Spherical Wave Channel and Analysis for Large Linear Array in LoS Conditions

Transcription

1 Spherical Wave Channel an Analysis for Large Linear Array in LoS Conitions Zhou, Zhou; Gao, Xiang; Fang, Jun; Chen, Zhi Publishe in: 5 IEEE Globecom Workshops (GC Wkshps) DOI:.9/GLOCOMW Link to publication Citation for publishe version (APA): Zhou, Z., Gao, X., Fang, J., & Chen, Z. (5). Spherical Wave Channel an Analysis for Large Linear Array in LoS Conitions. In 5 IEEE Globecom Workshops (GC Wkshps) [7444] IEEE--Institute of Electrical an Electronics Engineers Inc.. DOI:.9/GLOCOMW General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights. Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research. You may not further istribute the material or use it for any profit-making activity or commercial gain You may freely istribute the URL ientifying the publication in the public portal Take own policy If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim. L UNDUNI VERS I TY PO Box7 L un Downloa ate: 7. Nov. 8

2 Spherical Wave Channel an Analysis for Large Linear Array in LoS Conitions Zhou Zhou,XiangGao, Jun Fang an Zhi Chen National Key Laboratory on Communications, University of Electronic Science an Technology of China, Chengu, China Department of Electrical an Information Technology, Lun University, Sween Abstract Massive MIMO is consiere a key technology for the future wireless communication systems. The promising properties in terms of higher spectral an transmit-energy efficiency are brought by the large number of antennas at the base station (BS). As the number of antennas increases, the aperture of the BS antenna array may become much larger, as compare to toay s antenna arrays. In this case, mobile stations (MSs) an significant scatterers can locate insie the Rayleigh istance of large arrays, an spherical wavefronts rather than planar wavefronts are experience over the arrays. In this paper, we propose an analytical spherical-wave channel moel for large linear arrays, which is also compatible with conventional planewave moels. Base on the spherical-wave moel, we investigate how MSs can be spatially separate in simple line-of-sight (LoS) scenarios. The results theoretically explain the observation in experiments that spherical wavefronts help ecorrelate the MS channels more effectively than planar wavefronts. I. INTRODUCTION With the avent of massive MIMO [], [] for proviing a large spatial egrees of freeom, it comes to the questions of better analyzing an moeling the spatial properties of the raio channels. Classical MIMO channel moels are usually base on the planar wavefront assumption, an can provie performance preiction in both line-of-sight (LoS) an nonline-of-sight (NLoS) channel conitions [3]. However, as shown in recent massive MIMO stuies base on channel measurements, the plane-wave assumption oes not hol for physically-large arrays [4] [8]. When the number of antennas at the base station (BS) increases to hunres, the aperture of the antenna array may become much larger, as compare to conventional MIMO. In this case, mobile stations (MS) an scatterers most likely locate insie the Rayleigh istance of the large arrays. The raiation fiel of antennas is usually ivie into two regions, the near-fiel region (Fresnel zone) an the far-fiel region (Fraunhofer zone) [9]. The bounary between the two regions is approximately the Rayleigh istance [], Z = D λ, () where D is the maximum imension of the antenna or antenna array, an λ represents the wavelength. In conventional MIMO systems, for example, with up to 8 antennas in LTE [], we usually assume that the locations of MSs an scattering objects are far beyon the Rayleigh istance of the BS antenna array, i.e., Z, where is the istance between the MSs or scatterers an the BS. In far-fiel region, the plane-wave assumption is well-reasone for MIMO channel moels []. However, in massive MIMO with physically-large arrays, the Rayleigh istance Z becomes large as the array