IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, … · IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006 1433 Capacity Scaling Laws in MIMO Relay Networks

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006 1433

Capacity Scaling Laws in MIMO Relay NetworksHelmut Bölcskei, Senior Member, IEEE, Rohit U. Nabar, Member, IEEE, Özgür Oyman, Member, IEEE,

and Arogyaswami J. Paulraj, Fellow, IEEE

Abstract— The use of multiple antennas at both ends ofa wireless link, popularly known as multiple-input multiple-output (MIMO) wireless, has been shown to offer significantimprovements in spectral efficiency and link reliability throughspatial multiplexing and space-time coding, respectively. Thispaper demonstrates that similar performance gains can beobtained in wireless relay networks employing terminals withMIMO capability. We consider a setup where a designatedsource terminal communicates with a designated destinationterminal, both equipped with M antennas, assisted by K single-antenna or multiple-antenna relay terminals using a half-duplexprotocol. Assuming perfect channel state information (CSI) atthe destination and the relay terminals and no CSI at thesource, we show that the corresponding network capacity scalesas C = (M/2) log(K) + O(1) for fixed M , arbitrary (butfixed) number of (transmit and receive) antennas N at eachof the relay terminals, and K → ∞. We propose a protocolthat assigns each relay terminal to one of the multiplexed datastreams forwarded in a “doubly coherent” fashion (throughmatched filtering) to the destination terminal. It is shown that thisprotocol achieves the cut-set upper bound on network capacityfor fixed M and K → ∞ (up to an O(1)-term) by employingindependent stream decoding at the destination terminal. Ourprotocol performs inter-stream interference cancellation in acompletely decentralized fashion, thereby orthogonalizing the ef-fective MIMO channel between source and destination terminals.Finally, we discuss the case where the relay terminals do nothave CSI and show that simple amplify-and-forward relaying,asymptotically in K, for fixed M and fixed N ≥ 1, turns therelay network into a point-to-point MIMO link with high-SNRcapacity C = (M/2) log(SNR) + O(1), demonstrating that theuse of relays as active scatterers can recover spatial multiplexinggain in poor scattering environments.

Index Terms— Relay channel, Adhoc network, distributedorthogonalization, MIMO, fading channels

I. INTRODUCTION AND OUTLINE

THE use of multiple antennas at both ends of a wirelesslink, known as multiple-input multiple-output (MIMO)Manuscript received April 19, 2004; revised February 9, 2005 and July

2, 2005; accepted July 8, 2005. The associate editor coordinating the reviewof this paper and approving it for publication was H. Yanikomeroglu. Thispaper was presented in part at the Allerton Conference on Communications,Control, and Computing, Monticello, IL, U.S.A., Oct. 2003, and at the IEEEInternational Symposium on Information Theory (ISIT), Chicago, IL, U.S.A.,June 2004.

H. Bölcskei is with the Communication Technology Laboratory, ETHZurich, CH-8092, Zurich, Switzerland (e-mail: [email protected]).

R. U. Nabar is with the DSP Technology Group, Marvell Semiconductor,Inc., Santa Clara, CA, U.S.A. (e-mail: [email protected]).

Ö. Oyman is with the Corporate Technology Group at Intel Corporation,Santa Clara, CA, U.S.A. (e-mail: [email protected]).

A. J. Paulraj is with the Information Systems Laboratory, Stanford Univer-sity, Stanford, CA, U.S.A. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2006.04263.

Source Destination

Relays

Fig. 1. Dense wireless relay network with dead-zones around source anddestination terminals each equipped with M antennas. The relay terminalsemploy N ≥ 1 (transmit and receive) antennas.

wireless yields significant improvements in spectral effi-ciency and link reliability through spatial multiplexing [1]–[5] and space-time coding [6]–[9], respectively. While theinformation-theoretic performance limits of point-to-pointMIMO links are well understood [1]–[5] knowledge about theimpact of MIMO terminals in a multiuser or network contextis scarce. First results on MIMO broadcast and multiple-accesschannels have been reported in [10]–[18].

The aim of this paper is to quantify the impact of MIMOtechnology on large wireless relay networks, where a des-ignated multiantenna source terminal communicates with adesignated multiantenna destination terminal through a set ofmultiantenna or single-antenna relay terminals (see Fig. 1).We start by briefly reviewing MIMO gains in point-to-pointlinks.

A. MIMO Gains in Point-to-Point Links

In point-to-point wireless links, MIMO systems improvespectral efficiency through spatial multiplexing gain, linkreliability through diversity gain and coverage through arraygain:

Spatial multiplexing in MIMO systems yields a linear (inthe minimum of the number of transmit and receive antennas)increase in capacity for no additional power or bandwidthexpenditure [1]–[5]. The corresponding gain is realized bysimultaneously transmitting independent data streams in thesame frequency band. Under conducive channel conditions(i.e., rich scattering), the receiver exploits differences in thespatial signatures of the multiplexed data streams to separatethe different signals, thereby realizing a capacity gain.

1536-1276/06$20.00 c© 2006 IEEE

1434 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006

Diversity is a powerful technique to mitigate fading andincrease robustness to interference [19]. Diversity gain isobtained by transmitting the data signal over multiple (ide-ally) independently fading paths (time/frequency/space) andperforming proper combining at the receiver. Spatial (i.e.,antenna) diversity is particularly attractive when comparedwith time/frequency diversity since it does not incur an expen-diture in transmission time/bandwidth. Space-time coding [6]–[9] realizes spatial transmit diversity gain in point-to-pointMIMO systems without requiring channel knowledge at thetransmitter.

Array gain can be made available at the transmitter andthe receiver, requires channel knowledge, and results in anincrease in average receive signal-to-noise ratio (SNR) andhence improved coverage due to a coherent combining ef-fect [19].

B. Coherent and Noncoherent MIMO Relay Networks

Throughout this paper, we focus on dense relay networkswhere a designated source terminal equipped with M antennasspatially multiplexes data (i.e., transmits statistically indepen-dent data streams from different antennas) to a designated des-tination terminal with M antennas through K relay terminalswith N ≥ 1 transmit/receive antennas each. Communicationtakes place over two time slots using a two-hop half-duplexprotocol. We consider two scenarios:

• Coherent MIMO relay network: The source terminal hasno channel state information (CSI), the destination termi-nal has perfect knowledge of all channels (asymptoticallyin K , knowledge of the composite M ×M MIMO chan-nel between source and destination terminals will suffice,cf. Theorem 2). Each relay terminal maintains perfect CSIfor its N×M backward (i.e., channel between the sourceterminal and the relay terminal) and M×N forward (i.e.,channel between the relay terminal and the destinationterminal) channels.

