IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 4

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 4, APRIL 2012 2117

On Capacity Scaling of Multi-Antenna Multi-HopNetworks: The Significance of the Relaying

Strategy in the “Long Network Limit”Jörg Wagner, Student Member, IEEE, and Armin Wittneben, Member, IEEE

Abstract—The sum-capacity of a static uplink channel withsingle-antenna sources and an -antenna destination is known toscale linearly in , if the random channel matrix fulfills the con-ditions for the Marcenko-Pastur law: if each source transmits atpower , there exists a positive , such that

almost surely. This paper addresses the question to which ex-tent this result carries over to multi-hop networks. Specifically, an

-hop network with non-cooperative source antennas, fullycooperative destination antennas, and relay stages of (coop-erative or non-cooperative) relay antennas each is considered. Fourrelaying strategies are assessed based on the interrelationship be-tween two sequences. For each considered strategy , there existsa sequence , such that almostsurely, where denotes the supremum of the set of sum-ratesthat are achievable by the strategy over hops. This sequencedepends on the sequence , where corresponds to thepower of the source stage and each of the relay stages in an -hopnetwork. Results are summarized as follows:

• Decode & forward (DF): For , is constant withrespect to , if also is constant with respect to .

• Quantize & forward with (CF) and without (QF) Slepian & Wolfcompression: For , there exists a sequence ,such that is positive and constant with respect to .The corresponding sequence grows linearly withfor CF and exponentially with for QF.

• Amplify & forward (AF): Fix and . Then,(i) there exists , such that , if

and (ii) , if .

Index Terms—Amplify & forward, capacity scaling, compress &forward, decode & forward, MIMO uplink, multi-hop, multiple-input multiple-output, quantize & forward, random matrix theory,Slepian and Wolf compression.

I. INTRODUCTION

C ONSIDER wireless transmission from transmit an-tennas to an -antenna receive terminal over a random

and static frequency-flat channel. Assume that each transmitantenna transmits with power , where corresponds tothe sum-transmit power. If the channel coefficients betweenall pairs of transmit and receive antennas are identically and

Manuscript received May 27, 2010; revised October 09, 2010; accepted June15, 2011.

J. Wagner was with the Communication Technology Laboratory, ETHZurich, Sternwartstrasse 7, 8092 Zurich, Switzerland. He is now withSwisscom AG, Alte Tiefenaustrasse 6, 3050 Bern, Switzerland (e-mail:[email protected]).

A. Wittneben is with the Communication Technology Laboratory,ETH Zurich, Sternwartstrasse 7, 8092 Zurich, Switzerland (e-mail: [email protected]).

Communicated by L. Zheng, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2011.2177752

Fig. 1. non-cooperating source antennas transmit to a destination terminalwith antennas via stages of relay antennas. Relay antennas are coop-erative or non-cooperative depending on the relaying strategy.

independently distributed (i.i.d.) random variables with zeromean and nonzero variance, (sum-)capacity scales linearly in

almost surely, irrespective of whether transmit antennas cancooperate or not [1]. More precisely, the capacity of anpoint-to-point multiple-input multiple-output channel withwhite transmit covariance matrix, coincides with the sum-ca-pacity of an uplink channel with single-antenna transmitterminals and an -antenna receive terminal [2].

Envision the scenario that the transmit antennas are shad-owed from the receive terminal. Wireless connectivity can thenbe sustained through the installation of properly positioned in-termediate nodes that relay the source signals to the destina-tion via multiple hops (see Fig. 1). If the number of antennas ineach relay stage, , grows linearly with , then also the corre-sponding sum-capacity scales linearly in for any fixed numberof relay stages, . This result holds true even for non-cooper-ative relay stages1 [3]. Thus, as a generalization of the resultthat transmit antenna cooperation is not crucial in multi-antennasingle-hop networks, neither source nor relay antenna coopera-tion is crucial for linear sum-capacity scaling in multi-antennamulti-hop networks.

The above statement says nothing about the asymptotic con-stant of proportionality that fulfills almost surely

(1)

where denotes the supremum of the set of sum-rates thatare achievable through a certain relaying scheme “XF” in an

-hop network, except for the fact that it is strictly positive

1The term non-cooperative relay stage refers in this paper to the scenario thatjoint processing of receive signals of antennas within a stage is disabled, i.e.,each relay antenna is associated with a single-antenna node. Likewise, a coop-erative relay stage can be viewed as a single multi-antenna node.

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2118 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 4, APRIL 2012

for every . In particular, it does not preclude the scenario thatas . Accordingly, the central question of this

paper is, how evolves2 for increasing under various re-laying strategies. The provided answers reveal fundamental dif-ferences in this asymptotic behavior with respect to , not onlybetween cooperative and non-cooperative relaying strategies,but also between different non-cooperative strategies. Specifi-cally, the following four relaying strategies are investigated.

• Decode & forward: For this strategy, full cooperationamong antennas within relay stages is assumed. Relaystages can then decode messages from their precedingstage efficiently, and re-encode them prior to the for-warding. Thus, messages are regenerated in every relaystage and propagate from stage to stage until they reach thedestination. This strategy is optimal in multi-hop networksaccording to the data processing inequality [4].

• Pure quantize & forward (QF): This strategy can be exe-cuted in a completely decentralized fashion, i.e., withoutany relay antenna cooperation. The receive signal of eachrelay antenna is quantized. The index of the quantizationcodeword is then encoded and forwarded. For decoding,the destination recursively recovers the quantized relay re-ceive signals of each stage, until it can decode the sourcemessages based on the quantized receive signals of the firstrelay stage.

• Quantize & forward with Slepian & Wolf compression(CF): The above quantize & forward strategy QF can beenhanced through Slepian and Wolf compression [5] ineach relay stage. This additional step exploits the spa-tial correlation among relay receive signals, but requiresdissemination of channel state information within relaystages.

• Amplify & forward: This strategy operates in a completelydecentralized fashion, too. The receive signal of each relayantenna is amplified by a constant gain factor prior to trans-mission to the succeeding relay or destination stage. Thus,end-to-end transmission from source to destination stageoccurs over an equivalent single-hop channel. The strategyis particularly appealing due to its simplicity and low com-plexity.

In order to ensure a fair comparison, the sequencemust be considered together with a second sequence ,whose element corresponds to the per-stage transmit powerthat is allocated to the source antenna stage and each of the relaystages in an -hop network.

The results of this paper are outlined as follows:• Decode & forward: For , is constant with

respect to , if also is constant with respect to .• Quantize & forward with Slepian & Wolf compression:

For , there exists a linearly increasing sequence, such that is constant with respect to . This

is the best among the non-cooperative strategies.• Pure quantize & forward: For , there exists an

exponentially increasing sequence , such thatis constant with respect to .

2In this paper, the focus is not on the pre-log factor that is obviously incurred,if the source stage does not inject new signals into the network in every timeslot. Accordingly, it is not taken into account in and set to one.

• Amplify & forward: The convergence asis avoided for a linear growth of with , if and only ifalso the ratio grows at least linearly with .

Remarks: It is not surprising, that a constant does notsuffice for a constant , if a non-cooperative strategy is ap-plied. This is an immediate consequence of the inherent noiseaccumulation, which needs to be compensated by an increasedtransmit power. Moreover, the obtained results do not allow fora fair comparison between amplify & forward and pure quantize& forward. No statement is made about for and anexponential growth of . Vice versa, no statement is made onhow can benefit from more than relay nodes per stage.

