the probability density 1 1 1 · function (p.d.f.) of the SINR γ can be expressed as 1 where d , qr qt represents the length of the link. The con- 11 | 1− 1| r (n) ditional p.d.f. fPr, Q ,ε (x q1,e) can therefore be determined fγ (x)= f (n) r x q ,e 1 1 1 1 γ1 Q ,ε1 | | ··· | | by the p.d.f. of gεt as shown below. Z Z (8) 1 (n)  fQ(n),εr q ,e dq1 ...dqnde, 1 Proposition 2. Let f t (y), y θ /2 be the p.d.f. of beam ε1 m t | |≤ (n)   misalignment ε , the conditional p.d.f. fP Q ,εr (x q1,e) where (n) r q is the conditional p.d.f. of 1 r,1 1 1 fγ1 Q ,ε x ,e γ1 | | | 1 | given Q =q and εr=e is written as given Q(n),εr = q(n),e , with q(n) , (q , q ,..., q ). 1 1 1 1  1 2 n 1 x f r (x q ,e)= f , 1   Pr,1 Q1,ε1 1 gεt Here we assume line-of-sight (LOS) transmission in a short distance where | | PtL (d11) ge 1 PtL (d11) ge the “optical” LOS path provides the highest gain (i.e., lowest loss). Otherwise   the solid lines represent the logical LOS paths that provide the highest gain. where ge=G (e) as described by the radiation pattern (1) and 4 the p.d.f. f (x) for x [G , G ] can be written as Since the component interference I given Q(n)=q(n) and gεt s m j1 1 ∈ r ε1=e can be reformulated as 5 Gm ωf t ω log r ε1 6 10 x I n n r = P L (d ) G ϕ g t , (12) j1 Q( )=q( ),ε =e t j1 j1 ϕj1   (11) 1 · fgεt (x)= q . 1  6 Gm , r t  ln(10)x 5 log10 x where dj 1 q1 qj is the distance between TXj and RX1, , | −t | t q and gϕt G ϕ is a function of random variable ϕ , we Proof: Since 0 e θ /2, for g =G (e) we can j1 j1 j1 m e will establish in Lemma 2 the conditional p.d.f. of ϕt . derive from (1) that ≤ ≤  j1 (n) (n) t Lemma 2. Given Q = q , the departure angle ϕ 1 ω 10 Gm j ∈ e = log10 . [0, π] of the interfering link (TXj , RX1) can be written as 2 s 3 ge   t t t t t 2π ϕˆj1 εj , ϕˆj1 εj π, Note that the function ge = G(e) is differentiable within the ϕ = | − | − || | − |≥ (13) j1 Q(n)=q(n) t t otherwise interval 0 e<θm/2, we have ( ϕˆj1 εj , , ≤ | − | qr qt t , ∠ j − j 12 ln (10) 2 ln (10) 6 Gm where ϕˆj1 r t [ π, π) represents the signed q1 qj G′(e)= 2 ege = ge log10 , − ∈ − − 5ω − ω s5 ge angle2 under perfect beam alignment given Q(n) = q(n). Its   (n) conditional p.d.f. fϕt Q(n) q is given by and the p.d.f. of ge can be straightforwardly derived from the j1| ∗| p.d.f. fǫ(e) given in (5), as shown in (11). We can now apply (n)  fϕt Q(n) x q = 1 Fz Q(n) (2π) fz Q(n) (x +2π) (11) to (10) to conclude the proof. j1| | − | |   + Fz Q(n) (2π) FzQ(n) (π) fz Q(n) (2 π x) (14) | − | | − + Fz Q(n) (π) fz Q(n) (x) , B. Distribution of Interference Power | | 

