MULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO)

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY Caacity of MIMO Systems With Antenna Selection Andreas F. Molisch, Fellow, IEEE, Moe Z. Win, Fellow, IEEE, Yang-Seok Choi, Member, IEEE, and Jack H. Winters, Fellow, IEEE Abstract We consider the caacity of multile-inut multileoutut systems with reduced comlexity. One link-end uses all available antennas, while the other chooses the L out of N antennas that maximize caacity. We derive an uer bound on the caacity that can be exressed as the sum of the logarithms of ordered chi-square-distributed variables. This bound is then evaluated analytically and comared to the results obtained by Monte Carlo simulations. Our results show that the achieved caacity is close to the caacity of a full-comlexity system rovided that L is at least as large as the number of antennas at the other linkend. For examle, for L =3, N =8antennas at the receiver and three antennas at the transmitter, the caacity of the reducedcomlexity scheme is 2 bits/s/hz comared to 23 bits/s/hz of a full-comlexity scheme. We also resent a subotimum antenna subset selection algorithm that has a comlexity of N 2 comared to the otimum algorithm with a comlexity of N L. Index Terms Antenna arrays, information rates, MIMO systems. I. INTRODUCTION MULTIPLE-INPUT MULTIPLE-OUTPUT MIMO wireless systems are those that have antenna arrays at both transmitter and receiver. Early simulation studies that revealed the otentially large caacities of those systems were done in the 198s 1, and subsequent aers exlored the caacity analytically 2, 3. Since that time, interest in MIMO systems has exloded. Layered sace time ST receiver structures 4 6 and ST codes 7 make it ossible to aroach the caacity limits revealed in 2. Commercial roducts based on such codes are under develoment 8. Most imortantly, the standard for third-generation cellular hones 3rd Generation Manuscrit received June 25, 23; revised February 2, 24; acceted Aril 23, 24. The editor coordinating the review of this aer and aroving it for ublication is G. Leus. This work was suorted in art by an INGVAR grant of the Swedish Strategic Research Fund, a cooeration grant from the Swedish STINT, the Office of Naval Research Young Investigator Award N , the National Science Foundation under Grant ANI , and the Charles Stark Draer Endowment. Parts of this work were resented at ICC 21 and VTC fall 23. A. F. Molisch was with AT&T Laboratories-Research, Middletown, NJ 7748 USA. He is now with Mitsubishi Electric Research Laboratory MERL, Cambridge, MA 2139 USA and also at the Deartment of Electroscience, Lund University, Lund, Sweden Andreas.Molisch@ieee.org. M. Z. Win was with AT&T Laboratories-Research, Middletown, NJ 7748 USA. He is now with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 2139 USA win@ieee.org; moewin@mit.edu. Y.-S. Choi was with AT&T Laboratories-Research, Middletown, NJ 7748 USA. He is now with Intel, Inc., Hillsboro, OR USA yschoi@ ieee.org. J. H. Winters was with AT&T Labs-Research, Middletown, NJ 7748 USA. He is now with Motia Inc., Middletown, NJ 7748 USA jack. winters@ieee.org. Digital Object Identifier 1.119/TWC Partnershi Project 3GPP foresees the use of a simle ST code 9 with two transmit antennas and one or more receive antennas for circuit-switched communications and satial multilexing multile transmit data streams for high-seed downlink acket data access 1. In an earlier work, it was shown that the incremental gain of additional receive antennas is negligible if the total number of receive antennas N r is far larger than the number of transmit antennas 4. 1 This can be exlained by the fact that additional antennas do not rovide indeendent communication channels but just increase the diversity order. This motivates researchers to exlore the ossibility of relacing the maximal ratio diversity that is normally achieved in a such a MIMO system with selection diversity SD. Thus, in this aer, we roose a reduced-comlexity MIMO scheme that selects the L r best of the available N r antennas. Such a scheme can rovide the full number of indeendent communication channels, and additionally an SD gain. Comared to the use of all antennas, the antenna selection has the advantage that only L r instead of N r receiver RF chains are required. We still require the full number of antenna elements, but these are usually inexensive, as they are atch or diole antennas that can be easily roduced and laced. Antenna selection, or more recisely, the rincile of using L out of N antennas, was first studied in the context of antenna selection at one link-end, while only a single antenna is resent at the other link-end This is referred to as hybrid selection/maximum ratio combining MRC in the literature. Therefore, we will emloy the term hybrid selection/mimo H-S/MIMO for the more general case studied in this aer, namely antenna selection at one link-end, and multile antennas, all of which are used, at the other link-end. There has been considerable interest in H-S/MIMO in recent years. The case of antenna selection at the transmitter is treated in 15 using Monte Carlo simulations; this aer also develos a criterion for otimal antenna set selection for high signal-tonoise ratios SNRs; 16 extended this to the low-snr case. It has been shown that antenna selection is beneficial in a lowrank environment 17 and in interference-limited systems 18. A selection algorithm for minimizing the bit error robability of linear MIMO receivers is given in 19. The use of ST codes in combination with antenna selection was investigated in 2 and 21; the use of antenna selection in transmit receive diversity systems with channel knowledge at both link-ends was 1 Under certain circumstances, increasing that number can even lead to erformance degradation, as the channel estimation becomes more difficult and introduces estimation errors /$2. 25 IEEE