Related Work: Multiple-input multiple-output communica-tion over multiple hops has received a lot of attention recently.[3] establishes linear sum-capacity scaling in (the numberof antennas per stage) of amplify & forward multi-hop net-works for finite numbers of hops. This work is a generaliza-tion of a result on two-hop networks in [6]. [7] considers asystem that corresponds to an amplify & forward multi-hopnetwork with noiseless relays. For this network, it is shownthat the asymptotic constant of proportionality between sum-capacity and the number of nodes per stage (assumed not togrow with the number of hops) tends to zero as the numberof hops grows large. [8] studies the same setting for a finitenumber of hops with correlated fading and cooperative relay an-tennas that not only amplify, but also linearly combine the re-ceive signals within a stage. In contrast to the above works, [9]studies multi-hop networks for finite in terms of the achiev-able degrees of freedom in the limit of infinitely many hops.Moreover, the works [10] (decode & forward) and [11] (amplify& forward) study the diversity-multiplexing tradeoff of multi-antenna multi-hop networks. Multi-antenna multi-hop systemswith non-cooperative destination antennas have been studiedwith respect to achievable degrees of freedom in [12] and [13].

Notation: For the matrix , , , andstand for its determinant, Hermitian transpose, trace and nuclearnorm. Moreover, and denote its eigenvalueand the singular value (in descending order). Throughoutthe paper, all logarithms, unless specified otherwise, are to thebase 2. For the vector , denotes its Euclidean norm and

its element. The probability of the event is denotedby and the expectation with respect to the random variable

by . denotes the complement of set . Furthermore,the standard notation is used for the characteri-zation the asymptotic behavior of some function accordingto

Finally, the function is defined to be 1, if is true, andzero otherwise. The functions and denote Dirac deltaand Heaviside step, respectively.

Outline: Section II introduces the signal model and the ap-plied multi-hop communication protocol. Section III presents indetail the relaying strategies that are considered. Section IV con-stitutes the core of this work and provides four Theorems that

WAGNER AND WITTNEBEN: ON CAPACITY SCALING OF MULTI-ANTENNA MULTI-HOP NETWORKS 2119

characterize the scaling of the supremum of the set of achiev-able rates under each relaying strategy.

II. SIGNAL MODEL AND COMMUNICATION PROTOCOL

A stage of single-antenna source nodes, , aims totransmit data to a destination node that has access to the receivesignals of a cluster of antennas. Communication isassisted by relay stages of antennas each (see Fig. 1).Two cases are distinguished:

• Fully cooperative relay stages: All relay antennas in a stageare connected to a central node. This central node has ac-cess to the receive signals of all antennas in the stage, and,based on this knowledge, determines the transmit signalsof the relay antennas.

• Non-cooperative relay stages: Each relay antenna corre-sponds to a single node that has to determine its transmitsignal solely based on the knowledge of its own receivesignal.

The relay stages are labeled by . More-over, the antenna in a source, relay or destination stageis labeled by , and , respectively. The networkemploys a multi-hop protocol. More precisely, source signalstraverse all relay stages in descending order of indexes, i.e.,

, before they are received by the destinationstage. Transmission is divided into time slots, one dedi-cated to each hop, of symbol durations each. That is, trans-missions of different stages are orthogonal in time. Specifically,

• stage transmits to stage in time slot ,• stage transmits to stage in time slot ,

where ,• stage transmits to stage in time slot .

Channels between any two antennas are quasi-static and fre-quency-flat over the bandwidth of interest. The channel coef-ficient that corresponds to the link between transmit nodeand receive node is denoted by . With this notation, thechannel matrices are written as

if

if

if

(2)Furthermore, the sequence of signals that is transmitted byantenna in its dedicated transmit time slot is denoted by

. The vector of transmit signals of antennas within astage is denoted by

(3)

Likewise, the sequences of receive and additive noise signals,as they are observed by antenna in its dedicated receive timeslot, are denoted by and , respectively. Thereceive signal and noise vectors of a stageare denoted by

(4)

(5)

The transmission in time slot is thus described by the input-output (I-O) relation

(6)

(7)

(8)

Note that transmit antennas within the same stage are assumedto be symbol-synchronized.

The elements of the vectors , , andrepresent the thermal noise that is introduced at the receiverfront-ends. They are i.i.d. (both in space and time) circularlysymmetric complex Gaussian random variables of variance .The elements of the channel matrices ,are i.i.d. random variables that follow an arbitrary distributionwith zero-mean, unit variance and bounded fourth moment. Thetransmit signals of all antennas in source and relay stages aresubject to an average power constraint3. Specifically, source andrelay signals must fulfill almost surely

(9)

(10)

where corresponds to the sum-power of each stage. The pa-rameter will be treated as a sequence in .

It remains to specify, how each relay stage determines itstransmit signals from its receive signals, i.e., a map

(11)

where denotes the set of transmit signal vector sequences ofrelay stage . In the case of non-cooperative relay stages, thetransmit signal sequence of each antenna must solely depend onthe corresponding receive signal sequence of the antenna. Thatis, the map decouples into the following maps:

(12)

where denotes the set of transmit signals of relay .Three fundamentally different relaying techniques are investi-gated: (i) decode & forward, (ii) quantize & forward (with anoptional Slepian and Wolf compression), and (iii) amplify & for-ward. While the decode & forward strategy is optimal in termsof sum-capacity as an immediate consequence of the data pro-cessing inequality [4], it requires fully cooperative relay stages.The other forwarding strategies are known to be suboptimal, butdo not require cooperative relay stages. For all relaying strate-gies, transmitting nodes are assumed not to possess any transmitchannel state information. The amount of receive channel state

3This power constraint will be slightly relaxed in the case of amplify & for-ward relaying (see Section III-C).


TABLE IRECEIVE CHANNEL STATE INFORMATION

TABLE IIRATE FEEDBACK

information in the relay stages and at the destination node isspecified in Table I for each scheme individually. Moreover,transmitting nodes require certain rate feedback, which is pro-vided through perfect feedback links from the destination. Thisrate feedback is summarized in Table II for each scheme.

The three relaying techniques and corresponding achievablerates are revisited in the context of the considered multi-hopnetwork in the next section.

III. MULTI-HOP RELAYING TECHNIQUES

In the following, each source node chooses randomlyaccording to a uniform distribution a message out of themessage set with messages for transmission.The channel codebook of node has rate and is denotedby . Furthermore, for each node an encoding functionis defined as

(13)

A. Decode & Forward Relaying

Although decode & forward relaying can be applied to net-works with arbitrary , and , attention is restricted to thecase in this work. Decode & forward re-laying refers to the technique of decoding and re-encoding mes-sages in each relay stage. In order to enable coherent decodingin all relay stages and at the destination node, each relay stageas well as the destination node is assumed to possess perfectchannel state information of its preceding hop. The maps (11)can be split into two parts each, i.e., . First,the messages—one corresponding to each source node—astransmitted by the preceding stage are decoded based on the ob-

served sequence and the knowledge of the channelmatrix . The corresponding decoding function is

(14)

Then, the decoded messages are re-encoded as

(15)

where denotes the channel codebook of relay stage . Therate of this channel codebook is given by

for all .A sum-rate is achievable with the decode & forward

strategy, if it is supported by each individual hop. For thefirst hop between the source stage and the fully cooperativerelay stage , transmission occurs over an equivalent uplinkchannel with single-antenna transmit terminals and an -an-tenna receive terminal. A sum-rate is achievable over thefirst hop channel [2], if

(16)

Note that this work focuses on the achievable sum-rate anddoes not care about the individual rates . For all followinghops, transmission occurs over equivalent point-to-point chan-nels, since both transmit and receive stage are fully cooperative.Relay stage , , can transmit reliably to its suc-ceeding stage at a rate , if [1]

(17)

Here, a spatially white transmit covariance matrix is used dueto the absence of transmit channel state information. In order toachieve over the end-to-end channel, (16) and(17) need tobe fulfilled for . This corresponds to thecondition

(18)

B. Quantization-Based Relaying

Although quantization-based relaying can be applied to net-works with arbitrary , and , attention is restricted tothe case in this work. In this non-co-operative relaying scheme the maps , , asdefined in (12), are implemented in two steps, i.e.,

. In a first step, each relay quantizes its re-

ceive sequence . The quantized sequence is denoted

by , where denotes the quantiza-

tion codebook of the relay. The sequence of additive quantiza-

tion noise is defined as , the

quantization noise vector of stage is denoted by

(19)

The corresponding map is

(20)

where the quantization codebook has rate .