(n) n where fz Q ( ) and Fz Q(n) ( ) denote the conditional p.d.f. | | Let I1 = j=2 Ij1 be the sum interference power where Ij1 ∗ ∗ t t and c.d.f., respectively, of z = ϕˆj1 εj , with is the interference power from the jth concurrent transmission | − | P t t f (n) (x)= f t ϕˆ + x + f t ϕˆ x . (15) to RX1. In Lemma 1, we show that Ij1, j = 2, 3,...,n are z Q εj j1 εj j1 (n) (n) r | − conditional independent given Q = q and ε1 = e. Proof: In the absence of beam misalignment,  the angle Lemma 1. Let I , j = 2, 3,...,n, denote the interfer- t j1 ϕˆj1 that represents the angle-of-departure (AoD) at TXj ence power to RX1 from TXj , the conditional joint p.d.f. is determined by Q(n)=q(n). The AoD with misalignment, (n) f (n) r x ,...,x q ,e can be written as t t I21,...,In1 Q ,ε 2 n denoted as ϕ , is the sum of the deterministic ϕˆ and a | 1 | j1 j1 stochastic εt , as modeled in (13). Setting z= ϕˆt εt , its  (n) j j1 j (n) r | − | fI ,...,In Q ,ε x2,...,xn q ,e 21 1| 1 | conditional c.d.f. Fz Q(n) (t) can be expressed as n   | (n) t t r n P = fIj1 Q1,Qj ,ε (xj q1, qj ,e) , Fz Q( ) t q = ϕˆj1 εj t , | 1 | | j=2 | | − |≤ Y    from which the conditional p.d.f. fz Q n t q(n) can be where f r ( q , q ,e) , j = 2,...,n, is the condi- | ( ) Ij1 Q1,Qj ,ε1 1 j | | ∗| r obtained. Furthermore, we have tional p.d.f. of Ij1 given both Q1 = q1, Qj = qj and ε1 = e.  (n) P P (n) (n) r Fϕt Q(n) y q = (z 2π y) (z 2π) Proof: Given Q = q and ε1 = e, it is easy to obtain j1| | − ≤ ≥ that + P (2π z y)P (π z < 2π)+ P (z y) P (z < π) . − ≤ ≤ ≤ (n) (n) (n) r q fI21,...,In1 Q ,ε x2,...,xn ,e Taking the first derivative of Fϕt Q(n) y q with respect | 1 | j1| | n   to y leads to (14). (a) (n)  = f (n) r x x ,...,x , q ,e r Ij1 I(j+1)1 ,...,In1,Q ,ε j (j+1) n Likewise, the arrival angle ϕ [0, π] of the interfering | 1 | j1 j=2 (n) (n)∈ r   link (TXj , RX1) given Q =q and ε =e is written as Yn 1 (b) (n) r r (n) r = fIj Q ,ε xj q ,e 2π ϕˆ e , ϕˆ e π, 1| 1 | ϕr = | − | j1 − || | j1 − |≥ (16) j=2   j1 q(n),e ϕˆr e , otherwise, Yn ( j1 (c) | − | = fI Q ,Q ,εr (xj q1, qj ,e) , qt qr j1 1 j 1 r ,∠ 1− 1 | | where ϕˆj1 qt qr [ π, π) is the angle corresponding j=2 j − 1 ∈ − Y to the perfect beam alignment given Q(n)=q(n). where (a) applies the chain rule of conditional p.d.f. for multivariate random variables, (b) comes from the fact that (n) r ′ Ij1 (Q ,ε1) Ij 1 forms a markov chain for all j′ = j, −− −− (n) (n) 6 (c) is due to the dependence of I on Q = q only u j1 2Given two complex variables u1 and u2, the signed angle ∠ 1 denotes  u2  through the the pair (TXj , RX1), which reduces the condition the rotated angle from u1 to u2, which is defined to be negative if the rotation (n) (n) Q = q to Q1 = q1 and Qj = qj . occurs in the clockwise direction. 5

