Beamspace MIMO Channel Modeling andMeasurement: Methodology and Results at 28GHz

Akbar Sayeed and John BradyDepartment of Electrical and Computer Engineering

University of Wisconsin - Madison

Abstract—Millimeter-wave (mmW) wireless is as a promis-ing technology for meeting the Gigabit rate and millisecondlatency requirements of emerging 5G applications. This promisefundamentally rests on the large (GHz) bandwidths and high-gain/high-dimensional beamforming possible at mmW frequen-cies. While multi-beam operation is necessary for achievingspatial multiplexing, existing channel measurements are limitedto mechanically pointed horn antennas and single-beam phasedarrays of moderate sizes. In this paper, we report a new mea-surement methodology and initial results based on the conceptof beamspace MIMO channel modeling and communication. Themeasurement results use a novel multi-beam MIMO prototypesystem at 28GHz that uses a lens array for analog beamformingwith a beamwidth of about four degrees. Furthermore, it is thefirst mmW MIMO prototype that enables simultaneous multi-beam channel measurements (and communication). The uniquemeasurement capabilities of the multi-beam MIMO prototypeare discussed and the new modeling methodology is illustratedwith initial indoor measurements.

Index Terms—Millimeter-wave; channel measurements; chan-nel modeling; beamforming; MIMO

I. INTRODUCTION

Emerging millimeter wave (mmW) systems, operating inthe 30-300GHz band, represent a promising opportunity formeeting the growing wireless data rate requirements due toorders-of-magnitude larger available bandwidths and smallwavelengths that enable high-dimensional MIMO (multipleinput multiple output) operation. The large number of MIMOdegrees of freedom can be exploited for a number of criti-cal capabilities including: higher antenna/beamforming gain;higher spatial multiplexing gain; and highly directional com-munication with narrow beams. While the multiplexing gainof conventional MIMO systems has relied on rich multipathpropagation [1], [2], the narrow beams at mmW enable spatialmultiplexing primarily through multiuser point-to-multipoint(P2MP) operation via line-of-sight (LoS) and single-bouncemultipath propagation [3]. Thus, beamforming plays a keyrole in mmW wireless and beamspace channel modeling andsystem design is naturally applicable [4], [5].

There has been significant recent work on mmW chan-nel modeling and measurements at frequencies ranging from28GHz to 73GHz; see, e.g. [6]–[9]. However, these works are

This work is partly supported by the NSF under grants ECCS-1247583, IIP-1444962, ECCS-1548996, and the Wisconsin Alumni Research Foundation.The authors thank Usman Tayyab for his help in making the measurementsreported in this paper.

limited to measurements based on mechanically oriented hornantennas or single-beam phased arrays. While these worksprovide a wealth of useful information, it is widely recognizedthat more work is needed to develop more complete channelmodels. In particular, there is need for multi-beam channelmeasurements since exploiting the spatial multiplexing gainnecessarily requires multi-beam operation.

In this paper, we report initial channel measurement re-sults using a state-of-the-art multi-beam MIMO transceiverarchitecture, called CAP-MIMO, proposed by our group [4],[5]. A CAP-MIMO transceiver shown in Fig. 1(a) performsanalog multi-beamforming using a lens antenna array shownin Fig. 1(b) and offers a new methodology for beamspaceMIMO channel measurements. We have recently completedthe construction of a 28GHz CAP-MIMO prototype capableof simultaneously measuring four spatial channels by electron-ically selecting 4 out of 16 beams. This paper builds on recentresults for line-of-sight (LoS) and pathloss measurements thatwere reported in [10]. Sec. II provides a description of the28GHz CAP-MIMO prototype. A brief relevant review ofbeamspace MIMO theory is presented in Sec. III, followedby a beamspace system model for channel measurements inSec. IV. Sec. V presents the initial channel measurementresults. Concluding remarks are provided in Sec. VI.

II. PROTOTYPE SYSTEM DESCRIPTION

The 28GHz prototype system consists of a CAP-MIMOaccess point (AP) communicating with up to 2 single-antennamobile stations (MSs) as shown in Fig. 1(c). The CAP-MIMO transceiver at the AP uses a 6” circular lens with a16-feed square (4×4) array of antennas on its focal surfacerepresenting Nb = 16 distinct beams; see Fig. 1(b). The CAP-MIMO AP can simultaneous measure p = 4 signals fromthe 16-feed array through p = 4 receive mmW chains, asshown in Fig. 1(a). The feed antennas at the AP and the MSantenna are open-ended WR-15 waveguides. The system hasan RF bandwidth of 1GHz and a communication bandwidthof W = 125MHz.

