Rate of Channel Hardening of Antenna Selection Diversity Schemes and Its Implication on Scheduling

Transcription

1 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY Rate of Channel Hardening of Antenna Selection Diversity Schees and Its Iplication on Scheduling arxiv:cs/ v [cs.it] 5 Mar 2007 Dongwoon Bai, Patrick Mitran, Saeed S. Ghassezadeh, Robert R. Miller, and Vahid Tarokh Abstract For a ultiple antenna syste, we copute the asyptotic distribution of antenna selection gain when the transitter selects the transit antenna with the strongest channel. We use this to asyptotically estiate the underlying channel capacity distributions, and deonstrate that unlike ultipleinput/ultiple-output MIMO systes, the channel for antenna selection systes hardens at a slower rate, and thus a significant ultiuser scheduling gain can exist - O/log for channel selection as opposed too/ for MIMO, whereis the nuber of transit antennas. Additionally, even without this scheduling gain, it is deonstrated that transit antenna selection systes outperfor open loop MIMO systes in low signal-to-interference-plus-noise ratio SINR regies, particularly for a sall nuber of receive antennas. This ay have soe iplications on wireless syste design, because ost of the users in odern wireless systes have low SINRs. I. INTRODUCTION The use of ultiple transit antennas has been studied for wireless links because of its proise of high spectral efficiency. When the receiver has full channel state inforation CSI, the capacity of a MIMO channel is typically calculated under the assuption that either the transitter has full CSI closed loop MIMO or no CSI open loop MIMO []. In both cases, in order to achieve data rates close to capacity, the ipleentation of various signal processing and RF units is needed. The underlying costs Dongwoon Bai, Patrick Mitran, and Vahid Tarokh are with Harvard School of Engineering and Applied Sciences, Cabridge, MA 0238 USA eail: dbai@fas.harvard.edu; itran@deas.harvard.edu; vahid@deas.harvard.edu. Saeed S. Ghassezadeh and Robert R. Miller are with AT&T Labs - Shannon Laboratory, Florha Park, NJ USA eail: saeedg@att.co; rr@att.co. Manuscript subitted to the IEEE Transactions on Inforation Theory on February 20, 2007.

2 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 2 ay increase as the nuber of antennas increases. For soe applications, this cost is prohibitive and otivates the studies of alternative antenna technologies. Antenna selection schees are attractive, since they can reduce hardware costs draatically [2] [7]. Soe proposals consider the selection of ore than one antenna and require ore than one transit chain. Nonetheless, in this paper, we are ainly interested in a transit antenna selection schee that selects the best channel between transit antennas. We will deonstrate that under this schee, there is no channel hardening, and thus significant ultiuser scheduling gain can exist. This is unlike MIMO systes [8], where the asyptotic scheduling gain is zero and there is significant channel hardening. To this end, we copute the asyptotic distribution of the selection gain and use this to asyptotically estiate the underlying channel capacity distributions. We note that the exact distribution of the selection gain has been coputed in the literature and the channel capacity has been nuerically and explicitly as a series expansion calculated. However, these exact values are not insightful in predicting if channel hardening exists or not, let alone the rate of hardening. For this purpose, we will invoke the theory of extree order statistics assuing Rayleigh fading channels. The outline of the paper is given next. In Section II, we present our syste odel. In Section III, we calculate the asyptotic distribution of selection gains and outage capacity gains. In Section IV, we obtain upper and lower bounds for the ergodic capacity. These results deonstrate that channel hardening occurs at uch slower rate than MIMO, and thus significant ultiuser scheduling gain can exist. Additionally, we will show that even without this scheduling gain, transit antenna selection can outperfor an open loop MIMO syste in the low SINR regie for sall nuber of receive antennas. In Section V we copute the scheduling gain. Finally, in Section VI, we present our conclusions and final coents while ost of the proofs ay be found in the Appendix. II. THE SYSTEM MODEL We consider an n MIMO channel odel, with transit and n receive antennas. The input-output relation is given by y = Hs+w. The atrix H represents the channel atrix, and is assued to be known at the receiver. The coplex vector s is the transitted signal vector, the n vector y represents the received signal, and w is an n zero-ean i.i.d. circularly syetric coplex Gaussian noise vector, with covariance atrix E[ww ] = I n. Interference, if any, is assued to be absorbed in w. An average transit power constraint E[s s] ρ, is assued, where ρ 0.