Proposition 3. The p.d.f. of I given Q , Q ,εr is conditional c.d.f. P (γ x q ,e) as j1 1 j 1 1 ≤ | 1 f r (x q , q ,e) (17) Y Ij1 Q1,Qj ,ε1 1 j P , P | | (γ1 x q1,e) x , (18) 1 x ≤ | N0 + W ≤ = f q , q ,   gϕt Q1,Qj 1 j P PtL (dj1) gϕr j1 | PtL (dj1) gϕr , K+1 j1  j1  where W j=2 Wj . Denote the c.d.f. of Y and W

by FY ( )Pand FW ( ), respectively. FY (x) can be imme-P where, for x [Gs, Gm], we have ∗ P ∗ ∈ diately obtained byP applying Proposition 2, i.e., FY (x) = x 5 Gm (n) fY (t) dt. For the sum interference W with respect to ωfϕt Q(n) ω 6 log10 x q 0 j1| f x q(n) =  . K (K 1) interfering transmitters, fromP Proposition 4, we gϕt Q1,Qj q R ≥ j1  | | ln(10)x 6 log Gm know that   5 10 x x x K+1 q  Proof: By (12), (16), and Lemma 2, it is straightforward FW (x)= fW (t)dt = fWj (t)dt. to obtain the results by applying the similar method as shown P 0 P 0 j=2 Z Z O in Proposition 2. Instead of directly computing the convolution of p.d.f., We can now derive the conditional p.d.f. of I1 as follows. FW (x) can be alternatively obtained by P Proposition 4. The conditional p.d.f. of the sum interference 1 (n) (n) r 1 E given Q = q and ε1 = e is given by FW (x)= − exp sW (x) P L s − n  P  (n) 1 1  K n r r fI Q( ),ε x q ,e = fIj1 Q1,Qj ,ε (x q1, qj ,e) , = − fWj (s) (x) , 1| 1 | | 1 | L s L   j=2   O 1   where represents the convolution operator. where and − denote Laplace transform and its inversion, respectively,L andL s> 0. Proof:N Since Ij1, j=2, 3,...,n, are conditionally inde- We are ready to derive the upper and lower bounds using (n) (n) r pendent given Q =q and ε1=e, the p.d.f. of the sum of FY ( ) and FW ( ), shown in the following theorem. independent random variables equals the convolution of all the ∗ P ∗ individual probability functions. Theorem 1. Let FY ( ) and FW ( ) denote the c.d.f. of Y ∗ P ∗ Finally, the conditional p.d.f. of SINR in Proposition 1 and W , respectively, then we have P can be obtained by applying Proposition 2 and Proposition 4, Y (x) P x (x) , which is then used to compute the p.d.f. of SINR using (8). B ≤ N + W ≤ ≤ B  0  P where (x) and (x) are respectively given by IV. CDF OF SINR: UPPER BOUND AND LOWER BOUND B B (x) , sup FY ((N0 + t)x) FW (t) , It is rather involved to directly evaluate the SINR per- B t 0 − P ≥ formance based on the equations derived in the Sec. III, and  partially due to the convolution of p.d.f. in Proposition 4. For , scenarios where there are K interfering transmitters distributed (x) 1 + inf FY ((N0 + t)x) FW (t) . B t 0 − ≥ P uniformly at random around the receiving node RX1, whose r Proof: For the upper bound, for any t 0, we have location Q1 = q1 and beam misalignment ε1 = e are given, ≥ we derive upper and lower bounds on the c.d.f. of SINR for P Y P Y . According to Lemma 1, we know that, in the presence x = x, W t RX1 N0 + W ≤ N0 + W ≤ ≥ of the given Q and r, the component interference power ,    P  1 ε1 Ij1 P YP j 2,...,K+1 , can be treated as independent random vari- + P x, W t ∈{ } N0 + W ≤ ≤ ables. Furthermore, Ij1 are also identically distributed random  P  P W t +PP(Y (N +t) x) , variables due to the uniform deployments and orientations. ≤ ≥ ≤ 0 Thus, the interference can be viewed as independent and P Ij1 then, the upper bound (x) can be immediately obtained. identically distributed (i.i.d.) random variables. For the lower bound,B likewise, Following Lemma 1 and Proposition 3, we can obtain the Y Y conditional probability f r ( q ,e) by marginalizing Ij1 Q1,ε1 1 P x =P x, W t | ∗| N + W ≥ N + W ≥ ≥ out the variable Qj, which covers the location information  0   0 P  of the jth transmission pair. Since only Qt is required for the P YP j + P x, W t marginalization process, we have N + W ≥ ≤  0 P  P t t t P W t +P (Y (N0 +t) x) r t r t fIj1 Q1,ε (x q1,e)= fIj1 Q1,Q ,ε x q1, qj ,e fQ qj dqj . ≤ ≤ ≥ | 1 | | j 1 | j P Z holds for any t 0, and it subsequently  gives For notational simplicity, we use Y and W , 2 j K +1, ≥ j ≤ ≤ Y respectively to represent the random variables Pr,1 and Ij1 P x P (Y (N0 +t) x) P W t , r N0 + W ≤ ≥ ≤ − ≤ conditional on Q1 = q1 and ε1 = e. We can then rewrite the   P P  6