The MS’s serve as transmitters. Each MS is equipped with asingle mmW transmit chain consisting of a highly stable localoscillator (LO), an I/Q mixer (IQM), a power amplifier with16dB gain, and a bandpass filter. It can support an arbitrarydigital symbol stream at a rate of 125MS/s via an Altera DE4FPGA board and a 16-bit DAC with up to 8x oversampling

(a) (b)

(c) (d)Fig. 1. (a) A CAP-MIMO transceiver. (b) A 16-feed lens array used in theprototype. (c) A 28GHz CAP-MIMO prototype link. (d) Beams associatedwith each of the four channels.

for pulse shaping.At the CAP-MIMO AP receiver, four beams can be si-

multaneously measured, one in each quadrant of 4 feeds; seeFig. 1(d). Each measurement chain can be switched to one ofthe beams in each quadrant using a 1-to-4 switch, and consistsof a bandpass filter, a low-noise amplifier with 23dB gain, andan IQM followed by a 14-bit ADC operating at 125MS/s.All IQMs are driven by the same LO at the AP receiver.The four sampled received signals are fed to an Altera DE4FPGA board for data collection. One FPGA supports bothMS transmitters, and a second FPGA is associated with theAP receiver. Both FPGAs are connected to their own host PCsfor running MATLAB scripts that are used for configuring thetransmission frame for the MSs, measurements at the CAP-MIMO AP, and data extraction from the AP FPGA.

The raw sampled data from the 16 beams at the AP isprocessed offline to extract channel measurements. We reportmeasurements using a single MS. For each MS location, all16 feed locations at the AP are measured in four sets ofmeasurements, each set corresponding to a particular settingof switches for measuring the 4 feeds in each quadrant.

III. BEAMSPACE MIMO THEORY

It is well-known that an antenna aperture can be equivalentlyrepresented by its half-wavelength sampled version withoutany loss of information [5], [11]. The number of samples, n,represents the dimension of the 2D spatial signal space andalso determines the directivity or beamforming gain:

n =4A

λ2=πD2

lens

λ2= 636 , G = 10 log10(πn) = 33dBi (1)

where the numerical values are for the prototype with lensdiameter Dlens = 6”. Thus, the prototype lens aperture isequivalent to a half-wavelength spaced 2D array with 636elements.

In beamspace MIMO theory the antenna domain system isequivalently represented in the beamspace domain and the twoare related through a spatial Fourier transform [2], [4], [5]. The(crtically sampled) beamspace domain is related to the aperturedomain through a 2D Discrete Fourier Transform (DFT). LetU represent the n×n 2D DFT matrix whose columns representn orthogonal beam directions [4], [5]. Let x(f) denote the(vectorized) n-dimensional aperture domain sampled signalvector in the frequency domain. The corresponding beamspacesignal representation is given by

xb(f) = UHx(f)⇐⇒ x(f) = Uxb(f) . (2)

The critically-sampled aperture domain and beamspace do-main representations are equivalent since U is a unitarymatrix: UHU = UUH = I .

A. Beamspace MIMO System Modeling

The baseband system equation in aperture-frequency do-main is given by

r(f) = h(f)x(f) + w(f) , −W/2 ≤ f ≤W/2 (3)

where r(f) is the n× 1 vector of received frequency domainsignals on the critical samples of the antenna aperture, h(f)is the n × 1 vector of aperture domain frequency responsesfrom the MS to AP antenna aperture, x(f) is the transmittedfrequency-domain signal from the MS, and w(f) is the n× 1vector of measurement noise with power spectral density No.The noise at different critically sampled aperture elements isassumed to be independent.

The corresponding baseband system equation in beam-frequency is

rb(f) = UHr(f) = hb(f)x(f) + wb(f) (4)hHb = UHh(f), wb(f) = UHw(f) (5)

where rb(f) is the n×1 vector of received frequency domainsignal in the beamspace domain, hb(f) is the n× 1 vector ofbeamspace frequency responses from the MS to each of theAP feeds, and wb(f) is the n×1 vector of measurement noisewith power spectral density No. The noise in different beams isindependent since U is unitary. The beamspace MIMO systemrepresentation in (4) is completely equivalent to the aperturedomain MIMO system representation in (3) since U is unitary.

