IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY … · IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1 The Feasibility of Interference Alignment Over ... [18] to accommodate

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1

The Feasibility of Interference Alignment OverMeasured MIMO-OFDM Channels

Omar El Ayach, Student Member, IEEE, Steven Peters, Student Member, IEEE,and Robert W. Heath, Jr., Senior Member, IEEE,

Abstract—Interference alignment (IA) has been shown toachieve the maximum achievable degrees of freedom in theinterference channel. This results in sum rate scaling linearlywith the number of users in the high signal-to-noise-ratio (SNR)regime. Linear scaling is achieved by precoding transmittedsignals to align interference subspaces at the receivers, givenchannel knowledge of all transmit-receive pairs, effectively reduc-ing the number of discernible interferers. The theory of IA wasderived under assumptions about the richness of scattering in thepropagation channel; practical channels do not guarantee suchideal characteristics. This paper presents the first experimentalstudy of IA in measured multiple-input multiple-output orthogo-nal frequency-division multiplexing (MIMO-OFDM) interferencechannels. Our measurement campaign includes a variety ofindoor and outdoor measurement scenarios at The Universityof Texas at Austin. We show that IA achieves the claimed scalingfactors, or degrees of freedom, in several measured channelsettings for a 3 user, 2 antennas per node setup. In addition toverifying the claimed performance, we characterize the effect ofKronecker spatial correlation on sum rate and present two othercorrelation measures, which we show are more tightly related tothe achieved sum rate.

Index Terms—Channel measurements, interference alignment,multiple-inputmultiple- output (MIMO),software defined radio.

I. INTRODUCTION

INTERFERENCE alignment (IA) is a transmission strategyfor the interference channel that results in sum capacities

that scale linearly, at high signal-to-noise ratio (SNR), withthe number of users in the system [1]. Interference align-ment cooperatively aligns interfering signals over the time,space, or frequency dimensions. In multiple-input multiple-output (MIMO) interference channels, IA aligns signals in thespatial dimension by choosing transmit precoders such thatinterference at each receiver spans only a subspace of thereceive space. To achieve alignment and the maximum gains,however, certain dimensionality constraints need to be satis-fied; alignment is only possible for a certain number of usersif given a sufficient number of transmit and receive antennas.Moreover, guaranteeing the maximum degrees of freedom viaprecoding requires coding over infinitely many dimensions,

Copyright (c) 2010 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

The authors are with the Wireless Networking and Communications Group,Department of Electrical and Computer Engineering, The University ofTexas at Austin, Austin, TX 78712 USA (e-mail: omarayach, speters,[email protected]).

This work was supported by the DARPA IT-MANET program, GrantW911NF-07-1-0028 and the Office of Naval Research (ONR) under grantN000141010337.

made possible by using time or frequency extensions [1],which are not considered in this paper.

MIMO interference alignment, as well as alignment in otherdimensions, was first studied in [1]–[3]. Since then, IA hasbeen examined further from several angles. After derivingthe high SNR sum rate scaling of IA, distributed iterativealgorithms for constructing MIMO IA precoders were pre-sented in [4] and [5] with varying assumptions on reciprocityand channel knowledge. Other solutions were developed forsymmetric networks [6], cellular networks [7], single-inputsingle-output (SISO) networks [8], and SISO networks withlimited feedback [9]. More recent work addresses the feasi-bility of IA in terms of network structure and channel stateinformation requirements [10]–[12]. For example, [10] givesfeasibility conditions on the number of antennas needed pernode, while [11] examines the possibility of applying IA toa two user network with no channel state information at thetransmitter. Extending the IA concept to larger networks, [13]applies IA to large scale networks to derive new bounds onsum capacity. The work in [4]- [13] has helped theoreticallyquantify the gains of IA, however, feasibility and performancein real channels remains an open question.

The theoretical results in [4]- [13] were derived usingbaseband models with channels drawn independently from acontinuous distribution; this represents scattering too rich tobe observed in practical systems. As a result, performancemay be overestimated. Moreover, there are no comprehen-sive interference channel measurements suitable for studyingIA in practice. The only comparable results on multiuserMIMO channel measurements, not directly related to IA,target broadcast channels consisting of a single base stationand several receivers, and thus do not provide the requireddata on measured interference channels [14], [15]. Workdone in [16], for example, presents multiuser measurementsformed by concatenating separate single user measurements,claiming that the static measurement environment ensures thevalidity of the results. Related work on demonstrating IAin practice is limited to [17], which tested a hybrid versionof IA coupled with interference cancelation and successivedecoding in a single carrier narrowband MIMO wireless localarea network. The work in [17] does not provide insightinto the performance of the original MIMO IA solutionsin realistic wideband channels. Moreover, [17] downplaysthe importance of synchronization and other physical layerconcepts in the interference channel due to its narrowbandnature. Consequently, the viability of IA in measured channelshas not yet been evaluated.

arX

iv:0

911.

1849

v3 [

cs.I

T]

13

Oct

201

0

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2

In this paper, we establish the feasibility of MIMO IA inslowly time varying real world channels, with no frequencyor time extensions. To acquire suitable channel measurements,we implemented a MIMO-OFDM measurement testbed for the3-user 2 × 2 MIMO interference channel, using a softwaredefined radio platform [18]. We gave special attention tothe proper implementation of a synchronized MIMO-OFDMphysical layer, a consideration that was not emphasized in [14],[16], [17], to guarantee the validity of our measurements. Weaugment the system we implemented in [18] to accommodatemeasurement setups over large outdoor areas. We make chan-nel measurements for a variety of indoor and outdoor staticnode deployments and therefore extend the preliminary indoorresults derived in [18]1. We summarize the data collected anduse it to establish the true performance of IA in measuredwideband channels. We examine the average sum rate achievedversus signal-to-noise ratio and show that, as predicted intheory, IA outperforms time division multiple access (TDMA)as well as other MIMO techniques. We also show that IAachieves the maximum degrees of freedom in our setup. Wecharacterize the effect of non-ideal propagation channel char-acteristics, such as Kronecker spatial correlation, on achievedsum rate. Finally, we introduce two other correlation measures,matrix collinearity and subspace distance, and show that theyare more tightly related to the achieved sum rate.

In this paper we use the following notation: A is a matrix,and a is a scalar; A∗ denotes the conjugate transpose of A,‖A‖F is its Frobenius norm, trace(A) is its trace, span(A)is its column space, and null(A) its nullspace; ν(A) is anyeigenvector of A and νmax(A) is the dominant eigenvectorwhen eigenvalues are real; IN is the N ×N identity matrix;CN is the N -dimensional complex space.

This paper is organized as follows. Section II brieflypresents the MIMO-OFDM signal model in the presence ofinterference, Section III summarizes the basic idea of IA andintroduces several IA solutions as well as the algorithms usedfor comparison. Section IV details both the hardware andsoftware used in our measurement testbed. Sections V-A andV-B present and discuss the results obtained from our setupin indoor and outdoor environments respectively. We concludewith Section VI.

II. MIMO INTERFERENCE SIGNAL MODEL

Consider the K-user interference channel shown in Fig. 1with Mk transmit antennas at transmitter k and Nm receiveantennas at receiver m. All users send Ns streams of data usingorthogonal frequency division multiplexing (OFDM) with Nsubcarriers [19]. This is known as MIMO-OFDM, a widely de-ployed transmission technique in commercial wireless systemssuch as IEEE 802.11n and 802.16e [20]. In the interferencechannel in Fig. 1, each transmitter k communicates withits corresponding receiver k and interferes with all otherreceivers m 6= k. In this section we explain the MIMO-OFDMinterference signal model in this general case, though in the

1We also obtain larger data sets, in more indoor and outdoor antenna andnode configurations. The performance analysis is also extended to includevarious other interference channel algorithms.

