Chapter 6 GEOMETRIC MODELS FOR ANGLE AND TIME OF ARRIVAL 1. INTRODUCTION Array antennas are sometimes employed at the receiver for a variety of purposes such as to combat fading or to reduce co-channel interference.To evaluate the performance of a wireless communication system using antenna arrays it becomes necessary to have spatial channel models that describe the angle of arrival and time of arrival of the multipath components.Among the most widely used radio propagation models is the single bounce scattering geometric model,where propagation between the transmitting and receiving antennas is assumed to take place via single scattering from an intervening ob- stacle.The scattering cross section of the scatterers is assumed to be isotropic, but their density is umed to vary fro om locatio n to locat Currently,sev eral geometric models are available such as the ring model 53 discrete uni form model [118],circular scattering model [30],elliptical scattering model [981.Gaussian angle of arrival model [931.etc.with each model being ap- plicable to a s ecific environment type For example,the circular scatter ble to macro cell type of environments,whereas the elliptic model is suitable for a micro-or a pico-cell types of environment.Most of these models are 2D in nature in that they assume radio propagation to take place in a plane containing the transmitter and the receiver.In this Chapter we will consider the circular scattering model,the elliptical scattering model and the Gaussian density model and show statistics for the angle and time of arrival
Chapter 6 GEOMETRIC MODELS FOR ANGLE AND TIME OF ARRIVAL 1. INTRODUCTION Array antennas are sometimes employed at the receiver for a variety of purposes such as to combat fading or to reduce co-channel interference. To evaluate the performance of a wireless communication system using antenna arrays it becomes necessary to have spatial channel models that describe the angle of arrival and time of arrival of the multipath components. Among the most widely used radio propagation models is the single bounce scattering geometric model, where propagation between the transmitting and receiving antennas is assumed to take place via single scattering from an intervening obstacle. The scattering cross section of the scatterers is assumed to be isotropic, but their density is assumed to vary from location to location. Currently, several geometric models are available such as the ring model [53], discrete uniform model [118], circular scattering model [30], elliptical scattering model [98], Gaussian angle of arrival model [93], etc., with each model being applicable to a specific environment type. For example, the circular scattering model is applicable to macro cell type of environments, whereas the elliptical model is suitable for a micro- or a pico-cell types of environment. Most of these models are 2D in nature in that they assume radio propagation to take place in a plane containing the transmitter and the receiver. In this Chapter we will consider the circular scattering model, the elliptical scattering model, and the Gaussian density model and show statistics for the angle and time of arrival
164 RADIO WAVE PROPAGATION SMART ANTENNAS 2. GENERAL FORMULATION Figure 6.1 shows the geometry of scatterers distributed around the mobile station which is transmitting to a base station.The scatter region is considered convex in shape and can possibly extend to infinity.It is assumed that propa- gation takes place in the horizontal plane containing the tip of the transmitting and rec ving antennas. The global in this plane r edenoted b (x,y).The line joining the transmitter and receiver makes an angle with the x-axis and the coordinates of a point with respect to this line are (, The polar coordinates with respect to the base station are(r,B).The mobile station is assumed to be located at a distance D from the b estation along this line. The coordin ates of a poin with respect to the mobile station are denoted by (m,ym).The polar coordinates with respect to the mobile station are (rm,m).The relationships between these various coordinates are 6=TocosB=xcos+ysin;Im=rm cos om=-D (6.la) yo ro sin B=-xsin+ycos; hm=Tmsin中m=b (6.1b) r2=r话+D2-2rDcos 中=6+. (6.1c) The density of scatterers about the mobile station is given as a probabil- m Scatter Region y%↑ X BS D Figure 6.1.Geometrically based single bounce scattering model. ity density function pi(m,ym).Waves from the transmitter undergo single scattering from the scatter region before arriving at the receiver.The delay variable is related to the total path length traveled+m and the speed of light c via CT To+rm (6.2a) =r%+Vr号+D2-2 D cos B. (6.2b)