In a second step, the index of the quantized sequence is en-coded by the channel coder, i.e., mapped onto a codeword ofthe channel codebook whose rate is larger than or equal

to

(21)

The decoding in the destination stage is then performed ina successive fashion. In a first step, the messages sent by therelay stage are decoded based on the sequence of receivevectors and the knowledge of . Note that the trans-mission from to occurs over an uplink channel withsingle-antenna transmit terminals and a receive terminal with

antennas. Since the functions are not injective, and thusnot invertible, it is impossible to obtain a perfect reconstructionof the sequence of receive vectors . This ambiguityis accounted for by the sequence of additive quantization noisevectors . The element in the sequence of quantizedreceive vectors is then written in terms of the elementin the corresponding sequence of transmit vectors as

(22)

With the knowledge of the sequence and , the des-tination proceeds with decoding the messages of the nodes in

. These messages can be considered as being transmitted overa virtual uplink with single-antenna transmit terminals and areceive terminal with antennas.

Proceeding this way iteratively allows for tracing backthrough the relay chain stage by stage based on the sequencesof quantized receive vectors and the knowledge of the respec-tive channel matrix. In the iteration, the decoder obtains thesequence of quantized receive vectors by decodingthe messages of . These messages are transmitted over avirtual uplink channel with single-antenna transmit terminalsand a receive terminal with antennas. The effective I-Orelation between and can thus be written as

(23)

where is the element in the respective sequence of quan-tization noise vectors. In the -st iteration the decoder fi-nally arrives at the source stage, whose messages are transmittedover a virtual uplink channel with single-antenna transmit ter-minals and a receive terminal with antennas. They are decodedbased on the effective I-O relation

(24)

and the knowledge of and . The channel and quan-tization codebooks that are used later on render the quantiza-tion noise vectors , , i.i.d. (both in space andtime) circularly symmetric complex Gaussian distributed withvariance . Thus, for the end-to-end channel from to , a

sum-rate rate , where denotes the rateof the channel codebook of , is achievable, if it fulfills

(25)The rate of the channel codebook of relay node is de-noted by in the following. In this work, rates of channelcodebooks are required to coincide within each stage , i.e.,

. The rate will in each case deter-mine the quantization noise variance . For given the quanti-zation noise variances , , where , therate is achievable, if [1], [2]

(26)

Here, denotes the matrix, that collects allcolumns of that correspond to the nodes contained in .Moreover, the supremum of the set of rates that fulfill (26) isdenoted by in the sequel.

In order to achieve these rates, the source and relay nodesneed to generate their channel codebooks by choosing the en-tries and as independent realizations of circularly

symmetric complex Gaussian random variable and , re-spectively. The variances of these random variable are chosen tofulfill the average power constraints (9) and(10) with equality,i.e., and .

In the following, the quantization codebooks of the relaynodes and the resulting quantization noise variances are speci-fied. In this context, “pure quantization” and “quantization withSlepian and Wolf compression” is distinguished.

Pure Quantization: Pure quantization refers to a receivesignal quantization method that does not exploit the statisticalcorrelation among receive signals of relay nodes within thesame stage. Each relay node generates a quantizationcodebook of rate . The elements of the quantiza-tion noise vector are rendered i.i.d. circularly symmetriccomplex Gaussian with variance by choosing the entriesof the quantization codebook as statistically independent real-izations of a circularly symmetric complex Gaussian randomvariable of variance . Here, denotesthe average power of the desired part of the receive signal atnode . A codeword is declared to be the quantization ofthe observed sequence, if both are jointly typical. In order toensure that a quantization codeword for the observed sequenceis found based on a joint typicality check with probabilityone as , the mutual information between original andquantized observation of node must fulfill

(27)

or equivalently

(28)


This inequality must be fulfilled for all nodes in . Moreover,the rates of the quantization codebooks must be smallerthan or equal to the rate of the channel codebook, . For agiven , a quantization noise variance is achievable,if it fulfills

(29)

Quantization With Slepian and Wolf Compression: A moreefficient relaying method performs a Slepian and Wolf compres-sion of the receive signal upon quantization. Thus, the correla-tion in the receive signals of relay nodes within the same stagecan be efficiently exploited. As in the case of pure quantiza-tion, the elements of the quantization noise vector are ren-dered i.i.d. circularly symmetric complex Gaussian with vari-ance . The rate of the compressed quantization codebook ofrelay node is denoted by in the sequel. Rates of quan-tization codebooks within each stage are required to coin-cide, i.e., . The compression problemat hand has been studied in [14] in the context of a two-hop setupwith orthogonal second hop. The quantized observation vectorcan be reconstructed with probability one as , if for all

(30)

For a given quantization noise variance, the conditional mutualinformation expression can be written as (refer to the proof ofLemma 3)

(31)

(32)

Here, denotes the matrix that collects all rows

of which correspond to the nodes contained in . Again,the rate of the quantization codebook of each node is requiredto be smaller or equal to the rate of the channel codebook of thenode, i.e., . For a given , a quantization noisevariance is achievable, if it fulfills (30) for

(33)

C. Amplify & Forward

Amplify & forward refers to the technique of deriving atransmit signal from the receive signal through simple ampli-fication. This work focuses on the case that each relay withinstage applies the same gain factor , andsufficiently small, such that (10) is fulfilled. That is, asdefined in (12) is given by

(34)

Equivalently, the relay receive and transmit vectors are relatedas follows:

(35)

where the second equality follows from (6) and(7), respectively.The effective I-O relation from source to destination stage isthen obtained through recursive derivation of fromaccording to (35), and given by

(36)

This I-O relation represents an -user uplink channel withreceive antennas. The sum-capacity for a fixed channel realiza-tion is achieved by a Gaussian codebook, i.e., the are i.i.d.circular symmetric complex Gaussian with variance .Under this input-distribution a sum-rate is achievable, if itfulfills [1]

(37)

where and denote the covariance matrices of de-sired receive signal and accumulated noise at the destination,respectively

(38)

(39)

In the sequel, is chosen as . This choicedoes not fulfill the power constraints (10) exactly, but in thefollowing sense:

Proposition 1: Let andbe statistically independent random


matrices, each with i.i.d. elements of zero-mean, unit-varianceand bounded fourth moment, and fix the ratios and

. If , then• the sum-transmit power of each stage fulfills

(40)

• for every fixed and every stage ,

(41)

Remarks: Note that this proposition does not imply for agiven that

(42)The proof of the proposition is provided in Appendix C. Sinceit relies on notation and concepts introduced in Section IV-C, itis recommended to return to this proposition and its proof later.

IV. CAPACITY SCALING

This section establishes for each of the introduced relayingschemes the capacity scaling result as outlined above. Four the-orems, one for each relaying scheme, are provided and proved.Decode & forward is examined in Section IV-A, quantize &forward without and with Slepian and Wolf compression inSection IV-B-I and Section IV-B-II, and amplify & forward inSection IV-C.

A. Decode & Forward Networks

The following theorem characterizes the scaling of thesupremum of achievable sum-rates in decode & forwardmulti-antenna multi-hop networks:

Theorem 1: Let be statistically in-dependent random channel matrices, each with i.i.d. elementsof zero-mean, unit-variance and bounded fourth moment. Then,the supremum of the set of sum-rates that are achievable by thedecode & forward strategy, , fulfills for all

(43)

where

(44)

Note that the constant of proportionality that asymptoticallyrelates and is constant with respect to , if is constantwith respect to .

Proof of Theorem 1: Under the assumptions of the theoremthe following holds for every according to [15],[16] (see also [17, p. 10]):

(45)

Thus, there exists almost surely for every and arbitraryan , such that for all

(46)

For the supremum of the set of achievable sum-rates (18), thisimplies

(47)

This establishes the theorem.

B. Quantize & Forward Networks

Quantize & Forward Without Slepian and Wolf Compres-sion: The following theorem characterizes the scaling of thesupremum of achievable sum-rates in quantize & forwardmulti-antenna multi-hop networks that do not apply Slepianand Wolf compression:

Theorem 2: Let be statistically in-dependent random channel matrices, each with i.i.d. elementsof zero-mean, unit-variance and bounded fourth moment. Then,the supremum of the set of sum-rates that are achievable by thequantize & forward strategy, , fulfills for all

(48)

where , and is defined in (44).Moreover, the per-stage transmit power that is required forrendering constant with respect to grows exponentiallywith .

Note that the constant of proportionality that asymptoticallyrelates and is constant with respect to , if the SNR atthe destination stage, , is constant with respect to .

Proof of Theorem 2: It has to be shown that there existsa sequence , such that the SNR at the destination an-tennas, given by , is kept constant andbounded away from zero as . Then, the supremum of theset of achievable sum-rates as given by (25) fulfills [15], [16]

(49)

independently of . Since , this limit is strictlypositive.

First, a result on the multiple-input multiple-ouput (MIMO)uplink channel with single-antenna sources and an antenna


destination is stated. Under the assumptions on the fading distri-butions of the theorem, all sources can, in the limit of large , si-multaneously achieve a fraction of the supremum of achiev-able sum-rates. More precisely, the following lemma, whoseproof is given in Appendix A, holds:

Lemma 1: Consider an uplink channel with single-antennatransmit terminals with transmit power each, and an

-antenna receive terminal with spatial noise covariance matrix. The elements of the channel matrix are

distributed according to the assumptions of Theorem 2. Let

(50)

Then, there exists almost surely for every rate an ,such that for all the rate tuple

, where denotes the rate of the transmitterminal, is achievable. This lemma implies that in the limit oflarge the set of achievable is fully determined by the con-straint in (26) that corresponds to the set , since [15],[16]

(51)

where

(52)

Thus, there is for every fixed and given sequencealmost surely an , such that for all all rates

, , are achievable simultaneously, iffor all .

The next step, relies on the fact that as , the receivepower of all receive antennas in the various relay stages con-verges almost surely to . To make this precise, the fol-lowing lemma, whose proof is given in Appendix A, is stated:

Lemma 2: Let be a random matrix whose ele-ments are distributed according to the assumptions of Theorem2 and . Denote by the row of . Then

(53)

Thus, the in (29) fulfill for every fixed

(54)From the conclusions of Lemmata 1 and 2, we infer that there

is for every fixed almost surely an , such that for allthe sequence of quantization noise variances is

achievable according to (29), if for all

(55)

where the , , denote the infima ofachievable noise variances. Substitution of the asymptoticsuprema of achievable rates (52) into(55) yields a first-orderdifference equation in with . This difference equationis the starting point for the proof that any SNR value, ,can be sustained at the destination antennas for increasing byincreasing appropriately.

We apply the inequality4

(56)

which holds for , to upper-bound in (55) asfollows:

(57)

Hence, we conclude that the sequence of quantization noisevariances that is characterized by the following differ-ence equation is achievable almost surely in the limit :

(58)

The solution to this first-order difference equation is given by

(59)

Thus, for sustaining a certain SNR, ,and need to be coupled as follows:

(60)This implicit equation corresponds to an exponential growth of

with , since

(61)

In order to show that an exponential growth of with isnecessary for sustaining a constant destination SNR, a lower-

bound on , as defined in (55), is developed. There ex-ists for every destination SNR value a

, such that for arbitrarily large

(62)

4This inequality follows, since both sides evaluate to zero for, and the slope of the left-hand side is strictly larger than the

slope of the right-hand side for all


where . In order to prove this, is chosenas5

(65)

This is larger than one for positive , since takes on valueson the interval

(66)

(67)

(68)

Hence, we conclude that

(69)

With , this yields the following lower-bound on

:

(70)

That is, the power that is required for sustaining a constantSNR, , is lower-bounded according to

(71)

This lower-bound implies an exponential growth of the requiredwith .

Quantization With Slepian and Wolf Compression: Thefollowing theorem characterizes the scaling of the supremumof achievable sum-rates in quantize & forward multi-antennamulti-hop networks that apply Slepian and Wolf compression:

Theorem 3: Let be statistically in-dependent random channel matrices, each with i.i.d. elementsof zero-mean, unit-variance and bounded fourth moment. Then,the supremum of the set of sum-rates that are achievable by the

5Here, is the minimizer, since for every , andis monotonically increasing in

(63)

The positiveness of the derivative follows, since for ,and thus

(64)

quantize & forward strategy with additional Slepian and Wolfcompression, , fulfills for all

(72)

where , and is defined in (44).Moreover, the per-stage transmit power that is required forrendering constant with respect to grows linearly with .

That is, the constant of proportionality that asymptoticallyrelates and is constant with respect to , if the SNR atthe destination stage, , is constant with respect to .

Proof of Theorem 3: The proof is along the lines of theproof of Theorem 2. It has to be shown that there is for every

a , such that the SNR at the destination antennas, given by, is kept constant and nonzero. Then, the

supremum of the set of achievable sum-rates, as defined in (25),fulfills [15], [16]

(73)

independently of . This limit is strictly positive for positive.

Again, Lemma 1 implies that for every fixed and givensequence there exists almost surely an , such thatfor all all rates , , are achievablesimultaneously, if for all .

Next, the quantization noise variances are evaluated. The fol-lowing lemma serves as a starting point:

Lemma 3: Let be a Gaussian random vector with zero-mean and covariance matrix , where

is distributed according to the assumptions of The-orem 3 and . Let be the quantization of , which isobtained as , where the quantization noise vector

is a Gaussian random vector with zero-mean and covariancematrix . Let

(74)

where is defined in (44). Then, there exists for everytuple of rates of compressed quantization codebooks

with al-most surely an , such that for all the quantizationnoise variance is achievable in the sense of (30).

This lemma implies that, in the limit of large , the set ofachievable is fully determined by the constraint in (30) thatcorresponds to the set , since [15], [16]

(75)

where

(76)

Thus, there is for every fixed and given sequence ,where for all , almost surely an , such that for all


all quantization noise variances , ,that fulfill (76) are achievable simultaneously.

The conclusions of Lemmatas 1 and 3 allow to infer that thereexists for every fixed almost surely an , such that forall the sequence of quantization noise variances

is achievable according to (33), if for all

(77)

For a given the infimum of the achievable quantization

noise variances in stage is denoted by . It is conve-nient to define

(78)

The quantities , or, equivalently, are determined by the equations

(79)

Theorem 3 is established by showing that there exist constantsand , such that and fulfill

(77) for all , where can grow arbitrarily large.To this end, is upper-bounded by reducing the transmitpower of relay stage from to . This yieldsthe upper-bound , since

(i) can only increase for a fixed , when thetransmit power both of and are reduced from

to in a first step6.

