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Beach, MA

Bengtsson, M

Karlsson, PB

Mcnamara, DP

Ottersten, B

Yu, K

English

Herein, results from measurements conducted by the University of Bristol are presented. The channel characteristics of Multiple Input Multiple Output (MIMO) indoor systems at 5.2 GHz are studied. Our investigation shows that the envelope of the channel for non-line-of-sight (NLOS) indoor situations are approximately Rayleigh distributed and consequently we focus on a statistical description of the first and second order moments of the narrowband MIMO channel. Furthermore, it is shown that for NLOS indoor scenarios, the MIMO channel covariance matrix can be well approximated by a Kronecker product of the covariance matrices describing the correlation at the transmitter and receiver side respectively. A statistical narrowband model for the NLOS indoor MIMO channel based on this covariance structure is presented.</p

Yu, Kai,

Bengtsson, Mats,

Ottersten, Björn,

McNamara, Darren P.,

Karlsson, Peter,

Beach, M.,

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Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements

Kai Yu

Mats Bengtsson

Björn Ottersten

Darren Mcnamara

Peter Karlsson

Mark Beach

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Herein, results from measurements conducted by the University of Bristol are presented. The channel characteristics of multiple input multiple output (MIMO) indoor systems at 5.2 GHz are studied. Our investigation shows that the envelope of the channel for non-line-of-sight (NLOS) indoor situations are approximately Rayleigh distributed and consequently we focus on a statistical description of the first and second order moments of the narrowband MIMO channel. Furthermore, it is shown that for NLOS indoor scenarios, the MIMO channel covariance matrix can be well approximated by a Kronecker product of the covariance matrices describing the correlation at the transmitter and receiver side respectively. A statistical narrowband model for the NLOS indoor MIMO channel based on this covariance structure is presented