2 176 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 Fig. 1. Block diagram of the considered system. treated in A more detailed overview of the literature is given in 28. In this aer, we derive analytical bounds for the caacity distribution function of an H-S/MIMO system at one link-end. We show that an exact antenna selection algorithm requires high comutational comlexity and roose several alternative methods that have much lower comlexity while erforming almost as well as the exact selection criteria. The rest of the aer is organized as follows: In Section II, we set u the system model. Analytical bounds for the caacity are derived in Section III. Next, we resent a fast antenna selection algorithm in Section IV. Section V gives evaluations for the analytical bounds of H-S/MIMO and comares them to numerical simulation results. Conclusions and system design considerations are given in Section VI. II. SYSTEM MODEL We consider the case where the transmitter uses all available antennas while the receiver uses antenna selection. Fig. 1 exhibits a block diagram. At the transmitter, the data stream enters an ST encoder, whose oututs are forwarded to the transmit antennas. The signals are subsequently uconverted to assband, amlified by a ower amlifier, and filtered. For our model, we omit these stages, as well as their equivalents at the receiver, which allows us to treat the whole roblem in equivalent baseband. Note, however, that it is exactly these arts that are most exensive and make the use of reducedcomlexity systems desirable. From the antennas, the signal is sent through the mobile radio channel, which is assumed to be flat fading and quasi-static. By quasi-static, we mean that the coherence time of the channel is so long that a large number of bits can be transmitted within this time. More secifically, we assume that the data are encoded with near Shannon limit achieving codes. 3 It has been shown that LDPC codes with a block length of 1 aroach the Shannon limit within less than 1 db 3. For a data rate of 1 Mbits/s, such a block can be transmitted within 1 ms, which is shorter than the tyical 1 ms coherence time of wireless channels. Thus, each channel realization can be associated with 2 Parallel to our work see also 23 and 24, an alternative algorithm for the selection of antenna subsets was resented and a lower bound of the caacity was derived in 25 27; this algorithm will also be discussed in Section IV. 3 Such a code could be, e.g., the combination of ST rocessing 6 with a low-density arity check code 29. a Shannon-AWGN caacity value. The caacity thus becomes a random variable RV, rendering the concet of caacity cumulative distribution function and outage caacity meaningful erformance measures 2. We denote the N r matrix of the channel as h 11 h 12 h 1Nt h 21 h 22 h 2Nt H = h Nr 1 h Nr 2 h Nr If the channel is Rayleigh fading, the h ij are indeendent identically distributed i.i.d. zero-mean circularly symmetric comlex Gaussian RVs with unit variance, i.e., the real and imaginary arts have a variance of 1/2 each. Consequently, the ower carried by each transmission channel h ij is chi-square distributed with 2 degrees of freedom. The channel also adds white Gaussian noise, which is assumed to be indeendent among the N r receiver antenna elements. Following 2, we consider the case in which the h ij are indeendently fading, as this simlifies the theoretical analysis. More involved channel models are discussed, e.g., in The received signal, which is written as y = Hs + n = x + n 2 is received by N r antenna elements, where s is the transmit signal vector and n is the noise vector. A control algorithm to be discussed in Sections III and IV selects the best L r of the available N r antenna elements and downconverts their signals for further rocessing note that only L r receiver chains are required. ST encoder and decoder are assumed to be ideal so that the caacity can be achieved. We assume ideal knowledge of the channel at the receiver so that it is always ossible to select the best antennas. However, we do not assume any knowledge of the channel at the transmitter. This imlies that no waterfilling can be used and that the available transmitter ower is equally distributed among the transmit antennas. III. THEORY Let us first exlore the scenarios that are suited for H-S/MIMO. As shown in 2, the caacity is linearly roortional to minn r,. Any further increase of either N r or while keeing the other fixed only increases the diversity

3 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1761 order and ossibly the mean SNR, ossibly. Thus, if the number of antennas at one link-end is limited, e.g., due to sace restrictions, a further increase in the antenna number at the other link-end does not allow us to add statistically indeendent transmission channels which would imly linear increase in system caacity, but only rovides additional diversity. Since it is well known that SD has the same diversity order as that of MRC 34, we can anticiate that a hybrid scheme with N r > L r = will give a good erformance. In the next subsections, we will give a quantitative confirmations of this conjecture. A. Exact Exression for the Caacity The caacity of MIMO system using all antenna elements is given by 2 C full = log 2 det I Nr + ΓNt HH where I Nr is the N r N r identity matrix, Γ is the mean SNR er receiver branch, and suerscrit denotes the Hermitian transose. The receiver now selects those antennas that allow a maximization of the caacity, so that C select = max {log 2 det I Lr + Γ } H H S H where H is created by deleting N r L r rows from H, and S H denotes the set of all ossible H, whose cardinality is Nr L r. The otimum choice of antennas requires the knowledge of the comlete channel matrix. This may seem to necessitate the use of N r RF chains, which is in contrast with a lowcomlexity system. However, in a sufficiently slowly changing environment, the L r RF chains can be cycled through the N r antennas during the training bits. In other words, RF chains are connected to the first L r antennas during the first art of the training sequence, then to the second L r antenna during the next art, and so on. At the end of the training sequence, we ick the best L r antennas. Thus, we only need a few more training bits instead of additional RF chains and the decrease in the sectral efficiency due to those additional training bits is negligible, esecially in high-data-rate systems. B. Caacity Bound for L r An exact analytical solution for C select seems difficult. Thus, we derive analytical bounds in this subsection and verify them with Monte Carlo simulations in Section V. Our starting oint 3 4 is the uer caacity bound for the full-comlexity system with N r 2 C full log 2 1+ Γ γ i 5 i=1 where the γ i are indeendent chi-square-distributed