40 IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 2, MARCH 1997 Effects of Directional Antennas at the Base Station on the Doppler Spectrum Paul Petrus, Student Member, IEEE, Jeffrey H. Reed, Member, IEEE, and Theodore S. Rappaport, Senior Member, IEEE Abstract— In this letter we analyze the effect of directional antennas at the base station on the Doppler spectrum. It is well known that the Doppler spectrum is dependent on the probability density function (pdf) of the angle-of-arrival (AOA) of the multipath components at the mobile unit and the direction of the motion of the mobile. The Doppler spectrum is U-shaped as noted by Clark [1], when the pdf of the AOA of the multipath components at the mobile is uniform. Here, however, we study the Doppler fading at the base station and assume that the scatterers are uniformly located around the mobile within a circle and derive the pdf of the AOA of the multipath components at the mobile when directional antennas are used at the base station. I. CHARACTERIZING THE AOA AT MOBILE WHEN DIRECTIONAL ANTENNAS ARE USED AT BASE STATION H ERE we assume that the scatterers are uniformly located around the mobile ( ) within a circle of radius , as shown in Fig. 1. The base station is marked and is separated from the mobile by a distance . Also we assume that the signal undergoes a single bounce as it travels from the base station to the mobile, i.e., the signal from the mobile bounces off a scatterer and reaches the base station without the influence of other scatterers. Fig. 1 illustrates the condition when a flat-top directional antenna [2] with constant gain and beamwidth is used at the base station. Since the scatterers are confined to a circle around the mobile, the angle-of-arrival (AOA) of the multipath components at the base station is restricted to an angular region of . If , then the base station antenna will illuminate all of the scatterers and hence the probability density function (pdf) of the AOA at the mobile is uniform. But if , then the base station antenna will partially illuminate the scatterers and hence the pdf of the AOA of the multipath components at the mobile will not be uniform. In Fig. 1, the scatterer’s region illuminated by the base station antenna is marked EFGHIJ. Let us now derive the pdf of at the mobile, when . Let us consider only the region EFGMJ, where . Due to symmetry we get the same results for the region GHIJM, where . Let us divide the region EFGMJ into three distinct regions, JEM, Manuscript received December 6, 1996. This work was suppported by the DARPA under the GloMo project and the MPRG affiliates program. The associate editor cordinating the review of this letter and approving it for publication was Dr. A. Haimovich. The authors are with Mobile and Portable Radio Research Group (MPRG), Bradley Department of Electrical Engineering , Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA (e-mail: petrus@sol.mprg.ee.vt.edu). Publisher Item Identifier S 1089-7798(97)02828-7. Fig. 1. (Top view) Illustration of the AOA at the mobile when a directional antenna is used at the base station. EFM, and FGM. The values of , , and define the three regions. The pdf of the AOA can be derived by computing the area within a thin strip (shaded region) shown in Fig. 1. The area ( ) within a strip between and can be shown using calculus to be (1) where the value of for the three regions can be shown to be (2) Based on the geometry in Fig. 1, it can be shown that (3) Now and can be determined by substituting in (3). Since the scatterers are assumed to be uniformly distributed within the region EFGHIJ, the density of the area within the region is the reciprocal of the area within the region. The area density can be shown to be (4) Using (1) and (4), the cumulative distribution function (cdf) of the AOA, , can be expressed as (5) 1089–7798/97$10.00  1997 IEEE