Random-Phase Beamforming for Initial Access in Millimeter-Wave Cellular Networks

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1 Random-Phase Beamforming for Initial Access in Millimeter-Wave Cellular Networks Zohair Abu-Shaban, Henk Wymeersch, Xiangyun Zhou, Gonzalo Seco-Granados, Thushara Abhayapala The Australian National University, Australia. {zohair.abushaban, xiangyun.zhou, Chalmers University of Technology, Sweden. Universitat Autònoma de Barcelona, Spain. Abstract The utilization of the millimeter-wave frequency band mm-wave in the fifth generation 5G of mobile communication is a highly-debated current topic. Mm-wave MIMO systems will use arrays with large number of antennas at the transmitter and the receiver, implemented on a relatively small area. With the inherent high directivity of these arrays, algorithms to help the user equipment find the base station and establish a communication link should be carefully designed. Towards that, we examine two beamforming schemes, namely, random-phase beamforming RPBF and directional beamforming DBF, and test their impact on the Cramér-Rao lower bounds CRB of jointly estimating the direction-of-arrival, direction-of-departure, time-of-arrival, and the complex channel gain, under the line-of-sight channel model. The results show that the application of RPBF is more appropriate in the considered scenario as it attains a lower CRB with fewer beams compared to DBF. I. INTRODUCTION One of the enabling technologies of the fifth generation of mobile networks 5G is the millimeter-wave technology mmwave [], [2]. Mm-waves allow packing high number of antennas in a relatively small area due to their small wavelengths from mm to cm. In addition, the massively available spectrum of mm-wave would easily enable the transmission with extremely high data rates [3]. These two reasons reinforce the role that the mm-wave technology can play in 5G. Due to the special characteristics of the mm-wave channel, directional transmission with large-size arrays, at both the transmitter and the receiver, is going to be utilized [4]. However, when initiating a communication link, this high directionality is an issue for a user equipment UE trying to find the base station BS, and vise versa, especially that omni-directional transmission with antenna arrays is a challenge itself [5], [6]. The mm-wave initial access IA techniques with analog beamforming were reviewed in [7]. The three approaches compared therein include two direction-based methods, namely exhaustive search and iterative search. The third method is GPS-assisted algorithm. The disadvantage of these approaches is mainly the delay of finding the proper transmission direction. In contrast to the directional methods covered in [7], we focus on random beamforming. The concept of random beamforming is not new per-se. Randomly-directional beamforming is a method of opportunistic beamforming that was investigated in [8], [9] for mm-wave receivers. Under this scheme, the BS generates narrow beams with random directions and select the user with the highest signal-to-noise ratio. On the other hand, optimization-based random beamforming in [6] optimizes the beamforming weights so that the resulting beam pattern is omni-directional. The work therein focused on conventional multiple-input-multiple-output MIMO systems in the microwave band. Despite the advantage of omni-directional coverage in the AI phase, the cost function is complex and, for a high number of antennas, it is only solvable numerically. Another random beamforming investigated under conventional MIMO is the unitary beamforming, whereby the beamforming matrix is obtained by the singular value decomposition of the channel matrix, see for example []. This is not applicable in an IA scenario since the channel is