RVs with 2N r degrees of freedom. The equality alies in the unrealistic case when each of the transmitted comonents is received by a searate set of N r antennas in a manner where each signal comonent is received with no interference from the others 2. In our case, we select the best L r out of N r receive antennas, where L r. The uer bound can be obtained similar to 5, excet for exchanging the role of transmitter and receiver, and selecting those antennas whose instantaneous realizations of γ i are the largest. Since this equation is a crucial starting oint, let us elaborate on its hysical interretation. We consider a system where each of the N r receive antennas has its own set of size of transmit antennas. Naturally, this case is not feasible in ractice but must result in an uer bound of the caacity. Each set of transmit antennas corresonding to each of the N r receive antennas can carry one data stream. The maximum SNR which also achieves maximum caacity for this data stream can be obtained with maximal ratio transmission, which in turn results in chi-square-distributed SNR with 2 degrees of freedom at the receiver outut. Finally, we select those L r out of N r receive antennas that give the best SNR, and thus highest caacity. The caacity bound with antenna selection is thus L r C bound = log 2 1+ργi 6 i=1 where ρ = Γ/, and the γ i are ordered chi-squaredistributed variables with 2 degrees of freedom, out of a set of N r. 4 The joint statistics of the ordered SNRs γ i is shown in 7 at the bottom of the age 14, where Γ is Euler s Gamma function 35. Thus, the characteristic function of the caacity bound is γ 1 γ Nr 1 Φjν= N r! Γ N dγ r 1 dγ 2 dγ Nr L r N r ex jν log 2 1+ργi γ 1 i ex γ i. 8 i=1 i=1 4 We use {γ i } to denote the order set of {γ i }, i.e., γ 1 >γ 2 > γ Nr. Note that the ossibility of at least two equal γ i s is excluded as γ i γ j almost surely for continuous RVs γ i. γi γ1,γ 2,...,γ Nr = N r! N r i=1 1 Γ γ 1 i ex γ i, for γ1 >γ 2 > >γ Nr, otherwise 7

4 1762 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 First, we erform the integrations over the N r L r discarded antennas. As shown in the Aendix, these N r L r result in an exression of the form d N r L r + Nr L r =1 ex b N r L r γ Lr N r L r 1 k= c N r L r γ k L r. 9 The values of the coefficients b, d, and c are comuted via an iteration. We initialize with b = for all c = and then erform N r L r iterations b q for all, k d =1 1 = b q, for 1 q 11 b q q =1 12 see at the bottom of the age. For the next ste of the iteration, it is advantageous to rewrite 9 as N r L r = so that ex b N r L r γ Lr N r L r 1 k= c N r L r, = d N r L r c N r L r,k = c N r L r γ k+αnr Lr L r b N r L r = α N r L r =. 16 We then erform the next L r 1 integrations, which yield see Aendix an exression of the form N r L r = ex b N r 1 γ 1 M k= c N r 1 γ k+αnr where the arameters c N r,r, α Nr, and b N r are again comuted via a recursion. In each ste, we first comute bq = b q 18 α q = α q + ĉ q = jν ln2 c q 1, c q 1 + jν ρ ln2 cq, k = M 19 1 k<m. jν ρ ln2 cq, k = 2, otherwise 2 Then we can erform the second ste, which is obtaining coefficients for the next iteration ste with α q = α q 21 b q = b q 22 r 1 c q,r = ĉ q f q,r 1 k 23 f q,n = n i= k= bq n. 24 k + α q +i The final integration and incororation of constant multilicative factors yields Φ jν= ρ jνlr ln2 Nr! Γ N r N r L r = M r= ĉ N r 1,r Γ r + α Nr 1 N b. r+ α r 1 Nr 1 25 ĉ q = { c q 1, for q k 1, otherwise 13 d q = d q 1! + c q = q =1 q +2N t 1 k t= q +2 1 t= ĉ q,t ĉ q +t k+t! t b q b q t! t 14 k!, for 1 q d q 1! k!, for = q 15

5 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1763 The uer summation limit M is theoretically infinite, but the sum converges reasonably fast. In our comutations, M =5 roved to be sufficient for N r =8. Details about the derivation of the recursion relations for the coefficients can be found in the Aendix. The above equation yields the characteristic function of the caacity bound note that we have omitted the functional deendence of the arameters on ν for notational convenience. The robability density function df of the caacity bound is obtained by erforming an inverse Fourier transformation, which can be accomlished by a fast Fourier transform. C. Caacity Bound for L r > The bound derived above is quite tight for L r but tends to become rather loose for L r >. Esecially, this bound suggests an almost linear increase of the caacity with L r. 5 However, we have shown in Section III-A that we can only anticiate a logarithmic increase. We thus derive an alternative bound that reflects this fact. We consider the situation where each of the transmit antennas transmits an indeendent data stream. Furthermore, we assume the ractically imossible situation where none of the data streams interferes with each other. This is equivalent to having single inut multile outut SIMO systems each with searate N r receive antenna elements dedicated to the recetion of one such data stream. In each of the SIMO systems, we erform H-S/MRC, so that the normalized SNR of the jth SIMO system is given by L r i=1 γ i. 26 Assuming that none of the data streams interferes with any other, the total caacity is then L r C select log 2 1+ρ γ i = ξ j =Ψ 27 j=1 i=1 j=1 where the γ i are ordered chi-square-distributed variables with 2 degrees of freedom, taken from a set of N r available ones. Since the ξ j are i.i.d., the characteristic function of Ψ is finally j=1 C j ν = C 1 ν. 28 The comutation of C j ν, i.e., the characteristic function of ξ j, is similar to the method described in 22 and 36. It seems thus worthwhile to investigate subotimum algorithms with lower comutational comlexity. In this section, we resent a family of such algorithms that result in a small SNR enalty while drastically reducing comutation time. The determinant in 4 can be written as det I Lr + Γ H H = r 1+ ΓNt λ2k k=1 29 where r is the rank of the channel matrix and λ k is the singular value of H. Note that the rank and the singular values should be maximized for the maximum caacity. Suose there are two rows of H which are identical. Clearly, only one of these rows should be selected in H. Since these two rows carry the same information about the signal comonents, any one of these two rows may be deleted. In addition, if they have different owers i.e., square of the norm of the row, we select the row with the higher ower. When there are no identical rows, we choose two rows for the ossible deletion whose correlation is the highest and delete the one with the lower ower. In this manner, we can have the channel matrix H whose rows are minimally correlated and have maximum owers. The above argument leads to the following algorithm. 