aperture D increases, thus we may have <Z or Z. For example, in a semi-urban channel measurement campaign in the.6 GHz ban, as reporte in [6] an [7], a 7.4 m uniform linear array was eploye at the BS, giving the Rayleigh istance about 95 m. All the MSs an significant scatterers are within this istance. Spherical wavefronts over the large array were observe, by referring to the angular power spectrum over the array in the measure channels [4]. Therefore, planar wavefront assumptions for conventional MIMO channels are not suitable for physically-large arrays in massive MIMO scenarios. Motivate by the experimental results on massive MIMO channels, the characteristics of the near-fiel propagation, i.e., spherical wavefronts over large arrays, shoul be taken into consieration when analyzing an moeling massive MIMO channels. In [3], the early research on MIMO capacity consiers spherical-wave aware moels an briefly exploits the relationship between channel capacity an array geometries. In [4], the stuy empirically shows that spherical-wave moel is more accurate than plane-wave moel for short-range MIMO performance estimation, if the istance between the transmitter an receiver is below the Rayleigh istance. Besies shortrange communication, [5] an [6] also consier spherical wavefronts, an propose a technique for realizing high-rank channel matrix in pure LoS conitions an thus achieving high channel capacity, by optimizing antenna placement in uniform linear arrays. In this work, we propose an analytical channel moel for physically-large linear arrays, which takes spherical wavefronts into consieration. We characterize the spherical-wave channel steering vector by a group of geometrical parameters. The spherical-wave moel is formulate in a general form, an the close-form steering vector can be easily use for analytical stuy an extene to other types of antenna arrays. Base on this, we show that our moel is compatible with the conventional plane-wave moel. We theoretically stuy how the number of BS antennas, antenna spacing an MS spacing affect the spatial separability of the MSs in LoS conitions. We also emonstrate how spherical wavefronts help ecorrelate the MS channels, especially for closely-space MSs. This effect has been observe but is not explicit in previous stuies base on measure channels [4], [7].

3 The rest of the paper is organize as follows. In Sec. II, we propose an formulate our spherical-wave channel moel. Then Sec. III an IV present theoretical analysis an simulation results, respectively. Finally, Sec. V summarizes the paper. II. SPHERICAL-WAVE CHANNEL MODEL We first consier the channel from single-antenna MSs to multiple-antenna BS with LoS components only, then we make an extension by aing a reflecte path to the pure LoS channel. A. Basic Moel Fig. epicts the establishe coorinate system of the spherical-wave channel moel. We arbitrarily choose one antenna position on the array at the BS sie as the origin. The number of antennas is N r +, an the antenna inex i ranges from δn r to ( δ)n r, where δ> is efine as the antenna inex parameter an epens on the location of the origin. In the figure, θ [,π] is the angle of incience from the MS to the antenna at the origin, Δ r is the normalize BS antenna spacing in wavelength λ c, is the istance from the MS to the origin, an ϕ i (namely angle variation) is the inclue angle between the MS-origin irection an the irection of the MS to the ith-antenna. p = θ ϕ i θ ith antenna i> p Origin antenna ith antenna i< Mirror image for Origin antenna Mirror image for ith antenna ϕ i A Perfect Reflector Spherical Wave Front Fig.. Illustration of the spherical-wave moel. The effect of a perfect reflector is equivalent to having virtual BS antennas that are mirrore by the reflector. We first consier free-space transmission with only LoS components in the raio channel. This means that there is no scattering in the channel thus no small-scale or large-scale faing. Hence, the spherical-wave eigenmoe