Directional Beamforming for Millimeter-Wave MIMO Systems
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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.