unknown at that phase. The Cramér-Rao lower bounds CRB of a line-of-sight LOS mm-wave channel parameters were previously studied in []. Therein, the CRBs were provided as a function of the Fisher information matrix FIM, whose entries were given by high level expressions and are valid for uniform linear arrays ULA with directional beamforming DBF. In this paper, we investigate a random beamforming scheme, referred to as random-phase beamforming RPBF. Under this scheme, the beamforming vector is generated as a vector of complex exponentials with independent and identically distributed i.i.d. random phases. This scheme is used in [2] to initiate an iterative beamforming scheme which assumes full channel knowledge, and in [3], which assumes the directionof-departure DOD to be known. However, these two assumptions are not used in this paper. We look into the IA problem for a LOS mm-wave channel. Towards that, we study the effect of beamforming on the estimation bounds of the direction-ofarrival DOA, DOD, time-of-arrival TOA, and the complex channel gain. The current paper presents the CRB expressions for an arbitrary array geometry. Moreover, as a special case, we obtain expressions of CRB, using ULA, simpler than those in []. Although mm-wave may apply different array geometries, ULA is the standard array structure that is usually used to get initial insights. Finally, we investigate and assess the RPBF and DBF in terms of the CRB, as a function of the number of transmit antennas, receive antenna and transmit beams. II. PROBLEM FORMULATION Consider the scenario illustrated in Fig., where the receiver and the transmitter are equipped with arrays of and

2 y α UE NT θ T D 8 8 θ R BS Fig.. A schematic diagram of the considered scenario. Angles are positive clockwise. D is the transmitter-receiver separation distance. antennas, respectively. Without loss of generality, we consider the uplink and assume that the BS and UE are located in the same plane. Consequently, we consider a 2D approach. Moreover, similar to [], we assume that the UE and BS communicate via a LOS path only. In fact, the effect of NLOS links is limited, since the path loss exponent is much higher than the case of LOS, and 2 the absorption loss is high and results in weak reflections. These factors cause NLOS to have 2 db less power than LOS at 38 GHz [4]. Furthermore, we assume that the transmit antenna array is rotated by an unknown angle α. Finally, we assume that the arrays are narrow-band, i.e., the signal traverses the arrays apertures, both at the transmitter and the receiver, within a fraction of a symbol duration. That is A max cµt s, where A max is the maximum array aperture, c 3 8 m/s is the speed of light, T s is the symbol duration, and < µ <. Based on the above assumptions, the channel matrix is modeled by [4] H βa R θ R a H T θ T C, where β β R + jβ I C is the complex channel gain, θ R is the DOA, and θ T is the DOD. Under mm-wave channel, DOD is crucial in beamforming, since the knowledge of DOD would enable more efficient beamforming[2]. Moreover, a R θ R C is the receiver array response vector, and a T θ T C is the transmitter array steering vector. As an example, for the particular case of ULA, these two vector are given by 8 and 7. For notation simplicity, we drop the angle parameter from a R θ R and a T θ T. These two vectors are normalized so that a R 2 a T 2. Therefore, the received signal at the analog output of the array at a time instant t, can be written as rt [r t, r 2 t,..., r NR t] T, C, t [, T o ], Hxt τ + nt. where T o N s T s is the observation time and N s is the number of pilot symbols. Moreover, τ R + is the propagation delay of the transmitted signal and is related to the transmitter-receiver distance, D, by τ D /c. Furthermore, nt [n t, n 2 t,..., n NT