Analyses of Orthogonal and Non-Orthogonal Steering Vectors at Millimeter Wave Systems

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1 Analyses of Orthogonal and Non-Orthogonal Steering Vectors at Millimeter Wave Systems Hsiao-Lan Chiang, Tobias Kadur, and Gerhard Fettweis Vodafone Chair for Mobile Communications Technische Universität Dresden Helmholtzstrasse 8, Dresden, Germany {hsiao-lan.chiang, tobias.kadur, Abstract Beamforming is one of the most challenging roblems for millimeter wave communication. With limited codebook size, how to design the steering angles to comensate angles of arrival and dearture (AoAs/AoDs) is essential to beamforming erformance. Tyically, two categories of steering vector sets are commonly used. One is orthogonal steering vector set where the satial frequency indices of the steering angles are uniformly distributed in satial frequency domain. The other one is nonorthogonal steering vector set where the steering angles are uniformly distributed in angle domain. In this aer, analyses of these two designs are resented. Due to the fact that beamwidth are constant with resect to different satial frequency indices in satial frequency domain, if the satial frequency indices are uniformly distributed, one has the smallest deviation of the beamforming gain. Since the orthogonal steering vectors satisfy this condition that satial frequency indices are uniformly distributed, they can achieve higher data rates than the nonorthogonal ones when the AoAs are uniformly distributed over ( π, π ). Index Terms millimeter wave, satial frequency, analog beamforming, orthogonal steering vectors I. INTRODUCTION With the raid increase of data rate in wireless communication, bandwidth shortage is getting more critical. Therefore, there is a growing interest in using millimeter wave (mmw) for future wireless communications taking advantage of the enormous amount of available sectrum []. Measurements of most large and small scale arameters for mmw channels in urban areas at 73 GHz had been resented in []. It seems that ath loss in such environment is very severe, and in order to imrove caacity and service quality, mmw together with massive multile inut multile outut is a romising aroach [3]. Most of today s beamforming systems oerating in lower carrier frequency bands ( 6 GHz) are based on digital beamforming (DBF) architecture. In this architecture, each antenna has its own RF chain, including digital-to-analog converter (DAC) at the transmitter and analog-to-digital converter (ADC) at the receiver. However, in order to exloit huge available bandwidth in mmw bands, ADCs and DACs have to run at served Giga samles er second. It is nearly infeasible to equi each antenna with one RF chain due to high imlementation /6/$3. 6 Euroean Union cost and ower consumtion. Therefore, analog beamforming (ABF) [4], [] is adoted at mmw, which consists of multile hase shifters connecting to one RF chain. Comared to DBF, it rovides low cost solutions to imlementation requirements. Due to hardware imlementation cost and feedback overhead, the limited number of steering angles are redefined in the codebook. Larger size of the codebook means that more bits are needed to feedback the selected codeword [6] and more comlicated requirements on hase shifters. Therefore, with limited codebook size, how to design the steering angles is imortant. If angles of arrival (AoAs) are uniformly distributed in angle domain, intuitively, the design of steering angles are suosed to be discrete uniformly distributed in angle domain as well. However, the work in [7] shows different results that nonuniformly distributed steering angles achieve better data rate. The argument is that in satial frequency domain the steering satial frequencies and the satial frequencies of AoAs have nearly the same distribution. Nevertheless, this assumtion is only valid within some range of angles. In this aer, more convincing exlanations for the design of the steering vectors from different ersectives are resented. The work in [8] resented that beamwidth does not vary with satial frequency. According to this roerty, we find that when the designed steering satial frequencies are uniformly distributed in satial frequency domain, the maximum deviation of the beamforming gain is limited to 3 db. On the other hand, for the nonuniformly distributed steering satial frequencies, the maximum deviation of the beamforming gain could be greater or less than 3 db. In the simulated data rates, it shows that when the AoAs are uniformly distributed over ( π, π ), the averaged beamforming gain by non-orthogonal steering vectors is worse than the orthogonal ones because within some range of AoAs the beamforming gain is - db worse than -3 db gain. This aer is organized as follows: In Section II, we describe the system models. Section III elaborates the signals in angle and satial frequency domain. Also, orthogonal and nonorthogonal steering vectors are discussed in satial frequency domain. Simulation results are resented in Section IV, and we conclude our work in Section V.

