Back-of-the-envelope evaluation of the prevalence of RIMP or LOS propagation as a function of frequency

SUBMITTED TO JOURN AL 1 Back-of-the-en v elope e v aluation of the pre v alence of RIMP or LOS propagation as a function of frequenc y Aidin Razavi, Andrés Alayón Glazuno v , Senior Member , IEEE , Rob Maaskant, Senior Member , IEEE , and Jian Y ang, Senior Member , IEEE Abstract —The performance of 5G wireless communication sys- tems, employing Massive-MIMO at millimeter -wave fr equencies, is most likely measur ed only in Over -The-Air (O T A) setups. It is proposed to perf orm O T A measurements in two limiting en vironments of Rich Isotr opic MultiPath (RIMP) and Random Line-of-Sight (Random-LOS) instead of a typical or r epresen- tative channel. In the present paper , we present a back-of-the- en velope in vestigation of the impact of scattering on the frequency dependence of the signal fading statistics in the 500 MHz– 100 GHz band. W e intr oduce a simple model for a generic scattering envir onment by using randomly distributed resonant scatterers to in vestigate the impact of the size of the scattering en vironment, the scatterer density , and the number of scatterers on the signal variability in terms of the Rician K -factor as a function of frequency . The simplified model is also verified against full-wav e simulation using the Method of Moments (MoM). Index T erms —Propagation, Scattering, Scattering parameters, Antenna measur ements, F ading channels. I . I N T RO D U C T I O N 5 G wireless communication networks are being de veloped to meet the increasing demands for better quality-of-service, e.g., throughput in the Gbps range. The multi-user multiple- input multiple-output (MU-MIMO) technology employing very large array antennas at the radio base station (RBS), also known as Massi ve MIMO technology , is one of the k ey tech- nology enablers [1]. W ith this technology , the RBS will serv e many mobile stations simultaneously , and the data reaches each of the mobile antennas by beamforming the energy tow ards them [2]. The combination of large array antennas and many users turns out to be fav orable for data transmission. Fa vorable propagation (FP) conditions means that the channel vectors between the users and the RBS are nearly pairwise orthogonal. Therefore, the signal processing complexity can be considerably reduced since linear processing is very close to be optimal [3], [4]. This work has been supported by two projects from Sweden’ s innova- tion agency VINNO V A, one within the VINN Excellence Center Chase at Chalmers and another via the program Innovati ve ICT 2013, and by internal support from Chalmers. A. Razavi was with the Signals and Systems Department, Chalmers Univ ersity of T echnology at the time this work was conducted. He is now with Ericsson AB, Sweden. A. A. Glazunov is with the Department of Electrical Engineering, Uni versity of T wente, P .O. Box 217, 7500 AE Enschede, The Netherlands, R. Maaskant and J. Y ang are with the Electrical Engineering Department, Chalmers Univ ersity of T echnology , SE-41296 Gothenburg, Sweden (e-mail: aidin.razavi@tgeik.com; a.alayonglazunov@utwente.nl; rob .maaskant@chalmers.se; jian.yang@chalmers.se). A. A. Glazuno v is also affiliated with the Department of Electrical Engineering, Chalmers University of T echnology , (andres.glazunov@chalmers.se), R. Maaskant is also with the Department of Electrical Engineering, Eindhoven University of T echnology (TU/e), (r .maaskant@tue.nl). Another enabler is the use of very large portions of the electromagnetic spectrum. That is, in addition to operation in the traditional ultra high frequency (UHF) band, ne w very large contiguous bandwidths will be exploited at the extremely high frequency (EHF) band, i.e., the millimeter-w av e (mm- wa ve) frequencies. Currently , there is no complete understanding of the prop- agation characteristics at the frequency bands of interest, i.e., from 500 MHz to approximately 100 GHz for Massi ve MIMO. As is well-kno wn, channel models are indispensable tools for e v aluating, predicting or optimizing the performance of wireless systems [5]. Howe ver , it is in general not fully possible to know exactly what channel model is the typical, or the most representati ve one. On the other hand, it is often useful to look at phenomena in the limiting cases to model certain phenomena, e.g., static or high frequency limits. In wireless communications, the Rich Isotropic MultiPath (RIMP) propagation channel and the Line-of-Sight (LOS) propagation channel represent two limiting propagation en- vironments in terms of the spatial distribution of the angle- of-arriv al (AoA) or angle-of-departure (AoD). As argued in [6], [7], both channels are fav orable for the operation of Massiv e MIMO. Therefore, it is reasonable to expect that real propagation en vironments, which are likely to be in-between these extremes, w ould also be fa vorable. This has support from experimentally observed FP characteristics of Massive MIMO channels in real life [8]. This fact has a very profound implication, i.e., both the RIMP and the LOS environments may suf fice to characterize the O T A performance of wireless devices as suggested in [9]. The follo wing r eal-life hypothesis for O T A de vice characteri- zation has been formulated as: if a wir eless device is pr oven to work well in RIMP and Random-LOS, it will work well in all real-life en vir onments [10]. It is worthwhile to note that the Random-LOS propagation environment is a generalization of the LOS en vironment concept, where the randomness is a result of the unpredictable positions and orientations of antennas of the mobile stations, the deployment position of a RBS, or both. In Random-LOS, both the Angle-of-Arriv al (AoA) and the polarization of the LOS wa ve (i.e., the only wa ve present) are considered to be random v ariables. The RIMP and the Random-LOS en vironments are espe- cially relev ant to the O T A characterization of Massiv e MIMO RBS in 5G wireless systems. 