3 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 3 Assuing that a transit antenna selection syste chooses to transit only on the l-th antenna l {,,}, the capacity is given by CH,l = log 2 +ρ Let X l = n i= H il 2, then the best selection strategy is to choose antenna n H il 2. 2 i= l = argax[log 2 +ρx l ] = argaxx l 3 l l for transission. This schee is of interest, since it eliinates the need to feed back the channel atrix H. In fact, the receiver needs only to feed back the index of the best transit antenna to the transitter, requiring only log 2 bits of feedback inforation. We are interested to see if there is asyptotic channel hardening for such a syste, i.e. whether or not the underlying scheduling gain asyptotically goes to zero and the rate of which this occurs. To this end, we assue that H has independent zero-ean coplex Gaussian entries with variance /2 per real coponents. This is a flat fading channel odel. Thus X l has chi-square distribution with 2n degrees of freedo and X l is independent of X l whenever l l. For any set of i.i.d. rando variables Z,...,Z, we use the notation Z to denote ax i Z i. Using this notation, the expected received SINR for the transit antenna selection strategy is given by ρ X. The asyptotic distribution of X for large is of interest, since the ergodic capacity of the selection schee with the optial choice of transit antenna, is given by E H [CH,Ql ] = E H [log 2 +ρx l ] = E X [ log2 +ρx ]. 4 We note that other easures of perforance can also be derived based on the distribution of X. III. ORDER STATISTICS OF THE CHI-SQUARE DISTRIBUTION For any n, the probability density function pdf and the cuulative distribution function cdf of the chi-square rando variable X l 0 with 2n degrees of freedo are respectively given by x xn fx = e n! 5 and n Fx = e x for x 0. Clearly, the upper endpoint ωf sup{x Fx < } is infinity. k=0 x k k!, 6

4 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 4 and Let F and f denote the cdf and the pdf of X. It is well known that n F x F X x = F x = e x x k 7 k! f x f X x = F x fx n = e x x k x xn e k! n!, 8 respectively, for x 0 [9, pp. 9 ]. Clearly F x = 0 and f x = 0 for x < 0. The ergodic capacity of transit antenna selection can be coputed fro the above, by nuerical integration of 4. Analytical solutions see to be hard to obtain and offer no insights into channel hardening and scheduling gain for such a syste. k=0 k=0 A. A Key Convergence Result First, we have the following technical result whose proof ay be found in the Appendix. Lea : Let F and ωf be as defined above, then For any t > 0, the value Rt ωf t Fydy/{ Ft} is well defined and satisfies Rt <. As t,, if n =, Rt = +n t +O 9 t, if n F is in the doain of attraction of the Gubel distribution. That is to say, for all fixed real x, as, F a +b x Gx exp e x, 0 where Gx is the Gubel cdf, and the noralizing constants a and b can be selected to be a = q F / b = Rq. 2 In the above lea, other choices of a and b are also possible. We will study this next. Notation: For any two real-valued sequences c and d, we define c d if and only if li c d = 0; c = Θd if and only if 0 < li c /d <.

5 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 5 Lea 2: For F and G defined as above, li F a +b x = Gx if and only if the noralizing constants a and b satisfy a ln+n lnln lnn! 3 b. 4 Theore 3: The variance of X is bounded away for 0, i.e., the effective channel will exhibit significant fluctuations. Proof: It is known that the Gubel distribution has a ean γ = Euler s constant and variance π 2 /6 [9, p. 298]. Thus the ean and variance of X will approach to a +γ and b 2 π2 /6, respectively, as increases. Hence, we have E [ ] X a +γ, 5 Var [ ] π 2 X 6. 6 Because the variance is neither zero for any nor goes near zero as increases, the channel will fluctuate considerably. This suggest that the scheduling gain ay go to zero considerably slower than for MIMO. The scheduling gain of the above antenna-selection syste is coputed in Section V. B. Optiizing the Rate of Convergence In this subsection, we study the best pairs of noralizing constants a and b that provide an accurate fit to the Gubel distribution, even for relatively sall values of. For exaple, the siple choice of a = ln + n lnln lnn! and b = yields a poor estiate of the distribution for sall, and ay not yield accurate results for a realistic nubers of transit antennas. It is convenient to first introduce the sequence { α ax x 0 e x xn n! = }, 7 which is well defined for all greater than soe Mn. Clearly, α as. Theore 4: Let F and G be as given above. Consider the rate of convergence for fixed x of F a +b x Gx, 8