NI PXI-5670Signal Generator

NI PXI-5670Signal Generator

NI PXI-5670Signal Generator

NI PXI-5660Signal Analyzer

NI PXI-5660Signal Analyzer

NI PXI-5660Signal Analyzer

H11

H22

H33

H21

H23

H23

H12

H13

H31

PC

PC

PC

PC

NI PXI-6653 Synchronization ModulesNI PXI-6653 Synchronization Modules

Fig. 1. Simplified hardware block diagram.

remainder of this paper we specialize to the K = 3 user 2×2channel in which each user sends Ns = 1 stream of data.

The received signal at node k and subcarrier n for asufficiently slow fading channel is given by

yk[n] = Hk,k[n]Fk[n]sk[n]+∑m 6=k

Hk,m[n]Fm[n]sm[n]+vk[n],

(1)where yk is the Nk × 1 received signal vector, Hk,m is theNk×Mm channel matrix from transmitter m to receiver k withelements drawn i.i.d. from an arbitrary continuous distribution,Fk is the Mk × Ns precoding matrix used at transmitter k,sk is the Ns × 1 transmitted symbol vector at transmitter k,and vk is a complex vector of i.i.d. circularly symmetric whiteGaussian noise with covariance matrix E[vkv

∗k] = σ2INk

∀k.In this signal model, we assume perfect functioning of thecarrier recovery and symbol timing synchronization modules.We also assume that the impulse response of all the channelsis shorter than the cyclic prefix used, thus allowing us to writethe received signal as in (1). For simplicity, the transmit poweris assumed to be normalized to 1, and the effects of large scalefading are neglected.

Since the capacity region of the interference channel re-mains unknown, the performance of IA cannot be comparedto capacity until the latter has been established. We thereforeevaluate the performance of IA in comparison to other pre-coder designs by studying the achieved sum rate in bits/s/Hzaveraged over all subcarriers with uniform power allocation[21]. Network sum capacity is a point in the capacity region,and, more importantly, a metric that defines the total through-put of the network. The sum rate achieved by an optimalreceiver, assuming ideal decoding for all precoder designs, iscalculated as

Rsum =1

N

N∑n=1

K∑k=1

log2

∣∣∣INk+(σ2INk

+ Rk[n])−1

(Hkk[n]Fk[n]Fk[n]∗Hkk[n]∗)| , (2)

where

Rk[n] =∑m 6=k

Hk,m[n]Fm[n]Fm[n]∗Hk,m[n]∗

is the per-subcarrier interference covariance matrix [22]. SNRis emulated, in simulation, by varying the noise power whilekeeping the normalized channels constant. This normalizationof the measured channels when calculating sum rate is de-scribed in Section IV-C.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 3

III. INTERFERENCE ALIGNMENT AND OTHER TRANSMITTECHNIQUES

In this section we summarize several transmission strate-gies for the interference channel. The algorithms are runoffline using the measured channel data to demonstrate theexpected performance in practice. We start with the closed-form solution for interference alignment, which is valid onlyfor the three user system model when each user sends anumber of streams equal to half the number of transmit andreceive antennas. Then we summarize iterative interferencealignment and a signal-to-interference-plus-noise-ratio (SINR)maximizing solution. We also review TDMA and greedyinterference avoidance, which will be used for performancecomparison. While IA is optimal in terms of sum rate scaling,and while SINR maximization outperforms IA in the low SNRregime by considering SINR in the subspace chosen for thedesired signal, none of these strategies is yet proven to be sumrate optimal.

A. Closed Form Interference Alignment

IA, using enough antennas per node [5], aims at choosingthe set of precoding matrices Fk to force the receivedinterference at each of the K receivers to lie within a lowerdimensional subspace. Specifically, if receiver k intends ondecoding Ns independent data streams with no interference, itmust restrict interference to an Nk−Ns dimensional subspaceof the receive signal space, CNk .

Let Wk[n] be the Nk×Ns matrix describing the orthonor-mal basis for the interference free subspace used at node k andsubcarrier n. Prior to decoding, node k first projects on thebasis of the interference free subspace. Ignoring the AWGNnoise term, this yields

Wk[n]∗yk[n] = Wk[n]∗ (Hk,k[n]Fk[n]sk[n]+∑m6=k

Hk,m[n]Fm[n]sm[n]

. (3)

For alignment, the received interference must lie in the Nk −Ns dimensional nullspace of Wk[n]∗, which gives

span(Hk,m[n]Fm[n]) ⊆ null(Wk[n]∗), ∀m 6= k. (4)

In addition to satisfying (4), the interference alignment solu-tion must satisfy

rank(Wk[n]∗Hk,k[n]Fk[n]) = Ns (5)

to successfully decode all Ns streams with a linear receiver.This spatial alignment approach uses a finite number of

dimensions and is only proven to achieve the maximumdegrees of freedom for the 3 user channel with Ns equal tohalf the number of antennas per node [1], which is the casewe consider. We focus on Mk = Nm = 2, ∀k,m and Ns = 1.In this case, the conditions for interference alignment given in(4) and (5) are satisfied by choosing the precoding matricesas

F1[n] = ν((H3,1[n])−1H3,2[n](H1,2[n])−1

H1,3[n](H2,3[n])−1H2,1[n]), (6)

F2[n] = (H3,2[n])−1H3,1[n]F1[n], (7)

F3[n] = (H2,3[n])−1H2,1[n]F1[n]. (8)

The solution presented in (6), (7), and (8) is not unique. Infact, any IA solution can be rotated inside its subspace withoutdestroying alignment. Non-uniqueness can also be seen bythe ability to choose any eigenvector in (6), each resultingin different precoders and sum rate. Since the number of IAsolutions, and a method to finding the sum rate maximizingone, remains unknown, the solution space must be furtherinvestigated and non-uniqueness exploited to increase sumrate. While optimality is neither proven nor claimed, [23] is anexample of exploiting non-uniqueness to provide a modifiedIA algorithm that yields better sum rate performance than in[1], [4], [5]. Finally, such closed form solutions do not yetexist for networks with more than three users, except in thecase of symmetric channels and Ns = 1 presented in [6].

B. Iterative Interference Alignment

In [5], alignment in a K-user network is formulated in ageneral alternating minimization framework, alternating be-tween solving for the K precoders and the K interferencesubspaces. The alignment problem is viewed as minimizingthe “leakage” interference power over the set of precodersFk[n] and interference subspaces Ck[n]. This minimiza-tion problem is written as

minFm[n]∗Fm[n]=INs ,∀m

Ck[n]∗Ck[n]=INk−Ns ,∀k

K∑k=1

∑m=1m6=k

‖Hk,m[n]Fm[n]−

Ck[n]Ck[n]∗Hk,m[n]Fm[n]‖2F . (9)

The precoders Fm[n] are iteratively refined while keepingCk[n] fixed, and vice versa. As a result, the pseudo codefor such a minimization is

1) Choose the set Fm[n] randomly.2) Choose the columns of Ck[n] to be the Nk − Ns

dominant eigenvectors of∑m6=kHk,m[n]Fm[n]Fm[n]∗Hk,m[n]∗, ∀k.

3) Choose the columns of Fm[n] to be the Ns leastdominant eigenvectors of∑k 6=mHk,m[n]∗ (INk

−Ck[n]Ck[n]∗)Hk,m[n], ∀m.4) Repeat steps 2 and 3 until convergence.

In summary, the algorithm first finds the subspaces Ck[n]which are “closest” to the received interference, and thencalculates the precoders Fm[n] to align interference as closeas possible to the found subspaces. To cancel interferenceusing a linear receiver, for example, receiver k multipliesits received signal by the orthonormal basis of Wk[n] =INk−Ck[n]Ck[n]∗.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 4

Convergence is guaranteed by the fact that steps 2 and 3can only decrease the non-negative objective function. Thenon-convexity of (9), however, implies the potential presenceof multiple local optima. Thus, convergence to the globaloptimum is not guaranteed. To increase sum rate, this iterativealgorithm, and the maximum SINR algorithm of Section III-C,can be improved by performing several random initializationsand choosing the one that results in the highest sum rate [24].

