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2 Asymptotically Optimal Transmit Strategies for the Multiple Antenna Interference Channel Erik G Larsson, Danyo Danev Eduard A Jorswieck Abstract We consider the interference channel with multiple antennas at the transmitter We prove that at high signal-tonoise ratio SNR, the zero-forcing transmit scheme is optimal in the sum-rate sense Furthermore we prove that at low SNR, maximum-ratio transmission is optimal in the sum-rate sense We also provide a discussion of the connection to classical results on spectral efficiency in the wideb regime Finally, we propose a non-convex optimization approach based on monotonic optimization to solve the sum rate maximization problem BS h h h MS I INTRODUCTION The interference channel IFC is a classic object of study in information theory [], [] While its capacity is an open problem, a number of results on achievable rates were established a fairly long time ago [3], [4] In particular, it is known that strong interference should be decoded subtracted off the received data whenever possible, that weak interference should be treated as noise More recently, there has been renewed interest in the IFC, as witnessed by a number of contributions in the research literature [5], [6], [8], for example The principal driving motivation for this interest is that the IFC is a sound model for the spectrum sharing scenario in wireless communications, where multiple independent radio links coexist interfere in the same spectral b therefore interfere with each other In this work we are concerned with the IFC for the case when the transmitter has multiple antennas We refer to this as the multiple-input single-output MISO IFC A sketch of the MISO IFC, with two transmitter-receiver pairs, is given in Figure The importance of using multiple antennas is twofold First, it provides the usual rate diversity gains [9] Second, if partial channel state information is available at the transmitters, then this can be exploited to minimize the interference that one system generates to the other system Some previous work on the MISO IFC is available The MISO IFC is a special case of the multiple-input multipleoutput MIMO IFC [5], [6], hence many results therein can be specialized to the MISO case A characterization of the MISO IFC from a game-theoretic both non-cooperative cooperative point of view was presented by [] An E Larsson D Danev are with Linköping University, Dept of Electrical Engineering ISY, Division of Communication Systems, Linköping, Sweden {eriklarsson,danyo}@isyliuse E Jorswieck is with the Dresden University of Technology, Communications Laboratory, Chair of Communication Theory, D-6 Dresden, Germany jorswieck@ifnettu-dresdende This work was supported in part by the Swedish Research Council VR the Swedish Foundation for Strategic Research SSF E Larsson is a Royal Swedish Academy of Sciences KVA Research Fellow supported by a grant from the Knut Alice Wallenberg Foundation BS h MS Fig The two-user MISO interference channel under study illustrated for n transmit antennas explicit parameterization of the achievable rate region was given in [] Contribution: This paper is concerned with the characterization of sum-rate optimal transmit strategies for the MISO IFC We provide two main results First we prove that at high SNR, the zero-forcing ZF strategy is optimal in the sum-rate sense Second, we show that at low SNR, the sumrate-optimal strategy becomes maximal-ratio transmission MRT The results are discussed in the context of spectral efficiency in the wideb regime [] II MODEL We consider the -user MISO IFC in Figure We shall assume that transmission consists of scalar coding followed by beamforming, that all propagation channels are frequency-flat This leads to the following basic model for the matched-filtered, symbol-sampled complex baseb data received at MS MS : y h T w s + h T w s + e y h T w s + h T w s + e where s s are transmitted symbols, h ij is the complex-valued n channel-vector between BS i MS j, w i is the beamforming vector used by BS i The variables e, e are noise terms which we model as iid Gaussian with zero mean variance σ Single-stream transmission scalar coding followed by beamforming is optimal under certain circumstances, for example provided that BS i knows h ii MS, MS treat the interference as Gaussian noise [7], [9]