5G massiv e array antennas will, in practice, be only possible to measure in OT A setups due to the large number of ports. Moreover , as we go higher in frequency , e.g., for systems operating at mm-wa ve frequencies, SUBMITTED TO JOURNAL 2 there will be most likely no access to measurement ports at all. Here we therefore present a “back-of-the-env elope" in vesti- gation of the impact of scattering on the signal fading statistics as a function of frequency from 500 MHz to 100 GHz. W e neglect the water vapour and oxygen absorption effects as well as the propagation mechanisms leading to large-scale signal fading fluctuations. W e assume that these will occur on top of the presented scattering model. W e develop a model for signal fluctuations due to scattering, i.e., short-term f ading, as a function of frequenc y under v arious simplifying assumptions. The main idea is to in vestigate the impact of the size of the scattering environment, the scatterer density , the number of scatterers on the signal fading (i.e., signal variability) as a function of frequenc y . The model is generaland is not deriv ed for specific type of scatterers. Modeling the scatterers by resonant dipoles, the analytical model is compared to numeri- cal computations performed with functions inherited from the CAESAR code [11]. Under the assumptions used in this work, it is shown that as we go higher in frequency , the power in the LOS component gradually increases as compared to the power of the scattering contribution. This phenomenon has recently been observed in measurements as well [12]. Hence, for a fixed number of thin wire scatterers, the scattering en vironment behav es lik e the RIMP channel at lower frequencies, while at higher frequencies it becomes more like the (Random-)LOS channel. The paper is organized as follows. In Sect. II, the signal fading model and Rician K -factor are defined. Then, the scat- tering model and all assumptions for this model are presented in Sect. III. The detailed deriv ation of K -factor as a function of the av erage scattering cross-section of scatterers and therefore its dependence on the frequenc y are provided in Sect. IV . The results from the analytical K -factor computations and a MoM numerical simulations for dif ferent cases are compared in Sect. V , with discussions and analysis on the results. Finally , the paper is concluded in Sect. VI. I I . S I G NA L F A D I N G M O D E L T o study the scattering en vironment, let us assume a trans- mitter and a receiver antenna in the presence of scatterers as illustrated in Fig. 1. In order to model the single-port receive signal we introduce the complex random v ariable v = V oc 2 √ 2 R ar , (1) where V oc is the total open-circuit voltage induced at the receiv e antenna ports and R ar is the real part of the receiv e antenna input impedance. The recei ved power as a function of the open circuit voltage for the conjugate-matched load condition is P ar = | v | 2 = | V oc | 2 8 R ar . (2) Scattering, including physical phenomena such as reflec- tion, refraction and dif fraction, is the dominating propagation mechanism in wireless multipath contributions [5]. Objects (scatterers) surrounding the mobile antenna are assumed to Tx. antenna Rx. antenna r t r s r 0 Fig. 1. T ransmitter and receiv er antennas in the scatterering environ- ment. contribute the most to the fast fading fluctuations of the receiv ed signal [13]. Hence, we seek to estimate the impact of the scattering on the fast fading process as a function of frequency . The Rician probability distrib ution function (pdf) has gained widespread acceptance as a model of the continuous wa ve signal fluctuations caused by multipath propagation. A mea- sure of the severity of fluctuations is then giv en by the Rician K − factor defined as K = P LOS P RIMP = |h v i| 2 h| v | 2 i − |h v i| 2 , (3) where P LOS and P RIMP are the po wers of the LOS and the RIMP components, respectiv ely , h v i denotes the ensemble av erage of complex-v alued random variable v . The second parameter that defines the Rician pdf is the total receiv ed po wer P r = P LOS + P RIMP = h| v | 2 i . (4) The Rician distribution of the env elope of the complex signal amplitude receiv ed by the antenna | v | is given by [13], f | v | ( | v | ) = 2 (1 + K ) | v | P r exp  − K − ( K + 1) | v | 2 P r  × I 0   2 s K (1 + K ) P r | v |   , (5) where I 0 is the modified Bessel function of the first kind and zeroth order . The physical interpretation of (3) is that the recei ve signal fading, or se verity of fluctuations, is a function of the propor- tion between the power of the deterministic component of the receiv ed signal, i.e., gi ven by the numerator and the power in the stochastic component, i.e., gi ven by the denominator . The former can be interpreted as the LOS field component while the latter can be interpreted as the RIMP component. T wo limiting cases immediately arise, i.e., K → 0 and K → ∞ denoting the RIMP and the LOS channels, respecti vely . Thus, intermediate values of K will describe Rician propagation channels in between the RIMP and the LOS channels. It is worthwhile to note that in practice, the deterministic compo- nent can be just a strong reflected or diffracted wa ve, while the stochastic component not necessarily arises from an isotropic wa ve field distribution. For the sake of simplicity we will not consider this general case. Instead, we will look into the situation when there is a LOS field component in addition to a RIMP field component. SUBMITTED TO JOURNAL 3 I I I . S C A T T E R I N G M O D E L A N D G E N E R A L A S S U M P T I O N S In wireless communication channels, as well as in mi- crow av e sensing or radar applications, there is usually more than one single scatterer interacting with the receiv e and transmit antennas. Many scatterers hav e to be considered in order to completely define the propagation channel. Howe ver , in practice, full knowledge about the exact physical properties and positions of the scatterers is not av ailable. Here, we are mainly concerned with the statistical modeling of the channel; an approach that has gained widespread use in wireless com- munication [14] and remote sensing applications [15]. For this reason, the antennas and scatterers are assumed to be in the far -field region of each other and single scattering is assumed. T o fully and exactly e valuate the scattering contrib ution to the total receiv ed signal is a too complex task to accomplish ev en under the assumptions already stated above. W e need therefore to introduce further assumptions. The total open- circuit voltage entering (1) is a random v ariable due to the random nature of the scattered field component as sho wn fur - ther below . The randomness arises from the random positions of the scatterers relativ e to each other and the antennas. In turn, this results in the polarization of the scattered wav es behaving like a random process too. Also the phase difference can be considered random due to different path lengths traveled by the scattered wa ves. In a real scattering environment, the different scattering objects will appear to hav e different sizes depending on the frequency . Also the electrical distance between scatterers will increase with frequency . Moreover , dif ferent parts of a larger scatterer may be modeled by a set of smaller scatterers with no electrical coupling to each other under the approximations assumed herein. Small half-wa velength dipole antennas have been used to model the electromagnetic scattering in wireless channels in [16]. Herein, we adopt a similar approach. Before we proceed further , let us summarize our simplifying modeling assumptions: (I) W e consider a narrowband continuous wa ve signal. (II) Antennas and scatterers are in the far-field of each other where the far -field reference distance is giv en by R FF computed according to the criterion given in [17]. (III) For the sake of simplicity , we assume that the scat- terers are uniformly distributed within a spherical scattering volume. The probability distribution of a scatterer being located ρ from the center of a sphere of radius R s or at the angle θ is gi ven by p ( ρ ) = 3 ρ 2 R 3 s , (6a) p ( θ ) = sin θ 2 , (6b) respectiv ely , where the radius R s delimits the volume containing the scatterers. Furthermore, the recei ver antenna is located at the center of the volume. (IV) W e assume that the transmit antenna is far away from the scattering volume, while the receiv e an- tenna is within the scattering v olume at ρ = 0 . The communication scenario assumes a local cloud of scatterers surrounding the mobile user which is distant to the base station antenna and therefore illuminated uniformly by a plane wa ve. (V) Only single scattering is considered in our analyti- cal model. Ho wev er , in the later MoM simulations multiple scattering is included. (VI) The scattering contribution of the scatterers is deter- mined by their spatially av eraged scattering cross- section. (VII) The randomness in the positions of scatterers is assumed to originate from an ensemble of states originating from a random process. It is worthwhile to note that because our assumed channel model is simple in nature it can only predict dominant effects and thus general qualitativ e trends, such as the indication that the Rician K-factor increases with frequency , which is a phenomenon that has recently been observed in measurements as well [12]. Ho wev er , to predict more quantitati vely what happens in other communication scenarios as well as in more specific en vironments, we refer e.g. to [18]–[20]. I V . C O M P U TA T I O N O F K − FA C TO R A N D P r Consider the following