TABLE I SYSTEM PARAMETERS

Parameter Notation Value Wavelength λ 5 × 10−3 m Bandwidth W 500 MHz

Transmit Power Pt 1 mW π π Main-lobe Beamwidth θm  12 , 2  3dB-Beamwidth Ratio η (0, 1) 1 Misalignment Deviation ρ 0, 6 

Noise Power Density N0/W −114 dBm/MHz

Radius of Circular Hall R0 15 m Path Loss Exponent α 2.45 Fig. 2. Illustration of multiple interfering transmitters in the area of interest. Link Numbers n ≤ 30

which concludes the lower bound (x). 1 r B Sim.:θ =π/12 Note that given Q1 and ε , the outage probability can be 0.9 m 1 L.B.:θ =π/12 m expressed as the c.d.f. of γ , i.e., 0.8 U.B.:θ =π/12 1 m Sim.:θ =π/6 0.7 m , P Rth/W L.B.:θ =π/6 p1,out(Rth) (R1 < Rth)= Fγ1 2 1 , m U.B.:θ =π/6 − 0.6 m Sim.:θ =π/3   m where R denotes the rate threshold. Therefore tight bounds 0.5 th L.B.:θ =π/3 m on the c.d.f. are essential in evaluating the performance. c.d.f. F(x) U.B.:θ =π/3 0.4 m

0.3

V. PERFORMANCE EVALUATION 0.2

We consider a 60 GHz indoor wireless access network 0.1 within a circular space of radius R0 = 15 meters, as illustrated 0 0 5 10 15 20 25 in Fig. 2, where there are in total N = n concurrent SINR x (dB) transmissions. The receiving node in focus, RX1, is located at the center of a circular area and there are totally K = n 1 Fig. 3. The lower and upper bounds of c.d.f.s for γ1 and simulated results, − 1 interfering transmitters distributed uniformly at random within with respect to diverse θm, where n = 11, η = 0.4, and ρ = 20 . the area of interest, randomly oriented in a uniform manner. Results by numerical and Monte-Carlo methods are presented to investigate the accuracy of the bounds, and the sensitivity the derived bounds in Theorem 1 behave well for all groups, of outage probability and average throughput against beam which validates the feasibility of applying our upper and lower patterns and misalignment. To simplify the performance eval- bounds in analyzing the actual system performance. Note that uation, all nodes are assumed to be placed on the same hori- in all combinations we have considered here, the lower bound zontal plane. The common system parameters are summarized outperforms its upper counterpart. Furthermore, considerable in Table I and the p.d.f. of link lengths can be found in [25]. performance gain can be achieved by narrowing down the π Our evaluation consists of the two parts: beamwidth. For instance, when θm = 3 reduces to its half, π 1) Numerical results to validate the bounds for the fixed i.e., θm = 6 , there is roughly 4 dB gain, and there is another 3 dB gain when θ keeps going down to θ = π . typical receiver RX1, as depicted in Fig. 2. m m 12 2) Simulation results to evaluate the average performance In Fig. 4, we demonstrate the bound performance against π 1 of randomly deployed typical receivers. the factor η, where n = 21, θm = 6 and ρ = 20 . Again, both the upper and lower bounds are very tight. Despite of a π narrow beamwidth, i.e., θm = 6 , is employed, there is still A. Bounds for Fixed Typical Receiver at The Center huge performance difference for different η. As shown in the We validate the bounds derived in Theorem 1 for the fixed figure, a substantial gain can be achieved by decreasing η (i.e., typical receiver and investigate the impact of the main-lobe a faster attenuating main-lobe). For instance, the performance beamwidth θm and the half-power beamwidth ratio η on the gains roughly 6 dB when decreasing η from 0.6 to 0.5, while c.d.f. of SINR. The lower and upper bounds on the c.d.f. of roughly 10 dB gain can be achieved by η =0.4. This indicates SINR are illustrated in Fig. 3, where n = 11, η = 0.4, and the great importance of η in the antenna design. 1 ρ = 20 . To investigate the impact of θm and the associated The above results show that, both the main-lobe beamwidth factor η, we consider the following three distinct values of and the half-power beamwidth ratio are crucial factors that π π π beamwidth, i.e., θm = 12 , 6 , and 3 , respectively. In general, determine the performance. In what follows, we will consider 7

