Wireless Network Design: Optimization Models and Solution ...

3 Channel Models for Wireless Communication Systems 57
tion distance. It means that if the receiver has multiple antennas spaced equal to the
decorrelation distance, the signals received by the two antennas will be statistically
uncorrelated. Measurements have found that the decorrelation distance is approximately
equal to half the wavelength of the signal. This property of the channel is
used to exploit spatial diversity of the channel and signal processing techniques are
used to improve the performance of the receiver. It has also led to the deployment of
multiple antennas at transmitters and receivers of wireless communication systems.
3.5 Channel Models for Systems using Multiple Antennas
Consider a system where the transmitter is transmitting using multiple antennas and
the receiver is receiving signals using multiple antennas. The scattering environment
is assumed to scatter the signal and the receiving antenna is assumed to receive
several modified (in amplitude, phase, frequency and time) copies of the transmitted
signals. Using this scenario, several spatial channel models have been developed
using principles of geometry and a statistical framework [4, 12, 42, 43, 52].
3.5.1 Spatial Wireless Channel Models using Geometry
Consider a general representation of the wireless channel (extension of the FIR
model given in a previous section) when one mobile receiver (having one antenna)
is considered, as
h1(t,τ) =
L(t)−1
∑
l=0
Al,1(t) exp( j φl,1(t))δ(t − τl,1(t)) (3.18)
where L(t) is the number of multipath components, φl,k,Al,k(t),τl,k(t) are the phase
shift, amplitude and delay due to the lth path induced in the kth receiver. Note that
the above channel model assumes all parameters to be time varying. This model
can be extended to the case of using multiple antennas by including the Angles of
Arrival (AOA) into the model. It is given by
h1(t,τ) =
L(t)−1
∑
l=0
Al,1(t) exp( j φl,1(t))a(θl(t))δ(t − τl,1(t)) (3.19)
where a(θl(t)) is the array steering vector corresponding to the angle of arrival,
θl(t)) and is given by
a(θl(t)) = (exp(− j ψl,1), ψl,2, ··· , ψl,m ) T
(3.20)