scattering problem where the field radiated by a transmit antenna is scattered by a single linear scatterer . W e are then interested in computing the total field propagating tow ards the receive antenna. The geometry of the problem is illustrated in Fig. 1. The field radiated by the transmit antenna E d ( r t ) is defined in the far-field region by the far -field function G t ( ˆ r t ) radiated in direction ˆ r t = r t /r t as E d ( r t ) = G t ( ˆ r t ) e − j kr t r t + O ( r − 2 t ) as r t → ∞ , (7) where O ( x n ) stands for “order of" asymptotic. This is the field impinging on the scatterer . Similarly , in the far-field re gion, the scattered electric field E s is fully described by the far-field function G s scattered in direction ˆ r s = r s /r s as E s ( r s ) = G s ( ˆ r s ) e − j kr s r s + O ( r − 2 s ) as r s → ∞ , (8) where G s can be expressed in terms of the scattering matrix S ( ˆ r s , ˆ r t ) [21] G s ( ˆ r s ) = S ( ˆ r s , ˆ r t ) · E d ( r t ) , (9) where we hav e assumed that the amplitude of the plane wa ve incident at the scatterer from direction ˆ r t is giv en by E d ( r t ) . Hence, from (7)-(9), the scattered field can be expressed as E s ( r s ) = S ( ˆ r s , ˆ r t ) · G t ( ˆ r t ) e − j k ( r s + r t ) r s r t as r s , r t → ∞ (10) The total field impinging on the receiv e antenna will then be the sum of the scattered field (10) and the field radiated by the antenna (7) in the direction of vector r o E tot = E d ( r o ) + E s ( r s ) . (11) SUBMITTED TO JOURNAL 4 The open-circuit voltage induced at the receive antenna ports by an impinging wa ve is gi ven by [22] V tot oc = 2 λ j η I G r · E tot , (12) where G r is the far-field function of the receiv e antenna in the direction ˆ r for excitation current I and E tot the amplitude of the incident plane w av e field measured at the phase center of the recei ve antenna that we choose to be the origin of the coordinate system associated with that antenna. From (11) and (12) we see that the total open-circuit voltage can be expressed as the sum of two terms V tot oc = V d oc + V s oc , (13) where both terms are obtained from the corresponding terms in (11), i.e., induced by the direct field and the scattered field, respectiv ely . W e consider next the situation when many scatterers are present between the transmit and the recei ve antennas. The scatterers and the antennas have associated with them local coordinate systems, while their relative positions are defined relativ e to a common coordinate system. Single scattering (no coupling between the scatterers) is assumed, unlike in the later MoM simulations. Enforcing the above made assumptions giv es after some algebraic manipulations the following expres- sion for the open-circuit voltage induced by N s scatterers V s oc = 2 λ j η I N s X n =1 G r ( ˆ r r n ) · S n ( − ˆ r r n , ˆ r t n ) · G t ( ˆ r t n ) e − j k ( r r n + r t n ) r r n r t n , (14) where we use the radius-v ector notation introduced in Fig. 1. W e hav e also introduced the sub-index n in the scattering matrix to indicate that scatterers are in general different. The open-circuit voltage for the direct w ave is given by V d oc = 2 λ j η I G r ( − ˆ r o ) · G t ( ˆ r o ) e − j kr o r o . (15) Under the assumptions stated above, (13)-(15) provide a rather general description of the signal scattering model satisfying the abov e-stated assumptions. In order to e valuate (3), we need to ev aluate h V tot oc i and h| V tot oc | 2 i first. As can be seen from (14), h V tot oc i = h V s oc i + V d oc . Hence, we need to find h V s oc i = N s X n =1 D 2 λ j η I G r · S n · G t ED e − j k ( r r n + r t n ) r r n r t n E , (16) where we hav e omitted the arguments of some functions for the sake of simplicity . Observe that the ensemble av eraging has been factored into two terms: (i) a term that comprises the scattering matrix, which describes randomness of the scattered field polarization and AoA at the location of the recei ver antenna and, (ii) a term comprising the random positions of the scatterers. Both are independent random processes. W e immediately see that h V s oc i = 0 , (17) since both ensemble averages are zero as sho wn in Appendix A. Hence, we obtain that h V tot oc i = V d oc (18) Then from (2), (15) and the Friis equation [22] we arriv e at P LOS = |h V tot oc i| 2 8 R ar = | V d oc | 2 8 R ar =  λ 4 π r o  2 G or G ot P t , (19) where G or , G ot are the gains of the receiv e and transmit antennas in the LOS direction and P t is the transmit power . W e see from (13) and (17) that h| V tot oc | 2 i = h| V s oc | 2 i + | V d oc | 2 . (20) Hence, we need to find h| V s oc | 2 i = N s X n =1 N s X n 0 =1 D  2 λ η | I |  2 ( G r · S n · G t ) ( G r · S n 0 · G t ) ∗ E × D e − j k ( r r n + r t n − r r n 0 − r t n 0 ) r r n r t n r r n 0 r t n 0 E , (21) After taking into account the random position of scatterers and the antenna parameters con ventions and definitions in [22] we show in Appendix B that P RIMP = h| V s oc | 2 i 8 R ar = 3 N s 4 π R 2 s  λ 4 π r o  2 e r 2 G ot h σ s i P t , (22) where e r is the radiation ef ficiency of the receive antenna; all the other variables ha ve been defined abo ve. Combining (19) and (22) into (3) we obtain an estimate of the