6Proof: The derivative is obtained according to the implicitfunction theorem

where and . This derivative is

nonpositive for positive , , , and . This is seen as follows. Thedenominator is obviously positive. The numerator has zeros only at ,

and , which

implies that , , , and cannot be positive simultaneously ata zero, and, therefore, the numerator has the same sign for all positive tuples

. It thus remains to show that the numerator is negativefor an arbitrary choice of the positive tuple , e.g., for

, , , , it evaluates to .

(ii) the upper-bound on from (i) can again only in-crease for a fixed a transmit power of stageand fixed , when the transmit power of is reducedfrom to in a second step.

Equations (79) are rewritten in a first step as

(80)

where

(81)

and in a second step as

(82)

We take limits on both sides (Bernoulli l’Hospital) and obtain

(83)

This equation is solved for as follows:

(84)

Thus, there is for every an , such that for all

(85)

In order to ensure that

(86)

(87)

(88)


for all , we fix

(89)

which for small leads to the coupling ( by assumption)

(90)

Thus, we have shown that

(91)

It remains to show that there exists an , such that for all

(92)

To this end, we substitute and in (80) andtake to infinity. This results for into the first orderdifference equation (with )

(93)

or, equivalently

(94)

Thus, we conclude that there is for every an , suchthat for all there exists almost surely an , suchthat for all

(95)is achievable. In order to sustain a certain value of in thelarge limit, it is thus sufficient that

(96)

Since as , this leads to the following asymp-totically linear coupling between and :

(97)For the converse, i.e., the proof that any that scales slowerthan linearly with cannot sustain a constant SNR at the desti-nation stage, (79) serves as the starting point, again

(98)

The inequality7

(101)

implies the inequality

(102)

which leads to the following lower-bounds on and

(103)

(104)

Thus, the power required to sustain a constant SNR, , mustfulfill

(105)

C. Amplify & Forward Networks

The following theorem characterizes the scaling of thesupremum of achievable sum-rates in multi-antenna amplify &forward multi-hop networks.

Theorem 4: Let ,and be statistically independent random ma-trices, each with i.i.d. elements of zero-mean, unit-variance andbounded fourth moment, and fix and .Let (signal-to-noise ratio at destination stage) be a posi-tive constant. Fix

(106)

7Proof: Rewrite the inequality as

(99)

This inequality holds, since and is monotonouslyincreasing in

(100)


Let be the sequence, such thatalmost surely. Then,

(107)

where .Remark 1: The per-stage transmit power is chosen such

that the destination SNR is constant with respect to . Thistransmit power scales linearly with

(108)

Remark 2: Theorem 4 allows to conclude the following:• does not scale linearly in , if

scales less than linearly with .• The asymptotic sum-capacity of an single-hop

uplink channel (cf. [1], [16]) is approached for more thanlinear scaling of with .

Remark 3: The case of exactly linear scaling of with ,is not considered for the sake of brevity. In this case, isstill bound away from zero as . More precisely, if

(109)

The corresponding proof is along the very same lines as thesubsequent proofs for the two cases covered in Theorem 4.

For the proof of the theorem, the following notation is in-troduced. The empirical eigenvalue distribution (EED) of someHermitian matrix is defined in terms of the indicatorfunction as .The superscripts indicate parameters which the EEDdepends on. Whenever one of these parameters is taken to in-finity, the respective superscript is dropped. If isa random matrix with i.i.d. entries of zero-mean and unit-vari-ance, , and kept nonzero andfinite, the EED converges uniformly and almost

surely to an asymptotic EED , which

is referred to as Marcenko-Pastur law [15], [19]. Hence, thesupremum of the set of sum-rates that are achievable over asingle-hop uplink channel with single-antenna sources, an

-antenna destination and per-node transmit power ful-fills

(110)

The closed form solution to this integral corresponds to thelower expression in (107).

The expression for in (37) can be written in terms of theEED of (normalized by ) as follows:

(111)

where

(112)Note that the choice of and in the theorem ensures that

.In the proof of the theorem, limits are taken in (111), first with

respect to and then with respect to . The first step yieldsalmost sure convergence to a deterministic limit, the second stepdrives this limit either to zero in the case orto in the case . The essential effects behindthese results are the following:

• If , (almost) all eigenvalues of tend tozero, as grows large.

• If , the asymptotic EED of approachesthe one that corresponds to the single-hop case, as growslarge, while tends towards the identity matrix.

The following Propositions 2 and 3 make these effects preciseand form the basis for the subsequent proof of Theorem 4.

Proposition 2: Given the assumptions of Theorem 4, the EEDof converges uniformly as . That is, there is anasymptotic EED , such that

(113)

Moreover, the following statements hold:• If , the asymptotic EED of converges

in the limit pointwise to the Marcenko-Pastur law,i.e.,

(114)

• If , the asymptotic EED of convergesin the limit pointwise to a unit step at zero, i.e.,

(115)

Proposition 3: Given the assumptions of Theorem 4, the EEDof converges uniformly as : there is an asymptoticEED , such that

(116)

Moreover, for and , the asymptotic EEDof , (x), converges pointwise to

(117)


For a proof of Theorem 4, it remains to bring these two propo-sitions together in order to characterize the asymptotic EED of

in both cases. Specifically, the following is nontrivial:• If , the eigenvalues of do not die out

in the same way as those of .• If , the asymptotic EED of con-

verges pointwise to that of , i.e., to as .Proof of Theorem 4: The two cases are considered

separately. Case : It has to be shown that. Since is known to converge to

a nonrandom constant for every almost surely as[3], this is implied by .This, in turn, can be shown based on the following lemma:

Lemma 4: is for all upper-bounded according to

(118)

where is an matrix with orthonormal rows thatis obtained through the following sequence of singular valuedecompositions:

(119)

(120)

(121)...

The matrices and correspond to the first columns ofand , respectively, and for

.Remark: This upper-bound corresponds to the supremum of

sum-rates that are achievable by the amplify & forward strategyover an equivalent network with:

• noiseless relay stages ;• the relay stage replaced by an equivalent destina-

tion stage with antennas, whose preceding hop has achannel matrix ,

• white noise of power at the equivalentdestination stage (corresponds to the noise power at theoriginal destination stage due to the removed relay stages

, plus the noise power of the original desti-nation stage).

This lemma is applied for to obtain

(122)

where

(123)

and

(124)

with and constructed as in Lemma 4. Now, limits with re-spect to and can be taken on the right-hand side of (122).To this end, fix arbitrarily small, and fix suffi-ciently small, such that

(125)

and fix sufficiently small, such that

(126)

The two sums in (122) are considered individually. The first sumfulfills independently of

(127)

(128)

This bound is obtained in two steps:• Since the sum comprises no more than terms that are

upper-bounded by each, one obtains

(129)

• Since almost surely for all , one ob-tains

(130)

(131)

(132)

The last inequality follows, since.

For the second sum, there exists for every an , suchthat for all

(133)

For establishing this upper-bound, the following lemmata areused:


Lemma 5: Let be as in The-orem 4. Moreover, define an randommatrix whose elements follow a distribution in-dependent of the elements ofand fulfill . Then, the EEDs of

and

converge, as and and are fixed, uniformly andalmost surely to the same asymptotic EED, where isthe matrix that contains the first rows of .

Lemma 6: Let the matrices be as inTheorem 4. Moreover, let be an arbitrary randommatrix that fulfillsfor every fixed ratio almost surely. Define

. Then, for every pairof fixed ratios and and any

(134)

The bound (133) is obtained in four steps:• Due to the concavity of the -function, Jensen’s in-

equality can be invoked to establish:

(135)

(136)

(137)

(138)

where we choose as the maximal that fulfills.

