Wireless Network Design: Optimization Models and Solution ...

58 K. V. S. Hari
where m antennas are used and ψl,i = (2π/λ)[xi cos(θl(t)) + j sin(θl(t))], xi,yi are
the in-phase and quadrature phase components. The angular spread is dependent on
the scenario and the distribution is assumed accordingly.
Several models have been proposed. Lee [22] assumed that all the scatterers are
uniformly placed in a circular fashion. For this configuration, the correlation function
can be obtained [4]. But this was not found to be a satisfactory model. The
model was extended [42, 43] to add additional rings of scatterers. This model was
further modified [4] but all these models do not consider the delay spread or doppler
spectrum.
3.5.2 Geometrically Based Single-Bounce Statistical Models
The use of spatial scatter density functions to characterize the channel model is
developed using some geometry based principles and these models are called Geometrically
based single-Bounce Statistical Models (GBSB). Using these models, the
joint AOA and TOA probability density function is obtained (or numerically computed)
and the communication system performance is studied using these channel
models [20, 24].
3.5.2.1 Geometrically Based Single Bounce Models
The first model is called the Geometrically Based Single Bounce Circular Model
(GBSBCM) where the scatterers are assumed to be in a circular arrangement
around the transmitter and the other model is called the Geometrically Based Single
Bounce Elliptical Model (GBSBEM) which assumes that scatterers are uniformly
distributed within an ellipse, where the foci of the ellipse are the location of the
transmitter and receiver. The joint AOA-TOA probability density function at the
transmitter (base station) is given as
fτ,θT (τ,θT ) = (d2 − τ2c2 )(d2c + τ2c3 − 2τc2d cos(θT ))
(4πr2 R )(d cos(θT ) − τc) 3
f or d 2 − 2τcd cos(θT ) + τ 2 c 2 /(τc − d cos(θT )) ≤ 2rR
fτ,θ (τ,θT ) b = 0, else.
Similarly, the joint AOA-TOA probability density function at the receiver (mobile)
is given as
fτ,θR (τ,θR) = (d2 − τ2c2 )(d2c + τ2c3 − 2τc2d cos(θR))
(4πr2 R )(d cos(θR) − τc) 3
f or d 2 − 2τcd cos(θR) + τ 2 c 2 /(τc − d cos(θR)) ≤ 2rR
fτ,θR (τ,θR) = 0, else