Directional Beamforming for Millimeter-Wave MIMO Systems

Vasanthan Raghavan, Sundar Subramanian, Juergen Cezanne and Ashwin Sampath Qualcomm Corporate R&D, Bridgewater, NJ 08807 E-mail: {vraghava, sundars, jcezanne, asampath}@qti.qualcomm.com

Abstract—The focus of this paper is on beamforming in a that allows the deployment of a large number of antennas in millimeter-wave (mmW) multi-input multi-output (MIMO) set- a fixed array aperture. up that has gained increasing traction in meeting the high data- Despite the possibility of multi-input multi-output (MIMO) rate requirements of next-generation wireless systems. For a given MIMO channel matrix, the optimality of beamforming communications, mmW signaling differs significantly from with the dominant right-singular vector (RSV) at the transmit traditional MIMO architectures at cellular frequencies. The end and with the matched filter to the RSV at the receive most optimistic antenna configurations1 at cellular frequencies end has been well-understood. When the channel matrix can are on the order of 4 × 8 with a precoder rank (number of be accurately captured by a physical (geometric) scattering layers) of 1 to 4; see, e.g., [9]. Higher rank signaling requires model across multiple clusters/paths as is the case in mmW 2 MIMO systems, we provide a physical interpretation for this multiple radio-frequency (RF) chains which are easier to real- optimal structure: beam steering across the different paths with ize at lower frequencies than at the mmW regime. Thus, there appropriate power allocation and phase compensation. While has been a growing interest in understanding the capabilities of such an explicit physical interpretation has not been provided low-complexity approaches such as beamforming (that require hitherto, practical implementation of such a structure in a mmW only a single RF chain) in mmW systems [10]–[15]. system is fraught with considerable difficulties (complexity as well as cost) as it requires the use of per-antenna gain and phase On the other hand, smaller form factors at mmW frequencies control. This paper characterizes the loss in received SNR with an ensure3 that configurations such as 4 × 64 are realistic. Such alternate low-complexity beamforming solution that needs only high antenna dimensionalities as well as the considerably large per-antenna phase control and corresponds to steering the beam bandwidths at mmW frequencies result in a higher resolvabil- to the dominant path at the transmit and receive ends. While the loss in received SNR can be arbitrarily large (theoretically), this ity of the multipath and thus, the MIMO channel is naturally loss is minimal in a large fraction of the channel realizations sparser in the mmW regime than at cellular frequencies [16]– reinforcing the utility of directional beamforming as a good [18]. In particular, the highly directional nature of the channel candidate solution for mmW MIMO systems. ensures the relevance of physically-motivated beam steering at either end, which is difficult (if not impossible) at cellular I. INTRODUCTION frequencies. While this physical connection has been implicitly The ubiquitous nature of communications made possible by and intuitively understood, an explicit characterization of this the smart-phone and social media revolutions has meant that connection has remained absent so far. the data-rate requirements will continue to grow at an expo- We start with such an explicit physical interpretation in nential rate. On the other hand, even under the most optimistic this work by showing that the optimal beamformer structure assumptions, system resources can continue to scale at best at corresponds to beam steering across the different paths that a linear rate leading to enormous mismatches between supply capture the MIMO channel with appropriate power allocation and demand. Given this backdrop, many candidate solutions and phase compensation. We also illustrate the structure of this have been proposed [1]–[3] to mesh into the patchwork that power allocation and phase compensation in many interesting 1000 addresses the -X data challenge [4] — an intermediate special cases. Despite using only a single RF chain, the optimal stepping stone towards bridging this burgeoning gap. beamformer requires per-antenna phase and gain control (in One such solution that has gained increasing traction over general), which could render this scheme disadvantageous the last few years is communications over the millimeter- from a cost perspective. Thus, we study the loss in received wave (mmW) regime [5]–[8] where the carrier frequency is SNR with a simpler scheme that requires only phase control 30 300 in the to GHz range. Spectrum crunch, which is the and steers beams to the dominant path at either end. Our study major bottleneck at lower/cellular carrier frequencies, is less shows that this simpler scheme suffers only a minimal loss problematic at higher carrier frequencies due to the availability of large (either unlicensed or lightly licensed) bandwidths. 1In a downlink setting, the first dimension corresponds to the number of However, the high frequency-dependent propagation and shad- antennas at the user equipment end and the second at the base-station end. owing losses (that can offset the link margin substantially) 2An RF chain includes (but is not limited to) analog-to-digital and digital- complicate the exploitation of these large bandwidths. It is to-analog converters, power and low-noise amplifiers, mixers, etc. 3For example, a 64 element uniform linear array (ULA) at 30 GHz requires visualized that these losses can be mitigated by limiting an aperture of ∼ 1 foot at the critical λ/2 spacing — a constraint that can coverage to small areas and leveraging the small wavelengths be realized at the base-station end.