t] T C denotes zero-mean additive white Gaussian noise processes with noise power σn 2 W N /2, where N /2 is the noise spectral density, and W is the signal bandwidth. Furthermore, xt [x t, x 2 t,..., x NT t] T C is the transmit- x Fig. 2. Beam patterns generated using DBF left and RPBF right with 32 and N B 24. For directional beamforming, ψ l l. ted signal vector at the output of a beamforming matrix F [f, f 2,...f NB ] such that xt P t Fst and st [s t, s 2 t,..., s NB t] T, where N B is the number of transmitted beams and P t is the transmitted power. Moreover, s l t N s k a l,k pt kt s, l,..., N B where a l,k is the k th pilot symbol transmitted over the l th beam, and pt is a unit-energy pulse. To keep the transmitted power fixed, regardless of the number of antennas,, we normalize F such that Tr F H F, and E{sts H t} I. Matrix F is modeled in two ways: directional and random. These models and the differences between them are discussed in detail in the sequel of Section III. Our aim is to investigate the impact of beamforming schemes on the estimation lower bounds of φ [θ R, θ T, β, τ] T based on the observed signal, rt. III. RANDOM-PHASE BEAMFORMING AND DIRECTIONAL BEAMFORMING In this section, the two considered beamforming schemes are defined. The impact of these two schemes on the joint estimation of DOD, DOA, TOA and channel gain is investigated in subsequent sections. The first scheme we consider is DBF, which spatially steers the transmission beams towards the azimuth angles ψ l, l N B such that f l NB a T ψ l, 2 where d, and λ denote the inter-antenna spacing and the carrier wavelength, respectively. The DBF in 2 has been used in several works on mm-wave, for example, see []. On the other hand, the RPBF generate beams with uniformly distributed random phases such that f l NT N B [e jϑ l,,..., e jϑ l, ] T, 3 where ϑ l,n U, 2π. Although, practical phase shifters generate quantized phases, recently, phase shifters for mm-wave with 3.5 phase resolution were proposed [5], [6]. Note that in 3 the phase is random, in contrast to the scheme in [8], [9], which have a structure similar to 2, but with a random direction. Moreover, note that both 2 and 3 have f l 2 N B. This implies that increasing the number of beams,

3 will linearly scale down the power per beam to preserve the constant transmitted power condition. A nice feature of 3 is that the generated beams are not too narrow compared to those of DBF as shown in Fig. 2. Thus, RPBF exhibits a better spatial coverage, which is an essential feature when the initial direction of transmission is unknown. IV. CRAMÉR-RAO LOWER BOUND A widely used performance measures in estimation theory is the Cramér-Rao lower bound CRB [7]. CRB provides a lower bound for the variance of an unbiased estimator of a given parameter. We use the CRB to compare the best performance of RPBF and DBF in the scenario described in Section II. The CRB of estimating φ for an arbitrary array geometry is derived in Appendix A. Subsequently, for the case of ULA, we show in Appendix B that the CRBs for jointly estimating θ R, θ T, β and τ are given by 2 λ 2, CRBθ R 4 γnr 3 [Q], 2πd cos θ R [Q], λ 2, CRBθ T 5 γ detq 2πd cos θ T 2 β 2 2 CRBβ γ [Q], + + [Q],[Q] 2,2, 6 2 detq CRBτ γσpn 2. 