6 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 6 as. The optial sequences of constants a and b iniizing this rate is α, if n =, a = α +n α +O, if n 2, and α 2, if n =, b = +n α +O, if n 2. 2 For n =, the optial rate of convergence for any given x is F a +b x Gx = Θ, 2 and for n 2, α 2 F a +b x Gx = Θ log Corollary 5: For F, G, and q as given above, the choice of a = q and, if n =, b = +n q +O, if n 2 also satisfies 9 and 20, and this is therefore optial in the sense of iniizing the rate of convergence defined in 8. The proofs ay be found in the Appendix. Fro 9 and Corollary 5, we can check that the choice of noralizing constants in Lea a = q, b = Rq is optial. Fig. shows that the Gubel approxiation is an excellent fit with this choice of noralization coefficients. It can also be seen fro the figure that the variance of X stays bounded away fro zero as. q 2 23 C. Outage Capacity By using the above results, given a rate C 0, the corresponding outage probability P out C 0 can be approxiated by P out C 0 Pr { log 2 +ρx C 0 } G 2 C 0 ρ b a. 24

7 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 7 n= Prob{Selection Gain x 0 } 0.8 =2, Exact 0.6 =2, Gubel Approx. =5, Exact =5, Gubel Approx. 0.4 =0, Exact =0, Gubel Approx. 0.2 =20, Exact =20, Gubel Approx x 0 n=2 Prob{Selection Gain x 0 } 0.8 =2, Exact 0.6 =2, Gubel Approx. =5, Exact =5, Gubel Approx. 0.4 =0, Exact =0, Gubel Approx. 0.2 =20, Exact =20, Gubel Approx x 0 n=5 Prob{Selection Gain x 0 } 0.8 =2, Exact 0.6 =2, Gubel Approx. =5, Exact =5, Gubel Approx. 0.4 =0, Exact =0, Gubel Approx. 0.2 =20, Exact =20, Gubel Approx x 0 Fig.. Exact and Gubel-approxiate a = q, b = Rq distributions of the selection gain X for = 2,5,0,20 and n =,2,5. The outage capacity C out P 0 can then be approxiated as C out P 0 P out P 0 = log 2 [+ρf P 0 ] log 2 [+ρ{a b ln lnp 0 }]. 25 Fig. 2 shows 0% outage capacity of transit antenna selection and MIMO without feedback for SINR ρ = 5 db. The noralizing constants for the approxiations are chosen to be the sae as those in Fig.. The figure indicates that the above approxiations iprove as increases. Additionally, even ignoring the scheduling gain, the transit antenna selection schee outperfors full open loop MIMO schees in ters of outage capacity, when the nuber of receive antennas is sall.

8 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 8 6 ρ=5db 5 0% Outage Capacity bits/channel use n=, Tx Ant. Sel., Exact n=, Tx Ant. Sel., Approx. n=, MISO, Siulated n=2, Tx Ant. Sel., Exact n=2, Tx Ant. Sel., Approx. n=2, MIMO, Siulated n=3, Tx Ant. Sel., Exact n=3, Tx Ant. Sel., Approx. n=3, MIMO, Siulated Nuber of Transit Antennas Fig. 2. 0% outage capacity of transit antenna selection and MIMO versusfor n =,2,3 at ρ = 5dB. IV. ERGODIC CAPACITY A. Soe Useful Bounds Theore 6: For a rando variable X with cdf F as above, q E[X ] q eγ +, 26 where γ is Euler s constant e γ = , q is the quantile defined in, and q eγ + F /e γ +. The ergodic capacity Cρ E [ ] log 2 +ρ X also satisfies log 2 +ρ q Cρ log 2 +ρ qeγ +, 27 for any ρ > 0. The proof ay be found in the Appendix. Fro Jensen s inequality and the left inequality in 26, we obtain log 2 +ρ q Cρ log 2 +ρ E[X ] log 2 +ρ qeγ + 28