• Noncoherent MIMO relay network: In this case, the as-sumptions on CSI in the source and destination terminalsare the same as in the coherent case, the relay terminalshave no CSI.

C. Contributions and Relation to Previous Work

We investigate the information-theoretic performance limitsof large MIMO relay networks and quantify the correspondingmultiplexing, diversity, and array gains. Throughout the paperthe terminology distributed array/diversity gain will be usedto designate array/diversity gain resulting from the use of relayterminals. Our framework is based on deriving asymptotic (inthe number of relay terminals) network capacity, a conceptpioneered for single-antenna systems and AWGN channelsin [20], [21]. Results of the Gupta-Kumar [20], [22] andGastpar-Vetterli [21] type have recently been reported forsingle antenna fading channels in [23], for the case of nodemobility and large-scale fading in [24], for extended AWGNnetworks in [25], [26], and for multihop routing in [27].

Distributed space-time code design and information-theoretic performance limits for single-antenna fading relaychannels (with a finite number of nodes) have recently been

studied in [28]–[32]. Capacity results for MIMO relay chan-nels with a finite number of relays can be found in [31], [33],[34].

The detailed contributions reported in this paper can besummarized as follows:

• Generalizing the main result in [21] to the fading MIMOcase, we show that the coherent MIMO relay networkcapacity scales as C = (M/2) log(K) + O(1) for Mfixed, N arbitrary (but fixed), and K → ∞. In terms oftraditional MIMO gains this result states that a spatialmultiplexing gain of M/2, and a distributed per-streamarray gain of K are obtained.

• We propose a simple (half-duplex) protocol that achievesthe cut-set upper bound on coherent network capacity(up to an O(1)-term) in the large K limit, for fixed Mand for arbitrary (but fixed) N by employing matchedfiltering at the relay terminals and independent streamdecoding at the destination terminal. Our protocol assignseach relay terminal to one out of the M transmit-receiveantenna pairs (relay partitioning) and requires knowledge(at the relay) only of the corresponding N×1 and 1×Nbackward and forward channels, respectively. We showthat the proposed protocol effectively performs inter-stream interference cancellation, thereby orthogonalizingthe MIMO channel between source and destination in acompletely decentralized fashion.

• For noncoherent MIMO relay networks, we show thatsimple amplify-and-forward (AF) relaying, asymptoti-cally in K , for fixed M and fixed N ≥ 1, turns the relaynetwork into a point-to-point MIMO link with high-SNRcapacity C = (M/2) log(SNR) + O(1), demonstratingthat a spatial multiplexing gain of M/2 and no distributedarray gain are obtained. We find, however, that traditionalreceive array gain is realized. Our result also shows thatusing (even single-antenna) relays as active scattererscan recover spatial multiplexing gain in poor scatteringenvironments.

D. Notation

The superscripts T , H , and ∗ stand for transposition, con-jugate transpose, and element-wise conjugation, respectively.E denotes the expectation operator. det(A) stands for thedeterminant of the matrix A. ‖a‖ denotes the Euclidean normof the vector a. Im stands for the m × m identity matrix.For an m×n matrix A = [a1 a2 . . . an], we define themn × 1 vector vec(A) = [aT1 aT2 . . . aTn ]T. A ⊗ B standsfor the Kronecker product of the matrices A and B. |X | isthe cardinality of the set X . The notation u(x) = O(v(x))denotes that |u(x)/v(x)| remains bounded as x → ∞. Acircularly symmetric complex Gaussian random variable (RV)is a RV Z = X+ j Y ∼ CN (0, σ2), where X and Y are i.i.d.N (0, σ2/2). VAR(X) stands for the variance of the RV X .w.p. 1−−−→ denotes convergence with probability 1. Throughout the

paper all logarithms, unless specified otherwise, are to thebase 2.

BÖLCSKEI et al.: CAPACITY SCALING LAWS IN MIMO RELAY NETWORKS 1435

TABLE I

HALF-DUPLEX TWO-HOP PROTOCOL. COMMUNICATION TAKES PLACE

OVER TWO TIME SLOTS. (A → B SIGNIFIES COMMUNICATION FROMTERMINAL A TO TERMINAL B).

Time slot 1 Time slot 2

S → Rk, k = 1, 2, . . . , K Rk → D, k = 1, 2, . . . , K

E. Organization of the Paper

The rest of this paper is organized as follows. Section IIdescribes the channel and signal model. In Section III, wederive an upper bound on the capacity of the coherent MIMOrelay network. Section IV introduces a simple protocol thatachieves this upper bound asymptotically in K (up to an O(1)-term) and hence establishes (asymptotic) network capacity.Section V contains an analysis of a simple AF-based pro-tocol for noncoherent MIMO relay networks and derives thecorresponding asymptotic network capacity. We conclude inSection VI.

II. CHANNEL AND SIGNAL MODEL

A. General Assumptions

The discussion in the remainder of this section appliesto both coherent and noncoherent MIMO relay networks asdefined in Section I-B. We consider a wireless network con-sisting of K+2 terminals, with a designated source-destinationterminal pair and K relay terminals located randomly andindependently in a domain of fixed area (see Fig. 1). Thesource and destination terminals, equipped with M antennaseach, are denoted by S and D, respectively. The kth relayterminal employs N ≥ 1 transmit/receive antennas and isdenoted by Rk (k = 1, 2, . . . ,K). Throughout the paper, weassume that M and N are finite and consider the K → ∞capacity behavior of the network. We furthermore assume a“dead-zone” of non-zero radius around S and D [21] that isfree of relay terminals, no direct link between S and D (e.g.,due to large distance between S and D or due to D beinglocated deep inside a building and S outside the building),and transmission in spatial multiplexing mode (i.e., the sourceterminal transmits independent data streams from differentantennas) over two time slots using two hops in half-duplexmode (i.e., the terminals cannot transmit and receive simulta-neously). In the first time slot the relay terminals receive thesignal transmitted by the source terminal. After processing thereceived signals, the relay terminals simultaneously transmitthe processed data to the destination terminal during thesecond time slot while the source terminal is silent (seeTable I).

