Wireless Network Design: Optimization Models and Solution ...

3 Channel Models for Wireless Communication Systems 53
munication between the transmitter and receiver. This type of fading is also called
Flat fading.
Ricean channel: The amplitude of a channel which is modeled as a non-zero
mean complex Gaussian random process has a Ricean distribution. Such channels
are called Ricean channels and occur when there is Line-Of-Sight (LOS) communication
between the transmitter and receiver. The Ricean pdf is given in [31].
The above channels are characterized by the mean and variance of the channel,
which is modeled as a Gaussian random process. For the Ricean channel, an additional
factor called the k − f actor is defined as the ratio of total power in the LOS
path to the total scattered power received. If this ratio is zero, it implies Rayleigh
fading.
3.4.3 Doppler Spread Due to Relative Motion Between Transmitter
and Receiver and Coherence Time
Consider a mobile receiver moving at a velocity v and receiving multiple signals
from various directions (due to obstacles). Consider the case where there is no direct
LOS path and hence Rayleigh fading is considered. The signal is assumed to be a
narrowband signal centered around the carrier frequency fc. Considering the case
that the signal is received from all directions and assuming that they are all of equal
amplitude, the power spectrum of the Rayleigh fading channel is given by
S( f ) =
π fd
b0
� 1 − { f / fd} 2
(3.14)
where fd is the doppler frequency and the equation is valid for f < fd . The shape
of the spectrum is like a ‘bathtub with infinitely high steep walls’. Jakes [50] proposed
a technique to simulate a fading channel with the above doppler spectrum.
The technique involves considering several oscillators and appropriately combined
after suitable amplitude scaling. This is popularly known as the Jakes spectrum as
shown in Figure 3.1.
Consider the case where the receiver and transmitter are stationary. In this case
too, one gets a Doppler spread due to the movement of the scatterers (for example,
branches of trees, moving traffic) between the transmitter and receiver. The spectrum
of such a channel was obtained through measurements carried out at Stanford
University in the 2.5GHZ band [11]. A mathematical model, obtained from the measurement
data, is given by
�
S f ixed( f ) = 1 − { f / fm} 2 (3.15)
fm is the maximum doppler frequency (typically 2Hz to 4Hz). This spectrum shape
is an ‘inverted cup’ as shown in Figure 3.2.