, 9) is the probability distribution function of users around the antenna array. The integral limits of the above equation are as illustrated in Fig. 2.16, which shows an antenna array can steer its beam to any location Chapter 2 SMART ANTENNA ARRA YS AND CDMA CAPACITY 27 in a building group throughout the vertical angles [61,62], azimuth angles [^ \/,4b], and front to back [X1JC2] of the building group under investigation. X 2 Top V i e w S i d e V i e w Fig. 2.16 Integral Limits for User Probability Distribution Function From Eq. 2.31, the mean system capacity1 is K = N x2 ,a) W, building a = tan(^ ) (4.40) and the vertical angle of arrival to Array 3 is 6C2 = Y _ t a n \"nJJselCl (4.41) The distance components for Zone C3 are _c[0maJXo),Xo] d C3\\a d, C3\\b sin sin(#) W cross (^c)sin((9) dC3\\c - c{ #o)> from building groups E & J Compute total interference and capacity when steered to Increment Q< Reset 0o=0mJc{0o,Xo)} Fig. 5.1 Program Flowchart (See Fig. 4.1) Chapter 5 CAPACITY ESTIMATION AND RESULTS 82 5.2.3 Building Group D Intra-cell Interference Simulation Fig. 5.2 shows the steps for the simulation of intra-cell interference from building group D for Eq. 4.18. Given an interferer's location (x,<\/),d), Array l's normalized array factor, and the user PDF from Eq. 3.10, the weighted interference power is calculated. This is repeated for all interferers' locations within building group D to obtain the mean interference power from building group D. For efficient simulation, interference from (<\/>, 8) and (-Y, 8) are computed at the same time to save half of the integration steps. Set interferer's location to (x = X0,(f> = 0\u00b0,8 = 8mm[c(r,X0)\u00a7 Calculate power received, Pr\u00b0, by [ Array 1 from (0,8) and (-r,0) j Calculate weighted received power with the user distribution, ubuildjng(x, mmn{lx\u00ab\\d = 0m.x K^2^.)]) Compute interferer's AOA, { , 9) for the Zone I E3 interferer, and from (- max] 3 (x) No Reset ^ m a x l ( 2 X 0 ) Yes (E4) Fig. 5.9 Zones E3 and J3 Interference Simulation for Figs. 4.13 and 4.14 Chapter 5 CAPACITY ESTIMATION AND RESULTS 92 Set Zone E4 interferer's location to (x = 2XQ + X2,~~, of the obstacle. If the plane of incidence is defined as the plane containing both the incident and reflected waves, Figs. 4.2 and 4.3 show the two cases when the electric field is parallel and perpendicular to the plane of incidence, where El-and Er are the incident and reflected waves respectively. Fig. 4.2 Electric Field Parallel to Plane of Incidence Side View of City Block Chapter 4 MULTICELL INTERFERENCE 44 Fig. 4.3 Electric Field Normal to Plane of Incidence Top View of City Block As shown in both figures, even if the intended cell of service are the buildings across the street (left side on Figs. 4.2 and 4.3), the array is still susceptible to interference generated from buildings on its side of the street (right side on Figs. 4.2 and 4.3). Also shown is that the electric field components are vertically polarized, which is the orientation that the base station array able to receive. Because the interfering signal propagates both vertically and horizontally to the receiving antenna array, by bouncing Chapter 4 MULTICELL INTERFERENCE 45 off the buildings being served, the reflection coefficient must take into account both cases shown. Assuming that the medium defined by s2 is free space, we have (4.4) S2 \u2014 \u00a30 where so is the permittivity of free space and s\/r is the relative permittivity of the building material, tabulated in Table 4.1 for common materials. Material Glass Plexiglass Brick Limestone Permittivity (Sir) 4-7 3.45 4.44 7.51 Table 4.1 Permittivity of Common Building Materials From Figs. 4.2 and 4.3, assuming the front of buildings is reasonably smooth, we have C = C h r h ' (4.5) Er=YEi where Y = T\\\\ when Er and Ei are parallel to the plane of incidence, and T = Fx when Er and Et are perpendicular to the plane of incidence as illustrated in Figs. 4.1 and 4.2 respectively, and are defined as [26] - er sin C, +Js- cos2 C: r i l = , 2 (4-6) \u00a3 r S m & + 4 \u00a3 r - C 0 S Ci and Chapter 4 MULTICELL INTERFERENCE 46 sin - ^sr - cos2 C,.t s in^ ; + ^sr -cos 2 ^,. r , = (4.7) In addition to strictly horizontal or vertical propagation, Fig. 4.4 shows the case when a signal from an interfering user in building group F is reflected off building group D at an angle to Array 1. It shows that the normal and parallel axes of the propagating wave do not coincide with the horizontal and vertical spatial axes, and that the vertically polarized wave received by the array has components both normal and parallel to the plane of incidence. As such, both reflection coefficients of Eqs. 4.6 and 4.7 must be taken into account and superposition