978-1-4799-5952-5/15/$31.00 ©2015 IEEE relative to the optimal beamforming scheme in a large fraction Nr × 1 proper complex white Gaussian noise vector (that is, of the channel realizations, thus making it attractive from a n ∼CN(0, I)) added at the receiver. The symbol s is decoded practical standpoint. by beamforming at the receiver along the unit-norm Nr × 1 Notations: Lower- (x) and upper-case block (X) letters denote vector g to obtain vectors and matrices with x(i) and X(i, j) denoting the i-th √ s = gH y = ρ · gH Hfs + gH n. and (i, j)-th entries of x and X, respectively. x2 denotes the 2-norm of a vector x, whereas xH and xT denote the complex Let F2 denote the class of energy-constrained beamforming conjugate Hermitian and regular transposition operations of x, vectors. That is, F2 = {f : f2 ≤ 1}. Under perfect channel Z R R+ C respectively. We use , , and to denote the field of state information (CSI) (that is, H = H) at both the transmitter integers, real numbers, positive reals and complex numbers, and the receiver, optimal beamforming vectors fopt and gopt are respectively. The proofs of all the statements in this paper to be designed from F2 to maximize the received SNR [21], have not been provided here due to space constraints. defined as, II. SYSTEM SETUP | H H |2 · [|s|2] | H H |2 SNR ρ · g f E = ρ · g f . rx H 2 H Let H denote the Nr × Nt channel matrix with Nr receive E [|g n| ] g g and Nt transmit antennas. We assume an extended Saleh- Clearly, the above quantity is maximized with fopt2 =1, Valenzuela geometric model [19] for the channel where H otherwise energy is unused in beamforming. Further, a simple is determined by scattering over L paths and is denoted as application of Cauchy-Schwarz inequality shows that gopt is follows: a matched filter combiner at the receiver with gopt2 =1 L SNR = ρ · H HH H NrNt H resulting in rx f f. We thus have H = · α · u v (1) L Hv (HH H) =1 = v (HH H), = 1 , fopt 1 gopt Hv (HH H) (2) where α ∼CN(0, 1) denotes the complex gain, u denotes 1 2 Nr × 1 H the receive array steering vector, and v denotes where v1(H H) denotes a dominant unit-norm right singular N × 1 the t transmit array steering vector, all corresponding vector (RSV) of H. Here, the singular value decomposition of  H to the -th path. As a typical example of the case where a H is given as H = UΛV with U and V being Nr × Nr and ULA of antennas are deployed at both ends of the link (and Nt × Nt unitary matrices of left and right singular vectors, without loss of generality pointing along the X axis in a certain respectively, and arranged so that the corresponding leading global reference frame), the array steering vectors u and v diagonal entries of the Nr × Nt singular value matrix Λ are φ corresponding to angle of arrival (AoA) R, and angle of in non-increasing order. departure (AoD) φT, in the azimuth (assuming an elevation o angle θR, = θT, =90 )aregivenas III. EXPLICIT CONNECTION BETWEEN fopt, gopt AND 1 T PHYSICAL DIRECTIONS jkdR cos(φR,) j(Nr −1)kdR cos(φR,) u = √ · 1 e ··· e Nr A typical sparse mmW channel can be assumed to consist 1 T of a small number of dominant clusters (say, L =2or jkdT cos(φT,) j(Nt−1)kTd cos(φT,) v = √ · 1 e ··· e 3 Nt ) [7], [16]–[18], [22]. For example, a dominant line-of-sight (LOS) path with strong reflectors in the form of a few glass k = 2π λ where λ is the wave number with the wavelength windows of buildings in the vicinity of the transmitter or the of propagation, and dR and dT are the inter-antenna element receiver could capture an urban mmW setup. In the context spacing at the receive and transmit sides, respectively. To of such a sparse mmW channel H, the intuitive meaning simplify the notations and to capture the constant phase offset of fopt is to “coherently combine” (by appropriate phase (CPO)-nature of the array-steering vectors and the correspon- compensation) the energy across the multiple paths so as to dence with their respective physical angles, we will hence- maximize the energy delivered to the receiver. The precise CPO(φ ) CPO(φ ) forth denote u and v above as R, and T, , connection between the physical directions {φR,,φT,} in the d = d = λ respectively. With the typical R T 2 spacing, we have ULA channel model in (1) and fopt in (2) is now established. kdR = kdT = π. More general expressions can be written Towards this goal, a preliminary result is first provided. for the array steering vectors when the antennas are laid out = H according to other geometric configurations [13], [20]. Proposition 1. With H and the channel model in (1), HH H We are interested in beamforming (rank-1 signaling) over H all the eigenvectors of can be represented as linear 1, ··· , L opt with the unit-norm Nt × 1 beamforming vector f. The system combinations of v v . Thus, f is a linear combina- , ··· , model in this setting is given as tion of of v1 vL and gopt is a linear combination of √ u1, ··· , uL. y = ρ · Hfs + n It is important to note that while the right singular vectors where ρ is the pre-beamforming SNR, s is the symbol chosen of H (also, the eigenvectors of HH H) are orthonormal by from an appropriate constellation for signaling, and n is the construction, v1, ··· , vL need not be orthonormal. With this L · H HH H = β2|α |2 +(1− β2)|α |2 + β2|α |2 +(1− β2)|α |2 | H |2 + N N f f 1 2 2 1 v1 v2 t r H H 2 2 2 H 2|α1||α2|·|v v2|·|u u2|·cos (ν)+2β 1 − β · |α1| + |α2| ·|v v2|·cos (φ)+ 1 1 1 2 H H 2 2β 1 − β ·|α1||α2|·|u1 u2|· |v1 v2| · cos (ν + φ)+cos(ν − φ) (4)