• From Lemma 5, we know that Proposition 2 applies also,if is replaced by the random matrix . Note that

, if and only if also . Thus,there exists by Proposition 2 an , such that there existsfor all almost surely an , such that forall the fraction of eigenvalues of largerthan fulfills

(139)

• Moreover, Lemma 6 impliesalmost surely.

• Finally, we use again that almostsurely.

Thus, by combining (128) and (133), we have shown thatthere exists an , such that there exists for all almostsurely an , such that for all (122) fulfills

(140)

Case : Define the Hermitian matrixand rewrite it in terms of its eigenvalue decomposition

(141)

(142)

where denotes the eigenvector that corresponds to theeigenvalue. Let be an arbitrary positive semidefinite matrixwith an asymptotic EED that depends on and and fulfills

. We show that the asymptoticEEDs and coincide in the limit

(143)

(144)

(145)

(146)

(147)

(148)

(149)

The first inequality follows, since removing the negative/posi-tive definite part of can only increase/decrease each individualeigenvalue of . The second inequality follows, since alladded terms are nonnegative. With Proposition 3 we can thusconclude that there exists for every an , such that forall

(150)

which implies that converges pointwise to .


In the following, we use the identity

(151)

(152)

We consider the first term on the right-hand side and identifywith . Thus, we obtain

(153)

(154)

where the integration interval corresponds to the support of. We drop the sum, use the fact

converges pointwise to as , and thus obtainwith Proposition 2 that there exists for every an , suchthat for all almost surely

(155)

Next, we investigate the second term in (152). Application ofJensen’s inequality yields

(156)

(157)

(158)

Since , we obtain with (150) that there existsfor every an , such that for all

(159)We have shown that there exists for every an , suchthat for all

(160)

The integral evaluates to [16]

(161)

(162)

which is the sum-capacity of a single-hop uplink channel withsingle-antenna sources and an -antenna destination.

It remains to prove that the single-hop sum-capacity cannotbe exceeded. To this end, we can again consider , since

. We apply Lemma 4 forto obtain

(163)

where

(164)

(165)

with and constructed as in Lemma 4.Since almost surely for all , we obtain

(166)

Moreover, according to Lemma 5, the asymptotic EED ofcoincides with the asymptotic EED of , where con-tains the first rows of . Thus, we can write

(167)

(168)

(169)

which is the sum-capacity of a single-hop uplink channel. Here,we have taken the limit inside the definite integral according tothe bounded convergence theorem. The second term in (168)evaluates to zero due to the concavity of the -function andJensen’s inequality: we choose as the maxmimal , such that

and write

(170)

(171)


TABLE IIIILLUSTRATION OF : POSITION OF SET IN THE DECODING ORDER OF THE DECODING PHASE

(172)

(173)

From Lemma 6, we know thatalmost surely. Since as almostsurely—note that is not supported for

—the sum converges to zero almost

surely. We have thus shown that an achievable sum-rate cannotexceed the sum-capacity of a single-hop uplink channel.

V. CONCLUSIONS

In this work, it was shown that the choice of the relayingscheme is crucial in long multi-hop networks. While there isunsurprisingly an inherent performance gap between decode &forward and the investigated non-regenerative schemes due tonoise accumulation, the key contribution of this work is theidentification of the fundamental differences among the perfor-mances of non-regenerative relaying schemes in the regime oflong multi-hop networks.

APPENDIX APROOFS OF LEMMATA FOR THEOREMS 2 AND 3

Proof of Lemma 1: Consider the sum-capacity achievingminimum-mean-square error (MMSE) successive interferencecancellation (SIC) receiver structure [2]. We modify this re-ceiver as follows: rather than applying interference cancellationafter each decoded codeword, we decode codewords of mul-tiple transmit terminals simultaneously based on a given MMSEequalizer output signal. We group the transmit terminals intothe sets , . Codewords of transmit termi-nals within the same set are decoded simultaneously basedon the same MMSE equalizer output signal each, i.e., there are

interference cancellation steps in total. Since is not aninteger multiple of in general, we choose the cardinality ofthese sets as

ifelse.

(174)

We introduce the one-to-one map, where corresponds to the position

of set in the decoding order. E.g., implies thatset is decoded based on the MMSE output of the fourth“decoding phase”, when the transmit signals of three other setsare already canceled.

Let us denote the number of mutually interfering streams inthe decoding phase of set by and consider the ratio

(175)

Here, we denote by the inverse function of . This ratiois upper- and lower-bounded as follows:

(176)

(177)

(178)

(179)

Since both bounds converge to the same value as grows large,we conclude

(180)

In the following, we assume that transmission is divided intotime slots of symbol durations each. Each transmit ter-

minal transmits a sequence of codewords of length , one ineach time slot. We specify the decoding order for the codewordsof the set that are transmitted in the time slot throughthe function

, where

(181)

This function is one-to-one in for each fixed and illustratedin Table III. The codewords of the transmit terminals in setfor the time slot are selected from a common code whoserate is fully determined by the value of , i.e., therespective position in the decoding order for this time slot. Theaverage code rate over the codebooks of the transmit termi-nals in set is given by

(182)

Since (i) the codomain of with respect to is the same forall , and (ii) for each fixed the function is one to onein , this average rate is the same for all , i.e.,

(183)


In the following, we use a result [21, Lemma 3.1] on thesignal-to-interference-plus-noise ratios (SINRs), ,at the output of an MMSE receiver in an MIMO channelwith channel coefficients as assumed in this lemma. Ifas , then there exists a constant , such thatalmost surely

(184)

We apply this result, for each of the time slots, to each of thedecoding phases in the corresponding modified MMSE-SIC

receiver. Specifically, since is finite, we conclude that almostsurely

(185)

where denotes the SINR for the signal of transmitterminal in time slot , and is defined analogously to(180). Thus, in the limit of large , is achievable forall transmit terminals in in time slot almost surely, if

(186)

The average (over time slots) of the suprema of achievable ratesfor the transmit terminals in set is given by

(187)

According to (185), there exists for every almost surelyan , such that for all

(188)

where due to the choice of

(189)

(190)

Next, we show that there is for each an , such thatfor all

(191)

where fulfills [15]

(192)

Let us fix and denote by the SINR of the transmitterminal in time slot , when interference of each codeword iscanceled individually, once it is decoded. That is, we considerthe SINRs as seen in the capacity achieving structure. Thus, thefollowing relation holds [2]:

(193)

We fix the decoding order in this case, such that the codewordsof the transmit terminals in set are decoded in the decodingphases

(194)

Here, we denote by the inverse function of withrespect to . Since the SINR of each transmit terminal signalbecomes the larger the more interference signals are canceled,we have for all , such thatand and . Thus, we conclude

(195)

(196)

(197)

(198)

Here, the last inequality follows from (193). Taking the limitwith respect to in (195), normalized by , yields almostsurely

(199)

(200)


Likewise, we obtain for (198) almost surely

(201)

(202)

Since is finite almost surely according to [21,Lemma 3.1], there exists for every an , such that forall we have . Thus,we eventually conclude

(203)

which establishes (191). The second inequality follows, sinceis the normalized asymptotic sum-capacity of the channel.

We finally combine (188) and (191) in order to conclude thatthere exists for every almost surely an , such that forall

(204)

where we used the triangle inequality. Thus, the rate tupleis achievable almost

surely in the limit , if .Proof of Lemma 2: We decompose the matrix

into the matrices , , where

ifelse.

(205)

The matrices contain disjoint sections of , such that

(206)

With this notation, we obtain the following bounds on :

(207)

(208)

(209)

Here, we used that the maximum/minimum Eucledian norm ofany row of a matrix is upper-/lower-bounded by the maximum/minimum singular value of the matrix.