B. Channel and Signal Model

Throughout the paper, we assume that all channels arefrequency-flat block fading with the same block length, in-dependent realizations across blocks and the block lengthtaken to be an integer multiple of the duration of a timeslot. Transmission/reception between the terminals is perfectlysynchronized. The input-output relation for the S → Rk link1

1A → B signifies communication from terminal A to terminal B.

independent datastreams

first hop

Ek Pk

second hop

source terminal

K relay terminals

Hk+1

HkGk

Gk+1

destination terminal

Ek+1 Pk+1

Fig. 2. MIMO wireless relay network setup. The relay terminals employN ≥ 1 antennas.

is given by

rk =

√EkM

Hks + nk, k = 1, 2, . . . ,K (1)

where rk denotes the N × 1 received vector signal, Ek is theaverage energy received at the kth relay terminal over onesymbol period through the S → Rk link (having accountedfor path loss and shadowing), Hk =

[hk,1 hk,2 . . . hk,M

]is the N ×M random channel matrix corresponding to theS → Rk link, independent across relay terminals (i.e., inde-pendent across k) and consisting of i.i.d. CN (0, 1) entries, s =[s1 s2 . . . sM ]T is the M × 1 circularly symmetric complexGaussian transmit signal vector satisfying E{ssH} = IM(recall that the signaling mode is spatial multiplexing), and nkis an N×1 spatio-temporally white circularly symmetric com-plex Gaussian noise vector sequence, independent across k,with covariance matrix E{nknHk } = σ2nIN (k = 1, 2, . . . ,K).

Each relay terminal processes its received vector signal rkto produce the N × 1 vector signal tk, which is then trans-mitted to the destination terminal over the second time slot.The M × 1 vector signal received at the destination terminalis consequently given by

y =K∑k=1

√PkN

Gktk + z (2)

where Pk denotes the average signal energy over one symbolperiod contributed by the kth relay terminal (having accountedfor path loss and shadowing in the Rk → D link), Gk =[gk,1 gk,2 . . . gk,M

]T is the corresponding M ×N channelmatrix with i.i.d. CN (0, 1) entries, independent across k, andz =

[z1 z2 . . . zM

]T denotes an M × 1 spatio-temporallywhite circularly symmetric complex Gaussian noise vectorsequence satisfying E{zzH} = σ2nIM . The transmit signalvectors tk will in general depend both on the forward andbackward channels, Gk and Hk, respectively, and are chosento satisfy E{‖tk‖2} ≤ N thus imposing a per-relay averagepower constraint. We emphasize that assuming a total powerconstraint across relays and uniform (across relays) powerallocation does not change the main results of the papergiven by (17) and Theorem 3. All noise signals throughoutthe network are assumed independent of the transmit signals.The entire setup described above is summarized schematicallyin Fig. 2.


sourceterminal

relays

S

destinationterminal

Sc

Fig. 3. Applying a broadcast cut to the coherent relay network.

As already mentioned above, the path-loss and shadow-ing effects are described by {Ek}Kk=1 for the first hop and{Pk}Kk=1 for the second hop. We assume that these para-meters are independent RVs, strictly positive, bounded, andremain constant over the entire time period of interest. Therandomness of the Ek and Pk reflects the fact that therelay terminal locations are chosen randomly, strict positivityis a consequence of the assumption that the domain underconsideration is of fixed area, and the dead-zone assumptionimplies that the parameters Ek and Pk are bounded. Eventhough we do not specify the exact relay locations, as is donein geometrical network models such as those used in [26],the network topology and the propagation conditions areaccounted for through the statistics of the Ek and the Pk. Ifthe relay terminals are located randomly and uniformly withinthe domain of interest both the Ek and the Pk can be assumedto be independent. If the relay terminals are confined to certaingeographic areas (e.g., only at cell edge or only within acertain range from source or destination terminals) this canbe accounted for through different means in the Ek andthe Pk. The entire discussion in the paper applies to generalindependent, positive and bounded Ek and Pk. However, weshall see that the i.i.d. assumption on the Ek and the Pk oftenleads to significantly stronger results (cf. Theorems 2 and 3).

III. UPPER BOUND ON COHERENT MIMO RELAYNETWORK CAPACITY

In this section, we derive an upper bound on the capacityof coherent relay networks (as defined in Section I-B) inconjunction with two-hop relaying as described in Section II.In Section IV, we introduce a simple protocol for coherentMIMO relay networks that leads to a corresponding lowerbound on network capacity. This lower bound is then shown toasymptotically (up to an O(1)-term) approach the upper boundpresented in Theorem 1 below, which ultimately establishesthe asymptotic network capacity scaling behavior.

Theorem 1: The capacity of the coherent MIMO relaynetwork under two-hop relaying satisfies

C ≤ Cupper = M2 log(

1 +N

Mσ2n

K∑k=1

Ek

).

In the large relay limit K → ∞, defining μ =

(1/K)∑Kk=1 E{Ek}, we furthermore have

Cupperw.p. 1−−−→ C∞upper =

M

2log

(KNμ

Mσ2n

)

=M

2log(K) +O(1).

(3)

Proof: Separating the source terminal S from the rest ofthe network (see Fig. 3), using a broadcast cut [35], and apply-ing the cut-set theorem [35, Theorem 14.10.1], it follows thatthe capacity of the MIMO relay network is upper bounded by

Cu = E{Hk,Gk}Kk=1{

12I(s; r1, . . . , rK ,y | t1, . . . , tK)

}

where the factor 1/2 results from the fact that data istransmitted over two time slots. Applying the chain rule formutual information [35], we obtain

I(s; r1, . . . , rK ,y | t1, . . . , tK) =I(s; r1, . . . , rK | t1, . . . , tK)

+ I(s;y | r1, . . . , rK , t1, . . . , tK)(4)

where conditioning on the tk in the first term can be droppedsince neither s nor the rk depend on the tk. Conditioning onthe tk and assuming perfect knowledge of the Gk at the re-ceiver results in the second term on the right-hand-side (RHS)of (4) being equal to I(s; z) = 0 due to independence of sand z. We can therefore summarize our result as

Cu = E{Hk}Kk=1{

12I(s; r1, . . . , rK)

}.