9 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 9 Fro the fact that q eγ + q γ, we can see that log 2 +ρ qeγ + log2 +ρ q. 29 It follows that Cρ log 2 +ρ E[X ] 30 log 2 +ρ q +γ. 3 B. Asyptotic Analysis for Large Nuber of Receive Antennas A chi-square rando variable X with cdf F as in above is the su of 2n i.i.d. rando variables with ean and variance = 0.5. In order to study the change of F and q as functions of n, it will be convenient to write F n and q n respectively. As n increases, fro the central liit theore X n/ n converges to the Gaussian distribution with ean zero and variance one. Thus, for large n, q n = Fn = n+o n. 32 By Theore 6 and 32, E[X ] = n+o n, 33 Cρ = log 2 +ρ n+o n. 34 C. Nuerical Results Fig. 3 deonstrates the ergodic capacity of various systes for a few basic antenna configurations. It is observed that the transit antenna selection schee perforance is superior to an open loop MIMO schee in the low SINR regie. This ay have soe iplications on wireless syste design as ost of the users in odern wireless systes have low SINRs. In fact, the ergodic capacity of open loop MIMO is upper bounded by nlog 2 +ρ, while the ergodic capacity of transit antenna selection goes to infinity as, and is not upper bounded. In Figures 4 and 5, we study the ergodic capacity respectively as a function and n, for SINR ρ = 5 db. It is observed that our bounds in 27 and approxiation in 3 for the ergodic capacity of transit antenna selection schees are very good and becoe exact as increases. In ters of the ergodic capacity, it is also seen that transit antenna selection outperfors open loop MIMO when the nuber of receive antennas is sall. If ore than two receive antennas are deployed, it appears that open loop MIMO is better than transit antenna selection for a oderate nuber of transit antennas

10 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 0 Ergodic Capacity bits/channel use =, n=, SISO =, n=, AWGN =2, n=, MISO =2, n=, Tx Ant. Sel. =, n=2, SIMO =2, n=2, Tx Ant. Sel. =2, n=2, MIMO =, n=2, MIMO U.B SINR ρ db Fig. 3. Ergodic capacity versus ρ for =,2 and n =,2. =,,20 at SINR ρ = 5 db. Nonetheless, in lower SINR regies, transit antenna selection will outperfor open loop MIMO even for ore than two receive antennas. V. SCHEDULING For a single cell with ultiple users, any scheduling strategies have been proposed. Aong the, it is known that a greedy scheduling algorith axiizes the total syste capacity. In greedy scheduling, the base station selects the user with the best channel at any given tie. Only this user ay counicate with the base station. It is known that ultiple transit antennas in MIMO reduce channel fluctuations and thus the benefits of scheduling decrease as the nuber of transit antennas increases [8]. However, that is not necessarily the case for transit antenna selection because there are significant channel fluctuations even after deploying a large nuber of transit antennas. We copare the syste capacity of greedy scheduling for antenna selection to that of round robin scheduling for antenna selection as well as greedy and round-robin scheduling for MIMO. Our basic assuption for the analysis is that all users have the sae nuber of antennas.

11 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY ρ=5db AWGN n=, MISO n=, Tx Ant. Sel., Exact n=, Tx Ant. Sel., Approx. n=, Tx Ant. Sel., U.B. n=, Tx Ant. Sel., L.B. n=2, MIMO n=2, Tx Ant. Sel., Exact n=3, MIMO n=3, Tx Ant. Sel., Exact n=3, Tx Ant. Sel., Approx. n=3, Tx Ant. Sel., U.B. n=3, Tx Ant. Sel., L.B. Ergodic Capacity bits/channel use Nuber of Transit Antennas Fig. 4. Ergodic capacity versus for n =,2,3 at ρ = 5 db. We define the capacity of a scheduling algorith where there are K users and each user is equipped with n antennas in case of downlink or antennas in case of uplink as the average syste capacity after scheduling. The greedy scheduling capacity is then the sae as the ergodic capacity of transit-antenna-selection with K transit antennas, and it is given by E [ ] log 2 +ρ XK log2 +ρ q K +γ, 35 using 3. Fro 27, this is upper and lower bounded by log 2 +ρ q K and log 2 +ρ q eγ K+, respectively. Note that round robin scheduling has the sae capacity as the ergodic capacity of a pointto-point link with the sae nuber of transit antennas and is given by E [ ] log 2 +ρ X log2 +ρ q +γ, 36 and this is upper and lower bounded in 27. Fig. 6 shows the average syste capacity of transit-antenna-selection and MIMO versus. The approxiations and bounds in this figure are those given in 35, 36, and the discussions following