2 Figure : A diagram of one ABF system with four antenna elements at the receiver in a sarse mmw channel with two AoAs. II. SYSTEM MODELS Fig. shows a diagram of the ABF system with antennas at the receiver. An steering vector w k is selected from the codebook, which consists of N K steering vectors {w,, w NK }. Each steering vector can be exressed as w k [ ] T,e jπλ sinφ kδ d,,e jπλ sinφ k( )Δ d, () where k,,n K, φ k is the k th steering angle, Δ d λ is the distance between two antenna elements, and λ is the wavelength at the carrier frequency. In (), the term λ sinφ k is referred to as a steering satial frequency with resect to the steering angles φ k. ( ) T denotes transose of a matrix. Different to Rayleigh/Rician fading channel models, which are often assumed for centimeter wave (cmw) communication, mmw channel models show sarsity of AoAs and AoDs between transmitter and receiver due to severe ath loss []. Therefore, beam steering lays an imortant role at mmw systems. To deal with both AoDs and AoAs at the same time is a huge challenge because all the combinations of the beamformers at the transmitter and the receiver have to be considered. Accordingly, one can adot an omni-directional antenna at the transmitter, as introduced in IEEE 8.ad [9], to simlify the beamforming rocedure. Then the receiver, equied with a uniform linear array (ULA) antenna, only exeriences multile AoAs as shown in Fig., where two AoAs are assumed in the mmw environment. The received signal before RF chain can be formulated as N P r k s α wk H a + wk H z, () where w H k denotes Hermitian transose of w k, r k is the received signal with resect to the k th steering vector, s is the transmitted signal satisfying E[ s ], α C is the comlex attenuation coefficient for ath, and z is an additive white Gaussian noise (AWGN) vector with the element z nr CN(,σz), n r,. N P is the number of channel aths. a is an array roagation vector [] with resect to the th AoA φ, which can be reresented as a [ T,e jπλ sinφδ d,,e jπλ d] sinφ( )Δ, (3) where λ sinφ is the channel satial frequency with resect to the AoA φ. III. THE ANALYSES OF ORTHOGONAL AND NON-ORTHOGONAL STEERING VECTORS A. Satial Frequency Index By introducing a satial frequency scaling factor defined as Δ ν /( Δ d ) [], the satial frequency index (or normalized satial frequency) with resect to the steering angle φ k is defined as ν k }{{} satial frequency index sinφ k λ }{{ } Δ ν satial frequency sinφ k. (4) Given a satial frequency index ν k, the steering vector in () can be rewritten as w k [,e jπν k/,,e jπν k( )/ ] T. () B. Orthogonal and Non-Orthogonal Steering Vectors The steering vectors with uniformly distributed satial frequency indices are called orthogonal steering vectors, such as Butler matrix []. To be formal, one defines the orthogonality of any two steering vectors as w i, w j w i w i {, i j, i j, (6) where w i, w j denotes inner roduct of two vectors. Given the number of antennas, the N K (assume that N K ) satial frequency indices are shown as [] { ±., ±.,, ± ν k, if N K is an even number, ±, ±,, ±, if N K is an odd number. (7) The other way to design the steering vectors is having the uniformly distributed steering angles over the range of the scanning angles. From the relationshi between the steering angles and the corresonding satial frequency indices in (4), we know that the steering satial frequency indices {ν k k,,n K } are not uniformly distributed, which are different to (7). This kind of steering vectors belong to non-orthogonal steering vector set.