1 The channel vector h k is defined as the kth row of H, with k being an element of the set X = {1,...N r }. 2 For all k and l, k>l,inx, comute the correlation Ξk, l defined as Ξk, l = h k, h l, where a, b reresents an inner roduct between vector a and b. 3 Loo a Choose the k and l with k, l X,k >l that give the largest Ξk, l. If h k 2 h l 2, eliminate h l, otherwise, eliminate h k. b Delete l or k fromx. c Go to Loo until N r L r rows are eliminated. The method defined above shall be called the correlation based method CBM. It does not require the SNR value and it is based on the correlation of the rows of the channel matrix h k, h l, which can be aroximated by the correlation of the noisy estimates E{y k yl }. As an alternative method when the SNR is available, we suggest to use the mutual information between y k and y l. The zero-valued mutual information means that the kth receive antenna outut y k and the lth outut y l carry totally different information. This occurs when the corresonding channel vectors h k and h l are orthogonal. On the other hand, when the mutual information between y k and y l has a maximum value, y k and y l carry the same information so that we can delete one of them. The mutual information is defined as 37 IV. FAST ANTENNA SELECTION ALGORITHMS The otimum selection of the antennas requires N r L r comutations of determinants and is thus comutationally intensive. Iy k ; y l =Gy k +Gy l Gy k,y l 3 where G denotes the entroy. 6 5 Note that the increase is only almost linear because we are dealing with ordered stochastic variables. Thus, including more terms in the summation tends to give terms that have a lower SNR and thus a lower caacity. 6 We deviate from the usual entroy notation H to avoid confusion with the channel matrix H.

6 1764 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 In the MIMO system, the mutual information can be written as h k 2 Γ h l 2 Γ Iy k ; y l =log. h k 2 Γ h l 2 Γ h k, h l 2 Γ2 Nt 2 31 The above equation can be rewritten as Iy k ; y l = log h k 2 Γ log h k 2 Γ = log h k 2 Γ h l 2 Γ h k, h l 2 Γ2 Nt 2 h l 2 Γ log 1+ h k 2 Γ + h k 2 h l 2 Γ 2 h Nt 2 k, h l 2 Γ2 Nt 2. h l 2 Γ 32 Since h k 2 h l 2 h k, h l 2, the mutual information is uer bounded as follows: Similarly, we have Iy k ; y l log h k 2 Γ. 33 Iy k ; y l log h l 2 Γ. 34 Finally, the mutual information is uer bounded by { Iy k ; y l min log h k 2 Γ, log h l 2 Γ }. 35 We therefore define the normalized mutual information I y k ; y l = { Iy k ; y l } min log h k 2 Γ, log h l 2 Γ 36 as a measure of how close the two RVs are. We can also aly the mutual-information-based technique to x k, which is the signal comonent of y k, in order to avoid requiring the SNR value. Then the mutual information between the data comonents x k and x l is Ix k ; x l = log h k 2 h l 2 h k 2 h l 2 h k, h l Similarly, we define the normalized mutual information as I x k ; x l = Ix k ; x l min{ log h k 2, log h l 2 }. 38 The antenna selection algorithms based on mutual information then have a similar rogram structure as the one based on correlation CBM. All that is required is to relace Ξ by I as defined in 36 henceforth referred to as MIBM or 38 MIBM2. V. R ESULTS In this section, we evaluate the bounds derived in revious sections and comare them to Monte Carlo simulations. We first generate random realizations of mobile radio channels with transfer function h ij, which is an i.i.d. circularly comlex Gaussian RV with zero mean and a variance of 1/2 for the real and imaginary arts. From each realization of the matrix H, a comlete set S H of N r L r ossible matrices H are obtained by eliminating all ossible ermutations of N r L r rows from the matrix H. For each of the H, we comuted the caacity by 4, and selected the largest caacity from the set. Fig. 2 shows the cumulative distribution function of caacity for N r =8, =3, and various L r. The SNR is 2 db, and in the following, we consider the 1% outage caacity. With full exloitation of all available elements, 21.8 bits/s/hz can be transmitted over the channel. This number decreases gradually as the number of selected elements L r is decreased, reaching 18.2 bits/s/hz at L r =3.ForL r <, the caacity decreases drastically, since a sufficient number of antennas to rovide indeendent transmission channels is no longer available. These trends are well reflected in the bounds: the bound for the full-comlexity system is 22.7 bits/s/hz, decreasing to 2. bits/s/hz at L r =3. We also find that the bounds are tight for L r <, become looser for L r, and become tighter again for L r. This fact can be exlained as follows. As we have noted in Section III, the bound reflects the situation that each of the received signals has its own set of size of transmit antennas. This is fulfilled erfectly for L r =1and becomes a rogressively worse aroximation as L r increases. Note that this bound is used only u to L r. For larger L r, we bound the caacity by the case where we have indeendent data streams, none of which interferes with each other. Now it is well known 38 that N receive antennas can suress K interfering data streams while retaining a diversity order of N K for the remaining data streams. The bound is thus aroximately equivalent to a situation where we have L r + receive chains instead of the L r that are actually existing. The relative error thus becomes rogressively smaller as L r increases. Finally, Fig. 2 also shows the caacity of an L r full comlexity system. This shows us how much erformance we would lose when using for a fixed number of RF chains only the minimum number of antenna elements. Fig. 3 shows the influence of the SNR on the caacity. We lot the imrovement of the 1% outage caacity of an H-S/ MIMO system over a single-antenna system. We see that the caacity increase is very large at low SNRs factor of 25 at SNR = db, while for high SNRs, it tends to a fixed