for the channel coefficient h between the MS an the origin antenna is MS h e j π λ, () where / is the factor ue to the free-space pathloss. After some calculations in geometry, the corresponing sphericalwave channel vector can be expresse as h H [ ej π λc L δnr L δnr e j π λc L δnr+ e L δnr+... j π L λc ( δ)nr ], (3) L ( δ)nr where an L i = { Δr λ c i cos(θ ϕ i )+cos(ϕ i ), i Δ r λ c i cos(θ + ϕ i )+cos(ϕ i ), i < (4) { arctan( r i +c ϕ i (Δ r,,θ)= i ), i arctan( ri +c i ), i <, (5) r i =, c i =, an arctan( ) is the arctangent function efine in interval [,π]. Note that L i an ϕ i inicate the phase variation an angle variation over the array, respectively. Let us efine the steering vector as the normalize channel vector a(ϖ) = h h, (6) Δriλc sin(θ) Δriλc cos(θ) where ϖ (, θ, δ; N r, Δ r ). Note that δ an θ are not fully inepenent. It is interesting to compare this spherical-wave steering vector with the conventional MIMO steering vector that assumes planar wavefronts [], i.e., h i e jπδri cos(θ), (7) where i [,N r ]. Different from the plane-wave moel, the phase variation an pathloss variation over the array are nonlinear in the spherical-wave moel, as shown in (3). The spherical-wave moel is more general, an we will show in Sec. III that it can be approximate to the plane-wave moel when BS-MS istance is large ( Z). B. Choice of Origin With the efine spherical-wave steering vector in (6), one issue is how to choose the orinate origin. In general, for each MS, we can choose a ifferent antenna position on the array as the origin, an erive the relation between the parameters for ifferent MSs. For example, for two groups of parameters ϖ =(,θ,δ ; N r, Δ r ) an ϖ =(,θ,δ ; N r, Δ r ) of the same channel vector, i.e., h(ϖ )=h(ϖ ), assuming δ > δ, the angle variation ϕ,i an ϕ,i, where i [ δ k N r, ( δ k )N r ],k=,, follow the relation ϕ,i = ϕ,i+(δ δ )N r, (8) for i [ δ N r, ( δ )N r ]. In this paper, we have two choices of the coorinate origin for ifferent purposes of analysis. The two choices are as follows. ) The first antenna position is set as the origin, thus we have δ =an ϖ =(, θ, ; N r, Δ r ). With this setting, we emonstrate that the propose moel is compatible with the conventional plane-wave moel in Sec. III. ) We set the origin at the antenna position that is close to the point of the normal incience from the MS to the BS antenna array, so ϖ =(, π,δ ; N r, Δ r ), where δ is the corresponing antenna inex paramter. Without loss of generality, we assume there is always an antenna

4 position at the point of the normal incience. Therefore, using (4) an (5), we have L i = (Δ r λ c i) + sin(ϕ i + ω i ), (9) ϕ i = arctan( Δ rλ ci Δ ), ω i = arccos( rλ ci ), for (Δ rλ ci) + i [ N r δ,n r ( δ )]. Since ϕ i +ω i = π, (9) is simplifie to L i = (Δ r λ c i) +. () By using (), the analysis of the orthogonality between ifferent MS channels is tractable, an we present the analysis results in Sec. III. C. Moel Extension The propose moel can be easily extene from single-user to multi-user case. With N t single-antenna users, the channel matrix is H [ h(ϖ ) h(ϖ )... h(ϖ Nt ) ], () where h(ϖ k ) is the channel vector of the kth MS to the BS. We can also exten the moel to inclue a perfect reflector, as illustrate in Fig.. The effect of the reflection path is equivalent to having virtual BS antennas that are mirrore by the perfect reflector (see Fig. ). We can see this effect as increasing the number of BS antennas. Using the principle of superposition [], the channel vector becomes h κh(n r, Δ r, r,θ r )+h(n r, Δ r, r,θ r ), () where the subscript an inicate the actual BS antennas an the virtual BS antennas, respectively, an the first term is the channel of the LoS path, the secon term is the channel of the reflecte path. The factor κ is the power ratio between the LoS path an the reflecte path. Note that r an θ r are calculate accoring to the principle of specular reflection. III. MODEL ANALYSIS We consier multi-user MIMO (MU-MIMO) with two