background, Prop. 1 provides a non-unitary basis for the eigen- We start with a physical interpretation for the inner product H H space of H H when L ≤ Nt. As another ramification of this between u1 and u2 (a similar interpretation holds for v1 v2) fact, in the case where L>Nt, it is still true that the set , corresponding to CPO beams in two directions/paths. With H {v1, ··· , vL} spans the eigen-space of H H, however this set the assumption for u as earlier, we have is no longer a basis. These facts along with the fact that fopt sin (N π Δcos(φ )/2) is a dominant eigenvector of HH H also implies the following: H = ejμ · r R u1 u2 N sin (π Δcos(φ )/2) (5) H H H H r R SNRrx f H H f f H H f = max = max (3) π(N −1)·Δcos(φ ) ρ f : f =1 H H r R 2 f f f : f ∈ G f f where μ = 2 and Δcos(φR) cos(φR,2) − H cos(φR,1). Clearly, the largest magnitude of u1 u2 is 1 which L jθi + G : = e βi i βi ∈ R ,θi ∈ where f f i=1 v with is achieved when Δcos(φR)=0(or when the two paths [0, 2π),i=1, ··· ,L. Without loss in generality, we can set can be coherently combined in the physical angle space). L−1 2 0 βL = 1 − j=1 βj and θ1 =0in the definition of G Further, the smallest magnitude of is achieved in (5) when 2n Δcos(φR)= ,n ∈ Z\{0} to reduce the optimization in (3) to a 2(L − 1)-dimensional Nr . We denote this condition as optimization over G, defined as, electrical orthogonality (or simply, orthogonality) between the ⎧  ⎫ two paths, which is achievable with higher regularity in the ⎨ L−1  L−1 ⎬  physical angle space as Nr increases. G : = β + ejθi β + ejθL 1 − β2 . ⎩f f 1 v1 i vi j vL⎭ i=2 j=1 A. v1 and v2 are Orthogonal In other words, the optimization over the space of G should Proposition 2. When v1 and v2 are electrically orthogonal, result in the dominant right singular vector of HH H.Note the non-unit-norm version of fopt is given as that the constraint set in the optimization over G is the outer product of a (L − 1)-dimensional real sphere where H  j(∠α1−∠α2−∠u u2) 2 L−1 2 L fopt = βoptv1 + e 1 1 − βoptv2 i=1 βi ≤ 1 with a cuboid i=2 θi, 0 ≤ θi < 2π. We now consider the special case where L =2and perform this optimization and thus provide a physical interpretation of where H ⎡ ⎤ fopt and gopt. For this, note that H H simplifies in the L =2 1 |α |2 −|α |2 case to 2 ⎣ 1 2 ⎦ βopt = 1+ . L 2 2 2 2 2 2 H 2 H 2 H 2 H  (|α1| −|α2| ) +4|α1| |α2| ·|u1 u2| · H H = |α1| · v1v1 + |α2| · v2v2 + α1α2 · NtNr H H  H H The non-unit-norm version of gopt follows from expanding out (u1 u2) · v1v2 + α2α1 · (u2 u1) · v2v1 . H fopt and is not provided here. With β1 = β and θ2 = θ in the archetypical f from G,the 2 norm of f is given as While the structure of βopt is hard to visualize in general, H | 2| H 2 H Fig. 1(a) plots it as a function of u1 u for different choices f f =1+2β 1 − β ·|v1 v2|·cos(φ) |α1| K = 1 2 of |α2| .FromFig.1(a),ifu and u are orthogonal, H β 1 0 where φ θ + ∠v1 v2. Further, a tedious but straightforward we see that opt