7 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1765 Fig. 2. Exact caacity solid curves and bound dashed curves for N r =8, =3, SNR = 2 db. Dotted lines show caacity of =3, N r = L system. Note that for L =1, the solid and the dashed lines coincide, while for L =8, the solid and the dotted lines coincide. Fig. 3. Ratio of 1% outage caacity of a system with N r =8, L r =6, = 3, over that of a single-antenna system: bound dashed; exact dotted; and system with = L r =3solid. value of about 4. A factor of 3 in the caacity increase can be attributed to the number of indeendent communication channels between the transmitter and receiver. The remainder of the caacity increase is due to the diversity effect. Note also that Fig. 3 lots the imrovement in 1% outage caacity. If we were to consider the mean caacity, the influence of the SNR on the relative caacity increase would be significantly reduced. For standard N r = L r = systems, the relative mean caacity increase comared to a SISO system is ractically indeendent of the SNR. Another interesting oint is the comarison between antenna selection criteria based on caacity and antenna selection based on the subotimum algorithm that selects antennas with the highest owers. In our MC simulations, we also recorded for each channel realization the indices of those antennas that have the highest SNR. The indices of those antennas were then comared to those of the antennas that were chosen to maximize caacity. We found that only in about 5% of all channel realizations did the two selections agree with each other. The geometric interretation of this behavior is that for the deterministic case corresonding to one channel realization, the hase shifts between the antenna elements are the decisive factors for caacity, and are far more imortant than instantaneous SNR 39. Fig. 4 gives the caacities that are obtained by antenna selection based on an SNR criterion. We see that for L r N r, the 1% outage caacity decreases from 18.2 to 14.3 bits/s/hz at 2 db SNR when the SNR- instead of caacity- based criterion is used for antenna selection. This loss gets smaller as L r aroaches N r. The erformance of our fast antenna selection algorithms is detailed in Figs. 5 and 6. Again, the number of transmit and receive antennas is 3 and 8, resectively. For comarison, the ILM technique 25 is also evaluated. Each algorithm selects three receive antennas out of eight receive antennas. Among the roosed algorithms, the MIB methods outerform the CB technique. The ILM is shown to have a erformance that is very close to the exhaustive search. However, it requires Gram Schmidt orthogonalization and thus matrix inversion/multilications. The comlexity thus goes like N r L r N 3 t or N r N r L r N 3 t, whichever is the smaller 27. In contrast, the main comutational burden of our fast algorithms comes from the calculation of vector multilications h k, h l.

8 1766 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 slight erformance loss is offset by a considerable reduction in hardware costs. Instead of a full N r receiver chains, only L r receiver chains, lus an RF switch are required. We have also derived and comared several algorithms that allow the selection of the antennas without an exhaustive search over all ossible antenna combinations. These algorithms have a comlexity roortional to Nr 2, instead of the N r L r comlexity of otimum algorithms, while resulting in a caacity loss of less than 1 bit/s/hz at 1 db and 4 bits/s/hz at 3 db. Imortant alications for such systems are cellular and wireless local area network systems with MIMO caability. The necessity of selecting antennas at one link-end instead of using all of them stems from either comlexity or cost considerations. For examle, the number of transmit antennas foreseen for the sace time coder could be limited, as is already the case in the 3GPP standard. Furthermore, antenna selection can be esecially beneficial in low-rank and interference-limited systems. Thus, the results of this aer can serve as a guideline for designing reduced-comlexity MIMO cellular systems for third- and fourth-generation communications. Fig. 4. CDF of the caacity of a system with N r =8, =3. Selection of antenna by caacity criterion solid and by SNR criterion dotted. Each of those has a comlexity of, and we need N r N r + 1/2 of them. The comlexity thus goes as N r N r /2. The choice between ILM and MIBM2 or a similar algorithm is a tradeoff between erformance and comlexity. Assuming that ideal coding is emloyed, the outage robability when the bandwidth efficiency is 15 bits/s/hz 7 is shown in Fig. 7. The worst selection has 1 db loss at 1 3 outage robability. The MIBM has about 2 db loss while the correlation-based method exhibits around 6 db loss. The erformance of the fast algorithm MIBM2 is comarable to that of the MIBM at high outage robability. The MIBM2 has a good erformance overall while similarly to the CB methods it does not require the SNR value. Figs. 8 and 9 illustrate the outage caacities at 1% outage rate versus the number of receive antennas, N r, under fixed = L r =3at 1 db and 3 db SNR, resectively. VI. SUMMARY AND CONCLUSION We have investigated the behavior of MIMO systems that select a subset of available antennas at one link-end. In articular, we have derived uer bounds for the caacity of antenna selection, and we have also derived several algorithms that allow the selection of the antennas without an exhaustive search over all ossible antenna combinations. We comared the uer bounds to comuter simulation results and also comared the reduced-comlexity selection algorithms. For L r, selecting the best L r antennas gives almost the same caacity as the full-comlexity system. Caacity losses are less than 3.5 bits/s/hz for N r =8, =3, L r =3at 2 db SNR. This 7 That is, the robability that the caacity is smaller than 15 bits/s/hz. APPENDIX DERIVATION OF THE RECURSION RELATION The starting oint for the derivation is 8. We first solve the N r L r innermost integrals. 8 These integrals have the form y d q + q =1 ex b q x q 1 c q x k k= x Nt 1 ex xdx 39 where for readability we have substituted γ q x, γ q 1 y. The first art of the integral can be solved as 35 y d q x 1 ex xdx N t 1 = d N q t 1! ex y k= 1! y. k 4 k! Next, we ull out the summation over from the integral and consider the integrals y J q = ex b q x q N t 1 c q x k k= x 1 ex xdx Integrals of a similar form are also solved by the authors in 22. For convenience of the reader, we give here a short outline of the derivation.