12 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 2 ρ=5db 8 7 =2, Tx Ant. Sel., Exact =2, Tx Ant. Sel., Approx. =2, Tx Ant. Sel., U.B. =2, Tx Ant. Sel., L.B. =20, Tx Ant. Sel., Exact =20, Tx Ant. Sel., Approx. =20, Tx Ant. Sel., U.B. =20, Tx Ant. Sel., L.B. 6 Ergodic Capacity bits/channel use Nuber of Receive Antennas n Fig. 5. Ergodic capacity versus n for = 2,20 at ρ = 5 db. the. We can check that the capacity of greedy scheduling for transit-antenna-selection increases as the nuber of transit antennas increases while it decreases for MIMO. Thus, greedy scheduling works well with transit antenna selection in the presence of a large nuber of transit antennas. We define the scheduling gain as the increase of average syste capacity over that without scheduling. It can be approxiated as E [ ] [ ] log 2 +ρ XK E log2 +ρ X +ρ qk +γ q K q log 2 = log +ρ q +γ 2 +ρ +ρ q +γ log 2 + q K q log 2 + lnk = O = O q q log q The nuerically integrated values of 37 for 20 and the approxiated values of 38 for 2 20 are tabulated in Table I for 5dB ρ 0dB with 5dB increaent, n =, and K = 32. Note that the approxiations are at ost 0. bits away fro the exact values. Thus, 38 is a good

13 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 3 6 ρ=5db, K=32, n= 5 Average Syste Capacity bits/channel use Tx Ant. Sel., Round Robin, Exact Tx Ant. Sel., Round Robin, Approx. Tx Ant. Sel., Round Robin, U.B. Tx Ant. Sel., Round Robin, L.B. Tx Ant. Sel., Scheduling, Exact Tx Ant. Sel., Scheduling, Approx. Tx Ant. Sel., Scheduling, U.B. Tx Ant. Sel., Scheduling, L.B. MISO, Round Robin, Exact MISO, Round Robin, Siulated MISO, Scheduling, Siulated Nuber of Transit Antennas Fig. 6. Average syste capacity after scheduling versus for n = and K = 32 at ρ = 5dB. approxiation to 37. We define the fractional scheduling gain of greedy scheduling to be the greedy scheduling gain of as a fraction of the capacity of round robin scheduling. Since E [ log 2 +ρ X ] log2 +ρ q +γ, it is easy to see that E [ log 2 +ρ XK ] E [ log2 +ρ X ] E [ log 2 +ρ X ] = O log loglog. 4 VI. CONCLUSION In this work, we considered the use of transit antenna selection in ultiple antenna wireless systes. It was shown that for antenna selection systes unlike MIMO systes, the channel hardens at uch slower rate, and thus significant ultiuser scheduling gain can exist. Additionally, it was shown that even without this scheduling gain, transit antenna selection systes outperfor open loop MIMO systes at low SINR regies, particularly for a sall nuber of receive antennas. The iplications of these results on wireless syste design was briefly discussed.