164 RADIO WAVE PROPAGATION & SMART ANTENNAS 2. GENERAL FORMULATION Figure 6.1 shows the geometry of scatterers distributed around the mobile station which is transmitting to a base station. The scatter region is considered convex in shape and can possibly extend to infinity. It is assumed that propagation takes place in the horizontal plane containing the tip of the transmitting and receiving antennas. The global coordinates in this plane are denoted by (x, y). The line joining the transmitter and receiver makes an angle with the x-axis and the coordinates of a point with respect to this line are The polar coordinates with respect to the base station are The mobile station is assumed to be located at a distance D from the base station along this line. The coordinates of a point with respect to the mobile station are denoted by The polar coordinates with respect to the mobile station are The relationships between these various coordinates are The density of scatterers about the mobile station is given as a probabilFigure 6.1. Geometrically based single bounce scattering model. ity density function Waves from the transmitter undergo single scattering from the scatter region before arriving at the receiver. The delay variable is related to the total path length traveled and the speed of light c via
Geometric Models for Angle and Time of Arrival 165 The locus of all points having a constant value of is an ellipse with the base station and mobile station at its foci as shown in Figure 6.2.Waves arriving Ip +rm=ct Scatter Region A Figure 6.2.Waves arriving with delay less thanr will arrive from the intersection of the ellipse and the scatter region. from the scattering region that is enclosed by this ellipse will have delays less than Maximum delay TM occurs when the ellipse just encloses all points of the scatter region.If the scattering region extends to infinity,the maximun delay becomes infinity. The objective is to determine the marginal p.d.f.p(B)of the angle of arrival (AOA)and the marginal p.d.f.p()of the time of arrival (TOA)as seen from the given is assumed that the radiatio patterns of the transmitting and receiving antennas are omnidirectional in the xy plane.This is not a serious limitation as the transmitting antenna pattern could always be included n the definition of p(m,ym)and the receiving p().The ttem could always of random variables and p e problem from t lan sys tem (xm,ym)to the polar coordinate system (ro,B)or to (T,B).The joint density in (B)can be obtained from the standard Jacobian of the transfor- mation between Cartesian and polar coordinates relating the elemental area damdym==rodrodB.The result is p(Tb,B)=rop1(xm,ym) cing-D (6.3) A.PDFfor AOA. The marginal p.d.f.p(B)is obtained by including contribution from all points along the line B=constant.If this line intersects the scatter region at rbm()and roM()with0≤rm≤r%≤rbM<oo,then TAM(B) p()= p(r6,B)dra rom(B)
Geometric Models for Angle and Time of Arrival 165 The locus of all points having a constant value of is an ellipse with the base station and mobile station at its foci as shown in Figure 6.2. Waves arriving Figure 6.2. Waves arriving with delay less than will arrive from the intersection of the ellipse and the scatter region. from the scattering region that is enclosed by this ellipse will have delays less than Maximum delay occurs when the ellipse just encloses all points of the scatter region. If the scattering region extends to infinity, the maximum delay becomes infinity. The objective is to determine the marginal p.d.f. of the angle of arrival (AOA) and the marginal p.d.f. of the time of arrival (TOA) as seen from the base station given It is assumed that the radiation patterns of the transmitting and receiving antennas are omnidirectional in the xy plane. This is not a serious limitation as the transmitting antenna pattern could always be included in the definition of and the receiving antenna pattern could always be included in and The problem boils down to transformation of random variables from the Cartesian system to the polar coordinate system or to The joint density in can be obtained from the standard Jacobian of the transformation between Cartesian and polar coordinates relating the elemental area The result is A. PDF for AOA: The marginal p.d.f. is obtained by including contribution from all points along the line = constant. If this line intersects the scatter region at and with then