3 C. Beam Patterns Shown in Angle and Satial Frequency Domain Given the steering angle φ k and the AoA φ, the beam attern is defined as [] β,k w H k a N R n r e jπλ (sinφ sinφ k)n rδ d ejπλ Δ d( )(sin φ sin φ k ) sin(πλ Δ d (sin φ sin φ k )) sin(πλ Δ. (8) d(sin φ sin φ k )) Based on the received signals {r k k,,n K }, the goal of beamforming is to achieve the maximal S [7][]. Due to the assumtion that the noise is signal-indeendent, the codeword (or steering vector index) can be selected according to the following equations ˆk arg max r k k,...,n K arg max k,...,n K arg max k,...,n K NP s α wk H a NP s α β,k. (9) To clearly exlain the difference between the orthogonal and the non-orthogonal steering vectors, N P is assumed in (9) in order to analyzing the beam atterns by the orthogonal and the non-orthogonal steering vectors. Therefore, (9) becomes where ˆk arg max k,...,n K s α β,k arg max k,...,n K β,k,, () β,k sin (πλ Δ d (sin φ sin φ k )) sin (πλ Δ. () d(sin φ sin φ k )) () can also be reresented as a function of the satial frequency index, β,k sin (πλ Δ d(ν ν k )). () sin ( πλ Δ d (ν ν k )) Fig. (a) and (b) illustrate beam atterns in angle and satial frequency domain resectively with 6, range of AoAs is 9 o < φ < 9 o ( 8 < ν < 8), and the steering angles equal to φ k o, 6 o (ν k, 6.93). It is obvious that in Fig. (a) the beamwidth varies with steering angle. The beamwidth at φ k 6 o is wider than the beamwidth at φ k o. Considering a angle shift Δφ in (), 9 o < Δφ <9 o, the beam attern () becomes Beam attern β,k (db) Angle (degree) (a) Two beam atterns shown as the function of angle have different beamwidth. Beam attern β,k (db) Satial frequency index (b) Two beam atterns shown as the function of satial frequency index have the same beamwidth. Figure : Tyical examles of two beam atterns reresented in angle and satial frequency domain resectively. β,k sin (πλ Δ d (sin(φ +Δφ) sin(φ k +Δφ))) sin (πλ Δ, d(sin(φ +Δφ) sin(φ k +Δφ))) (3) which is not equal to β,k in () when Δφ. On the other hand, in Fig. (b), the beamwidth does not vary with satial frequency index [8]. Similarly, consider a satial frequency index shift Δν in (), 8 < Δν <8, the beam attern () becomes β,k sin (πλ Δ d((ν +Δν) (ν k +Δν))) sin ( πλ Δ d ((ν +Δν) (ν k +Δν))) sin (πλ Δ d(ν ν k )), Δν. (4) sin ( πλ Δ d (ν ν k )) Thus β,k in (4) is always equal to β,k in () with different value of Δν. It can be concluded that no matter what o o 6 o 6 o

4 Non-orthogonal steering vectors Orthogonal steering vectors Beam attern β,k (db) Beam attern β,k (db) Angle (degree) Angle (degree) (a) Steering angles (in degree): -9::9. (c) Steering angles (in degree): 7, 4, 43, 34, 6, 8,, 4, 4,, 8, 6, 34, 43, 4, 7. Beam attern β,k (db) max. deviation Beam attern β,k (db) max. deviation Satial frequency index Satial frequency index (b) Satial frequency indices: 8, 7.8, 7.3, 6.,.4, (d) Satial frequency indices: -7.::7.. 4,.,.8,.8,., 4,.4, 6., 7.3, 7.8, 8. Figure 3: Beam atterns of the non-orthogonal and the orthogonal steering vectors shown in angle and satial frequency domain with 6and N K 6. value of Δν is, the beam attern shifted to ν k +Δν always kees the same beamwidth. When φ U( π, π ), the robability density function sin φ (df) of ν (ν ) is given by (see the Aendix), ν < f ν (ν ), π ( ν N ) ν <, () R, ν which is not uniformly distributed. In [7], the argument that the orthogonal steering vectors show higher data rates is because the satial frequencies of the given AoAs and the orthogonal steering angles have nearly the same distribution, i.e., uniform distribution. However, this argument only holds under some assumtions. Different oinion that the orthogonal steering vectors show higher data rate is exlained as follows. In Fig. 3(b) and 3(d) the beam atterns of the non-orthogonal and the orthogonal steering vectors are shown in satial frequency domain. The total 3 beam atterns (6 in Fig. 3(b) and 6 in Fig. 3(d)) maing to different satial frequency indices have the same beamwidth values, as roven in (4). Due to the roerty of constant beamwidth in satial frequency domain, we analyze the difference between the orthogonal and the non-orthogonal steering vectors in satial frequency domain. In Fig. 3(d), the uniformly distributed satial frequency indices lead to the results that the beamforming gain of the orthogonal steering vectors in general is larger than or almost equal to -3 db gain. On the contrary, regarding the nonorthogonal cases in Fig. 3(b), the nonuniformly distributed satial frequency indices result in extremely low beamforming gain at some satial frequency indices, such as ν, ±.6. Although ν shows lower robability within <ν <, the overall data rates may degrade a lot due to low beamforming gain within this interval. More simulation results are shown in Section IV. The orthogonal and the non-orthogonal beam