9 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1767 Fig. 5. Outage robabilities of fast algorithms, N r =8, = L r =3, SNR = 1 db. Fig. 6. Outage robabilities of fast algorithms, N r =8, = L r =3, SNR = 3 db. By introducing this integral can be written as bq = b q for1 q 42 M =q ĉ q = { c q 1, for 1 k M, otherwise 44 y Emloying 4 M ĉ q xk e bq k= ex b q M x k= x e bq x dx = b q l= ĉ q xk dx. 45 M 1 l d l M l dx l b q k= ĉ q xk 46

10 1768 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 Fig. 7. Outage robability comarison, N r =8, = L r =3. Fig. 8. Outage robability comarison 1% as a function of the number of receive antennas = L r =3, SNR = 1 db. we get Introducing r = k l, we can write this as J q M e b q x = b q k= ĉ q k l= y 1 k! l k l! x k l. 47 bq J q M e b q x = b q r= M r x r ĉ q,r+t t= bq y 1 r + t! t. 48 r!

11 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1769 Fig. 9. Outage robability comarison 1% as a function of the number of receive antennas = L r =3, SNR = 3 db. The total integral thus is N t 1 d N q t 1! ex y + q 1 =1 M ĉ q bq,t t= M r= M r y r k= 1! y k k! t! e b t bq ĉ q,r+t t= bq q x bq 1 r + t! t. 49 r! Comaring this exression with the generic exression for the result of the q th integration q d q + =1 ex b q y q +2 1 k= c q 5 and matching coefficients, we get the recursion relations given in As mentioned in Section III-C, we erform this iteration N r L r times and write the result in the form N r L r = ex b N r L r γ Lr N r L r 1 k= y k c N r L r γ k+αnr Lr L r. 51 The integrals we have to solve for the next L r iteration stes are thus of the generic tye J q = y dx Nr L r = ex b q x M k= c q xk x 1 ex x1 + ρx jν ln2 52 where M =N r L r = M 1 for the first iteration ste and for the further stes note that since the series converges well, a finite number of terms is sufficient for the numerical comutations, and x = γ q, y = γ q 1. Since ρ usually has reasonably large values, and the behavior of the df is also mainly determined by the behavior of the characteristic function near ν =the nth moment is the nth derivative of the characteristic function at ν =,we aroximate 1 + ρx jν jν ln2 ρx ln2 1+ jν ln2. 53 ρx This aroximation was validated for the arameters used in Section V by comuting C bound by Monte Carlo simulations and comaring it to the analytical results based on the aroximation 53. Even for 1 Monte Carlo runs, the difference between analytical and numerical results was smaller than the uncertainty of the MC results. The integral J q can now be written as q = N r L r,...,n r y J q = ρ jν ln2 dx Nr L r = ex b q M x k= ĉ q xk+ αq 54

12 177 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 where bq = b q 55 α q = α q + jν ln2 56 α N r L r = 57 and c q 1, for k = M ĉ q = c q 1 + jν ρ ln2 cq, for 1 k<m. jν ρ ln2 cq, for k = 2, otherwise 58 Note that 2. Now from 4 y x k+ α ex axdx = a k+ α γ Euler k + α,ay 59 where γ Euler denotes here Euler s Gamma function of the second kind. Using its series exansion 35, the above integral becomes with a k+ α ex ay =ex ay f n = n= i= ay k+ α+n n k + α +i 6 y k+ α+n f n 61 n= a n n. 62 k + α +i i= Alying this result now to 54, we get J q = ρ jν ln2 Nr L r = ex b q M k= y ĉ q y k+ αq +n f n. 63 n= Comaring this to the generic form Nr L r = ex b q M x k= c q x αq +k 64 we find by comarison the coefficients in exressions By making use of the arameters b, ĉ, the final integral is of the form J N r = Nr L r = which yields 4 N r L r = r= ex b N r 1 ĉ N r 1,r k= x M ĉ N r 1 x k+ αnr 1 dx 65 Γ r + α Nr 1 bn. 66 r+ α r 1 Nr 1 ACKNOWLEDGMENT The helful suggestions of the reviewers are gratefully acknowledged. The authors also would like to thank H. Wang and J. L. Craig for critical reading of the manuscrit. REFERENCES 1 J. H. Winters, On the caacity of radio communications systems with diversity in Rayleigh fading environments, IEEE J. Sel. Areas Commun., vol. 5, no. 5, , Jun G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multile antennas, Wireless Pers. Commun., vol. 6, no. 3, , Feb I. E. Telatar, Caacity of multi-antenna Gaussian channels, Eur. Trans. Telecommun., vol. 1, no. 6, , Nov./Dec G. J. Foschini, Layered sace time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs Tech. J., vol. 1, no. 2, , Aug G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolniansky, Simlified rocessing for high sectral efficiency wireless communication emloying multi-element arrays, IEEE J. Sel. Areas Commun., vol. 17, no. 11, , Nov M. Sellathurai and S. Haykin, Further results on diagonal-layered sace time architecture, in Proc. Vehicular Technology Conf. VTC 21 Sring, Rhodes, Greece, V. Tarokh, N. Seshadri, and A. R. Calderbank, Sace time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inf. Theory, vol. 44, no. 2, , Mar M. Guillaud, A. Burg, E. Beck, M. Ru, and S. Das, Raid rototyingdesignofa4 4 BLAST-over-UMTS system, in Proc. 35th Asilomar Conf. Signals, Systems and Comuters, Pacific Grove, CA, 21, S. M. Alamouti, A simle transmit diversity technique for wireless communications, IEEE J. Sel. Areas Commun.,vol.16,no.8, , Oct GPP 3rd Generation Partnershi Project, UMTS Radio Interface, Mar N. Kong and L. B. Milstein, Combined average SNR of a generalized diversity selection combining scheme, in Proc. IEEE Int. Conf. Communications, Atlanta, GA, Jun. 1998, vol. 3, M. Z. Win and J. H. Winters, Analysis of hybrid selection/maximalratio combining of diversity branches with unequal SNR in Rayleigh fading, in Proc. 49th Annu. Int. Vehicular Technology Conf., Houston, TX, May 1999, vol. 1, , Analysis of hybrid selection/maximal-ratio combining in Rayleigh fading, IEEE Trans. Commun., vol. 47, no. 12, , Dec , Virtual branch analysis of symbol error robability for hybrid selection/maximal-ratio combining in Rayleigh fading, IEEE Trans. Commun., vol. 49, no. 11, , Nov. 21.