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                          Yu, K., Bengtsson, M., Ottersten, B., Mcnamara, D. P., Karlsson, P. B., &Beach, M. A. (2001). Second order statistics of NLOS indoor MIMOchannels based on 5.2 GHz measurements. In IEEE Globecom 2001. (Vol. 1,pp. 156 - 160). Institute of Electrical and Electronics Engineers (IEEE).10.1109/GLOCOM.2001.965098Link to published version (if available):10.1109/GLOCOM.2001.965098Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Second Order Statistics of NLOS Indoor MIMO ChannelsBased on 5.2 GHz MeasurementsKai Yu1, Mats Bengtsson1, Björn Ottersten1, Darren McNamara2, Peter Karlsson3 and Mark Beach21Department of Signals, Sensors and Systems,Royal Institute of Technology, Stockholm, Swedenemail: kaiyu@s3.kth.se2Center for Communication ResearchUniversity of Bristol, United Kingdom3Telia Research AB, Malmö, SwedenAbstract – Herein, results from measurements conducted by theUniversity of Bristol are presented. The channel characteristics ofMultiple Input Multiple Output (MIMO) indoor systems at 5.2 GHzare studied. Our investigation shows that the envelope of the chan-nel for non-line-of-sight (NLOS) indoor situations are approximatelyRayleigh distributed and consequently we focus on a statistical de-scription of the first and second order moments of the narrowbandMIMO channel. Furthermore, it is shown that for NLOS indoor sce-narios, the MIMO channel covariance matrix can be well approxi-mated by a Kronecker product of the covariance matrices describingthe correlation at the transmitter and receiver side respectively. A sta-tistical narrowband model for the NLOS indoor MIMO channel basedon this covariance structure is presented.I. INTRODUCTIONIt is well known that using antenna arrays at both the transmit-ter and receiver over a MIMO channel can provide very highchannel capacity as long as the environment has sufficientlyrich scattering. Under these circumstances, the channel matrixelements have low correlation leading to channel realizationsof high rank and consequently provide substantial channel ca-pacity increases. In [1] and [2] the channel capacity for MIMOsystems has been investigated theoretically. Some experimen-tal investigations have also been carried out trying to charac-terize the MIMO channel and the corresponding capacity, see[3, 4, 5, 6, 7].There is of course great interest in modeling of MIMOchannels. A so-called one-ring model has been proposed andinvestigated in [8]. In [9], a distributed scattering model hasThis work is conducted in part within SATURN (Smart Antenna Tech-nology in Universal bRoadband wireless Networks) funded by the EU ISTprogram.been proposed in order to explain the pinhole phenomenonthat may appear in outdoor situations. In [10], a model basedon channel power correlation coefficients is presented. How-ever, models based on experimental data are still rare. In [11],the initial results for the NLOS MIMO channel measurementshave been reported.This paper reports the results based on data measured inUniversity of Bristol as part of the SATURN project. An ini-tial statistical model based on the experimental data is also pre-sented. This paper is organized as follows: Section II gives abrief description of the measurement setup including the testenvironment and the test equipment. In Section III, a statisticalstructure is proposed and a least squares rank one approxima-tion method is described in order to separate a matrix into theKronecker product of two sub-matrices. Section IV presentsa statistical model based on this covariance structure and theresults from the model identification for different transmitterlocations. Finally we draw some conclusions in Section V.II. MEASUREMENT SYSTEMThe test site is the Merchant Venture’s Building of the Uni-versity of Bristol. The general layout of the test site includesoffice rooms, computer labs, corridors and open spaces. Theentire measurements include 15 transmitter locations and 3 re-ceiver locations. Both line-of-sight (LOS) and NLOS caseswere measured. Notice that in this paper, all the results arefrom NLOS cases as shown in Fig. 1. Here, the arrow at eachtransmitter location indicates the orientation of the transmit ar-ray. The transmitter was located in a computer lab and thereceiver was located in a large modern office with cubicles.The measurements were carried out using the Medav RUSKBRI vector sounder, which has an 8-element omnidirectional1560-7803-7206-9/01/$17.00 © 2001 