7 R [Q], where γ N s P t β 2 /σn 2 and σp 2 [ a H T FF H a T a H T D T FF H a T Q a H T D T FF H a T and D T diag,, 2,...,. A. DBF Analysis 2 pt dt, a H T D T FF H D T a T ], Define ϕ l 2πd/λ sin θ T sin ψ l. Then, for DBF [Q], NT 2N B N b l N b [Q] 2,2 2NT 2N B l cos ϕ l cos ϕ l, 8 NT cos ϕ cos ϕ l cos ϕ l 2, 9 Q r R{[Q],2 } [Q],, 2 Q i I{[Q],2 } NT 2N B N b l cos ϕ l sin ϕ l cos ϕ l sin ϕ l cos ϕ l 2. Note that 8 imply that the performance of DBF is highly governed by the difference between θ T and ψ l. Ideally, the difference should be zero to achieve the lowest CRB. Moreover, carefully inspecting 8, one can notice the averaging over N B. Thus, after a certain value of N B, increasing N B has a little effect on the CRBs, and a performance floor is reached. Since and N B will be typically very large, it is meaningful to analyze the limiting behavior of the elements and determinant of Q with. When considering the relationship between DBF CRB and, we can see that there are two components in 8 ; a short-term harmonic component represented by cos ϕ and a long-time trend represented by a polynomial of. Focusing on the long-term trend, it can be noticed that [Q], O, the real and imaginary parts of [Q],2 O, detq O, while [Q] NT 2 2,2 is constant. Consequently, we can deduce that CRBθ R O, CRBθ T O, CRBβ O, and CRBτ O. Due to the fixed transmit power constraint, higher means narrower beams, and higher received power in a certain direction as implied by 2. If that direction mismatches θ T, the CRBs of θ R, β, and τ tend to worsen when increases. B. RPBF Analysis For the RPBF case, since both and N B are typically high, we resort to the law of large numbers to compute the average CRB. Denoting the expected value by E{ }, we calculate E{[Q], }, E{[Q] 2,2 }, and E{detQ}, to obtain the limiting behavior of 4 7 as E{[Q], } a H T E{FF H }a T ah T a T, 2 NT i i 2 E{[Q] 2,2 } ah T D2 T a T E{[Q],2 } ah T D T a T E{detQ} 3 2 NT i i 3, 3 2, The results in 2 5 imply that CRBθ T O/, while CRBθ R, CRBβ, and CRBτ are constant in and N B. In contrast to DBF, increased does not decrease the spatial coverage. Thus, higher does not affect the received power in average, and the CRBs of θ R, β, and τ stay constant. V. SIMULATION AND NUMERICAL RESULTS With reference to the scenario illustrated in Fig., we consider a receiver equipped with a ULA lying in the x-axis with d λ/2, and covering a spatial field π/2, π/2. The transmitter, operating at f 38 GHz, is assumed to be tilted with an orientation angle α measured from the positive x- axis. Without loss of generality, we select α. The DODs and DOAs are measured clockwise from the line normal to the array. The BS is assumed to be located at the origin, while the UE is located at p 5, 25 meter. This results in θ R θ T.3. Similar to [8], the complex channel gain is computed using the free-space propagation model, leading to β β R + jβ I j Moreover, we consider pt to be a unit-energy ideal sinc pulse. Consequently, it follows from Parseval s theorem that σ 2 p 4 3 π2 W 2, 6

4 2 CRBθ R CRBθ T CRBβ CRBD -2-3 CRBθ R CRBθ T CRBβ CRBD -2-4 CRB CRB N B Fig. 3. CRB w.r.t N B using RPBF dashed line and DBF solid line with 64, 32. Directional beams are equally spaced over -π/2, π/2 where W R b /2b, and R b is the bit rate, and b is the number of bits per symbol. We present the results for Gbps bit rate and 6 QAM transmission, i.e., W 25 MHz. Furthermore, we assume σn 2 2 dbm, and P t W. To improve the results presentation, we provide the CRBτ in terms of CRBD. For DBF, the directions are chosen to be equally spaced to cover the region π/2, π/2, i.e., ψ l π 2 + πl N B +, l N B. Finally, in the following results, when referring to RPBF, CRB means the average CRB. RPBF plots are obtained by Monte-Carlo simulation averaged over iteration. A. Effect of N B on the CRB Fig. 3 illustrates the CRB as a function of N B with 32 and 64. It can be seen that, using RPBF, the CRB floor is reached at N B 8. Also, note that for N B < the average CRBS of θ R and D are less sensitive to N B compared to those of θ T, and β. On the other hand, the CRB under DBF is non-monotonic. Thus, N B should be carefully chosen for optimum estimation. However, since in the initial access phase the transmitter has no information on the direction of transmission, an optimal beam is not guaranteed. Comparing RPBF with