Recalling that s was assumed circularly symmetric complexGaussian with E{ssH} = IM , we get [36]

Cu = E{Hk}Kk=1

{12

log det

(IM +

1σ2n

K∑k=1

EkM

HHk Hk

)}

(5)which upon application of Jensen’s inequality [35] yields

Cu ≤ M2 log(

1 +N

Mσ2n

K∑k=1

Ek

)= Cupper

and hence completes the proof of the first part of the theorem.In order to prove the second part, we start by noting that

due to the assumption of the Ek (k = 1, 2, . . . ,K) beingbounded it follows that VAR(Ek) is bounded as well fork = 1, 2, . . . ,K , and hence the Kolmogorov condition

∞∑k=1

VAR(Ek)k2


yi =∑

k,Rk∈Xi

√PkN ‖gk,i‖

(√EkM

(‖hk,i‖2si +

∑Mj=1,j �=i h

Hk,ihk,jsj

)+ hHk,ink

)√

EkM (N +M) + σ

2n

+M∑

m=1,m �=i

⎛⎜⎜⎝

∑k,Rk∈Xm

√PkN g

Tk,ig

∗k,m

(√EkM

(‖hk,m‖2sm +

∑Mj=1,j �=m h

Hk,mhk,jsj

)+ hHk,mnk

)

‖gk,m‖√

EkM (N +M) + σ

2n

⎞⎟⎟⎠ + zi

(11)

Noting that M and N are finite and E{Ek} being finite fork = 1, 2, . . . ,K implies that μ is finite as well, we get

C∞upper =M

2log(K) +O(1) (8)

which completes the proof of Theorem 1.Discussion: It is straightforward to see that, opera-

tionally, the RHS of (5) results when all the relay terminalscan fully cooperate (and perform joint decoding) so as toeffectively form a point-to-point coherent (i.e., perfect receiveCSI) MIMO channel with M transmit and KN receiveantennas. Equation (8) explicitly reveals that for fixed Mnetwork capacity scales at most logarithmically in K withpre-log M/2. We finally note that the SNR in (8) has beenabsorbed in the O(1)-term. Throughout the discussion of thecoherent relaying architecture, we shall isolate the SNR in theO(1)-term in order to emphasize the scaling behavior as afunction of K .

IV. COHERENT RELAYING AND NETWORK CAPACITY

In this section, we present a simple relaying protocol,which does not require any cooperation between terminalsand employs single-antenna (i.e., independent) decoding at thedestination terminal. The capacity achieved by this protocolwill be shown to asymptotically (in K) approach the cut-set upper bound (up to an O(1)-term), thus establishing thecoherent MIMO relay network capacity scaling behavior.

A. The Relaying Protocol

The essence of the proposed protocol lies in assigning eachof the relay terminals to one out of the M transmit-receiveantenna pairs. Without loss of generality, we assume that theith transmit antenna and the ith receive antenna constitutethe ith pair. Our assignment policy needs to ensure that forincreasing K each transmit-receive antenna pair is servedby an increasing number of relay terminals. Consequently,the relay terminals are partitioned into groups with eachgroup being responsible for one of the M source-destinationantenna pairs (see Fig. 4). In the following, the set of relayterminals assigned to the ith transmit-receive antenna pair willbe denoted by Xi.

We start by assuming that the kth relay terminal is assignedto the lth transmit-receive antenna pair. Upon reception of rk,the relay performs matched filtering with respect to the as-

sourceterminal

destinationterminal

Tx 1

Tx M

Tx 2

Rx 1

Rx 2

Rx M

Tx 1 – Rx 1

Tx 2 – Rx 2

Tx M – Rx M

Fig. 4. Relay partitioning. Each transmit-receive antenna pair has anassociated group of relay terminals.

signed backward channel to obtain

uk = hHk,lrk

=

√EkM

⎛⎝‖hk,l‖2sl +

M∑j=1,j �=l

hHk,lhk,jsj

⎞⎠ + hHk,lnk.

(9)

Using2 E{|uk|2} = (Ek/M)(N2 +MN

)+ Nσ2n, the

processed received signal is scaled to unit average energy andtransmit matched filtering with respect to the assigned forwardchannel is performed so that

tk =uk√

EkM (N +M) + σ

2n

g∗k,l‖gk,l‖ (10)

which is easily seen to satisfy the transmit power con-straint E{‖tk‖2} = N .

Now, the signal at the ith receive antenna is given by

yi =M∑m=1

⎛⎝ ∑k,Rk∈Xm

√PkN

gTk,itk

⎞⎠ + zi, i = 1, 2, . . . ,M

which upon insertion of (10) and (9) yields (11) shown at thetop of the page.

2Note that we are averaging |uk|2 over the random channel, noise, andsignal.


Ii =12

log

⎛⎜⎝1 +

∣∣∣hsigi∣∣∣2

∑Mj=1,j �=i

∣∣hinti,j ∣∣2 + σ2n(1 +

∑k,Rk∈Xi ak,i +

∑Mm=1,m �=i

(∑k,Rk∈Xm bk,i,m

))⎞⎟⎠ (14)

We observe that yi consists of a contribution from thedesired data stream3 si, interference signal terms from thedata streams sm with m = i, forwarded noise from the relayterminals, and receiver noise. In order to set the stage forthe main result of this section (the capacity lower bound inTheorem 2), we shall next identify the signal, interference,and noise contributions to yi in detail. In particular, we seeka representation of the form

yi = hsigi si +

M∑j=1,j �=i

hinti,j sj +Ni, i = 1, 2, . . . ,M (12)

where hsigi denotes the effective scalar channel gain for thedata stream transmitted from the ith source terminal antenna,hinti,j stands for the effective channel seen by the interferencefrom sj to si, and Ni denotes the effective noise term at theith receive antenna.

Signal contribution: From (11) it follows by inspection that

hsigi =∑

k,Rk∈Xidk,i +

M∑m=1,m �=i

⎛⎝ ∑k,Rk∈Xm

fk,i,m

⎞⎠

where

dk,i =

√PkN

⎛⎝

√EkM ‖gk,i‖‖hk,i‖2√EkM (N +M) + σ

2n

⎞⎠

fk,i,m =

√PkN

⎛⎝

√EkM g

Tk,ig

∗k,mh

Hk,mhk,i

‖gk,m‖√

EkM (N +M) + σ

2n

⎞⎠ .

Interference contribution: Again, by inspection of (11), weget

hinti,j =∑

k,Rk∈Xixk,i,j +

∑k,Rk∈Xj

wk,i,j

+M∑

m=1,m �=i,j

⎛⎝ ∑k,Rk∈Xm

vk,i,j,m

⎞⎠

where

xk,i,j =

√PkN

⎛⎝

√EkM ‖gk,i‖hHk,ihk,j√EkM (N +M) + σ

2n

⎞⎠

wk,i,j =

√PkN

⎛⎝

√EkM g

Tk,ig

∗k,j‖hk,j‖2

‖gk,j‖√

EkM (N +M) + σ

2n

⎞⎠

vk,i,j,m =

√PkN

⎛⎝

√EkM g

Tk,ig

∗k,mh

Hk,mhk,j

‖gk,m‖√

EkM (N +M) + σ

2n

⎞⎠ .