frequency dependence of the K − factor as function of frequency K ( f ) = 8 π D or R 2 s 3 N s h σ s i , (23) where we have used the relationship between antenna gain and directivity G or = e r D or . Under the above assumptions, the frequency dependence of the K − factor is completely determined by the directivity of the receiver antenna and the type of scatterer used in the model, i.e., the correspond- ing h σ s i . Expression (23) describes the dependence of the K − factor on the radius of the scattering volume R s for constant N s . Clearly , in this case the scatterers will be further away from the receiving antenna if R s increases. This leads to a weaker contribution to the total scattered field po wer and therefore a predominance of the LOS component o ver the RIMP component, i.e., a larger K − factor . Let us now keep the scatterer density ρ s = N s /V s constant; where V s = 4 π R 3 s / 3 . Then, the K − factor can be written as K ( f ) = 2 D or ρ s R s h σ s i . (24) In this case, the trend is the opposite. Indeed, keeping the density constant, a larger radius of the scattering v olume will result in a smaller K − factor due to the lar ger contrib ution of the RIMP component to the total receiv ed power as compared to the LOS component. Results (23) and (24) are both obtained under the assump- tion that all scatterers and the antennas are in the far -field SUBMITTED TO JOURNAL 5 T ABLE I. Spatially av eraged scattering cross-section of dipole of length L L/λ 0 . 5 1 . 5 2 . 5 3 . 5 4 . 5 h σ dip /λ 2 i 0 . 1527 0 . 1835 0 . 2183 0 . 2510 0 . 2819 of each other . Now , for a fixed R s there will be a maximum number of scatterers N s that can be “packed” into this volume. T o obtain this estimate we use the far-field distance R FF to model the diameter of an imaginary sphere surrounding the scatterer . Hence, we need to estimate the number of spheres with volume V FF = π R 3 FF / 6 that can be packed into the volume containing the scatterers V s = 4 π R 3 s / 3 . This number is giv en by N s = η pack V s V FF = 8 η pack  R s R FF  3 , (25) where η pack ≈ 0 . 64 is the packing density of random close packing of spheres [23]. The corresponding scatterer density becomes ρ s = N s V s = 6 η pack π R 3 FF , (26) Thus, (23) and (24) both reduce to K ( f ) = π D or R 3 FF 3 η pack R s h σ s i , (27) which provides a lower bound on the K − factor in the scatter- ing propagation en vironment described above. It is worthwhile to note that the far -field distance R FF also depends on the frequency [17] R FF = 4 λG o π 2 r α E 1 − γ A , (28) where λ is the wa velength, G o is the antenna gain, α E = 0 . 06 is a fitting coefficient that is the same for all antennas and γ A is the antenna gain reduction factor defined by the user . The starting point of the corresponding far-field region for a required error magnitude of the antenna gain defined by 1 − γ A . V . R E S U LT S A. Specific assumptions For the results in this section, we specialize our scatterers to identical dipoles. The scatterers are assumed to be identical vertically polarized resonant dipoles. The spatially averaged scattering cross-section of dipoles is giv en as [24]: h σ dip i λ 2 = 1 . 178 L λ + 0 . 179 ln(22 . 368 L λ ) − 0 . 131 ln 2 (22 . 368 L λ ) , (29) where L = nλ/ 2 , n = { 1 , 2 , ... } , and λ is the free-space wa velength. V alues of the a veraged cross-section for different lengths of dipole that are used in the results in this section are summarized in T able I. Assuming R S = 15 m, the lo wer bound of the K -factor according to (27) is plotted in Fig. 2 vs. frequenc y , for dif ferent electrical lengths of the scatterers. It can be observed that with the increase in frequenc y , a lar ger number of scatterers can fit in the volume [according to (28)] which results in the increase of the scattered power and the decrease of the K -factor . On 10 9 10 10 10 11 − 30 − 20 − 10 0 10 20 30 40 50 60 N s = 10 3 R FF @500 MHz Low er b ound F requency [GHz] K-factor [dB] λ /2 3 λ /2 5 λ /2 7 λ /2 9 λ /2 Fig. 2. The Rician K -factor vs. frequency for different electrical lengths of the scatterers in a volume with R S = 15 m. Three different scenarios: (1) The lower bound, where the scattering volume is filled with largest possible number of scatterers where all of them are in the far-field region of each other, (2) the spacing between the scatterers is chosen based on the far-field distance ( R FF ) at the lowest frequency , and (3) a fixed number of scatterers randomly distributed in the scattering environment, regardless of the electrical size. the other hand, the lar ger electrical size of the scatterers leads to increased K -factor , since it means that a smaller number of scatterers can fit in the volume. Howe ver , if the number of scatterers is kept constant, the K -factor will increase with frequency . This is shown in Fig. 2 with the dash-dotted curves, where the spacing between the scatterers is chosen according to the R FF value at the lo west frequenc y , i.e., 500 MHz. Note that in this case N s is dependent on the electrical size of the scatterers. W e can also assume a case where a fix ed N s is chosen for all v alues of the electrical length. This case is shown in Fig. 2 with the solid lines