Beamforming in Interference Networks for Uniform Linear Arrays


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1 Beamforming in Interference Networks for Uniform Linear Arrays Rami Mochaourab and Eduard Jorswieck Communications Theory, Communications Laboratory Dresden University of Technology, Dresden, Germany Abstract We consider a multipleinput singleoutput interference channel where each transmitter is equipped with a uniform linear array. By controlling the geometry of the array, i.e. adapting the antenna spacing, the rotation of the array, and the number of antenna elements, we investigate whether the capacity of the channel can be achieved with single user decoding capabilities at the receivers. This objective is reached when it is possible for each transmitter to perform maximum ratio transmission to its intended receiver while simultaneously nulling the interference at all unintended receivers. We provide for the two and three user case the necessary antenna spacing and rotation angle of the array in closed form. For the four user case, an integer programming problem is formulated which additionally determines the required number of antennas. I. INTRODUCTION We consider a setting in which transmitterreceiver pairs utilize the same spectral band simultaneously. The transmitters are equipped with multiple antennas while the receivers use single antennas. This setting corresponds to the multipleinput singleoutput MISO interference channel IFC []. The interest in antenna arrays in mobile communications [], [3] is the ability to increase the antenna gain in desired spatial directions while reducing the interference gains in undesired directions. The degrees of freedom provided by an Nelement antenna array is the ability to modify the antenna pattern at N spatial directions. This mechanism is accomplished by adapting the phase and gain of the signal at each antenna element []. In [5], maximizing the power gain at intended directions while nulling the interference directions is achieved by controlling the phases of the antenna elements. In [6], an antenna array is represented by a Z transform, and synthesis of the array is performed to produce nulls in interference directions. Communication system performance can also be increased by altering the geometry of the array. The geometry of the array is determined by the number of antenna elements and the spacing between them. This method might not be practical however can lead to increase in the channel capacity, e.g., by changing the antenna spacing [7]. In [8], an upsampling algorithm is proposed for a uniform linear array where virtual antennas are added between existing antennas. The method increases the user rate and resolution of the array. Part of this work has been performed in the framework of the European research project SAPHYRE, which is partly funded by the European Union under its FP7 ICT Objective.  The Network of the Future. This work is also supported in part by the Deutsche Forschungsgemeinschaft DFG under grant Jo 8/. The contributions and outline of this paper are as follows. After describing the system model in Section II we shortly review efficient beamforming in the MISO IFC in Section III. We study the effects of the array geometry on the jointly achievable power gains at the receivers. In Section IV, we consider uniform linear arrays where each transmitter performs maximum ratio transmission MRT to its intended receiver. We assume that each transmitter can adapt the geometry of the array, i.e., can modify the number of antenna elements used, the spacing between the elements of the arrays, and also rotate the array mechanically. We consider the problem of achieving zero interference power gain while performing MRT. In this case, the capacity of the interference channel can be achieved since maximum power gain is achieved at the intended receiver without interference from unintended transmitters. For two and three user MISO IFC, the calculation of the rotation angle and the antenna spacing can be done in closed form requiring arbitrary number of antennas larger than two. For the four user case, an integer program is formulated to further calculate the required number of antennas. Notations: Column vectors and matrices are given in lowercase and uppercase boldface letters, respectively. a is the Euclidean norm of a,a C N. b is the absolute value of b,b C. T and H denote the transpose and the Hermitian transpose. The orthogonal projector onto the column space of Z is Π Z := ZZ H Z Z H. The orthogonal projector onto the orthogonal complement of the column space of Z is Π Z := I Π Z, where I is an identity matrix. II. SYSTEM MODEL We consider a Kuser MISO IFC. Each transmitter has an intended receiver and K unintended receivers situated in the farfield of the array. The number of antenna elements used by transmitter k is N k. A receiver i is situated at an angle ki with respect to the normal of the antenna array of transmitter k. The channel vector from transmitter k to receiver i is [9] h ki = α ki [,e jθ ki,...,e jn θ ki ] T, where α ki is the channel gain and θ ki = π d k λ sin ki + k is the electrical angle. The electrical angle is a function of the antenna spacing d k, the carrier wavelength λ, and the rotation angle of the array k [,π]. We assume that k can be controlled by mechanically rotating the antenna array. This mechanism is referred to as mechanical steering [].