1 70 Sim.:η=0.4 0.9 L.B.:η=0.4 θ =π/12 U.B.:η=0.4 60 m 0.8 = /6 Sim.:η=0.5 θm π L.B.:η=0.5 θ =π/3 0.7 U.B.:η=0.5 50 m θ =π/2 Sim.:η=0.6 m 0.6 L.B.:η=0.6 U.B.:η=0.6 40 0.5

c.d.f. F(x) 30 0.4

0.3 20 Average Throughput (Gbps) 0.2 10 0.1

0 0 -15 -10 -5 0 5 10 15 20 0 5 10 15 20 25 30 SINR x (dB) Number of Links n

Fig. 4. The lower and upper bounds of c.d.f.s for γ1 and simulated results, Fig. 6. The average sum throughput with varying n ≤ 30, where θm = π 1 π π π π 1 1 with respect to diverse η, where n = 21, θm = 6 and ρ = 20 . 12 , 6 , 3 or 2 , ρ= 20 , and η= 2.6 .

1 transmission interference is not significant except for users (θ ,η)=(π/12,0.2), n=30 0.9 m ( , )=( /12,0.2), n=1 with low signal power. If we then hold η=0.2 (i.e., fast θm η π 0.8 ( , )=( /2,0.2), n=30 θm η π attenuation beam) but increase the main-lobe beamwidth θm ( , )=( /2,0.2), n=1 π π (30) (1) θm η π 0.7 from to , the gap between Fγ1 (x) and Fγ1 (x), with a (θ ,η)=(π/12,0.8), n=30 12 2 m gap of around dB at -percentile and 1dB at -percentile. ( , )=( /12,0.8), n=1 5 10 90 0.6 θm η π ( , )=( /2,0.8), n=30 θm η π Interestingly, if we set η=0.8, i.e., the main-lobe has almost 0.5 (θ ,η)=(π/2,0.8), n=1 m a constant-gain top, the gap between upper and lower bounds c.d.f. F(x) 0.4 remains almost a constant of 8dB from 10-percentile up to 0.3 90-percentile, and the influence of the main-lobe beamwidth π 0.2 θm is very limited: less than 1 dB gap between θm = 12 and π . Therefore, when beam misalignment is small, the 0.1 θm = 2 main-lobe attenuation speed, quantified by η, dominates the 0 -100 -80 -60 -40 -20 0 20 40 60 sensitivity to concurrent transmission interference. SINR or SNR x (dB) In Fig. 6 we plot the average sum throughput as a function of the number of active links n ranging from 1 to 30, where Fig. 5. The c.d.f.s of SINR γ1 with diverse (θm, η) pairs for n = 1 or 30, 1 π π π π where ρ = 20 . the main-lobe beamwidth θm is set to 12 , 6 , 3 , and 2 , respectively, with fixed misalignment derivation to main-lobe 1 1 beamwidth ratio ρ= 20 and half-power beamwidth ratio η= 2.6 , the scenario where the typical receiver is randomly located. which is adopted from the experiment validation in [26]. As the number of active links increases, the average sum π throughput increases much faster for narrow beam θm= 12 B. Simulations for Randomly Located Typical Receiver π compared to wide beam θm= 2 , as determined by the slopes In contrast to the aforementioned scenario with fixed Q1 and of the curves. This is in line with our observations from Fig. 5 r ε1, we here focus on the situation where the typical receiver is where, when the main-lobe attenuates fast, the links with small randomly located, and the misalignment is not given. Besides main-lobe beamwidth are more or less noise/power limited the outage probability, in this section, the sum throughput and whereas the links with large main-lobe are interference limited. the average throughput are also evaluated. In Fig. 7 we investigate the sensitivity of the