## Wireless Communication## Chapter: Wireless Propagation Channels |

*Contributed by Peter E. Leuthold and
Pascal Truffer*

Several definitions exist for the scatter function of a multipath channel. Basically, these "parameters", "functions" and "profiles" describe the time and frequency dispersion of the channel. This contribution defines and discusses functions that defines delay spread and angle of arrival. For outdoor, vehicular channels, time dispersion is often a more useful parameter than angle of arrival, while for indoor systems with diversity, angle of arrival is more useful. Note however that if an antenna is in motion, there is a direct relation between angle of arrival and Doppler shift.

A description of the dispersive radio channel can be based on the electric field delay-direction spread vector defined in [4]. The vector denotes the MT antenna location, and and are the delay and incidence direction variables, respectively, where is determined by the azimuth and the coelevation in a spherical coordinate system.

The vector can be decomposed into a sum of M components each originating from a hypothetical impinging wave:

The notation means that the number of active paths or dominant waves varies with the location when the MT is moving along the trajectory. Under far field conditions the vector has two components which correspond to the vertical and horizontal polarization.

where is proportional to the field pattern of the MT antenna for the considered polarization.

We now assume that over a sufficiently small area *A* the wave incidence
constellation, i.e. the number of active paths, relative delay, angle
of arrival and amplitude, remains approximately constant.
Consequently, the spatial variations of the FDDSF mainly result from
the changes of the phase of the wave components.

Within the area *A* the wave incidence constellation is characterized by
the local* power delay-direction profile* (PDDP)

which may be presented in the SCRM as the product

In this equation, denotes the path loss for the distance
from the transmitter station to the area *A* and
is the local *delay-direction scattering function*
(DDScF) in *A*. Since the components in (1) are regarded as
independent we obtain with (4) the expression

that means each term is determined by the mean power , the mean
delay , the mean incidence direction and a
local scattering function which is considered to
be identical for all components. Obviously, the variables *M*,
, , and are random variables.

**Table 2:** Description of the primary random variables *M*, , (delay
spread )

Their specification is given in Table 2 [2] with the exception of which will be discussed later on.

**Figure:** Local PDDP generated with the SRCM

Figure 1 presents a realization of the local PDDP obtained with the SRCM. In this case as well as for the next Figures 2 to 6 the following parameters have been chosen: carrier frequency 5.2 GHz, MT velocity 1 m/s, delay spread . Such a situation is typical for a small room environment.

Taking the expectation of the local DDScF over a class *C* of
environments, i.e. a large room, yields the global DDScF

In accordance with the assumptions in Table 2 this function takes the form

where the global *delay scattering function*
(DScF) follows integrating with respect to
. It is reasonable to choose the global DScF as an
exponentially decaying function

that means the scattered power decreases with increasing delay. With regard to the channel simulation the global DDScF according to (7) and (8) can be considered as a target function which represents the expectation of Gaussian processes with the standard deviation and , respectively.

Figure 2 shows the successive approximation of the global DScF by the average of local DScFs.