168 RADIOWAVE PROPAGATION SMART ANTENNAS Figure 6.3 shows the geometry of the elliptical scattering model. M=Max.time delay Scatter Region Figure 6.3.Elliptical Scattering Model having two parameters D andM The model is appropriate for microcell environments where the antenna heights are relative low.With low antennas,the base station will receive multipath components from scatterers distributed around the mobile as well as itself.All angles in the azmuth plane are involved so that -开≤B≤π Furthermore,the maximum delay is set by the size of the ellipse.On physical grounds,the assumption of the model that delays greater than TM are non- existent may be justified by the fact that longer delays will have weaker strength owing to the longer path length traveled. The scatter density pi(m,ym)within this elliptical region is equal to pm(cm,n）=元 (6.17) 水eT6 For the elliptical scatter region the lowe integration and simplifying,the p.d.f.for the angle of arrival for the elliptical scattering model pe(B)is mu间=最-法品2a司 a1[1-(D/2a)212 -T≤B≤r(618) k1 [1-(D/2a)co892 (6.19) where
168 RADIOWAVE PROPAGATION & SMART ANTENNAS Figure 6.3 shows the geometry of the elliptical scattering model. Figure 6.3. Elliptical Scattering Model having two parameters D and The model is appropriate for microcell environments where the antenna heights are relative low. With low antennas, the base station will receive multipath components from scatterers distributed around the mobile as well as itself. All angles in the azimuth plane are involved so that Furthermore, the maximum delay is set by the size of the ellipse. On physical grounds, the assumption of the model that delays greater than are nonexistent may be justified by the fact that longer delays will have weaker strength owing to the longer path length traveled. The scatter density within this elliptical region is equal to For the elliptical scatter region the lower and upper limits in (6.4) are Substituting these and (6.17) into (6.4) and carrying out the integration and simplifying, the p.d.f. for the angle of arrival for the elliptical scattering model is where
Geometric Models for Angle and Time of Arrival 160 =品-(2) (6.20) (comadeanaii (6.21) Applying the above identity to (6.8)and simplifying,the Fourier cosine coef- ficients are (6.22) where 如A全会<1→6=am (6.23) Substituting (6.22)into (6.6),the standard deviationfor the elliptical scatter model is n=i n2 [π2 = /1+cos31 +4osn(1+co+m)十 (6.24) where the identity [43,1.513-4] n(1-x= was used in(6.24).Ther.ms.angular spread is.The Fourier coefficients can be substituted in (6.11)to yield the spatial cross correlation. The c.d.f.for the time of arrival for a given r is obtained by inserting (6.17)into (6.1).The ellipse with delay
Geometric Models for Angle and Time of Arrival 169 For this p.d.f., the mean angle is clearly zero. The Fourier cosine coefficients can be determined by the application of the identity [43, 3.616-7] Applying the above identity to (6.8) and simplifying, the Fourier cosine coefficients are where Substituting (6.22) into (6.6), the standard deviation for the elliptical scatter model is where the identity [43, 1.513-4] was used in (6.24). The r.m.s. angular spread is The Fourier coefficients can be substituted in (6.11) to yield the spatial cross correlation. The c.d.f. for the time of arrival for a given is obtained by inserting (6.17) into (6.12). The ellipse with delay will have a semi-major axis and a semi-minor axis Intersection of the
170 RADIOWAVE PROPAGATION SMART ANTENNAS ese with delaywith the scatter region is simply the ellipsetsc egral in (6.12)is simply equal to the area of ellipse Ar=ab.The c.d.f.in the time of arrival is -先-÷-婴 (6.25) Differentiating P()w.r.t.yields the p.d.f.pe(as 1a2r/rm)2-(D/2a)2D 1 o)=67m-o2a·a≤a (6.26) for the del be Yound from ( 2zx2-(D/2a)2 =M D/2a “公-o血 = 1-(②+（会） =w-2) (6.27) Likewise,the second moment 72can be found from(6.15) 7=品广2g-D2 D/2a Va-(D2a)de 孚p+2A+A@An(生器2》6测 The r.m.s.delay spread for the elliptical scattering model can be found from =72-72.Figure 6.4 shows the p.d.f.of the angle of arrival and
170 RADIOWAVE PROPAGATION & SMART ANTENNAS ellipse with delay with the scatter region is simply the ellipse itself. Because of the uniform density, the integral in (6.12) is simply equal to the area of ellipse The c.d.f. in the time of arrival is Differentiating w.r.t. yields the p.d.f. as The p.d.f. is peaked and has an integrable singularity at or at This peak corresponds to the line of sight path. The mean delay for the model can be found from (6.14) Likewise, the second moment can be found from (6.15) The r.m.s. delay spread for the elliptical scattering model can be found from Figure 6.4 shows the p.d.f. of the angle of arrival and