per-link aver- In Fig. 5, we illustrate the c.d.f. of γ1 with the fixed age throughput against ρ with a fixed half-power beamwidth misalignment derivation to main-lobe beamwidth ratio ρ= 1 , 1 π 20 ratio η = 2.6 . We investigate two groups with θm = 6 and where the main-lobe beamwidth θ is set to be π or π , and π m 12 2 3 , respectively, with the number of active links n = 10, 20 the half-power beamwidth ratio η is chosen as 0.2 or 0.8. or 30. For any given ρ, the per-link average throughput will (1) (30) We denote by Fγ1 (x) and Fγ1 (x) the outage probabilities decrease significantly as the main-lobe beamwidth θm and/or associated with n = 1 (hence no concurrent transmission the number of active links n increases, which clearly attributes π interference) and n = 30, respectively. For (θm, η)= 12 , 0.2 to the increase of the concurrent transmission interference. that corresponds to the scenario where the main-lobe is narrow Such per-link performance degradation (gap among different  and the beam attenuates fast, there is only a small gap between lines) decreases slightly as the misalignment increases. For (30) (1) Fγ1 (x) and Fγ1 (x). This is in line with our intuition fixed n and θm, the per-link average throughput remains stable that when the receive beam is very narrow and the beam for ρ < 0.05 and the degradation grows up to about 30% misalignment is small, the degradation caused by concurrent as ρ 1 . Regarding the practical significance, on the one → 6 8

respectively. Through the function of multivariate random 2.5 variables [27], we have f (y,z) f (x, v)= Y,Z , 2 X,V (y,z) |Jx,v | where x,v (y,z) is given by 1.5 J ∂x ∂x ∂y ∂z cz cy x,v (y,z)= ∂v ∂v = = cv, 1 θ =π/6,n=10 J 0 1 m ∂y ∂z θ =π/3,n=10 m θ =π/6,n=20 m thus we have 0.5

Per-Link Average Throughput (Gbps) θ =π/3,n=20 m 1 x = /6,n=30 θm π fX,V (x, v) = ( c v)− fY,Z , v . = /3,n=30 | | cv θm π 0   10-2 10-1 Finally, the p.d.f. fX (x) can be immediately obtained by beamwidth-dependent misalignment coefficient ρ the integral over all possible v. That is,

Fig. 7. Sensitivity of the per-link average throughput against ρ, where η = 1 x 1 π π fX (x)= ( c v)− fY,Z , v dv, 2.6 , θm = 6 or 3 , and n = 10, 20 or 30. | | cv v Z z ∈S   where z corresponds to the domain of marginal p.d.f. of Z, hand, it is beneficial to reduce beam misalignment, but the S namely, fZ (z). reward is diminishing as becomes smaller. Since a high ρ Particularly, if Y and Z are independent random variables, alignment precision indicates a high overhead/cost in practical we further have implementations, a quantitative evaluation of the performance 1 x loss is crucial to seek the proper trade-off between the fX (x)= ( c v)− fY fZ (v) dv. performance and cost. On the other hand, the performance | | cv v Z z   degradation caused by ρ remains almost the same as the main- ∈S π π lobe beamwidth increases from 6 to 2 , which clearly justifies our effort in quantifying the misalignment via ρ. REFERENCES

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