is applied to determine the amplitude of the reflected waves. For this, we need to determine the vector of the incident electric field, and decompose it into components that are perpendicular and parallel to the plane of incidence. For a vertically oriented antenna, the electric field unit vector in the far field is [26] E(e) = 6 (4.8) where \" denotes the unit vector and 6 is the vertical angle in Fig. 4.4. Illustrated in Fig. 4.5, the direction of the incident electric field vector, E-\u201e is therefore perpendicular to the direction of propagation, where EL and E^ are the components that are perpendicular and parallel to the plane of incidence. Fig. 4.4 3-D Reflection Chapter 4 MULTICELL INTERFERENCE 48 Suppose Z' = 1+ Z , we can relate the electric field components geometrically in Fig. 4.5, and the incident electric field have intensity of Ej = Z'sin(a:) (4.9) where z = cQ,*\u201e) tan (a) a = n -0. . (4.10) The electric field components that incident on the building's wall are therefore EL = Z'sin(v) where (4.11) n v = w 2 ur = tan\" Y = XQ tan(^) (4.12) and the reflected electric field intensities can be obtained using Eqs. 4.5 - 4.7 where the incident angle is \u00a3 = tan' ^Y2+Z2 (4.13) Fig. 4.5 Electric Field Components Chapter 4 MULTICELL INTERFERENCE The electric field amplitude loss associated with a reflection is L =K reflection r-. l E r . = E' and allow us to find the power of an interfering signal after a reflection in our model. 4.3.2 Diffraction Loss Diffraction occurs whenever a radio wave impinges upon a discontinuous surface such as a building corner, and provides coverage to shadowed regions where there is no LOS to the base station array. Unlike the macro cell scenario, where the base station antenna is raised many times higher than surrounding buildings and diffraction over roof tops is common, diffraction in a micro cell environment is primarily limited to building corners to provide coverage to perpendicular cross streets [28]. Niu and Bertoni [28] have performed a series of measurements in Manhattan and concluded that in an urban area with rectangular street grid and high rise buildings, diffraction accounts for 15 to 20 dB of power loss. Measurements in Stockholm [36] and Tampa [37] indicated that loss is in the range of 20 - 25 dB and concluded that 20 dB of loss is seen around a corner. Fig. 4.6 illustrates the shadowed regions in our urban model where interference to Array 1 is by means of diffraction. The value of 20 dB will be used as diffraction loss in our model. 50 (4.14) Chapter 4 MULTICELL INTERFERENCE 51 Fig. 4.6 Shadowed Regions 4.3.3 Distance Path Loss In [1] and [39], the use of a free space propagation path loss model was suggested between an external antenna and a user located within a building, without the use of an outdoor reference value. Regression analysis was also performed in [1] to compute various building parameters that best fits measurements made inside buildings and the following was suggested as a suitable path loss model PL(dB) = 32.44 + 20 log d + 20 log \/ + Bf (4.15) where d, f, and Bf are the distance between the user and the serving base station array, frequency of signal, and free space clutter factor with a mean value of 18.3 dB, respectively. Eq. 4.15 wil l therefore be used to model propagation loss in our model. 4.3.4 Building Penetration Loss Since our intended users are located in buildings, served by an external antenna array, the power loss associated with signals propagating through the external walls of a building must be taken into account. In [38], a series of measurements at 900 MHz, 1800 M H z , Chapter 4 MULTI CELL INTERFERENCE 52 and 2300 M H z were conducted and the average penetration loss was concluded to be 17 dB. Because all mobiles are subjected to a similar penetration loss of 17 dB, we can assume that the power control is able to compensate for this loss. Further, since measurements were made directly inside buildings in [1] of signal from an external transmitter, and 6\/ in Eq. 4.15 is very close to the average penetration loss reported in [38], it is assumed that 6\/is representative of the building penetration loss. 4.3.5 Element Factor From Eq. 2.28, it can be observed that the elemental factor need not be considered because perfect power control is assumed to be able to compensate for this in a single cell system. However, in a multi cell system, each base station has no power control ability on adjacent cells, and interfering signals from adjacent cell users arrive at different A O As to both the adjacent and in cell antenna arrays as illustrated in Fig. 4.10. If panel antenna elements are used, for an interfering user in building group F, served by Array 2, we let (~~