single-antenna MSs (N t =), as illustrate in Fig.. The analysis an obtaine results can be easily extene to the case of more MSs. In the uplink the receive signal at BS can be expresse as y = h x + h x + n, (3) where x, x are the complex{ transmitte symbols by the two MSs, with the same energy E x k } =E x. The instantaneous uplink channel capacity is N t C inst = log ( + E x σ N k), (4) k= where N is the complex Gaussian noise variance, an σ k is the kth singular value of matrix H =[h h ]. It can be prove that C inst reaches the maximum when σ = σ.if h an h have the same norm, this conition that σ = σ is calle favorable propagation conitions that the two MS channels are fully orthogonal [7]. Δ rλ{ c Horizontal Direction θ θ ith antenna i> Origin antenna ith antenna i< Spherical Wave Front Spherical Wave Front MS MS Δ λ u c Δ ω Vertical Direction Fig.. Illustration of the spherical-wave moel with two single-antenna MSs. Here we assume that the MS channels have the same pathloss, an the channel vectors are normalize, i.e., h k = a(ϖ k ),k =,. Denote the correlation coefficient between the two channels by f = a H (ϖ )a(ϖ ),wehave σ σ = +f f, (5) where σ /σ = if an only if f =. To reach higher channel capacity, we expect the channel correlation coefficient f to be close to. Next, we investigate the correlation between the MS channels in the propose spherical-wave moel. The channel correlation f(n r, Δ r, δ; θ,,, Δ ω ) is a function of a group of geometric parameters. Here we choose ifferent origins for the two MSs, an the geometric parameters for MS is use as a reference to erive the parameters for MS, as in this way our theoretical analysis can be simplifie. Thus, we have the antenna inex parameter δ =(δ,δ ),the BS-MS istance =, the angle of incience θ = θ, an is the spacing between MSs in wavelength, Δ ω is the inclue angle between MSs connecting irection an the vertical irection (see Fig. ). The geometric parameters for MSare an = +Δ uλ c + λ c sin(θ Δ ω ) (6) θ = π arctan( λ c cos(θ) sin(δ ω ) ). (7) λ c sin(θ) + cos(δ ω ) In the far-fiel region an when the MS spacing is too small, i.e., λ c, from (7) we have θ θ, where it is ifficult to spatially separate the MSs as the angles of arrival are similar. However, in the near-fiel region, even if the MS spacing is small, it is possible to spatially separate the MSs ue to spherical wavefronts are experience over the array. We show this effect later in this section.

5 Using the relation in (8), we can ecompose the channel correlation coefficient ( δ )N r f(n r, Δ r, δ; θ,,, Δ ω )= f i i= δ N r, (8) where f i = a,i a,i, a,i an a,i are the channel coefficients between the MSs an the ith BS antenna, an satisfies f i L,i L,i+(δ δ )N r, f i / L,i L,i+(δ δ )N r,for i [ δ N r, ( δ )N r ]. Note that f i inicates the variation of the phase ifferences between the two MSs over the array. With relatively large variation in phase ifference, we expect relatively better spatial separation of the MS channels. A. Angle an Phase Variations In the propose moel, we can see that the angle variation factor ϕ i in (5) etermines the phase variation over the array, see (3) an (4), an therefore the channel correlation f. We next iscuss the angle an phase variations over the array in two extreme cases when the MSs are very far from the BS an when the MSs are very close to the BS, as follows. If the MSs are very far from the BS, i.e., in the far fiel of the BS antenna array, Z, as iscusse in Sec. I, sufficiently we have Δ r λ c N r. In this case, in (5), r i an c i, thus there is neither angle variation nor pathloss ifference over the array as ϕ i (Δ r,,θ)=, L i. (9) The channel steering vector then becomes the conventional steering vector uner the assumption of planar wavefronts over the array, as in (7). If the MSs are very close to the BS, we have r i an c i, resulting in ϕ i (Δ r,,θ)=θ, () an L i Δ r λ c i. In this extreme case, there is again no angle variation over the array. For the phase variation, from () we see that when i (, ( δ )N r ], L i monotonically increases with respect to i, an when i [ δ N r, ), monotonically ecreases. Thus, L i is a quasi-convex function of the antenna inex i. Generally, for any antenna with incient angle θ, wehaveϕ i = θ π an L i = sin(θ) for some i > when θ> π. B. Phase Difference an Spatial Decorrelation As explaine by (8), the phase ifference f i etermines the channel correlation. Here we focus on the geometric factors relate to f i an try to preict the channel correlation. Specifically, we investigate the phase ifference f i in the two extreme cases when the MSs are very far from or very close to the BS, as follows. When Δ r λ c i an Δ r λ c i, using () we have πδ r (δ δ )N r, when i>(δ δ )N r f i πδ r (δ δ )N r, when i< () 4πΔ r i+πδ r (δ δ )N r, others. The channel correlation f in (8) can also be written as (δ δ )Nr ( δ )Nr f = f i + f i + f i i= δ Nr i= i=(δ δ )Nr+. () We see that if the antenna inex i is outsie the interval [, (δ δ )N r ], the phase ifference f i remains constant, see (). These antennas contribute little to reuce the channel correlation f. This inicates that we cannot always separate the MS channels by increasing the number of antennas in this extreme case. Hence, we call this phenomenon as a saturation of phase variation in large number of antennas, when MSs are very close to BS array. When Δ r λ c i an Δ r λ c i, wehave f i =π(cos θ cos θ )Δ r i, (3) where i=,..., N r, an θ is the angle of incience from MS to the first antenna, similarly θ. We can see that the ecorrelation of the MS channels relies on the ifference in the angles of incience. When θ = θ, i.e., two MSs are in the same vertical irection (Fig. ), f i =,it is thus impossible to spatially separate the MSs. In this extreme case that the MSs are very far from the BS, the channel correlation is given by f sin(π(n r + )(cos θ cos θ )Δ r ) sin(π(cos θ cos θ )Δ r ), (4) as iscusse in []. Generally, in between the two extreme cases, we have spherical wavefronts over the array. From (), the phase ifference f i over the array can be written as f i (Δ r λ c i) +( ) (5) (Δ r λ c (i +(δ δ )N r)) +( ). We evaluate the phase ifference f i as escribe in the theorem below. Theorem : If δ = δ, f i is a quasi-concave function of i an its value ranges from π λ c ( ) to. If =, f i monotonically increases with respect to i, an is boune in [ πδ r (δ δ )N r, πδ r (δ δ )N r ]. When δ = δ, it means that two MSs are locate in the same vertical irection perpenicular to the BS array, i.e., Δ ω =, an = inicates that two MSs are in the same horizontal irection parallel to the BS array, Δ ω =9, see Fig.. In the former case represents the physical spacing of the MSs in the vertical irection, an in the latter case (δ δ )N r is proportional to the MS spacing in the horizontal irection. Fixing the MSs spacing in the two cases, i.e., Δ r λ c (δ δ )N r =, from Theorem, we can see that the range of f i is larger in the latter case than in the former case, as πδ r (δ δ ) N r = π λ c. Thus it is more efficient to ecorrelate MS channels by increasing the number of antennas in the latter case, i.e., when the MSs are in the same horizontal irection parallel to the BS

6 array. Furthermore, the saturation on phase variation still exists in these two cases. We illuminate this phenomenon with simulations in Sec. IV. C. Pathloss Difference an Spatial Decorrelation Due to analytical tractability, we have f ( δ)n r i= δn r f i, an the equality hols if an only if L,i = L,i+(δ δ )Nr. Note that f i / L,i L,i+(δ δ )Nr represents prouct of pathloss from each MS to the ith antenna, an ( δ)n r i= δn r [ f i is the normalize inner ] pro- uct of the vectors L, δ L Nr,( δ an [ ] )Nr L, δ L Nr,( δ, which measures the ifference between two MSs pathloss over the array. In this section, )Nr our aim is to boun the channel correlation coefficient by this inner prouct an we have the results in the following. Theorem (Upper boun of correlation function): When δ δ an δ >δ,wehave ) f(n r, Δ r, δ; θ,,, Δ ω )<A+B, if(δ δ )>/N r, ) f(n r, Δ r, δ; θ,,, Δ ω )<A+B,if(δ δ )=/N r, 3) f(n r, Δ r, δ; θ,,, Δ ω )<B,if(δ δ )</N r, where A = C ( + G + +G ), B = Δ r λ c G (ln ( G G +) C ( ) C )+C 3, Δ r λ c Δ r λ c B = C (Δ r λ c ) + C 3, G = Δ r λ c (δ δ )N r, (6) an C, C, C an C 3 are constants. The term /N r is relate to the angular resolution of a linear array which escribes the ability of the array to resolve two rays. With spherical wave, if two MSs horizontal spacing is smaller than Δ r λ c, their channel correlation is etermine by their vertical spacing. For in B an B, by mean value inequality, when =, will be maximize subject to a fixe value of +. Thus ecreasing the average BS-MS istance + an at the same time increasing the vertical spacing of the MSs are beneficial to spatial ecorrelation of the MS channels. We also observe that increasing the BS antenna spacing Δ r λ c rather than the antenna number can ecrease the channel correlation. IV. SIMULATION RESULTS AND DISCUSSION In this section, we investigate the channel correlation of two single-antenna MSs in LoS, using the propose spherical-wave moel. We stuy two typical scenarios the inclue angle of two MSs ) Δ ω =9, where the MSs are on the line parallel to the BS antenna array, an ) Δ ω =, where the MSs are on the line perpenicular to the BS antenna array, see Fig.. In Fig. 3, we show the channel correlation of the MSs when the BS antenna spacing Δ r =/ wavelengths an the MSs are at the ege (Fig. 3(a) an (b)), an at the mile (Fig. 3(c) an ()) of a urban-macro cell. Accoring to 3GPP TR [8], we choose the urban-macro cell raius to be 5 m, so the BS-MS istance is 5 m at the ege an 5 m at the mile of the cell. We see that the channel correlation between the MSs reuces as we increase the MS spacing or the number of BS antennas. As expecte, the spatial separation is more ifficult in the secon scenario than in the first scenario, even with a very large number of antennas (up to a thousan), the correlation is quite high. Besies, when the MSs are at the mile of the cell, it is easier to separate them than when whey are at the ege (a) cell ege(5m), the first scenario (b) cell ege(5m), the secon scenario (c) mi cell(5m), the first scenario () mi cell(5m), the secon scenario MS spacing (unit of wavelength) Fig. 3. Channel correlation between two MSs in LoS conitions in the two stuie scenarios, when they are 5 m (at the ege of a cell) an 5 m (at the mile of a cell) away from the BS. Fig. 4 illustrates that the channel correlation reuces with an increasing number of antennas, for ifferent MS spacings. We can rely on increasing the number of antennas to separate the MSs before the saturation region is reache, as remarke in Theorem. We also observe ripples on the correlation coefficient as the antenna number increases. This is similar to the plane-wave moel where the correlation coefficient is a sinc function of the antenna number []. In Fig. 5(a), we see that the MS channel correlation increases as the BS-MS istance increases, an it becomes more an more ifficult to spatially separate the MSs, especially when their spacing is relatively small. In Fig. 5(b), we fix the aperture of the BS array to be 64λ c an increase the number of antennas. We can see that in this case aing antennas oes not help ecorrelate the MSs. Fig. 6 shows the MS channel correlation with LoS an one reflection path. The power ratio between the LoS path an the reflection path is set to κ B. Comparing with Fig. 3, the reflection path helps reuce the channel correlation, especially in the secon scenario that is particular ifficult to separate the MSs. The effect of the reflection path is equivalent to having virtual BS antennas (see Fig. ) which also contribute to the spatial separation