14 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 4 K = 32, n = ρ -5dB 0dB 5dB 0dB TABLE I. Exact and approxiated parenthesis scheduling gain in bits for various and ρ for n = and K = 32. and thus APPENDIX Proof of Lea : In the first part of the lea, for t > 0, t Rt Fydy = t Fydy Ft n k=0 t n = e t k=0 i=0 e yyk k! dy k = e t n k k=0 i=0 ti i! e t n k=0 tk k! t i i! < 42 <. 43

15 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 5 Fro 43, if n =, Rt =, and if n 2, This proves the first part of the lea. t n Fro 44, li t xrt/t = 0 for all real x and thus tn 2 n! +2n 2! Rt = +Otn 3 t n n! + tn 2 n 2! +Otn 3 = +n t +O t Ft+xRt li t Ft e t+xrt t+xrt n = li t e t t n = li t e xrt + xrt t n = e x. 45 For the second part, the result follows fro 45 and the following theore whose proof can be found in [0, Ch. 2]. Theore 7: Define the upper endpoint ωf sup{x Fx < }. Then F is in the doain of attraction of G = exp exp x if and only if there exists soe finite a < ωf such that and for all real x, where ωf a Fydy <, 46 Ft+xRt li = e x, 47 t ωf Ft Rt ωf t Moreover, the noralizing constants a and b in 0 can be chosen as { a = q inf x Fx } Fydy. 48 Ft 49 and b = Rq. 50 The following lea in [0, Ch. 2] will also prove useful in the proof of Lea 2. Lea 8: LetX be any sequence of rando variables such that, for soe constantsa, a, b > 0, b > 0, li Pr{X a +b x} = li Pr{X a +b x} = Gx, 5

16 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 6 for all continuity points of the distribution function Gx. Then 5 holds if and only if both of the following hold a a li = 0 b 52 b li =. b 53 are Proof of Lea 2: We prove 4 first. Fro Lea 8 necessary and sufficient conditions for b li b Rq = 54 Fro 9 in Lea we know that, if n =, Rt = +n t +O 55 t, if n 2, 2 and li q =. 56 Thus li b should be. Since the converse is also true, we have proved 4. Because of 4, 52 in Lea 8 reduces to li a a = 0 57 and the conditions for a and b can be separated. Since the reference a can be chosen to be q in this case, it suffices to prove that Fro the definition of q, For n =, li [q {ln+n lnln lnn!}] = n e q k=0 q k k! =. 59 q = ln, 60 and obviously 60 satisfies 58. Now for n 2 and large enough that q, n k=0 q k k! = < q n n 2 n! + k=0 q n n! +qn 2 q k k! n 2 k=0 k! q n n! +qn 2 e. 6

17 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 7 Thus, e q qn n! < e q Taking the logariths on both sides of 62 gives { } q n n! +e qn and it yields Define ε as q +n lnq lnn! ln [ q n < q +ln + en! n! q ] = q +n lnq lnn!+ln + en! q ln q n lnq +lnn! < ln+ln + en!. 64 q 63 q = ln+n lnln lnn!+ε 65 and then we ust prove li ε = 0. Fro 64, We can see li q ln = and li li lnq ln = ε = ln Again, by substituting 65 for the leftost q in 64 and defining δ lnq lnln, n δ ε < n δ +ln + en!. q Because we only need to show li δ = 0. For such that q ln, δ = lnq lnln by ean value theore. Fro 67 and 69, li ln + en! = 0, 68 q = q ln, for soe ζ q,ln or ln,q ζ q ln = ln+η q ln, for soe η 0, = n lnln lnn!+ε ln+η {n lnln lnn!+ε }, 69 li δ = 0. 70

18 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 8 Proof of Theore 4: First, note that we borrow soe of the proof techniques fro Hall s paper [2] and Galabos s book [0, Sec. 2.0]. We see that α satisfies As in the proof of Lea 2, it can be shown that α n lnα +lnn! = ln. 7 li [α {ln+n lnln lnn!}] = Then, we can express the general noralizing constants a and b as where δ and ε are sequences satisfy a = α +δ and b = +ε, 73 li δ = 0 and li ε = For n =, 2 and the choice of 9 and 20 are proved in [0, p. 42] because the chi-square distribution just becoe the exponential distribution. Hence, let us assue n 2, and define z x [ Fa +b x]. 75 We will shortly prove that z x e x as but assuing that this is the case, fro 75, [ F a +b x = z ] x, 76 and by the triangle inequality, F a +b x e zx e zx Gx F a +b x Gx Also fro [0, p. 8], for any z 0,/2, F a +b x e zx + e zx Gx. 77 e z z [e 2z2 ] < z e z. 78 For fixedx, becausez x e x, z x/ 0,/2 for large enough. By 78 with z = z x/, [ F a +b x e zx z ] [ ] x 2z 2 exp x ] 2z e [exp zx 2 x [ ] 2z = e zx 2 x +O 2. 79