for N s = 1000 . Unlike the previous cases, as expected, we observe in this case that larger electrical size will lead to lo wer K -factor . B. Comparison with numerical MOM-based simulations W e hav e used MoM in order to numerically simulate the model. Ef fects of multiple scattering and mutual coupling between the scatterers are included in the full-wav e simu- lations. The scatterers are modeled as half-wa ve PEC strips of λ/ 100 width, which are uniformly distributed in a cubic volume with side length of 30 m. Assuming the transmitter antenna is located far from the scatterers and the receiver antenna, it is modeled as a plane wav e impinging on the volume. Furthermore, in order to remov e the effect of the receiv er antenna’ s radiation pattern, it is assumed to be an ideal omnidirectional vertically polarized antenna. This implies that in the simulations, the vertical component of the field is studied and in the model we hav e D or = 1 . Finally , in order to reduce the computation time, Characteristic Basis Functions Method (CBFM) is employed for the resonant scatterers [11]. Fig. 3 shows the analytical and simulated K -factor for cases of 10 , 100 , and 1000 scatterers in the frequency range from 500 MHz up to 100 GHz. It is observed that the simulations and analytical formulas follow the same trend. Howe ver , the SUBMITTED TO JOURNAL 6 10 9 10 10 10 11 10 20 30 40 50 60 70 80 90 F requency [Hz] Rician K -factor [dB] N s = 10 N s = 10 2 N s = 10 3 Analytical Sim. - T x. out of the scattering volume Sim. - T x. within the scattering volume Fig. 3. Comparison of analytical and simulated K -factor vs. fre- quency , for different numbers of identical vertical resonant scatterers. Simulated K -factor for the case of transmitter antenna located in the scattering volume is plotted with dashed line. T ABLE II. Root-mean-square deviation of the K -factor in dB, for the analytical model compared to MoM results N 10 100 1000 Tx. out of the scattering volume 3 . 76 4 . 04 4 . 01 Tx. within the scattering volume 3 . 96 2 . 77 3 . 08 K -factor in the analytical model is slightly lar ger than the simulation results. This can be explained by the fact that the analytical model assumes only single scattering, whereas the effect of multiple scattering is included in the simulation results. Another source of error can be the fact that the density of the scatterers is lower in the cubic v olume than the spherical one. W e observe that for a fix ed number of scatterers, the K -factor increases with frequency , meaning that the LOS component becomes more dominant at the higher frequencies. So far , we ha ve made the assumption that the transmitter antenna is further away from the scattering v olume. Howe ver , we can also assume it to be located in the scattering volume among the scatterers. W e simulate the transmitter as a half wa ve dipole located at R t = R s / 2 = 7 . 5 m. The other details of the simulation are kept as before. The simulated K -factor for this scenario is also plotted in Fig. 3. W e observe that while the K -factor has slightly increased, the trend of its frequency dependence does not change. T able II summarizes the root- mean-square de viation of the K -factor v alues in dB, between the analytical model and the two cases in MoM simulations. In addition to the case of half-wav e PEC scatterers, we have considered two other cases in the MoM simulations for the sake of comparison. In one case, scatterers are loaded with a matched load in order to absorb part of the radiated field from the transmitter antenna and to reduce the scattered power . In the second case, a number of 2 λ × 2 λ PEC plates are also distributed among the half-wav e dipoles, in order to increase the scattered power . In this case, the number of PEC plates is 1, 5, and 20 in the cases where N S is 10, 100, and 1000, respectiv ely . 63 characteristic basis functions are used for each plate. The K factor is plotted vs. frequency for these three cases in Fig. 4. As expected, we observe in this figure that for 10 9 10 10 10 11 10 20 30 40 50 60 70 80 90 F requency [Hz] Rician K -factor [dB] N s = 10 N s = 10 2 N s = 10 3 Dip oles Loaded dipoles Dip oles and plates Fig. 4. The simulated Rician K -factor in the presence of PEC plate scatterers and match-loaded half-wave dipoles. the same N S , terminating the dipole in matched load will lead to higher K -factor , while the presence of the PEC plates leads to decreased K -factor . Howe ver , we observe that the general trend of the K -factor vs. frequency is the same for all cases. V I . C O N C L U S I O N S A simple model is introduced to inv estigate the frequency dependence of Rician K -factor in generic random scattering en vironments. The K -factor is deri ved analytically as a func- tion of the average scattering cross-section of the scatterers. The formulas are verified against full-wa ve MoM simulations which shows a good agreement between the two. The main contributing factors to the K -factor are sho wn to be the density of the scatterers ρ s , the radius of the scattering environment R s , the av erage bistatic cross section of the scatterers h σ s i , and the directi vity of the receiving antenna D or . Of these, h σ s i and D or are