C. Maximum SINR AlgorithmInterference alignment does not target maximizing sum

rate directly. It instead focuses on making the signal-to-interference ratio infinite at the output of the linear filtersWk; it does not attack other performance measures like thepost-processing signal-to-interference-plus-noise ratio. As aresult, perfect alignment often comes at the cost of lower post-alignment SNR or sum rate, the metric we are actually inter-ested in maximizing. We can thus consider another precoderdesign that maximizes other metrics, perhaps without aligninginterference perfectly. One such precoder design maximizesthe total SINR of the network, given by

S(Fk[n]) =K∑

k=1

‖Wk[n]Hk,k[n]Fk[n]‖2F

K∑k=1

( ∑m 6=k

‖Wk[n]Hk,m[n]Fm[n]‖2F + σ2‖Wk[n]‖2) , (10)

where Wk[n] is now the combiner used at receiver k [24].Instead of optimizing the sum rate or the sum SINR, which

is not quite tractable, this algorithm optimizes the sum signalpower over the sum interference-plus-noise power. Since thesets Fk[n] and Wk[n] are not independent, a closed-formsolution for this objective function is unlikely, however, it canbe solved via alternating minimization. For tractability, theprecoders are constrained to have columns of equal norm, thussatisfying Fk[n]∗Fk[n] = 1

NsINs ,∀k [24]. By fixing all Fm[n]

we can solve for Wk[n] as

Wk[n] = νmax

∑m 6=k

Hk,m[n]Fm[n]Fm[n]∗Hk,m[n]∗

+σ2INk

)−1Hk,k[n]Fk[n]Fk[n]∗Hk,k[n]∗

). (11)

Conversely, by fixing all Wk[n], we can solve for the pre-coders as

Fm[n] = νmax

∑k 6=m

Hk,m[n]∗Wk[n]∗Wk[n]Hk,m[n]

−1

Hm,m[n]∗Wm[n]∗Wm[n]Hm,m[n]) . (12)

This results in an algorithm pseudo-code given by1) Choose the set Fm[n] randomly.2) Choose the columns of Wk[n] as given by (11).3) Choose the columns of Fm[n] as given by (12).4) Repeat steps 2 and 3 until convergence.

Maximum SINR can be generalized to multiple streams inbigger networks [4], where the system must be solved for eachcolumn of each matrix, resulting in non-orthogonal solutions.

D. Other Transmit Strategies

For comparison, we consider other transmission schemes,such as TDMA and greedy interference avoidance. In anetwork employing TDMA, transmissions from different usersare orthogonal in time, meaning that only one user transmitsin any given time slot. TDMA systems can take advantage ofmultiuser diversity by scheduling, in every time slot, the userwith the most favorable channel, in terms of instantaneousrate. This requires channel information to be known at thetransmitter, to make the selection process possible, and thus isa fair comparison to IA which also requires this knowledge.Note that TDMA is conceptually equivalent to other orthog-onal resource allocation techniques such as FDMA, whereorthogonality is in the frequency domain.

We also consider greedy interference avoidance, a beam-forming strategy for the interference channel [25]. In SVDbeamforming for the point-to-point channel [26], beamformingvectors are chosen as Fk = νmax (Hk,k), neglecting theinterference covariance matrix. In the presence of interferenceas in (1), however, the rate achieved by each user dependson the matrix

(σ2INk

+ Rk[n])−1/2

(Hk,k[n]). In greedyinterference avoidance [25], users align their signals alongthe most dominant eigenmode of this matrix, thus sendingin the direction they receive the least interference in. Sincein this approach the choice of a user’s precoder affects theinterference subspaces observed in the network, this precoderselection is done for many iterations, in hope of reaching afixed point, but the algorithm does not always converge [27].

IV. SYSTEM IMPLEMENTATION AND TECHNICALAPPROACH

In this section, we present the main software and hardwareparts of the measurement testbed developed. We discuss themain concepts in our MIMO-OFDM system implementationsuch as training, channel estimation, and carrier recovery. Wealso introduce the system parameters used to collect channelmeasurements. We then discuss the main tools and metricsused in our performance analysis, as well as introduce thepreliminary calculations such as the normalization neededbefore further processing the acquired data.

A. Software Implementation

Our MIMO-OFDM testbed software, implemented in Na-tional Instruments’ LabVIEW [28], uses the parameter valuesindicated in Table I for all communicating users in the net-work. We use OFDM modulation with an FFT size of 256and a 64 sample guard interval. The total signal bandwidthused in our measurement setup is 16 MHz, which results inan effective OFDM symbol time of 20µs. Communicationis done at a carrier frequency of 2.4 GHz in the industrial,scientific, and medical (ISM) band. Data on each subcarriercan be modulated using BPSK or M-QAM.

To faithfully predict the performance of interference align-ment at high SNR, we pay special attention to our pilotstructure and channel estimation implementation. Users se-quentially send two OFDM symbols of frequency domainpilots [19] that are known to all receivers. This makes training

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 5

Fig. 2. Picture of the measurement testbed implemented showing theantennas, RF front end equipment, as well as a sample of the softwareinterface.

Time

Freq

uenc

y (S

ubca

rrie

rs)

Dat

a

User 1

Dat

a

User 2

Dat

a

User 3

Antenna 1 Pilot Symbols

Antenna 2 Pilot Symbols

Variable Amount of Data

Fig. 3. Simplified pilot and data example for measurement.

from each user orthogonal in time. During each user’s trainingphase, the first symbol contains pilots from the first antennaon even subcarriers only, and the second antenna on oddones. In the second OFDM training symbol this subcarrierassignment is reversed as shown in Fig. 3. The use of equallyspaced pilots that are orthogonal across antennas is provento be optimal in [29]. We use each user’s OFDM trainingsymbols to estimate the wideband time domain channel exactlyas in [29] assuming proper functioning of our frequency andtime synchronization modules. We also assume that the cyclicprefix length is greater than the number of channel taps inall channels. After obtaining these estimates, the optional“payload data” sent after each user’s training is equalizedusing a frequency domain equalizer, similar to narrowbandequalizers [26]. Payload data is used to verify correct receptionand estimation. For brevity, we omit the details of the channelestimation implementation in this paper and refer the readerto [29] and the references therein.

We briefly discuss, however, the resulting mean square error(MSE) of our estimates, as this is a critical issue when predict-ing performance at SNR levels higher than our measurements’raw SNR. In a preliminary measurement campaign, we noticethat our measured channels have a maximum length L = 5channel taps. With Mk = 2 transmit antennas ∀k and L = 5

TABLE IMIMO-OFDM SYSTEM PARAMETERS

Carrier Freq. 2.4 GHzTransmit Power 6 dBm

Bandwidth 16 MHzFFT Size 256

Subcarrier Spacing 6.25 kHzGuard Interval 64 samples

Total Symbol Duration 20µs

taps, a minimum of LMk = 10 pilot subcarriers are neededto estimate the channel. For the pilot structure in [29], itwas shown that the resulting MSE = σ2

P where P is thetotal power spent on training. In this case, this is the powerper subcarrier multiplied by the number of pilot subcarriers.This implies that, with enough training, channel estimates canbe made arbitrarily accurate. We now note that our systemsends two full OFDM training symbols, instead of the neededLMk = 10 in practice. Therefore, after discounting null tones,we get 400 pilot subcarriers, which is forty times the neededtraining. This results in a channel estimate MSE which is16dB lower than in practical systems, which often use closeto minimal training. This makes the quality of our channelestimates that of a practical system functioning at a 16dBhigher SNR. As a result, over training our channels by a factorof 40, allows us to faithfully predict IA performance at SNRlevels 16dB higher than the measurement’s raw SNR. Furtherdetails on optimal pilot structure, estimation, and mean squareerror can be found in [29].

Since we are mostly interested in measuring the channel,we send two pilot symbols for every OFDM payload datasymbol. This puts minimum payload data in between the pilotsfrom different users, thus keeping the measurement time in themicrosecond range. Sending and equalizing payload data inthe measurement exercise is recommended to verify correctreception and decoding, which ensures that the recordedchannel measurements correspond to successful transmissions.