is either or with full power allocated to calculation shows that f H HH Hf is as in (4) at the top of the the strongest path. In addition, a straightforward calculation H H page, where ν ∠v1 v2 −∠u1 u2 +∠α1 −∠α2. Observe that shows that the phase term ν captures the phase misalignment between the |α |2 H | H |→1=⇒ β2 → 1 . two paths since |ui H vj| is maximized for all {i, j}∈1, 2 u1 u2 opt 2 2 |α1| + |α2| when ν =0(coherent phase alignment). We now consider many special cases in terms of the physical B. u1 and u2 are Orthogonal model to study the performance of the optimal beamforming Proposition 3. If u1 and u2 are electrically orthogonal, the scheme. For this, we define the normalized received SNR  non-unit-norm version of fopt is given as (denoted as SNRrx): H H SNRrx 1 f H Hf −j∠vH v SNR = · . = β + e 1 2 1 − β2 rx H fopt optv1 optv2 NtNr · ρ NtNr f f where is important to characterize the performance achievable with √ A+ B if |α |≥|α | beamforming along the dominant path in benchmarking the β2 = 2C√ 1 2 with performance of the optimal scheme. opt A− B if |α | < |α | 2C 1 2 It is straightforward to see that this scheme results in the (|α |2 −|α |2)2 SNR A = 1 2 +2|α |2 · (|α |2 + |α |2) following received : | H |2 1 1 2 v1 v2 SNR 2 2 4 2 2 rx (|α1| −|α2| ) 4|α1| |α2| 2 2 2 B = + · |α1| −|α2| 1 2 2 H 2 2 2 H 2 | H |4 | H |2 = max |α1| + |α2| |v v2| , |α2| + |α1| |v v2| v1 v2 v1 v2 L 1 1 2 2 1 2 4|α | |α | 1 C = 1+ · |α |2 + |α |2 − 1 2 . + · 2|α ||α |·| H |·| H |·cos (ν) . H 2 1 2 H 2 1 2 v1 v2 u1 u2 |v1 v2| |v1 v2| L 2 2 2 2 2 The non-unit-norm version of gopt follows from expanding out With max(|α1| , |α2| )=K and min(|α1| , |α2| )=1 H fopt and is not provided here. where K ≥ 1,wehave 2 H  Fig. 1(b) plots βopt as a function of |v1 v2| for different SNRrx |α1| 1 choices of K = |α | . As before, Fig. 1(b) shows that βopt 2 H 2 H H 2 = · K + |v1 v2| +2K ·|v1 v2|·|u1 u2|·cos (ν) converges to 0 or 1 as v1 and v2 become more orthogonal. A L (a) straightforward calculation also shows that 1 2 H 2 H H ≤ · K + |v1 v2| +2K ·|v1 v2|·|u1 u2| |α |4 L | H |→1=⇒ β2 → 1 . v1 v2 opt 4 4 |α1| + |α2| with equality achieved in (a) in the most optimistic scenario of coherent phase alignment (ν =0). Clearly, the upper bound is C. v1 and v2 are Parallel H H increasing in K, |v1 v2| and |u1 u2|. Under favorable channel 1 2 H H If v and v are parallel (or nearly parallel), we can use conditions ({|v1 v2|, |u1 u2|} ≈ 1), beamforming along a | H |≈1 SNR (K+1)2 v1 v2 to rewrite rx as in (6) at top of the next page. single path can yield , corresponding to a case where β θ L Clearly, this objective function is independent of and . the two paths coherently add at the receiver to increase the Therefore, any power allocation scheme across the two paths is signal amplitude. When only one of the paths is strong (K optimal. This conclusion is not surprising since the transmitter 1) or when v1 and v2 are electrically orthogonal, this