DBF in Fig. 3, it can be inferred that the RPBF scheme attains a lower floor at a low N B than the DBF, except when the latter happens to have a beam close to the receiver direction. To see why the CRB floors, notice the averaging effect in 8, and the independence of N B in 2 5. Intuitively, recall that with a fixed the receiver beam-width about θ R is fixed. Consider the RPBF case shown in Fig. 2. When N B is small, there is a limited chance that a random transmit beam will hit θ R with a suitable gain, but as N B goes up, this chance enhances and the CRB starts to decrease. As N B grows significantly, the resultant beams becomes almost omni-directional and, regardless of how high N B is, the CRB becomes fixed. On the other hand, consider the DBF case in Fig. 2, where beams have a comb-like shape and the CRB mainly depends on the difference between the transmit beams and the receive beam. As N B grows higher, more transmit beams fall within the the receive beam-width. However, since f l 2 /N B, the received power stays fixed, and the CRB floor is reached Fig. 4. CRB w.r.t using RPBF dashed line and DBF solid line with N B 8, 32. Directional beams are equally spaced over -π/2, π/2 B. Effect of on the CRB The CRBs as function of are provided in Fig. 4. In both schemes, CRBs of θ T, β, and τ decrease as /. On the other hand, CRB of θ R decreases by three orders of magnitude, when increases by one order of magnitude. In addition, these results confirm the conclusion made using Fig. 3 in that RPBF provides better bounds than DBF does, when N B is fixed. C. Effect of on the CRB Fig. 5 illustrates the results of investigating the CRB in terms of. Considering DBF, it is hard to draw any conclusion for scaling factors in terms of due to the high non-linearity observed and represented by 8. However, there is an average trend that can be seen, as discussed in Section IV, whereby CRBθ T decreases with /NT 2 while the other CRBs increase with. As for RPBF, it can be seen from the figure that only the estimation of θ T improves when increasing. Note that due to the power normalization discussed in Section II, the CRBs of β and θ R are not affected by an increased. D. Summary of Results From Figs. 3 5, it can be inferred that RPBF outperforms DBF in terms of the channel parameters CRBs, except in some cases shown in Fig. 3, where a few N B choices can results in a better DBF performance. Note that in these cases a beam or more are close enough to the receiver direction, hence the better CRB. However, since at the IA stage the receiver location is unknown, a careful choice of N B cannot be made for DBF. As a result, RBPF is more reliable in this case during AI phase. VI. CONCLUSION Mm-wave MIMO is a very promising technology for future mobile communication. In this paper, we have studied the impact of two beamforming schemes on the CRB of estimating DOA, DOD, TOA, and complex channel gain. RPBF has shown better CRB floor at a smaller number of beams than DBF. Thus, as shown by the numerical results, it would be favorable to use RPBF in the initial access phase. Table I summarizes the CRB scaling factors in terms of, and N B. As future work, we will consider mm-wave channels with multi-path propagation and carry out an analytical investigation of RPBF and DBF.