Pilot symbols are used to estimate frequency offsets betweeneach transmit-receive pair. For proper MIMO communication,the transmit chains corresponding to the 2 transmit antennasper user are synchronized to justify the assumption of a singlefrequency offset per transmit-receive pair. For the sake of ourchannel measurements, however, and since we want to testthe performance of IA in the absence of such impairments,we synchronize all users’ transmit chains. Our measurementsshow that the use of the onboard high precision oscillatorsto synchronize all transmit RF chains results in frequencyoffsets within a 100 Hz of each other which are estimatedand further corrected in software via MIMO-OFDM synchro-nization techniques presented in [30]. Software correction isdone in stages starting with a coarse time synchronization,fractional frequency offset estimation, integral frequency offsetestimation and finally fine time synchronization [30].

B. Hardware Description

Our hardware setup consists of five National InstrumentsPXI-1045 chassis connected to 3 PCs [31]. The first PCcontrols 2 PXI chassis, containing the three users’ transmit

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 6

chains. The remaining three PXI-1045 chassis each house thereceive chains of one of the users, two of which are connectedto the same PC to make the testbed more mobile. In addition tothe RF hardware installed, each PXI-1045 chassis holds a NIPXI-6653 module for timing and synchronization. A simplifiedhardware block diagram is shown in Fig. 1.

Each transmitter, or RF signal generator, named PXI-5670,consists of two physical units, an arbitrary waveform gener-ator, NI PXI-5421, and an upconverter, NI PXI-5610 [32].The arbitrary waveform generator produces an intermediatefrequency signal which is later modulated to RF via theupconverter. Each receiver, or RF signal analyzer, named NIPXI-5660, constitutes a downconverter, NI PXI-5600, and adigitizer, NI PXI-5620 [33]. On the receive side, the down-converter downconverts the signal to an intermediate frequencyafter which the digitizer takes over and samples the waveformwhich is then sent to the PC for processing using the LabVIEWsoftware blocks. Note that each user consists of two transmitand two receive chains, totaling six transmit chains and sixreceive chains for our overall network setup.

Similar software defined setups have been used in paperssuch as [34] to implement single user MIMO communication.Our system, however, is significantly more complex to supportmultiple users whose hardware components are housed indifferent chassis and controlled by different PCs. Moreover,software implementation differs greatly in the methods usedfor training and channel estimation as well as carrier recovery.

To support the cross chassis synchronization needed forthis multi-user prototype, we install NI PXI-6653 timing andsynchronization modules in each PXI-1045 chassis [35]. Thismodule has a high stability reference oven-controlled-crystal-oscillator (OCXO) which can be exported to other chassis,thus enabling synchronization. Locking all the transmitters’phase locked loops to this high precision OCXO helps ensureminimal carrier frequency offsets between transmitters. Al-though frequency offset correction is implemented in software,synchronizing the transmitters in hardware further strengthensthe validity of the obtained measurements. Note that due tohardware limitations, the PXI-6653 and PXI-5610 will onlyallow us to synchronize the intermediate frequency signal,and therefore the RF local oscillators remain independent.Our measurements indicate that the difference in frequencyoffsets between transmitters, when locked into the IF referencesignal from the PXI-6653, is below 100 Hz at a carrier of 2.4GHz. This remaining frequency offset is then estimated andcorrected in software [30] and the MIMO links perform asexpected.

In addition to synchronizing clocks, the PXI-6653 allowsus to export the trigger generated when the master user beginssignal generation. This digital signal is then used to triggergeneration at the other transmitters. While digitally triggeredacquisition, by connecting the receivers to the transmitters’PXI-6653, is possible in small-scale indoor setups such asthose presented in [18], our outdoor measurement setupstretches over distances of about 250ft, making digital trig-gering impossible. Therefore, the receivers are not connectedto this reference trigger signal. To accommodate these outdoorsetups, acquisition is triggered via analog edge triggers. Under

this type of triggering, the receiver starts recording sampleswhenever the received signal level exceeds a predefined thresh-old. We discard any measurement that has been corruptedby the ambient interference in the 2.4 GHz ISM band andthus retain only valid interference free measurements. Thisis done by automatically checking the known payload datafor errors and discarding any transmission with a very highbit error rate since they correspond to frames in whichthe synchronization and estimation blocks malfunctioned dueto interference. Therefore, only measurements coming fromtransmissions that have been correctly received by all receiversare automatically recorded. Triggered acquisition and clocksynchronization ensure that our measurements include onlychannel effects, and are thus free of any timing impairments.

C. Technical ApproachWe now introduce three tools that will be essential for the

performance analysis that follows in Section V. We first dis-cuss how channels are normalized and sum rate is calculatedfor our measurement scenarios. We then discuss how Kro-necker spatial correlation is calculated for our measurements, aconcept which we will link to the performance of IA in SectionV-A. We then introduce two correlation metrics which we latershow are more tightly related to IA and signal subspaces.

1) Calculating Sum Rate: Before evaluating the sum rateperformance of interference alignment over measured chan-nels, we must first obtain normalized channel matrices, H.We normalize the measured channels over the full data set, i.e.no time windowing is applied. For fair comparison with thesimulated Rayleigh channels, we normalize our measurementsto have elements of unit variance and, thus, an averageFrobenius norm of four [36],

Hk,m(ω) = 2Hk,m(ω)√

1|Ω|∑ω′∈Ω ||Hk,m(ω′)||2F

, (13)

where Ω is the set of all measurements collected in the scenarioconsidered, i.e. when normalizing a matrix obtained whend = 1λ, Ω would be the set of all channel measurementsobtained in that configuration (in our measurements |Ω| = 50as indicated in Table II).

2) Kronecker Spatial Correlation: The channel’s spatialcorrelation, calculated according to the Kronecker model, isgiven by

RRX =1

|Ω|∑ω∈Ω

H(ω)H(ω)∗, (14)

RTX =1

|Ω|∑ω∈Ω

H(ω)∗H(ω), (15)

where Ω is the set of all measurements, H(ω), taken in theconsidered configuration. When calculating correlation, thechannel matrices H(ω) are individually normalized to haveunit Frobenius norm, H(ω)/ ||H(ω)||F . For our multiusercase, when calculating receive correlation for user k, for exam-ple, we average over the channels Hk,` ∀` which should havesimilar receive Kronecker correlation due to the separabilityof the model. Similarly, for the transmit correlation of user `,we consider the channels Hk,` ∀k.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 7

TX 1

TX 2

TX 3

d

d

d

d

d

RX 1

RX 2

RX 3

d

d

d

d

d

Fig. 4. Schematic of an example indoor measurement configuration. Antennasare placed at varying distance d apart. Also note that not only are thenodes placed at fixed positions, but all objects in this room remain in theirfixed locations throughout the duration of the measurement campaign. Indoormeasurement details are further summarized in Table II.

3) Matrix Collinearity and Subspace Distance: IA perfor-mance can be closely linked to the signal spaces in the networkwhich the Kronecker model does not fully capture. Therefore,we propose two other distance metrics that we show in SectionV-B can be used to predict performance.

Channel matrix collinearity is a typical correlation measureconsidered in practice [37]. The collinearity between twomatrices A and B is defined as [38]

c(A,B) =|trace (AB∗)|‖A‖F ‖B‖F

. (16)

To adopt a simple correlation measure, inspired by traditionalminimum distance metrics, we define maximum collinearitybetween cross channels as

cmax(H) = max(k,`) 6=(m,n)

c(Hk,`,Hm,n). (17)

Considering maximum collinearity captures the worst case,most aligned, channels that negatively affect the IA solutionthe most.