coherent sees two equivalent paths and exciting these paths with any K2 gain is lost and the beamforming gain is L . We now elaborate weight should be optimal. on these statements by studying D. u1 and u2 are Parallel  SNRrx Proposition 4. If u1 and u2 are parallel, the non-unit-norm ΔSNR RSV  version of fopt is given as SNRrx Dom. path H j(∠α1−∠α2−∠u u2) 2 fopt = βoptv1 + e 1 1 − βoptv2 in the different special cases considered in Sec. III. where A. v1 and v2 are Orthogonal |α |2 A simple calculation using the structure of βopt in Prop. 2 β2 = 1 . opt 2 2 shows that ΔSNR is as in (7) at the top of the next page. |α1| + |α2| H Clearly, ΔSNR is increasing in |u1 u2| with The non-unit-norm version of gopt follows from expanding out min |α |2, |α |2 H fopt and is not provided here. 1 ≤ ΔSNR ≤ 1+ 1 2 2 2 2 max (|α1| , |α2| ) Note that βopt mimics a maximum ratio combining solution, allocating power to each path in proportion to the gain of that where the lower bound is realized when u1 and u2 are path. orthogonal and the upper bound is realized when they are parallel. The above relationship clearly shows that the worst- IV. SNR LOSS WITH DIRECTIONAL BEAMFORMING case performance loss with beamforming along a single path We now consider a scheme where the entire power at the is 3 dB. This SNR loss (in dB) is plotted in Fig. 2(a) as a |α1| H K = | 2| transmit end is directed along only one path (the dominant function of |α2| for different choices of u1 u . one): either v1 (β =1)orv2 (β =0). Note that this scheme is amenable to analog (RF) beamforming as it can B. u1 and u2 are Orthogonal be implemented with analog phase shifters alone. As a result, In terms of loss with respect to beamforming along the this scheme is of low-complexity and is advantageous in mmW dominant path, a simple calculation shows that ΔSNR is as MIMO systems. In contrast, fopt requires digital beamforming in (8) at the top of the next page, where βopt is as in the (in general) — a higher complexity implementation — as it statement of Prop. 3. While this expression is also difficult K = |α1| needs both phase shifters and gain control stages. Thus, it to visualize, Fig. 2(b) plots it as a function of |α2| for |α |2 + |α |2 +2|α ||α |·| H |·cos (ν)+2β 1 − β2 · cos(φ) · |α |2 + |α |2 +2|α ||α |·| H |·cos(ν)  1 2 1 2 u1 u2 1 2 1 2 u1 u2 SNRrx = L · 1+2β 1 − β2 · cos(φ) 1 = · |α |2 + |α |2 +2|α ||α |·| H |·cos(ν) . L 1 2 1 2 u1 u2 (6) 2 2 4 4 2 2 H 2 |α1| + |α2| + |α1| + |α2| +2|α1| |α2| · 2|u1 u2| − 1 ΔSNR = 2 · max (|α |2, |α |2) (7) 1 2 2 2 2 2 2 2 H 2 βopt·|α2| +(1−βopt)·|α1| |α1| + |α2| − 1 −| 2| · √ v1 v 1+2β 1−β2 ·|vH v | ΔSNR = opt opt 1 2 2 2 H 2 2 2 H 2 (8) max |α1| + |α2| |v1 v2| , |α2| + |α1| |v1 v2| 2 2 H |α1| + |α2| +2|α1||α2| cos(ν) ·|v v2| ΔSNR = 1 (9) max |α |2 + | H |2|α |2, |α |2 + | H |2|α |2 +2|α ||α || H | cos(ν) 1 v1 v2 2 2 v1 v2 1 1 2 v1 v2 2 H 2 |α2| · 1 −|v v2| =1+ 1 2 H 2 2 H (10) |α1| + |v1 v2| |α2| +2|α1||α2||v1 v2|·cos(ν)