166 RADIOWAVE PROPAGATION SMART ANTENNAS ToM(B) ropi(ro CosB-D,rosinB)dro (6.4) rom(B) It is of interest to detemine the mean angand the variancefor a given distribution.It is reasonable to assume that the scatter density is symmetrically distributed about the line joining the base station and the mobile.In this case the p.d.f.p()will be an even function in B and the mean angle =0.The 2=B2p()d (6.5) Equation (6.5)can be cast in a different form by making use of Parseval's theorem for Fourier series.Expressing Band p()as a Fourier series in the interval (,)equation (6.5)can be expressed as 3+m∑(-1)马 (6.6) where the even p.d.f.p()is expressed as Fourier cosine series 间=云+B,on0 (6.7) n=1 with coefficients Bn 2 p(B) (6.8) Equation (6.6)is very convenient if the Fourier cosine coefficients of p(B) can be found in a closed form.The r.m.s.angular spread in a scattering environment is defined as 2(-B). In the study of array antennas,it is also of interest to determine the complex spatial correlation p(kod)=(e-jkodco (6.9) between the field received at two points located along the x-axis and sepa- rated by a distance d.The quantity ko is the usual wavenumber in free-space and angle brackets denote expectation with respect to the angular variable =+B.In Appendix B.it is shown that the expression for (kod)for
166 RADIOWAVE PROPAGATION & SMART ANTENNAS It is of interest to determine the mean angle and the variance for a given distribution. It is reasonable to assume that the scatter density is symmetrically distributed about the line joining the base station and the mobile. In this case the p.d.f. will be an even function in and the mean angle The variance for a symmetric distribution is Equation (6.5) can be cast in a different form by making use of Parseval’s theorem for Fourier series. Expressing and as a Fourier series in the interval equation (6.5) can be expressed as where the even p.d.f. is expressed as Fourier cosine series with coefficients Equation (6.6) is very convenient if the Fourier cosine coefficients of can be found in a closed form. The r.m.s. angular spread in a scattering environment is defined as In the study of array antennas, it is also of interest to determine the complex spatial correlation between the field received at two points located along the x–axis and separated by a distance d. The quantity is the usual wavenumber in free-space and angle brackets denote expectation with respect to the angular variable In Appendix B, it is shown that the expression for for
Geometric Models for Angle and Time of Arrival 167 symmetric distributions,with p(-)=p(),can also be put in the form: p(kod)= dco(dsin sinB)p(B)d (6.10) Jo(kod)+2>(-j)"Jn(kod)Bn cosno, (6.11) n-1 where Jn()is the Bessel function of the first kind of order n. B.PDF for TOA: The marginal p.d.f.for the tune of arrival could,in principle,be obtained by first finding the joint density p(,B)and integrating over all possible values of B(r).However,a simpler approach is to first find the c.d.f.P(r)and then to take its derivative w.r.t.r to get p(r)[301.To determine the c.d.f.for a to this.Denoting the intersection of the scatter region with the ellipse as Ar. the c.d.f and the p.d.f.for the time of arrival are P(r)= (6.12) a p(r)= aT JJA. p(cm,hm)dzmdym (6.13) The mean delay and the second moment of the delay 72 can be determined from 干=rp()d (6.14) 不=2pr)dn (6.15) 3.ELLIPTICAL SCATTERING MODEL In this model,first described in [78],the scatter density is assumed to be estation and the mobile. he ellipse SCTM/2 the semi-minor axis6is equal to The area of the ellipse is Ae=mab.The equation of the ellipse in polar coordinates is 4a2-D2 Tbe=4a-2D Cos B (6.16)