While maximum matrix collinearity is a practical correlationmeasure that can be linked to the performance of IA, it issensitive to the ordering of the channels’ columns. Sum rate,however, is directly linked to the SNR after projection ontothe interference free space and, thus, to the distance betweenthe subspaces spanned by the effective channels Hk,mFm,which collinearity does not directly measure. To show thisrelationship in Section V-B, we must first define the projectionF-norm distance [39] between two subspaces with orthonormalbasis U and V as

dpF (U,V) =1√2||UU∗ −VV∗||F . (18)

To incorporate the distances between all channels, as wellas signal and interference subspaces present in the network,we define two projection F-norm based distances, namely

TX 2TX 3

dRX 2

RX 3

RX 1

TX 1

d

d

dd

d

Fig. 5. Schematic of an example indoor measurement configuration. Transmitantennas are all placed in the middle to model a system with co-locatedtransmitters such as a group of access points or base stations. The receiversare then placed as shown.

the average subspace distance between the set of effectivechannels Hk,mFm as

d (HF) =

√√√√√ ∑k 6=m

dpF (Ψ(Hk,kFk),Ψ(Hk,mFm))2

(KK−1

) ,

(19)and the average column space distance between the set ofchannels Hk,m

d (H) =

√√√√√ ∑k 6=m

dpF (Ψ(Hk,k),Ψ(Hk,m))2

(KK−1

) , (20)

where Ψ (A) is the operator that extracts the orthonormalbasis for the column space of A. In the special case of (19),where Hk,mFm are vectors, Ψ (Hk,mFm) simply normalizesHk,mFm.

V. RESULTS

In this section we present the main results on the perfor-mance of IA over measured channels. We divide the sectioninto two subsections corresponding to our two measurementcampaigns, indoor and outdoor.

A. Indoor Results

We made measurements in the Wireless CommunicationLab in the Engineering Science Building at The Universityof Texas at Austin. Transmitter-receiver pairs were placed atdistances ranging from 1 to 6 meters apart. With a wavelengthof 12.5cm, all node placements are, therefore, in the far fieldof the other nodes’ antennas [40]. The measurement campaigndetails are summarized in Figs. 4 and 5, and Table II. Allomnidirectional antennas are placed in the same horizontalplane with the antenna arrays placed parallel to each other asshown in the figures. SNR is kept above 25dB allowing us to

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 8

05

1015

20

050

100150

200−5

0

5

10

15

Packet NumberSubcarrier

Cha

nnel

Mag

nitu

de (d

B)

Fig. 6. The temporal evolution of ‖H2,1‖ in our static indoor environment.H21 exhibits very limited time selectivity due to the static placement of thenodes and limited motion in the environment.

predict performance up to 41dB as discussed in Section IV.Fig. 6 shows an example temporal evolution of the channel‖H2,1‖ over 20 packet transmissions. Fig. 6 shows two char-acteristics of indoor channels: limited frequency selectivity andhigh temporal correlation. Our measurements indicate that thechannel correlation after 200 ms remains above 97%.

Interference alignment relies on the assumption that ele-ments of channel matrices are drawn independently at randomfrom a continuous distribution. This assumption, however, isnot likely to be satisfied in real channels that exhibit spatialcorrelation, thus introducing dependence in the matrix ele-ments. While (4) and (5) will still be almost surely satisfiablewith correlated channels, thus not influencing the feasibil-ity of alignment, SNR after alignment may be significantlydecreased due to aligned signal spaces. Our measurementresults, therefore, give insight into the performance of thistheoretically attractive transmit strategy in realistic channelswith complexities that are not entirely captured by the simplei.i.d. models used in proving theoretical results.

To systematically study performance and link it to spa-tial correlation, we arrange nodes as shown in Fig. 4in which all antennas and users are placed a distance dapart. We make measurements for variable values of d ∈λ/2, λ, 2λ, 3λ, 4λ, 5λ to study the effect of spatial cor-relation. Fig. 7 shows the magnitude of the off-diagonalelements of RRX and RTX for the configuration in Fig. 4. Wealso conduct measurements with co-located transmitters andreceivers placed on the vertices of a triangle, as shown in Fig.5. In this arrangement, we position node pairs at a distance 1mapart, and antennas of the same node at a distance of λ/2, 1λ,and 3λ, yielding a receive correlation for user 2, for example,of 0.267, 0.117, and 0.034 respectively.

Given our measured data and the results on spatial corre-lation, we now turn to characterizing the performance of IAand verifying its ability to provide the maximum achievabledegrees of freedom in our three user interference channel. By

TABLE IIINDOOR MEASUREMENT DETAILS

Tx-Rx Spacing ∼ 6mAntenna Type 2.4 GHz omnidirectional

Antenna Spacing d ∈ 0.5λ, 1λ, 2λ, · · · , 5λConfigurations 1. Equidistant nodes & antennas

2. Triangle configuration# of Measurements 50 for each configuration

& antenna spacingMeasurement Duration 180µs

Time Between Measurements ∼ 45sReceive SNR > 25dB

Mobility Fixed nodes & environment

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Distance (in λ)

Cor

rela

tion

Mea

sure

RX Corr. User 1RX Corr. User 2RX Corr. User 3TX Corr. User 1TX Corr. User 2TX Corr. User 3

Fig. 7. Channel’s spatial correlation vs. antenna spacing in the indoor wirelessenvironment.

doing so, we will have verified the optimality of interferencealignment in the high SNR regime. To that end, Fig. 8 and9 plot the average sum rate, as defined in (2), achieved inour three user network for closed form IA and the transmitstrategies presented in Section III-D.

The indoor performance results summarized in Fig. 8, aregenerated in the antenna configuration shown in Fig. 4. Fig. 9,is generated using a configuration with co-located transmitters,as shown in Fig. 5, which may model systems having basestations or access points in a given location. As anticipatedfrom theory, Fig. 8 and 9 verify that IA outperforms greedyinterference avoidance, TDMA, and its equivalent transmissionschemes. Iterative IA performs identically and, therefore, is notshown. Moreover, we note that the throughput gain from IA islargest in the high SNR regime, which is the case of claimedoptimality. Comparing the rate at which network throughputincreases with SNR, we observe that IA benefits more froma marginal increase of SNR, thus achieving more degreesof freedom than TDMA. The slope of the curves, relativeto log2(SNR), is approximately 1.8 in TDMA and 2.8 forIA, thus confirming that IA provides the maximum achievabledegrees of freedom, which in this case is 3.

Fig. 8 and 9 show constant differences between the variousmeasurement scenarios at high SNR. This is due, primarily,to varying degrees of spatial correlation. Measurements withclosely spaced antennas, such as the case of d = 0.5λ in

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 9

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

SNR (dB)

Sum

Rat

e b/

s/H

zd=0.5λd=2λd=5λSimulated Rayleigh

IA

TDMA

Interference Avoidance

Fig. 8. Network sum rate vs. SNR for the configuration in Fig. 4 with severalantenna spacings in an indoor environment. This confirms that IA outperformsTDMA and other transmit strategies, and achieves the predicted 3 degrees offreedom for this 3 user network. We only plot a subset of scenarios since theother values of distance d perform as expected and lie close to the curvesof d = 2λ and d = 5λ. Note: The transmission scheme for each line isannotated directly on the figure. The different measurement scenarios appliedto each scheme can be identified by the markers as shown in the legend.

both node configurations, exhibit significantly more spatialcorrelation across antennas, as shown in Fig. 7. This resultsin more aligned channels than the simulated i.i.d. Rayleighchannels, which decreases SNR after alignment. The orderingof curves in Fig. 8 reveals that IA benefits from increasedantenna and user spacing with diminishing returns as usersbecome more widely spread. This trend is consistent with thedecreasing correlation shown in Fig. 7. This is also reflectedin Fig. 9 which shows the performance of IA with co-locatedtransmitters where d = 3λ outperforms d = 1λ and d = 0.5λ.

Though the trend of increasing sum rate with antennaspacing is noticeable, this is less evident at low levels ofcorrelation. For example, in Fig. 8, d = 5λ outperformsd = 2λ, though the latter has lower Kronecker correlation.In reality, the performance of IA, is more tightly related tothe distances between the signal and interference subspacesin the system. The direct link subspaces and IA performancemakes antenna spacing and traditional correlation measuresonly a crude tool for comparison. We also note the differencein performance between the configurations of Fig. 4 and 5when antenna spacing is fixed at 0.5λ. As a result, the relativeimportance of antenna vs. user spacing requires further study.We later discuss other correlation measures that are shown, bysimulation and measurement, to be more closely related to theperformance of IA and can help us characterize the relativeimportance of both antenna and user spacing.