B. Beamspace MIMO Channel Modeling

In general the aperture-domain vector of channel frequencyresponses can be modeled as

h(f) =

Np∑`=0

β`an(θ`)e−j2πτ`f (6)

where β` is the complex path attenuation, τ` is the path delay,and θ` = (θaz,`, θel,`) ∈ [−0.5, 0.5]× [−0.5, 0.5] is the pair ofspatial frequencies in azimuth and elevation associated withthe `-th path. Np denotes the total number of paths and the` = 0 path corresponds to the LoS path. We note that inthis model, τ` are all relative to the delay of the LoS (or

strongest) path which we assume to be τ0 = 0 without lossof generality. The spatial frequencies θ = (θaz, θel) are inturn related to the physical angles in azimuth and elevation,φ = (φaz, φel) ∈ [−π/2, π/2]× [−π/2, π/2], as

θaz = 0.5 sin(φaz) , θel = 0.5 sin(φel) . (7)

In (6), an(θ) represents the array response vector for a pointsource in the far-field located in the direction θ ↔ φ. Foruniform linear arrays (ULAs) an(θ) takes the form of adiscrete spatial sinusoid with frequency θ, and for criticallysampled 2D uniform planar arrays (UPAs), an(θ) is the kro-necker product of the steering vectors of ULAs in azimuth andelevation: an(θ) = anaz

(θaz)⊗ anel(θel) where naz and nel

are the dimensions of the array in azimuth and elevation andn = naz ×nel. For 2D UPAs the unitary beamforming matrixUn for mapping the aperture domain into the beamspacedomain is given by: Un = Unaz

⊗Unel.

The beamspace channel frequency response vector is relatedto the aperture domain frequency response vector as

hb(f) = UHn h(f) (8)

=

Np∑`=0

β`UHnaz

anaz (θaz,`)⊗UHnel

anel(θel,`)e

−2πτ`f .

Specifically, the element of hb(f) corresponding to the i-thfixed angle in azimuth and k-th fixed angle in elevation is

hb,i,k(f) =1√n

Np∑`=0

β`fnaz

(θaz,` −

i

naz

)fnel

(θel,` −

k

nel

)e−j2πτ`f (9)

where fn(θ) = sin(nπθ)/ sin(πθ) is a Dirichlet sinc functionwith a peak of n at θ = 0 and zeros at multiples of 1/n. Thecorresponding impulse response is given by

gb,i,k(τ) =1√n

Np∑`=0

β`fnaz

(θaz,` −

i

naz

)fnel

(θel,` −

k

nel

)sinc

(W(τ − τ`

W

))(10)

where sinc(x) = sin(πx)/(πx) is the sinc function. Therelationship (10) states that gb,i,k(τ) is peaky in the beamdirections and delay corresponding to strong paths. Thus, weexpect gb,i,k(τ) to exhibit sparse peaks in angle-delay primar-ily due to LoS and single-bounce multipath since multiplebounces incur significant attenuation.

IV. SYSTEM MODEL FOR CHANNEL MEASUREMENTS

In CAP-MIMO, the front-end lens array computes an ap-proximate 2D Fourier transform of the aperture domain signal,and the elements of the beamspace signal vector xb(f) areapproximated by critically sampling the focal surface of thelens [4], [5]. Critical sampling in beamspace is related to theangular beamwidth which is approximately given by [4], [5]

δφ =λ

Dlens(radians) (11)

and is about 4o for the 6” lens in the prototype. The currentCAP-MIMO prototype samples the focal surface with a 4×416-element array centered on broadside, see Fig. 1(b) and1(d), spanning an angular spread of about ∆φ = 16o in bothelevation and azimuth. A larger angular spread can be coveredby either moving the 16-element array along the focal surfaceor by using a larger feed array.

Orthogonal Frequency Division Multiplexing (OFDM)modulation corresponding to N = 64 tones is used forchannel measurements. A constant modulus chirp signal inthe frequency domain, known both at the MS and the AP, isused for probing the channel. At the receiver (AP), the knownprobing signal is used for time-frequency offset correctionbetween the MS and the AP as well as for channel estimation.For each MS location, M = 100 measurements are made forall of the Nb = 16 beamspace feeds of the AP. The samplinginterval and the OFDM symbol duration are

Ts =1

W= 8ns and Tofdm = NTs = 0.512µs . (12)

The baseband system equation in beam-frequency is

rb(f) = hb(f)x(f) + wb(f) , −W/2 ≤ f ≤W/2 (13)

where rb(f) is the Nb×1 vector of received frequency domainmeasurements for the Nb = 16 beamspace feeds, hb(f) is theNb × 1 vector of beamspace frequency responses from theMS to each of the AP feeds, x(f) is the transmitted signalfrom the MS, and wb(f) is the Nb×1 vector of measurementnoise with power spectral density No. The noise in differentfeeds/beams is assumed to be independent. The OFDM-basedprobing of the channel results in a sampled representation ofthe system equation in (13) with ∆f = W/N = 1.95MHz:

rb[k] = hb[k]x[k] + wb[k] , k = −N2, · · · , 0, · · · , N

2− 1

(14)where rb[k] = rb(k∆f) is the sampled version of thereceived frequency-domain signal, and hb[k], x[k], and wb[k]are defined similarly.

