Frequency-Nonselective Fading Channels
Note from (9.3) and (9.4) that we have
Equation 9.7
where X(f) = F{x(t)} and G(f,t) = Ft{g(t, t)}. Assume that the
channel fading is frequency-nonselective (flat) (i.e., W < Bc);
then the channel frequency response G(f,t) is approximately constant over the signal
bandwidth [i.e., G(f,t) = G(t)]. In this case, (9.7) can be written as
Equation 9.8
Hence the effect of a flat-fading channel can be modeled as a
time-varying multiplicative distortion. Note that since g (t, t) is assumed to be
a complex Gaussian process, G(t) is also a complex Gaussian process. When the fading
is Rayleigh, we have E{G(t)} = 0. For mobile
communications, the autocorrelation function of G(t) is typically
modeled by the Jakes model [216]:
Equation 9.9
where P is the average power of
the fading process (i.e., P = E{|G(t)|2}) J0(·) is the Bessel function of the first
kind and zeroth order. The corresponding Doppler power spectrum of the channel
is then given by
Equation 9.10
9.2.2 Frequency-Selective Fading Channels
Now assume that the transmitted baseband signal has a bandwidth
of W and that W
> Bc (i.e., the channel exhibits
frequency-selective fading). By the sampling theorem, we have
Equation 9.11
Equation 9.12
Hence the noiseless received signal is given by
Equation 9.13
Let 
; then for practical purposes we can
use the following truncated tapped-delay-line model to describe the
frequency-selective fading channel [396]:
Equation 9.14
where 
, and 
conprises
independent complex Gaussian processes.
Sections