In addition to confirming interference alignment’s theoret-ical achievements, our measurements give insight into thefeasibility of adopting iterative algorithms in static indoordeployments. Though these algorithms may require manyiterations to converge, static channels allow the precoding ma-trices to be used over many successive packet transmissions.This fact minimizes the relative overhead incurred by using

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

SNR (dB)

Sum

Rat

e b/

s/H

z

d=0.5λd=1λd=3λSimulated Rayleigh

IA

TDMA

Interference Avoidance

Fig. 9. Network sum rate for IA and TDMA Vs. SNR for the configurationwith co-located transmitters shown in Fig. 5 in an indoor environment. Note:The transmission scheme for each line is annotated directly on the figure. Thedifferent measurement scenarios applied to each scheme can be identified bythe markers as shown in the legend.

Fig. 10. Area surrounding the Engineering Science Building where the out-door measurements were taken. The TX/RX hexagons outline the approximateareas in which the 6 transceivers were placed in different configurations anddo not imply actual co-location (i.e. antennas were placed around that regionwith sufficient antenna and user spacing).

iterative algorithms.

B. Outdoor Results

We conduct our outdoor experiments in the area surroundingthe Engineering Science Building2 shown in Fig. 10. Theenvironment contains several buildings of steel reinforcedconcrete, two aluminum annexes, as well as other impedingand reflective objects normally present in a typical outdoorenvironment making it a good representative area to study theperformance of IA in.

Transmitters and receivers are placed approximately 200ftapart in both line-of-sight and non-line-of-sight arrangements.To support this long range transmission we use 500mW poweramplifiers to maintain a receive SNR of 25dB, which allowsus to predict performance up to an SNR of 41dB as shownin Section IV. Fig. 11 shows an example frequency plot of

2The image is taken from Google Earth ( c©2009 Tele Atlas).

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 10

0 50 100 150 200−4

−2

0

2

4

6

8

10

Subcarrier Index

Cha

nnel

Mag

nitu

de (d

B)

Fig. 11. A sample magnitude plot of an element H1,1 in an outdoor NLOSarrangement. This plot shows significantly more selectivity than the channelof Fig. 6.

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

SNR (dB)

Sum

Rat

e b/

s/H

z

LOS d=1λLOS d=4λNLOS d=4λSimulated Rayleigh

IA

TDMA

Interference Avoidance

Fig. 12. Network Sum Rate versus SNR for IA, TDMA, and interferenceavoidance in our outdoor measurement scenarios. Note: The transmissionscheme for each line is annotated directly on the figure. The differentmeasurement scenarios applied to each scheme can be identified by themarkers as shown in the legend.

the first element of H1,1 and verifies that the outdoor channelindeed exhibits more multipath than the indoor one. Examiningthe power delay profile reveals the presence of 5 channel taps,resulting in a wideband channel with a coherence bandwidthof 3.2 MHz.

Fig. 12 confirms all conclusions drawn from the previousindoor results. Observing the curves in Fig. 12, we again noticethat lower correlation yields better sum rates. The NLOSscenarios, with a receive correlation coefficient for user 2,for example, of 0.105, performs significantly better than theLOS scenarios with antenna separations of 1λ and 4λ, whichshow receive correlations of 0.166 and 0.138 respectively.Moreover, due to an increased reliance on multipath, NLOS

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

SNR (dB)

Sum

Rat

e (b

/s/H

z)

IA Indoor d=0.5λIA Indoor d=2λIA Outdoor NLOS d=4λIA Simulated RayleighMAX SINR Indoor d=0.5λMAX SINR Indoor d=2λMAX SINR Outdoor NLOS d=4λMAX SINR Sim. Rayleigh

Fig. 13. Sum rate Vs. SNR plots for interference alignment and the MAXSINR Algorithm. While both algorithms converge at high SNR, MAX SINRoutperforms IA at low SNR when noise is a significant limiting factor in thenetwork.

channels vary more in space, resulting in lower correlationacross users. The dependence of sum rate on correlation acrossusers makes the Kronecker correlation coefficients not the besttool for comparison since they say very little about the distancebetween the signal of interest and the interference subspaces.We later show that matrix collinearity and subspace distanceare more tightly related to the performance of IA and, thus,can be used to better characterize IA’s performance.

Fig. 12 also confirms that IA outperforms TDMA andachieves the maximum degrees of freedom in the three userinterference channel and is optimal in the high SNR regime.In addition to outperforming orthogonal techniques such asTDMA, Fig. 12 shows that when interference power is equal tothe received signal power, IA outperforms greedy interferenceavoidance [25]. In many realistic ad hoc network deployments,however, communicating nodes are likely to be positionedclose to one another, thus receiving different signal and inter-ference powers. This fact makes the comparison of IA, whichdoes not benefit from low interference power, to interferenceavoidance, which clearly benefits from lower interference, in asetup where all channel gains are equal, an unfair comparison.For example, one can show that at an SIR level of about 10dB,IA only outperforms interference avoidance at SNR valueshigher than 20dB. Therefore, depending on the received SIRlevels, a network may choose to align or avoid interference.When the network is noise limited, interference is insignificantand can be avoided, but as SNR increases, communicationbecomes interference limited and IA dominates. IA’s subopti-mality at low SNR also motivates the MAX SINR algorithmdescribed in Section III-C.

Fig. 13 plots the sum rates achieved by the SINR maximiz-ing algorithm in selected indoor and outdoor scenarios. Asexpected, the SINR maximizing algorithm outperforms inter-ference alignment, which is oblivious to the signal power inthe interference free space. Also as expected, this performancegap is most noticeable in the low-to-medium SNR regime and

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 11

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

SNR (dB)

Itera

tions

Indoor d=0.5λIndoor d=2λOutdoor LOS d=4λSimulated Rayleigh

Fig. 14. Number of iterations needed for MAX SINR to outperformIA. As SNR increases, the number of iterations the algorithm takes tooutperforms closed form IA increases superlinearly. We note, however, thatour measurements indicate that Rayleigh simulations may overestimate thenumber of iterations needed to outperform IA by as much as a factor 3.

decays as we transition to increasingly higher SNR. In additionto smaller improvement at high SNR, the cost of this algorithmin this regime increases as well. Fig. 14 clearly shows theincreasing number of iterations needed for MAX SINR tostart outperforming closed form IA. The same can be saidabout absolute convergence: as SNR increases, the numberof iterations needed for convergence increases. While theseobservations can be seen without measurements, Figs. 13 and14 show more.

From Fig. 13, we see that highly correlated channels such asthe case of d = 0.5λ exhibit a bigger increase in sum rate byusing the MAX SINR algorithm instead of IA. This furtherhighlights the suboptimality of IA when the i.i.d. channelassumption is farther from reality. Moreover, Fig. 14 indicatesthat not only does MAX SINR outperform IA more in corre-lated channels, but it also does so faster. While an average ofabout 150 iterations are needed till MAX SINR outperformsIA in simulated Rayleigh channels at SNR=40dB, only 55 areneeded when channels are highly correlated, approximately a3 fold difference. As a result, the performance of MAX SINRis underestimated in simulation. The convergence analysis ofMAX SINR and the reason why it appears to be faster andbetter in correlated channels is an open problem.