7 Correlation Coefficient.5 = = = (a) cell ege(5m), the first scenario = = =3.5 = = = (b) cell ege(5m), the secon scenario = = =3 V. CONCLUSION AND FUTURE WORK Base on the propose spherical-wave moel, we investigate the channel correlation of two MSs in ifferent settings. We illustrate how the spherical wavefronts can help ecorrelate the MS channels. The spherical-wave moel is compatible with the conventional plane-wave moel, an can be extene to other types of array an the case of more MSs. ACKNOWLEDGEMENT The authors woul like to acknowlege the financial support from the Fun of High Level Acaemic Conference Program in the Grauate School of UESTC uner Grant A (c) mi cell(5m), the first scenario () mi cell(5m), the secon scenario Fig. 4. Channel correlation as the number of antennas increases, for antenna spacing Δ r =/ wavelengths an ifferent MS spacings. Correlation Coefficient = = =3 = The ratio between MS-BS istance an Rayleigh istance (a) Δ r =/, N r = 8, the first scenario Correlation Coefficient =3 =7 = (b) Array aperture is 64λ c, = 5 m, the first scenario Fig. 5. (a) Channel correlation of the MSs with the BS-MS istance. (b) Channel correlation of the MSs with a fixe BS array aperture an varying number of antennas (a) cell ege(5m), the first scenario (b) cell ege(5m), the secon scenario (c) mi cell(5m), the first scenario () mi cell(5m), the secon scenario MS spacing (unit of wavelength) REFERENCES [] F. Rusek, D. Persson, B. K. Lau, E. Larsson, T. Marzetta, O. Efors, an F. Tufvesson, Scaling up MIMO: Opportunities an challenges with very large arrays, IEEE Signal Process. Mag., vol. 3, no., pp. 4 6, Jan. 3. [] E. Larsson, O. Efors, F. Tufvesson, an T. Marzetta, Massive MIMO for next generation wireless systems, IEEE Commun. Mag., vol. 5, no., pp , Feb. 4. [3] D. Gesbert, H. Bolcskei, D. GORE, an A. Paulraj, Outoor MIMO wireless channels: moels an performance preiction, IEEE Trans. Commun., vol. 5, no., pp , Dec.. [4] S. Payami an F. Tufvesson, Channel measurements an analysis for very large array systems at.6 ghz, in Antennas an Propagation (EUCAP), 6th European Conference on, Mar., pp [5] X. Gao, F. Tufvesson, O. Efors, an F. Rusek, Channel behavior for very-large MIMO systems - initial characterization, in COST IC4, Bristol, UK, Sept.. [6], Measure propagation characteristics for very-large MIMO at.6 ghz, in Conference Recor of the Forty Sixth Asilomar Conference on Signals, Systems an Computers (ASILOMAR), Nov., pp [7] X. Gao, O. Efors, F. Rusek, an F. Tufvesson, Massive MIMO performance evaluation base on measure propagation ata, IEEE Trans. Wireless Commun., vol. 4, no. 7, pp , July 5. [8] X. Gao, O. Efors, F. Tufvesson, an E. Larsson, Massive MIMO in real propagation environments: Do all antennas contribute equally? IEEE Trans. Commun., vol. PP, no. 99, pp., 5. [9] J. D. Kraus an R. J. Marhefka, Antenna for all applications, Upper Sale River, NJ: McGraw Hill,. [] A. F. Molisch, Wireless Communications. John Wiley & Sons, 7. [] Requirements for Further Avancements for Evolve Universal Terrestrial Raio Access (EUTRA) (LTE-Avance), Mar. 9. [] D. Tse an P. Viswanath, Funamentals of Wireless Communication. Cambrige University Press, 5, ch. 7, pp [3] P. Driessen an G. Foschini, On the capacity formula for multiple inputmultiple output wireless channels: a geometric interpretation, IEEE Trans. Commun., vol. 47, no., pp , Feb [4] Spherical wave moel for short-range MIMO, IEEE Trans. Commun., vol. 53, no. 5, pp , May 5. [5] F. Bohagen, P. Orten, an G. Oien, Design of optimal high-rank lineof-sight MIMO channels, IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 4 45, Apr. 7. [6], On spherical vs. plane wave moeling of line-of-sight MIMO channels, IEEE Trans. Commun., vol. 57, no. 3, pp , Mar. 9. [7] H. Q. Ngo, E. Larsson, an T. Marzetta, Aspects of favorable propagation in massive mimo, in 4 Proceeings of the n European Signal Processing Conference (EUSIPCO), Sept. 4, pp [8] Stuy on 3D channel moel for LTE, 3GPP St. TR , Rev. V.., 4. Fig. 6. Channel correlation in LoS conition with one reflection path, for the two stuie scenarios, when the MSs are 5 m (at the mile of a cell) an 5 m (at the ege of a cell) away from the BS.