IEEEvoidRoomEntrancePortercorridor ScienceComputerScienceComputer(computers)Computer RoomComputerScienceTx13ArrayofficeRoomArrayRx3voidvoid20 mFigure 1. Measurement scenario for NLOS indoor MIMOchanneluniform linear array (ULA) at the transmitter side and an 8-element ULA with 120◦ beamwidth at the receiver side (forpictures, see [7]). The distance between two neighboring an-tenna elements is 0.5λ for both arrays. There is a feedbackfrom the receiver to the transmitter by a cable in order to syn-chronize the transmitter and receiver.The measurements were centered at 5.2 GHz. A periodicmultifrequency signal with 120 MHz bandwidth was sent outby the transmitter and captured by the receiver. The chan-nel impulse response was then sampled and saved in the fre-quency domain. The maximum expected channel excess de-lay was 0.8µs, corresponding to 97 frequency subchannels.For each transmit element, one ‘vector snapshot’ (one mea-surement from each receive element) is taken by the receiverthrough switching control circuits. The sampling time for oneMIMO snapshot (8 vector snapshots) is 102.4µs, which is wellwithin the coherence time. One complete measurement in-cludes 199 blocks with 16 MIMO snapshots within each block,therefore there are 3184 complete MIMO snapshots in totalfor each frequency subchannel. The time delay between twoneighboring blocks is 26.624ms. This means the total time forone complete measurement is 5.3s. During the measurements,people were moving around both at the transmitter and receiverside.III. MEASUREMENT ANALYSIS METHODA. Data Model and Second Order StatisticsAssume there are m transmit elements and n receive elements.For a narrowband MIMO channel, the input-output relation-ship could be expressed in the baseband asy(t) = Hmn s(t) + n(t), (1)where s(t) is the transmitted signal, y(t) is the received signaland n(t) is additive white Gaussian noise. The channel matrixHmn here is an n by m matrix.We define the transmitter, receiver and channel covariancematrices asRTxH = E[(h∗i hi)T ] for i = 1, . . . , n (2)RRxH = E[hjhj∗] for j = 1, . . . , m (3)RH = E[vec(Hmn )vec(Hmn )∗], (4)wherevec(Hmn ) = [(h1)T , (h2)T , . . . , (hm)T ]T , (5)hi is ith row of Hmn , hj is jth columns of Hmn , (·)∗ is com-plex conjugate transpose, (·)T is transpose and E[·] denotes theexpected value.In [10], it is claimed that the correlation between the powerof two subchannels could be modeled by the product of thecorrelations seen from the transmitter and receiver. Here we tryto verify whether this structure could be extended to describealso the phase of the complex valued amplitudes. Notice thatfor the normalized channel matrix Hmn , this structure can alsobe expressed asRH = RTxH ⊗RRxH , (6)where RH is the channel covariance matrix, RTxH and RRxHare the covariance matrices at the transmitter and receiver sidedefined in (4), (2), (3) respectively and ‘⊗’ denotes the Kro-necker product.B. Normalization MethodIt is well known that when the transmitted power is equallyallocated to each transmit element, the channel capacity maybe expressed as in [2]C = log2det(In +ρmHH∗), (7)where H is the normalized channel matrix and ρ is the aver-age signal-to-noise-ratio (SNR) at each receiver branch. Tonormalize the channel matrix from data, we use the same nor-malization factor for all realizations such that1NN∑i=1‖ Hi ‖2F = nm, (8)where ‖ · ‖F denotes the Frobenius norm.157C. Least Squares Kronecker FactorizationFitting the channel covariance matrix RH optimally into theKronecker product of two Hermitian matrices X and Y boilsdown to solving the following problemminX,Y‖ RH −X⊗Y ‖F . (9)The least squares rank one approximation method in [12]can be used to solve this problem. The main idea is to re-arrange the elements of the covariance matrix, RH , and X⊗Yto get a least squares problem of the formmin ‖ Rtran − x(yc)∗ ‖F , (10)wherex = vec(X), (11)y = vec(Y), (12)(·)c means complex conjugate and Rtran is the transformed ma-trix. To obtain the transformed matrix Rtran, we use the permu-tation matrix T, defined such thatT vec(X⊗Y) = vec(xyT ) (13)for all matrices X and Y. The matrix Rtran is then defined byvec(Rtran) = T vec(RH). (14)Note that (9) and (10) are equivalent since T is orthonormal.The solution to the least squares rank one approximation ofRH in (10) is easily calculated using the singular value decom-position (SVD) [13]. Let λmax as the largest singular value ofRtran, and umax and vmax as the left and right singular vectorcorresponding to λmax. Then, x and y can be