Geometric Models for Angle and Time of Arrival 167 symmetric distributions, with can also be put in the form: where is the Bessel function of the first kind of order n. B. PDF for TOA: The marginal p.d.f. for the tune of arrival could, in principle, be obtained by first finding the joint density and integrating over all possible values of However, a simpler approach is to first find the c.d.f. and then to take its derivative w.r.t. to get [30]. To determine the c.d.f. for a particular it is essential to consider all delays that are less than All points of the scatter region that lie within the ellipse will contribute to this. Denoting the intersection of the scatter region with the ellipse as the c.d.f and the p.d.f. for the time of arrival are The mean delay and the second moment of the delay can be determined from 3. ELLIPTICAL SCATTERING MODEL In this model, first described in [78], the scatter density is assumed to be uniform inside an elliptical region of maximum delay with foci at the base station and the mobile. The semi-major axis a of the ellipse is and the semi-minor axis 6 is equal to The area of the ellipse is The equation of the ellipse in polar coordinates is
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
Geometric Models for Angle and Time ofArrival 171 Figure 6.5 shows a plot of rMpe(r)for the elliptical scattering model for D 1000 m and TM =5 us.For these parameters,the r.m.s.angular spread from (6.24)is 2 x 55=110.The mean delay from 6.27)is 4.0741 us and second mome nt in the delay variable from (68)is.7The r.ms. delay spread is5s .100 0.6 100 0.6 D-1 km,tu=5us 0.5 0.5 0.4 0.4 0.3 a 0.2 0.2 0.1 .100 100 0 Figure64.PDF of the angle of arrival for the elliptical scattering model Standard deviation in angle is=55°. Figure 6.6 shows the variation ofthe r.m.s.angular spread and r.m.s.delay spread with increasing size of scatter region.The size of the elliptical scatter that while both the angula spread and the delay spread increase with the ellipse size,the delay spread increases more or less linearly with increasing TM. 4. CIRCULAR SCATTERING MODEL The next model we consider is the circular scattering model where the scatter density is uniformly distributed inside a circular region of radius R
Geometric Models for Angle and Time of Arrival 171 Figure 6.5 shows a plot of for the elliptical scattering model for D = 1000 m and For these parameters, the r.m.s. angular spread from (6.24) is 2 x 55° = 110°. The mean delay from 6.27) is 4.0741 and the second moment in the delay variable from (6.28) is The r.m.s. delay spread is 0.523 Figure 6.4. PDF of the angle of arrival for the elliptical scattering model. Standard deviation in angle is Figure 6.6 shows the variation of the r.m.s. angular spread and r.m.s. delay spread with increasing size of scatter region. The size of the elliptical scatter region is increased by increasing and maintaining the distance between the transmitter and receiver fixed at 1 km. It is seen that while both the angular spread and the delay spread increase with the ellipse size, the delay spread increases more or less linearly with increasing 4. CIRCULAR SCATTERING MODEL The next model we consider is the circular scattering model where the scatter density is uniformly distributed inside a circular region of radius R
172 RADIOWAVE PROPAGATION SMART ANTENNAS 950.70.750.80.850.90.95 6.5 D=1 km,Tu=5 us 3.5 3.5 3 25 0.7 0.750.8 0.850.90.95 2.5 /t about the mobile.Such a model is appropriate for macrocell environments where the base station antenna height is relatively high that no significant scatterers lie around it.It is assumed in the following that R<D although the converse situation can also be handled similarly. geometry of the circ ular Ac=mR2.The p.d.f.of the scatter density is p1(em,n）=在 (6.29) SubstitutingR,the points of intersection of the line constant with the circle are found to be Tom DCosB-D2 cos2B-D2+R2 (6.30) ToM DCosB+D2 Cos2B-D2+R2 (6.31)
172 RADIOWAVE PROPAGATION & SMART ANTENNAS Figure 6.5. Plot of versus for the elliptical scattering model. The mean delay is and the second moment is RMS delay spread is 0.523 about the mobile. Such a model is appropriate for macrocell environments where the base station antenna height is relatively high that no significant scatterers lie around it. It is assumed in the following that R < D although the converse situation can also be handled similarly. Figure 6.7 shows the geometry of the circular scattering model. The area of the scatter region is The p.d.f. of the scatter density is Substituting the points of intersection of the line = constant with the circle are found to be