After presenting the immediate conclusions that can bedrawn from our measurements, we return to the effect ofchannel correlation on sum rate. While a general trend ofincreasing performance with lower Kronecker correlation wasobserved in Fig. 8, 9 and 12, this is common to most MIMOtechniques. IA, however, in addition to being affected by thecondition of each user’s channel, relies on cross channelsfor alignment. Therefore, correlation across users is likely toaffect IA’s performance even more. We study the effect of twoproposed channel “correlation” measures which are shown, viameasurement and simulation, to more closely influence theperformance of IA: channel matrix collinearity and subspace

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10

5

10

15

20

25

30

35

40

Maximum Matrix Collinearity

Sum

Rat

e b/

s/H

z

Simulated RayleighMeasured Channels

Fig. 4d=0.5λ

Fig. 5d=0.5λ

Fig. 15. The effect of channel collinearity on IA performance (SNR =40dB). As collinearity increases, column spaces are more aligned, which inturn reduces SNR after alignment. Our measurements indicate that as usersget closer, channel collinearity increases and sum rate decreases.

distance.Fig. 15 shows the performance of IA versus channel

collinearity. It can be seen that all our measurements, indoorand outdoor exhibit decreasing performance with collinearity.To confirm this relationship, and since our measurementscannot span the entire range of channel collinearity, we alsoplot the performance over simulated Rayleigh channels withvarying levels of collinearity. The match between our mea-surements and the simulated trend makes channel collinearitya simple feature that can be used to predict performance.Moreover, smaller values of the user spacing, d, result inhigher collinearity and thus lower sum rate. Comparing theresults for d = 0.5λ, we notice that the configuration ofFig. 5 outperforms that of Fig. 4. In this configuration thereceivers remain separated, exhibiting lower collinearity. Therelationship between sum rate and collinearity is not withoutreason. The factor directly controlling the achieved sum rateis SNR after alignment, which is a function of the distancebetween the chosen signal and interference subspaces. Whilecollinearity is affected by the ordering of the columns of ma-trices, it is a measure of the similarity of column spaces. Highcollinearity translates into highly aligned signal subspacesand, consequently, lower SNR after alignment. For example,considering the worst case of perfectly aligned channels, wesee that the precoders in (6), (7), and (8) yield signals thatall lie in the same subspace, which drastically decreases theachieved sum rate.

As stated earlier, collinearity is sensitive to matrix columnordering and does not directly measure subspace distance.This motivates the use of the distances defined in (19) and(20). Fig. 16 plots the average sum rate achieved by IA overmeasured channels vs. the distances defined in eqs. (19) and(20). Again, to demonstrate the validity of this relationship,we plot the performance of simulated Rayleigh channels withvarying subspace distance. We see that as the signal and

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

Average Projection F−norm Distance

Sum

Rat

e b/

s/H

z

Simulated Channel Distance (Eq. 20)Simulated Effective Channel Distance (Eq. 19)Measured Effective Channel Distance (Eq. 19)Measured Channel Distance (Eq. 20)

Fig. 4d=0.5λ

Fig. 5d=0.5λ

Fig. 16. The effect of the average channel and effective channel distance asdefined in (20) and (19) respectively, on IA performance. Our measurementsindicate that as users come closer, so do the channel subspaces. As a result,both SNR after alignment and sum rate decrease.

interference subspaces become farther apart, IA’s performanceincreases. This can be explained by considering a linear IAreceiver, which projects onto the basis of the interference freespace to decode. The more aligned the signal and interferencespaces are, the smaller the signal component in the interferencefree subspace will be, diminishing post-projection SNR andsum rate.

A similar monotonic relationship between sum rate and thedistance between the subspaces of the channels themselves,defined in (20), confirms the intuition that closer aligned chan-nels result in closer aligned signal and interference subspaces.The metrics used in Fig. 16, though more complicated, arebetter estimators of the performance than matrix collinearity.This can be seen by noting that the curves in Fig. 16,have a well behaved derivative over a larger subset of theirdomain, i.e. have a less pronounced cut off behavior for highlycorrelated channels. As a result, slight changes in subspacedistance, even close to zero, can more accurately estimateincremental changes in sum rate. We end by saying thatin summary we have showed that “cross-user correlation”influences sum rate more directly. Therefore, IA is expectedto perform better when minimizing cross-correlation betweenchannels takes priority over the typical antenna separation.Although the connection between user spacing and collinearityor subspace distance is not in the scope of this paper, ourresults indicate that users positioned close to each other, ingeneral, result in worse channels, from an IA perspective.

VI. CONCLUSION AND FUTURE WORK

In this paper, we have presented the first MIMO interferencechannel testbed programmed using a flexible software definedradio base. We have presented indoor and outdoor networkchannel measurements collected in our university environment,and then processed them to evaluate the performance gainsof IA. We showed that the observed gains are in agreement

with those found through theory and simulation. We alsocharacterized the effect of channel imperfections on the sumrate achieved by IA. In subsequent work we will extend themeasurement setup beyond three users as well as implementa real time closed loop IA system. This effort will be amajor step in transforming IA from a theoretical concept, toa solution for large scale ad hoc networks.

REFERENCES

[1] V. Cadambe and S. Jafar, “Interference alignment and degrees offreedom of the K-user interference channel,” IEEE Trans. Inf. Theory,vol. 54, no. 8, pp. 3425–3441, August 2008.

[2] M. Maddah-Ali, A. Motahari, and A. Khandani, “Signaling overMIMO multi-base systems: combination of multi-access and broadcastschemes,” Proc. of IEEE International Symposium on Information The-ory, Seattle, WA, pp. 2104–2108, July 2006.

[3] S. Jafar and M. Fakhereddin, “Degrees of freedom for the MIMOinterference channel,” Proc. of IEEE International Symposium on In-formation Theory, Seattle, WA, pp. 1452–1456, July 2006.

[4] K. Gomadam, V. Cadambe, and S. Jafar, “Approaching the capacity ofwireless networks through distributed interference alignment,” Proc. ofIEEE Global Telecommunications Conference, New Orleans, LA, pp.1–6, December 2008.

[5] S. W. Peters and R. W. Heath, Jr., “Interference alignment via alternatingminimization,” Proc. of IEEE International Conference on Acoustics,Speech, and Signal Processing, pp. 2445 –2448, April 2009.

[6] R. Tresch, M. Guillaud, and E. Riegler, “On the achievability of inter-ference alignment in the K-User constant MIMO interference channel,”Proc. of IEEE Workshop on Statistical Signal Processing, pp. 277–280,September 2009.

[7] C. Suh and D. Tse, “Interference alignment for cellular networks,” Proc.of Allerton Conference on Communication, Control, and Computing,Monticello, IL, Sept. 2008.

[8] S. W. Choi, S. Jafar, and S.-Y. Chung, “On the beamforming designfor efficient interference alignment,” Communications Letters, IEEE,vol. 13, no. 11, pp. 847 –849, nov. 2009.

[9] I. Thukral and H. Bolcskei, “Interference alignment with limited feed-back,” Proc. of IEEE International Symposium on Information Theory,Seoul, Korea, July 2009.

[10] C. Yetis, T. Gou, S. Jafar, and A. Kayran, “Feasibility conditions forinterference alignment,” Global Telecommunications Conference, 2009.GLOBECOM 2009. IEEE, pp. 1 –6, nov. 2009.

[11] C. Huang, S. A. Jafar, S. Shamai, and S. Vishwanath, “On degreesof freedom region of MIMO networks without CSIT,” 2009. [Online].Available: http://www.citebase.org/abstract?id=oai:arXiv.org:0909.4017

[12] R. Tresch and M. Guillaud, “Cellular interference alignment withimperfect channel knowledge,” IEEE International Conference on Com-munications (ICC), Workshop on LTE Evolution, Dresden, Germany,June 2009.

[13] M. Aldridge, O. Johnson, and R. Piechocki, “Asymptotic sum-capacityof random gaussian interference networks using interference alignment,”Information Theory Proceedings (ISIT), 2010 IEEE International Sym-posium on, pp. 410 –414, jun. 2010.

[14] J. Koivunen, P. Almers, V.-M. Kolmonen, J. Salmi, A. Richter, F. Tufves-son, P. Suvikunnas, A. Molisch, and P. Vainikainen, “Dynamic multi-linkindoor MIMO measurements at 5.3GHz,” Proc. of European Conferenceon Antennas and Propagation, Edinburgh, UK, pp. 1–6, November 2007.

[15] F. Kaltenberger, M. Kountouris, D. Gesbert, and R. Knopp, “On thetradeoff between feedback and capacity in measured MU-MIMO chan-nels,” IEEE Trans. Wireless Commun., vol. 8, no. 9, pp. 4866–4875,September 2009.