2 The beamforming vector used at transmitter k is w k C N which contains the complex weighting components on each antenna element. Each transmitter has a total power constraint which we set to unity. Thus, the feasible beamforming vectors for a transmitter k satisfy w k. The signal at a receiver i can be written as K y i = h ki H w k +n i, i =,...,K, k= where n i is additive noise which we model as i.i.d. complex Gaussian with zero mean and variance σ. The signal to interference plus noise ratio SINR at receiver i is hii H w i γ i w,...,w K = k i h ki H w k 3 +σ. The SINR region is the set composed of all achievable SINR tuples, defined as Γ := {γ w,...,w K,...,γ K w,...,w K R K : w k,k =,...,K. The outer boundary of this region consists of Pareto optimal points in which it is impossible to improve the SINR at one receiver without simultaneously decreasing the SINR at least one other receiver. Next, we characterize the beamforming vectors that are relevant for Pareto optimal operation. III. PARETO OPTIMAL BEAMFORMING We review the characterization of efficient beamforming in MISO IFC. The framework used for this purpose considers a single transmitter s power gain region []. A. Power Gain Region and Efficient Beamforming The power gain region of a transmitter k is the set of power gains simultaneously achievable at the receivers. This set is Ω k := { hk H w k,..., hkk H w k : wk. 5 This region is a Kdimensional set which has a convex boundary []. The boundary points of the gain region are of interest since they characterize extreme power gains achievable at the receivers. At these points, the transmitter cannot increase the power gain at one receiver without changing the power gain at another. Next, we formalize the boundary parts Ω k. For this purpose, we first need the following definition. Definition : A vector x dominates a vector y in direction e, written as x e y, if x l e l y l e l for all l, l n, and the inequality has at least one strict inequality. Definition : A point y R n + is called upper boundary point of a compact convex set C in direction e if y C while the set { y R n + : y e y R n + \C. We denote the set of upper boundary points in direction e as e C. An illustration of a twodimensional power gain region of transmitter is given in Fig.. The direction vectors e,e, and e 3 refer to three different parts of the boundary. The point in e Ω which achieves maximum power gain at receiver h H w h e 3 Ω h H w ZF e e h H w MRT h H w Fig.. An illustration of a twodimensional gainregion and its upper boundaries in directions e = [,],e = [, ], and e 3 = [,]. corresponds to MRT. The point in e Ω which achieves zero power gain at receiver corresponds to ZF transmission. It can be observed that boundary parte which includes MRT and ZF transmissions is relevant for efficient operation of the transmitter in the network. Since each transmitter seeks to maximize the power gain at its receiver, the direction vector is not of interest since it describes the boundary part that minimizes the power gain at all receivers simultaneously. Thus, we define the set of feasible direction vectors as E := {, K \. Theorem : [, Theorem ] All upper boundary points of the set Ω k in direction e k E can be achieved by K w k λ k = v max λ k,i e k,i h ki h ki, H 6 i= where λ k Λ := {λ [,] K : K i= λ i =. From Theorem, the beamforming vectors that achieve a boundary part of the gain region are characterized by K realvalued parameters each between zero and one. The next result characterizes the beamforming vectors for each transmitter that are relevant for Pareto optimal operation. Theorem : [, Theorem ] All points Pareto optimal points of the SINR region Γ in can be reached by the beamforming vectors in 6 and direction vectors specified as e k,k = + and e k,i = for i k. The boundary part of a transmitter s gain region specified in Theorem includes as special beamforming vectors the MRT and ZF to each unintended receiver [, Section V.A]. For the special twouser MISO IFC case, efficient beamforming is proven to be linear combination of MRT and ZF []. For transmitter analogously transmitter, the efficient beamforming vectors are w λ = λ Π hh Π hh + λ Π h h Π h h, 7 where λ [,λ MRT ],λ MRT = Π hh / h []. The beamforming vectors in 7 are identical to the ones in Theorem, i.e., the parametrization in 7 achieves