3 Under these assumptions the following rates are achievable: R w, w log + wt h w T h + σ for the link BS MS, R w, w log + wt h w T h + σ for BS MS For fixed channels {h ij }, under the power constraint w i, i,, 3 we define the achievable rate region as R R w, w, R w, w w,w, w i III MAIN RESULTS Define the sum-rate as follows Rw, w R w, w + R w, w Note that R is a function of h ij w i where the notation makes the latter dependence explicit Proposition : At high SNR, ZF is sum-rate optimal More precisely lim σ argmax w, w Rw, w w ZF, wzf, where w ZF Π h h Π hh w ZF Π h h Π hh Proposition : At low SNR, MRT is sum-rate optimal More precisely lim σ argmax w, w Rw, w w MRT, wmrt, where w MRT h w MRT h h h Proof: of Propositions We note first from Corollary in [] that for any rate point on the Pareto boundary of the achievable region R, we must have Π h w α h Π h h + Π α hh Π h h 4 Π h w α h Π h h + Π α hh Π h h, 5 where α, α are real-valued α, α Hence, the beamforming vectors corresponding to the optimal sum-rate point have the form of 4 5 Also, the point α, α, corresponds to zero-forcing: w w ZF w w ZF Furthermore, α, α Π h h h, Π h h h corresponds to maximum-ratio transmission: w w MRT w w MRT Let Π h h, Then we can write Π h h, ξ Π h h, ξ Π h h, λ hh h Π h, h λ hh h Π h h Rw, w Rα, α log + α + α ξ σ + α λ + log + α + α ξ σ + α λ 6 Applying Lemma see Appendix proves Proposition Next, observe that h h Π h h + Π h h Π h h + Π h h Applying Lemma 3 see Appendix proves Proposition IV ILLUSTRATION To illustrate the results, we make use of the following variant of the Pareto boundary parameterization This parameterization is proven in [] as explained therein it also has a game-theoretic interpretation Proposition 3: Any point on the Pareto boundary is achievable with the beamforming strategies w λ w λ λ w MRT + λ w ZF λ w MRT + λ w ZF λ w MRT + λ w ZF λ w MRT + λ w ZF for some λ, λ The parameterization of the Pareto boundary of the twouser MISO IFC in Proposition 3 can be interpreted in the following sense: The strategy λ corresponds to a completely selfish behavior On the other h, λ corresponds to a completely altruistic behavior By choosing a certain λ, the transmitter can choose its level of selfishness or altruism Note that there is a one-to-one mapping between λ k 7

4 α k in Section III In particular, for the parameterization in 7, λ k corresponds to ZF transmission while λ k corresponds to MRT In the numerical results, we will use the parameterization in 7 to illustrate the complete achievable rate region In this region, we mark the ZF MRT point As the SNR is increased for a certain set of channel realizations, we can observe that the ZF MRT rate points behave as predicted theoretically in the last section In Figure, we show the achievable rate region, along with the MRT ZF points for a two-user MISO IFC with two antennas at each transmitter, for fixed but romly chosen channel realizations Illustrations are provided for different SNRs in the range { 3,,,, 3} db The asymptotic optimality of MRT for small SNR ZF for high SNR can be clearly observed The path of these two operating modes cross at an SNR of about db V DISCUSSION We provide an interpretation alternative derivation of Proposition using the results of [] Specifically, [] analyzed the low-snr regime for a communication link introduced two performance measures, namely the min the wideb slope S Reference [] then showed that the system bwidth B, the transmission rate R, the transmit power P the spectral efficiency C E b satisfy the fundamental limit R B C The function C capacity expression CSNR, ie C the SNR which solves 8 is directly related to the common E b CSNR SNR CSNR for At low SNR, the function C E b can be expressed see [] as C S db E b db, 9 3dB min with min log e Ċ S ] [Ċ C the better is the approxima- min