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 2 opt 2 opt 0.5 0.5 β β

0.4 0.4 K = 0.25 K = 0.25 K = 0.50 K = 0.50 0.3 K = 0.75 0.3 K = 0.75 K = 0.99 K = 0.99 0.2 K = 1.01 0.2 K = 1.01 K = 1.25 K = 1.25 0.1 K = 1.50 0.1 K = 1.50 K = 1.75 K = 1.75 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 |uH u | |vH v | 1 2 1 2 (a) (b)

2 |α1| β K = 1 2 1 2 1 Fig. 1: opt as a function of |α2| for different choices of: a) u and u when v and v are orthogonal, and b) v and v2 when u1 and u2 are orthogonal.

H |α1| different values of |v1 v2|. With K = |α | ≥ 1, note that above expression is maximized at K =1. Substituting K =1, 2 2 1 ΔSNR can be rewritten as we have βopt = 2 and H ⎡ ⎤ 1+|v1 v2| 1 −| H |2 · 1 − β2 ΔSNR = . v1 v2 opt 1+| H |2 ΔSNR =1+⎣ ⎦ · v1 v2 2 H 1+2βopt 1 − βopt ·|v1 v2| It is easy√ to see that the above expression is maximized√ at H 2+1 ⎡ ⎤ |v1 v2| = 2−1 with a maximum value of ΔSNR = 2 = 2 2 H 1 − βopt · (1 − K )+2βopt|v1 v2| 0.8175 dB. Thus, beamforming along the dominant path is ⎣ ⎦ . 2 H 2 no worser than 0.8175 dB in terms of optimal beamforming K + |v1 v2| performance. This trend is reinforced by the ΔSNR plot in K = |α1| Fig. 2(b) as a function of |α2| for different values of While optimizing the above expression in terms of K is H |v1 v2|. difficult given the complicated functional involvement of K in the above expression, by treating βopt as a fixed quantity, it is C. v1 and v2 are Parallel straightforward to see that the above expression is decreasing A simple corollary of the observation that any power alloca- in K. Without being rigorous, this argument suggests that the tion scheme across the two paths is optimal is that ΔSNR =1. 3 0.8 |uHu | = 0.1 |vHv | = 0.1 1 2 1 2 |uHu | = 0.3 0.7 |vHv | = 0.3 2.5 1 2 1 2 |uHu | = 0.5 |vHv | = 0.5 1 2 0.6 1 2 |uHu | = 0.7 |vHv | = 0.7 1 2 1 2 2 |uHu | = 0.9 |vHv | = 0.9 1 2 0.5 1 2 SNR) SNR) Δ ( Δ ( 10

1.5 10 0.4 10 log 10 log 0.3 1

0.2

0.5 0.1

0 0 1 2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5 4 4.5 5 K K (a) (b) Fig. 2: ΔSNR between the optimal beamforming scheme and beamforming along the strongest path as a function of K:(a) as given by (7) when v1 and v2 are orthogonal, and (b) as given by (8) when u1 and u2 are orthogonal.