3Recall that single-antenna decoding is performed at the receiver and theith receive antenna is designated to decode the signal transmitted from theith source terminal antenna.

Noise contribution: The total noise term Ni follows from(11) as

Ni =∑

k,Rk∈Xiñk,i +

M∑m=1,m �=i

⎛⎝ ∑k,Rk∈Xm

q̃k,i,m

⎞⎠ + zi

where ñk,i| {Ek, Pk,Hk,Gk}Kk=1 ∼ CN (0, ak,iσ2n) andq̃k,i,m| {Ek, Pk,Hk,Gk}Kk=1 ∼ CN (0, bk,i,mσ2n) with

ak,i =Pk‖gk,i‖2‖hk,i‖2

N(EkM (N +M) + σ

2n

)

bk,i,m =Pk‖hk,m‖2|gTk,ig∗k,m|2

N(EkM (N +M) + σ

2n

) ‖gk,m‖2 .

B. Lower Bound on MIMO Relay Network Capacity

Based on (12), we are now ready to derive the asymptotic(in K) capacity achieved by the relaying protocol describedin Section IV-A. Our main result is summarized as follows.

Theorem 2: For a fixed number of source-destination an-tenna pairs M and fixed N , in the K → ∞ limit such that4|X1| = |X2| = · · · = |XM | = K/M , assuming perfectknowledge of {Ek, Pk,Hk,Gk}Kk=1 at each of the receiveantennas, the relay network capacity scales at least as

C∞lower =M

2log(K) +O(1). (13)

For i.i.d. {Ek} and i.i.d. {Pk} knowledge of hsigi and hinti,j (j =1, 2, . . . ,M, j = i) at the ith receive antenna is sufficientfor (13) to hold.

Proof: Recalling that we assumed independent decodingfor each of the multiplexed data streams, and using (12), weget5

Clower =M∑i=1

E{Hk,Gk}Kk=1{Ii}

where Ii is given in (14) shown at the top of the page. Here, weused the fact that perfect knowledge of {Ek, Pk,Hk,Gk}Kk=1implies circularly symmetric complex Gaussianity of Ni andresults in perfect knowledge of hinti,j (i = 1, 2, . . . ,M, j =1, 2, . . . ,M, j = i), which renders the noise plus interferencecontribution in (12) circularly symmetric complex Gaussian.Next, multiplying and dividing the argument of the logarithmin (14) by (K/M)2, we obtain (15) [see following page].Using the assumption that the Ek and Pk are positive andbounded ∀k, it is straightforward but tedious to show thatthe variances of each of the RVs, dk,i, fk,i,m, xk,i,j , wk,i,j ,

4The assumption of equal size relay clusters is made for the sake ofsimplicity of exposition only. The scaling result (13) is valid more generallyfor |Xi| (i = 1, 2, . . . , M) proportional to K .

5Note that the Ek and Pk were assumed to remain constant over the entiretime period of interest so that ergodic capacity is obtained by averaging overthe Hk and Gk only.


Ii =12

log

⎛⎜⎝1 +

∣∣∣ hsigiK/M∣∣∣2

∑Mj=1,j �=i

∣∣∣ hinti,jK/M∣∣∣2 + σ2nK/M

(1

K/M +∑

k,Rk∈Xiak,iK/M +

∑Mm=1,m �=i

(∑k,Rk∈Xm

bk,i,mK/M

))⎞⎟⎠ (15)

vk,i,j,m, ak,i, and bk,i,m are bounded ∀ i, j, k,m. Hence,the Kolmogorov condition (6) is satisfied for each of thesesequences (as a function of k) and it follows from [37,Theorem 1.8.D] that

∑k,Rk∈Xi

dk,iK/M

−∑

k,Rk∈Xi

E{dk,i}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xm

fk,i,mK/M

−∑

k,Rk∈Xm

E{fk,i,m}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xi

xk,i,jK/M

−∑

k,Rk∈Xi

E{xk,i,j}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xj

wk,i,jK/M

−∑

k,Rk∈Xj

E{wk,i,j}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xm

vk,i,j,mK/M

−∑

k,Rk∈Xm

E{vk,i,j,m}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xi

ak,iK/M

−∑

k,Rk∈Xi

E{ak,i}K/M

w.p. 1−−−→ 0

∑k,Rk∈Xm

bk,i,mK/M

−∑

k,Rk∈Xm

E{bk,i,m}K/M

w.p. 1−−−→ 0

as K → ∞. Next, noting that E{fk,i,m} = E{xk,i,j} =E{wk,i,j} = E{vk,i,j,m} = 0 ∀ i, j, k,m, and applying [37,Theorem 1.7], we obtain

Iiw.p. 1−−−→ 1

2log

(1 +

Kδ2iMσ2n

(MK + αi + βi

))

(16)

for K → ∞, where

δi =∑

k,Rk∈Xi

E{dk,i}K/M

αi =∑

k,Rk∈Xi

E{ak,i}K/M

βi =M∑

m=1,m �=i

⎛⎝ ∑k,Rk∈Xm

E{bk,i,m}K/M

⎞⎠ .

Hence, the total capacity is given by

Cw.p. 1−−−→ 1

2

M∑i=1

log

(1 +

Kδ2iMσ2n

(MK + αi + βi

))

and for K → ∞C∞lower =

M

2log(K) +O(1).

This concludes the proof of (13).The proof for i.i.d. {Ek} and i.i.d. {Pk} starts by noting that

i.i.d. {Ek}Kk=1, {Pk}Kk=1, {Gk}Kk=1, and {Hk}Kk=1 impliesthat ñk,i and q̃k,i are i.i.d. (across k) ∀i, which, combined withthe fact that the corresponding variances are bounded, allows

100 200 300 400 500 600 700 800 900 10004

5

6

7

8

9

10

11

12

13

14

number of relaysca

paci

ty (b

it/s/

Hz)

upper bound

lower bound

Fig. 5. Capacity upper and lower bounds vs. number of relays for thecoherent case.

to apply the Lindeberg-Levy theorem [37, Theorem 1.9.1.A]to conclude that asymptotically in K the noise terms Ni arecircularly symmetric complex Gaussian for all i. Since hsigiand hinti,j (j = 1, 2, . . . ,M, j = i) are assumed perfectlyknown at the ith receive antenna, the receive signal yi is cir-cularly symmetric complex Gaussian. Applying steps similarto those leading to (16) and the steps thereafter completes theproof.