expressed asx = γumax, (15)yc = γ−1λmaxvmax, (16)where γ is an arbitrary scalar and its value depends on what Xand Y should be. Then it is straightforward to transform thetwo vectors x and yc into X and Y inversely.It can be shown that the solution X and Y will alwaysbe Hermitian as long as RH is Hermitian, therefore it is notnecessary to force that structure on the solution explicitly.0 0.5 1 1.5 2 2.5 3 3.5 400.0050.010.0150.020.0250.03envelope of one channel coefficientFigure 2. Histogram of the envelope of one channel coeffi-cient for NLOS indoor MIMO scenario, Tx13-Rx3 and the fit-ted Rayleigh distribution envelope (normalized)IV. MEASUREMENT RESULTSAs stated in section II, the measurements use an 8-elementtransmitter and receiver. In this paper, pairs of 2 and 3 neigh-boring elements at both the transmitter and receiver have beenused as an example to get 2x2 and 3x3 MIMO channel realiza-tions that are used to analyze the second order statistics of thechannel matrix. It is interesting to find that even though peoplewere moving around during the measurements, the time varia-tions on a single link of each narrowband subchannel are stillnot enough to give an average of zero, indicating a quite sta-tionary scenario. Therefore both the frequency and spatial do-main have been averaged, i.e. all the snapshots from differentfrequency subchannels and spatial 2x2/3x3 pairs (at the sameTx and Rx locations) have been seen as different channel real-izations in order to get sufficient data to study both the channeldistribution and second order statistics. Reasonably good re-sults have been found for other setups with up to 5 elementsat each side. The data are not enough to calculate setups withmore elements due to the insufficient spatial average. In thefollowing section, the data measured at Tx13 – Rx3 are used,see Fig. 1. Similar results have been found in the other fourtransmitter locations, see Fig. 1.A. Distribution of the channel coefficientThe distribution of the channel coefficient has been investi-gated by plotting the histogram of the envelope of one channelcoefficient and the cumulative density function (CDF) of thephase of one channel coefficient. Fig. 2 and Fig. 3 shows the158−200 −150 −100 −50 0 50 100 150 20000.10.20.30.40.50.60.70.80.91Cumulative Density FunctionPhase (degree)Measured Data             Ideal Uniform DistributionFigure 3. CDF of the phase of one channel coefficient and theCDF of the uniform distribution within [-180◦,180◦]result of the envelope distribution of one channel coefficientand the CDF of the phase of one channel coefficient comparingwith the fitted Rayleigh distribution and uniform distributionon [−pi, pi] respectively. We concluded that the channel coef-ficient for the NLOS indoor MIMO situation is approximatelyzero-mean complex Gaussian.B. Second order statisticsIt is well known that a complex Gaussian distributed randomvariable is completely characterized by its first and second or-der moments. Although the individual elements of the chan-nel matrix appear Gaussian distributed, we have not examinedthat they are jointly Gaussian. To investigate the second orderstatistics for this NLOS indoor scenario, we define the modelerror, Ψ to evaluate the difference between two matrices A andBΨ(A,B) =‖ A−B ‖F‖ A ‖F. (17)From the measured data, two model errors have been in-vestigated, the results are listed in the first two rows of Ta-ble 1, where the (ˆ·) indicates a sample covariance estimate ofthe corresponding quantities in equations (2), (3), (4) respec-tively and the matrices X and Y are calculated using the leastsquares Kronecker factorization method. It is also interestingto compare the differences between the matrices X, Y and thecovariance matrices RTxH , RRxH in the last two rows of Table1. Here since γ is an arbitrary scalar, the least squares methodhas been used to find γ which fits all elements between twomatrices.Table 1. List of Model Errors2x2 3x3Ψ(RˆH ,X⊗Y) 0.76 % 4.52 %Ψ(RˆH , RˆTxH ⊗ RˆRxH ) 0.86 % 4.79 %Ψ(RˆTxH ,X) 0.40 % 1.74 %Ψ(RˆRxH ,Y) 0.03 % 1.60 %From Table 1, it is clearly shown that the channel covari-ance matrix could be well modeled as shown in equation (6),notice that this structure could explain above 95% of the re-ceived signal power in both cases.C. Statistical Model and SimulationsIf the channel coefficients are complex Gaussian, it is easy toshow from equation (6), as in [8], thatH = (RRxH )1/2G[(RTxH )1/2]T , (18)where G is a stochastic M by N matrix with independent andidentically