The closer E b gets to tion in 9 Note, that the first second derivative in are taken of the function common capacity function CSNR The low SNR approximation was recently extended to the multiple access networks in [3] in terms of its robust slope region For the multiple access channel, there is basically no difference in terms of minimum E b between MAC single-user channels The following result generalizes this to the MISO interference channel Define SNR ρ Rate R Rate R Rate R Rate R Rate R 5 x Rate R x Rate R Rate R Rate R Rate R Fig Examples of achievable rate regions for the two-user two-antenna MISO IFC The red points correspond to the parameterization in 7 The red squares are the two single-user points The blue circles correspond to ZF beamforming rates The blue crosses correspond to the MRT beamforming rates The SNR is { 3,,,, 3} db

5 Proposition 4: Let C k SNR log + SNR h T kkw k for k Then, even in interference channels, the following minimum E b is achievable [ ] E b Ck min SNR Proof: We follow closely the proof of [, Theorem 8] start with an upper bound on the achievable rate where we assume that the receiver knows the codewords of the other users perfectly subtract them before decoding the intended user The capacity for user k, k, in this genie-aided setting is C k SNR log + SNR h T kkw k, whose derivative at SNR is equal to the expression in To lower-bound the capacity we apply a receiver which treats the interference of the other user as additive noise The lower bound is C k SNR log + SNR ht kk w k σ + I j SNR, 3 where I j h T jkw j is the interference caused by the other user The function in 3 has a derivative at SNR which is identical to that in This completes the proof because upper lower bound converge to the same minimum E b One interesting observation is that the optimal in terms of minimum E b receiver at the mobiles is the receiver which treats the interference simply as additional additive noise With this background we are able to restate Proposition provide an alternative proof Proposition 5: The beamforming vectors which optimize the minimum E b correspond to the MRT beamformers Proof: The result follows directly from the characterization in because E b min, h T w E b min, h T w which is minimized by the MRT beamforming vectors VI SUM-RATE MAXIMIZATION BY MONOTONIC OPTIMIZATION In applications, one is typically interested in finding specific points on the Pareto boundary, a notable example being the sum-rate point The main difficulty with solving the sum-rate maximization problem is that the problem is non-convex Using the parameterization in Proposition 3, this maximization problem can be posed as max {R λ + R λ}, 4 λ,λ where λ [λ, λ ] In this section, we shall illustrate that problem 4 can be solved using polyblock algorithm for monotonic optimization [4] Effectively the approach is to transform the original nonconvex objective function 4 into a strictly increasing function over a constraint set that is normal The price to pay is that the dimension of the variable space must be enlarged from two corresponding to λ, λ to three We give a brief description in what follows; more details are available in [5] The first result relates the sum-rate maximization problem to the area of monotonic optimization Proposition 6: The maximum sum-rate problem max λ [,] {R λ + R λ} is a difference of monotonic functions dm programming problem Proof: The result follows as a corollary of Lemma 4 in Appendix II, because the objective function can be rewritten as R λ + R λ [f λ g λ] + [f λ g λ] f λ + f λ [g λ + g λ], }{{}}{{} φλ ψλ where the functions f i λ g i λ, i, are defined in Appendix II By Lemma 4 both functions φ ψ are monotonically increasing As a consequence of Proposition 6, problem 4 can be formulated as the following general dm problem max λ [,] {φλ ψλ} 5 with strictly increasing functions φ ψ Next, we substitute ψλ ψ t in 5 obtain the equivalent programming problem with x [λ, λ, t] max{φx + ψx 3 } }{{} st x D 6 Φx with the constraint set { D x R 3 + : x, x, x 3 ψx }, x 7 ψ Two key observations allow us to proceed First, we