3 the points on the boundary specified by e in Fig.. In case λ [λ MRT,], the parametrization in 7 achieves the boundary in direction e in Fig.. This boundary part is important in a multicast setting between transmitter and both receivers for achieving all Pareto optimal points [], [3]. B. Effects of Array Geometry on Gain Region In order to gain insights on the effect of antenna geometry on the power gain region, we use the parametrization in 7 and calculate the power gains at intended and unintended receivers. The interference power at receiver as a function of λ is h H w λ h H Π hh = λ Π hh 8 = λ h = λ N α. 9 The interference power gain for a fixed λ does not depend on the antenna spacing or the rotation of the array. However, the parameter λ can depend on the antenna geometry if the transmitter maintains MRT transmission which corresponds to λ MRT = Π hh / h. The power gain at the intended receiver is h H wλ = λ A+, λ B where A = Π hh and B = Π h h. The terms A and B are calculated next. A = h H h / h N = e jiθ θ α = e jnθ θ α N i= e jθ θ N = ejnθ θ e jnθ θ α 3 e jθ θ e jθ θ N = cosn θ θ cosθ θ α N, where the second equality in comes from the exponential sum formula and from [,.3.]. B = h h H h h 5 = N α cosn θ θ cosθ θ α N. 6 The power gain at the intended receiver in depends on the array geometry as can be observed in A and B. In Fig., plots of the gain region for different values of d /λ ranging between. and in a. step length are given. The shape of the gain region changes with the antenna spacing, and it can be observed that the union of all gain regions gives a box shaped region. Hence, the interference power on receiver when transmitter performs MRT can be altered between its maximum value and zero only by changing the antenna spacing. In Fig. 3, the gain region is plotted for changing N between and 5. The gain region increases in size with increasing N. Interestingly, the increase in shape is not uniform. h H w 5 3 d/λ =. d/λ =. d/λ =.3 d/λ =. d/λ = h H w Fig.. Power gain region for different values of d /λ. N =, α = α =, =, =, and =. h H w h H w Fig. 3. Power gain region for different values of antennas between and 5. d /λ =.5, α = α =, =, =, and =. IV. ACHIEVING THE CAPACITY OF MISO IFC In this section, we restrict all transmissions to MRT: w MRT k = h kk h kk = [,e jθ kk,...,e jn k θ kk ] T. Nk 7 MRT achieves maximum power gain of hkk H w MRT k = α k at the intended receiver. Note that with MRT, the amplitudes of the signals at each antenna element are equal which leads to a uniform linear array [5]. The capacity of the channel is achieved when all transmitters can perform ZF on unintended receivers while maintaining MRT to intended receivers. For convenience, we choose to perform the analysis on transmitter which is analogously performed for all transmitter. A. Effects of Antenna Geometry on Interference We study the effect of changing d /λ, the rotation of the array, and the number of antennas N on the interference level at an unintended receiver. The transmitter performs MRT as in 7. In Fig., the interference gain at the unintended receiver is plotted for changing d /λ and number of antennas. While maintaining maximum power gain at the intended receiver, the interference is shown to be zero for specific values of the parameters. In Fig. 5, the effect of the rotation angle and antenna spacing on the interference is plotted for the same setting. Through the choice of d /λ and, it is also possible to null the interference gain.
4 h H w Fig.. Effects of d /λ and number of antennas on interference. =, = 35, and =. d /λ N 6 where t/n / N. Choosing the antenna spacing as in 5 achieves maximum gain at the intended receiver and no interference gain at the unintended receiver. In Fig. 6, the transmitter modifies the antenna spacing according to 5. h H w / N Fig. 6. The antenna array pattern where =, = 5,N =, =,α =, and α =. The optimal antenna spacing w.r.t. wavelength calculated from 5 is d /λ =.77 where t =. Fig. 5. Effects of d /λ and array rotation on interference. =, =, and N =. B. Two User MISO IFC Given the number of antennas N and rotation of the array, we calculate d /λ such that the interference gain is zero. This problem is formulated as = h H w MRT = h H h / h 8 = cosn θ θ cosθ θ α α N α 9 = cosπn d λ sin + sin + α cosπ d λ sin, + sin + N where the step from 8 to follows from . The numerator of should be zero while the denominator should not be zero giving the conditions cosπn d λ sin + +sin + =, cosπ d λ sin + +sin +. These conditions lead to the following solution πn d λ sin + sin + = tπ, 3 where t is an integer greater than and not a multiple of N. From 3 we get N d λ sin + sin + = t, d λ = t/n sin + +sin, 5 C. Three User MISO IFC We show in this section that transmitter has to adapt d /λ as well as the rotation angle in order to achieve ZF on both unintended receivers while maintaining MRT. For this purpose, we reformulate 5 to d λ = t/n sin cos + +, 8 where we used the identity sinz cosz = sinz z + sinz +z from [,.3.33]. The conditions for ZF on both unintended receivers is d λ = t /N sin /cos + /+, 9 d λ = t 3 /N sin 3 /cos + 3 /+. 