From [20], it is known that almost surely

(210)

(211)

Since is finite, we can conclude that also almost surely

(212)

(213)

In the following, we choose for a given sufficientlylarge, such that

(214)

and

(215)

Then, there exists for every almost surely an , such thatfor all

(216)

The first inequality follows from (207), the second one from(212), and the third one from (214). Likewise, there exists forevery almost surely an , such that for all .

(217)

(218)

Again, the first inequality follows from (207), the second onefrom (213), and the third one from (215). The combination ofthe bounds (216) and (218) yields, that there also exists for every

almost surely an , such that for all

(219)

which establishes the Lemma.


Proof of Lemma 3: We rewrite the considered conditionalmutual information as follows:

(220)

(221)

(222)

(223)

Here, we used the fact that is independent of whenconditioned on in order to obtain (221). The three entropyexpressions are evaluated as follows:

(224)

(225)

(226)

(227)

Here, denotes the matrix that contains the columnsof whose indexes are contained in . Thus, we obtain ac-cording to (223)

(228)

Next, we use the following corollary that follows fromLemma 1:

Corollary 1: Let the elements of the random matrixbe distributed according to the assumptions of Theorem

2. Let and be positive constants. Then, there exists almostsurely an , such that for all :

(229)

where denotes the matrix that contains the columns ofwhose indexes are contained in .

The proof of this corollary is provided subsequent to thisproof. According to Corollary 1, there exists almost surely an

, such that for all

(230)

We apply this result to (228), and conclude that there is almostsurely an , such that for all

(231)

(232)

where (232) converges to almost surely as , i.e.,there exists almost surely for every tuple of rates of com-pressed quantization codebooks

with an , such that for all thequantization noise variance is achievable in the sense of(30).

Proof of Corollary 1: Lemma 1 implies that there is for allalmost surely an , such that for all

(233)

Since almost surely

(234)

this is equivalent to (229).

APPENDIX BPROOFS OF PROPOSITIONS AND LEMMATA FOR THEOREM 4

Proof of Proposition 2: The uniform and almost sure con-vergence of to forfollows from the following result in [7]:

Lemma 7: Let bestatistically independent random matrices that fulfill the condi-tions for the Marcenko-Pastur law and have elements of unit-variance. Define . Then, the EED of

(235)


converges uniformly and almost surely as : there existsan asymptotic , such that

(236)

Moreover, the Stieltjes transform of this asymptotic EED8 ful-fills the implicit equation

(237)

Remark: Note that is normalized with respect tohere, while it is normalized with respect to in [7].

The Stieltjes transform therein relates to as.

For the setting considered in this work, is identified with, with , with , and , with for all .

This yields for our setting the implicit equation

(238)

where

(239)

The asymptotic analysis for can now be performed inthe Stieltjes domain, since the EED converges topointwise with respect to , if and only if converges to

pointwise [18, Corollary 1]. The basis for this asymp-totic analysis is the following lemma, whose proof is providedsubsequently in this Appendix:

Lemma 8: Let and be some function.

• Then, for all and

(240)

• Then, for all negative and

(241)

Lemma 8 is applied to , where is identified with andwith . In the limit , the implicit (238) simplifies

as follows:1) If , then , and thus

(242)

8The Stieltjes transform of some EED is defined as[17]. This definition is adopted from [7] here, while it is generally

more common to define the Stieltjes transform with a minus sign in the denom-inator. The EED is uniquely determined by its Stieltjes transform.

The solution to this equation is the Stieltjes transform ofthe Marcenko-Pastur law with parameter .

2) If , then for . Lemma 8applies, since for . Notethat the Stieltjes transform is positive for positive , andthus the left-hand side of (238) would be larger than one,if , cf. proof of [7, Theorem 4]). Thus

(243)

Since the Stieltjes transform is an analytic function onits full domain, (243) holds for all . The correspondingasymptotic EED is .

Proof of Proposition 3: The matrix is defined as, where . The

EED of converges uniformly and almost surely to anasymptotic EED. This and the corresponding implicit equationfor the Stieltjes transform of the asymptotic EED offollows again from Lemma 7, where is identified with ,

with and with . Thus, we obtain the followingequation for the Stieltjes transform:

(244)

with

(245)

Again, we assume . Once more, the obtained limitingStieltjes transform generalizes to its full domain, since it is ananalytic function. Since , the followingstatements apply:

• For every fixed and , the following inequality holdsfor all and :

(246)

• If , then

(247)

This implies that the terms converge uniformly as andtend to infinity:

(248)

which in turn yields

(249)

In a next step, it is concluded that also , if. To this end, the following Lemmata 9 and 10

are stated. The corresponding proofs are provided subsequentlyin this Appendix.


Lemma 9: Let be a sequence of posi-tive semidefinite random matrices with parameter whose EEDconverges uniformly to the asymptotic EED almostsurely. Assume for all i) and ii)

almost surely. Then, thefollowing types of convergence are equivalent9:

1) , where fulfills for every :

2) .3) .

Lemma 10: fulfills the assumptions of Lemma 9 whenthe parameter is identified with .

As a consequence of that, there exists for every an ,such that there exists for all almost surely an ,such that for all

(250)

(251)

(252)

(253)

(254)

(255)

(256)

(257)

Expression (253) is obtained by applying the triangle inequalityand using the homogeneity of the nuclear norm. Equality be-

9 denotes the nuclear norm.

tween (253) and (254) is established by the following chain ofidentities for a positive semidefinite matrix :

(258)

(259)

(260)

(261)

(262)

Equation (255) follows by adding and substractingand repeated application of the triangle inequality. In (256), theindividual integrals are upper-bounded by the largest ones. Inthe final step, both terms in (256) are upper-bounded by . Forthe first term, one can fix an , such that this upper-bound holdsfor all by (249) and Lemma 9. For fixed (and thus

), one can finally choose an large enough, such thatfor all the second term is smaller than . Notethat for fixed almost surelyaccording to Lemma 10. Therefore, the limit can be taken in-side the integral due to the uniform convergence of

to ). We have thus shown that convergespointwise to as . Since the eigenvalues of the in-verse of are the inverse eigenvalues of , i.e.,

, one can conclude that also convergespointwise to .

Proof of Lemma 4: We define the matrices

(263)and

(264)

Note that corresponds to the noise co-variance matrix at the destination antennas under the assumption


of noiseless relay stages to . Therefore, we obtainthe following upper-bound:

(265)

(266)

(267)

(268)

In (265), we use the singular value decomposition, where is an diagonal

matrix with all nonzero singular values on its diagonal andan matrix derived from by deleting the rows thatcorrespond to the zero singular values.

Equation (266) follows from the following chain of identities,where we define :

(269)

(270)

(271)

and the fact that

(272)

Equation (267) follows from the fact thatis concave

on the set of positive definite matrices according to Lemma11 on p. 42, and, thus, for a given trace maximized by therespective matrix proportional to the identity [1]. In (268), weintroduce the matrix , which fulfills

We finally obtain by induction

(273)

(274)...

(275)

(276)

(277)

Proof of Lemma 5: We prove the statement by showing thatthe S-transforms of the asymptotic EEDs of and coincide.The S-transform of is given by [7, Theorem 2]

(278)We derive the S-transform of in the fol-

lowing. First, we obtain the S-transform offrom [7,

Theorem 2] as

(279)


Using that the S-transform of is given by(e.g., [17]) we obtain the S-trans-

form ofas [7, Theorem 1]

(280)

(281)

Finally, we obtain the S-transform of the asymptotic EED ofas [7, Eq. (15)] as

(282)

(283)

which coincides with (278).