note that the function Φx is strictly increasing Second, we have the following result about the constraint set Lemma : The set D defined in 7 is normal Proof: Choose the vector x D choose any vector y such that y x We need to verify that y D First note that y x y x Also, since ψ is strictly increasing in x, x, we have ψx, x ψy, y Since y 3 x 3, it follows that y D, too Furthermore, the constraint set is compact, bounded, connected We have found that the programming problem in 5 is equivalent to maximization of a strictly increasing function over a normal set This is a monotonic optimization problem in stard form [4] Therefore, we can apply the outer polyblock approximation algorithm [4] to solve the sumrate maximization problem 4 For a detailed description of the monotonic optimization framework how it can be applied to solve the sum-rate maximization problem 4, see [5]

6 VII CONCLUSIONS The parameterization of the achievable rate region of the two-user MISO IFC leads to a simple structured but nonconvex optimization problem In this work, we try to underst the asymptotic behavior of the optimal beamforming solution at high low SNR It turns out that for small SNR, MRT is optimal the interpretation in terms of minimum E b justifies the analysis For high SNR values, the optimal strategy is ZF beamforming APPENDIX I LOW AND HIGH SNR RESULTS Lemma : Let,, ξ, ξ, λ λ be positive real numbers For any α i [, ], i any positive number σ, consider Rα, α in 6 We have that lim arg maxr σ α, α,, σ S where S is the unit square in R defined as S {α, α : α, α } Proof: Consider the function f σ α, α Rσα,α σ +ξ σ +ξ σ 4 [σ +λ α +α+ξ α ] σ +λ α σ +λ α σ +λ α +α+ξ α 8 Since the function x is continuous strictly increasing we have that We define arg min S R σα, α arg min S f σα, α a maxλ α S + α + ξ α a max λ α S + α + ξ α Since the numerator in the function f σ α, α is always non-negative, we obtain f σ α, α σ + ξσ + ξσ + λ α σ + λ α σ 4 σ + a σ + a We observe that f σ, compare the value of f σ α, α with the value f σ, Consequently we get f σ α, α σ +ξ σ +ξ σ +λ α σ +λ α σ 4 σ +a σ +a ξ +ξ +λα +λα a aσ σ +a σ +a + ξ +ξ λα +λα +ξ ξ +λλα α aa σ +a σ +a + ξ ξ λα +λα +ξ +ξ λλα α σ σ +a σ +a + ξ ξ λλα α σ 4 σ +a σ +a a+aσ σ +a σ +a a a σ +a σ +a + ξ ξ λα +λα σ σ +a σ +a a+aσ4 a a σ +ξ ξ λα +λα σ σ +a σ +a For an arbitrarily chosen positive number ε we can find a σ ε > such that a + a σ 4 + a a σ ε/ for all σ, σ ε ] Now for every point α, α S such that λ α + λ α εξ ξ every σ, σ ε ] we have f σ α, α a+aσ4 a a σ +ξ ξ λα +λα σ σ +a σ +a ε/+ε σ σ +a σ +a ε σ σ +a σ +a > Thus for any σ, σ ε ] the minimum of f σ α, α at a point of S within the elipse defined by the equation λ α + λ α εξ ξ Since, is within this ellipse for each ε > its diameter tends to zero when ε goes to zero, we have that lim argmin f σα, α,, σ S which proves the lemma Lemma 3: Let,, ξ, ξ, λ λ be positive real numbers For any α i [, ], i any positive number σ, consider Rα, α in 6 We have that lim arg maxr σ α, α,, σ + S + ξ + ξ where S is the unit square in R defined as S {α, α : α, α } Proof: As in the proof of Lemma we consider the function f σ α, α defined by Equation 8 Clearly, for any positive σ the function f σ α, α is continuous on S differentiable infinitely many times on ints which is the set of all interior points of S We can calculate the first partial derivatives to be α α, α f σ α, α λ α α +ξ α σ +λ α [σ +λ α +α+ξ α ] α+ξ α ξ α / α σ +λ α +α+ξ α α α, α f σ α, α λ α α +ξ α σ +λ α [σ +λ α +α+ξ α ] α+ξ α ξ α / α σ +λ α +α+ξ α Suppose now that σ > that x, x ints is a local optimum of f σ α, α We must then have that α x, x α x, x