5 Data rate (bit/s/hz).. Max. data rate Non orthogonal, φ ~U( o, o ) Non orthogonal, φ ~U( o,3 o ) Non orthogonal, φ ~U(3 o,4 o ) Non orthogonal, φ ~U(4 o,6 o ) Non orthogonal, φ ~U(6 o,7 o ) Non orthogonal, φ ~U(7 o,9 o ) Data rate (bit/s/hz).. Max. data rate Orthogonal, φ ~U( o, o ) Orthogonal, φ ~U( o,3 o ) Orthogonal, φ ~U(3 o,4 o ) Orthogonal, φ ~U(4 o,6 o ) Orthogonal, φ ~U(6 o,7 o ) Orthogonal, φ ~U(7 o,9 o ) S (db) Figure 4: Comarison of achievable data rates by the erfect and the non-orthogonal steering vectors with different range of AoAs S (db) Figure : Comarison of achievable data rates by the erfect and the orthogonal steering vectors with different range of AoAs. atterns shown in angle domain are illustrated in Fig. 3(a) and 3(c) for reference. IV. SIMULATIOESULTS In the simulations, 6, N K 6, and N P are considered. The distribution of AoA are uniformly distributed over ( π, π ) and the channel gain is normalized to, α. The simulated codebooks (consisting of N K satial frequency indices) for the orthogonal and the non-orthogonal steering vectors are introduced in Fig. 3(b) and 3(d), and the otimal orthogonal and non-orthogonal steering vectors are selected according to (). Fig. 4 illustrates the data rates by erfect and nonorthogonal steering vectors, where the erfect steering vector is assumed to exactly comensate the AoA, i.e., φ φ k. Thus we have the maximal data rate shown as ( ) R max log + α σz. (6) In Fig. 4, the data rates vary a lot with different range of AoAs because in Fig. 3(b) the 6 beam atterns are not equally distributed in satial frequency domain. The case of φ U( o, o ) ( ν.7), is even bit/s/hz worse than the case of φ U(7 o, 9 o ) (7.73 ν 8). Consider the channel bandwidth of.6 GHz at mmw systems [3], the data rates would dramatically decrease if the AoA changes from 8 o to o when considering moving user terminal. Owing to the beam atterns are symmetric at o, only the cases with φ are simulated. Then comaring Fig. 3(d) with Fig., it can be seen that the data rates with different range of AoAs kee constant as we conclude in the revious section. The results in Fig. are very close to the data rate by orthogonal steering vectors with φ U( π, π ) in Fig. 6 because the deviation of the beamforming gain is almost u to 3 db. In contrast, the averaged data rate by the non-orthogonal steering vectors with φ U( π, π ) in Fig. 6 shows worse result since the data rates within < φ < o (.7 ν.7) are quite low. Data rate (bit/s/hz) Max. data rate Orthogonal, φ ~U( π/,π/) Non orthogonal, φ ~U( π/,π/) S (db) Figure 6: Comarison of achievable data rates by the erfect, the orthogonal, and the non-orthogonal steering vectors with uniformly distributed AoA over ( π, π ). The strategy of the design of the steering vectors can be extended to other ractical alications, such as N K <. For examle, 6and N K 4, we can design the satial frequency indices as ν 6,,, 6 (the corresonding steering angles are φ 48.6, 4., 4., 48.6). Designing the steering vectors in this way ensures the smallest deviation of the beamforming gain. V. CONCLUSIONS This aer elaborates why orthogonal steering vectors show better data rates than non-orthogonal ones by analyzing the steering angles, the steering satial frequency indices, and beam atterns in satial frequency domain. Due to the fact that the constant beamwidth values in satial frequency domain, if the beam atterns are designed with uniformly distributed satial frequency indices, which can ensure the smallest deviation of the beamforming gain. Orthogonal steering vectors satisfy this condition so that they can achieve higher data rates than non-orthogonal ones.

