ctor channel. An example of this configuration is given below for the case of a 2D antenna array [4]: exp (jpm exp (jpm) exp(j P)p(2) where atx (On,m,ZoD, On,m,AOD) and arx (On,m,ZoA, On,m,AoA) are the transmit and receive antenna array impulse response vectors, respectively, corresponding to ray me {1, ,M} in cluster ne{1,..., N} given by the following expression: = exp An,m ---- = exp An,m where to is the wavelength of carrier frequency fo; ,, and r rx(On,m,ZoA, are the corresponding angular 3D spherical unit vectors of the transit and receive, respectively; Wix and Wrx denote the location matrices of the transmit and receive antenna elements in 3D Cartesian coordinates. The location matrices in the vectored impulse response are provided for an antenna configuration that is a uniform rectangular array consisting of cross-polarized antenna elements shown in Fig. 3.4. The Doppler shift generally depends on the time-variance of the channel as it is defined as the derivative of the channel phase over time. It can result from transmitter, receiver, or scatterers movement. The general form of the exponential Doppler component is given as follows [24]: where ,,(t) denotes the normalized vector that points to the direction of the incoming wave as seen from the receiver at time t. The velocity vector of the receiver at time t is 312 Chapter 3 denoted by v(t) while to denotes a reference point in time that defines the initial phase ==0. The above expression is only valid for time-invariant Doppler shift, satisfying Spatial correlation is often said to degrade the performance of multi-antenna systems and imposes a limit on the number of antennas that can be effectively fit in a small mobile device. This seems intuitive as the spatial correlation decreases the number of independent channels that can be created by precoding. When modeling spatial correlation, it is useful to employ the Kronecker model, where the correlation between transmit and receive antennas are assumed independent and separable. This model is reasonable