V. CHANNEL MEASUREMENTS AND INITIAL RESULTS

Indoor measurements were collected in two set ups: i) LoSpropagation, and ii) non LoS (NLoS) with metal reflectors tocreate additional multipath. The MS served as the transmitter,and for each MS position M = 100 measurements, rmb [k],m = 1, · · · ,M , of the received signal vector rb[k] weretaken. These measurements were then processed in MATLABto extract M = 100 beam-frequency response vectors hmb [k],which were then averaged to yield a 16 × 1 average beam-frequency response vector for each MS location:

hb[k] =1

M

M∑m=1

hmb [k] =[hb,1[k], · · · , hb,16[k]

]T(15)

where hb,i[k] = hb,i(k∆f) represents the k-th sample of theaverage frequency response for the i-th beam. The beam-impulse response vectors, gmb [n], are obtained from hmb [k]

through an N -point FFT operation, and the correspondingaverage channel impulse response vectors are given by

gb[n] = [gb,1(n), · · · , gb,16[n]]T, n = 0, · · · , N − 1 (16)

where gb,i[n] = gb,i(nTs) is the sampled version of theaveraged impulse response for the i-th beam/channel; see (10).

A. Power Spectral Density and Power Delay Profile

For each MS location, the average power spectral density(PSD) and the average path delay profile (PDP) for the i-thbeam are calculated as

PSDi[k] = |hb,i[k]|2 , PDPi[n] = |gb,i[n]|2 (17)

and represent the distribution of channel power in beam-frequency or beam-delay, respectively. For each MS location,the aggregate PSD and the aggregate PDP (over all beams)are given by

PSD[k] =

Nb∑i=1

PSDi[k] , PDP[n] =

Nb∑i=1

PDPi[n] . (18)

Similarly, the average channel power in the i-th beam(spatial power density/profile) is given by Pave,i =∑N/2−1k=−N/2 PSDi[k] =

∑N−1n=0 PDPi[n] and the total aver-

age channel power is given by Pave =∑Nb

i=1 Pave,i =∑i,k PSDi[k] =

∑i,n PDPi[n].

B. Correlation Between Multi-Beam LoS Measurements

One of the unique capabilities afforded by the multi-beamCAP-MIMO prototype is the ability to make simultaneousmeasurements across multiple beams. This capability canbe leveraged for measuring channel statistics. For example,simultaneous measurements across two beams can be usedfor measuring correlation between the beams or second-orderstatistics. In general, with a p-beam measurement capability,p-th order statistics can be measured in beamspace.

In this paper, we illustrate the measurement of correlationacross beams. Let A and B denote the two most powerfulbeams for a given MS position. We are interested in esti-mating the correlation between A and B for small or micromovements (on the order of a few wavelengths) of the MS inthe neighborhood of a given location. Suppose that we makeM measurements for the two beams for each of the Q microoffset locations around a given MS position

hm,qb,A , hm,qb,B , m = 1, · · · ,M , q = 1, · · · , Q . (19)

The frequency-domain cross-correlation matrix between thetwo beams can be estimated as

ΣAB = E[hb,AhHb,B ] ≈ 1

MQ

M∑m=1

Q∑q=1

hm,qb,Ahm,qHb,B . (20)

If a multi-beam system is used to collect the measurementssimultaneously for both beams as the MS moves (in somerandom order) over the Q locations, we can expect ourestimate to be closer to the true cross correlation between thebeams, which would be non-zero if the beams are correlated.

On the other hand, if a single-beam system is used, themeasurements for the two beams will necessarily have to bemade sequentially while the MS is randomly moving over theQ locations in the micro region. In this case, we argue thatour correlation estimate is:

ΣAB = E[hb,AhHb,B ] ≈ E[hb,A]E[hHb,B ] ≈ 0 .

This is due to the fact that most of the intrinsic channelvariation is going to be due to the changes in the phases of thepath gain β` in (6) which can be correlated for simultaneousmeasurements and uncorrelated for sequential measurements.

The PSDs and cross PSDs (X-PSDs) PSDA, PSDB ,PSDAB can be obtained from the diagonal entries of ΣAA,ΣBB and ΣAB respectively.

(a)

(b)

Fig. 2. Spectral correlation matrices and PSDs for simultaneous versussequential measurements with micro movements: (a) Normalized spectralcorrelation matrices. (b) Unnormalized PSDs and X-PSDs.