C. Network Capacity and Discussion of Results

Network capacity: We can now combine Theorems 1and 2 to state that asymptotically in K the coherent networkcapacity (under the assumptions in Theorem 2) is given by

C =M

2log(K) +O(1). (17)

We hasten to add that the O(1)-terms in the upper and lowerbounds in Theorems 1 and 2 are in general different so that we“sandwich” the exact network capacity up to an O(1)-term.Consequently, our relaying protocol is optimal in the sensethat it achieves the right capacity scaling law, or equivalently,it is optimal in the sense that it achieves network capacityup to an O(1)-term. The difference between the O(1)-termsin the lower and upper bounds can be significant (see, e.g.,Fig. 5). Moreover, (17) implies that the proposed protocolasymptotically turns the network into a point-to-point MIMOlink with multiplexing gain M/2 and distributed per-streamarray gain K . From (16) we can furthermore conclude that inthe large K limit each of the M single-input single-output(SISO) channels (with corresponding input-output relation(12)) made up of the individual transmit-receive antenna pairsapproaches an AWGN channel. This effect can be attributedto distributed diversity gain. The factor 1/2 penalty in the


multiplexing gain comes from the fact that communicationtakes place over two time slots (“half-duplex protocol”). Thedependence of network capacity on SNR in (17) is throughthe quantities δi, αi, and βi, and has been absorbed in theO(1)-term.

Channel state information: We note that for our relayingprotocol to be asymptotically (in K) optimal (up to an O(1)-term) we need perfect knowledge of {Ek, Pk,Hk,Gk}Kk=1 ateach of the receive antennas in the general case, and perfectknowledge of hsigi and h

inti,j (j = 1, 2, . . . ,M, j = i) at the ith

receive antenna in the case of i.i.d. {Ek} and i.i.d. {Pk}. Inthe general case, perfect knowledge of {Ek, Pk,Hk,Gk}Kk=1is a sufficient condition for (13) to hold. More generally,we need to ensure that the conditions for the Lindeberg-Feller theorem [37, Theorem 1.9.2.A] to hold are satisfied.Acquiring knowledge of {Ek, Pk,Hk,Gk}Kk=1 at each of thereceive antennas will be challenging in practice and will leadto significant training overhead. Knowledge of the hsigi andhinti,j (j = 1, 2, . . . ,M, j = i) at the ith receive antennacan be obtained through standard multiple-input single-output(MISO) training schemes since the overall effective channelfor each receive antenna becomes a MISO channel. Acquiringknowledge of the relays’ backward channels can be donethrough standard training approaches; obtaining knowledgeof the forward channels is more challenging and equivalentto acquiring transmit CSI in point-to-point wireless links. Atfirst sight it may appear that accurate CSI at the relays iscritical for the scaling law (17) to hold. However, it wasshown recently [38] (for N = 1) that (17) remains validunder very mild conditions on the knowledge of the phasesof the backward and forward channels; knowledge of themagnitudes of the fading coefficients is not required. It wasfound, however, that the price to be paid for inaccurate channel(phase) knowledge at the relays is an increase (w.r.t. the perfectCSI case) in the number of relays needed to achieve a givennetwork capacity [38].

Distributed orthogonalization: Besides realizing multi-plexing gain, distributed array gain, and distributed diver-sity gain, the proposed relaying protocol also orthogonalizesthe effective MIMO channel between source and destinationterminals and hence multistream interference is completelyeliminated without cooperation between the relay terminals.We can therefore conclude that even though multistreaminterference at the relay terminals is not canceled (recallthat our result holds for any N ≥ 1) as was done in [39]through zero-forcing, in the large relay limit multistreaminterference cancellation is achieved in a completely decen-tralized fashion by exploiting differences in the (distributed)spatial signatures of the interfering data signals across therelay terminals. The resulting coherent combining of thedata signals at their intended destination terminal antennasraises the received power of the desired signal by a factorof K over the noise and interference (distributed array gain)so that asymptotically in K all M source-destination pairscan communicate as though only one pair was active. Sincethe overall MIMO channel is orthogonalized, single-antenna(i.e., independent) decoding at the receive terminal achievescapacity in the large K limit. This implies that there is no needfor sophisticated joint decoding schemes such as successive

cancellation [40], [41] or sphere decoding [42]. In contrast, inan M×M point-to-point MIMO link, achieving a multiplexinggain of M requires cooperation (i.e., joint processing) betweenthe receive antennas.

Relation to interference channels: Since no cooperationis required between antennas at the transmitter (spatial mul-tiplexing) and between antennas at the receiver (orthogonal-ization), the overall system can also be regarded as a networkwhere M single-antenna source destination pairs communicateconcurrently through a set of K common relays, as wasdone in [43]. The asymptotic (in K) sum-capacity of thisnetwork is given by (17). In the absence of relay termi-nals (assuming a direct link between source and destination)the overall system consisting of M single-antenna sourceterminals and M single-antenna noncooperating destinationterminals constitutes an interference channel [44]. Assum-ing that the output signals of this interference channel arestatistically equivalent [44], which is the case if the scalarchannels between the source and the destination terminalsare i.i.d., it follows that the sum capacity of this interferencechannel equals the sum capacity of any one of its multiple-access channels [44]. Consequently, in the absence of relaysthe lack of cooperation at the destination terminals resultsin a multiplexing gain of 1. We can therefore conclude thatfor M > 2 coherent relaying yields a higher multiplexing gainthan that obtained in the no-relay case.

Comments on path loss and shadowing: We concludethis section with a few comments on the assumptions madeon the scalar RVs Ek and Pk incorporating the effects oflarge-scale fading and path loss. Positivity of the Ek and Pkimplies that as the network grows large the relay terminalspopulate a domain of fixed area (i.e., the network is dense).While this assumption may seem restrictive, we emphasizethat the Ek and Pk can be arbitrarily small (but positive),which implies that the domain can be large. On the other hand,investigation of the upper bound in (3) reveals that increasingthe size of the domain will result in a decrease of μ =(1/K)

∑Kk=1 E{Ek} and hence in reduced network capacity.

The asymptotic growth rate, however, remains unchanged aslong as we have an infinite number of relay terminals withpositive Ek and positive Pk. Finally, the assumption of the Ekand Pk being bounded implies that the source and destinationterminals cannot communicate with the relay terminals in alossless fashion. Clearly, relaxing this assumption can onlyincrease network capacity.

D. Numerical Example

In the following, we provide a numerical result for M = 2,N = 1, and6 Ek/σ2n = Pk/σ

2n = 10dB (k = 1, 2, . . . ,K).