6 VI. APPENDIX Given φ U( π, π ), the cumulative distribution function sin φ (CDF) of ν (ν ) can be written as F ν (ν )P (ν ν ) ( ) sin φ P ν ( ( )) P φ sin ν N ( ( )) R F φ sin ν ˆ sin (ν /) ˆ sin (ν /) π dφ π/ π dφ, ( ( ) ) ν < π sin ν + π, ν <, ν By the rule for derivative of sin (x) / x, one has the df of ν, [] R, Kohno, et al., Array Antenna Beamforming Based on Estimation of Arrival Angles Using DFT on Satial Domain, IEEE PIMRC, 99, [] J. L. Butler, et al., Beam forming matrix simlifies design of electronically scanned antennas, Electronic Design, Vol. 9, Ar. 96, [] D. Ramasamy, et al., Comressive Adatation of Large Steerable Arrays, IEEE Information Theory and Alications Worksho,, [3] IEEE Std 8..3cTM-9, Part.3: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Secifications for High Rate Wireless Personal Area Networks (WPANs) Amendment : Millimeterwave- based Alternative Physical Layer Extension, IEEE Comuter Society, Oct. 9., ν < f ν (ν ), π ( ν N ) ν < R, ν. ACKNOWLEDGMENT The research leading to these results has received funding from the Euroean Union Seventh Framework Programme (FP7/7-3) under grant agreement n 6963 (MiWaveS). REFERENCES [] T. S. Raaort, et al., Wireless Communications: Princiles and Practice, nd Edition, Pearson Education, Uer Saddle River, NJ,. [] T. A. Thomas, et al., 3D mmwave Channel Model Proosal, IEEE VTC Fall, 4,. 6. [3] T. S. Raaort, et al., Millimeter Wave Mobile Communications for G Cellular: It Will Work!, IEEE Access, Vol., May 3, [4] A. Hajimiri, et al., Integrated Phased Array Systems in Silicon, IEEE Proceedings, Vol. 93, Se., [] J. C. Liberti, et al, Smart Antennas for Wireless Communications: IS-9 and Third Generation CDMA Alications, Prentice Hall, NJ, 999. [6] L. Zhou and Y. Ohashi, Efficient codebook-based MIMO beamforming for millimeter-wave WLANs, IEEE PIMRC,. 9. [7] D. Yang, et al., DFT-Based Beamforming Weight-Vector Codebook Design for Satially Correlated Channels in the Unitary Precoding Aided Multiuser Downlink, IEEE ICC,,.. [8] D. H. Johnson, et al, Array Signal Processing: Concets and Techniques, Prentice Hall, 993. [9] IEEE Std 8.ad-. Part : Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Secifications Amendment 3: Enhancements for Very High Throughut in the 6 GHz Band, Dec..

Low-Complexity Spatial Channel Estimation and Hybrid Beamforming for Millimeter Wave Links

Low-Complexity Spatial Channel Estimation and Hybrid Beamforming for Millimeter Wave Links Hsiao-Lan Chiang Tobias Kadur Wolfgang Rave and Gerhard Fettweis Technische Universität Dresden Email: {hsiao-lanchiang