13 MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION R. Nabar, D. Gore, and A. Paulraj, Selection and use of otimal transmit antennas in wireless systems, in Proc. Int. Conf. Telecommunications ICT, Acaulco, Mexico, IEEE, S. Sandhu, R. U. Nabar, D. A. Gore, and A. Paulraj, Near-otimal selection of transmit antennas for a MIMO channel based on Shannon caacity, in Proc. 34th Asilomar Conf. Signals, Systems and Comuters, Pacific Grove, CA, 2, D. Gore, R. Nabar, and A. Paulraj, Selection of an otimal set of transmit antennas for a low rank matrix channel, in Proc. Int. Conf. Acoustics, Seech and Signal Processing ICASSP 2, Istanbul, Turkey, R. S. Blum and J. H. Winters, On otimum MIMO with antenna selection, in Proc. Int. Conf. Communications ICC 22, NewYork, R. W. Heath, A. Paulraj, and S. Sandhu, Antenna selection for satial multilexing systems with linear receivers, IEEE Commun. Lett., vol. 5, no. 4, , Ar D. Gore and A. Paulraj, Statistical MIMO antenna sub-set selection with sace time coding, IEEE Trans. Signal Process., vol. 5, no. 1, , Oct A. Ghrayeb and T. M. Duman, Performance analysis of MIMO systems with antenna selection over quasi-static fading channels, in Proc. IEEE Int. Sym. Information Theory, Lausanne, Switzerland, 22, A. F. Molisch, M. Z. Win, and J. H. Winters, Reduced-comlexity transmit/receive diversity systems, IEEE Trans. Signal Process., vol. 51, no. 11, , Nov , Caacity of MIMO systems with antenna selection, in IEEE Int. Conf. Communications, Helsinki, Finland, 21, Y.-S. Choi, A. F. Molisch, M. Z. Win, and J. H. Winters, Fast antenna selection algorithms for MIMO systems, resented at the Vehicular Technology Conf. Fall 23 Invited Paer, Orlando, FL. 25 A. Gorokhov, D. Gore, and A. Paulraj, Performance bounds for antenna selection in MIMO systems, in Proc. Int. Conf. Communications ICC 3, Anchorage, AK, , Receive antenna selection for MIMO flat-fading channels: Theory and algorithms, IEEE Trans. Inf. Theory, vol.49,no.1, , Oct , Receive antenna selection for MIMO satial multilexing: Theory and algorithms, IEEE Trans. Signal Process., vol. 51, no. 11, , Nov A. F. Molisch and M. Z. Win, MIMO systems with antenna selection, IEEE Microw. Mag., vol. 5, no. 1, , Mar R. G. Gallager, Low-density arity check codes, IRE Trans. Inf. Theory, vol. 8, no. 1, , Jan T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, Design of caacity-aroaching irregular low-density arity-check codes, IEEE Trans. Inf. Theory, vol. 47, no. 2, , Feb K. Yu and B. Ottersten, Models for MIMO roagation channels A review, J. Wireless Commun. Mob. Comut., vol. 2, no. 7, , Nov A. F. Molisch, A generic model for MIMO wireless roagation channels, IEEE Trans. Signal Process., vol. 52, no. 1, , Jan A. F. Molisch and F. Tufvesson, Multiath roagation models for broadband wireless systems, in CRC Handbook of Signal Processing for Wireless Communications, M. Ibnkahla, Ed. Boca Raton, FL: CRC Press, M. Z. Win, N. C. Beaulieu, L. A. She, B. F. Logan, and J. H. Winters, On the SNR enalty of msk with hybrid selection/maximal ratio combining over i.i.d. Rayleigh fading channels, IEEE Trans. Commun., vol. 51, no. 6, , Jun M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, A. F. Molisch, M. Z. Win, and J. H. Winters, Reduced-comlexity transmit/receive-diversity systems, in Proc. IEEE Vehicular Technology Conf. Sring 21, Rhodes, Greece, T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, J. H. Winters, Otimum combining in digital mobile radio with co-channel interference, IEEE J. Sel. Areas Commun., vol. 2, no. 4, , Jul P. F. Driessen and G. J. Foschini, On the caacity formula for multile inut-multile outut wireless channels: A geometric interretation, IEEE Trans. Commun., vol. 47, no. 2, , Feb I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, Andreas F. Molisch S 89 M 95 SM F 5 received the Dil. Ing., Dr. techn., and habilitation degrees from the Technical University TU Vienna, Vienna, Austria, in 199, 1994, and 1999, resectively. From 1991 to 2, he was with the TU Vienna, becoming an Associate Professor there in From 2 to 22, he was with the Wireless Systems Research Deartment at AT&T Laboratories- Research, Middletown, NJ. Since then, he has been a Senior Princial Member of Technical Staff with Mitsubishi Electric Research Laboratories, Cambridge, MA. He is also Professor and Chairholder for radio systems at Lund University, Lund, Sweden. He has done research in the areas of SAW filters, radiative transfer in atomic vaors, atomic line filters, smart antennas, and wideband systems. His current research interests are MIMO systems, measurement and modeling of mobile radio channels, and ultrawide bandwidth UWB. He has authored, coauthored, or edited four books, eight book chaters, some 85 journal aers, and numerous conference contributions. Dr. Molisch is an editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, co-editor of a recent secial issue on MIMO and smart antennas in Journal on Wireless Communications Mobile Comuting, and coeditor of an ucoming IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATONS secial issue on UWB. He has articiated in the Euroean research initiatives COST 231, COST 259, and COST273, where he is chairman of the MIMO channel working grou. He is also vice chairman of Commission C signals and