frequency-dependent and contribute to the frequency dependence of the K -factor . In the simulations two scenarios are in vestigated with the transmitter antenna both within the scattering v olume and out of it. It is observed that for thin wire scatterers and planar scatterers, the K -factor increases quadratically with the frequency . Although we have used resonant dipoles in the model, the formulas hav e the flexibility to accommodate other types of scatterers as long as the av erage cross-section is known. As a final remark we may say that the model presented here needs further study to take into account other specific propagation scenarios of great relev ance kno wn as obstructed line-of-sight, and the more general ones described as non- line-of-sight scenarios. Specific applications such as vehicle- to vehicle, or massive multiple-input multiple-output systems need to be studied too. A P P E N D I X A Although the scatterers are assumed to have same orienta- tion, and due to their random positions, the polarization vector ˆ p s of the scattered field impinging on the receive antenna will SUBMITTED TO JOURNAL 7 be randomly mismatched to its polarization vector ˆ p r , where 6 ˆ p r ˆ p s is the mismatch angle. W e have D 2 λ j η I G r · S n · G t E = D 2 λ j η I | G r ||S n · G t | cos( 6 ˆ p r ˆ p s ) E = D 2 λ j η I | G r ||S n · G t | ED cos( 6 ˆ p r ˆ p s ) E = 0 , (30) since h cos( 6 ˆ p r ˆ p s ) i = 0 for 6 ˆ p r ˆ p s uniformly distributed between 0 and 2 π . In order to show that the second ensemble average is also zero we now make further assumptions that simplify our computations r t n ≈ r o , (31a) r r n = ρ n , (31b) where (31a) states that the transmit antenna is at a much larger distance from both the receive antenna and the scatterer as compared to the radius R s that delimits the volume containing the scatterers. (31b) is just a substitution of variables. Hence, we can now write D e − j k ( r t n + r r n ) r r n r t n E ≈ 1 r o D e − j kr t n e − j kρ n ρ n E . (32) W ithout losing generality , we assume the local coordinate sys- tem on the receiv er antenna is chosen such that the transmitter antenna is located at θ = 0 and the n -th scatterer is located at θ n . Then, (32) is approximated as 1 r o D e − j kr t n e − j kρ n ρ n E ≈ 1 r o D e − j k ( r o − ρ n cos θ n ) e − j kρ n ρ n E = e − j k ( r o ) r o D e − j kρ n (1 − cos θ n ) ρ n E , (33) where, using (6b), it can be sho wn after straightforw ard algebraic manipulations that D e − j kρ n (1 − cos θ n ) ρ n E = Z R s ρ =0 Z π θ =0 e − j kρ (1 − cos θ ) ρ 3 ρ 2 R 3 s sin θ 2 d θ d ρ = 0 + O ( R − 2 s ) as R s → ∞ . (34) In practice, it suffices that R s  λ 2 π . A P P E N D I X B Let’ s first recall some antenna parameters following the definitions in [22]. The directivity of an antenna is defined as D o ( ˆ r ) = 4 π | G ( ˆ r ) | 2 2 η P rad , (35) where G ( ˆ r ) is the far-field function satisfying the normaliza- tion integral I D o ( ˆ r )dΩ = I 4 π | G ( ˆ r ) | 2 2 η P rad dΩ = 4 π . (36) The total radiated power is P rad = e rad P t , (37) where e rad is the radiation efficiency of the antenna and P t is the input po wer to the antenna, which in turn is related to the current at the antenna input port | I | (in (12)) as P t = 1 2 R ar | I | 2 , (38) The antenna gain is gi ven by G o = e rad D o . (39) Follo wing the assumption that the polarization of the scat- tered field at the position of the receiv er antenna is random, we write the ensemble av erage of | V s oc | 2 as h| V s oc | 2 i = N s X n =1 N s X n 0 =1 D  2 λ η | I |  2 ( G r · S n · G t ) ( G r · S n 0 · G t ) ∗ E × D e − j k ( r r n + r t n − r r n 0 − r t n 0 ) r r n r t n r r n 0 r t n 0 E = N s X n =1 D  2 λ r o η | I |  2 | G r · S n · G t | 2 ED 1 ρ n 2 E , (40) where we hav e used the results in Appendix A, i.e., for n 6 = n 0 the ensemble averages are null. All the terms in (40) are identical since the scatterers are identical. Let’ s denote the first ensemble average in (40) by X , then X = D  2 λ r o η | I |  2 | G r · S n · G t | 2 E = D  2 λ r o η | I |  2 | G r | 2 |S n · G t | 2 cos 2 ( 6 ˆ p r ˆ p s ) E = D  2 λ r o η | I |  2 | G r | 2 ED |S n · G t | 2 ih cos 2 ( 6 ˆ p r ˆ p s ) E , (41) where the factorization of the ensemble a verage has been performed by grouping terms that are statistically independent. The bistatic scattering cross section is defined as σ ( − ˆ r r n , ˆ r t n ) = 4 π |S n ( − ˆ r r n , ˆ r t n ) · G t | 2 | G t | 2 . (42) Then, inserting (42) into (41) giv es X = D  2 λ r o η | I |  2 | G r | 2 E h σ i 4 π D | G t | 2 ED cos 2 ( 6 ˆ p r ˆ p s ) E . (43) W e now compute the ensemble av erage terms in (41). Since we assume that 6 ˆ p r ˆ p s is uniformly distrib uted between 0 and 2 π , then h cos 2 ( 6 ˆ p r ˆ p s ) i = 1 2 . (44) Giv en assumption (31a), we use (35), (37) and (39) to arrive at h| G t | 2 i ≈ | G t | 2 = 2 η G ot P t 4 π . (45) SUBMITTED TO JOURNAL 8 Combining (36), (37) and (38) into the first ensemble a verage in (43) we get D  2 λ r o η | I |  2 | G r | 2 E = D  2 λ r o η  2 e rad R ar | G r | 2 2 P rad E , (46a) = 8 R ar e rad 2 η  λ r o  2 D | G r | 2 2 η P rad E , (46b) = 8 R ar e rad 2 η  λ r o  2 I | G r | 2 2 η P rad dΩ 4 π , (46c) = 8 R ar e rad 4 π 2 η  λ r o  