Statistical Modeling of Multipath Fading Channels
We first describe the statistical modeling of mobile wireless
channels. We follow [396] closely. For a typical
terrestrial wireless channel, we can assume the existence of multiple
propagation paths between the transmitter and the receiver. With each
transmission path we can associate a propagation delay and an attenuation
factor, which are usually time-varying due to changes in propagation conditions
resulting primarily from transceiver mobility. In the absence of additive noise,
the received complex baseband signal in such a channel is given by
Equation 9.1
where x(t) is the transmitted
baseband signal; an(t) and tn(t) are, respectively, the path attenuation and the
propagation delay for the signal received on the nth path; and fc is the carrier frequency. By inspecting
(9.1), we can see that we can model the
multipath fading channel by a time-varying linear filter with impulse response
g(t,t) given by
Equation 9.2
For some mobile channels, we can further assume that the
received signal consists of a continuum of multipath components. Accordingly,
for these channels, (9.1) is modified as
follows:
Equation 9.3
where a(t,t) denotes the
attenuation factor associated with a path delayed by t at time instant t. The corresponding baseband time-varying impulse
response of the channel is then
Equation 9.4
By the central limit theorem, assuming a large enough number of
paths between the transmitter and the receiver, and by further assuming that the
associated attenuations per path are independent and identically distributed,
the impulse response g(t, t) can be modeled by a complex-valued Gaussian random
process. If the received signal r(t) has only a
diffuse multipath component, g(t, t) is characterized by a zero-mean complex Gaussian
random variable (i.e., |g(t,t)| has a Rayleigh distribution). In this case the
channel is called a Rayleigh fading channel.
Alternatively, if there are fixed scatterers or signal reflections in the
medium, g(t, t) has a nonzero
mean value and therefore |g(t, t)| has a Rician distribution. In this case the channel
is a Rician fading channel.
We will assume that the fading process g(t, t) is wide-sense
stationary in t, and define its corresponding
autocorrelation function as
Equation 9.5
A further reasonable assumption for most mobile communication
channels is that the attenuation and phase shift associated with path delay
t1 are
uncorrelated with the corresponding attenuation and phase shift associated with
a different path delay t2. This situation is known as uncorrelated scattering. Thus (9.5) can be expressed as
Equation 9.6
where Rg(t, Dt) represents the average
channel power as a function of the time delay t and the difference Dt in observation time. The multipath spread of the channel, Tm, is the range of values of the path delay
t for which Rg(t, 0) is essentially constant. Let Sg(f, Dt) = Ft{Rg(t, Dt)} [i.e., the Fourier transform of Rg(t, Dt) with respect to t]. Then Sg(f, Dt) is essentially the
frequency response function of the linear time-varying channel. The coherence bandwidth of the channel, Bc, is the range of values of frequency
f for which Sg(f, 0) is
essentially constant. Hence the multipath delay spread Tm and the coherence bandwidth Bc are related reciprocally (i.e., Bc 
1/Tm). Roughly speaking, the channel
frequency response remains the same within the coherence bandwidth Bc. Let W be
the bandwidth of the transmitted signal. When W
> Bc, the channel is called frequency-selective fading; and when W < Bc,
the channel is called frequency nonselective
fading or flat fading.
We can also take the Fourier transform of Rg(t, Dt) with respect to Dt to obtain the scattering
function Sg(t, l)
= FDt{Rg(t, Dt)}. The Doppler spread
of the channel, Bd, is the range of
values of frequency l for which Sg(0, l) is
essentially constant. The channel coherence time
is given by Tc 1/Bd.
Roughly speaking, the channel time response remains the same within the
coherence time Tc. Let T be the symbol interval of the transmitted signal.
When T < Tc (i.e., small Doppler), the channel is
said to be time-nonselective fading or slow fading; and when T
> Tc (i.e., large Doppler), the
channel is said to be time-selective fading or
fast fading.
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