systems of URSI International Union of Radio Scientists, chairman of the IEEE a channel modeling grou, and reciient of several awards. Moe Z. Win S 85 M 87 SM 97 F 4 received theb.s.degreemagna cum laude from Texas A&M University, College Station, in 1987, and the M.S. degree from the University of Southern California USC, Los Angeles, in 1989, both in electrical engineering. As a Presidential Fellow at USC, he received both the M.S. degree in alied mathematics and the Ph.D. degree in electrical engineering in Dr. Win is an Associate Professor at the Laboratory for Information and Decision Systems LIDS, Massachusetts Institute of Technology, Cambridge. Prior to joining LIDS, he sent 5 years at AT&T Research Laboratories and 7 years at the Jet Proulsion Laboratory. His main research interests are the alication of mathematical and statistical theories to communication, detection, and estimation roblems. Secific current research toics include measurement and modeling of time-varying channels, design and analysis of multile antenna systems, ultrawide bandwidth UWB communications systems, otical communications systems, and sace communications systems. Dr. Win has been involved actively in organizing and chairing sessions, and has served as a member of the Technical Program Committee in a number of international conferences. He served as the Technical Program Chair for the IEEE Communication Theory Symosia of ICC-24 and Globecom-2, as well as for the IEEE Conference on Ultra Wideband Systems and Technologies in 22, the Technical Program Vice Chair for the IEEE International Conference on Communications in 22, and the Tutorial Chair for the IEEE Semiannual International Vehicular Technology Conference in Fall 21. He is the current Chair and ast Secretary for the Radio Communications Committee of the IEEE Communications Society. He currently serves as Area Editor for Modulation and Signal Design and Editor for Wideband Wireless and Diversity, both for IEEE TRANSACTIONS ON COMMUNICATIONS. He served as the Editor for Equalization and Diversity from July 1998 to June 23 for the IEEE TRANSACTIONS ON COMMUNICATIONS, and as a Guest-Editor for the 22 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS Secial Issue on Ultra-Wideband Radio in Multiaccess Wireless Communications. He received the International Telecommunications Innovation Award from Korea Electronics Technology Institute in 22, the Young Investigator Award from the Office of Naval Research in 23, and the IEEE Antennas and Proagation Society Sergei A. Schelkunoff Transactions Prize Paer Award in 23. In 24, he was named Young Aerosace Engineer of the Year by the AIAA and received the Fulbright Foundation Senior Scholar Lecturing and Research Fellowshi, the Institute of Advanced Study Natural Sciences and Technology Fellowshi, the Outstanding International Collaboration Award from the Industrial Technology Research Institute of Taiwan, and the Presidential Early Career Award for Scientists and Engineers from the White House. He is an IEEE Distinguished Lecturer and elected Fellow of the IEEE, cited for contributions to wideband wireless transmission.

14 1772 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 25 Yang-Seok Choi S 91 M 1 received the B.S. degree from Korea University, Seoul, South Korea, in 199, the M.S.E.E. degree from the Korea Advanced Institute of Science and Technology, Taejon, South Korea, in 1992, and the Ph.D. degree from Polytechnic University, Brooklyn, NY, in 2, all in electrical engineering. From 1992 to 1996, he was with Samsung Electronics, Co., Ltd., Suwon, Korea, where he develoed 32 QAM modem for HDTV and QPSK ASIC for DBS. During 2 summer he held a Summer intern osition at AT&T Labs-Research Shannon Lab, Florham Park, NJ. In 2, he joined National Semiconductor, East Brunswick, NJ, where he was involved in the develoment of W-CDMA. During 21 22, he was a Senior Technical Staff Member at AT&T Labs-Research, Middletown, NJ where he researched on MIMO systems, OFDM systems and information theory. From 22 to 24 he had been with ViVATO, Inc., Sokane, WA, working on MIMO OFDM systems, smart antenna systems, and antenna/beam selection techniques. He researched on Smart antenna alications to CSMA rotocol and co-invented Comlementary Beamforming. In 24, he joined Intel Cororation, Hillsboro, OR where he studies on Broadband Wireless communications systems. His research interests include MIMO, OFDM, MC-CDMA, smart antenna, blind identification/equalizer, carrier/timing recovery, sace-time coding, cross-layer design and caacity of time-varying multiath channel. He holds seven U.S. atents. Jack H. Winters S 77 M 81 SM 88 F 96 received the Ph.D. degree in electrical engineering from The Ohio State University, Columbus, in He is the Chief Scientist at Motia, Inc., where he is involved with smart antennas for wireless systems. In 1981, he joined AT&T and worked in the research area for more than 2 years. At AT&T, he was Division Manager of the Wireless Systems Research Division of AT&T Labs-Research, Middletown, NJ. He is an IEEE Distinguished Lecturer for both the IEEE Communications and the Vehicular Technology Societies, Area Editor for Transmission Systems for the IEEE TRANSACTIONS ON COMMUNICATIONS, and a New Jersey Inventor of the Year for 21.