2 , (46d) where we ha ve used the assumption that the AoA of the scat- tered wa ves are isotropically distrib uted since the scatterers are uniformly distributed within the spherical scattering v olume. For the second ensemble a verage in (40) and by using (6b) we straightforwardly obtain that D 1 ρ 2 n E = Z R s 0 1 ρ 2 3 ρ 2 R 3 s d ρ = 3 R 2 s . (47) Combining (43), (44), (45), (46d), (47) into (40) provides the result in (22). R E F E R E N C E S [1] F . Boccardi, R. W . Heath, A. Lozano, T . L. Marzetta, and P . Popovski, “Fiv e disruptive technology directions for 5G, ” IEEE Communications Magazine , vol. 52, no. 2, pp. 74–80, 2014. [2] E. G. Larsson, O. Edfors, F . Tufv esson, and T . L. Marzetta, “Massive MIMO for next generation wireless systems, ” IEEE Communications Magazine , vol. 52, no. 2, pp. 186–195, 2014. [3] T . L. Marzetta, “Noncooperative cellular wireless with unlimited num- bers of base station antennas, ” IEEE T ransactions on W ireless Commu- nications , vol. 9, no. 11, pp. 3590–3600, 2010. [4] F . Rusek, D. Persson, B. K. Lau, E. G. Larsson, T . L. Marzetta, O. Edfors, and F . Tufv esson, “Scaling up MIMO: Opportunities and challenges with very large arrays, ” IEEE Signal Pr ocessing Ma gazine , vol. 30, no. 1, pp. 40–60, 2013. [5] A. A. Glazuno v , V .-M. K olmonen, and T . Laitinen, “MIMO over -the- air testing, ” LTE-Advanced and Next Generation Wir eless Networks: Channel Modelling and Propagation , pp. 411–441, 2012. [6] H. Q. Ngo, E. G. Larsson, and T . L. Marzetta, “ Aspects of fa vorable propagation in massiv e MIMO, ” in Proceedings of the 22nd Eur o- pean Signal Processing Conference (EUSIPCO 2014) , Lisbon, Portugal, September 2014, pp. 76–80. [7] E. Björnson, E. G. Larsson, and T . L. Marzetta, “Massive MIMO: T en myths and one critical question, ” IEEE Communications Magazine , vol. 54, no. 2, pp. 114–123, 2016. [8] X. Gao, O. Edfors, F . Rusek, and F . T ufvesson, “Massi ve MIMO performance ev aluation based on measured propagation data, ” IEEE T ransactions on Wir eless Communications , vol. 14, no. 7, pp. 3899– 3911, 2015. [9] P .-S. Kildal, “Rethinking the wireless channel for OT A testing and network optimization by including user statistics: RIMP, pure-LOS, throughput and detection probability , ” in Pr oceedings of the 2013 International Symposium on Antennas and Pr opagation (ISAP) , Nanjing, China, October 2013. [10] P . Kildal and J. Carlsson, “New approach to OT A testing: RIMP and pure-LOS reference en vironments & a hypothesis, ” in Proceedings of the 7th Eur opean Conference on Antennas and Pr opagation (EuCAP 2013) , Gothenburg, Sweden, April 2013, pp. 315–318. [11] R. Maaskant, “ Analysis of large antenna systems, ” Ph.D. dissertation, T echnische Universiteit Eindhov en, 2010. [12] K. Haneda, S. L. H. Nguyen, A. Karttunen, J. Järveläinen, A. Bamba, R. D’Errico, J. Medbo, F . Undi, S. Jaeckel, N. Iqbal, J. Luo, M. Ry- bako wski, C. Diakhate, J.-M. Conrat, A. Naehring, S. Wu, A. Goulianos, and E. Mellios, “Measurement results and final channel models for preferred suitable frequency ranges, ” T ech. Rep. H2020-ICT -671650- mmMA GIC/D2.2, May 2017. [Online]. A vailable: http://5g- mmmagic. eu/ [13] W . Jakes, Micr owave mobile communications , ser . IEEE Press classic reissue. IEEE Press, 1974. [14] A. Molisch, Wir eless Communications , ser. W iley - IEEE. Wiley , 2012. [15] L. Tsang, J. Kong, and K. Ding, Scattering of Electr omagnetic W aves, Theories and Applications , ser . Scattering of Electromagnetic W aves. W iley , 2004. [16] M. E. Bialkowski, P . Uthansakul, K. Bialkowski, and S. Durrani, “In- vestigating the performance of MIMO systems from an electromagnetic perspectiv e, ” Microwave and Optical T echnology Letters , vol. 48, no. 7, pp. 1233–1238, 2006. [17] I. Kim, S. Xu, and Y . Rahmat-Samii, “Generalised correction to the friis formula: quick determination of the coupling in the Fresnel region, ” IET Micr owaves, Antennas & Propa gation , v ol. 7, no. 13, pp. 1092–1101, 2013. [18] M. Riaz, N. M. Khan, and S. J. Nawaz, “ A generalized 3-D scattering channel model for spatiotemporal statistics in mobile-to-mobile com- munication environment, ” IEEE T ransactions on V ehicular T echnology , vol. 64, no. 10, pp. 4399–4410, Oct 2015. [19] M. Ghoraishi, J. T akada, and T . Imai, “Identification of scattering objects in microcell urban mobile propagation channel, ” IEEE T ransactions on Antennas and Propa gation , vol. 54, no. 11, pp. 3473–3480, Nov 2006. [20] M. K. Samimi and T . S. Rappaport, “3-D millimeter-wa ve statistical channel model for 5G wireless system design, ” IEEE T ransactions on Micr owave Theory and T echniques , vol. 64, no. 7, pp. 2207–2225, July 2016. [21] J. V an Bladel, Electr omagnetic Fields , ser . IEEE Press Series on Elec- tromagnetic W ave Theory . John W iley and Sons, 2007. [22] P .-S. Kildal, F oundations of Antennas: A Unified Approac h for Line- of-Sight and Multipath . Kildal Antenn AB, 2015, A vailable at www .kildal.se. [23] H. M. Jaeger and S. R. Nagel, “Physics of the granular state, ” Science , vol. 255, no. 5051, p. 1523, 1992. [24] P . Z. Peebles, “Bistatic radar cross sections of chaff, ” IEEE T ransactions on Aer ospace and Electronic Systems , no. 2, pp. 128–140, 1984.