Multipath Propagation - Mathematical Modeling

Mathematical Modeling

The mathematical model of the multipath can be presented using the method of the impulse response used for studying linear systems.

Suppose you want to transmit a single, ideal Dirac pulse of electromagnetic power at time 0, i.e.

At the receiver, due to the presence of the multiple electromagnetic paths, more than one pulse will be received (we suppose here that the channel has infinite bandwidth, thus the pulse shape is not modified at all), and each one of them will arrive at different times. In fact, since the electromagnetic signals travel at the speed of light, and since every path has a geometrical length possibly different from that of the other ones, there are different air travelling times (consider that, in free space, the light takes 3 μs to cross a 1 km span). Thus, the received signal will be expressed by

where is the number of received impulses (equivalent to the number of electromagnetic paths, and possibly very large), is the time delay of the generic impulse, and represent the complex amplitude (i.e., magnitude and phase) of the generic received pulse. As a consequence, also represents the impulse response function of the equivalent multipath model.

More in general, in presence of time variation of the geometrical reflection conditions, this impulse response is time varying, and as such we have

Very often, just one parameter is used to denote the severity of multipath conditions: it is called the multipath time, and it is defined as the time delay existing between the first and the last received impulses

In practical conditions and measurement, the multipath time is computed by considering as last impulse the first one which allows to receive a determined amount of the total transmitted power (scaled by the atmospheric and propagation losses), e.g. 99%.

Keeping our aim at linear, time invariant systems, we can also characterize the multipath phenomenon by the channel transfer function, which is defined as the continuous time Fourier transform of the impulse response

where the last right-hand term of the previous equation is easily obtained by remembering that the Fourier transform of a Dirac pulse is a complex exponential function, an eigenfunction of every linear system.

The obtained channel transfer characteristic has a typical appearance of a sequence of peaks and valleys (also called notches); it can be shown that, on average, the distance (in Hz) between two consecutive valleys (or two consecutive peaks), is roughly inversely proportional to the multipath time. The so-called coherence bandwidth is thus defined as

For example, with a multipath time of 3 μs (corresponding to a 1 km of added on-air travel for the last received impulse), there is a coherence bandwidth of about 330 kHz.