5 CRB CRBθ R CRBθ T CRBβ CRBD Fig. 5. CRB w.r.t using RPBF dashed line and DBF solid line with N B 8, 32. Directional beams are equally spaced over -π/2, π/2 APPENDIX A CRBφ FOR AN ARBITRARY GEOMETRY Consider the received signal model in. Let µφ Hxt τ P t βa R a H T Fst τ. Using arbitrary array geometries at the transmitter and receiver, the corresponding steering vectors are given by a T NT e jkt θ T v nt, nt, 2,...,. 7 a R NR e jkt θ R u nr, nr, 2,...,, 8 where k T θ 2π λ [sinθ, cosθ] is the wavenumber vector, and u n and v n are vectors of the Cartesian coordinates of the n th antenna at the receiver and the transmitter, in meters, respectively. We start by computing FIM [Iφ] p,q T { } o µ H φ µφ σn 2 R dt, 9 φ p φ q where φ p is the p th element in φ, p, q 5. Consequently, µφ j P tβ mk Ra Ra H T Fst τ, θ R 2 µφ j P tβ ma Ra H T K T Fst τ, θ T 2 µφ P ta Ra H T Fst τ, β R 22 µφ j P ta Ra H T Fst τ, β I 23 µφ st τ P thf, 24 k T θ R K R diag [u, u 2,..., u NR ], θ R k T θ T K T diag [v, v 2,..., v NT ]. θ T Define the following matrices Σ T o st τs H t τ dt N s I, TABLE I SCALING EFFECT ON CRBS FOR HIGH AND N B. CRBθ R CRBθ T CRBβ RPBF DBF N B N R 3 constant O constant O O constant constant O constant CRBτ constant O constant T o Σ st τ sh t τ dt, T o st τ s H t τ Σ 2 dt pt τ 2 dtn s I σ 2 pn s I. Note that the derivative of an even function is odd, and the product of an even and an odd functions is odd. The integral of an odd function over its domain is zero. Consequently, Σ. From Section II, φ θ R. Thus, substituting 2 into 9, and using a H bc H d c H da H b, it is easy to show that [Iφ], N s P t β 2 σn 2 a H R K 2 Ra R a H T FF H a T. 25 Define ζ N s P t /σ 2 n. Subsequently, it can be shown that [Iφ] 2,2 ζ β 2 a H T K T FF H K T a T, 26 [Iφ] 3,3 [Iφ] 4,4 ζ a H T FF H a T, 27 [Iφ] 5,5 P tn s σp 2 σn 2 TrHFF H H H, 28 [Iφ],2 ζ β 2 a H R K R a R R{a H T K T F F H a T }, 29 [Iφ],3 ζ β I a H R K R a R a H T FF H a T, 3 [Iφ],4 ζ β R a H R K R a R a H T FF H a T, 3 [Iφ] 2,3 ζ R{jβ a H T FF H K T a T }, 32 [Iφ] 2,4 ζ R{β a H T FF H K T a T }, 33 [Iφ],5 [Iφ] 2,5 [Iφ] 3,5 34 [Iφ] 4,5 [Iφ] 3,4. 35 Now that we obtained FIM, the CRB can be derived from [7] CRBφ diag I φ. 36 From 25 35, it can be seen that Iφ is block diagonal. Thus, for φ [θ R, θ T, β R, β I ] T, 36 can be written as [ diag I CRBφ φ ] [Iφ]. 37 5,5 To compute I φ, we start by defining the following [ ] [ ] R a H T R K R a a H T FFH a T R R 2 a H R K2 R a, T 2 a H T K T FF H a T, R T 3 a H T K T FF H K T a T

6 [ ] [ ] S βi T R + β R T I, T S 2 β R T r β I T 2 T R + jt I, I and γ N s P t β 2 /σ 2 n. Consequently, from 25 35, I φ γn [ ] r A U β 2 U T γn r C β 2 38 β 2 R 2 T β 2 R T R β I R T β R R T β 2 R T R β 2 T 3 S S 2 β I R T S T β R R T S 2 T By block-matrix inversion, [ ] θr [ CRB β 2 A diag UC U T ], 39 θ T γn [ ] R βr [ CRB β 2 diag C U T A UC U T β I γ UC + C ] 4 We now proceed to evaluate 39 and 4. Directly from 38, [ ] TR A UC U T β 2 2 R 2, 4 T T 3 T 2 2 T C U T A UC U T UC + C β 2 T βi 2 R2 S + R 2 R 2 T T 3 T 2 + β 2 β I β R R S S 2 R 2 R 2 T T 3 T 2 2 β I β R R 2 + S S 2 β R 2 R 2 T T 3 T 2 2 R R2 S + 2 R 2 R 2 T T 3 T 2 + β 2 2 For a ULA, APPENDIX B CRB FOR A ULA R 2πd/λ cosθ R a H R D R a R πd/λ cosθ R, 43 R 2 2πd/λ 2 cos 2 θ R a H R D 2 Ra R 2 3 πd/λ2 cos 2 θ R 2, 44 where D R diag,, 2,...,. Moreover [ ] [ 2πd ] T T 2 [Q], λ cosθr[q],2 T 2 T 2πd 2πd 2 3 λ cosθr[q],2 λ cos 2 θ R[Q] 2,2 [ a H where Q T FF H a T a H T D T FF H ] a T a H T D T FF H a T a H T D T FF H, D T a T D T diag,, 2,...,, and [Q],2 Q R + jq I. Substituting in 4, then in 39 yields 2 λ 2, CRBθ R γnr 3 [Q], 2πd cos θ R [Q], λ 2. CRBθ T γ detq 2πd cos θ T 45 Note that since β R and β I are independent, the CRB of β can be simplified to CRBβ CRBβ R + CRBβ I. Thus, substituting in 42, and taking the trace, yields 2 β 2 CRBβ γ [Q], [ 2NR + + [Q],[Q] 2,2 2 detq ]. Finally, following from 28 and 37, it is easy to see that CRBτ γσpn 2. R [Q], ACKNOWLEDGMENT This work is partly supported by the Horizon22 project HIGHTS High precision positioning for cooperative ITS applications MG-3.5a , the Australian Research Council s Discovery Projects funding scheme under Project DP433, and the Spanish Grant TEC R. REFERENCES [] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. 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