7 This is equivalent to ξ x λxx+ξ x x +ξ x σ +λ x 9 σ +λ x +x+ξ x σ +λ x +x+ξ x ξ x λxx+ξ x x +ξ x σ +λ x σ +λ x +x+ξ x σ +λ x +x+ξ x Let us now define a i max ix i + ξ i i x i [,] b i min ix i + ξ i i x i [,] for i, For the right h side of Equation 9 we have that Since λ x x + ξ x + ξ σ + λ x σ + λ x + x + ξ σ + λ x + x + ξ λ x x + ξ x + ξ σ + λ x σ + λ x + x + ξ σ + λ x + x + ξ λ a σ + λ + a b σ σ + b λ a lim σ + λ + a σ + b σ σ + b we have that for an arbitrary ε >, there exists σ ε that λ a σ + λ + a b σ σ + b ε thus ξ x ε, such whenever σ σ ε In a similar fashion we conclude that there exists σ ε such that ξ x ε, whenever σ σ ε Choosing σ ε to be equal to max{σ ε, σ ε }, we obtain that for σ σ ε we have i ξ ix i i ε, i, Since the functions i ξ i x/ are continuous for x [,, we have that for a sufficiently small ε, the inequalities are equivalent to i i + ξ i gε x i i, i, i + ξi Here the function gε is such that lim ε gε This simply means that for an arbitrary δ > we can choose ε > consequently σ ε such that i i + ξ i δ x i i, i,, i + ξi for all σ σ ε We observe that when σ σ ε we have i α, α >, if α i > α i i + ξi α, α <, if α i < i α i i + ξi δ, for i, Since for any σ >, the function f σ α, α is continuous on S, this implies that for an arbitrary point α, α S, there exists a point x, x in the square defined by such that f σ α, α f σ x, x Moreover, this inequality is strict if the point α, α does not belong to the square defined by Thus if x, x is a point, where a global minimum of f σ α, α is achieved, then x x satisfy for any sufficiently large σ This shows that lim arg min f σα, α,, σ + S + ξ + ξ This completes the proof of the lemma APPENDIX II MONOTONIC OPTIMIZATION RESULTS Let us define the following quantities for further use h, h H Π h, h, h H Π h Obviously, the inequalities 3 hold Define further the functions f λ log σ n + w λ T h + w λ T h, f λ log σ n + w λ T h + w λ T h, g λ log σ n + w λ T h, g λ log σ n + w λ T h Finally, let fλ f λ+f λ gλ g λ+g λ Lemma 4: The functions f λ, f λ, fλ as well as g λ, g λ, gλ are strictly increasing in λ λ

8 Proof: All six functions depend on λ or λ via the following terms α λ w T λ h λ w MRT + λ w ZF T h λ w MRT + λ w ZF λ h + λ Π h h hh Π h h λ + λ + λ λ hh Π h h λ + λ λ λ Similarly, we obtain α λ w T λh λ + λ λ λ, β λ w T λh λ λ λ, β λ w T λh λ λ λ Next, the first derivatives with respect to λ or λ are computed directly as dα λ dλ λ + λ λ λ λ, 4 where the last inequality follows from 3 The monotonicity of α λ follows similarly The first derivative of β λ with respect to λ is given by dβ λ λ 3 λ + λ 5 dλ λ λ Since fλ, λ gλ, λ can be expressed as fλ logσ n + α λ + β λ + logσ n + α λ + β λ gλ logσ n + β λ + logσ n + β λ the result in Lemma 4 follows from 4 5 REFERENCES [] R Ahlswede, The capacity region of a channel with two senders two receivers, Ann Prob, vol, pp 85 84, 974 [] A B Carleial, Interference channels, IEEE Trans on Inf Theory, vol 4, no, pp 6 7, Jan 978 [3] T Han K Kobayashi, A new achievable rate region for the interference channel, IEEE Trans on Information Theory, vol 7, no, pp 49 6, Jan 98 [4] M H M Costa, On the Gaussian interference channel, IEEE Trans on Inf Theory, vol 3, pp 67 65, 985 [5] X Shang, B Chen, M J Gans, On the achievable sum rate for MIMO interference channels, IEEE Trans on Inf Theory, vol 5, pp , 6 [6] S A Jafar M Fakhereddin, Degrees of freedom for the MIMO interference channel, IEEE Trans on Inf Theory, vol 53, pp , 7 [7] X Shang B Chen, Achievable rate region for downlink beamforming in the presence of interference, Proc IEEE Asilomar, 7 [8] S Vishwanath S A Jafar, On the capacity of vector Gaussian interference channels, IEEE ITW, 4 [9] D Tse P Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 5 [] E Jorswieck, E G Larsson D Danev, Complete characterization of the Pareto boundary for the MISO interference channel, IEEE Transactions on Signal Processing To appear [] E G Larsson E A Jorswieck, Competition cooperation on the MISO interference channel, IEEE Journal on Selected Areas in Communications, vol 5, no 7, pp 59 69, Sept 8 [] S Verdú, Spectral efficiency in the wideb regime, IEEE Trans on Information Theory, vol 48, no 6, pp , June [3] T Muharemovic, A Sabharwal, B Aazhang, Policy-based multiple access for decentralized low power systems, IEEE Trans on Wireless Communication To appear [4] H Tuy, Monotonic optimization: Problems solution approaches, SIAM Journal on Optimization, vol, no, pp , [5] E Jorswieck E G Larsson, Monotonic optimization framework for the two-user MISO interference channel, in preparation, 8