Fig. 5 shows the upper bound on coherent network capacityin Theorem 1 and the corresponding lower bound achievedby the matched-filtering based relaying protocol described inSection IV-A. While our relaying protocol can indeed be seento exhibit logarithmic capacity scaling in K , we can also seethat there is a significant gap between the bounds, which canbe attributed to the differences in the O(1)-term. This gap is

6Note that the Ek and Pk were chosen to be deterministic for simplicity.


independent of K (as seen in Fig. 5) and becomes insignificantonly for very large values of K .

V. NONCOHERENT MIMO RELAY NETWORKS

So far, we considered coherent relay networks, where eachrelay terminal knows its assigned backward and forwardchannels perfectly. In the following, we relax this assumptionand study networks with no CSI at the relay terminals, i.e.,noncoherent relay networks as defined in Section I-B. In par-ticular, we investigate a simple amplify-and-forward protocol,where relay terminals simply forward a scaled version of thereceived signal without additional processing.

A. The AF Protocol

We assume that the kth relay terminal knows the averagereceived signal plus noise energy Ek+σ2n at each of its N an-tennas and performs the normalization tk = (Ek+σ2n)

−1/2rk.This ensures that the power constraint7 E{‖tk‖2} = N issatisfied. It now follows from (2) that the signal received atthe destination terminal D is given by

y =K∑k=1

√Pk

N(Ek + σ2n)Gkrk + z. (18)

Inserting (1) into (2), dividing the RHS and left-hand side(LHS) of (18) by

√K, and simplifying, we get

y√K

=1√K

K∑k=1

(√PkEk

NM(Ek + σ2n)GkHk

)

︸︷︷︸A

s

+1√K

K∑k=1

(√Pk

N(Ek + σ2n)Gknk

)+

z√K︸︷︷︸

b

or equivalently y/√K = As+b. In the following, similar to

Theorem 2, again we need to distinguish the cases of arbitraryindependent Ek and Pk and the case of i.i.d. {Ek}Kk=1 andi.i.d. {Pk}Kk=1. In the general case, we need to assume that thereceiver has perfect knowledge8 of {Ek, Pk,Gk}Kk=1. In thei.i.d. case knowledge of the compound channel matrix A inthe receiver is sufficient. In order to simplify the exposition,we shall provide the detailed derivation for the i.i.d. case only.The general case requires minor modifications to establish thei.i.d. nature of the compound channel matrix A conditionedon perfect knowledge of {Gk}Kk=1. In the K → ∞ limit,applying the Lindeberg-Feller theorem [37, Theorem 1.9.2.A]and [37, Theorem 1.8.D] in conjunction with the Kolmogorovconditions

∞∑k=1

VAR(PkEkEk+σ2n

)k2


0 5 10 15 20 25 300

1

2

3

4

5

6

capa

city

(bit/

s/H

z)

number of relays

M = 2

M = 4

CAF∞

CAF∞

Fig. 6. Capacity vs. number of relays for the AF protocol.

not have channel knowledge. For N = 1, for example,each of the relay terminals induces an overall channelbetween the source and destination that is pin-hole [46]like (up to the forwarded noise contribution) and henceof rank 1 w.p. 1. This implies that each of the relayssupports only one spatial data stream between source anddestination. Asymptotically in K due to the independenceof the individual Hk and Gk across k, the collection ofrelays ensures that all M data streams are being “served”.This result demonstrates that employing relays as activescatterers can recover spatial multiplexing gain in poorscattering environments. Numerical evidence of this facthas been provided in [47].

• The absence of channel knowledge at the relay terminalsresults in a lack of distributed array gain reflected by thefact that C∞AF is independent of K . This is in stark con-trast to the coherent case, where a distributed per-streamarray gain of K was obtained. Note, however, that (20)implies traditional receive array gain and diversity gain(independent of K , i.e., there is no distributed diversitygain as in the coherent case) present in coherent point-to-point MIMO links (recall that the destination terminalknows the compound channel matrix A perfectly).

• Unlike the coherent case, the effective MIMO channelbetween S and D is not orthogonalized so that jointdecoding at the destination terminal is crucial to achievecapacity.

• The capacity result obtained for the noncoherent case isnot as strong as the result for the coherent case. Theexpression in (20) corresponds to the capacity achievedby a particular relaying strategy, namely amplify-and-forward. In contrast, in the coherent case, the cut-setbound provides the ultimate capacity upper bound achiev-able through any single-letter processing strategy at therelays.

B. Numerical Example

The purpose of this numerical example is to demonstratethat convergence (w.r.t. increasing K) to C∞AF depends crit-

ically on the number of source-destination terminal antennapairs M and is generally very fast. Fig. 6 shows the capacity(obtained through Monte Carlo simulation) of the AF protocoldescribed above as a function of the number of relays Kfor N = 1 and M = 2, 4 along with the large K ca-pacity limit as predicted by (20). For finite K , knowledgeof {Ek, Pk,Hk,Gk}Kk=1 is assumed at the destination ter-minal, rendering the noise term conditionally Gaussian. Wefurthermore assume that Ek/σ2n and Pk/σ

2n are drawn inde-

pendently from a truncated log-normal distribution (modelingmacroscopic fading) with mean 10 dB, standard deviation (ofthe underlying log-normal distribution) 4 dB, truncated suchthat Pk/σ2n and Ek/σ

2n are bounded above by 15 dB and

below by 5 dB. We can see that the capacity of the AF protocolconverges very quickly to C∞AF and that increasing M requireslarger K to achieve the same fraction of (the larger) C∞AF.Fig. 6 also shows that for small K the capacity increaseslinearly in K , which is due to the fact that the correspondingeffective channel matrix A is building up rank and hence mul-tiplexing gain (reflected by a pre-log increase). For large K ,once A is full-rank with high probability, the curve flattensout and increasing K mainly has the effect of rendering theelements of A Gaussian.

VI. CONCLUSIONS

We studied capacity scaling laws in MIMO relay networksunder two-hop half-duplex relaying. For M antennas at thesource and destination terminals and perfect channel stateinformation at the relays, we showed that asymptoticallyin the number of relay terminals K , independently of thenumber of relay terminal antennas N , network capacity scalesas C = (M/2) log(K) +O(1). In particular, we showed thatthis capacity can be realized by relay partitioning, backwardand forward matched filtering at the relay terminals, andindependent stream decoding (thanks to distributed orthogo-nalization) at the destination terminal. This is possible becausethe relays orthogonalize the effective MIMO channel in adistributed fashion. For noncoherent networks, we investigateda simple amplify-and-forward protocol and showed that forany N ≥ 1, the large K capacity limit is given by C =(M/2) log(SNR)+O(1). In the noncoherent case, cooperationbetween the receive terminal antennas is crucial to achievecapacity.