Proof of Lemma 6: The proof relies on a result of [25], [26](also [17, Theorem 2.39]): Let be a randommatrix, where:

• is an arbitrary nonnegative definite randommatrix whose EED converges uniformly to a non-random distribution almost surely as and satisfies

almost surely;• is an random matrix, independent of , with i.i.d.

elements of unit variance.Then, for every fixed ratio

(284)

Since the trace of a product is invariant under cyclic permu-tation of the factors, we can write

(285)

If is identified with , where and, we obtain

(286)

Repeating the argument for

(287)

with yields

(288)

(289)

Finally, we obtain with

(290)

which completes the proof.

Proof of Lemma 8: The lemma follows from the fact thatthe limit can be taken inside a continuous function, which allowsus to write

(291)

(292)

(293)

(294)

From the rule of Bernoulli-l’Hospital, we know that

(295)

where and are positive constants.If , there exists by definition some ,

such that the absolute value of the argument of the exponentialfunction in (294) can be upper-bounded according to

(296)

Evaluating (295) for renders this upper-bound zero,which establishes that also the left-hand side of (296) becomeszero and (294) evaluates to one in this case.

For the proof of the other two cases (note that is real andnegative then), we write analogously to (294)

(297)


If , there exists some , such thatthe argument of the exponential function in (297) can be upper-bounded according to

(298)

The limit (295) does not exist for implying that bothsides of (298) evaluate to minus infinity, and (294) to zero inturn.

Proof of Lemma 9: We establish the following chain ofidentities:

(299)

(300)

(301)

(302)

(303)

(304)

(305)

In (300), we write the nuclear norm of the matrix in termsof its singular values . Since the matrix is normal,i.e., , itssingular values coincide with the absolute values of its eigen-values. In (302), we arrange the terms, such that they can berelated to the EED of . In (305), we take the limit insidethe integral. This is justified, since the maximum eigenvalue of

converges to some bounded constant by assumption. Thus,the integration is over the compact interval , where weintegrate over a uniformly convergent sequence (in ) of func-tions. This establishes the equivalence between 1) and 3). It re-mains to establish that

(306)

For the forward part, consider for, i.e., . Fix any . Since

is monotonically increasing on the interval of interest, wecan write

(307)

Thus, if does not tend to zero for all , the integralcannot tend to zero. The same reasoning can be applied for theinterval .

For the backward part, we break the integration in (305) intotwo parts. The first integral is from zero to some constant ,

. is a sequence of Riemann integrablefunctions that is uniformly bounded and pointwise convergent.By the bounded convergence theorem for the Riemann integral(e.g., [24]) we can take the limit inside the integral. Thus, thelimit of this first integral is zero. The second part of the integralis from to . Here, the limit cannot be taken inside the inte-gral in general (note that might be unbounded).However, we can write

(308)

The second term on the right-hand side of (308) evaluates toone, since the limit can be taken inside the integral. Again, this isjustified by the bounded convergence theorem (e.g., [24]). Thefirst integral on the right-hand side evaluates to one for every

, since otherwise there would be a contradiction between thefollowing two statements:

(309)

(310)

The first statement corresponds to the assumptionalmost surely for each , the

second statement follows by taking the limit inside the integral,which is fine for every fixed value of .

Proof of Lemma 10: We go through each of the assump-tions required for Lemma 9:

1) To show that the EED of converges uniformly to anonrandom distribution irrespective of the values of and

, we use a result from [25], [26] (also [17, Theorem2.39]):Let , where• is an arbitrary nonnegative definite random

matrix whose EED for every fixed ratio converges


uniformly to a nonrandom distribution almost surely as;

• is an random matrix, independent of , withi.i.d. elements of unit variance.

Then, the EED of converges uniformly to a nonrandomdistribution function almost surely as . If is iden-tified with and with

(311)

uniform and almost sure convergence of the EED offollows by induction.

2) For the trace condition, we obtain for arbitrary andby application of Lemma 6:

(312)

(313)

Note that and are fixed here, and thus the sum isfinite.

3) For the condition on the maximum eigenvalue, we use thetriangle inequality and the submultiplicativity of the spec-tral norm:

(314)

(315)

(316)

(317)

where we used that the nonzero eigenvalues of the matrixproducts in (315) and (316) coincide. Repeated applicationof steps (316) and (317) finally yields

(318)

(319)

(320)

where each of the multipliers in (318) is bounded accordingto [20]. The ratio and are fixed, such that also theresulting product is finite.

Lemma 11: Let and be positive definite matrices.Then, the function

(321)

is concave on the set of positive semidefinite matrices .Proof of Lemma 11: Define , where

is positive semidefinite, is symmetric, and . We proofconcavity of in by proving that is convexin for all such that is positive semidefinite. We computethe first and second derivative of with respect to as

(322)

(323)

and

(324)

(325)

(326)


This expression is nonpositive due to [27, Lemma 2.3], whichstates that for a positive definite matrix , a positive semidefinitematrix , and a Hermitian matrix

(327)

In (326), we identify with , withand with .

APPENDIX CPROOF OF PROPOSITION 1

The sum-transmit power of stage is given by

(328)

Each of the traces converges to one almost surely. This followsby repeated application of the following result from [25], [26]:Let be a random matrix, where:

• is an arbitrary nonnegative definite random ma-trix whose EED converges uniformly to a nonrandomdistribution almost surely as and satisfies

almost surely;• is an random matrix, independent of , with i.i.d.

elements of zero-mean and unit variance.Then, for every fixed ratio , almostsurely.

Due to the convergence of the traces, the following holds:

(329)

(330)

where the right-hand side fulfills for

(331)

For the second part, the transmit power of relay is writtenas

(332)

where denotes the column of and

First, a result from [28, Theorem 1] yields, that for every

(333)

(334)

(335)

where the right-hand side fulfills

(336)

for . [28, Theorem 1] requires the tobe positive semi-definite (obvious) and to fulfill almost surely

(follows by the almost sure con-vergence of the traces in ).

Thus, there exists for every relay an , such that forall

(337)

Next, consider for each relay the smallest such as arandom variable. The distribution of these i.i.d. random vari-ables fulfills for all

(338)

Now, fix for arbitrarily close to one, sufficiently large,such that . Then, for all

(339)

Consider now the inequality for all

(340)


where the lower-bound fulfills due to (339) and the strong lawof large numbers

(341)

This establishes the proposition.

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Jörg Wagner received both his M.Sc. degree in Information Technology andElectrical Engineering and his Ph.D. degree (Dr. Scientiae) from ETH Zurich,Switzerland, in 2006 and 2011, respectively.

From 2006 to 2011, he was a research and teaching assistant with the Com-munication Technology Laboratory of ETH Zurich, Switzerland. Since 2011 hehas been with Swisscom (Schweiz) AG.

Armin Wittneben received his M.Sc. and Ph.D. degrees in Electrical Engi-neering from the Technical University of Darmstadt, Germany, in 1983 and1989 respectively. In 1989 he joined the central research unit ASCOM Tech ofASCOM Ltd., Switzerland, as postdoctoral researcher. In 1992 he became headof the Wireless Technology and Security Department. In 1996 he also servedas co-director of ASCOM Tech. In 1997 he received the Venia Legendi (Ha-bilitation) in Communication Technology from the TU-Darmstadt. In 1998 heaccepted a position as full professor of Telecommunication at the Saarland Uni-versity, Germany. Since 2002 he has been full professor of Wireless Technologyat ETH Zurich, Switzerland and is serving as head of the Communication Tech-nology Laboratory. His research interests include all aspects of wireless com-munication with special emphasis on communication theory, signal processing,cooperative wireless signaling and protocols, distributed MIMO systems andUltra-Wideband/Ultra-Low Power communication, imaging and position loca-tion.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 4

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