3 Equating 9 and 3 we get t 3 /N sin 3 /cos + 3 /+ t /N = sin /cos + /+, 3 t 3 sin /cos + /+ = t sin 3 /cos + 3 /+. 3 Equation 3 can be expanded to 6. Collecting sin and cos we get the following expression tan = t 3 sin cos + t sin 3 cos +3, t 3 sin sin + t sin 3 sin which is the rotation angle necessary to achieve ZF simultaneously on the two unintended receivers. With in 33
5 t 3 sin /[cos + /cos sin + /sin ] = t sin 3 /[cos + 3 /cos sin + 3 /sin ] 6 t 3 sin cos + t sin 3 cos +3 t 3 sin sin + t sin 3 sin +3 = t sin cos + t sin cos + t sin sin + t sin sin + 7 h H w / N h H w / N Fig. 7. The antenna array pattern where =, =, 3 = 6,N =,α =,α =, and α 3 =. The antenna spacing w.r.t. wavelength calculated from 9 is d /λ =.879 where t = and the rotation from 33 is = 5. and the antenna spacing in 8, ZF can be achieved on the unintended receivers while maintaining maximum power gain on the intended receiver. This result is shown in Fig. 7. Note that this is achieved for arbitrary N. D. Four User MISO IFC In order to derive the conditions to achieve ZF at the three unintended receivers while performing MRT, we use the calculations for the rotation angle in 33 to achieve ZF at two unintended receivers simultaneously. We derive the rotation angle for ZF at receivers and 3 and also the rotation angle to achieve ZF at receivers and and equate these to get the equation in 7. Cross multiplying 7 and arranging the terms for the coefficients t,t 3,t we get = t sin 3 sin sin 3 S +t 3 sin sin sin S 3 +t sin 3 sin sin 3. S 3 The above condition leads to an optimization problem to calculate t,t 3,t. Since, these variables influence the number of required antennas as N = maxt,t 3,t +, we formulate the following integer programming problem min t +t 3 +t s.t. t S +t 3 S 3 +t S = t i N. 35 Fig. 8. The antenna array pattern where =, =, 3 = 5, = 7,α =,α =,α 3 =, and α =. From the optimization problem in 35 we get t =,t 3 =,t = which gives N = 3. The antenna spacing w.r.t. wavelength calculated from 9 isd /λ = 3.33 and the rotation from 33 is = The above problem can be solved numerically to obtain the result in Fig. 8. REFERENCES [] S. Vishwanath and S. Jafar, On the capacity of vector Gaussian interference channels, in Proc. of ITW, Oct., pp [] L. Godara, Applications of antenna arrays to mobile communications. i. performance improvement, feasibility, and system considerations, Proc. IEEE, vol. 85, no. 7, pp. 3 6, 997. [3], Application of antenna arrays to mobile communications. ii. beamforming and directionofarrival considerations, Proc. IEEE, vol. 85, no. 8, pp. 95 5, 997. [] W. Gabriel, Adaptive arrays An introduction, Proc. IEEE, vol. 6, no., pp. 39 7, 976. [5] F. R. Castella and J. R. Kuttler, Optimised array antenna nulling with phaseonly control, Radar and Signal Processing, IEE Proceedings F, vol. 38, no. 3, pp. 6, 99. [6] D. Gaushell, Synthesis of linear antenna arrays using Z transforms, IEEE Trans. Antennas Propag., vol. 9, no., pp. 75 8, 97. [7] A. Sayeed, Deconstructing multiantenna fading channels, IEEE Trans. Signal Process., vol. 5, no., pp , Oct.. [8] T. Rouphael and J. Cruz, A spatial interpolation algorithm for the upsampling of uniform linear arrays, IEEE Trans. Signal Process., vol. 7, no. 6, pp , Jun [9] S. Haykin, Adaptive Filter Theory, 3rd ed. PrenticeHall, 996. [] R. Mochaourab and E. A. Jorswieck, Optimal beamforming in interference networks with perfect local channel information, IEEE Trans. Signal Process.,, accepted for publication. [] E. A. Jorswieck, E. G. Larsson, and D. Danev, Complete characterization of the Pareto boundary for the MISO interference channel, IEEE Trans. Signal Process., vol. 56, no., pp , Oct. 8. [] E. A. Jorswieck, Beamforming in interference networks: Multicast, MISO IFC and secrecy capacity, in Proc. IZS,, invited. [3] D. Tomecki and S. Stanczak, On feasible SNR region for multicast downlink channel: Two user case, in Proc. ICASSP, Mar., pp [] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, 97. [5] S. Schelkunoff, A mathematical theory of linear arrays, Bell Syst. Tech. J, vol., pp. 8 7, 93.
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