Wireless Network Design: Optimization Models and Solution ...

3 Channel Models for Wireless Communication Systems 59
where θT ,θR are the angles of arrival relative to the line of sight of the array at the
transmitter and receiver respectively, c is the velocity of light and τR is the maximum
delay at the receiver.
3.5.2.2 Gaussian Wide Sense Stationary Uncorrelated Scattering model
The next model is called the Gaussian Wide Sense Stationary Uncorrelated Scattering
model (GWSSUS). This model assumes several clusters in space around the
transmitter. Each cluster will have several scatterers. The scatters in each cluster are
chosen/assigned such that the difference in delays with the scatterers are not resolvable
with the signal transmission bandwidth. Each cluster has a mean AOA, θ0k for
the cluster. One can define
vk,b =
Nk ∑ αk,i exp( j φk,i)a(θ0k − θk,i) (3.21)
i=1
where Nk is the number of scatterers in the kth cluster, αk, j,φk, j,θk, j are the amplitude,
phaseshift and AOS of the jth scatterer in the kth cluster. Assuming that the
scatterers are assigned to a cluster at least for b data bursts, the received signal over
b data bursts is given by
x b =
Nc
∑
k=1
v k,b s(t − τk) (3.22)
Nc being the number of clusters. If there are a large number of clusters, the Gaussian
nature of v k,b can be assumed and the mean and covariance matrix of this vector can
be obtained easily.
3.5.2.3 Gaussian Angle Arrival Model
A special case of GWSSUS is when Nc = 1 and the density function of the mean
AOA of the clusters is assumed to be Gaussian. The array covariance matrix is given
by
R(θ0,σθ ) = Pra(θ0)a H (θ0) ⊙ B(θ0,σθ ) (3.23)
where the klth element of the matrix is given by Bkl(θ0,σθ ) = exp(−2(πds(k −
l)) 2 σ 2 θ cos2 θ), with Pr as the received signal power, ds is the element spacing, ⊙
is the Schur (element-wise) product of matrices. This model is called the Gaussian
Angle Arrival (GAA) model. The performance of the AOA estimation algorithms
using this model have been studied in [18] [45].