[16] G. Bauch, J. Bach Andersen, C. Guthy, M. Herdin, J. Nielsen, J. Nossek,P. Tejera, and W. Utschick, “Multiuser MIMO channel measurementsand performance in a large office environment,” Proc. of IEEE WirelessCommunications and Networking Conference, Hong Kong, pp. 1900–1905, March 2007.

[17] S. Gollakota, S. D. Perli, and D. Katabi, “Interference alignment andcancellation,” SIGCOMM Computer Communication Review, vol. 39,no. 4, pp. 159–170, 2009.

[18] O. El Ayach, S. W. Peters, and R. W. Heath, Jr., “Real world feasibilityof interference alignment using MIMO-OFDM channel measurements,”Proc. of IEEE Conference on Military Communications, Boston, MA,pp. 1–6, October 2009.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 13

[19] G. Stuber, J. Barry, S. McLaughlin, Y. Li, M. Ingram, and T. Pratt,“Broadband MIMO-OFDM wireless communications,” Proceedings ofthe IEEE, vol. 92, no. 2, pp. 271–294, Feb 2004.

[20] A. van Zelst and T. Schenk, “Implementation of a MIMO OFDM-basedwireless lan system,” IEEE Trans. Signal Process., vol. 52, no. 2, pp.483–494, Feb. 2004.

[21] H. Bolcskei, D. Gesbert, and A. Paulraj, “On the capacity of wirelesssystems employing OFDM-based spatial multiplexing,” IEEE Trans.Commun., vol. 50, pp. 225–234, February 2002.

[22] R. Blum, “MIMO capacity with interference,” IEEE Journal on SelectedAreas in Communications, vol. 21, no. 5, pp. 793–801, 2003.

[23] I. Santamaria, O. Gonazales, R. W. Heath, Jr., and S. W. Peters, “Max-imum sum-rate interference alignment algorithms for mimo channels,”to appear in Proc. of IEEE Global Telecommunications Conference,Miami, Fl, Dec. 2010.

[24] S. W. Peters and R. W. Heath, Jr., “Cooperative algorithms for MIMOinterference channels,” CoRR, vol. abs/1002.0424, 2010.

[25] C. Rose, S. Ulukus, and R. Yates, “Wireless systems and interferenceavoidance,” IEEE Trans. Wireless Commun., vol. 1, July 2002.

[26] A. Paulraj, R. Nabar, and D. Gore, Intro. to Space-Time WirelessCommunications. New York, NY, USA: Cambridge University Press,2008.

[27] S. Ye and R. Blum, “Optimized signaling for MIMO interferencesystems with feedback,” IEEE Trans. Signal Process., vol. 51, no. 11,pp. 2839–2848, Nov 2003.

[28] G. Johnson and R. Jennings, LabVIEW graphical programming.McGraw-Hill Professional, 2006.

[29] I. Barhumi, G. Leus, and M. Moonen, “Optimal training design forMIMO OFDM systems in mobile wireless channels,” IEEE Trans. SignalProcess., vol. 51, no. 6, pp. 1615–1624, 2003.

[30] E. Zhou, X. Zhang, H. Zhao, and W. Wang, “Synchronization algorithmsfor MIMO OFDM systems,” Proc. of IEEE Wireless Communicationsand Networking Conference, New Orleans, LA, vol. 1, pp. 18–22 Vol.1, March 2005.

[31] National Instruments, “NI PXI-1045 Data Sheet,” 2009. [Online].Available: http://www.ni.com/pdf/products/us/pxi1045.pdf

[32] ——, “NI PXI-5670 Data Sheet,” 2009. [Online]. Available: http://www.ni.com/pdf/products/us/5670 datasheet.pdf

[33] ——, “NI PXI-5660 Data Sheet,” 2009. [Online]. Available: http://www.ni.com/pdf/products/us/4mi469-471.pdf

[34] A. Gupta, A. Forenza, and R. W. Heath, Jr., “Rapid MIMO-OFDMsoftware defined radio system prototyping,” Proc. of IEEE Workshop onSignal Processing Systems, October 2004.

[35] National Instruments, “NI PXI-6653 Data Sheet.” [Online]. Available:http://www.ni.com/pdf/products/us/pxi665x pxie6672 datasheet.pdf

[36] J. Wallace, M. Jensen, A. Swindlehurst, and B. Jeffs, “Experimentalcharacterization of the MIMO wireless channel: Data acquisition andanalysis,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 335–343,2003.

[37] N. Czink, B. Bandemer, G. Vilar, L. Jalloul, and A. Paulraj, “Can multi-user MIMO measurements be done using a single channel sounder?”COST 2100, TD (08), Lille, France, October 2008, vol. 621.

[38] G. Golub and C. Van Loan, Matrix computations. Johns Hopkins UnivPr, 1996.

[39] A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithmswith orthogonality constraints,” SIAM Journal on Matrix Analysis andApplications, vol. 20, no. 2, pp. 303–353, 1998.

[40] C. A. Balanis, Antenna Theory: Analysis and Design. New York, NY:Wiley, 1997.

Omar El Ayach (S08) received his B.E. degree incomputer and communications engineering from theAmerican University of Beirut, Lebanon, in 2008.He completed his M.S. in electrical engineeringdegree at The University of Texas at Austin in 2010.

He is now a Ph.D. student in the Wireless Net-working and Communication Group (WNCG) at TheUniversity of Texas at Austin under the supervisionof Prof. Robert W. Heath, Jr. in the Wireless Systemsand Innovation Lab (WSIL) group. His researchinterests are in the broad area of MIMO signal

processing and information theory, in particular interference managementtechniques in wireless systems and network sciences.

Steven W. Peters is a Ph.D. student at the Univer-sity of Texas at Austin and Co-Founder and CEOof Kuma Signals, LLC. He received B.S. degreesin electrical engineering and computer engineeringfrom the Illinois Institute of Technology in 2005 andan M.S.E. degree in electrical engineering from theUniversity of Texas at Austin in 2007. From 2005–2007 he was a research assistant at the AppliedResearch Laboratories, where he worked on phys-ical layer and coding design for ground-wave highfrequency transhorizon communication systems. He

has served as a consultant to several companies working on wireless systemdesign and standards. His research interests include interference mitigationtechniques in wireless networks, cooperative communication, and MIMO.

Robert W. Heath, Jr. (S96-M01-SM06) receivedthe B.S. and M.S. degrees from the University ofVirginia, Charlottesville, in 1996 and 1997, respec-tively, and the Ph.D. degree from Stanford Univer-sity, Stanford, CA, in 2002, all in electrical engi-neering.

From 1998 to 2001, he was a Senior Member ofthe Technical Staff then a Senior Consultant withIospanWireless Inc., San Jose, CA, where he workedon the design and implementation of the physicaland link layers of the first commercial MIMO-

OFDM communication system. In 2003, he founded MIMO Wireless Inc., aconsulting company dedicated to the advancement of MIMO technology. SinceJanuary 2002, he has been with the Department of Electrical and ComputerEngineering, The University of Texas at Austin, where he is currently anAssociate Professor and Associate Director of the Wireless Networking andCommunications Group. His research interests include several aspects ofMIMO communication: limited feedback techniques, multihop networking,multiuser MIMO, antenna design, and scheduling algorithms, as well as 60-GHz communication techniques and multimedia signal processing.

Prof. Heath has been an Editor for the IEEE TRANSACTIONS ON COM-MUNICATIONS and an Associate Editor for the IEEE TRANSACTIONS ONVEHICULAR TECHNOLOGY. He is a member of the Signal Processing forCommunications Technical Committee in the IEEE Signal Processing Societyand is the Vice Chair of the IEEE COMSOC Communications TechnicalTheory Workshop, is a general organizer for the 2009 CAMSAP Conference,and was a Technical Co-Chair for the 2010 IEEE International Symposiumon Information Theory. He is the recipient of the David and Doris LybargerEndowed Faculty Fellowship in Engineering. He is a licensed Amateur RadioOperator and is a registered Professional Engineer in Texas.