Fig. 2 shows the multi-beam correlation results for a LoSpropagation set up as shown in Fig. 1(c) so that we do notexpect significant multipath. The two dominant channels wereselected as A and B; in this case, A = 7 and B = 11, ver-tically adjacent beams. We collected M = 100 simultaneousmeasurements for A and B for Q = 9 locations in a squareof size about 1.5λ, centered on the original position of the

MS. Auto-correlation and cross-correlation matrices were cal-culated for A and B. To emulate sequential measurements, wesimply permuted the order of the Q positional measurementsfor B in computing the cross-correlation. Fig. 2(a) shows theimages of normalized correlation and cross-correlation matri-ces (normalized by their respective largest absolute values).Fig. 2(b) shows the corresponding unnormalized PSDs/X-PSD. As evident, for simultaneous measurements, the twodominant channels show very significant correlation, whereasfor sequential measurements they exhibit much diminishingcorrelation, as expected in this case since the two adjacentbeams are illuminated by the same common source (MS).

C. NLoS Multipath Measurements

The second set of illustrative measurements are for NLoSpropagation in which the MS waveguide antenna is facingaway from the AP and two metal reflectors are introducedin the environment to introduce multipath, as shown in Fig. 3.In this case, M = 100 measurements were collected for all 16

Fig. 3. NLoS multipath set up. The MS is facing towards reflectors awayfrom the AP.

beams for Q = 3 micro locations of the MS around a givenlocation. Figs. 4, 5, and 6 plot the individual PSDi[k] and thePDPi[n], defined in (17), for the Nb = 16 beams for threedifferent positions of the MS. The 16 plots correspond to thephysical location of the 16 feed antennas for the 16 beams.The 16 beams are indexed 1, · · · , 16 in a row-wise fashion(1st row: 1, 2, 3, 4; 2nd row: 5, 6, 7, 8, and so on). The x-axiscorresponds to frequency in MHz for the PSDs, and distancein meters for the PDPs, where the distance is calculated forthe n-th PDP sample as

distance(n) = Tsnc = 2.4n , n = 0, · · · , N − 1 (21)

where c represents the speed of light, Ts is the samplinginterval defined in (12), and the n = 0 PDP sample representsthe (dominant) multipath component. The four strongest beamsin descending order of power are color coded as: red, purple,green, cyan. Fig. 7 plots the aggregate PSDs and PDPs overall beams for the three MS positions.

There are two things worth noting. First, the PSDs and PDPsshow significant variation across the three positions due tothe multipath propagation. In comparison, the PSDs plotted inFig. 2 for the LoS set up show significantly less variability overfrequency. Second, the dominant channels in the 16 channels,

(a)

(b)

Fig. 4. NLoS PSDi and PDPi for 16 channels with the TX in the centerposition.

showed in colored plots, change for the different MS locations.Finally, we note in Fig. 7(b) that there is a small multipath peakat a distance of about 16 meters from the main peak whichcorresponds to a reflection from a metal shelf (not shown)behind the AP.

VI. CONCLUDING REMARKS

In this paper we have outlined a new multi-beam method-ology for mmW MIMO channel measurements and presentedinitial measurement results using a 28GHz CAP-MIMO proto-type. Building on the LoS and pathloss measurements reportedin [10], here we have reported additional results that showcasethe multi-beam approach in two aspects: estimation of multi-beam correlation, and the variation in NLoS beam-frequencychannel responses as a function of the transmitter location inthe presence of multipath.

The multi-beam CAP-MIMO prototype opens new possi-bilities for mmW channel measurements that are not possiblewith existing sounders. Two key characteristics of CAP-MIMOare particularly relevant. First, it enables spatial resolutionsthat are challenging to achieve in phased array systems. Forexample, the resolution of the current CAP-MIMO prototypeis comparable to a conventional 2D array with over 600half-wavelength-spaced elements. Second, simultaneous multi-beam measurements enable new capabilities that are not possi-

(a)

(b)

Fig. 5. NLoS PSDi and PDPi for the 16 channels with the TX in thenorthwest corner position.

ble with existing single-beam systems, including measurementof channel statistics.

Additional measurements are needed for “calibrating” thechannel measurements using the new prototype. In particular,as in any other sounder, the measurements represent a com-bination of the true underlying propagation environment aswell as the intrinsic beam-frequency response of the hardware,including the lens array. VNA-based measurements of thebeam-frequency response of the CAP-MIMO sounder will beuseful in this regard. Furthermore, while the basic beamspaceMIMO theory has been developed for planar arrays, we planto extend the results to circular arrays.

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