19 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 9 The rate of convergence of F a +b x e zx is doinated by the / ter, which will turn out to be uch faster than that of e zx Gx. Now, consider the rate of convergence of e zx Gx. Fro the definition of Fx in 6, we can easily see that as x, [ x xn Fx = e + n +O n! x Fro 7 and noting that δ,ε,/α 0 as, and e a+bxa +b x n n! αn α = e n! e x e δ+εx x 2 + x + δ + ε n x α α α = e x[ δ +ε x+oδ 2 +ε2 ] [ +n x +O α α 2 + δ + ε ] α α = [ e x δ +ε x+ n ] x+o +δ 2 +ε 2 α + n a +b x +O a +b x 2 Because Fa +b x is equal to the product of 8 and 82, α 2 ] = + n +O α α Fa +b x = [ e x δ +ε x+ n ] +x+o α α 2 +δ 2 +ε Fro this, it is clear that as, Using the fact that Gx = exp e x, e zx Gx z x = [ Fa +b x] e x. 84 = Gx expe x z x = Gx e x z x+o e x z x 2 = Gxe x δ +ε x n +x+o α Obviously, to cancel out n x+/α, δ = n α +O α 2 α 2 +δ 2 +ε

20 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 20 and ε = n +O α α Therefore, 9 and 20 ust be satisfied to optiize the rate of convergence. Moreover, if δ and ε are chosen as 86 and 87, by the second order expansion, it can be shown that the ters of O/α 2 cannot be canceled out. Thus Because e zx Gx = Θ α α = ln+n lnln lnn!+o, 89 /α 2 0 is uch slower than / 0 as. Hence, cobining 88 and 79 into 77 yields F a +b x Gx = Θ. 90 Fro 89, F a +b x Gx = Θ α 2 log 2. 9 Proof of Corollary 5: For n =, q = α = ln and b =. Therefore, 9 and 20 hold. Let us now take n 2. Fro the definition of α and q, = αn e α n! = qn n + e q n! where n P k = n!/n k!. By taking logariths, α +n lnα lnn! = q +n lnq lnn!+ln k= + n P k q k n k= n P k q k, Define ε as q = α + ε. We can see that α and ε 0 as because of 72. Substituting α +ε for q yields Define ε = n ln + ε +ln + α β n k= n k= n P k α +ε k. 94 n P k α +ε k. 95 It is obvious that ε /α 0 and β 0 as. Thus for large enough, [ ε ] ε 2 ε = n +O +β +Oβ α α

21 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 2 However, since 96 becoes Then ε = β = = n k= n P k α +ε k = n +O α α 2, 97 ε = n ε + n +O α α α n α +O α 2 n α [ n α +O α 2 = n α +O ][ + n α 2 +O α α 2 ]. 99 Clearly, q satisfies 9. We can obtain 20 by substituting 9 for q in 23. Proof of Theore 6: We first introduce a siple convex ordering result by van Zwet [, Ch. 2]. Assue X and Y be arbitrary rando variables, whose cdfs are F X and F Y respectively. Van Zwet showed that if F Y F Xx is convex, then F X E[X] F Y E[Y] and F X E[X ] F Y E[Y ], provided the expectations exist. If F Y F Xx is concave, then the inequalities are reversed. Now, we return to the chi-square distribution, where the rando variable X follows the cdf Fx in 6 and consider a rando variable Y with the cdf F Y y = /y < y <. We can easily see that F Y E[Y ] =. 00 Because FY Fx = /Fx, we only need to show that /Fx is convex. If so, /Fx is then concave and E[X ] F = q. 0 Then, the lower bound of 26 will be proved. Because F0 = 0, we can assue X > 0. It will be sufficient to show that d 2 [ ] dx 2 = f xfx 2{fx} 2 Fx {Fx} 3 > 0 02 for x > 0. However, this can be verified explicitly when n =. For n 2, it follows fro f xfx {fx} 2 = n n! k+ n n k +n! xk k= < n n. 03