The aim of this paper was to introduce the idea of distrib-uted multistream interference cancellation and to study theimpact of CSI at the relay terminals on the network capacityscaling behavior. A number of interesting topics remain opensuch as: (i) the investigation of more sophisticated relayingstrategies such as successive cancellation and Costa precodingin the coherent case, and applying nonlinear functions in thenoncoherent case possibly also exploiting the spatial nature ofthe channel; (ii) an analytic investigation of the convergenceproperties of network capacity in the large K limit, and inparticular the impact of (i) on this convergence behavior;(iii) the extension to more general channel models, such asfrequency-selective fading.


VII. ACKNOWLEDGMENT

The authors would like to thank S. Visuri for bringingreference [37] to their attention.

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[47] A. Wittneben and B. Rankov, “Impact of cooperative relays on thecapacity of rank-deficient MIMO channels,” in Proc. 12th IST Summiton Mobile Wireless Communications, Aveiro, Portugal, June 2003, pp.421–425.


Helmut Bölcskei (M’98–SM’02) was born in Aus-tria on May 29, 1970. He received the Dr. techn.degree in electrical engineering from Vienna Uni-versity of Technology, Vienna, Austria, in 1997.

In 1998, he was with Vienna University of Tech-nology. From 1999 to 2001, he was a PostdoctoralResearcher with the Information Systems Labora-tory, Department of Electrical Engineering, StanfordUniversity, Stanford, CA. During that period, hewas also a Consultant for Iospan Wireless, Inc., SanJose, CA. From 2001 to 2002, he was an Assistant

Professor of Electrical Engineering at the University of Illinois at Urbana-Champaign, Urbana. Since February 2002, he has been an Assistant Professorof Communication Theory at the Swiss Federal Institute of Technology(ETH) Zurich, Zurich, Switzerland. He was a Visiting Researcher at PhilipsResearch Laboratories, Eindhoven, The Netherlands, ENST Paris, France, andthe Heinrich Hertz Institute, Berlin, Germany. His research interests includecommunication and information theory with special emphasis on wirelesscommunications and signal processing.

Prof. Bölcskei received a 2001 IEEE Signal Processing Society YoungAuthor Best Paper Award, the 2006 IEEE Communications Society LeonardG. Abraham Best Paper Award, and was an Erwin Schrödinger Fellow (1999–2001) of the Austrian National Science Foundation (FWF). He served as anAssociate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING, theIEEE Transactions on Wireless Communications, and the EURASIP Journalon Applied Signal Processing. Currently he is on the Editorial Board ofFoundations and Trends in Networking.

Rohit U. Nabar (S’02–M’03) was born in Bombay,India on Dec. 18, 1976. He received the B.S. degree(summa cum laude) in electrical engineering fromCornell University, Ithaca, NY in 1998 and the M.S.and Ph.D. degrees in electrical engineering fromStanford University, Stanford, CA in 2000 and 2003respectively. At Stanford, he was a member of theSmart Antennas Research Group and a recipient ofthe Dr. T. J. Rodgers Stanford graduate fellowship.His doctoral research focused on signaling for real-world MIMO channels.

From 2002 to 2004, Dr. Nabar was a postdoctoral researcher in theCommunication Theory Group at the Swiss Federal Institute of Technology(ETH) in Zurich, Switzerland. Between 2004 and 2005, he was a facultymember in the Communications and Signal Processing Research Group atthe Department of Electrical and Electronic Engineering, Imperial College,London, U.K. Since July 2005, Dr. Nabar has been a member of theDSP technology group at Marvell Semiconductor, Inc. in Santa Clara, CA.His research interests include signal processing and information theory forwireless communications and networks.

Dr. Nabar currently serves as an Associate Editor for the IEEE COMMU-NICATIONS LETTERS. He has coauthored (with A. Paulraj and D. Gore) thetextbook “Introduction to Space-Time Wireless Communications,” publishedby Cambridge University Press, U.K. in 2003.

Özgür Oyman was born in Istanbul, Turkey, onMarch 5, 1979. He received the B.S. (summa cumlaude) degree in electrical engineering from CornellUniversity, Ithaca, NY in 2000 and the M.S. andPh.D. degrees in electrical engineering from Stan-ford University, Stanford, CA, in 2002 and 2005,respectively. He is currently a research scientist inthe Corporate Technology Group at Intel Corpora-tion in Santa Clara, CA, USA.

During his doctoral studies, he worked with theSmart Antennas Research Group within the Infor-

mation Systems Laboratory at Stanford University. He was a visitor at theCommunication Theory Group at the Swiss Federal Institute of Technology(ETH) in Zürich, Switzerland in summer 2003. His work experience includestwo summer internships at Qualcomm Inc. (2001) and Beceem Communica-tions Inc. (2004). His research interests are in communication and informationtheory for wireless communications with special focus on multiple-inputmultiple-output (MIMO) antenna systems and adhoc wireless networks.

He was the recipient of the Benchmark Stanford Graduate Fellowship. Heis a member of Tau Beta Pi, Eta Kappa Nu and the IEEE.

Arogyaswami J. Paulraj (F’91) is a Professor in theDept. of EE, Stanford University, where he super-vises the Smart Antennas Research Group workingon applications of space-time techniques for wirelesscommunications.

Paulraj was educated at the Naval EngineeringCollege and the Indian Institute of Technology, India(Ph.D. 1973). His nonacademic positions includedHead, Sonar Division, Naval Oceanographic Labo-ratory, Cochin; Director, Center for Artificial Intelli-gence and Robotics, Bangalore; Director, Center for

Development of Advanced Computing; Chief Scientist, Bharat Electronics,Bangalore, and Chief Technical Officer and Founder, Iospan Wireless Inc.

Paulraj’s research has spanned several disciplines, emphasizing estimationtheory, sensor signal processing, parallel computer architectures/algorithmsand space-time wireless communications. His engineering experience includeddevelopment of sonar systems, massively parallel computers, and morerecently broadband wireless systems.

Paulraj has won several awards for his engineering and research contribu-tions. He is the author of over 280 research papers and holds eight patents.Paulraj is a Member of the National Academy of Engineering, a Fellow ofthe Institute of Electrical and Electronics Engineers (IEEE) and a Member ofthe Indian National Academy of Engineering.

Documents

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, … · IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006 1433 Capacity Scaling Laws in MIMO Relay Networks