distributed (IID) CN (0, 1) elements. Here (·)1/2denotes any matrix square root such that R1/2(R1/2)∗ = R.Note that this model is also a special case of the model sug-gested in [9].Monte-Carlo computer simulations have been used to gen-erate 1000 channel realizations and the CDF of the simulatedchannel capacity is compared with that from the measured data.The results for the 2x2 and 3x3 cases are given in Fig. 4.As a reference, the capacity for the IID channel is also in-cluded. From the figure, it is shown that the simulated CDFfits the CDF of measured data quite well, which agrees withour model.V. CONCLUSIONSWe access experimental MIMO channel measurements at 5.2GHz. Our investigation shows that for NLOS indoor MIMOscenarios, the elements of the narrowband channel matrix arezero-mean complex Gaussian. Furthermore, it is shown thatthe channel covariance matrix can be well approximated by theKronecker product of the covariance matrices at the transmit-ter and receiver side respectively. We also introduce a narrow-band model for the NLOS indoor MIMO channel based on thissecond order statistical structure. Monte-Carlo simulations ofthe channel capacity have shown good agreement between themeasured data and the statistical model.REFERENCES[1] I. Emre Telatar, “Capacity of multi-antenna Gaussianchannels,” European Transactions on Telecommunica-tions, vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999.1595 10 15 20 2510−210−1100MIMO Channel Capacity (bits/s/Hz)Cumulative Density FunctionStatistical ModelMeasured Data    IID Channel      2x2 3x3 Figure 4. Cumulative density function of channel capacity formeasured data, statistical model and IID MIMO channel. TheSNR at the receiver side is 20dB. Power is equally allocated tothe transmit elements[2] G. J. Foschini and M. J. Gans, “On limits of wireless com-munications in a fading environment,” Wireless PersonalCommunications, vol. 6, pp. 311–335, 1998.[3] Jean Philippe Kermoal, Laurent Schumacher, Preben E.Mogensen, and Klaus I. Pedersen, “Experimental inves-tigation of correlation properties of MIMO radio channelsfor indoor picocell scenarios,” in Proceedings IEEE Ve-hicular Technology Conference. IEEE VTC Fall, 2000, pp.14–21.[4] Carol C. Martin, Jack H. Winters, and Nelson R. Sollen-berger, “Multiple-Input-Multiple-Output (MIMO) radiochannel measurements,” in Proceedings IEEE VehicularTechnology Conference. IEEE VTC Fall, 2000, pp. 774–779.[5] Rickard Stridh and Björn Ottersten, “Spatial characteriza-tion of indoor radio channel measurements at 5 GHz,” inProceedings IEEE Sensor Array and Multichannel SignalProcessing Workshop, March 2000.[6] Rickard Stridh, Peter Karlsson, and Björn Ottersten,“MIMO channel capacity on a measured indoor radiochannel at 5.8 GHz,” in Proceedings of the Asilomar Con-ference on Signals, Systems and Computers, October 2000.[7] D. P. McNamara, M. A. Beach, P. N. Fletcher, and P. Karls-son, “Initial investigation of multiple-input multiple-output channels in indoor enviroments,” in Proceed-ings IEEE Benelux Chapter Symposium on Communica-tions and Vehicular Technology, Leuven, Belgium, October2000.[8] Da-Shan Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn,“Fading correlation and its effect on the capacity of mul-tielement antenna systems,” IEEE Transactions on Com-munications, vol. 48, no. 3, pp. 502–513, March 2000.[9] D. Gesbert, H. Bölcskei, D. Gore, and A. Paulraj, “MIMOwireless channels: capacity and performance,” in Proceed-ings Global Telecommunications Conference, November2000, vol. 2, pp. 1083–1088.[10] K. I. Pedersen, J. B. Andersen, J. P. Kermoal, and P. Mo-gensen, “A stochastic Multiple-Input-Multiple-Output ra-dio channel model for evaluation of space -time codingalgorithms,” in Proceedings IEEE Vehicular TechnologyConference. Fall, 2000, pp. 893–897.[11] Kai Yu, Mats Bengtsson, Björn Ottersten, Peter Karls-son, Darren McNamara, and Mark Beach, “Measurementanalysis of NLOS indoor MIMO channels,” in IST Mo-bile Communications Summit, Barcelona, Spain, Septem-ber 2001.[12] Peter Strobach, “Low-rank detection of multichan-nel Gaussian signals using block matrix approximation,”IEEE Transactions on Signal Processing, vol. 43, no. 1,pp. 233–242, January 1995.[13] Gene H. Golub and Charles F. Van Loan, Matrix Compu-tations, The Johns Hopkins University Press, London, 3rdedition, 1996.160

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Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements

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