D. u1 and u2 are Parallel 2.45 dB and 3.4 dB. Thus, this study suggests that directional K = |α1| ≥ 1 SNR beamforming could serve as a useful low-complexity scheme With |α2| ,the loss can be written as in (9)-(10) at the top of the previous page. This SNR loss with good performance in the mmW regime. K = |α1| term is plotted in Fig. 3 as a function of |α2| for V. C ONCLUDING REMARKS H different choices of |v1 v2| and ν. From this study, we see that This paper developed an explicit mapping and dependence ΔSNR can be significantly larger than 3 dB provided that both of the optimal beamformer structure on the different aspects paths are approximately similar in terms of gain and are also of the sparse channel that characterize propagation in the essentially parallel, but with opposite phases (characterized by mmW regime. This study showed that the optimal beamformer o ν = 180 ). In this setting, the right singular vector combines approaches dominant path (directional) beamforming as either the gains in both paths by appropriate phase compensation. the AoDs or AoAs of the paths become more (electrically) or- On the other hand, beamforming along only the strongest thogonal. In general, if the AoDs or AoAs are not orthogonal, path leads to destructive interference of the signal from the optimal beamforming entails appropriate power allocation and sub-dominant path resulting in significant performance loss. phase compensation across the paths. While specific channel Theoretically speaking, ΔSNR can be as arbitrarily large as realizations can be constructed to ensure that directional possible. Nevertheless, barring such extreme conditions, this beamforming can suffer significantly relative to the optimal study also shows that the performance loss is similar to the 3 scheme, in a distributional sense, the loss in received SNR dB characterization in other settings. is expected to be minimal. Furthermore, this small additional gain in received SNR with optimal beamforming comes at the E. Directional Beamforming at Both Ends cost of tight phase synchronization across paths, an onerous While we have so far considered the case of directional task at mmW frequencies especially since relative motion on beamforming at the transmitter, the receiver uses a matched the order of the wavelength (a few millimeters) can render the filter corresponding to such a scheme, which may not be optimal beamformer unuseable in practice. These conclusions directional. We now consider the case of directional beam- on small losses with directional beamforming as well as its forming at both ends. In Fig. 4, we plot the complementary robustness relative to the optimal scheme provides a major cumulative distribution function (CCDF) of the loss in SNRrx fillip to the search for good directional learning approaches, with such a bi-directional scheme relative to the optimal a task that has received significant and increasing attention in beamforming scheme for different choices of L.Thegainsof the literature. the paths as well as their directions are chosen independently REFERENCES and identically distributed (i.i.d.) from a certain path loss model and over the 120o field-of-view of the arrays. From [1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and this figure, we note that for a large fraction of the channel challenges with very large arrays,” IEEE Sig. Proc. Magaz., vol. 30, no. realizations, beamforming along the dominant direction only 1, pp. 40–60, Jan. 2013. results in a small performance loss. In particular, the median [2] N. Bhushan, J. Li, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic, L =2, 3 5 0.3 0.95 R. T. Sukhasvi, C. Patel, and S. Geirhofer, “Network densification: The losses in the three cases ( and )are dB, dominant theme for wireless evolution into 5G,” IEEE Commun. Magaz., dB and 1.85 dB, and the 90-th percentile losses are 1.8 dB, vol. 52, no. 2, pp. 82–89, Feb. 2014. ν = 135o ν = 180o 4 14 H |vHv | = 0.1 |v v | = 0.1 1 2 1 2 H 3.5 |vHv | = 0.3 12 |v v | = 0.3 1 2 1 2 H |vHv | = 0.5 |v v | = 0.5 1 2 1 2 3 H |vHv | = 0.7 10 |v v | = 0.7 1 2 1 2 H |vHv | = 0.9 |v v | = 0.9 2.5 1 2 1 2 SNR) SNR) 8 Δ Δ ( ( 10 2 10 6 10log 10log 1.5

4 1

2 0.5

0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 K K (a) (b) Fig. 3: ΔSNR between the optimal beamforming scheme and beamforming along the strongest path as a function of K when o o u1 and u2 are parallel for different choices of ν:(a)ν = 135 , and (b) ν = 180 .

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