# Rethinking MIMO for Wireless Networks: Linear Throughput Increases with Multiple Receive Antennas

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4 Proof : The outage probability upper bound is derived by rewriting the outage probability as the tail probability of random variable SINR and then applying Markov s inequality: ] S P SINR β] =P I k + β Ik + = P ] S β a) Ik + ] βe S b) = βe I k + ] ] E S c) = βe I k + ] N k ) d) < β πd λ ) μ k ) ) +, N k where a) holds due to Markov s inequality, b) is due to the independence of random variables I k and S, c) holds because S is χ N k), and d) follows from Lemma 3. Setting this bound equal to ɛ and then solving for λ yields the associated lower bound to λ ɛ. ) It is worthwhile to note that the N k β ɛ term in the λ ɛ lower bound is the density increase due to array gain i.e., increased signal power), while the k ) term is the density increase due to interference cancellation. IV. LINEAR DENSITY SCALING Having established a lower bound to λ ɛ as a function of k, the number of interferers cancelled by the partial-zf receiver, we can now show that linear density scaling is achievable. If we choose the number of cancelled interferers k = θn for some constant 0 <θ<, the density lower bound becomes: ) ɛ μ λ ɛ β πd θ) θ N + ) β ɛ N θ ).) θ This lower bound divided by N converges to a constant, thereby showing that linear density scaling is achieved for any θ satisfying 0 <θ<. 3 This scaling result can be intuitively understood by examining how the signal and aggregate interference power increase with N. Choosing θ< ensures that the signal power, which is χ θ)n, increases linearly with N. Based on the upper bound in Lemma 3 we can see that the condition θ>0 ensures that the interference power increases only linearly with N if λ is linear in N. These linear terms are offsetting, and thus 3 This derivation proves that λ ɛ scales at least linearly with N, but does not preclude super-linear scaling. We strongly believe that super-linear scaling is not achievable by either PZF or the optimal MMSE receiver, and a rigorous proof of this is under investigation. Density Partial ZF k = N/3) Partial ZF Lower Bound MRC Full ZF k = N ) Receive Antennas N) Fig.. Maximum density versus N for ɛ =., β =, =3, θ = 3, d =. allow an approximately constant SINR to be maintained as λ is increased linearly with N. If an MRC receiver k =0) or PZF receiver with a constant value of k is used, the signal power increases linearly with N as desired but the interference power increases too quickly with the density as λ ), thereby limiting the density growth to N. At the other extreme, if full zero forcing k = N )is used or k = N κ for any positive integer κ, the interference power scales appropriately with the density but the signal power is χ κ and thus does not increase with N, thereby limiting the density increase to N. A. Optimal Fraction of Cancelled Interferers The parameter θ determines the fraction of the available receive degrees of freedom that are used to perform interference cancellation. Although any θ 0, ) achieves linear scaling, we can find the optimal value of θ by maximizing the λ ɛ lower bound in ) in the limit of large N. This is clearly equivalent to maximizing the quantity θ) θ. By simply setting the derivative w.r.t. θ) to zero and solving, we find the optimal value of θ to be: θ =. ) For the degrees of freedom should be used to boost signal power rather than to cancel interference i.e., θ 0, because far-away interference is not much weaker than nearby interference when the path loss exponent is small and thus there is little benefit in cancelling only the nearby interferers. On the other hand, as the path loss exponent increases θ the power from nearby interferers begins to dominate and thus it is preferable to use the antennas for interference cancellation rather than signal boosting. V. NUMERICAL RESULTS In Figures and the numerically computed maximum density λ ɛ and the lower bound to λ ɛ given in ) are plotted versus N for ɛ =., β =, d =for =3Fig. ) and

6 APPENDIX II PROOF OF LEMMA 3 First we have ] E X i H i = E X i ] H i = E X i ] where E X i H i ]=E X i ] E H i ]=E X i ] due to the independence of X i and H i and the fact that EH i ]=. Because the ordered squared distances X, X,... are a -D PPP with intensity πλ, random variable πλ X i is χ i and thus has PDF fx) = xi e x i )!. Therefore Xi E ) ] / = πλ) x / xi e x i )! dx 0 = πλ) x i e x dx i )! 0 = πλ) Γ ) i. Γi) This quantity is finite only for i>, and thus the expected power from the nearest uncancelled interferer is finite only if k +>. To reach the upper bound, we use the inequality derived in 4] which uses Kershaw s inequality to the gamma function): Γ ) i < Γi) i ) where is the ceiling function and we require i>. Therefore Γ ) i Γi) < = k ) i x ) dx ) k ), where the inequality in the second line holds because x is a decreasing function. REFERENCES ] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Commun. : Kluwer Academic Press, no. 6, pp , 998. ] S. Govindasamy, D. Bliss, and D. H. Staelin, Spectral efficiency in single-hop ad-hoc wireless networks with interference using adaptive antenna arrays, IEEE Journal on Sel. Areas in Communications, vol. 5, no. 7, pp , Sept ] A. Hunter, J. G. Andrews, and S. Weber, The transmission capacity of ad hoc networks with spatial diversity, IEEE Trans. on Wireless Communications, To Appear, ] K. Huang, J. G. Andrews, R. W. Heath, D. Guo, and R. Berry, Spatial interference cancellation for multi-antenna mobile ad hoc networks, IEEE Trans. on Info. Theory, Submitted arxiv: ). 5] S. Weber, J. G. Andrews, and N. Jindal, Transmission capacity: Applying stochastic geometry to uncoordinated ad hoc networks, IEEE Journal on Sel. Areas in Communications, Submitted arxiv: ). 6] M. Haenggi, On distances in uniformly random networks, IEEE Trans. on Info. Theory, pp , Oct ] P. Gupta and P. Kumar, The capacity of wireless networks, IEEE Trans. on Info. Theory, vol. 46, no., pp , Mar ] R.J.Muirhead,Aspects of Multivariate Statistical Theory. Wiley, 98.

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