Editing Multipath propagation (section)

==Mathematical modeling==
[[File:Multipath impulse response.png|thumb|right|320px|Mathematical model of the multipath impulse response.]]
The mathematical model of the multipath can be presented using the method of the [[impulse response]] used for studying [[linear system]]s.

Suppose you want to transmit a single, ideal [[Dirac delta function|Dirac pulse]] of [[electromagnetism|electromagnetic]] power at time 0, i.e.

:<math>x(t)=\delta(t)</math>

At the receiver, due to the presence of the multiple electromagnetic paths, more than one pulse will be received, 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&nbsp;μs to cross a 1&nbsp;km span). Thus, the received signal will be expressed by

:<math>y(t)=h(t)=\sum_{n=0}^{N-1}{\rho_n e^{j\phi_n} \delta(t-\tau_n)}</math>

where <math>N</math> is the number of received impulses (equivalent to the number of electromagnetic paths, and possibly very large), <math>\tau_n</math> is the time delay of the generic <math>n^{th}</math> impulse, and <math>\rho_n e^{j\phi_n}</math> represent the [[complex amplitude]] (i.e., magnitude and phase) of the generic received pulse. As a consequence, <math>y(t)</math> also represents the impulse response function <math>h(t)</math> 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

:<math>\tau_n=\tau_n(t)</math>

:<math>\rho_n=\rho_n(t)</math>

:<math>\phi_n=\phi_n(t)</math>

Very often, just one parameter is used to denote the severity of multipath conditions: it is called the '''[[multipath time]]''', <math>T_M</math>, and it is defined as the time delay existing between the first and the last received impulses

:<math>T_M=\tau_{N-1}-\tau_0</math>

[[File:Multipath transfer function.png|thumb|320px|Mathematical model of the multipath channel transfer function.]]
In practical conditions and measurement, the multipath time is computed by considering as last impulse the first one which allows receiving 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 <math>H(f)</math>, which is defined as the continuous time [[Fourier transform]] of the impulse response <math>h(t)</math>

:<math>H(f)=\mathfrak{F}(h(t))=\int_{-\infty}^{+\infty}{h(t)e^{-j 2\pi f t} d t}=\sum_{n=0}^{N-1}{\rho_n e^{j\phi_n} e^{-j2 \pi f \tau_n}}</math>

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

:<math>B_C \approx \frac{1}{T_M}</math>

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