22 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 22 We now show the upper bound of 26. If a rando variable Y has a cdf F Y y = e y 0 < y <, then For x > 0, define E[Y ] = k= k. 04 hx F Y Fx = ln[ Fx]. 05 Then, for n =, h x = 0, and for n 2, h x = d [ ] fx dx Fx x n 2 n 2! + n k= n kxk k! = n k=0 xk k! 2 > Therefore, by convex ordering, FE[X ] F Y E[Y ] = F Y k= = exp k k= [ { = + exp k= k }] k ln+. 07 For, we can easily show that k= k ln+ is increasing as a function of and by the definition of γ as Euler s constant, [ li k= k ln+ Thus k= k ln+ γ and it yields ] = li E[X ] F [{ + k= } ] k ln+ = γ e γ = q + eγ Hence, 26 is proved. The upper bound of 27 can be deduced fro the upper bound of 26 by Jensen s inequality because ρ > 0 and log 2 + ρ is then a concave function. Now, only the proof for the lower bound of 27 reains. Define Z log 2 +ρx. Then Z = log 2 +ρx. The cdf of Z is 2 z F Z z = F. 0 ρ

23 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 23 We will show that /F Z z is convex. Let /Fx be Qx and then d 2 [ ] [ 2 dz 2 = Q z 2 z 2 z ] 2 z F Z z ρ ρ +Q ρ ρ ln22. If we substitute 2 z /ρ for x, then we only need to show Q x x+ +Q x 0, 2 ρ for x > 0, and 2 becoes [ [ f {Fx} 3 xfx+2{fx} 2] x+ ] fxfx ρ [[ f {Fx} 3 xfx+2{fx} 2] x fxfx ], 3 because f xfx+2{fx} 2 0 by 03. We clai [ f xfx+2{fx} 2] x fxfx 0. 4 This is because [ n 2x xn e x n e x n! k=0 ] x k + xn 0. 5 k! n! Thus /F Z z is concave. By the convex ordering with the cdf F Y y = e y 0 < y <, 2 E[Z] F Z E[Z ] = F 6 ρ and therefore = 2E[Z] ρ F = q, 7 Cρ = E[Z ] log 2 +ρ q. 8 REFERENCES [] I. E. Telatar, Capacity of ulti-antenna Gaussian channels, European Transactions on Telecounications, vol. 0, no. 6, pp , Nov [2] M. Z. Win and J. H. Winters, Analysis of Hybrid Selection/Maxial-Ratio Cobing in Rayleigh Fading, IEEE Transactions on Counications, vol. 47, no. 2, pp , Dec [3] R. S. Blu and J. H. Winters, On Optial MIMO With Antenna Selection, IEEE Counications Letters, vol. 6, no. 8, pp , Aug [4] S. Sanayei and A. Nosratinia, Asyptotic Capacity Analysis of Transit Antenna Selection, in Proc. of International Syposiu on Inforation Theory, p. 242, June July 2002.

24 SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 24 [5] N. Shara and L. H. Qzarow, A Study of Opportunis for Multiple-Antenna Systes, IEEE Transactions on Inforation Theory, vol. 5, no. 5, pp , May [6] A. F. Molisch, M. Z. Win, Y. S. Choi, and J. H. Winters, Capacity of MIMO Systes With Antenna Selection, IEEE Transactions on Wireless Counications, vol. 4, no. 4, pp , July [7] Z. Chen, J. Yuan, and B. Vucetic, Analysis of Transit Antenna Selection/Maxial-Ratio Cobining in Rayleigh Fading Channels, IEEE Transactions on Vehicular Technology, vol. 54, no. 4, pp , July [8] B. M. Hochwald, T. L. Marzetta, and V. Tarokh, Multi-antenna channel hardening and its iplication for rate feedback and scheduling, IEEE Transactions on Inforation Theory, vol. 50, no. 9, pp , Sept [9] H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed., New Jeresy: John Wiley & Sons, [0] J. Galabos, The Asyptotic Theory of Extree Order Statistics, 2nd ed., Florida: Robert E. Krieger, 987. [] W. R. van Zwet, Convex Transforations of Rando Variables, Asterda: Matheatical Centre Tracts, 964. [2] P. Hall, On the rate of convergence of noral extrees, Journal of Applied Probability, vol. 6, no. 2, pp , June 979.